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Two-particle pseudorapidity correlations, measured using charged particles with $p_{\mathrm{T}} >$ 0.5 GeV and $|\eta| <$ 2.4, from $\sqrt{s_{NN}}$ = 2.76 TeV Pb+Pb collisions collected in 2010 by the ATLAS experiment at the LHC are presented. The correlation function $C_N(\eta_1,\eta_2)$ is measured for different centrality intervals as a function of the pseudorapidity of the two particles, $\eta_1$ and $\eta_2$. The correlation function shows an enhancement along $\eta_- \equiv \eta_1 - \eta_2 \approx$ 0 and a suppression at large $\eta_-$ values. The correlation function also shows a quadratic dependence along the $\eta_+ \equiv \eta_1$ + $\eta_2$ direction. These structures are consistent with a strong forward-backward asymmetry of the particle multiplicity that fluctuates event to event. The correlation function is expanded in an orthonormal basis of Legendre polynomials, $T_n(\eta_1)T_m(\eta_2)$, and corresponding coefficients $a_{n,m}$ are measured. These coefficients are related to mean-square values of the Legendre coefficients, $a_n$, of the single particle longitudinal multiplicity fluctuations: $a_{n,m} = \langle a_na_m \rangle$. Significant values are observed for the diagonal terms $\langle a_n^2 \rangle$ and mixed terms $\langle a_na_{n+2}\rangle$. Magnitude of $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ is the largest and the higher order terms decrease quickly with increase in $n$. The centrality dependence of the leading coefficient $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ is compared to that of the mean-square value of the asymmetry of the number of participating nucleons between the two colliding nuclei, and also to the $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ calculated from HIJING.
Heavy-ion collisions, forward-backward correlations, multiplicity fluctuations
§ INTRODUCTION
Ultra relativistic heavy-ion collisions at RHIC and LHC create hot dense matter of deconfined quarks and gluons. The initial density distributions in the collisions fluctuate event to event and so a proper understanding of these fluctuations is necessary to describe and study the subsequent evolution of the produced matter. Study of the multiplicity correlations in the transverse direction and its fluctuations event by event have helped place important constraints on the transverse density fluctuations in the initial state <cit.>. Multiplicity correlations in the longitudinal direction are sensitive to the initial density fluctuations in pseudorapidty ($\eta$). These density fluctuations influence the early time entropy production and generate long-range correlations (LRC) which appear as correlations of the multiplicity of produced particles separated by large $\eta$ difference <cit.>. For example, EbyE differences between the number of nucleon participants in the target and the projectile may lead to a long-range asymmetry in the longitudinal multiplicity distribution <cit.>. Longitudinal multiplicity correlations can also be generated during the final state, such as from resonance decays, jet fragmentation and Bose-Einstein correlations, which are typically localized over a smaller range of $\eta$ difference, and are commonly referred to as short-range correlations (SRC).
Previous studies of the longitudinal multiplicity correlations have focussed on the forward-backward correlations of the particle multiplicity in two symmetric $\eta$ windows around the center of mass of the collision system <cit.>. Recently, a more general method which uses on the correlation function in the full $\eta_1$, $\eta_2$ space has been proposed <cit.>. The orthogonal modes of the correlation function provides information on the orthogonal modes in EbyE single particle longitudinal multiplicity fluctuations and their magnitudes. The method had been applied to HIJING <cit.> and AMPT <cit.> models to extract different shape components of the multiplicity fluctuation. In this proceedings, the correlation functions and the extracted magnitudes of the shape components of longitudinal multiplicity fluctuation, measured in Pb+Pb collisions at 2.76 TeV at the LHC using the ATLAS <cit.> detector are presented.
§ METHOD AND ANALYSIS PROCEDURE
The two particle correlation function in pseudorapidity is defined as <cit.>:
\begin{equation}
C(\eta_1,\eta_2) = \frac{\langle N(\eta_1)N(\eta_2) \rangle}{\langle N(\eta_1)\rangle\langle N(\eta_2)\rangle}
\label{eq:meth1}
\end{equation}
where $N(\eta) \equiv \frac{dN}{d\eta}$ is the multiplicity density at $\eta$ in a single event and $\langle N(\eta) \rangle$ is the average multiplicity at $\eta$ for a given event class.
In principle, the averages in Eq. <ref> should be calculated over event-classes defined using narrow centrality intervals, so that it contains only dynamical fluctuations that decouple from any residual centrality dependence of the average shape, $\langle N(\eta) \rangle$, which would lead to a modulation of the projections of the correlation function along the $\eta_1$ or $\eta_2$ axes. But due to experimental limitations and finite statistics, this modulation from the change of average shape cannot be completely removed in $C(\eta_1,\eta_2)$ as defined in Eq.<ref>. However, these modulations can be removed by a simple redefinition of the correlation function <cit.>:
\begin{equation}
C_N(\eta_1,\eta_2) = \frac{C(\eta_1,\eta_2)}{C_P(\eta_1)C_P(\eta_2)} ,
\label{eq:meth2}
\end{equation}
where $C_P(\eta_{1(2)}) = \frac{\int C(\eta_1,\eta_2)d\eta_{2(1)}}{2Y}$, with $Y$ being the maximum value of $\eta_1$ and $\eta_2$. The resulting distribution is then renormalized such that the average value of $C_N(\eta_1,\eta_2)$ in the $\eta_1$ and $\eta_2$ plane is one.
Following <cit.>, the correlation function is expanded into the orthonormal polynomials,
\begin{equation}
C_N(\eta_1,\eta_2) = 1+ \sum_{n,m=1}^{\infty} a_{n,m}\frac{T_n(\eta_1)T_m(\eta_2) + T_n(\eta_2)T_m(\eta_1)}{2},
\label{eq:meth3}
\end{equation}
where $T_n(\eta) = \sqrt{n+\frac{1}{2}}P_n(\eta/Y)$ and $P_n(x)$ are the Legendre polynomials. The $a_{n,m}$ coefficients in the expansion are related to the magnitudes of the shape fluctuations in the EbyE distribution, $a_{n,m} = \langle a_na_m \rangle$ where $a_n$ are the coefficients in the expansion of the single particle ratio, $\frac{N(\eta)}{\langle N(\eta) \rangle} = 1 + \sum a_nT_n(\eta)$.
In the analysis the correlation functions are calculated in 5% centrality bins <cit.>, defined using the transverse energy distribution in the ATLAS Forward Calorimeters (FCal). The analysis uses charged particle tracks reconstructed in the ATLAS inner detector to construct the correlation functions. The tracks are required to have $p_{\mathrm{T}} >$ 0.5 GeV and $|\eta| <$ 2.4. The correlation function is constructed as the ratio of distributions of same-event pairs ($S(\eta_1,\eta_2) \propto \langle N(\eta_1)N(\eta_2)\rangle$ and mixed-event pairs ($B(\eta_1,\eta_2) \propto \langle N(\eta_1)\rangle\langle N(\eta_2)\rangle$), $C(\eta_1,\eta_2) = \frac{S(\eta_1,\eta_2)}{B(\eta_1,\eta_2)}$. The events used for constructing the mixed event pairs are required to have similar total number of reconstructed tracks, $N_{\mathrm{ch}}^{\mathrm{rec}}$, (matched within 0.5%) and $z$-coordinate of the collision vertex (matched within 2.5 mm). The events are also required to be close to each other in time to account for possible time-dependent variation of the detector conditions. To correct $S(\eta_1,\eta_2)$ and $B(\eta_1,\eta_2)$ for detector inefficiencies, the tracks are weighted by the inverse of their tracking efficiencies. Remaining detector effects largely cancel in the same to mixed event ratio. The systematic uncertainties in the correlation function are evaluated to be in the range 2$-$8.5% depending on the centrality interval. These uncertainties are propagated into $C_N(\eta_1,\eta_2)$ and other derived quantities, as the $\langle a_na_m \rangle$. More details of the analysis and systematic uncertainties along with a complete set of results can be found in this reference <cit.>.
§ RESULTS
The top panels of Figure.<ref> shows the correlation function $C_N(\eta_1,\eta_2)$ in a mid-central and a peripheral event class. The correlation functions show a broad ”ridge-like” shape along $\eta_1 = \eta_2$, and a depletion in the large $\eta_-$ region around $\eta_1 = -−\eta_2 \approx \pm 2.4$. The magnitude of the ridge structure is larger in peripheral event classes than in central event classes and could have a significant contribution from SRC. The depletion in the large $\eta_-$ region reflects the contribution from the LRC. The lower panels of Figure.<ref> shows the extracted $\langle a_na_m \rangle$ coefficients for the two centrality classes. Non-zero values are observed for the diagonal terms, $\langle a_n^2 \rangle$ and also for mixed coefficients of the form $\langle a_na_{n+2} \rangle$. The first six diagonal terms and first five mixed terms are shown in the figure. Magnitude of $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ is much larger than the other coefficients and the magnitude of higher order terms decrease quickly with increasing $n$. The magnitudes of the coefficients are larger in the peripheral event classes. Also the magnitude of the higher order coefficients decrease less rapidly with increasing $n$ in the peripheral event class compared to the central event class. A significant contribution to $\langle a_na_m \rangle$, particularly the higher order terms, could be from short-range correlations which contribute to coefficients of all orders.
Correlation function ($C_N(\eta_1,\eta_2)$) (top row) and the extracted $\langle a_na_m \rangle$ coefficients (bottom row) for 20-25% and 60-65% centrality classes <cit.>.
To further analyse the features of the correlation function, the correlation function is expressed in terms of $\eta_- = \eta_1-\eta_2$ and $\eta_+=\eta_1+\eta_2$. The resulting correlation function, $C_N(\eta_{-−},\eta_+)$ is then projected onto the $\eta_{-−}$ ($\eta_+$) axis in narrow ranges of $\eta_+$ ($\eta_{−-}$). The shape of the projections along $\eta_-$ are more sensitive to the SRC as the SRC have a strong dependence on $\eta_-$. The shape of the projections along $\eta_+$ on the other hand is more sensitive to the long-range correlations. If the first-order coefficient $a_1$ is dominating, then the correlation function in $\eta_+$ and $\eta_-$ can be written as
\begin{align}
\label{eq:res1}
C_N(\eta_{-},\eta_+) \sim 1 + \langle a_{\mathrm{1}}^{\mathrm{2}} \rangle\frac{3}{8Y^2}(\eta_{+}^{2} - \eta_{-}^{2}) \\ \nonumber
\approx 1+0.065\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle(\eta_+^2 - \eta_-^2)
\end{align}
Since the SRC are expected to have a weak dependence on $\eta_+$, a quadratic dependence of the projection, $C_N(\eta_+)$, on $\eta_+$ can be expected.
$C_N(\eta_−)$ for different $\eta_+$ slices (top row) and $C_N(\eta_+)$ for different $\eta_-$ slices (bottom row) for 20-25% and 60-65% centrality classes <cit.>.
The projections, $C_N(\eta_{-−})$ and $C_N(\eta_+)$, are shown in Figure.<ref> for different $\eta_+$ and $\eta_-$ slices respectively for the two centrality classes. Along the $\eta_-$ direction, the projections for all $\eta_+$ slices peak at $\eta_{−-}$ = 0 and decrease quickly for $\lvert \eta_{-}\rvert <$ 1, followed by a much weaker decrease beyond that. The quick decrease in $\lvert \eta_{-}\rvert< $ 1 is consistent with the dominance of short-range correlations, which are mostly centred around $\eta_- \approx$ 0. The weaker decrease at large $\lvert \eta_{-}\rvert$ could be related to the $-\eta_-^2$ term in Eq. <ref>.
The projections along $\eta_+$ direction show a clear quadratic dependence on $\eta_+$ for all $\eta_-$ slices used for projection. This reflects the dominant $a_1$ component of the long-range correlation. The quadratic dependence is quantified by fitting the $C_N(\eta_+)$ data with the function $C_N(\eta_+) = 0.065\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle\eta_+^2 + b$, where $b$ is a constant. The function fits the data quite well and are shown as solid lines in the figure. The $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ values are extracted from the fit for each $\eta_-$ window used for projection.
$\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ values calculated directly from the $C_N(\eta_1,\eta_2)$ (solid circles), obtained from fits shown in Fig. <ref> (open symbols), HIJING model (solid line), as well as the RMS values of $A_{Npart}$ from Eq. <ref> (dashed line). The shaded bands or error bars are the total uncertainties <cit.>.
Figure.<ref> shows the centrality dependence (in terms of the total number of participating nucleons, $N_{\mathrm{part}}$) of the RMS value of the first order coefficient, $a_1$. The values calculated from the Legendre expansion (Eq. <ref>) as well as those obtained from the fit to projections in Figure.<ref> are shown. The magnitude of $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ increase towards peripheral event classes. The values from the fits are always smaller than those from direct calculation by 2 $-–$ 20% depending on the centrality interval and $\eta_{-}$ slice used for projection. This could be due to the contribution from the SRC to the values calculated using Legendre expansion.
Figure.<ref> also shows the centrality dependence of the RMS value of the asymmetry between number of forward going and backward going participants, $A_{N_{\mathrm{part}}}$ defined as
\begin{equation}
A_{N_{\mathrm{part}}} = \frac{N_{\mathrm{part}}^{\mathrm{F}} - N_{\mathrm{part}}^{\mathrm{B}}}{N_{\mathrm{part}}^{\mathrm{F}} + N_{\mathrm{part}}^{\mathrm{B}}},
\label{eq:res2}
\end{equation}
where $N_{\mathrm{part}}^{\mathrm{F}}$ and $N_{\mathrm{part}}^{\mathrm{B}}$ denote the number of forward going and backward going participants respectively. The $A_{N_{\mathrm{part}}}$ values are calculated from a Monte-Carlo Glauber model <cit.>. The RMS values, $\sqrt{\langle A_{N_{\mathrm{part}}}^2 \rangle}$, are scaled down by an arbitrary factor of 0.4 to approximately match the $\sqrt{\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle}$ values. The centrality dependence of the RMS values of $A_{N_{\mathrm{part}}}$ quite well match the centrality dependence of the RMS values of $a_1$ in the mid-central classes, but show a stronger decrease in the most central event classes and a weaker increase in the more peripheral event classes. The good match between the centrality dependence of the RMS values of $a_1$ and $A_{N_{\mathrm{part}}}$ over a large centrality range suggest that $a_1$ modulations are driven by $A_{N_{\mathrm{part}}}$ and is consistent with the observation in <cit.> that EbyE, $a_1$ is strongly correlated with $A_{N_{\mathrm{part}}}$. Also shown in the figure are $\sqrt{\langle a_{\mathrm{1}}^2 \rangle}$ values calculated from HIJING. The values from HIJING over-estimate the data except in the most peripheral event classes.
§ SUMMARY AND CONCLUSIONS
Two-particle pseudorapidity correlation functions $C_N(\eta_1,\eta_2)$ are measured as a function of centrality for charged particles with $p_{\mathrm{T}} >$ 0.5 GeV and $|\eta| <$ 2.4, for Pb+Pb collisions at 2.76 TeV. The correlation function shows a ”ridge like” enhancement along the $\eta_1 \approx \eta_2$, and suppression at $\eta_1 \approx −\eta_2 \sim \pm 2.4$. These structures are further investigated by projecting the 2-D correlation function as function of $\eta_-$ and $\eta_+$ in narrow ranges of $\eta_−$ and $\eta_+$ respectively. The $C_N(\eta_-)$ projections show strong contribution from the short-range correlations, particularly in the region $|\eta_-| <$ 1, where the magnitude of the correlation decrease quickly with increase in $|\eta_-|$. The $C_N(\eta_+)$ distribution shows a clear quadratic dependence, characteristic of a forward-backward asymmetry induced by the asymmetry in the number of participating nucleons in the two colliding nuclei.
The correlation function is decomposed into a sum of products of Legendre polynomials those describe the different shape components, and the coefficients $\langle a_na_m \rangle$ are calculated. Significant values are observed for $\langle a_n^2 \rangle$ and $\langle a_na_{n+2} \rangle$. Magnitude of $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ is much larger than that of other coefficients and the magnitude of higher order terms decrease with increase in $n$. These coefficients are observed to increase for peripheral collisions, consistent with the increase of the multiplicity fluctuation for smaller collision systems. The centrality dependence of $\langle a_{\mathrm{1}}^{\mathrm{2}} \rangle$ is compared with the centrality dependence of the coefficient of the quadratic term in the fits to $C_N(\eta_+)$. The fit results are 2$–-$20% smaller than that obtained from a Legendre expansion, which could be due to the stronger influence of short-range correlations on the values calculated directly from the Legendre expansion.
§ ACKNOWLEDGEMENT
This research is supported by NSF under grant number PHY–1305037
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1511.00558
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Center for Epigenetics, Van Andel Research Institute, Grand Rapids Michigan 49503, USA.
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1511.00163
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Numerical Algorithms
Kassem Mustapha, [email protected]
Khaled Furati, [email protected]
Department of Mathematics and Statistics,
King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.
We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order $\mu\in (0,1)$ with variable coefficients. For the spatial discretization, we apply the standard piecewise linear continuous Galerkin method. Well-posedness of the fully discrete scheme and error analysis will be shown.
For a time interval $(0,T)$ and a spatial domain $\Omega$, our analysis suggest that the error in
$L^2\bigr((0,T),L^2(\Omega)\bigr)$-norm is of order $O(k^{2-\frac{\mu}{2}}+h^2)$ (that is, short by order $\frac{\mu}{2}$
from being optimal in time) where $k$ denotes the maximum time step, and $h$ is the maximum diameter of the
elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal $O(k^{2}+h^2)$ error bound in the stronger $L^\infty\bigr((0,T),L^2(\Omega)\bigr)$-norm. Variable time steps are used to compensate the singularity of the continuous solution near $t=0$.
§ INTRODUCTION
In this paper, we investigate a numerical solution that allows a time discontinuity for solving time fractional diffusion equations with variable diffusivity. Let $\Omega$ be a bounded convex polygonal domain in $\R^d$ ($d=1,2,3$),
with a boundary $\partial \Omega$, and $T > 0$ be a fixed time.
Then the fractional model problem is given by:
\begin{equation}\label{eq: ivp}
\begin{aligned}
{\cD}^{\mu} u(x,t)-\nabla \cdot (\A(x,t) \nabla u(x,t)) &= f(x,t) \quad &&{\rm on
}~~\Omega\times (0,T],\\
u(x,0)&= u_0(x) &&{\rm on }~~ \Omega,\\
u(x,t)&= 0 &&{ \rm on
} ~~\partial \Omega\times (0,T],
\end{aligned}
\end{equation}
where we assume that $\A \in \C^1([0,T],L^\infty(\Omega))$ and satisfies
\begin{equation}\label{eq: A positive}
0<a_{\min} < \A (x,t) < a_{\max}<\infty\quad { \rm on
} ~~\overline \Omega\times [0,T].\end{equation}
Here, ${\cD}^{\mu}$ is the Caputo's fractional derivative defined by
\[
{\cD}^{\mu} v(t)=I^{1-\mu}
\omega_{1-\mu}(t):=\frac{t^{-\mu}}{\Gamma(1-\mu)},
\]
where throughout the paper, $0<\mu <1$. Noting that, $I^{1-\mu}$ is the Riemman Liouville fractional integral operator, and $v'$ denotes the time partial derivative of $v$.
Over the past few decades, researchers have observed numerous biological, physical and financial systems in which some key underlying random motion conform to a model where the diffusion is anomalously slow (subdiffusion) and not to the classical model of diffusion. For instance, the fractional diffusion model problem (<ref>) is known to capture well the dynamics of subdiffusion processes, in which the mean square variance grows at a rate slower than that in a Gaussian process, see <cit.>. Fractional diffusion has been successfully used to describe diffusion in media with fractal geometry <cit.>, highly heterogeneous aquifer <cit.>,
and underground environmental problem <cit.>. Two distinct approaches can be used for modelling fractional sub-diffusion. One based on fractional Brownian motion and Langevin equations <cit.>, this leads to a diffusion equation with a varying diffusion coefficient exhibiting a fractional power law scaling in time <cit.>. The other based on continuous time random walks and master equations with power law waiting time densities which leads to a diffusion equation with fractional order temporal derivatives operating on the spatial Laplacian <cit.>.
The innovation of this paper is to investigate the piecewise linear time-stepping discontinuous Galerkin (DG) method combined with the standard finite elements (FEs) in space for solving numerically time fractional models with
variable diffusion coefficients of the form (<ref>). Since their inception in the early 1970s, DG methods have found numerous applications <cit.>, including for the time discretization of fractional diffusion and fractional wave equations, <cit.>. Their advantages include excellent stability properties and suitability for adaptive refinement based on a posteriori error estimates <cit.> to handle problems with low regularity. The present work is motivated by an earlier paper <cit.>. There in, the first author and McLean considered a piecewise-linear DG method for a fractional diffusion problem with a constant diffusivity:
\begin{equation}
\label{eq: reimann} u'(x,t) - {^{R}{\rm D}}^{1-\mu} \Delta u(x,t) = f(x,t) \quad\mbox{ for } (x,t)\in \Omega\times (0,T],
\end{equation}
where $^{R}{\rm D}^{1-\mu} u:=\frac{\partial }{\partial t}( I^\mu u)$ (Riemann–Liouville fractional derivative). Recently, high order $hp$-DG methods with exponential rates of convergence for fractional diffusion (<ref>) and also for fractional wave equations were studied in <cit.>. Noting that, when $\A$ is constant and $f\equiv 0$ in (<ref>), one may look at (<ref>) as an alternative representation of (<ref>).
Numerical solutions for model problems of the form (<ref>) with constant diffusion parameter $\A$ have attracted considerable interest in recent years. The case of variable coefficients is indeed very
interesting and also practically important. However, due to the additional difficulty in this case, there are only few papers in the existing literature. With $\Omega=(0,L)$, Alikhanov <cit.> constructed a new difference analog of the time fractional Caputo derivative with the order of approximation $O(k^{3-\mu})$. Difference schemes of order $O(h^q+k^2)$ (with $q\in\{2,4\}$) were proposed and analyzed assuming that $u$ is sufficiently regular, where $k$ is the temporal grid size and $h$ is the spatial grid size. For a time independent diffusivity, Zhao and Xu <cit.> proposed a compact difference scheme for (<ref>). Stability and convergence properties of the scheme were proved. For time fractional convection-diffusion problems, Cui <cit.> studied a compact exponential scheme. The stability and the convergence analysis were showed assuming that the coefficients of the model problem are constants. For time independent coefficients, Saadatmandi et al. <cit.> investigated the Sinc-Legendre collocation method.
For one-dimensional spatial domains and constant diffusion parameter $\A$, Murio <cit.> and Zhang et al. <cit.> studied two classes of finite difference (FD) methods. Stability properties were provided. Another FD scheme in time (with $L1$ approximation for the Caputo fractional derivative) combined with the spatial fourth order compact difference approach was studied by Ren et al. <cit.>. Convergence rates of order $O(k^{1+\mu}+ h^4)$ were proved. Murillo and Yuste <cit.> presented an implicit FD method over non-uniform time steps. An adaptive procedure was described to choose the size of the time meshes. Lin and Xu <cit.> combined a FD scheme in time and a spectral method in space. Accuracy of order $ O(k^{1+\mu}+ r^{-m})$ was proved, where $r$ is the spatial polynomial degree, and $m$ is related to the regularity of the exact solution $u$. Later, Li and Xu <cit.> developed and analyzed a time-space spectral method. Zhao and Sun <cit.> combined an order reduction approach and $L1$ discretization of the fractional derivative.
A box-type scheme was constructed and a convergence rate of order $ O(k^{1+\mu} + h^{2})$ had been proved. Finite central differences in time combined with the FE method in space was studied by Li and Xu <cit.>. For a smooth $u$, a convergence rate of order $O(k^2+h^{\ell+1})$ was achieved where $\ell$ is the degree of the FE solutions in space. Recently, a similar convergence rate was shown by Zeng et al. <cit.> where the fractional linear multistep method was used for the time discretization. For a high-order local DG method for space discretization, we refer to the work by Xu and Zheng <cit.>.
For two- or three-dimensional spatial domains with $\A=1$ in (<ref>), Brunner et al. <cit.> used an algorithm that couples an adaptive time
stepping and adaptive spatial basis selection approach for the numerical solution of (<ref>). A semi-discrete piecewise linear Galerkin FE and lumped mass Galerkin methods were studied by Jin et al.
<cit.>. An optimal error with respect to the
regularity error estimates was established for $f\equiv 0$ and non-smooth initial data $u_0$. Cui <cit.> studied the convergence analysis of compact alternating direction implicit (ADI) schemes for sufficiently smooth solutions of (<ref>). For three-dimensional spatial domains,
a fractional ADI scheme was proposed and analyzed by Chen et al. <cit.>.
Mustapha et al. <cit.> proposed low-high order time stepping discontinuous
Petrov-Galerkin methods combined with
FEs in space. Using variable time meshes, $O(k^{m+(1-\mu)/2}+h^{r+1})$ convergence rates were shown, where $m$ and $r$ are the degrees of approximate solutions in the time and spatial variables, respectively. Optimal convergence rates in both variables were demonstrated numerically. In <cit.>, a hybridizable DG method in space was extensively studied by Mustapha et al..
The outline of the paper is as follows. Section <ref> introduces a fully discrete DG FE scheme. In
Section <ref>, we prove the stability of the discrete solution and provide a remark about the existence and uniqueness of the numerical solution. Section <ref> is devoted to introduce time and space projection operators that will be used later to show the convergence of the numerical scheme. The error analysis is given in
Section <ref>. Using suitable refined time-steps (towards $t=0$) and quasi-uniform spatial meshes, in the $L^2((0,T),L^2(\Omega))$-norm,
convergence of order $O(k^{2-\frac{\mu}{2}}+h^2)$ is achieved. Section <ref> is dedicated to present a sample
of numerical test which illustrate that our error bounds are
pessimistic. For a strongly graded time mesh, in the stronger $L^\infty((0,T),L^2(\Omega))$-norm, we observe optimal convergence rates, that is, error of order $O(k^{2}+h^2)$.
§ THE NUMERICAL METHOD
To describe our fully discrete DG FE method, we introduce a time partition of the interval $[0,T]$ given by the points:
$0=t_{0}<t_{1}<\cdots<t_{N}=T\,.$ We set $I_n=(t_{n-1},t_n)$ and $k_n=t_n-t_{n-1}$ for $1\le n\le N$ with $k:=\max_{1\le n\le
N}k_n$. Let $S_h\subseteq H_0^1(\Omega)$ denotes the space of continuous, piecewise polynomials
of total degree $\le 1$ with respect to a quasi-uniform partition of $\Omega$ into
conforming triangular finite elements, with maximum diameter $h$.
Next, we introduce our time-space finite dimensional DG FE space:
\[
\W=\{w \in L^2((0,T), S_h):~~w|_{I_{n}}\in
\p_1(S_h)~{\rm for}~1\le n\le N\} \]
where $\p_1(S_h)$ denotes the space
of linear polynomials in the time variable $t$, with coefficients in $S_h$.
We denote the left-hand limit, right-hand limit and
jump at $t_n$ by
\[
w^n:=w(t_n)=w(t_n^-),\quad w^n_+:=w(t_n^+),\quad [w]^n:=w^n_+-w^n,
\]
respectively. The weak form of the fractional diffusion equation in (<ref>) is
\begin{equation}\label{eq: weak In}
\int_{I_n}\bigl[\iprod{{\cD}^{\mu} u,v}+a\bigl(t,
=\int_{I_n}\iprod{f,v}\,dt,\quad \forall ~v\in L^2\bigl(I_n, H^1(\Omega)\bigr)\,.
\end{equation}
Throughout the paper, $\iprod{\cdot,\cdot}$ denotes the $L^2$-inner product and $\|\cdot\|$ is the associated norm, and $\|\cdot\|_m$ ($m\ge 1$) denotes the norm on the Sobolev space $H^m(\Omega)$. We use $\| \cdot\|_{L^q(Y)}$ ($q\ge 1)$ to denote the norm on
$L^q((0,T),Y(\Omega))$ for any Sobolev space $Y(\Omega)$.
For each fixed $t \in (0,T]$, $a(t,\cdot, \cdot):H_0^1 (\Omega)\times H_0^1(\Omega)\rightarrow\mathbb{R}$
is the bilinear form
$$ a(t,v,w)= \iprod{\A(\cdot,t) \nabla v,\nabla w}= \int_{\Omega} \A (x,t)\nabla v(x)\cdot \nabla w(x)\,dx$$
associated with the operator $\nabla \cdot (\A (\cdot,t) \nabla)$ which is symmetric and positive definite (by (<ref>)), that is,
there exist positive constants $c_0$ and $c_1$ such that
\begin{equation}\label{pd}
c_0\| v(t)\|_1^2 \leq |v(t)|_1^2:=a(t,v,v) \leq c_1 \|v(t)\|_1^2\quad \forall ~~v(t) \in H_0^1(\Omega)\,.
\end{equation}
The DG FE approximation $U\in \W$ is
defined as follows: Given $U(t)$ for $0\le t\le t_{n-1}$, the
solution $U\in \p_1(S_h)$ on $I_n$
is determined by requesting that for $1\le n\le N$,
\begin{multline*}
\int_{I_n}\bigl[\iprod{{\cD}_{dg}^{\mu} U+
\sum_{j=0}^{n-1}\omega_{1-\mu}(t-t_j)\,[U]^j,X}
+a\bigl(t, U,X\bigr)\bigr]\,dt
=\int_{I_n}\iprod{f,X}\,dt,~ \forall~X\in \p_1(S_h),
\end{multline*}
with $U^0_+=U^0 \in S_h$ is a suitable approximation of the initial data $u_0$, where
\[ {\cD}^{\mu}_{dg} U(t):=\sum_{j=1}^n\int_{t_{j-1}}^{\min\{t_j,t\}}\omega_{1-\mu}(t-s)\,U'(s)\,ds\quad {\rm for}~~t\in I_n\,.\]
\begin{equation}\label{eq: Dmu U}
\begin{aligned}
{\rm ^R D}^\mu U(t)&:= \frac{\partial}{\partial
+\sum_{j=1}^{n-1}\omega_{1-\mu}(t-t_j)\,[U]^j\quad {\rm for}~~t\in I_n,
\end{aligned}
\end{equation}
our scheme can be rewritten in a compact form as follows: for $1\le n\le N$,
\begin{multline}\label{eq: DGFM}
\int_{I_n}\bigl[\bigiprod{{\rm ^R D}^\mu U,X}
+a\bigl(t, U,X\bigr)\bigr]\,dt
=\int_{I_n}\iprod{f+\omega_{1-\mu}(t)U^0,X}\,dt\quad \forall~X\in \p_1(S_h).
\end{multline}
Noting that, since the DG FE scheme (<ref>) amounts to a square linear system, the existence of the numerical solution $U$ follows from its uniqueness. The uniqueness follows immediately from the above stability property in Theorem <ref>.
§ STABILITY OF THE NUMERICAL SOLUTION
To show the stability of the DG FE scheme (<ref>), we claim first the identity: $v(t)=I^{\mu}({\rm ^R D}^\mu v)(t)$ for any $v \in \W.$ If $v$ is an absolutely continuous function in the time variable, this identity follows by applying the fractional integral operator $I^\mu$ to both sides of the equality ${\rm ^R D}^\mu v(t)={\cD}^{\mu}v(t)+\omega_{1-\mu}(t)v(0)$ and then changing the order of integrals and using the identity: $\int_{s}^{\tilde t}
\omega_{1-\mu}(t-s)
\omega_{\mu}(\tilde t -t)\,dt=1.$
If $v\in \W$, then
$$v(t)=I^{\mu}({\rm ^R D}^\mu v)(t)\quad {\rm for~~} t \in I_n~~{\rm with}~~1\le n\le N.$$
Since $v$ has possible discontinuities at the time nodes $t_0,t_1,\cdots, t_{j-1}$, from (<ref>),
\begin{equation}\label{eq:formula for Dalpha}
{\rm ^R D}^\mu v(s)=\omega_{1-\mu}(s)v^0_+
+\sum_{i=1}^{j-1}\omega_{1-\mu}(s-t_i)\,[v]^i+{\cD}^{\mu}_{dg} v(s) ~~{\rm for}~~s \in I_j\,.
\end{equation}
Applying the operator $I^{\mu}$ to both sides and using $I^{\mu}({\cD}^{\mu} v)(t)=\int_0^t v'(s)\,ds$, we observe
\begin{multline*}
I^{\mu}({\rm ^R D}^\mu v)(t)=v^0_++\sum_{j=2}^{n-1}\int_{I_j}
\omega_{\mu}(t-s)
\sum_{i=1}^{j-1}\omega_{1-\mu}(s-t_i)\,[v]^i\,ds\\+\int_{t_{n-1}}^{t}
\omega_{\mu}(t-s)
\sum_{i=1}^{n-1}\omega_{1-\mu}(s-t_i)\,[v]^i\,ds+\int_0^t v'(s)ds~~{\rm for}~~t \in I_n\,.
\end{multline*}
Now, changing the order of summations and rearranging the
terms yield
\begin{multline*}
I^{\mu}({\rm ^R D}^\mu v)(t)=v^0_++\sum_{i=1}^{n-2}\sum_{j=i+1}^{n-1}\int_{I_j}
\omega_{\mu}(t-s)
\omega_{1-\mu}(s-t_i)\,[v]^i\,ds\\+\sum_{i=1}^{n-1}\int_{t_{n-1}}^{t}
\omega_{\mu}(t-s)
\omega_{1-\mu}(s-t_i)\,[v]^i\,ds+\sum_{j=1}^{n}\int_{t_{j-1}}^{\min
\{t,t_j\}} v'(s)ds \\=v^0_++\sum_{i=1}^{n-2}\int_{t_{i}}^{t}
\omega_{\mu}(t-s)
\omega_{1-\mu}(s-t_i)\,[v]^i\,ds\\+\int_{t_{n-1}}^{t}
\omega_{\mu}(t-s)
\omega_{1-\mu}(s-t_{n-1})\,[v]^{n-1}\,ds+\sum_{j=1}^{n}\int_{t_{j-1}}^{\min
\{t,t_j\}} v'(s)ds\,.
\end{multline*}
Integrating and simplifying, then we have
\begin{align*}
I^{\mu}({\rm ^R D}^\mu v)(t)&=v^0_++\sum_{i=1}^{n-1}\,[v]^i+\sum_{j=1}^{n-1}\,(v^j-v^{j-1}_+)+v(t)-v^{n-1}_+=v(t)~~{\rm for}~~t \in I_n\,.\quad \Box
\end{align*}
In the next lemma we state some important properties of the Riemman Liouville factional operators. These properties will be used to show the stability of the numerical scheme, as well as, in our error analysis in the forthcoming section.
For $\ell \in \{0,1\}$, we let $ \C^\ell(J_n,L^2(\Omega))$ ($J_n:=\cup_{j=1}^n I_j$) denote the space of functions $v: J_n \rightarrow L^2(\Omega)$ such that
the restriction $v|_{I_j}$ extends to an $\ell$-times continuously differentiable function on the
closed interval $\overline I_j$ for $1 \le j \le n.$ For later use, we let
\[\|v\|_{I_j}:=\sup_{t\in I_j}\|v(t)\|\quad{\rm and}\quad \|v\|_{J_n}:=\max_{j=1}^n \|v\|_{I_j},\]
where we drop $n$ when $J_n=J_N$.
For $1 \le n \le N$ and for $0<\alpha<1,$ we have
(i) The operator $^{R}{\rm D}^\alpha$ satisfies: for $v \in \C^1(J_n,L^2(\Omega))$,
$$\int_0^{t_n} \iprod{^{R}{\rm D}^\alpha v,v}\,dt \ge \frac{(\pi \alpha)^\alpha}{(1+\alpha)^{1+\alpha}}\cos\Big( \frac{\alpha\pi}{2}\Big) t_n^{-\alpha} \int_0^{t_n}\|v(t)\|^2\,dt\,.$$
(ii) The integral operator $I^{\alpha}$ satisfies: for $\, v,\,w \in \C^0(J_n,L^2(\Omega))$
$$ \left|\int_0^{t_n}\iprod{I^\alpha v,w}\,dt\right|^2
\le \sec^2\Big( \frac{\alpha\pi}{2}\Big)\,\int_0^{t_n}\iprod{I^\alpha v,v}\,dt \int_0^{t_n}\iprod{I^\alpha w,w}\,dt\,.
The property (i) was proven in <cit.> by using the Laplace transform and Plancherel Theorem.
For the proof of the property (ii), see <cit.>. $\quad \Box$
The next theorem shows the stability of the DG FE scheme.
Assume that $U^0 \in L^2(\Omega)$ and $f \in L^2((0,T),L^2(\Omega))$. Then,
\[
\int_0^T\|U\|_1^2\,dt
\leq C T^{1-\mu}\|U^0\|^2
\]
Choosing $ X=U$ in the DG FE scheme (<ref>),
and then summing over $n$, we obtain
\[
\int_0^{T}\bigl[\bigiprod{{\rm ^R D}^\mu U,U}
=\int_0^{T} \iprod{ f+\omega_{1-\mu}(t) U^0,U}\,dt.
\]
Since $a(\cdot,U,U)\ge c_0\|U\|_1^2$ by (<ref>) and $\iprod{f,U}\le \frac{1}{2c_0}\|f\|^2+\frac{c_0}{2}\|U\|^2$, we have
\[
\int_0^{T}\bigl[\bigiprod{{\rm ^R D}^\mu U,U}
\le \int_0^{T} \Big(\iprod{\omega_{1-\mu}(t) U^0,U}+\frac{1}{2c_0}\|f\|^2\Big)\,dt\,.
\]
Using the identity $U(t)=I^{\mu}({\rm ^R D}^\mu U)(t)$ from Lemma <ref>, Lemma
<ref> (ii), the inequality $ab\le \frac{a^2}{4}+b^2$, and the identity $I^{\mu} \omega_{1-\mu}(t)=1$, yield
\begin{multline}\label{eq: U0 RL}
\int_0^{T} \iprod{\omega_{1-\mu}(t) U^0,U}\,dt=\int_0^{T} \iprod{ \omega_{1-\mu}(t) U^0,I^{\mu}({\rm ^R D}^\mu U)}\,dt
\\ \leq \frac{1}{4}\int_0^{T}\bigiprod{{\rm ^R D}^\mu U,U}\,dt+\sec^2(\mu
\pi/2)
\int_0^{T}\omega_{1-\mu}(t)(I^{\mu} \omega_{1-\mu})(t)\,dt\|U^0\|^2 \\ \leq \frac{1}{4}\int_0^{T}\bigiprod{{\rm ^R D}^\mu U,U}\,dt+C\,T^{1-\mu}\|U^0\|^2\,.
\end{multline}
To complete the proof, we combine the above two equations and use the positivity property of the operator ${\rm ^R D}^\mu$ given by Lemma
<ref> (i).$\quad \Box$
§ PROJECTIONS AND ERRORS
In this section, we introduce time and space projections, and then derive some bounds
and errors properties that will be used later in our convergence analysis.
§.§ Projection in space
For each $t\in [0,T]$, the elliptic projection operator $\,R_h:H^1_0(\Omega) \to S_h$ is defined by
\begin{equation}\label{eq: Ritz projection} a(t,R_h v-v, \chi)=0\quad \forall~~ \chi\in S_h\,.\end{equation}
By the assumption $\A \in \C^1([0,T],L^\infty(\Omega)),$ for each $t\in (0,T)$, the projection error $\xi:=R_h u -u$ has the well-known approximation property:
\begin{equation}\label{eq:estimate ritz projector 1} \| \xi(t)\|+h\|\nabla \xi(t)\|\leq C\,%\frac{h^{\min\{s,r\}+1}}{r^{s+1}}
h^2\|u(t)\|_2\quad {\rm for}~~u(t)\in H^2(\Omega) \cap H^1_0(\Omega)\,.%\frac{h^{\min\{s,r\}+1}}{r^{s+1}}
\end{equation}
Moreover, we have
\begin{equation}\label{eq:estimate ritz projector 2} \| \xi'(t)\|\leq C\,h^2(\|u(t)\|_2+\|u'(t)\|_2)\quad {\rm for}~~u(t),\,u'(t) \in H^2(\Omega) \cap H^1_0(\Omega)\,.%\frac{h^{\min\{s,r\}+1}}{r^{s+1}}
\end{equation}
To show (<ref>), we decompose
$\xi'$ as: $\xi'=g_h +(R_h u'-u')$ where for each $t\in (0,T)$, $g_h(t)=(R_h u)'(t)-R_h u'(t)\in S_h$. Since $\|(R_h u'-u')(t)\|\le C\,h^2\|u'(t)\|_2$ by the approximation property (<ref>) applied to $u'$,
it remains to derive a similar bound for $g_h$. To do so, we use the Nitsche’s trick: for each $t\in (0,T)$, there is $\phi \in H^2(\Omega)$ such that
\[-\nabla \cdot (\A(t) \nabla \phi)=g_h(t)\quad{\rm in}~~\Omega,\quad \phi=0~~{\rm on}~~\partial \Omega\]
with $\|\phi\|_2\le C\|\nabla \cdot (\A(t) \nabla \phi)\|$ which holds for convex polyhedral domains. Taking the inner product with
$g_h(t) \in S_h$, and then using the orthogonality
property of $R_h$,
\[\|g_h(t)\|^2=a(t,\phi, g_h(t))=a(t,R_h\phi, g_h(t))=a(t,g_h(t),R_h\phi)\,.\]
But, by the definition of $R_h$, for each $t\in (0,T)$,
\begin{equation}\label{eq: g_h}
\begin{aligned}
a(t, g_h(t), \chi)&=a(t, (R_h u)' (t), \chi)-a(t, u' (t), \chi)\\
&=\frac{d}{dt}a(t, \xi (t), \chi)-\iprod{\A'(t) \nabla \xi(t),\nabla \chi}\\
&=-\iprod{\A'(t)\nabla \xi(t),\nabla \chi}\quad \forall~~ \chi\in S_h\,.\end{aligned}
\end{equation}
\begin{multline*}
\|g_h(t)\|^2=
-\iprod{\A'(t)\nabla \xi(t),\nabla R_h \phi}
=\iprod{\A'(t)\nabla \xi(t),\nabla (\phi-R_h \phi)}+\iprod{ \xi(t),\nabla\cdot(\A'(t)\nabla \phi)}\,.\end{multline*}
Finally, using the Cauchy-Schwarz inequality, the approximation property in (<ref>), and the inequality $\|\phi\|_2\le C\|\nabla \cdot (\A(t) \nabla \phi)\|$, we observe
\[\|g_h(t)\|^2\le C\|\nabla (\phi-R_h \phi)\|\,\|\nabla \xi(t)\|+C\|\xi(t)\|\,\|\phi\|_2\le C\,h^2\|g_h(t)\|\,\|u(t)\|_2\,.\]
The proof of (<ref>) is completed now.
§.§ Projection in time
The local $L^2$-projection operator $\pw : L^2( I_{n},L^2(\Omega)) \to
\mathcal{C}( I_{n},\p_1(L^2(\Omega))$ defined by:
\[\int_{I_n} \iprod{\pw v-v,\,w}\,dt=0~~\forall~~w \in \p_1(L^2(\Omega))\quad{\rm for}~~1\le n\le N,\]
where $\p_{1}(L^2(\Omega))$ is the space
of linear polynomials in the time variable $t$, with coefficients in $L^2(\Omega)$ . Explicitly,
\[
\pw v(t)=\frac{12}{k_n^3}(t-t_{n-\frac{1}{2}})\int_{I_n}(s-t_{n-\frac{1}{2}})\,v(s)\,ds+\frac{1}{k_n}\int_{I_n}v(s)\,ds
\quad\text{for $t\in I_n$\,},
\]
where $t_{n-\frac{1}{2}}:=(t_{n-1}+t_n)/2.$ Hence, for $v'\in L^1( I_{n},L^2(\Omega))$,
\[
(\pw v)'(t)=\frac{12}{k_n^3}\int_{I_n}(s-t_{n-\frac{1}{2}})\,v(s)\,ds=\frac{6}{k_n^3}\int_{I_n}(t_n-t)(s-t_{n-1})\,v'(s)\,ds\,.
\]
Thus, for $1\le n\le N$, we have
\begin{equation}\label{(ii)}
\|\pw v(t)\| \leq \frac{4}{k_n}\int_{I_n}\|v(s)\|\,ds\quad{\rm and}\quad \|(\pw v)'(t)\| \leq \frac{3}{2k_n}\int_{I_n}\|v'(s)\|\,ds\,.
\end{equation}
Setting $\eta_v=\pw v-v$, we have the well-known projection error bound: for $t\in I_n$,
\begin{equation}\label{projection error}
\|\eta_v(t)\|+k_n\,\|\eta_v'(t)\|\le C\, k_n^{\ell-1}\int_{I_n}\Big\|\frac{\partial^\ell v}{\partial t^\ell} (s)\Big\|\,ds\quad{\rm for}~~\ell=1,\,2.
\end{equation}
Next, we show an error bound property of $\pw$ that involves the operator ${\rm ^R D}^\mu.$
Let $\frac{\partial^\ell v}{\partial t^\ell} \in L^1((0,t_n),L^2(\Omega))$ for $\ell\in \{1,\,2\}$. We have
\begin{align*}
\int_{I_n} \bigiprod{{\rm ^R D}^\mu \eta_v,\eta_v}\,dt &\leq
C\,k_n^{1-\mu} \max_{j=1}^{n} k_j^{2\ell-2}\Big(\int_{I_j}\Big\|\frac{\partial^\ell v}{\partial t^\ell} \Big\|\, dt\Big)^2\quad{\rm for}~~1\le n\le N.
\end{align*}
We integrate by parts and notice that
\begin{equation}\begin{aligned}
\int_{I_n}\bigiprod{{\rm ^R D}^\mu \eta_v,\eta_v}\,dt
&= \iprod{I^{{1-\mu}}\eta_v(t), \eta_v(t)}\bigg|_{t_{n-1}^+}^{t_n^-} - \int_{I_n}\iprod{I^{{1-\mu}}\eta_v, \eta_v'}\,dt\\
&= \iprod{\I^n(t_n), \eta_v(t_n)}- \int_{I_n}\iprod{\I^n(t), \eta_v'(t)}\,dt,
\end{aligned}
\end{equation}
where for $t\in I_n$,
\begin{align*}\I^n(t)&:=I^{{1-\mu}}\eta_v(t)-I^{{1-\mu}}\eta_v(t_{n-1})\\
[\omega_{{1-\mu}}(t-s)-\omega_{{1-\mu}}(t_{n-1}-s)] \eta_v(s)\,ds+ \int_{t_{n-1}}^t
\omega_{{1-\mu}}(t-s)\eta_v(s)\,ds\,.\end{align*}
Simplifying then integrating, we observe
\begin{align*} \|\I^n(t)\|
\Big(\int_0^{t_{n-1}}
[\omega_{{1-\mu}}(t_{n-1}-s)-\omega_{{1-\mu}}(t-s)] \,ds+ \int_{t_{n-1}}^t
\omega_{{1-\mu}}(t-s)\,ds\Big) \|\eta_v\|_{J_n}\\
&\le 2\,\omega_{2-\mu}(k_n) \, \| \eta_v\|_{J_n}\quad{\rm for}~~t\in I_n\,.\end{align*}
Therefore, an application of the Cauchy-Schwarz inequality gives
\begin{align*}
\int_{I_n}|\bigiprod{{\rm ^R D}^\mu \eta_v,\eta_v}|\,dt &\le 2\,\omega_{2-\mu}(k_n)\,\| \eta_v\|_{J_n}\left(\| \eta_v(t_n)\|+\int_{I_n}\| \eta_v'\|\,dt\right),
\end{align*}
and hence, using the error projection in (<ref>), we obtain the desired bound.
$\quad \Box$
Since $^R{\rm D}^{\mu} v(t)=\omega_{1-\mu}(t)v(0)+I^{{1-\mu}} v'(t)$ and since $\|\eta_v\|_{I_n}\le 5\|v\|_{I_n}$ by the triangle inequality and the first inequality in (<ref>), we have
\[\int_{I_n}
|\bigiprod{^R{\rm D}^{\mu} v,\eta_v}|\,dt\le
(\omega_{1-\mu}(t)\|v(0)\|+\|I^{{1-\mu}} v'\|)\,dt.\]
Summing over $n$ gives
\[
\int_0^T
|\bigiprod{^R{\rm D}^{\mu} v,\eta_v}|\,dt\le
5\,\|v\|_J(\omega_{2-\mu}(T)\|v(0)\|+\int_0^T\omega_{2-\mu}(T-s)\| v'(s)\|\,ds)\,.\]
On the other hand, noting that
\[\int_0^T
\bigiprod{^R{\rm D}^{\mu} \pw v,v}\,dt=
\iprod{I^{{1-\mu}} \pw v(T),v(T)}-\int_0^T
\iprod{I^{{1-\mu}} \pw v,v'}\,dt,\]
and hence, by the Cauchy-Schwarz inequality and the first inequality in (<ref>),
\[
\begin{aligned}\Big|\int_0^T
\bigiprod{^R{\rm D}^{\mu} \pw v,v}\,dt\Big|&\le
\|\pw v\|_{J}\int_0^T\Big[\omega_{1-\mu}(T-t)\|v(T)\|+
\int_0^t\omega_{1-\mu}(t-s)\,ds\|v'(t)\|\Big]dt\\
\]
We combine the above two inequalities and use that $\|v\|_J\le \|v(0)\|+\int_0^T\|v'\|\,dt$, we obtain the bound below that will be used to show the convergence of our scheme,
\begin{equation}\label{eq: bound 2}
\Big|\int_0^T
\bigiprod{^R{\rm D}^{\mu} v,\eta_v}\,dt\Big|+
\Big|\int_0^T
\bigiprod{^R{\rm D}^{\mu} \pw v,v}\,dt\Big| \le C\,T^{1-\mu}\,\Big(\|v(T)\|+\int_0^T\|v'\|\,dt\Big)^2\,.\end{equation}
§ ERROR ESTIMATES
This section is devoted to investigate the convergence of the DG FE scheme, (<ref>). We decompose the error as follows:
\begin{equation}\label{eq: U-u=theta+eta}
U-u=\zeta+\pw \xi+\eta_u\quad {\rm with}~~\zeta=U-\pw R_hu\,.\end{equation}
Recall that $\xi=R_h u-u$ and $\eta_u=\pw u-u$. The main task now is
to estimate $\zeta$.
Choose $U^0=R_h u_0$. For $1\le n\le N,$ we have
\[
\begin{aligned}
\|\zeta\|_{L^2(H^1)}^2
&\le C\Big(h^4C_1(k,u)+
\end{aligned}\]
\begin{equation}\label{eq: C1C2}
\begin{aligned}
\Big(k_n^{-\frac{\mu}{2}}\int_{I_n}\| u'\|_2\, dt\Big)^2+\Big(\|u_0\|_2+\|u'\|_{L^1(H^2)}\Big)^2,\\
C_2(k,u)&=\max_{n=1}^N \,k_n^{2\ell-2-\mu}\Big(k_n^{-\mu}\Big(\int_{I_n}\Big\|\frac{\partial^\ell u}{\partial t^\ell} \Big\|\,dt\Big)^2+\Big(\int_{I_n}\Big\|\nabla\frac{\partial^\ell u}{\partial t^\ell} \Big\|\,dt\Big)^2\Big)\,.\end{aligned}
\end{equation}
We start our proof by taking the inner product of (<ref>) with $\zeta$, using the identity ${\cD}^{\mu} u(t)={^R}{{\rm D}}^{\mu}u(t)-\omega_{1-\mu}(t)u_0$, and then integrating over the time subinterval $I_n$,
\[
\int_{I_n}\bigl[\bigiprod{{\rm ^R D}^\mu u,\zeta}
+a\bigl(t, u,\zeta\bigr)\bigr]\,dt
The above equation, the DG FE scheme (<ref>) and the decomposition in (<ref>) imply
\begin{multline}\label{eq: intermdiate equation 1}
\int_0^T\left(\bigiprod{^R{\rm D}^{\mu} \zeta ,\zeta}+|\zeta|_1^2\right)dt=\int_0^T\iprod{\omega_{1-\mu}(t)\,\xi(0),\zeta}\,dt\\
-\int_0^T\left[\bigiprod{^R{\rm D}^{\mu}
(\pw\xi+\eta_u), \zeta}+a(t, \pw\xi+\eta_u , \zeta)\right] dt\,. \end{multline}
Now, using the continuity property in Lemma <ref> $(ii)$, we notice that
\[
\begin{split}
\Big|\int_0^T\bigiprod{^R{\rm D}^{\mu} \eta_u,\zeta}\,dt\Big| &\le C\int_0^T \bigiprod{^R{\rm D}^{\mu}
\eta_u,\eta_u}\,dt+\frac{1}{4}\int_0^T\bigiprod{^R{\rm D}^{\mu} \zeta,\zeta}\,dt,\\
\Big| \int_0^T\bigiprod{^R{\rm D}^{\mu} \pw \xi,\zeta}\,dt\Big| &\le
C\int_0^T \bigiprod{^R{\rm D}^{\mu} \pw \xi,\pw \xi}\,dt+\frac{1}{4}\int_0^T\bigiprod{^R{\rm D}^{\mu}
\zeta,\zeta}\,dt\,.\end{split}\]
In addition, following the steps in (<ref>), we observe
\[
\int_0^{T} \iprod{\omega_{1-\mu}(t)\, \xi(0),\zeta}\,dt
\leq \frac{1}{4}\int_0^{T}\bigiprod{{\rm ^R D}^\mu \zeta,\zeta}\,dt+C\,T^{1-\mu}\|\xi(0)\|^2
\]
Inserting the above three inequalities in (<ref>), then simplifying, and using the positivity property of ${\rm ^R D}^\mu$, Lemma <ref> (i), yield
\begin{multline}\label{eq: intermdiate equation 2}
\int_0^T|\zeta|_1^2dt\le C\,T^{1-\mu}\|\xi(0)\|^2
+C\int_0^T\left(\bigiprod{^R{\rm D}^{\mu} \eta_u,\eta_u}+
\bigiprod{^R{\rm D}^{\mu} \pw\xi,\pw\xi}\right)dt\\+\sum_{n=1}^N \Big|\int_{I_n} a(t, \pw\xi+\eta_u , \zeta) dt\Big|\,. \end{multline}
From the definitions of the time projection $\pw$ and the space projection $R_h$,
\begin{multline*}
\int_{I_n}\iprod{\A(t_n)\nabla (\pw\xi+\eta_u), \nabla \zeta}\,dt
=\int_{I_n}\iprod{\A(t_n)\nabla \xi , \nabla \zeta}\,dt
=\int_{I_n}\iprod{[\A(t_n)-\A(t)]\nabla \xi , \nabla \zeta}\,dt
\end{multline*}
and so,
\begin{align*}
\Big|\int_{I_n}a(t, \pw\xi+&\eta_u , \zeta)dt\Big|\\
&=\Big|\int_{I_n}\iprod{\A(t_n)\nabla (\pw\xi+\eta_u) +[\A(t)-\A(t_n)]\nabla (\pw\xi+\eta_u) , \nabla \zeta}\,dt\Big|\\
&=\Big|\int_{I_n}\iprod{[\A(t)-\A(t_n)]\nabla (\eta_\xi+\eta_u) , \nabla \zeta}\,dt\Big|\\
&\le Ck_n \int_{I_n}\|\nabla (\eta_\xi+\eta_u)\|\,\| \nabla \zeta \|\,dt\,. \end{align*}
Thus, by the inequality $\|\nabla \eta_\xi(t)\|\le \|\nabla \xi(t)\|+4k_n^{-1}\int_{I_n}\|\nabla \xi(s)\|\,ds$ (follows from the triangle inequality and the first property of $\Pi_k$ in (<ref>)) for $t\in I_n$, and property (<ref>),
\[
\Big|\int_{I_n}a(t, \pw\xi+\eta_u , \zeta)dt\Big|
\le C\,k_n^2\int_{I_n}(\|\nabla \xi\|^2+\|\nabla\eta_u\|^2)\,dt+\frac{1}{2}\int_{I_n} |\zeta|_1^2\,dt
\,.\]
Inserting this in (<ref>) and using (<ref>) for $t=0$, we get
\begin{multline*}\int_0^T|\zeta|_1^2dt\le C\,h^4\|u_0\|_2^2 \\
+C\sum_{n=1}^N\int_{I_n}\Big(\bigiprod{^R{\rm D}^{\mu} \eta_u,\eta_u}+
\bigiprod{^R{\rm D}^{\mu} \pw\xi,\pw\xi}+ k_n^2(\|\nabla \xi\|^2+\|\nabla\eta_u\|^2)\Big)\,dt\,.\end{multline*}
But, for $t\in I_n$ and for $\ell \in \{1,\,2\}$,
\begin{align*}
\int_{I_n}\bigiprod{^R{\rm D}^{\mu} \eta_u,\eta_u}\,dt&\le C k_n\max_{j=1}^{n} k_j^{2\ell-2-\mu}\Big(\int_{I_j}\Big\|\frac{\partial^\ell u}{\partial t^\ell} \Big\|\, dt\Big)^2\quad {\rm by~Lemma}~ \ref{lem: estimat of Dalpha eta},\\
\|\nabla \xi(t)\|&\le C\,h\|u(t)\|_2\quad {\rm by~the~elliptic~projection~error}~ \eqref{eq:estimate ritz projector 1},\\
\|\nabla\eta_u(t)\|&\le C\, k_n^{\ell-1}\int_{I_n} \Big\|\nabla \frac{\partial^\ell u}{\partial t^\ell} \Big\|\,ds \quad {\rm by~the~time~projection~error}~ \eqref{projection error},
\end{align*}
where in the first inequality we also used the non-increasing time step assumption. So,
\begin{equation}\label{eq:orthogonality2}
\begin{aligned}
\int_0^T|\zeta|_1^2dt
&\le C\,h^4\|u_0\|_2^2+C\int_0^{T}
\bigiprod{^R{\rm D}^{\mu} \pw\xi,\pw\xi}dt+C\,h^2k^2\int_0^T\|u\|_2^2dt\\
& +C\max_{n=1}^N \,k_n^{2\ell-2-\mu}\Big(\Big(\int_{I_n}\Big\|\frac{\partial^\ell u}{\partial t^\ell} \Big\|\,dt\Big)^2+k_n^{\mu}\Big(\int_{I_n}\Big\|\nabla \frac{\partial^\ell u}{\partial t^\ell} \Big\|\,dt\Big)^2\Big)\,.
\end{aligned}
\end{equation}
It remains to estimate $\int_0^T
\bigiprod{^R{\rm D}^{\mu} \pw\xi,\pw\xi}\,dt$. From the decomposition:
\begin{equation}\label{eq: decomposition pw xi}
\int_{I_n}
\bigiprod{^R{\rm D}^{\mu} \pw\xi,\pw\xi}\,dt=
\int_{I_n}
\left[\bigiprod{^R{\rm D}^{\mu} \eta_\xi,\eta_\xi}+
\bigiprod{^R{\rm D}^{\mu} \xi,\eta_\xi}+
\bigiprod{^R{\rm D}^{\mu} \pw\xi,\xi}\right]\,dt\,.\end{equation}
By Lemma <ref>,
\[
\int_{I_n} \bigiprod{{\rm ^R D}^\mu \eta_\xi,\eta_\xi}\,dt \leq
\Big(\int_{I_j}\| \xi'\|\, dt\Big)^2\le C\,k_n\max_{j=1}^{n}
\Big(k_j^{-\frac{\mu}{2}}\int_{I_j}\| \xi'\|\, dt\Big)^2\,.\]
Inserting the above bound in (<ref>), then summing over $n$ and using the achieved bound in (<ref>), we obtain
\begin{equation}\label{eq: estimate of pi xi}
\int_0^T
|\bigiprod{^R{\rm D}^{\mu} \pw\xi,\pw\xi}|\,dt\le
\Big(k_n^{-\frac{\mu}{2}}\int_{I_n}\| \xi'\|\, dt\Big)^2+C\Big(\|\xi(0)\|+\int_0^T\|\xi'\|\,dt\Big)^2.\end{equation}
Finally, to complete the proof, we combine (<ref>) and (<ref>). $\quad \Box$
In the next theorem we show our main convergence results of the DG FE solution.
the exact solution $u$ of problem (<ref>) satisfies the finite regularity assumptions:
\begin{equation} \label{eq:countable-regularity v1}
\| u'(t)\|_2+t\| u''(t)\|_1\le {\bf M}\,\, t^{\sigma-1}\quad {\rm for}~~~t>0,
\end{equation}
for some positive constants ${\bf M}$ and $\sigma$. Due to the singular behaviour $u$ near $t=0$, we employ a family of non-uniform meshes, where the
time-steps are graded towards $t=0$; see <cit.>. More
precisely, for a fixed parameter $\gamma\ge1$, we assume that
\begin{equation}\label{eq: tn standard}
t_n=(n/N)^\gamma T\quad\text{for $0\le n \le N$.}
\end{equation}
One can easily see that the sequence of time-step sizes $\{k_j\}_{j=1}^N$ is nondecreasing, that is, $k_i\le k_j$ for $1\le i\le j\le N$. One can also show the following mesh property:
\begin{equation}\label{eq: time mesh property}
k_j\le \gamma k t_j^{1-1/\gamma}\,.\end{equation}
Let $u$ be the solution of (<ref>) satisfying the regularity property (<ref>) with $\sigma>\mu/2$. Let $U$ be the DG FE solution defined by (<ref>). Then, we have
\[
\int_0^T \|U-u\|^2 \,dt \le C\,(h^4+
k^{\gamma(2\sigma-\mu)})\quad{\rm for}\quad
1\le \gamma \le \frac{4-\mu}{2\sigma-\mu}\,
\]
where $C$ is a constant that depends on $T$, $\mu$, $\gamma$, $\sigma$, and on ${\bf M}$.
From the decomposition of the error in (<ref>), the triangle inequality, the bound in Theorem <ref>, the inequality $\|\pw \xi\|_{L^2(L^2)}\le \|\xi\|_{L^2(L^2)}$ by (<ref>), the elliptic projection error
(<ref>), the error from the time projection (<ref>), we have
\[
\int_0^T\|U-u\|^2\,dt \le C\Big(h^4C_1(k,u)+
\]
By the definitions of $C_1(k,u)$ and $C_2(k,u)$ in (<ref>), the regularity assumption (<ref>), and the inequality $h^2k^2\le \frac{1}{2}(h^4+k^4)$, we observe
\[
\begin{aligned}
\int_0^T\|U-u\|^2\,dt
&\le Ch^4\max_{n=1}^{N}
\Big(k_n^{-\frac{\mu}{2}}\int_{I_n}t^{\sigma-1}\, dt\Big)^2+Ch^4\Big(1+\int_0^Tt^{\sigma-1}\,dt\Big)^2\\
& +C\,k_1^{-\mu}\left(\int_{I_1}t^{\sigma-1}\,dt\right)^2+C\max_{n=2}^N \,k_n^{2-\mu}\left(\int_{I_n}t^{\sigma-2}\,dt\right)^2+C\,h^2k^2\\
&\le C(h^4\max_{n=1}^{N}
k_n^{2\sigma-\mu}+h^4 +k_1^{2\sigma-\mu}+\max_{n=2}^N \,k_n^{4-\mu}t_n^{2\sigma-4}+k^4)\\
&\le C\,(h^4+k^{\min\{\gamma(2\sigma-\mu),4-\mu\} })
\end{aligned}\]
where in the last inequality, by the mesh property (<ref>), we used
\begin{align*}
k_n^{4-\mu}{t_n}^{2(\sigma-2)}&\le C\,k^{4-\mu} t_n^{2(\sigma-2)+4 -\mu-(4-\mu)/\gamma}
\le C\,k^{\min\{\gamma(2\sigma-\mu),4-\mu\}}.\quad \Box\end{align*}
§ NUMERICAL RESULTS
We present a sample of numerical tests using a model problem in one space dimension, of the form (<ref>) with $\Omega=(0,1)$, $[0,T]=[0,1],$ and $\A(x,t)=1+t^{3/2}$. We choose $u_0(x)=\sin(\pi x)$ for the initial data and choose the source term $f$
so that
\begin{equation}\label{eq:num ex3}
u(t)= (1+t^{1-\mu}) \sin(\pi x)\,.
\end{equation}
One easily verifies that the regularity
condition (<ref>) holds for $\sigma=1-\mu$.
The numerical tests below reveal faster rates of convergence than those suggested by Theorem <ref>, and that our regularity assumptions are more restrictive than is needed in practice. More precisely, Theorem <ref> shows suboptimal (in time) convergence of order $O(k^{2-\frac{\mu}{2}}+h^2)$ for sufficiently graded time meshes in the time-space $L^2$-norm. However, we demonstrate numerically optimal (in both time and space) rates of convergence in the stronger $L^\infty(L^2)$-norm. To this end, We introduce a finer mesh
\begin{equation}\label{eq: fine grid}
\G^{m}=\{\,t_{j-1}+\ell k_j/m:\text{$j=1$, 2, \dots, $N$ and
$\ell=0$, 1, \dots, $m$}\,\},
\end{equation}
and define the discrete maximum norm $\|v\|_{\G^m}=\max_{t\in\G^{m}}\|v(t)\|$, so that, for sufficiently large values of $m$, $\|U_h-u\|_{\G^m}$ approximates the uniform error $\|U_h-u\|_{L^\infty(L^2)}$. In all tables, we choose $m=10.$
For the numerical illustration of the convergence rates in time, we choose $M$ (the number of uniform spatial subintervals) to be sufficiently large such that the spatial error is negligible compared to the error from the time discretization. We employ a time mesh of the form (<ref>). Tables <ref>, <ref> and <ref> show the error (in the stronger $L^\infty(L^2)$) and the rates of convergence when $\mu=0.3$, $0.5$, $2/3$ and $0.7$ respectively, for various choices of $N$ and $\gamma$. We observe optimal rates of order $O(k^{\gamma\sigma})$ for various choices of $1\le \gamma \le \frac{2}{\sigma}$ which is faster than the rate $O(k^{\frac{\gamma}{2}(2\sigma-\mu)})$ for $1\le \gamma \le \frac{4-\mu}{2\sigma-\mu}$ predicted by our theory in Theorem <ref>. Noting that, in Table <ref>, $\sigma\le \mu$ and thus the assumption $\sigma >\mu/2$ in this theorem is not sharp.
Errors and time convergence rates with $\mu=0.3$ for various choices of $\gamma$.
$N$ 2c|$\gamma=1$ 2c|$\gamma=2$
10 5.8997e-03 1.1252e-03 9.9332e-04
20 3.5981e-03 0.71339 4.1163e-04 1.4507 2.5524e-04 1.9604
40 2.1827e-03 0.72111 1.5008e-04 1.4556 6.4530e-05 1.9838
80 1.3208e-03 0.72468 5.4700e-05 1.4562 1.6137e-05 1.9996
160 7.9804e-04 0.72692 1.9995e-05 1.4519 4.0085e-06 2.0092
320 4.8168e-04 0.72840 7.3478e-06 1.4443 9.9164e-07 2.0152
Errors and time convergence rates with $\mu=0.5$ for various choices of $\gamma$.
$N$ 2c|$\gamma=1$ 2c|$\gamma=2$
2c|$\gamma=3$ 2c|$\gamma=4$
10 1.149e-02 3.262e-03 1.560e-03 1.882e-03
20 7.641e-03 0.589 1.619e-03 1.011 5.972e-04 1.385 4.869e-04 1.951
40 5.151e-03 0.569 8.037e-04 1.010 2.192e-04 1.446 1.209e-04 2.009
80 3.641e-03 0.500 3.997e-04 1.008 7.867e-05 1.478 2.933e-05 2.044
160 2.570e-03 0.503 1.992e-04 1.005 2.797e-05 1.492 7.011e-06 2.064
320 1.812e-03 0.504 9.940e-05 1.003 9.908e-06 1.497 1.774e-06 1.982
Errors and time convergence rates for various choices of $\gamma$.
$N$ 2c|$\gamma=1$ 2c|$\gamma=2$
2c|$\gamma=4$ 2c|$\gamma=6$
10 1.677e-02 7.579e-03 3.416e-03 3.261e-03
20 1.327e-02 0.338 4.677e-03 0.696 1.393e-03 1.294 9.087e-04 1.843
40 1.044e-02 0.346 3.036e-03 0.623 5.553e-04 1.327 2.471e-04 1.879
80 8.191e-03 0.350 1.940e-03 0.646 2.205e-04 1.332 6.435e-05 1.941
160 6.427e-03 0.350 1.229e-03 0.658 8.753e-05 1.333 1.643e-05 1.970
$N$ 2c|$\gamma=1$ 2c|$\gamma=3$
2c|$\gamma=5$ 2c|$\gamma=7$
10 1.792e-02 5.149e-03 3.625e-03 3.991e-03
20 1.446e-02 0.309 2.905e-03 8.258e-01 1.318e-03 1.459 1.121e-03 1.832
40 1.160e-02 0.318 1.577e-03 8.810e-01 4.673e-04 1.496 3.052e-04 1.877
80 9.290e-03 0.321 8.479e-04 8.955e-01 1.652e-04 1.499 7.981e-05 1.935
160 7.447e-03 0.319 4.547e-04 8.989e-01 5.843e-05 1.500
Next, we test the performance of the spatial FEs discretizaton of the scheme (<ref>). A uniform spatial
mesh that consists of $M$ subintervals where each is of width $h$ will be used. We refine the time mesh such that
the spatial error is dominating. By Theorem <ref>, a convergence of order $O(h^2)$ is expected. We illustrate
these results in Table <ref>.
Errors and convergence rates in space with $\mu=0.3, 0.5$ and $0.7$.
$M$ 2c|$\mu=0.3$ 2c|$\mu=0.5$
10 1.2156e-02 1.2780e-02 1.2563e-02
20 3.1130e-03 1.9653 3.2743e-03 1.9646 3.1768e-03 1.9836
40 7.8803e-04 1.9820 8.2897e-04 1.9818 7.9873e-04 1.9918
80 1.9826e-04 1.9909 2.0864e-04 1.9903 2.0029e-04 1.9956
160 4.9724e-05 1.9954 5.2355e-05 1.9946 5.1065e-05 1.9717
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1511.00402
|
Dedicated to Professor Tony J. Puthenpurakal
Mafi] Amir Mafi
On the computation of the Ratliff-Rush closure]
On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length
A. Mafi, Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj,
Iran. [email protected]
[2000]13A30, 13D40, 13H10.
Let $(R,\fm)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\fm$-primary ideal of $R$ and $J$ be a minimal reduction of $I$. In this paper we show that if $\widetilde{I^k}=I^k$ and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde{I^n}=I^n$ for all $n\geq k$. As a consequence, we can deduce that if $r_J(I)=2$, then $\widetilde{I}=I$ if and only if $\widetilde{I^n}=I^n$ for all $n\geq 1$. Moreover, we recover some main results of [<ref>] and [<ref>]. Finally, we give a counter example for Question 3 of [<ref>].
§ INTRODUCTION
Throughout this paper, we assume that $(R,\fm)$ is a Cohen-Macaulay local ring of positive dimension $d$, infinite residue field and $I$ an $\fm$-primary ideal of $R$. An ideal $J\subseteq I$ is called a reduction of $I$ if $I^{n+1}=JI^n$ for some $n\in\mathbb{N}$. A reduction $J$ is called a minimal reduction of $I$ if it does not properly contain a reduction of $I$. The least such $n$ is called the reduction number of $I$ with respect to $J$, and denoted by $r_J(I)$. These notions were introduced by Northcott and Rees [<ref>], where they proved that minimal reductions of $I$ always exist if the residue field of $R$ is infinite. Recall that $x\in I$ is a superficial element of $I$ if there exists $k\in\mathbb{N}_0$ such that $I^{n+1}:x=I^n$ for all $n\geq k$. A set of elements $x_1,...,x_d$ is a superficial sequence of $I$ if $x_i$ is a superficial element of $I/{(x_1,...,x_{i-1})}$ for all $i=1,...,d$. A superficial sequence $x_1,...,x_d$ of $I$ is called tame if $x_i$ is a superficial element of $I$, for all $i=1,...,d$. Elias [<ref>] defined and proved the tame superficial sequence exists (see also [<ref>]). Swanson [<ref>] proved that if $x_1,...,x_d$ is a superficial sequence of $I$, then $J=(x_1,...,x_d)$ is a minimal reduction of $I$. It is known that every minimal reduction can be generated by superficial sequence (see [<ref>] or [<ref>]).
The Ratliff-Rush closure of $I$ is defined as the ideal
$$\widetilde{I}=\cup_{n\geq 1}(I^{n+1}:I^n).$$
It is a refinement of the integral closure of $I$ and $\widetilde{I}=I$ if $I$ is integrally closed (see [<ref>]). The Ratliff-Rush filtration $\widetilde{I^n}$, $n\in\mathbb{N}_0$, carries important information on the associated graded ring $G(I)=\bigoplus_{n\geq 0}{I^n}/{I^{n+1}}$. For example, Heinzer, Lantz and Shah [<ref>] showed that the $\depth G(I)\geq 1$ if and only if $\widetilde{I^n}=I^n$ for all $n\in\mathbb{N}_0$.
The aim of this paper is to compute the Ratliff-Rush closure in some senses and as an application, we shall reprove some main results of [<ref>], [<ref>] and [<ref>]. Finally, we reprove Theorem 1 of [<ref>] and Theorem 1.6 of [<ref>] with a much easier proof, and we also give a counter example for Question 3 of [<ref>]. This example also says that Theorem 1.8 of [<ref>] does not hold in general. For any unexplained notation or terminology, we refer the reader to [<ref>] and [<ref>].
§ RATLIFF-RUSH CLOSURE, ASSOCIATED GRADED RING
Let $d=2$, $x_1,x_2$ be a superficial sequence of $I$ and $J=(x_1,x_2)$. Let $k\in\mathbb{N}_0$ such that $J\cap I^n=JI^{n-1}$ for all $n\geq k+1$. Then $\widetilde{I^n}=I^n$ for all $n\geq 1$ if and only if $I^n:x_1=I^{n-1}$ for $n=1,...,k$.
$(\Longrightarrow)$ immediately follows by [<ref>, Corollary 2.7].
$(\Longleftarrow)$. By [<ref>, Corollary 2.7], it is enough for us to prove
$I^n:x_1=I^{n-1}$ for all $n\geq k$. By using induction on $n$, it is enough to prove the result for $n=k+1$. For this, firstly we prove that $JI^k:x_1=I^k$. But this is an elementary fact that $JI^k:x_1=(x_1I^k+x_2I^k):x_1=I^k+(x_2I^k:x_1)$ and also $x_2I^k:x_1=x_2I^{k-1}$. Hence $JI^k:x_1=I^k$. Therefore, by our assumption, we have $(J\cap I^{k+1}):x_1=I^k$ and so we have $I^{k+1}:x_1=I^k$, as desired.
The following result immediately follows by Proposition 2.1.
Let $d=2$, $x_1,x_2$ be a superficial sequence of $I$ and $J=(x_1,x_2)$. Let $k\in\mathbb{N}_0$ such that $r_J(I)=k$. Then $\widetilde{I^n}=I^n$ for all $n\geq 1$ if and only if $I^n:x_1=I^{n-1}$ for $n=1,...,k$.
Let $d=2$, $x_1,x_2$ be a superficial sequence of $I$ and $J=(x_1,x_2)$ such that $r_J(I)=2$. Then $\widetilde{I^n}=I^n$ for all $n\geq 1$ if and only if $I^2:x_1=I$.
The Hilbert-Samuel function of $I$ is the numerical function that measures the growth of the length of $R/I^n$ for all $n\in\mathbb{N}$. For all $n$ large this function $\lambda(R/I^n)$ is a polynomial in $n$ of degree $d$
where $e_0(I),e_1(I),...,e_d(I)$ are called the Hilbert coefficients of $I$.
Let $A=\bigoplus_{m\geq 0}A_m$ be a Notherian graded ring where $A_0$ is an Artinian local ring, $A$ is generated by $A_1$ over $A_0$ and $A_{+}=\bigoplus_{m>0}A_m$. Let $H_{A_{+}}^i(A)$ denote the i-th local cohomology module of $A$ with respect to the graded ideal $A_+$ and set $a_i(A)=\max\{m\vert\ \ [H_{A_{+}}^i(A)]_m\neq 0\}$ with the convention $a_i(A)=-\infty$, if $H_{A_{+}}^i(A)=0$. The Castelnuovo-Mumford regularity is defined by $\reg (A):=\max\{a_i(A)+i\vert\ \ i\geq 0\}$
Let $d=2$ and $J$ be a minimal reduction of $I$ such that $r_J(I)=2$. If $\widetilde{I}=I$, then we have the following:
$\reg{ G(I)}=2$.
(ii) $e_2(I)=\lambda(I^2/{JI})$.
The case $(i)$ follows by Corollary 2.3 and [<ref>, Theorem 2.1 and Corollay 2.2] and the case $(ii)$ follows by Corollary 2.3 and [<ref>, Theorem 3.1].
Let $d=2$, $\widetilde{I}=I$ and $J$ be a minimal reduction of $I$. If $\reg{ G(I)}=3$, then by [<ref>, Lemma 1.2 and Corollary 2.2], [<ref>, Proposition 3.2] and Proposition 2.4 we have $r_J(I)=3$.
The following result is an improvement of [<ref>, Theorem 2.11] and [<ref>, Proposition 16].
Let $d=2$, $\widetilde{I}=I$ and $J$ be a minimal reduction of $I$. Then $r_J(I)=2$ if and only if $P_I(n)=H_I(n)$ for $n=1,2$, where $H_I(n)$ and $P_I(n)$ are
the Hilbert-Samuel function and the Hilbert-Samuel polynomial respectively.
$(\Longrightarrow)$ let $r_J(I)=2$. Then by Corollary 2.3, $\widetilde{I^n}=I^n$ for all $n\geq 1$ and so by [<ref>, Proposition 16] we have $H_I(n)=P_I(n)$ for all $n=1,2$.
$(\Longleftarrow)$ is clear by [<ref>, Proposition 16].
Let $J$ be a minimal reduction of $I$, $x_1\in J$ and $\overline{I}=I/{(x_1)}$, $\overline{J}=J/{(x_1)}$. Then, by definition of reduction number, we have
If $r_{\overline{J}}(\overline{I})=k$ and $I^{k+1}:x_1=I^k$, then $r_J(I)=k$.
(ii) If $d=2$ and $I^2:x_1=I$, Then $r_{\overline{J}}(\overline{I})\leq 2$ if and only if $r_J(I)\leq 2$.
Let $d=2$ and $J$ be a minimal reduction of $I$ such that $J\cap I^n=JI^{n-1}$ for $n=1,...,t$. If $r_{\overline{J}}(\overline{I})=k$ and $\lambda(I^{n+1}/{JI^n})=\lambda({\overline{I}}^{n+1}/{{\overline{J}}{\overline{I}}^n})$ for $n=t,...,k-1$. Then $I^{n+1}:x_1=I^n$ for $n=0,...,k-1$.
By [<ref>, Proposition 1.7(ii)], $(x_1)\cap I^n=x_1I^{n-1}$ for $n=1,...,t$ and so $I^n:x_1=I^{n-1}$ for $n=1,...,t$. Now, consider the exact sequence
$$0\longrightarrow{I^{n+1}:x_1}/{JI^n:x_1}\longrightarrow I^{n+1}/{JI^n}\longrightarrow{\overline{I}}^{n+1}/{{\overline{J}}{\overline{I}}^n}\longrightarrow 0.\ \ (\dagger)$$
By our assumption, $I^{n+1}:x_1=JI^n:x_1$ for $n=t,...,k-1$. Assume that $yx_1\in JI^{t}$. Then we have $yx_1=\alpha_1x_1+\alpha_2x_2$ for some $\alpha_1,\alpha_2\in I^t$. Hence $(y-\alpha_1)x_1=\alpha_2x_2\in x_2I^t$ and since $x_1,x_2$ is a regular sequence, we obtain $y-\alpha_1=sx_2$ for some $s\in R$. Since $(y-\alpha_1)x_1=sx_1x_2\in x_2I^t$ and $x_2$ is a non-zerodivisor, it follows that $sx_1\in I^t$ and so $s\in I^t:x_1$. Therefore $s\in I^{t-1}$ and so $y\in I^t$. Thus by repeating this argument, we obtain $I^{n+1}:x_1=I^n$ for $n=0,...,k-1$, as desired.
The following result was proved in [<ref>, Theorem 2.4], [<ref>, Theorem 3.10] and [<ref>, Theorem 3.7], and we give a simplified proof.
Let $J$ be a minimal reduction of $I$ such that $J\cap I^n=JI^{n-1}$ for $n=1,...,t$ and $\lambda(I^{t+1}/{JI^t})\leq 1$. Then
$\depth G(I)\geq{d-1}$.
By using Sally's descent, we may deduce the problem to the case of $d=2$. Set $r_{\overline{J}}(\overline{I})=k$. Then, by using the exact sequence $(\dagger)$, we have $\lambda({\overline{I}}^{n+1}/{{\overline{J}}{\overline{I}}^n})=\lambda(I^{n+1}/{JI^n})\leq 1$ for $n=t,...,k-1$. By Lemma 2.8, we have $I^{n+1}:x_1=I^n$ for $n=0,...,k-1$. By [<ref>, Proposition 1.1], we know that
$\sum_{n\geq 0}\lambda(\widetilde{I^{n+1}}/{J\widetilde{I^n}})=e_1(I)=e_1(\overline{I})=\sum_{n=0}^{k-1}
\lambda(I^{n+1}/{JI^n})=\sum_{n=0}^{t-1}\lambda(I^{n+1}/{JI^n})+k-t$. Therefore by [<ref>, Theorem 1.3], we have $r_J(I)\leq k$. Thus by Lemma 2.8 and Corollary 2.2, we obtain $\widetilde{I^n}=I^n$ for all $n\geq 1$. Hence $\depth G(I)\geq 1$, as required.
Let $d=2$ and $J=(x_1,x_2)$ a minimal reduction of $I$ such that $J\cap I^n=JI^{n-1}$ for all $n\geq 3$. If either $I^2:x_1=I$ or $I^2:x_2=I$, then $\widetilde{I^n}=I^n$ for all $n\geq 1$. In particular $\depth G(I)\geq 1$.
By using the same argument that was used in the proof of proposition 2.1, the result immediately follows.
Let $d=2$ and $J=(x_1,x_2)$ be a minimal reduction of $I$ such that $\lambda(J\cap I^2 /JI)\leq 1$. Then either $I^2:x_1=I$ or $I^2:x_2=I$.
If $\lambda(J\cap I^2/{JI+I^2\cap(x_1)})=1$, then $I^2\cap(x_1)\subseteq JI$ and so $I^2\cap(x_1)\subseteq[x_1I+x_2I]\cap(x_1)$. Therefore $I^2\cap(x_1)={x_1}I$ and so $I^2:x_1=I$. If $\lambda(J\cap I^2/{JI+I^2\cap(x_1)})=0$, then $I^2\cap(x_1)+Ix_2=J\cap I^2$. Hence $I^2\cap(x_1x_2)+Ix_2=I^2\cap(x_2)$ and so $Ix_2=I^2\cap(x_2)$. Thus $I^2:x_2=I$.
The following result was proved in [<ref>, Theorem 3.2] and [<ref>, Corollary 1.5] and we give an easier proof
Let $J$ be a minimal reduction of $I$ such that $J\cap I^n=JI^{n-1}$ for all $n\geq 3$. If $\lambda(J\cap I^2/IJ)\leq 1$, then $\depth G(I)\geq{d-1}$.
By Sally's descent, we may assume that $d=2$. Now, by using Lemmas 2.11 and 2.10 the result follows.
Let $d\geq 3$ and $k\in\mathbb{N}_0$ such that $\widetilde{I^k}=I^k$. If $x_1,...,x_d$ is a tame superficial sequence of $I$ and $J=(x_1,...,x_d)$ such that $J\cap I^n=JI^{n-1}$ for all $n\geq{k+2}$, then ${\fa}^mI^n:x_1={\fa}^mI^{n-1}$ for all $n\geq{k+1}$ and all $m\in\mathbb{N}_0$, where $\fa=(x_2,...,x_d)$. In particular, $\widetilde{I^n}=I^n$ for all $n\geq k$.
We will proceed by induction on $n$. Assume $n=k+1$. Then by [<ref>, Lemma 2.7] and our assumption we have ${\fa}^mI^{k+1}:x_1\subseteq{\fa}^m\widetilde{I^{k+1}}:x_1={\fa}^m\widetilde{I^k}=
{\fa}^mI^k$. Therefore ${\fa}^mI^{k+1}:x_1={\fa}^mI^k$ for all $m\in\mathbb{N}_0$. Assume $n\geq{k+1}$ and that for all $t$ with ${k+1}\leq t\leq n$ and all $m\in\mathbb{N}_0$, ${\fa}^mI^t:x_1={\fa}^mI^{t-1}$. We show that for all $m\in\mathbb{N}_0$, ${\fa}^mI^{n+1}:x_1={\fa}^mI^n$. Let $yx_1$ be an element of ${\fa}^mI^{n+1}$. Then $yx_1\in{\fa}^m$ and by using [<ref>, Lemma 2.1] we obtain $y\in{\fa}^m$. Therefore we can write the expression, $y=\sum_{i_2+...+i_d=m}r_{i_2...i_d}x_2^{i_2}...x_d^{i_d}$. Since the element $yx_1$ belongs to ${\fa}^mI^{n+1}$ too, we obtain the following equalities
where $s_{i_2...i_d}\in I^{n+1}$ for all $i_2,...,i_d$ such that $i_2+...+i_d=m$. As $x_1,...,x_d$ is a regular sequence in $R$, by equating coefficients in the previous expressions, we get $r_{i_2...i_d}x_1-s_{i_2...i_d}\in(x_2,...,x_d)$ for all $i_2,...,i_d$ such that $i_2+...+i_d=m$. Hence $s_{i_2...i_d}\in J\cap I^{n+1}$ and by our assumption we obtain $s_{i_2...i_d}\in JI^n$ for all $i_2,...,i_d$ such that $i_2+...+i_d=m$. Hence, going back to the equalities we wrote for $yx_1$, we obtain $yx_1\in{\fa}^mJI^n={\fa}^{m+1}I^n+x_1{\fa}^mI^n$. Therefore we have
$${\fa}^mI^{n+1}\cap (x_1)\subseteq{\fa}^{m+1}I^n\cap{(x_1)}+x_1{\fa}^mI^n=x_1({\fa}^{m+1}I^n:x_1)+x_1{\fa}^mI^n.$$
By applying the inductive hypothesis we get ${\fa}^mI^{n+1}\cap{(x_1)}\subseteq x_1{\fa}^{m+1}I^{n-1}+x_1
{\fa}^mI^n=x_1{\fa}^mI^n$. This proves that ${\fa}^mI^{n+1}:x_1\subseteq{\fa}^mI^n$ and so ${\fa}^mI^{n+1}:x_1={\fa}^mI^n$ for all $m\in\mathbb{N}_0$. In particular, if we set $m=0$, then $I^{n+1}:x_1=I^n$ for all $n>k$ and so by [<ref>, Corollary 2.7], $\widetilde{I^n}=I^n$ for all $n\geq k$, as desired.
The following result easily follows by Theorem 2.13.
Let $x_1,...,x_d$ be a tame superficial sequence of $I$ and $J=(x_1,...,x_d)$.
(i) If $\widetilde{I}=I$ and $J\cap I^{n}=JI^{n-1}$ for all $n\geq 3$, then $\widetilde{I^n}=I^n$ for all $n\geq 1$. In particular $\depth G(I)\geq 1$.
(ii)If $r_J(I)=2$, then $\widetilde{I}=I$ if and only if $\depth G(I)\geq 1$.
Let $k\in\mathbb{N}_0$ such that $r_J(I)=k+1$ and $\widetilde{I^k}=I^k$. Then $\widetilde{I^n}=I^n$ for all $n\geq k$.
The following example shows that the equality of Corollary 2.14(ii) maybe happen.
Let $K$ be a field, $R=K[\![ x,y]\!]$, $I=(x^6,x^4y^2,x^3y^3,x^2y^4,xy^5,y^6)$ and $J=(x^6,y^6+x^4y^2)$. Then $r_J(I)=2$, $\depth G(I)=1$ and so $G(I)$ is not C.M.
§ INVARIANCE OF A LENGTH
Let $J=(x_1,...,x_d)$ be a minimal reduction of $I$. In [<ref>] Wang defined the following exact sequence for all $n,k$
$$0\longrightarrow T_{n,k}\longrightarrow \oplus^{{k+d-1}\choose{d-1}} I^n/{JI^{n-1}}\overset{\phi_k}{\longrightarrow} J^kI^n/{J^{k+1}I^{n-1}}
\longrightarrow 0,\ \ \ (*)$$
where $\phi_k=(x_1^k,x_1^{k-1}x_2,...,x_1^{k-1}x_d,...,x_d^k)$ and $T_{n,k}=\ker(\phi_k)$. He also showed that $T_{1,k}=0$ for all $k$ and if $d=1$, then $T_{n,k}=0$ for all $n,k$. By using the exact sequence $(*)$, we drive the following easy lemma and we leave the proof to the reader.
Let $t\in\mathbb{N}_0$ and $J=(x_1,...,x_d)$ be a minimal reduction of $I$. Then we have the following:
(i) If $J\cap I^n=JI^{n-1}$ for $n=1,...,t$, then $T_{n,k}=0$ for $n=1,...,t$ and all $k$.
(ii) If $I$ is integrally closed, then $T_{2,k}=0$ for all $k$. In particular, if $I=m$, then $T_{2,k}=0$ for all $k$.
The following lemma is known see the proof of [<ref>, Proposition 2.1].
Let $J=(x_1,...,x_d)$ be a minimal reduction of $I$. Then
$\lambda(I/J)=e_0(I)-\lambda(R/I)$ and $\lambda(I^{n+1}/{J^{n}I})=e_0(I){{n+d-1}\choose{d}}+\lambda(R/I){{n+d-1}\choose{d-1}}-\lambda(R/I^{n+1})$ for $n\geq 1$ which are independent of $J$.
In [<ref>], Puthenpurakal proved that $\lambda({\fm}^3/{J{\fm}^2})$ is independent of the minimal reduction $J$ of $\fm$ and subsequently Ananthnarayan and Huneke [<ref>] extend it for $n$-standard admissible $I$-filtrations.
The following result was proved in [<ref>, Theorem 1] and [<ref>, Theorem 3.5]. We reprove it with a much easier proof.
Let $t\in\mathbb{N}_0$ and $J=(x_1,...,x_d)$ be a minimal reduction of $I$. If $J\cap I^n=JI^{n-1}$ for $n=1,...,t$, then $\lambda(I^{n+1}/{JI^n})$ is independent of $J$ for $n=1,...,t$.
We have $\lambda(I^{n+1}/{JI^{n}})=\lambda(I^{n+1}/{J^{n}I})-\sum_{k=1}^{n-1}
\lambda(J^kI^{n+1-k}/{J^{k+1}I^{n-k}})$. Therefore by Lemma 3.1 and the exact sequence $(*)$, we obtain $\lambda(I^{n+1}/{JI^{n}})=\lambda(I^{n+1}/{J^{n}I})-\sum_{k=1}^{n-1}{{k+d-1}\choose{d-1}}\lambda(I^{n+1-k}/{JI^{n-k}})$. Now by using Lemma 3.2 and using induction on $n$, the result follows.
The following example is a counterexample for Question 3 of [<ref>] and it also says that Theorem 1.8 of [<ref>] does not hold in general.
The computations are performed by using Macaulay2 [<ref>], CoCoA [<ref>] and Singular [<ref>].
Let $K$ be a field and $S=K[\![ x,y,z,u,v ]\!]$, where $I=(x^2+y^5,xy+u^4,xz+v^3)$. Then $R=S/I$ is a Cohen-Macaulay local ring of dimension two, ideals $J_1=(y,z)R$ and $J_2=(z,u)R$ are minimal reduction of $\fm=(x,y,z,u,v)R$ and $\lambda({\fm}^4/{J_1{\fm}^3})=17$, $\lambda({\fm}^4/{J_2{\fm}^3})=20$.
This paper was done while I was visiting the University of Osnabruck. I would like to thank the Institute of Mathematics of the University of Osnabruck for hospitality and
partially financial support and I also would like to express my very great appreciation to Professor Winfried Bruns for his valuable and constructive suggestions during the planning and development of this research work. Moreover, I would like to thank deeply grateful to the referee for the careful reading of the manuscript and the helpful suggestions.
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1511.00060
|
Long Short-Term Memory (LSTM) networks, a type of recurrent neural
network with a more complex computational unit, have been
successfully applied to a variety of sequence modeling tasks. In
this paper we develop Tree Long Short-Term Memory (), a neural network model based on LSTM, which is
designed to predict a tree rather than a linear sequence.
TreeLSTM defines the probability of a sentence by estimating
the generation probability of its dependency tree. At each time
step, a node is generated based on the representation of the
generated sub-tree. We further enhance the modeling power of
by explicitly representing the correlations
between left and right dependents. Application of our model to the
MSR sentence completion challenge achieves results beyond the
current state of the art. We also report results on dependency
parsing reranking achieving competitive performance.
§ INTRODUCTION
Neural language models have been gaining increasing attention as a
competitive alternative to . The main idea is to
represent each word using a real-valued feature vector capturing the
contexts in which it occurs. The conditional probability of the next
word is then modeled as a smooth function of the feature vectors of
the preceding words and the next word. In essence, similar
representations are learned for words found in similar contexts
resulting in similar predictions for the next word. Previous
approaches have mainly employed feed-forward
<cit.> and recurrent neural networks
<cit.> in order to map the
feature vectors of the context words to the distribution for the next
word. Recently, RNNs with Long Short-Term Memory (LSTM) units
<cit.> have emerged as a popular architecture due to
their strong ability to capture long-term dependencies. LSTMs have
been successfully applied to a variety of tasks ranging from machine
translation <cit.>, to speech recognition
<cit.>, and image description generation
Despite superior performance in many applications, neural language
models essentially predict sequences of words. Many NLP tasks,
however, exploit syntactic information operating over tree structures
(e.g., dependency or constituent trees). In this paper we develop a
novel neural network model which combines the advantages of the LSTM
architecture and syntactic structure. Our model estimates the
probability of a sentence by estimating the generation probability of
its dependency tree. Instead of explicitly encoding tree structure as
a set of features, we use four LSTM networks to model four types of
dependency edges which altogether specify how the tree is built. At
each time step, one LSTM is activated which predicts the next word
conditioned on the sub-tree generated so far. To learn the
representations of the conditioned sub-tree, we force the four LSTMs
to share their hidden layers.
Our model is also capable of generating trees just by sampling from a
trained model and can be seamlessly integrated with text generation
Our approach is related to but ultimately different from recursive
neural networks <cit.> a class of models which operate on
structured inputs. Given a (binary) parse tree, they recursively
generate parent representations in a fashion, by
combining tokens to produce representations for phrases, and
eventually the whole sentence. The learned representations can be
then used in classification tasks such as sentiment analysis
<cit.> and paraphrase detection
<cit.>. tai:ea:2015 learn distributed
representations over syntactic trees by generalizing the LSTM
architecture to tree-structured network topologies. The key feature of
our model is not so much that it can learn semantic representations of
phrases or sentences, but its ability to predict tree structure and
estimate its probability.
Syntactic language models have a long history in NLP dating back to
Chelba:Jelinek:2000 (see also
roark2001probabilistic and
charniak2001immediate). These models differ in how grammar
structures in a parsing tree are used when predicting the next word.
Other work develops dependency-based language models for specific
applications such as machine translation
<cit.>, speech
recognition <cit.> or sentence completion
<cit.>. All instances of these models apply
Markov assumptions on the dependency tree, and adopt standard n-gram
smoothing methods for reliable parameter
estimation. Emami:ea:2003 and
sennrich2015modelling estimate the parameters of a
structured language model using feed-forward neural networks
<cit.>. mirowski-vlachos:2015 re-implement
the model of gubbins-vlachos:2013:EMNLP with RNNs. They view
sentences as sequences of words over a tree. While they ignore the
tree structures themselves, we model them explicitly.
Our model shares with other structured-based language models the
ability to take dependency information into account. It differs in
the following respects: (a) it does not artificially restrict the
depth of the dependencies it considers and can thus be viewed as an
infinite order dependency language model; (b) it not only estimates
the probability of a string but is also capable of generating
dependency trees; (c) finally, contrary to previous dependency-based
language models which encode syntactic information as features, our
model takes tree structure into account more directly via representing
different types of dependency edges explicitly using LSTMs. Therefore,
there is no need to manually determine which dependency tree features
should be used or how large the feature embeddings should be.
We evaluate our model on the MSR sentence completion challenge, a
benchmark language modeling dataset. Our results outperform the best
published results on this dataset. Since our model is
a general tree estimator, we also use it to rerank the top $K$
dependency trees from the (second order) MSTPasrser and obtain
performance on par with recently proposed dependency parsers.
§ TREE LONG SHORT-TERM MEMORY NETWORKS
We seek to estimate the probability of a sentence by estimating the
generation probability of its dependency tree. Syntactic information
in our model is represented in the form of dependency
In the following, we first describe our definition of dependency path
and based on it explain how the probability of a sentence is
§.§ Dependency Path
Generally speaking, a dependency path is the path between root
and $w$ consisting of the nodes on the path and the edges connecting
To represent dependency paths, we introduce four types of edges which
essentially define the “shape” of a dependency tree. Let $w_0$
denote a node in a tree and $w_1, w_2, \dots, w_n$ its left
dependents. As shown in Figure <ref>, Left edge is the
edge between $w_0$ and its first left dependent denoted as $(w_0,
w_1)$. Let $w_k$ (with ) denote a non-first left
dependent of $w_0$. The edge from $w_{k-1}$ to $w_k$ is a edge (Nx stands for Next), where $w_{k-1}$ is the right adjacent sibling of
$w_k$. Note that the Nx-Left edge $(w_{k-1}, w_k)$ replaces edge
$(w_0, w_k)$ (illustrated with a dashed line in
Figure <ref>) in the original dependency tree. The
modification allows information to flow from $w_0$ to $w_k$
through rather than directly from $w_0$
to $w_k$. Right and Nx-Right edges are defined analogously
for right dependents.
[scale=.5,->,>=stealth',thick,main node/.style=circle,fill=blue!20,draw,inner sep=0pt,minimum size=3mm]
[main node][label=right:$w_0$] (L0) at (-1, 3) ;
[main node][label=[xshift=-2pt]$w_1$] (L1) at (-3, 0) ;
[main node][label=[xshift=-4pt]$w_{k-1}$] (L2) at (-5.5, 0) ;
[main node][label=[xshift=3pt]$w_k$] (L3) at (-9.5, 0) ;
[main node][label=$w_n$] (L4) at (-12, 0) ;
(L0) edge node[below right= 1pt] Left (L1);
(L1) edge[-,line width=1.2pt,style=dotted] (L2);
(L2) edge node[below = 5pt] Nx-Left (L3);
(L0) edge[bend right = 17, style=dashed] (L2);
(L0) edge[bend right = 13, style=dashed] (L3);
(L0) edge[bend right = 11, style=dashed] (L4);
(L3) edge[-,line width=1.2pt,style=dotted] (L4);
Left and Nx-Left edges. Dotted line between $w_1$
and $w_{k-1}$ (also between $w_k$ and $w_n$) indicate that there may
be nodes inbetween.
Given these four types of edges, dependency paths (denoted as $\mathcal{D}(w)$) can be defined as
follows bearing in mind that the first right dependent of root
is its only dependent and that $w^p$ denotes the parent of $w$. We use $( \dots )$ to denote a sequence, where $()$ is an empty sequence and $\Vert$ is an operator for concatenating two sequences.
* if $w$ is root, then $\mathcal{D}(w) = ()$
* if $w$ is a left dependent of $w^p$
* if $w$ is the first left dependent, then
$\mathcal{D}(w) = \mathcal{D}(w^p) \Vert ( \langle w^p, \text{\sc Left}\rangle )$
* if $w$ is not the first left dependent and
$w^s$ is its right adjacent sibling, then
$\mathcal{D}(w) = \mathcal{D}(w^{s}) \Vert ( \langle w^s,
\text{\sc Nx-Left}\rangle )$
* if $w$ is a right dependent of $w^p$
* if $w$ is the first right dependent, then
$\mathcal{D}(w) = \mathcal{D}(w^p) \Vert ( \langle w^p, \text{\sc Right}\rangle
* if $w$ is not the first right dependent and
$w^s$ is its left adjacent sibling, then
$\mathcal{D}(w) = \mathcal{D}(w^s) \Vert ( \langle w^s, \text{\sc
Nx-Right}\rangle )$
A dependency tree can be represented by the set of its dependency
paths which in turn can be used to reconstruct the original tree.[Throughout this paper we assume all dependency trees are projective.]
Dependency paths for the first two levels of the tree in Figure <ref> are as follows
(ignoring for the moment the subscripts which we explain in the next
section). $\mathcal{D}(\text{sold}) = ( \langle\text{\sc root},
\text{\sc Right}\rangle )$ (see definitions (1) and (3a)),
$\mathcal{D}(\text{year})=\mathcal{D}(\text{sold}) \Vert
( \langle\text{sold}, \text{\sc Left}\rangle )$ (see (2a)),
$\mathcal{D}(\text{manufacturer}) = \mathcal{D}(\text{year}) \Vert
( \langle\text{year}, \text{\sc Nx-Left}\rangle )$ (see (2b)),
$\mathcal{D}(\text{cars}) = \mathcal{D}(\text{sold}) \Vert
( \langle\text{sold}, \text{\sc Right}\rangle )$ (see (3a)),
$\mathcal{D}(\text{in}) = \mathcal{D}(\text{cars}) \Vert
( \langle\text{cars}, \text{\sc Nx-Right}\rangle )$ (according
to (3b)). =-1
§.§ Tree Probability
The core problem in syntax-based language modeling is to estimate the
probability of sentence $S$ given its corresponding tree $T$, $P(S | T)$.
We view the probability computation of a
dependency tree as a generation process. Specifically, we assume
dependency trees are constructed top-down, in a breadth-first
manner. Generation starts at the root node. For each node at
each level, first its left dependents are generated from closest to
farthest and then the right dependents (again from closest to
farthest). The same process is applied to the next node at the same
level or a node at the next level. Figure <ref> shows the
breadth-first traversal of a dependency tree.
Under the assumption that each word $w$ in a dependency tree is
only conditioned on its dependency path, the probability
of a sentence $S$ given its dependency tree $T$ is:
\begin{equation}
\label{eq:treeprob}
P(S | T) = \prod_{w \in \text{BFS}(T) \setminus \textsc{root}}^{} P(w|\mathcal{D}(w))
\end{equation}
where $\mathcal{D}(w)$ is the dependency path of $w$. Note that each
word $w$ is visited according to its breadth-first search order
(BFS(T)) and the probability of root is ignored since
every tree has one. The role of root in a dependency tree is the
same as the begin of sentence token (BOS) in a sentence. When
computing $P(S|T)$ (or $P(S)$), the probability of root (or
BOS) is ignored (we assume it always exists), but is used to predict
other words. We explain in the next section how TreeLSTM estimates $P(w|\mathcal{D}(w))$. =-1
edge from parent/.style=draw,thick,-stealth,
child node sold1
child node[xshift=-9pt] manufacturer3
child node The9
child node luxury8
child node auto7
child [missing]
child [missing]
child [missing]
child node year2
child node last6
child node cars4
child node 1,21410
child node in5
child node U.S.11
child node the12
Dependency tree of the sentence The luxury auto manufacturer
last year sold 1,214 cars in the U.S. Subscripts
indicate breadth-first traversal. root has only one
dependent (i.e., sold) which we view as its first right
§.§ Tree LSTMs
[scale=.75,->,>=stealth',thick,main node/.style=ellipse,fill=blue!20,draw,font=,inner sep=0pt,minimum size=2.5mm]
[main node] (w0) at (0, 2) $\mathbf{w_0}$;
[main node] (w1) at (-2.5, 0) $\mathbf{w_1}$;
[main node] (w2) at (-6, 0) $\mathbf{w_2}$;
[main node] (w3) at (-9.5, 0) $\mathbf{w_3}$;
[main node] (w4) at (2.5, 0) $\mathbf{w_4}$;
[main node] (w5) at (6, 0) $\mathbf{w_5}$;
[main node] (w6) at (9.5, 0) $\mathbf{w_6}$;
(w0) edge[bend right=25,line width=1pt] (w1);
(w0) edge[bend right=15,line width=1pt] (w2);
(w0) edge[bend right=10,line width=1pt] (w3);
(w0) edge[bend left=25,line width=1pt] (w4);
(w0) edge[bend left=15,line width=1pt] (w5);
(w0) edge[bend left=10,line width=1pt] (w6);
[line width=2.5pt] (0, -1) – node[left = 5pt, yshift=-2pt] Generated by four LSTMs node[right = 5pt] with tied $\mathbf{W}_e$ and tied $\mathbf{W}_{ho}$ (0, -2.5);
[scale=.75,->,>=stealth',thick,main node/.style=rectangle,rounded corners=3pt,fill=blue!10,draw,font=,inner sep=0,minimum size=2.5mm,minimum width=4mm,minimum height=1.5cm,path picture=
[fill=blue!50!black] (0, -0.5) circle (1mm);
[fill=blue!50!black] (0, 0) circle (1mm);
[fill=blue!50!black] (0, 0.5) circle (1mm);
[main node] (w0) at (0, 2) ;
[main node] (w1) at (-2.5, 0) ;
[main node] (w2) at (-6, 0) ;
(y0) at (0, -0.5) $\mathbf{w_0}$;
(y1) at (-2.5, -2.5) $\mathbf{w_1}$;
(y2) at (-6, -2.5) $\mathbf{w_2}$;
[main node] (w3) at (-9.5, 0) ;
(y3) at (-9.5, -2.5) $\mathbf{w_3}$;
(x1) at (-2.5, 2.5) $\mathbf{w_0}$;
(x2) at (-6, 2.5) $\mathbf{w_1}$;
(x3) at (-9.5, 2.5) $\mathbf{w_2}$;
[main node] (w4) at (2.5, 0) ;
[main node] (w5) at (6, 0) ;
[main node] (w6) at (9.5, 0) ;
(y4) at (2.5, -2.5) $\mathbf{w_4}$;
(y5) at (6, -2.5) $\mathbf{w_5}$;
[main node] (w3) at (-9.5, 0) ;
(y6) at (9.5, -2.5) $\mathbf{w_6}$;
(x4) at (2.5, 2.5) $\mathbf{w_0}$;
(x5) at (6, 2.5) $\mathbf{w_4}$;
(x6) at (9.5, 2.5) $\mathbf{w_5}$;
[line width=1.5pt,blue] (w0) – node[left = 5pt, above = 25pt, rotate=-50] Gen-L (w1);
[line width=1.5pt] (w0) – (y0);
[line width=1.5pt,red] (w1) – node[above = 20pt] Gen-Nx-L (w2);
[line width=1.5pt,blue] (x1) – (w1);
[line width=1.5pt] (w1) – (y1);
[line width=1.5pt,red] (w2) – node[above = 20pt] Gen-Nx-L (w3);
[line width=1.5pt,red] (x2) – (w2);
[line width=1.5pt] (w2) – (y2);
[line width=1.5pt,red] (x3) – (w3);
[line width=1.5pt] (w3) – (y3);
[line width=1.5pt,blue!70!black] (w0) – node[right = 5pt, above = 25pt, rotate=50] Gen-R (w4);
[line width=1.5pt,red!70!black] (w4) – node[above = 20pt] Gen-Nx-R (w5);
[line width=1.5pt,blue!70!black] (x4) – (w4);
[line width=1.5pt] (w4) – (y4);
[line width=1.5pt,red!70!black] (w5) – node[above = 20pt] Gen-Nx-R (w6);
[line width=1.5pt,red!70!black] (x5) – (w5);
[line width=1.5pt] (w5) – (y5);
[line width=1.5pt,red!70!black] (x6) – (w6);
[line width=1.5pt] (w6) – (y6);
Generation process of left ($w_1, w_2, w_3$) and right ($w_4,
w_5, w_6$) dependents of tree node $w_o$ (top) using four LSTMs
(Gen-L, Gen-R, Gen-Nx-L and Gen-Nx-R). The
model can handle an arbitrary number of dependents due to Gen-Nx-L and Gen-Nx-R.
A dependency path $\mathcal{D}(w)$ is subtree which we denote as a
sequence of $\langle$word, edge-type$\rangle$
tuples. Our innovation is to learn the representation of
$\mathcal{D}(w)$ using four LSTMs.
The four LSTMs (Gen-L, Gen-R, Gen-Nx-L and Gen-Nx-R) are used to represent the four types of edges (Left, Right, Nx-Left and )
introduced earlier. Gen, Nx, L and R are
shorthands for Generate, Next, Left and Right. At each
time step, an LSTM is chosen according to an edge-type; then the
LSTM takes a word as input and predicts/generates its dependent or
sibling. This process can be also viewed as adding an edge and a node
to a tree. Specifically, LSTMs Gen-L and Gen-R are used to
generate the first left and right dependent of a node ($w_1$ and $w_4$
in Figure <ref>). So, these two LSTMs are responsible for
going deeper in a tree. While Gen-Nx-L and
generate the remaining left/right dependents and therefore go wider in
a tree. As shown in Figure <ref>, $w_2$ and $w_3$ are
generated by , whereas $w_5$ and $w_6$ are
generated by . Note that the model can handle any
number of left or right dependents by applying or
multiple times.
We assume time steps correspond to the steps taken by the
breadth-first traversal of the dependency tree and the sentence has
length $n$. At time step $t$ ($1 \leq t \leq n$), let $\langle
w_{t'}, z_t \rangle$ denote the last tuple
in $\mathcal{D}(w_t)$. Subscripts $t$ and $t'$ denote the
breadth-first search order of $w_t$ and $w_{t'}$, respectively. $z_t
\in \{ \text{\sc Left}, \text{\sc Right}, \text{\sc Nx-Left},
\text{\sc Nx-Right} \}$ is the edge type (see the definitions in
Section <ref>). Let $\mathbf{W}_e \in \mathbb{R}^{s \times
|V|}$ denote the word embedding matrix and $\mathbf{W}_{ho} \in
\mathbb{R}^{|V|\times d}$ the output matrix of our model, where
$|V|$ is the vocabulary size, $s$ the word embedding size and $d$ the
hidden unit size. We use tied $\mathbf{W}_e$ and tied
$\mathbf{W}_{ho}$ for the four LSTMs to reduce the number of
parameters in our model.
The four LSTMs also share their hidden states. Let $\mathbf{H} \in
\mathbb{R}^{d \times (n+1)}$ denote the shared hidden states of
all time steps and $e(w_t)$ the one-hot vector of $w_t$. Then,
$\mathbf{H}[:, t]$ represents $\mathcal{D}(w_t)$ at time step $t$, and
the computation[We ignore all bias terms for notational
simplicity.] is:
\begin{align}
\mathbf{x}_t &= \mathbf{W}_e \cdot e(w_{t'}) \\
\mathbf{h}_t &= \text{LSTM}^{z_t}(\mathbf{x}_t, \mathbf{H}[:,t']) \\
\mathbf{H}{[:, t]} &= \mathbf{h}_t \\
\mathbf{y}_t &= \mathbf{W}_{ho} \cdot \mathbf{h}_t
\end{align}
where the initial hidden state $\mathbf{H}[:, 0]$ is initialized to a
vector of small values such as 0.01. According to
Equation (<ref>b), the model selects an LSTM based on edge
type $z_t$. We describe the details of $\text{LSTM}^{z_t}$ in the next
paragraph. The probability of $w_t$ given its dependency
path $\mathcal{D}(w_t)$ is estimated by a softmax function:
\begin{equation}
\label{eq:softmax}
P(w_t|\mathcal{D}(w_t)) = \frac{\exp(\mathbf{y}_{t,w_t})}{\sum_{k'=1}^{|V|} \exp(\mathbf{y}_{t,k'})}
\end{equation}
We must point out that although we use four jointly trained LSTMs to
encode the hidden states, the training and inference complexity of our
model is no different from a regular LSTM, since at each time step only
one LSTM is working.
We implement $\text{LSTM}^{z}$ in Equation (<ref>b) using
a deep LSTM (to simplify notation, from now on we write $z$ instead
of $z_t$). The inputs at time step $t$ are $\mathbf{x}_t$ and
$\mathbf{h}_{t'}$ (the hidden state of an earlier time step $t'$) and
the output is $\mathbf{h}_t$ (the hidden state of current time step).
Let $L$ denote the layer number of $\text{LSTM}^{z}$ and
$\hat{\mathbf{h}}_t^{{l}}$ the internal hidden state of the $l$-th
layer of the $\text{LSTM}^{z}$ at time step $t$, where $\mathbf{x}_t$
is $\hat{\mathbf{h}}_t^{0}$ and $\mathbf{h}_{t'}$ is
$\hat{\mathbf{h}}_{t'}^L$. The LSTM architecture introduces multiplicative gates and memory
cells $\hat{\mathbf{c}}^l_t$ (at $l$-th layer) in order to address the
vanishing gradient problem which makes it difficult for the
standard RNN model to learn long-distance correlations in a
sequence. Here, $\hat{\mathbf{c}}^l_t$ is a linear combination of the
current input signal $\mathbf{u}_t$ and an earlier memory
cell $\hat{\mathbf{c}}^l_{t'}$. How much input information
$\mathbf{u}_t$ will flow into $\hat{\mathbf{c}}^l_t$ is controlled by
input gate $\mathbf{i}_t$ and how much of the earlier memory cell
$\hat{\mathbf{c}}^l_{t'}$ will be forgotten is controlled by forget
gate $\mathbf{f}_t$. This process is computed as follows:
\begin{align}
\mathbf{u}_t &= \tanh( \mathbf{W}^{z,l}_{ux} \cdot \hat{\mathbf{h}}_t^{l-1} + \mathbf{W}^{z,l}_{uh} \cdot \hat{\mathbf{h}}_{t'}^l ) \\
\mathbf{i}_t &= \sigma( \mathbf{W}^{z,l}_{ix} \cdot \hat{\mathbf{h}}_t^{l-1} + \mathbf{W}^{z,l}_{ih} \cdot \hat{\mathbf{h}}_{t'}^l ) \\
\mathbf{f}_t &= \sigma( \mathbf{W}^{z,l}_{fx} \cdot \hat{\mathbf{h}}_t^{l-1} + \mathbf{W}^{z,l}_{fh} \cdot \hat{\mathbf{h}}_{t'}^l ) \\
\hat{\mathbf{c}}^l_t &= \mathbf{f}_t \odot \hat{\mathbf{c}}^l_{t'} + \mathbf{i}_t \odot \mathbf{u}_t
\end{align}
where $\mathbf{W}^{z,l}_{ux} \in \mathbb{R}^{d \times d}$
($\mathbf{W}^{z,l}_{ux} \in \mathbb{R}^{d \times s}$ when $l=1$) and
$\mathbf{W}^{z,l}_{uh} \in \mathbb{R}^{d \times d}$ are weight
matrices for $\mathbf{u}_t$, $\mathbf{W}^{z,l}_{ix}$ and
$\mathbf{W}^{z,l}_{ih}$ are weight matrices for $\mathbf{i}_t $ and
$\mathbf{W}^{z,l}_{fx}$, and $\mathbf{W}^{z,l}_{fh}$ are weight
matrices for $\mathbf{f}_t$. $\sigma$ is a sigmoid function and
$\odot$ the element-wise product.
Output gate $\mathbf{o}_t$ controls how much information of the
cell $\hat{\mathbf{c}}^l_t$ can be seen by other modules:
\begin{align}
\mathbf{o}_t &= \sigma( \mathbf{W}^{z,l}_{ox} \cdot \hat{\mathbf{h}}_t^{l-1} + \mathbf{W}^{z,l}_{oh} \cdot \hat{\mathbf{h}}_{t'}^l ) \\
\hat{\mathbf{h}}^l_t &= \mathbf{o}_t \odot \tanh(\hat{\mathbf{c}}^l_t)
\end{align}
Application of the above process to all layers $L$, will
yield $\hat{\mathbf{h}}^L_t$, which is $\mathbf{h}_t$. Note that in
implementation, all $\hat{\mathbf{c}}^l_t$ and $\hat{\mathbf{h}}^l_t$
($1 \leq l \leq L$) at time step $t$ are stored, although we only care
about $\hat{\mathbf{h}}^L_t$ ($\mathbf{h}_t$).
§.§ Left Dependent Tree LSTMs
[scale=.75,->,>=stealth',thick,main node/.style=rectangle,rounded corners=3pt,fill=blue!10,draw,font=,inner sep=0,minimum size=2.5mm,minimum width=4mm,minimum height=1.5cm,path picture=
[fill=blue!50!black] (0, -0.5) circle (1mm);
[fill=blue!50!black] (0, 0) circle (1mm);
[fill=blue!50!black] (0, 0.5) circle (1mm);
[main node] (w0) at (0, 2) ;
[main node] (w1) at (-2.5, 0) ;
[main node] (w2) at (-6, 0) ;
[main node] (w3) at (-9.5, 0) ;
(y0) at (0, -0.5) $\mathbf{w_0}$;
(x1) at (-2.5, 2.5) $\mathbf{w_0}$;
(x2) at (-6, 2.5) $\mathbf{w_1}$;
(x3) at (-9.5, 2.5) $\mathbf{w_2}$;
[main node] (w4) at (2.5, 0) ;
[main node] (w5) at (6, 0) ;
[main node] (w6) at (9.5, 0) ;
(y4) at (2.5, -2.5) $\mathbf{w_4}$;
(y5) at (6, -2.5) $\mathbf{w_5}$;
[main node] (w3) at (-9.5, 0) ;
(y6) at (9.5, -2.5) $\mathbf{w_6}$;
(x4) at (2.5, 2.5) $\mathbf{w_0}$;
(x5) at (6, 2.5) $\mathbf{w_4}$;
(x6) at (9.5, 2.5) $\mathbf{w_5}$;
[line width=1.5pt,blue] (w0) – node[left = 5pt, above = 25pt, rotate=-50] Gen-L (w1);
[line width=1.5pt] (w0) – (y0);
[line width=1.5pt,red] (w1) – node[above = 20pt] Gen-Nx-L (w2);
[line width=1.5pt,blue] (x1) – (w1);
[line width=1.5pt,red] (w2) – node[above = 20pt] Gen-Nx-L (w3);
[line width=1.5pt,red] (x2) – (w2);
[line width=1.5pt,red] (x3) – (w3);
[line width=1.5pt,blue!70!black] (w0) – node[right = 5pt, above = 25pt, rotate=50] Gen-R (w4);
[line width=1.5pt,red!70!black] (w4) – node[above = 20pt] Gen-Nx-R (w5);
[line width=1.5pt,blue!70!black] (x4) – (w4);
[line width=1.5pt] (w4) – (y4);
[line width=1.5pt,red!70!black] (w5) – node[above = 20pt] Gen-Nx-R (w6);
[line width=1.5pt,red!70!black] (x5) – (w5);
[line width=1.5pt] (w5) – (y5);
[line width=1.5pt,red!70!black] (x6) – (w6);
[line width=1.5pt] (w6) – (y6);
[main node] (w1b) at (-2.5, -2.5) ;
[main node] (w2b) at (-6, -2.5) ;
[main node] (w3b) at (-9.5, -2.5) ;
(wb) at (-11, -2.5) ;
(y1) at (-2.5, -5) $\mathbf{w_1}$;
(y2) at (-6, -5) $\mathbf{w_2}$;
(y3) at (-9.5, -5) $\mathbf{w_3}$;
[line width=1.5pt,green!80!black] (wb) – (w3b);
[line width=1.5pt,green!80!black] (w3b) – node[above = 15pt] Ld (w2b);
[line width=1.5pt,green!80!black] (w2b) – node[above = 15pt] Ld (w1b);
[line width=1.5pt,blue!70!black] (w1b) – (w4);
[line width=1.5pt,green!80!black] (y1) – (w1b);
[line width=1.5pt,green!80!black] (y2) – (w2b);
[line width=1.5pt,green!80!black] (y3) – (w3b);
(w1) edge[bend right=25,line width=1.5pt] (y1);
(w2) edge[bend right=25,line width=1.5pt] (y2);
(w3) edge[bend right=25,line width=1.5pt] (y3);
Generation of left and right dependents of node $w_0$
according to LdTreeLSTM.
TreeLSTM computes $P(w|\mathcal{D}(w))$ based on the dependency
path $\mathcal{D}(w)$, which ignores the interaction between left and
right dependents on the same level. In many cases, TreeLSTM will
use a verb to predict its object directly without knowing its subject.
For example, in Figure <ref>, TreeLSTM uses
$\langle$root, Right$\rangle$ and to predict cars. This
information is unfortunately not specific to cars (many
things can be sold, e.g., chocolates,
candy). Considering manufacturer, the left dependent
of sold would help predict cars more accurately.
In order to jointly take left and right dependents into account, we
employ yet another LSTM, which goes from the furthest left dependent
to the closest left dependent (Ld is a shorthand for left
dependent). As shown in Figure <ref>, Ld LSTM
learns the representation of all left dependents of a node $w_0$; this
representation is then used to predict the first right dependent of
the same node. Non-first right dependents can also leverage the
representation of left dependents, since this information is injected
into the hidden state of the first right dependent and can percolate
all the way. Note that in order to retain the generation capability of
our model (Section <ref>), we only allow right
dependents to leverage left dependents (they are generated before right dependents).
The computation of the LdTreeLSTM is almost the same as in TreeLSTM except when . In this
case, let $\mathbf{v}_t$ be the corresponding left dependent sequence
with length in
Figure <ref>). Then, the hidden state ($\mathbf{q}_k$)
of $\mathbf{v}_t$ at each time step $k$ is:
\begin{align}
\mathbf{m}_k &= \mathbf{W}_e \cdot e(\mathbf{v}_{t,k}) \\
\mathbf{q}_k &= \text{LSTM}^{\text{\sc Ld}}(\mathbf{m}_k, \mathbf{q}_{k-1})
\end{align}
where $\mathbf{q}_K$ is the representation for all left dependents. Then, the
computation of the current hidden state becomes (see
Equation (<ref>) for the original computation):
\begin{align}
\mathbf{r}_t &= \begin{bmatrix}
\mathbf{W}_e \cdot e(w_{t'}) \\
\mathbf{q}_K
\end{bmatrix} \\
\mathbf{h}_t &= \text{LSTM}^{\text{\sc Gen-R}}(\mathbf{r}_t, \mathbf{H}[:,t'])
\end{align}
where $\mathbf{q}_K$ serves as additional input for
$\text{LSTM}^{\text{\sc Gen-R}}$. All other computational details are
the same as in TreeLSTM (see Section <ref>).
§.§ Model Training
On small scale datasets we employ Negative Log-likelihood (NLL) as our
training objective for both TreeLSTM and LdTreeLSTM:
\begin{equation}
\label{eq:nllobj}
\mathcal{L}^{\text{NLL}}(\theta) = -\frac{1}{|\mathcal{S}|} \sum_{S \in \mathcal{S}} \log P(S | T)
\end{equation}
where $S$ is a sentence in the training set $\mathcal{S}$, $T$ is the
dependency tree of $S$ and $P(S | T)$ is defined as in
Equation (<ref>).
On large scale datasets (e.g., with vocabulary size of 65K), computing
the output layer activations and the softmax function with NLL would
become prohibitively expensive. Instead, we employ Noise Contrastive
Estimation (NCE; gutmann2012noise, MnihTeh2012)
which treats the normalization term $\hat{Z}$ in
$\hat{P}(w|\mathcal{D}(w_t)) = \frac{\exp(\mathbf{W}_{ho}[w,:] \cdot
\mathbf{h}_t)}{\hat{Z}}$ as constant. The intuition behind NCE is
to discriminate between samples from a data distribution
$\hat{P}(w|\mathcal{D}(w_t))$ and a known noise distribution $P_n(w)$
via binary logistic regression. Assuming that noise words are
$k$ times more frequent than real words in the training set
<cit.>, then the probability of a word $w$ being from our
model $P_d(w, \mathcal{D}(w_t))$ is
$\frac{\hat{P}(w|\mathcal{D}(w_t))}{ \hat{P}(w|\mathcal{D}(w_t)) + k
P_n(w) }$.
We apply NCE to large vocabulary models with the following training
\begin{equation*}
\begin{split}
\mathcal{L}^{\text{NCE}}(\theta) = &
-\frac{1}{|\mathcal{S}|} \sum_{T \in \mathcal{S}} \sum_{t=1}^{|T|} \bigg( \log P_d(w_t, \mathcal{D}(w_t)) \biggr. \\
& \biggl. + \sum_{j=1}^k \log [ 1 - P_d(\tilde{w}_{t,j}, \mathcal{D}(w_t)) ] \biggr)
\end{split}
\end{equation*}
where $\tilde{w}_{t,j}$ is a word sampled from the noise
distribution $P_n(w)$. We use smoothed unigram frequencies
(exponentiating by 0.75) as the noise distribution $P_n(w)$
<cit.>. We initialize $\ln \hat{Z} = 9$ as suggested
in chen2015recurrent, but instead of keeping it fixed we
also learn $\hat{Z}$ during training <cit.>. We
set $k=20$.
§ EXPERIMENTS
We assess the performance of our model on two tasks: the Microsoft
Research (MSR) sentence completion challenge
<cit.>, and dependency parsing reranking. We also
demonstrate the tree generation capability of our models. In the
following, we first present details on model training and then present
our results. We implemented our models using the Torch library
<cit.> and our code is available at
§.§ Training Details
We trained our model with back propagation through time
<cit.> on an Nvidia GPU Card with a mini-batch
size of 64. The objective (NLL or NCE) was minimized by stochastic
gradient descent. Model parameters were uniformly initialized in
$[-0.1, 0.1]$. We used the NCE objective on the MSR sentence
completion task (due to the large size of this dataset) and the NLL
objective on dependency parsing reranking. We used an initial learning
rate of 1.0 for all experiments and when there was no significant
improvement in log-likelihood on the validation set, the learning rate
was divided by $2$ per epoch until convergence
<cit.>. To alleviate the exploding gradients
problem, we rescaled the gradient $g$ when the gradient norm $||g|| >
5$ and set $g = \frac{5g}{|| g ||}$
<cit.>. Dropout
<cit.> was applied to the 2-layer TreeLSTM
and LdTreeLSTM models.
The word embedding size was set to $s = d
/ 2$ where $d$ is the hidden unit size.
§.§ Microsoft Sentence Completion Challenge
The task in the MSR Sentence Completion Challenge
<cit.> is to select the correct missing word for
1,040 test sentences when presented with five
candidate completions. The training set contains 522 novels from the
Project Gutenberg which we preprocessed as follows. After removing
headers and footers from the files, we tokenized and parsed the
dataset into dependency trees with the Stanford Core NLP toolkit
<cit.>. The resulting training set contained 49M
words. We converted all words to lower case and replaced those
occurring five times or less with UNK. The resulting vocabulary size
was 65,346 words. We randomly sampled 4,000 sentences from the
training set as our validation set.
The literature describes two main approaches to the sentence
completion task based on word vectors and language models. In
vector-based approaches, all words in the sentence and the five
candidate words are represented by a vector; the candidate which has
the highest average similarity with the sentence words is selected as
the answer. For language model-based methods, the LM computes the
probability of a test sentence with each of the five candidate words,
and picks the candidate completion which gives the highest
probability. Our model belongs to this class of models.
Model $d$ $| \theta |$ Accuracy
4lWord Vector based Models
LSA — — 49.0
Skip-gram 640 102M 48.0
ivLBL 600 96.0M 55.5
4lLanguage Models
KN5 — — 40.0
UDepNgram — — 48.3
LDepNgram — — 50.0
RNN 300 48.1M 45.0
RNNME 300 1120M 49.3
depRNN+3gram 100 1014M 53.5
ldepRNN+4gram 200 1029M 50.7
LBL 300 48.0M 54.7
LSTM 300 29.9M 55.00
LSTM 400 40.2M 57.02
LSTM 450 45.3M 55.96
Bidirectional LSTM 200 33.2M 48.46
Bidirectional LSTM 300 50.1M 49.90
Bidirectional LSTM 400 67.3M 48.65
4lModel Combinations
RNNMEs — — 55.4
Skip-gram + RNNMEs — — 58.9
4lOur Models
TreeLSTM 300 31.6M 55.29
LdTreeLSTM 300 32.5M 57.79
TreeLSTM 400 43.1M 56.73
LdTreeLSTM 400 44.7M 60.67
Model accuracy on the MSR sentence completion task. The results
of KN5, RNNME and RNNMEs are reported in
mikolov2012thesis, LSA and RNN in
zweig2012computational, UDepNgram and LDepNgram in
gubbins-vlachos:2013:EMNLP, depRNN+3gram and
depRNN+4gram in mirowski-vlachos:2015, LBL in
MnihTeh2012, Skip-gram and Skip-gram+RNNMEs in
Mikolov:ea:2013b, and ivLBL in
NIPS2013_5165; $d$ is the hidden size and
$|\theta|$ the number of parameters in a model.
Table <ref> presents a summary of our results together with
previoulsy published results. The best performing word vector model is
ivLBL <cit.> with an accuracy of 55.5, while the
best performing single language model is LBL <cit.>
with an accuracy of 54.7. Both approaches are based on the
log-bilinear language model <cit.>. A combination of
several recurrent neural networks and the skip-gram model holds the
state of the art with an accuracy of 58.9 <cit.>. To
fairly compare with existing models, we restrict the layer size of our
models to 1. We observe that LdTreeLSTM consistently outperforms
TreeLSTM, which indicates the importance of modeling the
interaction between left and right dependents. In fact, LdTreeLSTM ($d=400$) achieves a new state-of-the-art on this task,
despite being a single model. We also implement LSTM and bidirectional
LSTM language models.[LSTMs and BiLSTMs were also trained with
NCE (; hyperparameters were tuned on the
development set).] An LSTM with $d=400$ outperforms its smaller
counterpart ($d=300$), however performance decreases with $d=450$. The
bidirectional LSTM is worse than the LSTM (see MnihTeh2012
for a similar observation). The best performing LSTM is worse than a
LdTreeLSTM ($d=300$). The input and output embeddings
($\mathbf{W}_e$ and $\mathbf{W}_{ho}$) dominate the number of
parameters in all neural models except for RNNME, depRNN+3gram and
ldepRNN+4gram, which include a ME model that contains 1 billion sparse
n-gram features <cit.>. The
number of parameters in TreeLSTM and LdTreeLSTM is not
much larger compared to LSTM due to the tied $\mathbf{W}_e$
and $\mathbf{W}_{ho}$ matrices.
§.§ Dependency Parsing
2*Parser 2c|Development 2cTest
UAS LAS UAS LAS
MSTParser-2nd 92.20 88.78 91.63 88.44
TreeLSTM 92.51 89.07 91.79 88.53
TreeLSTM* 92.64 89.09 91.97 88.69
LdTreeLSTM 92.66 89.14 91.99 88.69
NN parser* 92.00 89.70 91.80 89.60
S-LSTM* 93.20 90.90 93.10 90.90
Performance of TreeLSTM and LdTreeLSTM on
reranking the top dependency trees produced by the 2nd order MSTParser
Results for the NN and S-LSTM parsers are reported in
chen-manning:2014:EMNLP2014 and
dyer-EtAl:2015:ACL-IJCNLP, respectively. *
indicates that the model is initialized with pre-trained word vectors.
In this section we demonstrate that our model can be also used for
parse reranking. This is not possible for sequence-based language
models since they cannot estimate the probability of a tree. We use
our models to rerank the top $K$ dependency trees produced by the
second order MSTParser
<cit.>.[http://www.seas.upenn.edu/ strctlrn/MSTParser]
We follow closely the experimental setup of
chen-manning:2014:EMNLP2014 and
dyer-EtAl:2015:ACL-IJCNLP. Specifically, we trained TreeLSTM and LdTreeLSTM on Penn Treebank
sections . We used section 22 for development and
section 23 for testing. We adopted the Stanford basic dependency
representations <cit.>; part-of-speech tags were
predicted with the Stanford Tagger <cit.>. We
trained TreeLSTM and LdTreeLSTM as language models
(singletons were replaced with UNK) and did not use any POS tags,
dependency labels or composition features, whereas these features are used in
chen-manning:2014:EMNLP2014 and
dyer-EtAl:2015:ACL-IJCNLP. We tuned $d$, the number of
layers, and $K$ on the development set.
Table <ref> reports unlabeled attachment scores (UAS) and
labeled attachment scores (LAS) for the MSTParser, TreeLSTM
($d=300$, 1 layer, $K=2$), and LdTreeLSTM ($d=200$, 2 layers,
). We also include the performance of two neural
network-based dependency parsers;
chen-manning:2014:EMNLP2014 use a neural network classifier
to predict the correct transition (NN parser);
dyer-EtAl:2015:ACL-IJCNLP also implement a transition-based
dependency parser using LSTMs to represent the contents of the stack and buffer in a continuous space. As can be seen, both TreeLSTM and LdTreeLSTM outperform the baseline MSTParser,
with LdTreeLSTM performing best. We also initialized the word
embedding matrix $\mathbf{W}_e$ with pre-trained GLOVE vectors
<cit.>. We obtained a slight improvement over TreeLSTM (TreeLSTM* in Table <ref>; $d=200$, 2
layer, $K=4$) but no improvement over LdTreeLSTM. Finally,
notice that LdTreeLSTM is slightly better than the NN parser in
terms of UAS but worse than the S-LSTM parser. In the future, we would
like to extend our model so that it takes labeled dependency
information into
§.§ Tree Generation
Generated dependency trees with LdTreeLSTM trained on the PTB.
This section demonstrates how to use a trained LdTreeLSTM to
generate tree samples. The generation starts at the root
node. At each time step $t$, for each node $w_t$, we add a new edge
and node to the tree. Unfortunately during generation, we do not know
which type of edge to add. We therefore use four binary classifiers
(Add-Left, Add-Right, Add-Nx-Left and Add-Nx-Right) to predict whether we should add a Left, Right, Nx-Left or Nx-Right edge.[It is
possible to get rid of the four classifiers by adding START/STOP
symbols when generating left and right dependents as in
<cit.>. We refrained from doing this for
computational reasons. For a sentence with $N$ words, this approach
will lead to $2N$ additional START/STOP symbols (with one START and
one STOP symbol for each word). Consequently, the computational
cost and memory consumption during training will be three times as
much rendering our model less scalable. ] Then when a classifier
predicts true, we use the corresponding LSTM to generate a new node by
sampling from the predicted word distribution in
Equation (<ref>). The four classifiers take the previous
hidden state $\mathbf{H}[:,t']$ and the output embedding of the
current node $\mathbf{W}_{ho} \cdot e(w_{t})$ as
features.[The input embeddings have lower dimensions and
therefore result in slightly worse classifiers.] Specifically, we
use a trained LdTreeLSTM to go through the training corpus and
generate hidden states and embeddings as input features; the
corresponding class labels (true and false) are “read off” the
training dependency trees. We use two-layer rectifier networks
<cit.> as the four classifiers with a hidden size
of 300. We use the same LdTreeLSTM model as in
Section <ref> to generate dependency trees. The
classifiers were trained using AdaGrad <cit.> with a
learning rate of 0.01. The accuracies of Add-Left, Add-Right, Add-Nx-Left and Add-Nx-Right are 94.3%,
92.6%, 93.4% and 96.0%, respectively. Figure <ref>
shows examples of generated trees.
§ CONCLUSIONS
In this paper we developed TreeLSTM (and LdTreeLSTM), a
neural network model architecture, which is designed to predict tree
structures rather than linear sequences. Experimental results on the
MSR sentence completion task show that LdTreeLSTM is superior to
sequential LSTMs. Dependency parsing reranking experiments highlight
our model's potential for dependency parsing. Finally, the ability of
our model to generate dependency trees holds promise for text
generation applications such as sentence compression and
simplification <cit.>. Although our experiments
have focused exclusively on dependency trees, there is nothing
inherent in our formulation that disallows its application to other
types of tree structure such as constituent trees or even taxonomies.
§ ACKNOWLEDGMENTS
We would like to thank Adam Lopez, Frank Keller, Iain Murray, Li Dong,
Brian Roark, and the NAACL reviewers for their valuable feedback.
Xingxing Zhang gratefully acknowledges the financial support of the
China Scholarship Council (CSC). Liang Lu is funded by the UK EPSRC
Programme Grant EP/I031022/1, Natural Speech Technology (NST).
|
1511.00372
|
Precision QCD for LHC Physics: The nCTEQ15 PDFs
Fredrick I. Olness
[Work supported in part by the U.S. Department of Energy under grant
] Southern Methodist University, Dallas, TX 75275, USA
based on work in collaboration with D. B. Clark,
E. Godat,
T. Je¸o,
C. Keppel,
K. Kovařík,
A. Kusina,
F. Lyonnet,
J.G. Morfín,
P. Nadolsky,
J.F. Owens,
I. Schienbein,
J.Y. Yu
Searches for new physics at the LHC will increasingly depend on identifying
deviations from precision Standard Model (SM) predictions. At the
higher energy scales involved for the LHC Run 2, the heavy quarks
play a more prominent role than at the Tevatron. Recent theoretical
developments improve our ability to address multi-scale problems and
properly incorporate heavy quark masses across the full kinematic
range. These developments are incorporated into the new nCTEQ15 PDFs,
and we review these developments with respect to sample Run 2 measurements,
and identify areas where additional effort is required.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
Kovarik:2015cma, Kusina:2012vh, Kovarik:2010uv, Schienbein:2009kk
The leading-order (LO) differential cross section ($d\sigma/dy$)
for $W^{+}$production at the Tevatron (2 TeV) and the LHC (14 TeV)
as a function of rapidity. The partonic contributions are also displayed
for $\{u\bar{d},c\bar{s},u\bar{s},c\bar{d}\}$. The vertical scales
are logarithmic.
The strange quark contribution (yellow) as a fraction of
the total $d^{2}\sigma/dM/dy$ in pb/GeV for $pp$ to $W^{+}$ (left),
$W^{-}$ (center), $Z$ (right) production at the LHC for 14 TeV
with CTEQ6.6 using the VRAP program at NNLO. C.f. Ref. <cit.>
for details.
Our field has seen major discoveries in recent years from a variety
of experiments, large and small, including a number recognized with
Nobel Prizes. The recent performance of the LHC has exceeded expectations
and produced an unprecedented number of events to be analyzed. On
the Intensity Frontier, Fermilab is advancing a number of high-precision
experiments (Muon $g-2$, Mu2e), as well as expanding its neutrino
program. Thus, there is a wealth of data to explore, and a comprehensive
analysis requires the most advanced and innovative tools.
As the accuracy of the experimental measurements increases, it is
essential to improve the theoretical calculations to match. If we
can make detailed predictions of W/Z/Higgs production (for example),
then we have the ability to distinguish a “new physics” signal
from an uncertain SM background process. To determine if the newly
discovered Higgs boson is that of the Standard Model (SM) or a more
exotic type, we must study both the production cross section and various
decay channels to make its proper characterization. In a complementary
manner, the Fermilab high-intensity high-statistics experiments force
us to reexamine previous assumptions (nuclear corrections, isospin
and lepton-flavor symmetries) and require us to extend our calculations
to increasingly high orders including subtle electroweak corrections.
The key step for all the above analyses is to make accurate predictions,
including realistic estimates of the underlying theoretical uncertainty.
The PDFs are at the heart of this program.
§ PDF FLAVOR DETERMINATION & HEAVY TARGETS
The computed nuclear correction ratio, $F_{2}^{Fe}/F_{2}^{D}$
as a function of $x$ for $Q^{2}=20\, GeV^{2}$. Figure-a) shows the
basic dimuon process $\nu N\to\mu^{+}\mu^{-}X$. Figure-b) shows the
fit using the $\nu N$ DIS data (fit A2) compared with parameterizations
of the neutral current lepton ($\ell^{\pm}N$) DIS data (KP, SLAC/NMC,
HKN07). The data are from the NuTeV experiment. See Ref. <cit.>
for details.
The nCTEQ15<cit.> PDFs for selected nuclei
for the gluon and strange PDF at $Q=10\, GeV$.
The objective of the nCTEQ project is to obtain the most precise set
of Parton Distribution Functions (PDFs) to facilitate measurements
and interpret hadronic processes at both fixed-target experiments,
HERA, RHIC, Tevatron, and the LHC. The project began when it was realized
that a limiting factor on the proton PDF precision was the nuclear
corrections used for the wealth of nuclei data—particularly the
DIS data which is crucial for flavor differentiation.
As the bulk of the data used in the global analyses of the PDFs comes
from Deeply Inelastic Scattering (DIS) processes, much of this is
measured on heavy targets (e.g., iron or lead) where nuclear corrections
must be taken into account. This data is very important for distinguishing
the separate flavor components in the proton. Surprisingly, the strange
quark PDFs have a large influence on LHC “benchmark” processes.
In Fig. <ref> we note that the heavy quark initiated
contributions ($c\bar{s}$) at the LHC can be 30% or more of the
total cross section, whereas it is only a few percent at the Tevatron.
Furthermore, the larger $\sqrt{s}$ energy of the LHC probes a much
broader range in rapidity $y$, and hence a broader range in the partonic
$x$. While the LO illustration of Fig. <ref> is instructive,
in Fig. <ref> we show the high-precision results of
the NNLO calculation for $\{W^{\pm},Z\}$ using the VRAP program;<cit.>
if we are to make full use of this very precise NNLO result, we must
improve the precision of the strange PDF.
The primary constraint on the strange quark PDF comes from neutrino-induced
DIS dimuon production ($\nu N\to\mu^{+}\mu^{-}X$) on heavy targets.
[New data from LHC are beginning to provide information the strange
quark at larger $Q$ and smaller $x$; cf. Refs. <cit.>.
] Fig. <ref>-a) shows the basic dimuon process used to
constrain $s(x)$; the anti-neutrino process can constrain $\bar{s}(x)$.
As the neutrino experiments use heavy nuclear targets (typically iron
or lead), we need to know the nuclear correction to relate this information
back to the proton data.
Fig. <ref>-b) shows the extracted nuclear correction factor
for the neutrino DIS ($\nu N$) processes (fit A2) as compared with
that for charged lepton ($\ell^{\pm}N$) DIS processes (KP, SLAC/NMC,
HKN07), and we observe some significant differences. As was demonstrated
in Ref. <cit.>, if we properly incorporate the experimental
correlated errors in the global PDF fit, we are unable to find a nuclear
correction which is compatible with both the $\nu N$ and $\ell^{\pm}N$
data simultaneously;
[Note that this difference was present only if we imposed
the full constraints of the experimental correlated systematic errors;
if the systematic and statistical errors were added in quadrature,
a common correction factor was obtained. This observation highlights
the importance of the experimental error treatment in the fits, and
resolves a number of questions regarding the compatibility of these
data sets.
] thus, we must account for this when we extract the strange PDF and
include an additional uncertainty.
§ THE NCTEQ15 PDFS
The nCTEQ15 PDFs showing the uncertainty bands for selected
partons ($g,s)$. For comparison, we also show bands for HKN07,<cit.>
EPS09,<cit.> and DSSZ.<cit.>
Correlation measures ($\cos\phi$) for $u_{val}$ (left)
and $d_{val}$ (right) for lead at $Q=10\, GeV$. Eight selected experiments
are highlighted with symbols. To emphasize the anti-correlation between
$u_{val}$ and $d_{val}$ we have flipped the $d_{val}$ plot vertically.
See Ref. <cit.> for details.
The nCTEQ framework allows the nuclear correction factors to be integrated
dynamically into the fit to better identify tensions between
data sets, and to extract more accurate PDFs when using data from
nuclear targets. We have now released the nCTEQ15 PDFs with error
sets which provide our results of the global analysis for all nuclear
$A$ values.
[These are available on-line at the HepForge repository: http://ncteq.hepforge.org/
]<cit.> In addition to the Deep Inelastic Scattering
(DIS) and Drell-Yan (DY) processes, we also include inclusive pion
production data to help constrain the gluon PDF. Within our framework
we are able to obtain a good fit to all data.
Fig. <ref> displays selected partons for a range of nuclear
$A$ values. We have determined the uncertainties using the Hessian
method with an optimal rescaling of the eigenvectors to accurately
represent the uncertainties for the chosen tolerance criteria. In
Fig. <ref> we compare the nCTEQ15 PDF uncertainty bands
with other sets from the literature. While the general features are
similar, there are some important differences. For example, the nCTEQ15
parameterization allows different correction factors for the up and
down quarks. To investigate which data sets are driving this difference,
we examine the correlation of the data sets with specific flavor components,
and asses the impact of individual experiments. Fig. <ref>
shows the correlation $\cos\phi$ for the up and down valence as a
function of $x$ for the lead PDFs at $Q=10$ GeV. Selected experiments
are highlighted with symbols. To emphasize the fact that the up and
down valence are relatively anti-correlated, we have vertically flipped
the plot for the down valence to make the correspondence between the
two plots readily apparent. We find that the fit exploits the additional
freedom to reduce the $\chi^{2}$ by an additional $\sim10\%$. While
these are interesting observations, work still remains to definitively
distinguish parameterization effects from the underlying physics.
In view of the differences, the true nPDF uncertainties should be
obtained by combining the results of all analyses and their uncertainties.
§ CONCLUSIONS
The nCTEQ15 PDFs represent the first complete analysis of nuclear
PDFs with errors in the CTEQ framework. The framework used for the
nCTEQ15 fit can combine data from both proton and nuclear targets
into a single coherent analysis; thus, it can yield more accurate
PDFs when using data from nuclear targets.
All in all we find relatively good agreement between different nPDF
sets. Most of the noticeable differences occur in regions without
any constraints from data and so they can be attributed to different
assumptions such as parameterization of the nuclear effects.
Using the nCTEQ15 fit as a reference, it will be interesting to include
the upcoming LHC data as we continue to investigate the relations
between the proton and the nuclear PDFs.
I am pleased to thank D. B. Clark,
E. Godat,
T. Je¸o,
C. Keppel,
K. Kovařík,
A. Kusina,
F. Lyonnet,
P. Nadolsky,
J.G. Morfín,
J.F. Owens,
I. Schienbein,
J.Y. Yu for helpful discussions and collaboration.
|
1511.00109
|
∫ω/2πf(ω- μ)
$^1$ Department of Physics, Nanoscience Center, FIN 40014,
University of Jyväskylä, Jyväskylä, Finland
$^2$ Dipartimento di Fisica, Università di Roma Tor
Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy
$^3$ Laboratori Nazionali di Frascati, Istituto Nazionale di
Fisica Nucleare, Via E. Fermi 40, 00044 Frascati, Italy
$^4$ European Theoretical Spectroscopy Facility (ETSF)
We discuss an extension of our earlier work on the time-dependent Landauer–Büttiker formalism for noninteracting electronic transport. The formalism can without complication be extended to superconducting central regions since the Green's functions in the Nambu representation satisfy the same equations of motion which, in turn, leads to the same closed expression for the equal-time lesser Green's function, i.e., for the time-dependent reduced one-particle density matrix. We further write the finite-temperature frequency integrals in terms of known special functions thereby considerably speeding up the computation. Simulations in simple normal metal – superconductor – normal metal junctions are also presented.
§ INTRODUCTION
The process of Andreev reflection<cit.> (AR) occurring at the interface between a normal metal (N) and a superconductor (S) is of great importance with applications in spintronics and quantum computing. An incoming electron from N to S produces a Cooper pair in S and a reflected hole in N<cit.>. In an NSN junction normal metal electrodes are spatially separated by a superconducting central region and an entangled electron–hole pair can be transported. This can be seen when the junction separation is of the order of the superconducting coherence length for the studied material, and when the incident electron energies are less than the superconducting gap for the AR process to occur<cit.>.
The quantum transport problems are typically time dependent; there is no guarantee that the system would in an instant relax to a steady-state configuration once the junction is “switched on” (as in connecting different devices or driving them out of equilibrium by an external perturbation). In contrast, there are transient effects depending on, e.g., the system's geometry<cit.>, its predisposition to external perturbations<cit.>, the physical properties of the transported quanta and their mutual interactions<cit.>. Even if the transport mechanisms were discussed in an idealized noninteracting setting, it is therefore important to consider a fully time-dependent description of the studied processes.
The Landauer–Büttiker formalism is simple to understand as it relates to an intuitive physical picture of charge transport in a multiterminal junction<cit.>. Including the transient description to the formalism by studying the nonequilibrium Green's function approach does not complicate the final result<cit.>; the physical picture is still clear and intuitive as different features of the transport setup can be directly linked to the time-dependence<cit.>. In this paper, we present an extension to earlier results for both superconducting junctions and arbitrary temperatures (Sec. <ref>). We present a formula for the time-dependent one-particle reduced density matrix (TD1RDM) as such since it is a closed, analytic expression which can readily be implemented for numerical model systems. Further details of the derivation are to be found in another work<cit.>. In Sec. <ref> we illustrate the features of the formula by studying transients in simple NSN junctions.
§ BACKGROUND AND NAMBU REPRESENTATION
We consider a quantum transport setup similar to one studied in the previous volume of this conference series<cit.>. In this setup, a noninteracting central region is connected between metallic leads, and the Hamiltonian takes the form
Ĥ = Ĥ_leads + Ĥ_central + Ĥ_coupling
= ∑_kασ[ϵ_kα+θ(t)V_α]ĉ_kασ^†ĉ_kασ + ∑_mnσ T_mnĉ_mσ^†ĉ_nσ + ∑_mkασ(T_mkαĉ_mσ^†ĉ_kασ + ) .
The operators $\hat{c}^{(\dagger)}$ annihilate (create) electrons from (to) a region specified by the subscript indices: $k\alpha$ is the $k$-th basis element of the $\alpha$-th lead, $m,n$ label the basis elements of the central region, and $\sigma\in\{\uparrow,\downarrow\}$ is a spin-$\frac{1}{2}$ index. These operators moreover obey the fermionic anticommutation relations $\{\hat{c}_{x\sigma},\hat{c}_{y\sigma'}^\dagger\}=\delta_{xy}\delta_{\sigma\sigma'}$. The Hamiltonian structure is determined by the single-particle levels in the leads $\epsilon_{k\alpha}$ and the tunneling matrices $T$ between the states of the central region ($T_{mn}$) and between the states of the central region and the leads ($T_{mk\alpha}$). The system is driven out of equilibrium for times $t>0$ by a sudden shift of the lead energy levels by $V_\alpha$. In addition to a sudden bias, it is also possible to include time-dependent bias profiles without complicating the following derivations<cit.>. In order to describe a superconducting island, we will now add a pairing field operator $\hat{\varDelta}$ to the Hamiltonian of the central region by<cit.>
Ĥ_central →∑_mnσ T_mnĉ_mσ^†ĉ_nσ + ∑_m _m ĉ_m↑^†ĉ_m↓^†+ ∑_m _m^* ĉ_m↓ ĉ_m↑ .
The one-electron Green's function in the above setup can be defined via the Nambu spinor $\underline{\hat{\varPhi}}_m = (\hat{\varPhi}_{m}^1 , \hat{\varPhi}_{m}^2)^{T} = (\hat{c}_{m\uparrow} , \hat{c}_{m\downarrow}^\dagger)^{T}$, which obeys the anticommutation relation in a tensor product sense $\{\hat{\varPhi}_m^\mu,\hat{\varPhi}_n^\nu\} = \delta_{mn}\delta^{\mu\nu}$, as a contour-ordered product<cit.>
G_rs(z,z') = -⟨𝒯_γ[_r(z) ⊗_s^†(z')] ⟩where the contour-ordering operator $\mathcal{T}_\gamma$ is taken for the variables $z,z'$ on the Keldysh contour $\gamma$<cit.>. When the product in Eq. (<ref>) is expanded, the elements in the resulting $2\times 2$ Nambu matrix are the normal and the anomalous components of the Green's function<cit.>. As already denoted above, we put an underline for quantities in the Nambu space. The matrix elements in the Green's function label the transport setup in the following block form [notice that the indices $r,s$ in Eq. (<ref>) may belong to any block and that the number of leads is arbitrary]
= h_11 0 ⋯ h_1 C
0 h_22 ⋯ h_2C
⋮ ⋮ ⋱ ⋮
h_C1 h_C2 ⋯ h_CC ; = G_11 G_12 ⋯ G_1 C
G_21 G_22 ⋯ G_2C
⋮ ⋮ ⋱ ⋮
G_C1 G_C2 ⋯ G_CC
(h_αα')_kk'(t) = [ϵ_k+θ(t)V_]δ_αα'δ_kk' 0
0 -[ϵ_k+θ(t)V_]δ_αα'δ_kk' ,
(h_CC)_mn = T_mn _mδ_mn
_m^*δ_mn -T_mn,
(h_Cα)_mkα = T_mk 0
0 -T_mk , (h_αC)_kαm = T_km 0
0 -T_km
for the leads, central region and couplings, respectively. It is important to notice that even though only the central region is superconducting, all the blocks in the Hamiltonian are written in the form of Bogoliubov–de Gennes<cit.>. Including the pairing field in the Hamiltonian of the central region adds no extra complication to the evolution of the Green's function<cit.>; the only difference, compared to the earlier work in Refs. <cit.>, is in the interpretation of the matrices in Nambu space. It is also possible to include non-local pairing field $\varDelta_{mn}$ with arbitrary spin-coupling leading to $4$-component Nambu spinors which includes the possibility to study also Majorana fermions<cit.>. The Hamiltonian and the Green's function are connected via the equation of motion (with the boundary condition that the Green's function is antiperiodic along the contour, i.e., the Kubo–Martin–Schwinger boundary conditions<cit.>)
[/z - (z)](z,z') = δ(z,z')
and the corresponding adjoint one. We describe the leads within the wide-band approximation (WBA), where the electronic levels of the central region are in a narrow range compared to the lead bandwidth which gives for the retarded embedding self-energy
_,mn^R() = ∑_k (h_C)_mk1/-_k-V_+η (h_C)_kn ≈-_,mn/2 ,
where the bandwidth matrices satisfy $\underline{\varGamma} = \sum_\a \underline{\varGamma}_\a$. For the lead Green's function between the coupling Hamiltonians in Eq. (<ref>) the structure is similar to that of Eq. (<ref>). Approximating the embedding self-energy this way, as a purely imaginary constant, closes the equation of motion (<ref>), and it can then be solved analytically<cit.>. As the solution we get the lesser Green's function (in region $CC$) in the equal-time limit and the TD1RDM by
(ρ_CC)_mn(t) = -(G_CC^<)_mn(t,t) = -(G^<_CC,↑)_mn(t,t) (-F^>_CC)_nm(t,t)
(F̅^<_CC)_mn(t,t) (-G^>_CC,↓)_nm(t,t)
= ⟨ĉ_n↑^†(t)ĉ_m↑(t)⟩ ⟨ĉ_n↓(t)ĉ_m↑(t)⟩
⟨ĉ_n↑^†(t)ĉ_m↓^†(t)⟩ ⟨ĉ_n↓(t)ĉ_m↓^†(t)⟩
where the normal ($G^{\lessgtr}_\sigma$) and anomalous ($F^{\lessgtr}$) components<cit.> follow by expanding the product in Eq. (<ref>). It is also possible to solve the equations of motion analytically for the two-time Keldysh components of the Green's function; this offers the possibility to extract not only densities and currents but other physical quantities such as noise from the solution<cit.>.
Expressed in the left eigenbasis, $\bra{{\varPsi}^{\text{L}}}\underline{h}_{\text{eff}} = {\eps}\bra{{\varPsi}^{\text{L}}}$ (the eigenvalues $\eps$ are in general complex), of the nonhermitian effective Hamiltonian $\underline{h}_{\text{eff}} = \underline{h}_{CC} - \im\underline{\varGamma}/2$ the matrix elements of the TD1RDM take the explicit form<cit.>
⟨_j^L | ρ_CC(t) | _k^L ⟩
= ∑_α{_α,jk _α,jk + V_α_α,jk[_α,jk(t)+_α,kj^*(t)]+V_α^2_α,jk^-(ϵ_j-ϵ_k^*)t_α,jk} ,
_α,jk = ⟨_j^L | _α| _k^L ⟩,
_α,jk = /ϵ_k^* - ϵ_j{1/^β(ϵ_k^*-μ_α)+1 + 1/2π[ψ(1/2-β(ϵ_k^*-μ_α)/2π)-ψ(1/2-β(ϵ_j-μ_α)/2π)]} ,
_α,jk(t) = /(_k^* - _j)(_k^* - _j - V_α){^-(_j - _k^*)t/^β(_k^* - μ_α)+1 + ^-πt /β^-(_j - μ_α)t×.
. [𝔉(_k^*-μ_α,t,β) + _k^*-_j - V_α/V_α 𝔉(_j-μ_α,t,β) - _k^* - _j/V_α 𝔉(_j-μ,t,β)]} ,
_α,jk = /^β(_k^*-μ)+1-1/2π[ψ(1/2 - β(_j-μ_α)/2π) - ψ(1/2 - β(_k^*-μ)/2π)]/(_k^*-_j)(_k^* - _j + V_α)V_α
- /^β(_k^*-μ_α)+1-1/2π[ψ(1/2 - β(_j-μ)/2π) - ψ(1/2 - β(_k^*-μ_α)/2π)]/(_k^*-_j)(_k^* - _j - V_α)V_α ,
where $\beta$ is the inverse temperature, $\mu$ is the chemical potential and we also denoted the electro-chemical potential as $\mu_\a = \mu + V_\a$. In the above expressions $\psi$ is the digamma function<cit.> and we defined another special function by $\mathfrak{{F}}(z,t,\beta) \equiv \frac{1}{\im\beta z + \pi} \ \hypf\left(1, \frac{1}{2}+\frac{\im\beta z}{2\pi}, \frac{3}{2}+\frac{\im\beta z}{2\pi}, \ex^{-2\pi t/\beta}\right)$ with $\hypf$ being the hypergeometric function<cit.>.
Evaluating the TD1RDM in a physically relevant basis, say, the localized site basis of the central region, is then readily done as a basis transformation from the left eigenbasis, $\{\ket{{\varPsi}^{\text{L}}}\}$, to the desired one, $\{\ket{{\varPhi}}\}$, by
_mρ_CC(t)_n = ∑_jk _m_j^R/_j^L_j^R
⟨_j^L | ρ_CC(t) | _k^L ⟩,
where $\{\ket{{\varPsi}^{\text{R}}}\}$ are the right eigenvectors of $\underline{h}_{\text{eff}}$; this form in Eq. (<ref>) follows from the biorthogonality of the left/right eigenvectors of a nonhermitian matrix.
Studying the asymptotic behaviour of the digamma and hypergeometric functions the results in Eqs. (<ref>), (<ref>) and (<ref>) can be shown to reduce to those in Ref. <cit.> in the zero-temperature limit ($\beta\to\infty$)<cit.>.
§ AN INTRODUCTORY EXAMPLE
Let us motivate the discussion for the NSN setup by means of a simple example. Consider a single dot connected to two leads for which the Hamiltonian can be separated in parts for leads, tunneling and dot, respectively as
Ĥ = ∑_kσ_kĉ_kσ^†ĉ_kσ + ∑_kσt_k0ĉ_kσ^†ĉ_0σ + ∑_kσt_k0^* ĉ_0σ^†ĉ_kσ + _0∑_σĉ_0σ^†ĉ_0σ
+ _0 ĉ_0↑^†ĉ_0↓^†+ _0^* ĉ_0↓ ĉ_0↑
with $\eps_{k\a}$ giving the level structure of the leads $\a\in\{L,R\}$, $t_{k\a 0}$ corresponding to the tunneling strength between the leads and the dot, and $\eps_0,\varDelta_0$ being the energy and the pairing strength in the dot, respectively. Let us introduce a new set of operators $\hat{\tilde{c}}_{x\sigma} = \hat{c}_{x\sigma}^\dagger$ obeying the fermionic anticommutation relation. The Hamiltonian can now be rewritten in terms of the new and old operators as
Ĥ = ∑_k_kĉ_k↑^†ĉ_k↑ + ∑_k(-_k)ĉ̃̂_k↓^†ĉ̃̂_k↓ + ∑_k_k
+ ∑_k(t_k0ĉ_k↑^†ĉ_0↑ + t_k0^*ĉ_0↑^†ĉ_k↑) + ∑_k[(-t_k0)ĉ̃̂_0↓^†ĉ̃̂_k↓ + (-t_k0^*)ĉ̃̂_k↓^†ĉ̃̂_0↓]
+ _0 ĉ_0↑^†ĉ_0↑ + (-_0)ĉ̃̂_0↓^†ĉ̃̂_0↓ + _0 + _0ĉ_0↑^†ĉ̃̂_0↓ + _0^* ĉ̃̂_0↓^†ĉ_0↑
where two constant shifts $\sum_{k\a}\eps_{k\a}$ and $\eps_0$ occur due to the anticommutation relations. Each term in Eq. (<ref>) has a similar structure, $\hat{c}^\dagger\hat{c}$, and we may model the dot part as in Fig. <ref>
The dot viewed as a two-level system for different spins.
Couplings between the lead and the dot in a transport setup.
where the matrix is of the form of Eq. (<ref>) and the corresponding eigenvalues are $\eps_{\pm} = \pm \sqrt{\eps_0^2 + |\varDelta_0|^2}$. The transport setup corresponding to Eq. (<ref>) can then be viewed through the energy diagram in Fig. <ref> where we notice the nature of the constant shifts in Eq. (<ref>); they could be regarded as a chemical potential. The energy levels for the lead sector are raised so that the energy level continuum for the spin-up particles goes up from $\sum_{k\a}\eps_{k\a}$ and the energy level continuum for the spin-down particles goes down from $\sum_{k\a}\eps_{k\a}$. Similarly, for the dot sector we have the energy levels raised by $\eps_0$. The coupling terms $t_{k\a 0}$ connect separately the spin-up and spin-down particles between the leads and the dot, and the pairing strength term $\varDelta_0$ acts as a hopping term flipping the spins within the dot.
According to this picture for the NSN setup and the Nambu structure in Eq. (<ref>) we will evaluate the local bond currents for the more general structure in Eq. (<ref>) by
J_mn(t) = -∑_σ[T_mn(G_CC,σ^<)_nm(t,t) + ]
and the Cooper pair density by
P_m(t) = (F_CC^>)_mm(t,t)^2T_mmt
satisfying the continuity equation<cit.>
/t n_m(t) = ∑_n J_mn(t) - 4 [_m^* P_m(t)^-2T_mm t]
where the site density is the expectation value $n_m = \langle \hat{n}_m \rangle$ of $\hat{n}_m = \sum_\sigma \hat{c}_{m\sigma}^\dagger \hat{c}_{m\sigma}$. In the Nambu representation of the lesser Green's function the diagonal blocks therefore give rise to the bond current whereas the off-diagonal blocks correspond to the Cooper pair density. In the continuity equation (<ref>) the two different terms on the right-hand side can also be identified as the normal current and the super current.
§ TD RESPONSE OF A SUPERCONDUCTING JUNCTION
As a first study we show a numerical confirmation of the presented formula; in the limit when the superconducting gap $\varDelta$ and the temperature $1/\beta$ vanish we should recover equal results with the formula in Ref. <cit.>. For the sake of simple interpretation of the transients let us take, as an example, a $2$-site tight-binding dimer coupled to two semi-infinite one-dimensional leads. Let the hopping parameter between the sites be equal everywhere: $t_\a = t_{\a C} = t_C \eqqcolon \epsilon_0$ (for $\a = L,R$) leading to the tunneling rate $\varGamma_\alpha = 2t_{\a C}^2/|t_\a| = 2\epsilon_0$. This parameter is the strength of the bandwidth matrix elements in the form of Eq. (<ref>). In fact, WBA is not a very good approximation in this case, as the resonances are comparatively rather wide, but we are only comparing different formulae within the WBA, so the results should only be taken as comparative. Let us also bias the leads symmetrically to $V_L = -V_R = \epsilon_0$ with respect to the chemical potential $\mu=0$. We calculate, from the TD1RDM, the local bond current between the two sites in the dimer using Eq. (<ref>). (Due to equal hopping parameters through the setup this is equal to the current through the lead interfaces modulo a minor time delay.)
In Fig. <ref>(a) we compare normal central region to a superconducting one by evaluating the TD1RDM from a normal Hamiltonian without the pairing field (non-Nambu), and from a Nambu Hamiltonian with varying pairing field strength at zero temperature. We see how the N and S ($\varDelta=0$) cases are on top of each other, and increasing the value for $\varDelta$ decreases the absolute value of the current through the central region as the energy levels of the central region get raised by $\sqrt{\varDelta^2 + \epsilon_0^2}$. In Fig. <ref>(b) we compare normal central regions at varying temperatures. In this comparative benchmark of varying temperature, we do not consider superconducting central regions as the temperature effects for the pairing field $\varDelta(T)$ should also be taken into account according to the self-consistent gap equation<cit.>. (In further simulations, also these parameters are considered in more detail.) The zero-temperature limit, $\beta \to \infty$, is evaluated from the results in Ref. <cit.> which roughly agrees with an evaluation with $\beta=10/\epsilon_0$. (Increasing $\beta$ even more would naturally bring the curves exactly on top of each other.) Because the level structure of the studied system is symmetric around the chemical potential, increasing the temperature $1/\beta$ decreases the current due to broadening of the distribution function close to the Fermi level. In general, however, there is a possibility of enhancing the current by increasing the temperature if, for instance, the electronic levels were all above the Fermi level. In that case, the lead-states with energy higher than the energy of the levels get occupied, leading to an enhanced current.
(Color online) Transient current through a $2$-site dimer; comparison between different formulae and parameters. (a) Normal vs. superconducting central region at zero temperature; (b) normal central region at varying temperatures.
After the comparisons presented above we can confidently conclude that the formulation of the NSN transport setup and the implementation of the formula for the TD1RDM is working properly. Now, we turn to a more concrete and physically relevant example, and we analyze the transient features in more detail. We consider a superconducting island made of a benzene-like molecule belonging to the class of quasi-one-dimensional polyacene chains<cit.>. Transport in simple island setups has been studied, e.g., in Refs. <cit.>, in a single-electron-tunneling level, where Coulomb blockade region is explored, and it is shown how the superconducting gap $\varDelta$ strongly and non-trivially affects the tunneling process. In polyacene samples (and in other carbon based materials, such as graphene) the superconductivity could be induced, e.g., by charge injection, chemical doping or using the proximity effect leading to critical temperatures ranging from $1$ to $10$ K<cit.>. Our setup is shown schematically in Fig. <ref>. We model the benzene molecule in a single $\pi$-orbital tight-binding framework with the hopping parameter $t_C = -2{.}7$ eV<cit.>, and relate other energies to this scale. We also saturate the molecule's edges (longitudinally, in the transport direction) by hydrogen with modified tight-binding parameters for hydrogen on-site energies and hydrogen–carbon hopping<cit.>, respectively, so that there is no band gap in equilibrium. This condition is set because we want to isolate the effects from the superconducting gap $\varDelta$ without complicating the spectrum with the semiconducting gap. The coupling strength between the molecule and the leads and the lead hopping are chosen so that we are in weak coupling regime $\varGamma = 0{.}2$ eV.
(Color online) Transport setup in an NSN junction. Normal metal leads of continuum states undergo a level shift due to the bias voltage $V_{L/R}$ with respect to the chemical potential $\mu=E_{\text{F}}$ (Fermi level). The discrete level structure of the central region is determined by the tight-binding and gap $\varDelta$ parameters, and it is also broadened due to the coupling ($\varGamma$). Possible transition mechanisms are shown as CT, AR and CAR; see text for description.
Looking at the setup in Fig. <ref> more closely suggests different transition mechanisms depending on the parameters. The simplest case is when the bias voltage $V_\alpha$ is larger than the superconducting gap $\varDelta$. In this case, all the levels inside the bias window act as transport channels, and transitions through the superconducting states are disrupted since the energy for the incoming electrons is high enough to break possible Cooper pairs (CP); this is referred to as normal tunneling (NT). If the bias voltage is smaller than the superconducting gap, this opens a possibility for the formation of a CP in the central region. In this case, it is possible to observe Andreev reflection (AR) between an electron and a hole in the source (or drain) lead forming the CP in the center, or to observe a crossed Andreev reflection (CAR) where an electron from the source (drain) lead is coupled to a hole in the drain (source) lead through the CP in the center. Also, direct tunneling of an electron via the CP, referred to as cotunneling (CT), is a possible transmission channel. Next, we simulate these different processes by a suitable parameter choice.
(Color online) Transient currents (top panel) and pair densities (bottom panel) in the molecule when varying $\varDelta$. The inset shows the absolute value of the Fourier-transformed current.
(Color online) Transient currents (top panel) and pair densities (bottom panel) in the molecule when varying $V$. The inset shows the absolute value of the Fourier-transformed current.
We start with a familiar example by simulating NT: The condition is such that the bias window is larger than the gap. We will, in addition, fix the temperature, $\beta=100/|t_C|$, well below the critical temperature so that the gap can be approximated as the (constant) value at zero temperature $\varDelta(T=0)$. The sample is a benzene molecule consisting of $8$ atomic sites ($6$ carbon and $2$ hydrogen), coupled to the leads from four sites overall, see Fig. <ref>. The bias voltage is symmetrically set to $V_L = -V_R = 3|t_C| / 2$ and the gap $\varDelta$ is varied but kept smaller than or equal to this value. The transient currents through the sample [calculated by summing the bond currents from Eq. (<ref>) transversally in the middle of the molecule], the corresponding Fourier transforms and the pair densities [calculated by summing the pair densities within the molecule: $P(t) = \sum_m P_m(t)$ from Eq. (<ref>)] can be seen in Fig. <ref>. Increasing the gap $\varDelta$ decreases the overall current as the conducting states are being pushed away from the bias window. This also leads to shifts in the transient frequencies seen in the Fourier spectrum. By looking also at the spectral function $A(\w) = -\frac{1}{\pi} \Im \Tr [G^{\text{R}}(\w)]$, where $G^{\text{R}}$ is the normal Nambu component of the retarded Green's function, plotted in Fig. <ref> we can further identify the transitions.
In Fig. <ref>, when $\varDelta=0$, we see two intramolecular transitions at frequencies $\w=|t_C|$ and $\w=2|t_C|$ which move a little when the gap is increased to $\varDelta=|t_C|/2$ corresponding to the shifted energy levels in the spectral function. We also observe two lead–molecule transitions at $\w=|t_C|/2$ and $\w=5|t_C|/2$ when $\varDelta=0$; these frequencies shift with the peaks in the spectral function corresponding to the fixed bias window at $V=3|t_C|/2$. The reason why we do not see a lead–molecule transition at $\w=3|t_C|/2$ when $\varDelta=0$, even though there is a zero-energy state in the molecule, is due to the fact that this state corresponds to the wavefunction's nodal planes being located exactly at the lead interface, and therefore it is an inert state not taking part to the transient dynamics<cit.>. Also, as the conditions are for NT, we observe the pair density within the molecule going to zero from its equilibrium value when $\varDelta < V$; this means that there are no out-of-equilibrium CPs forming in the central region, and we see no AR or CAR processes. When we set the gap equal to the bias window, we notice, first of all, that the steady-state current goes to zero since there are no transport channels within the bias window. Some transient oscillations are still present due to the states in the vicinity of the resonant window, which is seen as an intramolecular transition at $\w\sim 7|t_C|/2$.
(Color online) Transient currents (top panel) and pair densities (bottom panel) in the molecule when varying $V$. The inset shows the absolute value of the Fourier-transformed current.
(Color online) Spectral functions of the coupled benzene molecule when varying $\varDelta$.
Next, we will adjust the transport conditions to visualize the AR and CAR processes. We have the same benzene molecule as the central region at the same temperature, $\beta=100/|t_C|$. In Fig. <ref> we choose the gap as $\varDelta=|t_C|/2$, and in Fig. <ref> we, on the other hand, choose the gap as $\varDelta=3|t_C|/2$. When $V\leq \varDelta$ we observe, in both cases, the transient current oscillating towards a zero steady-state current. The oscillation frequencies seen in the Fourier spectrum can be interpreted from the spectral function in Fig. <ref> mainly as intramolecular transitions (around $\w=|t_C|$ and $\w=2|t_C|$). Interestingly, in Fig. <ref> also, when we increase the bias voltage above the superconducting gap $V=|t_C|$ there still are no other states within the bias window except the inert state split by $\varDelta$. This resonant window does not add anything to the transient dynamics (due to the inert state), but the static part of the density matrix is modified leading to nonzero steady-state current. This is interpreted as CT where, in addition to AR and CAR, also an electron is transferred from one lead to another via the CP. This is also confirmed by looking at the pair density as it remains nonzero also for $V=|t_C|$. With the larger gap in Fig. <ref> and for $V\leq\varDelta$ we observe CP formation within the molecule but for $V>\varDelta$ the pair density goes to zero. For the smaller voltages we mainly find the first intramolecular transition at around $\w=7|t_C|/2$. For larger voltages we also see the lead–molecule transitions at lower frequencies, and we recover again the NT regime as the bias voltage is high enough for breaking the CPs within the molecule.
In this transport setup, it is not, in general, easy to distinguish between AR and CAR processes as they involve multiple steps. One possible case is when an electron hops from the lead to the central region (molecule–lead transition), then “transfers” from the spin-up sector in the central region to the spin-down sector for which the probability is given by the pairing strength (“intramolecular” transition between the two branches of states split by $\varDelta$), and then finally hops back to the lead as a hole (molecule–lead transition). As all these transitions are visible in the transient oscillations, we may only conclude whether AR and CAR processes are present or not.
§ CONCLUSIONS AND OUTLOOK
We presented an extension to the time-dependent Landauer–Büttiker formalism, discussed in Refs. <cit.>, to include superconducting central region in the transport setup, and to evaluate the TD1RDM at arbitrary temperatures. The derived formulae are analytic and closed expressions involving known special functions, and they can readily be implemented to study various quantum transport problems very efficiently and also at large temporal and spatial scales.
As an application of the presented formalism we simulated transport in a superconducting benzene-like molecule attached to two-dimensional normal metal leads. Assigning a proper parameter set for the transport window and the superconducting gap, we observed formation of Cooper pairs within the central molecule leading to Andreev reflection processes.
R.T. thanks the Väisälä Foundation of The Finnish Academy of Science and Letters for financial support. R.v.L. thanks the Academy of Finland for support. E.P. and G.S. acknowledge funding by MIUR FIRB Grant No. RBFR12SW0J.
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1511.00430
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rvt]A. Calzona
focal]M. Carrega
focal]G. Dolcettocor1
rvt,focal]M. Sassetti
[cor1]Corresponding author
[rvt]Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy
[focal]CNR-SPIN, Via Dodecaneso 33, 16146 Genova, Italy
We analyze the time evolution of spin-polarized electron wave packets injected into the edge states of a two-dimensional topological insulator. In the presence of electron interactions, the system is described as a helical Luttinger liquid and injected electrons fractionalize. However, because of the presence of metallic detectors, no evidences of fractionalization are encoded in dc measurements, and in this regime the system do not show deviations from its non-interacting behavior. Nevertheless, we show that the helical Luttinger liquid nature emerges in the transient dynamics, where signatures of charge/spin fractionalization can be clearly identified.
Topological insulators, Luttinger liquids, transient dynamics
§ INTRODUCTION
The concept of single quasi-particle fails when applied to interacting electrons in one dimension. Indeed, the presence of two distinct Fermi points implies that the low energy excitations are represented by collective charge and spin density waves with bosonic nature <cit.>.
In the presence of electron interactions, a variety of peculiar quantum phenomena emerges. Among them, charge fractionalization represents one of the most striking signature <cit.>, being a manifestation of Luttinger liquid (LL) behavior <cit.>: an electron injected into a LL is fractionalized by interactions into counter-propagating density waves with fractional charges $(1\pm K) e /2$, $e$ being the electron charge and $K$ the Luttinger parameter accounting for the strength of electron interactions ($K=1$ in the non-interacting case). This mechanism is reflected in a sequence of multiple reflections of charge density waves at the interfaces between interacting and non-interacting regions.
First evidences of charge fractionalization has been reported in dc tunneling experiments in cleaved-edge-overgrown samples, exploiting momentum-resolved tunneling and multi-terminal geometries <cit.>.
Here from the knowledge of the current asymmetry and the strength of interactions it was possible to determine the degree of fractionalization.
Also carbon nanotubes attracted a lot of interest in this perspective. Indeed they represent the archetype of non-chiral LLs, characterized by two counter-propagating modes at the Fermi energy. Since their non-Fermi liquid behavior was observed in a variety of transport experiments, in good agreement with theoretical predictions, they have been proposed as ideal playgrounds to explore the phenomenon of charge fractionalization <cit.>.
Alternative setups can be created by exploting the chiral edge states of integer quantum Hall bars: if the boundaries of the bar are close enough, electron interactions between the counter-propagating edge states are not negligible, and the full system can be viewed as a non-chiral LL <cit.>.
Most of the theoretical works studying charge fractionalization have focused on noise measurements in carbon nanotubes and quantum Hall bars. However, from an experimental point of view, clear signatures of fractionalization in noise experiments are still lacking. Indeed, in general, only high-frequency noise carries information about fractionalization, whose detection can be hardly achievable.
Very recently, time-resolved experiments have confirmed the physical picture of charge density waves being fractionalized at the interfaces between non-interacting and interacting regions, using integer quantum Hall channels <cit.>.
To perform such experiments two ingredients are crucial: time-resolved measurement and absence of inelastic scattering. The former allows to follow the dynamic evolution of the charge density waves propagating throughout the system, while the latter guarantees that the phase coherence is preserved.
As far as the first issue is concerned, current measurements with time resolution of $\sim 1$ ps have been performed <cit.>, allowing to detect the real-time dynamics of edge plasmons <cit.>.
On the other hand, to overcome the problem of dephasing, topologically protected edge states can be used, these being characterized by long ($>\mu$m) coherence lengths.
Here we investigate evidences of fractionalization and LL physics in a new paradigm of the one-dimensional world: the helical Luttinger liquid (hLL)<cit.>.
This state can be created at the edge of a two-dimensional topological insulator (2DTI), where a pair of counter-propagating one-dimensional channels appears <cit.>. Crucially, the chirality of each channel is intimately connected to its spin-polarization, meaning that in a helical liquid spin-up and spin-down excitations counter-propagate, giving rise to the so-called spin-momentum locking <cit.>.
After their discovery in HgTe/CdTe quantum wells (QWs) <cit.>, evidences of helical edge states were found also in InAs/GaSb QWs <cit.> and other 2D materials <cit.>.
Very recently, the hLL has been observed in a InAs/GaSb QW, where non-Fermi liquid transport properties have been measured <cit.>.
In the presence of time reversal symmetry elastic backscattering from one channel to the counter-propagating one is inhibited <cit.> and non-local transport properties have been experimentally confirmed in multi-terminal geometries <cit.>. Helical edge states offer a promising platform to study quantum phenomena in one dimension, ranging from topological superconductivity <cit.>, majorana fermions <cit.>, and spin textures <cit.>. For example, in hLL, because of spin-momentum locking, one can exploit a spin-polarized tip to inject right-moving or left-moving electrons in the system simply by adjusting the spin-polarization of the tip <cit.>.
Then, because of interactions, the injected electrons fractionalize and propagate into the system in the form of charge and spin density waves.
Remarkably, helicity bounds the charge and spin degrees of freedom together <cit.>, so that charge fractionalization is intimately linked to spin fractionalization <cit.>. Therefore, if spin-polarized non-interacting electrons enter an hLL, both charge and spin fractional collective excitations are created.
Injection from a spin-polarized tip into an hLL has already been studied <cit.>: it was shown that, because of interactions, the injection of spin-up, i.e. right-moving, electrons induces fractional charge and spin excitations propagating in both the two directions, in sharp contrast with the non-interacting case.
Fractional excitations can be detected via current measurements
which, however, are in general performed via metallic contacts used as detector, whose presence drastically affects the behavior of the whole system <cit.>.
Here we study how the presence of metallic contacts affects the visibility of fractionalization phenomena in transport measurement in hLL. In agreement with previous works on standard LL <cit.>, we confirm that dc currents collected at the detectors do not encode information about neither charge nor spin fractionalization, thus apparently preventing the observation of this effect.
A naive explanation of this phenomenon relies on the observation that, because of charge conservation, all the injected charges of a right (left) moving wave packet will be collected at the right (left) contact. Therefore, to observe fractionalization phenomena in hLL physics, alternative detection schemes have to be conceived.
Hybrid setup based on topological insulators and capacitive charge sensors, involving quantum dots, single-electron-transistors and high-electron-mobility-transistors, have been proposed, in order to avoid the need of Fermi-liquid contacts <cit.>.
Here we reconsider spin-polarized injection in an hLL coupled with metallic
contacts showing that, by means of time-resolved measurement, evidence of fractionalization can be recovered in the transient regime. The time evolution
of such an inhomogeneous system is peculiar since fractionalization arise not only when electrons are injected from the tip into the hLL, but also when
density waves arrive at the interfaces between the hLL and the contacts. Indeed, the charge/spin excitations which reach the interface between an interacting
and a non-interacting region are partially reflected and partially transmitted <cit.>, leading to multiple and subsequent fractionalizations.
By solving the equation of motion for the collective density fields we predict the time evolution of an injected wave packet, finding clear evidences of charge and spin fractionalizations at the interfaces with the contacts. Furthermore, we study the currents detected at the terminals when the tip is biased.
Although dc currents
don't display any signatures of fractionalization, we demonstrate that it is still possible to extract information about hLL physics and fractionalization by studying the transient dynamics.
The paper is organized as follows. In Sec. <ref> we provide the theoretical description of the setup with the inhomogeneous hLL model. In Sec. <ref> we introduce the equation of motion approach used to evaluate the space-time evolution of the injected wave packets, present the main results and discuss the experimental feasibility. Finally Sec. <ref> is devoted to the conclusions.
§ MODEL
The helical Luttinger liquid
The Hamiltonian density of the interacting helical fermions appearing on the edge of a 2DTI is
\hat{\mathcal{H}} = \hat{\mathcal{H}}_0 + \hat{\mathcal{H}}^{(int)}
$ where
\begin{equation}
\hat{\mathcal{H}}_0 = -i v_F \left( \hat{\psi}^{\dagger}_{\uparrow}\partial_x\hat{\psi}_{\uparrow} - \hat{\psi}^{\dagger}_{\downarrow}\partial_x\hat{\psi}_{\downarrow}\right)
\end{equation}
is the free part ($v_F$ is the Fermi velocity) and, introducing the electron density on each channel $ \hat{\rho}_{\sigma}=\colon \hat{\psi}^{\dagger}_{\sigma}\hat{\psi}_{\sigma}\colon$,
\begin{equation}
\hat{\mathcal{H}}^{(int)} = \frac{1}{2} \sum_{\sigma=\uparrow,\downarrow} \left(g_{4_\parallel} \hat{\rho}_{\sigma}\hat{\rho}_{\sigma} + g_{2_\perp} \hat{\rho}_{\sigma}\hat{\rho}_{-\sigma} \right)
\end{equation}
takes into account the electron-electron interaction via the parameters $g_{4_\parallel}$ and $g_{2_\perp}$, under the assumption of short range interaction <cit.>.
The presence of interactions can be treated exactly within the bosonization formalism <cit.>, which consists in rewriting the electron operator as $\hat{\psi}_{\sigma}(x)=e^{-i\sqrt{2\pi}\hat{\phi}_{\sigma}(x)}/\sqrt{2\pi a}$, with $a$ a short distance cut-off. Note that, in writing the previous field operator we have omitted the so-called Klein factors <cit.> which are irrelevant in this context.
The scalar field $\hat{\phi}_{\sigma}$ describes particle-hole excitations and is directly related to the particle density of the relative channel as
$\hat{\rho}_{\uparrow/\downarrow}(x)=\mp\frac{1}{\sqrt{2\pi}}\partial_x\hat{\phi}_{\uparrow/\downarrow}(x)$. Since the electron density is linear in the scalar fields, $\hat{\mathcal{H}}^{(int)}$ can be straightforwardly diagonalized.
By introducing $\hat{\phi}=\frac{1}{\sqrt{2}}\left (\hat{\phi}_{\downarrow}-\hat{\phi}_{\uparrow}\right ), \hat{\theta}=\frac{1}{\sqrt{2}}\left (\hat{\phi}_{\downarrow}+\hat{\phi}_{\uparrow}\right )$
satisfying $[\partial_x\hat{\phi}(x),\hat{\theta}(x')]={-}i\delta(x-x')$, the total Hamiltonian density assumes the standard form of a hLL <cit.>
\begin{equation}\label{eq:Hll}
\hat{\mathcal{H}}=\frac{v}{2}\left [\frac{1}{K}\left (\partial_x\hat{\phi}\right )^2+K\left (\partial_x\hat{\theta}\right )^2\right ]
\begin{equation}
v=v_F\sqrt{\left (1+\bar{g}_{4\parallel}\right )^2-\bar{g}_{2\perp}^2}
\end{equation}
the velocity of collective excitations and
\begin{equation}
\end{equation}
the interaction Luttinger parameter ($K\leq 1$), with $\bar{g}_{4\parallel(2\perp)}=\frac{g_{4\parallel(2\perp)}}{2\pi v_F}$. In the absence of interactions $K=1, v=v_F$ and the system behaves as a Fermi-liquid, with spin-polarized excitations $\rho_\uparrow(x)$ and $\rho_\downarrow(x)$ propagating to the right and to the left respectively.
In the presence of interactions, spin-up and spin-down density waves no longer represent the chiral excitations, which in turn are given by their superpositions
\begin{equation}\label{eq:frac}
\hat{\rho}_{\pm}(x)=\frac{1\pm K}{2}\hat{\rho}_{\uparrow}(x)+\frac{1\mp K}{2}\hat{\rho}_{\downarrow}(x)
with $+ (-)$ excitations propagating to the right (left).
Injection and fractionalization
What happens when electrons are injected into the interacting system crucially depends on the electron interaction through the Luttinger parameter $K$. Consider for example tunneling of spin-up electrons from a nearby tip. In the absence of interactions, spin-up collective excitations are created in the liquid, which, due to spin-momentum locking, propagate to the right with velocity $v_F$. On the other hand, in the presence of electron interactions both right-moving and left-moving collective excitations are created, as shown in Eq. (<ref>), so that a fraction $(1+K)/2$ of the injected flow propagate to the right with velocity $v$, while a smaller amount $(1-K)/2$ propagate to the left with velocity $-v$.
Therefore the presence of electron interactions strongly affect the physical observables (both spin and charge), due to the fractionalization mechanism.
In particular, one could be tempt to conclude that, by measuring the current at the left and the right side of the injection point, clear evidence of charge and spin fractionalizations could be accessed: the measurement of a current at the left of the injection point when the tip is spin-up polarized seems to be a conclusive manifestation of the presence of electron interactions. Unfortunately, this naive expectation is made much more complicate by the presence of metallic contacts in a real measurement.
The role of metallic contacts
Experimentally, metallic contacts must be attached at some points on the edge, so that the current carried by the interacting helical edge states can be measured. These contacts are macroscopic objects behaving as non-interacting Fermi-liquid.
A standard way to theoretically keep into account their presence is by means of the so-called inhomogeneous Luttinger model <cit.>.
Here we apply this model to the case of a helical liquid. It corresponds to model the Fermi-liquid contacts as one-dimensional systems with vanishing interactions: formally, one assumes that the interaction parameters $g_{4\parallel}$ and $g_{2\perp}$ are non-vanishing for $|x|<L/2$ only, with $L$ the distance between the contacts, while the interaction is absent for $|x|>L/2$.
(Color online) Schematic view of the setup. A tip injects spin-polarized wave packets inside the hLL ($hLL$), that fractionalize and are detected by the left ($C_l$) and right ($C_r$) contacts.
The system, schematically depicted in Fig. (<ref>) (left contact $C_l$, interacting helical liquid $hLL$, right contact $C_r$) is then described by the inhomogeneous hLL Hamiltonian
\begin{equation}\label{eq:Hinh}
\hat{H}=\int~dx~\frac{v(x)}{2}\left [\frac{1}{K(x)}\left (\partial_x\hat{\phi}\right )^2+K(x)\left (\partial_x\hat{\theta}\right )^2\right ]
Here the velocity of propagation and the Luttinger parameter acquire a space dependence and are given by:
\begin{equation}\label{eq:parameters}
v(x)=\left \{\begin{array}{cc}
v_F & |x|>L/2 \\ v & |x|<L/2
\end{array}\right ., \ \ \ \ \ \ K(x)=\left \{\begin{array}{cc}
1 & |x|>L/2 \\ K & |x|<L/2
\end{array}\right .
In the following we study how the scenario depicted in the presence of spin-polarized tunneling changes due to the presence of the metallic contacts, and what informations about hLL physics can be still measured.
§ TIME-RESOLVED DYNAMICS OF SPIN-POLARIZED INJECTION
In the inhomogeneous Luttinger liquid model, the change in the excitations velocities and Luttinger parameters Eq. (<ref>) makes the interfaces between the hLL and the metallic contacts to effectively behave as potential barriers for the collective excitations: a chiral density wave incoming at one of the interfaces is separated into a transmitted component and into a reflected one.
Note that, because of helicity, both charge and spin density waves undergo scattering phenomena.
Only the transmitted component is finally measured in the contact, while the reflected one propagate toward the other contact, where again it can be either transmitted or reflected back, and so on and so forth.
It is thus important to describe the time evolution of the injected electron wave packet. This can be properly done within the equation of motion approach <cit.>.
Equation of motion
The charge density (from now on in units $e$) can be expressed in terms of the scalar field $\hat \phi(x,t)$ as $\hat\rho(x,t)=\hat\rho_\uparrow(x,t)+\hat\rho_\downarrow(x,t)=\frac{1}{\sqrt{\pi}}\partial_x\hat\phi(x,t)$. By means of the continuity equation it is also possible to define the charge current $\hat j(x,t) = - \frac{1}{\sqrt{\pi}}\partial_t\hat\phi(x,t)$.
Note that, because of helicity, charge current and density are related to spin density $\hat{\rho}_s(x,t)=\frac{1}{v(x)K(x)}\hat{j}(x,t)$ and spin current $\hat{j}_s(x,t)=\frac{v(x)}{K(x)}\hat{\rho}(x,t)$ respectively (from now on in units $\hbar/2$). This implies that, in a helical liquid, fractionalization manifests both in the charge and in the spin sector <cit.>.
In order to determine the explicit dynamics of the averaged density $\rho(x,t)$ and current $j (x,t)$ it is necessary to solve the equation of motion for the scalar field. Recalling that $\partial_t\hat{\phi}(x,t)=v(x)K(x)\partial_x\hat{\theta}(x,t)$ and $\partial_t\hat{\theta}(x,t)=\frac{v(x)}{K(x)}\partial_x\hat{\phi}(x,t)$, one finds
\begin{equation}\label{eq:motion}
\partial_t^2\hat\phi(x,t)=v(x)K(x)\partial_x\left [\frac{v(x)}{K(x)}\partial_x\hat\phi(x,t)\right ]
which can be solved by imposing the continuity of $\hat\phi(x,t)$ and $\frac{v(x)}{K(x)}\partial_x\hat\phi(x,t)$ at the interfaces at $x=\pm L/2$.
In particular, we are interested in studying the dynamic evolution of average density and current after a sudden injection of electrons in the hLL at $t=0$. We thus assign the initial condition, specifying the charge density profile $\rho^{(0)}(x)\equiv \rho(x,t=0)$, with $Q=\int~\rho^{(0)}(x)~dx$ the total injected charge. Electrons are injected from a tip whose spin-polarization forms an angle $\theta$ with the spin quantization axis of the helical fermions <cit.>.
Note that, because of helicity, assigning the initial injected charge density $\rho^{(0)}(x)$ and its spin-polarization (related to the angle $\theta$) is equivalent to assigning initial conditions on both charge and spin densities and currents.
Therefore at time $t=0$ chiral density waves <cit.>
\begin{equation}\label{eq:rho0}
\rho^{(0)}_{\pm}(x)=\frac{1\pm K\cos\theta}{2}\rho^{(0)}(x)
\end{equation}
are created in the interacting region[In the following we consider injections of Gaussian electron wave packets
$\rho^{(0)}(x) = \frac{Q}{\sqrt{2\pi} \sigma_x} \exp\left[\frac{-x^2}{2 \sigma_x^2} \right]$,
with $Q$ the injected charge and $\sigma_x$ accounting for its spatial distribution, that are localized near the center of the hLL far from the contacts.].
Note that in the non-interacting case ($K=1$), for $\theta=0$ ($\pi$) only right(left)-moving excitations are created, as expected.
We now consider separately the dynamic of the right-moving chiral component of the injected density and of the left-moving one, see Eq. (<ref>). Solving the equation of motion Eq. (<ref>) with the initial conditions Eq. (<ref>), we find
\begin{equation}
\label{eq:FrCa:rho2+}
\rho_{{\rm p}}(x,t) =
\begin{cases}
\begin{split}
- \frac{2}{1+K} &\zeta\sum_{n=0}^{+\infty}\gamma^{2n+1} \\ &\rho_{+}^{(0)} \left(-\zeta(x+ v_Ft) + (2n+2)L - \frac{L}{2} \left(\zeta+1\right)\right) \end{split} & x \text{ in } C_l \\
\begin{split} \sum_{n=0}^{+\infty} \gamma^{2n} &\rho_{+}^{(0)}\left(x- v t + 2n L\right) \\&- \sum_{n=0}^{\infty} \gamma^{2n+1}\,\rho_{+}^{(0)}\left(-x- v t+(2n+1)L\right) \end{split} & x\text{ in } hLL\\ \begin{split}
\frac{2}{1+K}&\zeta \sum_{n=0}^{+\infty} \gamma^{2n} \\ &\rho_{+}^{(0)}\left(\zeta(x- v_Ft) + 2nL- \frac{L}{2} \left(\zeta-1\right) \right)\end{split} & x\text{ in } C_r.
\end{cases}
\begin{equation}
\label{eq:FrCa:rho2-}
\rho_{\rm m}(x,t) =
\begin{cases}
\begin{split}
\frac{2}{1+K}&\zeta \sum_{n=0}^{+\infty} \gamma^{2n} \\ &\rho_{-}^{(0)}\left(\zeta(x+ v_Ft) - 2nL+ \frac{L}{2} \left(\zeta-1\right) \right)\end{split}
& x \text{ in } C_l \\
\begin{split} \sum_{n=0}^{+\infty} \gamma^{2n} &\rho_{-}^{(0)}\left(x+ v t - 2n L\right) \\&- \sum_{n=0}^{\infty} \gamma^{2n+1}\,\rho_{-}^{(0)}\left(-x+ v t-(2n+1)L\right) \end{split} & x\text{ in } hLL\\
\begin{split}
- \frac{2}{1+K} &\zeta\sum_{n=0}^{+\infty}\gamma^{2n+1} \\ &\rho_{-}^{(0)} \left(-\zeta(x- v_Ft) - (2n+2)L + \frac{L}{2} \left(\zeta+1\right)\right) \end{split} & x\text{ in } C_r.
\end{cases}
with $\gamma=\frac{1-K}{1+K}$ and $\zeta = \frac{v}{v_F}$. In the above expression $\rho_{\rm p}(x,t)$ and $\rho_{\rm m}(x,t)$ represent the evolved density profiles associated to the initial chiral density $\rho^{(0)}_+(x)$ and $\rho^{(0)}_-(x)$ respectively. Noteworthy $\rho_{\rm p}(x,t)$ and $\rho_{\rm m}(x,t)$ are no more chiral since they take contributions from all the multiple reflections with the contacts. The time evolution of the total charge density is then given by
\begin{equation}
\label{eq:rho}
\rho(x,t)=\rho_{\rm p}(x,t)+\rho_{\rm m}(x,t).
\end{equation}
Once the excitations have entered the contacts, they propagate with velocity $v_F$. The currents in the left ($j_l$) and in the right ($j_r$) contacts, moving away from the hLL region, can thus be calculated as
\begin{align}
\label{eq:jl} j_{l}(x,t)&= v_F \rho(x,t)\quad & x &\in C_l\\
\label{eq:jr} j_{r}(x,t)&= v_F \rho(x,t)\quad & x &\in C_r
\end{align}
where, for sake of definition, currents entering in the contacts have positive sign.
In Fig. <ref> we report the evolution of a Gaussian electron wave packet injected in the center of the hLL, see Fig. <ref>(a), from a tip spin-polarized in the $\hat{z}$ direction ($\theta=0$).
(Color online) Charge density $\rho$, spin-up polarized density $\rho_{\uparrow}$ and spin-down polarized density $\rho_{\downarrow}$ profiles at different times. (a) At $t=0$ a spin-up polarized wave packet is injected in the center of the hLL. (b) The fractionalized density waves propagate away from the center of the system toward the contacts. (c) At the contacts they can be either transmitted, and measured, or reflected. (d) The reflected density waves propagate back toward the center of the system. Parameters are: $\theta=0, K=0.5, v_F=10^5$ m/s, $v=v_F/K$, $L=20 \mu$m, $\delta t=35$ ps.
Because of interactions, the injected spin-up charge $Q$ splits into two chiral excitations with charge $\frac{1+K}{2}Q$ and $\frac{1-K}{2}Q$, propagating toward right and left contacts respectively, see Fig. <ref>(b).
At the interfaces, some incident charge is transmitted to the contacts, see Fig. <ref>(c), where, according with Eqs. (<ref>, <ref>), it can be measured as a current signal.
The reflected density waves propagate toward the opposite contact (Fig. <ref>(d)), where, again, they are partially transmitted and partially reflected.
Therefore, the fractionalization mechanism can be probed by means of time-resolved current dynamics, as shown in Fig. <ref>, where the currents arriving at the contacts as a function of time are reported.
Here, the contacts measure the different fractions of the injected wave packet after multiple reflections. Consider for example the current measured at the right contact $j_{r}$ as a function of time (similar arguments hold for the behavior of $j_{l}$).
The first peak (at short times) corresponds to the component of the injected wave packet being directly transmitted through the right interface, while the first dip corresponds to the injected wave packet being reflected at the left interface, then transmitted through the right one and finally detected. Multiple reflections, detected at larger times, give smaller contributions, as argued also from Eqs. (<ref>, <ref>).
(Color online) Currents $j_r(x=L,t)$ and $j_l(x=-L,t)$, see Eqs. (<ref>, <ref>), collected at the right and left contacts respectively as a function of time, after that a Gaussian wave packet has been injected in the center of the interacting region at $t=0$. Currents entering the contacts have positive sign. Two different polarizations of the tip are shown: $\theta=0$ (black full) and $\theta=\pi/2$ (red dashed). Parameters are: $K=0.5, v_F=10^5$ m/s, $v=v_F/K$, $L=20$ $\mu$m.
Crucially, evidences of fractionalization are lost in simple dc measurements. Indeed, the total charge detected at the contacts $Q_\alpha=\int_0^{+\infty}j_{\alpha}(x,t)~dt$ can be evaluated from Eqs. (<ref>, <ref>, <ref>) and reads
\begin{equation}\label{eq:Qdc}
Q_l=\frac{1-\cos\theta}{2}Q, \ \ \ \ \ \ Q_r=\frac{1+\cos\theta}{2}Q
independently of the Luttinger parameter $K$.
This result already emerges from Fig. <ref>.
If spin-up electrons are injected ($\theta=0$), all the fractionalized contributions sum up to the total injected charge $Q$ in the right contact, while they cancel out exactly in the left one. On the other hand, if the injected charge density is unpolarized in the $\hat{z}$ direction ($\theta=\pi/2$), the right and left contacts collect the same charge $Q/2$, in agreement with Eq. (<ref>).
Note that noise measurements represent an alternative to current detection to investigate the fractionalization phenomenon. However, just like dc current measurements mask the Luttinger liquid behavior, the zero-frequency noise does not carry information about fractionalization in the setup <cit.>. Therefore, the noise at finite frequency should be investigated, which however could be challenging to measure, since a wide window spectral range at tens of GHz should be resolved.
So far, we have studied the evolution consequent to the sudden injection of a single localized bunch of electrons at $t=0$, described by an initial density $\rho^{(0)}(x)$. However, the formalism we developed can be used also to study the injection of a generic current in the interacting region, with an arbitrary time-dependence $I^{(inj)}(t)$. Let's consider a series ($n=0,1,2, \dots$) of instantaneous injections of packets, each with associated charge $Q_n$ and subsequent time separation $\Delta t$. Exploiting the linearity of the equation of motion (Eq. (<ref>)), the overall density at time $t$ is given by
\begin{equation}
\label{eq:subs_inj}
\rho(x,t) = \sum_{n=0}^{m} Q_n \; \tilde\rho(x,t-n~\Delta t)
\end{equation}
with $m$ the integer part of $\frac{t}{\Delta t}$. The function $\tilde\rho(x,t)$ is the time evolution, obtained by means of Eqs. (<ref>-<ref>), of an initial wave packet $\tilde\rho^{(0)}(x)$ with normalized shape ($\int~\tilde{\rho}^{(0)}(x)~dx=1$). Identifying the current $I^{(inj)}(n~\Delta t)=Q_n / \Delta t$ and considering the limit $\Delta t \to 0$ one obtains from Eq. (<ref>)
\begin{equation}\label{eq:rhotildeOK}
\end{equation}
Note that for a tip biased with a very sharp pulse, $I^{(inj)}(\tau)=Q^{(inj)}\delta(\tau)$, one recovers the previous result for $\rho(x,t)$ given in Eq. (<ref>).
We now use the expression of Eq. (<ref>) to study the dynamic evolution of the system in the experimentally relevant case corresponding to injection from a biased tip. We assume that at $t=0$ a constant voltage is imposed to the tip, that starts to inject electron wave packets into the interacting region. Contrary to the previous case of sudden injection, now the tip continuously injects trains of wave packets. The injected current can be modeled as
\begin{equation}
\label{eq:I(t)}
I^{(inj)}(t) = \frac{I_0}{2}\mathrm{Erf}\left (\frac{t}{\Delta \tau}-2\right ),
\end{equation}
with $\mathrm{Erf}$ being the Gaussian errors function and $\Delta \tau$ representing the time interval needed by the tip to be polarized by the battery[We could have used the simplified model $I^{(inj)}(t)=I_0\Theta(t)$, with $\Theta$ the Heaviside step function, instead of Eq. (<ref>), and we would have obtained similar results. However, Eq. (<ref>) keeps into account the finite time interval $\sim\Delta \tau$ needed by the tip to be polarized by the battery.].
The time-dependence of the injected current Eq. (<ref>) is shown in the insets of Fig. (<ref>).
For sake of convenience, we have chosen $\tilde{\rho}^{(0)}(x)$ to be very sharply peaked ($\tilde\rho^{(0)}(x)=\delta (x)$), thus modeling the injection from a narrow tip in the center of the hLL. We underline that the results are not affected if one chooses a broader spatial distribution.
From Eqs. (<ref>, <ref>, <ref>), it is possible to predict the currents measured at the left and right terminals, due to the presence of the injected current $I^{(inj)}(t)$ in Eq. (<ref>).
The results are shown in Fig. <ref>.
(Color online) Currents $j_l$ and $j_r$, see Eqs. (<ref>, <ref>), detected at the left and right contacts respectively due to injection from a biased spin-polarized tip, with the injected current modeled by Eq. (<ref>) and shown in the insets. Currents entering the contacts have positive sign. We have assumed a narrow tip localized at the center of the hLL ($\tilde \rho^{(0)}(x) = \delta(x)$).
(a) Case $\theta=0$. In the absence of interactions (dashed black), all the injected current flows to the right contact, while no current is detected by the left contact. In the presence of interactions both the right (solid blue) and left (dotted red) contacts detect a non-vanishing current in the transient regime. (b) Case $\theta=\pi/2$. Both the left and right contacts detect an equal current in the absence of interactions (dashed black). However, in the presence of interactions (solid magenta) additional features are observed in the transient regime, due to fractionalization. Parameters are: $K=0.5, v_F=10^5$ m/s, $v=v_F/K$, $L=20 \mu$m, $\Delta \tau=25$ ps
Let us analyze the non-interacting case first, represented by the black dashed lines.
Initially ($t\apprle 0.2$ ns), no current is measured by the contacts, since the charge, injected from the tip into the center of the hLL (see the insets in Fig. <ref>), needs some time $\sim\delta/v_F$ to reach the contacts, with $\delta$ the distance of the detection points from the injection point. In particular, if $\theta=0$, Fig. <ref>(a), the left contact measures no signal and all the injected current is detected by the right one, while if $\theta=\pi/2$, Fig. <ref>(b), the injected current is equally detected by both the left and the right contacts ($j_l = j_r$).
Considering the presence of interactions at large times no deviations from the non-interacting behavior are present: if $\theta=0$, all the current is detected by the right terminal, while the injected current is equally distributed in the left and right contacts if $\theta=\pi/2$. This result is in agreement with Eq. (<ref>): dc measurements cannot provide information about charge fractionalization. On the other hand, differences between the non-interacting and the interacting regimes are captured by transient effects.
First, a non zero signal is measured in advance with respect to the non-interacting case, because the time of flight $\sim\delta/v$ is reduced as the velocity is renormalized ($v>v_F$).
The transient regime in the Figure corresponds to $t\apprle 0.6$ ns. Consider the case $\theta=0$ first, corresponding to Fig. <ref>(a). In this interval, in the presence of interactions the left contact detects a finite current, which vanishes at larger times ($t \apprge 0.6 \, ns$). This result differs from the non-interacting case, where, also in the transient regime, no current is measured by the left contact.
Similar arguments hold for the case $\theta=\pi/2$ shown in Fig. <ref>(b), where signatures of interactions are found in the transient regime only.
Therefore, sudden injection is not mandatory in order to detect evidences of fractionalization and hLL physics. Indeed, we have shown that these features can be found in the transient regime of “dc-like” injection.
These arguments clearly show that, despite dc measurements do not encode information about hLL physics in our setup, time-resolved dynamics and transient effects are able to provide evidences of charge and spin fractionalization.
We remark that our formalism allows to describe a wide range of injection modalities (arbitrary spin-polarization, sudden or adiabatic injection) that allow to investigate fractionalization in hLLs in a variety of setups.
Finally, we briefly discuss the experimental feasibility of the proposed setup.
The ability to inject localized wave packets into the edge states of a topological system <cit.> has been recently improved by Kamata et al. <cit.>, who were also able to perform time-resolved measurements with an accuracy of $\sim 1$ ps.
In order to observe time-resolved dynamics and transient effects, the multiple reflected wave packets must be resolved in time. Let $\sigma_t$ be the full-width-at-half-maximum of the time distribution of the wave packets (see Fig. <ref>). The time interval between two consecutive wave packets incoming in each terminal is $\sim L/v$, so the condition to resolve two consecutive wave packets is $L\apprge v\sigma_t$. By taking $\sigma_t\sim 20 $ps <cit.> and $v\sim 10^5 $m/s <cit.>, one finds $L\apprge 2 \mu$m, which represents a lower bound for the possible observation of the fractionalized wave packets. On the other hand, an upper bound is represented by the inelastic mean free path or phase-coherence length $\lambda_{in}$, since inelastic processes can induce backscattering even in the presence of time reversal symmetry, destroying the quantum coherence of the excitations <cit.>.
Then, in order to probe the fractionalization mechanism discussed in this work the condition
\begin{equation}\label{eq:exp}
v\Delta t\apprle L\apprle \lambda_{in}
\end{equation}
must be satisfied.
The one-dimensional channels used in Ref. <cit.> were integer quantum Hall edge states characterized by an inelastic mean free path of the order of $\apprge 100$ $\mu$m. Up to now, in the case of 2DTIs, ballistic transport has been observed in shorter samples only, with an inelastic mean free path of the order of few tens of $\mu$m at best <cit.>.
Therefore it may be considered that a $2$ $\mu$m $\apprle L\apprle$ $20$ $\mu$m long hLL should satisfy the requirement Eq. (<ref>).
In order to fulfill Eq. (<ref>) one can work either to reduce the product $v\Delta t$ or to increase the inelastic mean free path $\lambda_{in}$.
On the one hand, InAs/GaSb quantum wells are expected to have edge states with slower propagation velocity ($v_F\sim 10^4$ m/s) with respect to HgTe/CdTe quantum wells <cit.>. On the other one, larger value for the inelastic mean free path can be achieved by improving the quality of the samples, thus reducing possible sources of inelastic scattering such as the presence of quasi-2D charge puddles in the bulk <cit.>.
§ CONCLUSIONS
In this work we have investigated the fractionalization phenomenon in helical Luttinger liquids, where helicity can lead to the fractionalization of both charge and spin degrees of freedom.
In particular, we have studied the injection of spin-polarized electrons into the interacting edge state of a 2DTI, contacted with metallic detectors. We have analyzed both the cases of sudden injection, that allows to clearly follow the time dynamics of the injected wave packets, and the case of injection from a constant biased tip. In both situations, evidences of non-Fermi-liquid physics are lost in the dc regime, and despite the presence of electron interactions, the helical edge states appear as non-interacting at all. However, we have discussed how signatures of charge and spin fractionalization can be recovered by studying the time-resolved dynamics of the injection processes. As far as the case of sudden injection is concerned, real-time current detection allows to observe the fractionalized wave packet undergoing multiple reflections. On the other hand, we have found evidences of Luttinger liquid physics in the case of injection from a biased tip as well, provided the transient regime is analyzed. We have also inspected the experimental feasibility, showing that the proposed setup could in principle be implemented.
§ ACKNOWLEDGMENTS
We acknowledge the support of the MIUR-FIRB2012 - Project HybridNanoDev (Grant No.RBFR1236VV), EU FP7/2007-2013 under REA grant agreement no 630925 – COHEAT, MIUR-FIRB2013 – Project Coca (Grant No. RBFR1379UX), the COST Action MP1209 and the CNR-SPIN via Seed Project PGESE003.
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1511.00323
|
Interventions in Networks
Steven K. Thompson$^{1}$,
Department of Statistics and Actuarial Science, Simon Fraser
University, Burnaby, BC, Canada
$\ast$ E-mail: [email protected]
§ ABSTRACT
Interventions are made in networks to change the network or its values
in a desired way. The intervention strategies evaluated in the study
described here use network sampling designs to find units to which
interventions are applied. An intervention applied to a network node
or link can change a value associated with that unit. Over time the
effect of the intervention can have an effect on the population that
goes beyond the sample units to which it is directly applied. This
paper describes the methods used for this study. These include a
variety of link-tracing sampling designs in networks, a number of
types of interventions, and a temporal spatial network model in which
the intervention strategies are evaluated. An intervention strategy
is associated with an agent and different intervention strategies
interact and adapt to each other over time. Some preliminary results
are summarized regarding potential intervention strategies to help
alleviate the HIV epidemic.
§ AUTHOR SUMMARY
§ INTRODUCTION
The purpose of the project described here is to create a sampling
design and simulation system for evaluating the effectiveness of
potential strategies for interventions in networks. The motivating
problem for this work has been the effort to understand and reduce the
spread of HIV. An effective intervention is one that helps
in reducing the epidemic at least locally in an area of focus.
The approach taken uses a network or spatial sampling design to find
units to which to make interventions. These designs in many cases
involve tracing of links from sample units to add new units to the
sample. The population network itself is the result of choices of
individuals, and consensus of pairs of individuals, in forming and
dissolving links between them. The tendencies of individuals in this
process are viewed also as sampling designs.
A typical design in this study might use spatial sampling to find
initial candidate nodes for selection. Subsequent selections can use
link tracing to find additional candidate nodes. The sampling process
includes a selection process for adding units to the sample and an
attrition process by which nodes are removed from the sample.
An intervention strategy is the combination of a sampling design and
the intervention made to its sample units. We think of a design or
strategy as having an agent behind it. Sexually transmissible
infections are often countered by public health agencies with seek and
treat designs which follow sexual links from infected persons and test
and treat their recent partners. A person choosing not to take on
additional partners once he or she has a partner, on he other hand, is
exercising individual agency. In a choice of a safer sex practice such
as using a condom in a given situation, agency may be associated with
the pair of individuals and their negotiations.
A natural agent such as a virus has its own sampling design. In the
case of HIV it is a link-tracing design, reaching new individuals
through sexual and blood-exposure links. The intervention is
infection of the individual.
One intervention strategy interacts with another. An intervention
design that follows links from individuals who test positive for HIV
is influenced in where it goes by the pattern of virus spread. An
intervention such as an antiretroviral prescription or, potentially, a
vaccine or a cure, affects the virus by reducing its spread.
Behavioral changes of individuals and pairs in cooperation
affect the spread of the virus. The virus affects the population by
increasing mortality rate of infected individuals and, as described
later in this paper, affecting the link pattern or temporal network
geometry of the population. Additionally, two types of virus such as
HIV and herpes simplex virus 2 can catalyze the spread of each
The sampling designs are adaptive in that the probability of following
a link or adding a unit to the sample can depend on the values of the
originating or candidate nodes, on values associated with the links
between, and on surrounding network conditions. In addition, a deeper
adaptivity changes design parameters as virus strains evolve and
people learn.
Because of the multiple agents, the interactions between designs, the
adaptations to interventions, and effects of interventions unfolding
over time, the births and deaths of nodes and the formings and
dissolvings of links, the problem is inherently complicated and it is
helpful to find simplifications. A number of the designs, human and
natural, have common features. Some variables in the overall process
have stochastic stability properties given specific intervention
strategies. In such cases we can view the summary effect of an
intervention as the change in its equilibrium distribution once the
intervention strategy is put into effect.
§.§ Background
Design-based inference methods using Markov Chain Monte Carlo sampling
from a conditional distribution given a sufficient sample statistic
are used in link-tracing designs in networks
<cit.>. The
current approach to likelihood-based inference with link-tracing
designs most often assumes design ignorability in the sense of
<cit.>. The likelihood approach to inference for
link-tracing designs in networks is described
<cit.>, developed for exact Bayes inference with a
simple model/design combination in <cit.>, and for
computational Bayes inference with adaptive web sampling with a
stochastic block network model in Kwanisai, M. (2005, Estimation in
Link-Tracing Designs with Subsampling. Ph.D. thesis, The Pennsylvania
State University, University Park, PA, USA]. This approach was
developed farther and for more complex network models in
<cit.> and subsequent papers by those authors. In
<cit.> the Bayes inference deals with both a
network model and a network sampling design that is nonignorable.
Issues of inference in network epidemic models are additionally
discussed in <cit.>.
<cit.> describe empirical likelihood based
confidence intervals for adaptive cluster sampling.
<cit.> analyze weighting systems for estimation
in indirect sampling, a class of adaptive network designs.
Approaches combining design and model based methods include the random
walk asymptotics-based estimators used with respondent driven sampling
<cit.>, <cit.>, and
<cit.>. A different
approach is used in <cit.> and
<cit.>. A different approach still
combining design and model based methods is developed in
<cit.>. Current methods in respondent driven
sampling are assessed in <cit.>,
<cit.> and <cit.>.
<cit.>, <cit.>, <cit.>, and other
authors have looked into the combined effect of experimental
treatment-assignment designs and unit-selecting
sampling designs on inference.
A sample of units is selected from a population by some sampling
design. This sample obtains the experimental units. An experimental
design is then used to assign the experimental units to treatments.
The experimental design enables inference regarding the effects of
treatments to the experimental units, and the sampling design enables
inferring how this potential effect extends to the population as a
Recently a number of papers have come out with approaches to extending
static network models to the dynamic situation.
and <cit.>
developed an interesting class of network evolution models based on
behavioral characteristics of actors/nodes such as tendencies toward
reciprocity, transitivity, homophily, and assortative matching. At
the same time he developed inference methods to estimate model
parameters from incomplete longitudinal data. A summary of this
work together with a review of other approaches to dynamic network
modeling is contained in <cit.>.
<cit.> present a dynamic network model based on
exponential random graph models. A recent summary of statistical
network models is provided by <cit.>.
Latent space network models were introduced in
<cit.> for static networks. A dynamic extension of that
approach in which nodes move by small random increments was developed
by <cit.>.
In this paper I use the latent space approach with an underlying
temporal spatial point processes having grouping and clustering
tendencies and births and deaths of point-objects, which serve as the
network nodes. The temporal network model on top of that has links
forming and dissolving over time, allowing for link probabilities to
depend on social distances between nodes, node characteristics
including including sex of a potential partner, group identities, and
various network properties such as degrees and component memberships
of each potential partner. This model is then used for the spread and
adaptations of virus strains and evaluations of effectiveness of
different sampling designs and intervention strategies.
§ METHODS
In this project we are primarily interested in evaluating the
effectiveness of sampling designs for bringing interventions to units
in a population. The intervention is then applied to units in the
sample, according to an intervention design. From this point of view,
an experiment set in a population is an intervention with the purpose
of inferring a cause and effect relationship between the intervention
and its effect on sample units. For an experiment it is considered
desirable that an intervention or treatment applied to one unit does
not affect the response of any other unit. More broadly though, we are
particularly interested in interventions that, when applied to a
sample of units, have a desirable effect on the population as a whole,
including units to which the intervention has not been applied.
The methods of the study consist largely of devising sampling designs
and intervention strategies for implementation in spatial, temporal,
and network settings. An intervention strategy consists of a sampling
design for selecting the units to receive the intervention and a
procedure for assigning interventions to units in the sample.
Application of an intervention to a unit changes the value of some
variable associated with the unit.
We think of the population as a network or graph structure $G$
consisting of nodes and edges and having values $y$ associated with
nodes and $x$ associated with edges. The network and its values
are stochastic and change over time, so that we have a stochastic process
\[
\{G_t, t \in T\}
\]
where $T$ is the time index set, which we will take to be discrete.
In more detail the population at time $t$ is
\[
G_t = \{U_t, E_t, y(U_t), x(E_t)\}
\]
where $U_t$ is the set of units or nodes, $E_t$ is the set of edges or links
between nodes, $y(U_t)$ represents the variables of interest
associated with nodes, and $x(E_t)$ represents variables associated with
the links. Note that $U_t$ and $E_t$ are random sets, containing
different elements at different time point, with insertions and
deletions of nodes and insertions and deletions of links taking place
over time.
To make interventions in this population we need a sampling design to
reach nodes or edges on which to make the interventions. At time $t$
the sample is a subset of the network and its values:
\[
s_t \subset \{U_t, E_t\}
\]
and we have a sampling process through time:
\[
\{s_t, t \in T\}
\]
The sampling design is the probability of selecting the sample, which
at time $t$ typically depends on the sample, the network and values at
times up to $t$,
\[
{\rm P}(s_t \,|\, s_{t'}, G_{t'}, \phi_{t'}, t' \le t)
\]
where $ \phi_{t}$ are design parameters, which also may change with
time. Once the sample at time $t$ is selected the intervention can
change values $y$ and $x$ associated with nodes or links in the sample.
Note that over time the sampling process can select units to add to
the sample and units to drop from the sample.
An intervention to a sample unit is a procedure that changes a value
$y_u$ associated with a unit $u \in s$, or a value $x_{uv}$ associated
with an edge $(u,v) \in s$ to a new value $y'_u$ or $x'_u$. For
example if the intervention is prescribing a medication to person $u$,
an indicator variable $y_u$ might change from 0 to 1. The
intervention is thus distinguished from the effects of an intervention
program, which develop over time and might affect nodes and edges outside
the sample as well as inside.
The set of nodes $U_t$ is a stochastic point process. We model it as
a spatial temporal point process with variable amounts of clustering
in space and motion over time.
Specifically, we construct a spatial temporal Poisson cluster process
as follows. A set of $k$ group center locations are initially selected
independently and uniformly over a study region, which we take as the
unit square.
Let $c_{t}$ be center location at time $t$ for group a given group. At each
time step $t$ it is perturbed as follows. The most recent
displacement was $\delta_{t-1} = c_{t-1}- c_{t-2}$. The new
displacement is $\delta_{t} =\delta_{t-1} + \epsilon_t$. Assuming
the spatial study region of interest is two dimensional, the random
perturbation $\epsilon_t$ is bivariate normal with means 0 a small standard
deviation equal in each direction and zero correlation. For a
process with directional motion one can
add to the means or have unequal standard deviations and nonzero correlation.
This produces clusters that drift around independently and have a
momentum tendency in their ever changing direction and speed.
Further, this drift is kept in its initial distribution by using a
Markov Chain Monte Carlo (Metropolis-Hastings) selection step for each
group at each time step.
The numbers of nodes per group are initially distribution
Poisson($\lambda$). The expected number $\lambda$ can be constant or
selected independently for each group from a lognormal distribution,
producing more uneven sized clusters.
The locations of nodes are initially distributed with a bivariate
normal distribution around group center locations. Relative to the
group center, a node gets at each time step a random displacement
according to an autoregressive process or order 2, with the parameters
chosen to keep it in the initial bivariate normal distribution
relative to its group center. Given the group centers, nodes move
independently to each other.
Nodes are stochastically deleted over time depending on birth and
immigration rates and deleted depending on mortality and emigration
rates. Mortality rates depend on things like ages and stages of
nodes. Insertion rates are deliberately set to keep
the population and the group sizes in fluctuating but stochastically
stable distributions. There are many ways to do this, as is done in
population dynamics models. Here we simply imagine that as nodes die
or move out of the population, there is a tendency for new nodes to
move in over time to maintain a relatively stable population, so that
expected insertions is set accordingly. Actual insertions are Poisson
with that mean value, and assignments of new nodes to groups is with
probabilities proportional to target group sizes compared to current
Network links are inserted and deleted by the following process. An
individual node has a selection function, centered on it and
decreasing with distance, for making tentative selections of nearby
nodes with whom to form a partnership. Probability of selection
decreases with distance. A simple choice is a normal kernel
\[
g(d) = a_i e^{-d^2/2 \sigma_i^2}
\]
where $g$ is the probability of tentative selection by node $i$ of a
node $j$ at distance $d$ from it and $\sigma$ is a spread parameter which
can be node specific. Further we set $g(d) = 0$ for some $d$ greater
than some maximum reach
distance such as $3 \sigma_i$. Other selection function options
include the logistic function, which has an additional shape parameter
and the disk step function with is a constant out to the reach radius
and 0 beyond it.
Meanwhile node $j$ may be making a parallel selection decision on node
$i$ and a link tentatively forms with probability
\[
h(d) = a_ia_j e^{-d^2/2 (1/\sigma_i^2 + 1/\sigma_j^2)}
\]
In addition, the selection probability is further modified by
dependence on values of the two nodes, including the current degree
already of each, and by network values such as sizes of the components
each node is in and whether they are already in the same component or
not. Dependence on a value such as degree can be compensatory, as
when a node will not take on a new partner if either it or the
potential partner has already one or more partners already, or
preferential attachment, when a node with high degree makes it more
likely to take on another relationship.
In this way, the network formation process depends on $N_t$ designs,
where $N_t$ is the number of nodes at time $t$ and links are formed by
consensus or negotiations between pairs of nodes. Deletion of a link
is at the discretion of either node in a partnership, and the
probability of deletion can similarly depend on current values.
In this way the design that creates, maintains, and changes the
network is decentralized, with agency largely residing with
individuals and pairs.
The sampling designs with which we reach into the network population
can in some cases use a conventional frame such as a list of nodes and
in other cases use spatial sampling techniques. A Bernoulli
sample in a population of $N_t$ units has $N_t$ independent trials,
selecting unit $i$ with probability $p_i$, for $i = 1, ..., N_t$. The
sample size is random, with expected value $\sum_{i=1}^{N_t}p_i$. A
conventional design for sampling with replacement from $N_t$ units has
$n$ independent trials with unit $i$ having probability of selection
$p_i$ on each of the trials. A random sample without replacement has
equal probability of selection for each possible combination of $n$
distinct units, giving probability $n/N_t$ probability that unit $i$
is included in the sample.
Our focus of interest
in this paper is on the link-tracing designs. These can be started
with any of the simple designs above. In many cases of interest, the
sampling process attains over time a stochastically stable
distribution regardless of the initial design by which it started.
Also nodes are being
Consider a sampling process that at time $t-1$ has a sample $S_{t-1}$.
The sample $S_t$ is the result of following links out from $S_{t-1}$,
supplemented by new nodes selected at random, spatially, or other
design not relying on links. An edge going from node $i$ to node $j$
is a pair $(i,j)$ in the current edge set $E_t$, the pair being
ordered if the link is directional.
A flexible type of link tracing design
selects a Bernoulli sample of links out from the current sample to add new
nodes to the sample. The probability $p_{(i,j)}$ of following link
$(i,j)$ to add node $j$ to the sample, for $(i,j)\in E_{t-1}$,
$i \in S_{t-1}$, and $j \notin S_{t-1}$ can depend on values associated with
the origin node $i$, values associated with the destination node $j$,
and values associated with the link $(i,j)$ between them. For a unit
$i$ outside the sample, the probability it is added at time $t$
through link tracing is
\[
p(i \in s_t \,|\, i \notin s_{t-1}) = 1 - \prod_{\{j:j\in s_{t-1}, e(j,i) \in E_{t-1}\}} (1-p_{(i,j)})
\]
In addition
nodes may be added on occasion through direct Bernoulli sampling or
random sampling without replacement.
Nodes are removed from the sample through Bernoulli removals. Thus
\[
p(i \notin s_t \,|\, i \in s_{t-1}) = q_i
\]
The probability $q_i$ for removing node $i$ from the sample may depend
on values associated with the node such as how long it has been in the
sample. In many cases the
Bernoulli removals may be independent from one sample node to another. In other cases, however,
dependence results from limited resources or designs which increase or
decrease probabilities of removal depending on current sample size. In addition a node $i$ is removed from the sample when it is
deleted from the population.
Different variations of this type of design can select a random sample of
without replacement of $n_t$ links out from the sample, where $n_t$
may have a fixed target value but be constrained to be no greater than
the number of links out. If the number of links out is less than
$n_t$, the additional desired units can be selected by simple random
sampling from those not in the sample. In another variation links out are selected
with replacement using $n_t$ independent trials with link $(i,j)$
having probability of selection $p_{(i,j)}$
for $(i,j)\in E_{t-1}$,
$i \in S_{t-1}$, and $j \notin S_{t-1}$. Further variations include
designs of a type used in respondent driven sampling of members of a
hidden population in which each person in the sample is given a set
number $k$ of recruitment coupons with which she can recruit up
to $k$ individuals with whom she is linked.
In the temporal setting, sampling without replacement has more than
one potential meaning. It can mean that we do not select a unit that
is already in the current sample, and keeping track of the number of
times a unit is selected or else only keeping track of the set of
distinct units selected. Alternatively, without-replacement sampling
can mean, after a unit has been removed from the sample, we will not
later select it back into the sample. Further we can generalize the
concept of replacement to a continuous variable ranging from 0
to 1. In that case we multiple the usual selection probability $p_i$
for unit i times $r$, so that $r=1$ corresponds to complete
replacement and $r=0$ is strictly without-replacement. An
intermediate value of $r$ means that selection probability is damped
for a unit previously in the sample. In addition $r$ can depend on
how long it has been since the unit was last in the sample.
A random walk in a graph is a simple design that has received much
study in the static network setting but little in the temporally
stochatic graph setting.
A simple random walk design has one unit in the sample at a time. One
link is followed at random to find the new sample unit, and the
current one is dropped. Letting $d_{ti}$ be the degree, or number of
links out from node $i$ at time $t$,
\[
p(s_t = \{i\} \,|\, s_{t-1} = \{j\}) = 1/d_{ti}
\]
This is often supplemented with a probability $p$ of taking a random
jump to a different unit in the population, giving
\[
p(s_t = \{i\} \,|\, s_{t-1} = \{j\}) = p(1/d_{ti}) + (1-p)(1/(N_t-1)))
\]
so that the walk does not get stuck in a single component of the
network. The classical random walk is with replacement, so that in
the static graph setting it forms a Markov chain with the current
state of the process being the node in the current sample. This
viewpoint runs into some complications in the temporal stochastic
graph setting of interest here. The nodes themselves are transient
entities, because of the birth and death process, so that they can not
form recurrent states of a stochastic process. Further, we can follow
a link to a node, only to have the link be deleted at that time step,
so we may have the sample stuck for a long time on a single isolated
node until a new link might connect to it. Thus the design might be
modified to take a random jump with some higher probability if that
happens. Still further, the current node may itself be deleted while
it is the sample node, and the sampling process will have ended unless
we again modify the design to start a new random walk from a randomly
selected node. Also, as network links change the connected components
of the graph change, and a simple random walk that is stuck in one
component can move to a different component when they transiently
The population process together with the sampling process gives us a
complex stochastic process. Even though what would appear to be basic
entities of the process, like nodes, links, and sample members are
transient, we find that under a wide range of conditions stochastic
stability properties of some variables are apparent. By stochastic
stability we refer to variables that have stationary distributions or
ergodic properties over time. A variable in such a distribution is
forever varying, or fluctuating, but it's distribution stays the same,
once equilibrium is reach.
Although it is not necessary to have stationary of limiting
distribution of variables in order to study the effectiveness of
intervention strategies, it is convenient in a number of ways. With
stochastic stability, population values will stay in a predictable
distribution in the absence of any interventions. If we make an
intervention, a simple measure of its effect is the equilibrium
distribution that results over time, in comparison with the
equilibrium distribution without the intervention.
In many cases an agent we make an intervention against, adapts over
time to counter our intervention. More generally, one intervention
design interacts with another. The effect of the intervention is
most simply measured as the equilibrium that results after all
adaptations and interactions attain their new equilibrium
In more detail, the effect of an intervention is the distribution of
sample paths over time, compared to the distribution of sample paths
without the intervention, regardless of whether the processes are in
equilibrium. In terms of a simulation study, an advantage of
processes that attain equilibrium distributions of effects over time
is that the distributions can be determined from a single realization
over a long time span, or a few such realizations, rather than in
every case needing to be run for a large number of realizations.
Because the overall stochastic process is complex and changes in
values of input parameters can change a process from stable to
non-stable, we examining stability properties empirically. Helpful
tools for doing this include time series plots of variables of
interest, cumulative mean functions of such functions, cumulative
histograms, and cumulative empirical characteristic functions (ECF) of key
variables. The empirical characteristic function of a stochastic
process variable $X_t$ is defined as
\[
c_x(a) = \frac{1}{t}\sum_{t'=0}^te^{iaX_{t'}} = \frac{1}{t}\sum_{t'=0}^t
\left[\cos(aX_{t'}) + i \sin(aX_{t'})\right]
\]
We are interested not only in when the process is stochastically
stable, but when it is changing. In particular, once we initiate an
intervention strategy, we would like to see early signs that it is
changing. The tail of the ECF is sometimes described as reflecting
the roughness of a distribution. In many cases when a process starts
to change, as the distribution starts to change in the direction of
the new equilibrium, the tail of the ECF appears to go wild, writhing
like a snake that has become restless, calling attention to the
Many parameters of the network model and designs are individual, by
node. They can be changed for one individual in the course of a
simulation run. In this case we can ask, what is the benefit to that
a single individual of making this change. Individual parameters
include tendencies in forming links, compensatory changes for high
degrees of self or other, or preferential attachment tendencies.
In designing interventions against a virus spreading in a human
network, we would like to be able to anticipate or infer where the
virus might spread next, and which units to distribute our
intervention to in order to have the most beneficial effect. To do
this we devise a simple type of design based inference in the
network. This is done as follows.
Consider the link tracing design that was described above that samples
with replacement regarding units that were previously in the sample
and without replacement regarding units currently in the sample.
Bernoulli tracing of links, independently with probabillity $p$ for
each link leading out of the sample, is supplemented by some small
chance for selecting random units, which is done with independent
Bernoulli selections from the units outside the sample with small
probability $p_r$. Given the current sample size $n_t$ after link
tracing additions and a target sample size $n_{\rm target}$, units are
removed from the sample by independent Bernoulli removals, each having
probability $q$, where $q = (n_t - n_{\rm target})/n_t$ if $n_t -
n_{\rm target} > 0$ and $q = 0$ if $n_t - n_{\rm target} \le 0$. If
we give the design a high rate $p$ of tracing links out, the sample
moves very fast through the population. Its sample size varies but
stays in a stable distribution around the target size.
Hazard function based on Bernoulli (p) per time step has expected time
to event $E(x)=1/p$.
Hazard function based on a Weibull distribution discrete approximation,
\[
f(x) = \lambda\beta \lambda^{(\beta - 1)} e^{(-x\lambda)^\beta}
\]
\[
E(x) = \Gamma(1 + 1/\beta)/\lambda
\]
set $E(x) = 1/p$ and solve for $\lambda$, giving
\[
\lambda = p \, \Gamma(1.0 + 1.0/\beta)
\]
\[
h(x) = (\lambda \beta) (\lambda x)^{(\beta - 1)}
\]
A daily probability $p$ of an event such as mortality based on a
longer term rate $p_a$ such as probability per year is calculated from
p_a = 1 - (1-p)^k
or its inverse
\[
p = 1 - (1-p_a)^{1/k}
\]
where $k$ is the number of time steps in the longer time rate, such as
365 days in a year.
§.§.§ HIV Epidemic
The HIV epidemic serves as a motivating example for the methods of
this project. We focus in particular on the heterosexual epidemic.
The virus uses a link-tracing design to select a sample of people. It
makes an intervention by infecting each person in its sample. The
links followed are sexual partnership links, and transmissions provide
the link tracings. The probability of tracing is very low per contact
event, often but not always less than one one-thousandth. We model
the design as without replacement, though that is a simplification
since in some cased there may be multiple infection of an individual
with different strains of virus. Since there is currently no
practical cure available for HIV, the design is without replacement.
Removal of a node from the sample occurs only with deletion of the
node from the population, at death or emigration out of the study
population. The probability of tracing depends on values of the
origin node, the destination node, and the link between them. For
example, the probability of transmission of HIV per sexual contact
event depends on the stage of infection in the infected individual,
the susceptibility to infection of the exposed individual, and on the
details of the type of contact in that event.
For the virus' design parameters, there is a tradeoff in which a high
tracing rate is associated with a decrease in survival time of its
host. Letting $\alpha$ represent virulence or host mortality rate and
$\beta$ represent transmission rate, a simple form of function
characterizing the tradeoff is (Boyker, Fraser)
$\beta = c \alpha ^{1/\gamma}$
where $\gamma > 1$. Thinking of virus modifying its transmission rate
by small increments over time, with the per-time-step mortality rate
of the host human being affected as a result, the relationship can be
\[
\alpha_t = a \beta_t^{\gamma}
\]
where $a$ is a constant
related to $c$ and $\gamma$ is a curvature parameter.
Existing and potential
interventions to reduce the epidemic include behavior changes
affecting patterns of relationships between individuals, safer sexual
practices used strategically, antiretroviral drug combination
treatments, vaccines, cures, pre-exposure prophylactic treatments for
partners of infected individuals, and control of catalyzing infections
such as Herpes simplex virus 2 (HSV2). In counter-response, HIV
adapts to interventions against it with mutations, recombinations, and
their selections.
§.§.§ Network epidemic dynamics and rates background
Rates of heterosexual transmission of HIV per coital act and frequency of coital acts are
estimated in <cit.> in a study of 171 monogamous
couples in which one member was HIV positive, in Rakai, Uganda.
Probability of transmission to
the other partner in the longitudinal study was estimated as a function
of viral load. Rates were about 4 times higher is the presence of
genital ulceration disease (GUD).
<cit.> have estimates for stage specific rates of
HIV transmission based on serodiscordant heterosexual couples in Rakai,
Uganda. They estimate a 26 times for early stage and 7 times for late
stage compared to chronic stage. The estimate early stage infection
lasts about three months.
For this study I've used initial estimates of HIV transmission rates with and without
catalyzing genital ulceration disease (GUD) such as herpes simplex
virus type 2
(HSV-2) from the metastudy <cit.>.
Effectiveness of condom use in reducing heterosexual HIV transmission
is reviewed in <cit.>.
Infectious agents typically run into a tradeoff in the evolution of
transmission rates. Higher transmission rate is associated with
increased virulence, increasing the mortality rate of the host which
in turn slows down the spread of the pathogen strain in the
population. Tradeoff functions have the general form
that transmission rate is a decelerating function of
In this study I use a simple tradeoff function from
<cit.>, who uses rate data from
<cit.> for HIV. Other functions with similar tradeoff
properties have been used by other authors such as
<cit.> presents
opposing views of different researchers on the relative importance of
early stage in the transmission of HIV. <cit.> presents
opposing views of different researchers on the relative importance of
early stage in the transmission of HIV.
Dynamic epidemic models
showing selective advantage of virulent strains in early stage of an
epidemic and advantage shifting to less virulent strains, which allow
their host to survive longer, as the epidemic matures were compared to
laboratory studies with colonies two competing strains of bacteria
<cit.>, finding agreement with the model predictions.
<cit.> presents
opposing views of different researchers on the relative importance of
early stage in the transmission of HIV.
<cit.> describes some of the challenges in developing
an HIV cure based on antiretrovirals together with an agent for
releasing the virus from latency.
Properties of networks in which
relationships shift preferentially that are missed by static network
epidemic models are examined in <cit.>. The
stability of casual contact and close contact patterns over time was
studied <cit.> using diary based methods with 49
volunteers, finding that the close contacts tended to be more stable
than casual contacts.
Network structure and change patterns in relation to individual
behaviors were investigated in <cit.> for
syphilis among young people and HIV among drug users, finding in both
studies that spread of disease was associated with network cohesion,
in the form of separation of components or local density of
The importance of concurrent relationships in the spread of the HIV
epidemic is investigated in <cit.> and
<cit.>. Interaction of early transmission rate
and concurrency in sexual links is discussed in
<cit.>. A modeling approach combining network
models and compartment models is used in <cit.>
to evaluate the importance of concurrency.
<cit.> describes a modeling approach based on a set of
partial differential equations with the addition of some temporal network
aspects such as degree distributions in which contacts change,
describing mean behavior for infinite network size and approximate
behavior for moderate size.
<cit.> and <cit.> describe a
network of 50,185 sexual contacts between 6,642 escorts and 10,106 sex
buyers as reported on a Web discussion forum.
<cit.> (and especially its Supplement 1) modeled the individual variability of
adherence over time for an individual as well as the variability
between individuals. Their model, used for the purpose of estimating
parameters, is individual based but not network based except to the
extent of having assumed degree distributions. This study uses an
inference method that appears to essential be approximate Bayesian
An approach bringing network effects to epidemic compartment models by
having different groups with different network degrees is described in
<cit.>. Epidemic threshold properties of simple dynamic
network models in which degree says constant but neighbors exchange,
that is, identity of partners change instantaneously at random times
are described in <cit.>. Compartmental models are
modified to add some network effects in <cit.>.
<cit.> examine network effects in
compartmental models by including a contact or mixing matrix into the
model, comparing assortative mixing patterns, in which individual's
contacts tend to be within their own group and dissortative patterns,
in which contacts tend to be between groups. Early work in modeling
the dynamics of the HIV epidemic includes
<cit.> and
The models for dynamic spatial network populations and interacting
designs developed in work also have some relationship to the
literature on evolutionary dynamics. <cit.> provides a summary
of models of species interactions, epidemics, and selection based on
systems of partial differential equations. <cit.>
describe recent work on stochastic point process models built on top
of that approach. Among the many cases of ecological systems
exhibiting the sort of spatial, temporal, and network patchiness
addressed by the models and designs of this paper, a good example
is described by <cit.>.
§ RESULTS
In this section we look at some results of using the network sampling
approach to evaluate the effectiveness of intervention designs, using
the HIV epidemic as our test example. The first type of result we
look at are what are the characteristics of the virus design and how
does it respond to human network activity over time. The second
result we look at is what can one individual or two in cooperation
accomplish to reduce the risk to themselves or others. And the third
result we look at involves network effects of two types of ideal
interventions we hope are available at a future date, namely
treatments that cure or clear an infected individual's body of HIV.
§.§.§ Virus adaptation to temporal
network geometry
Most link-tracing designs through networks select nodes having more
links in to them with higher probability than nodes with fewer links.
This is true of simple random walk designs, snowball designs of
various types, and adaptive web sample designs among others. It is
true of a natural link-tracing design such as HIV uses in selecting
people following sexual links, and which we model here as independent
Bernoulli selections, without-replacement, with unequal probabilities
of transmission tracing depending on node and like characteristics.
The only designs I am aware of for which this is not true are targeted
random walk designs in which a random walk in a graph is modified
using Markov chain Monte Carlo techniques in order to achieve desired
long term selection probabilities including having them equal for all
Combined with with network clustering tendencies in which certain sets
of nodes are at least temporarily more highly connected than average,
the link-tracing design of the virus leads to a pattern of spread in
the population over time characterized by local explosions, even if
the explosions occur in relative slow motion, followed by longer
periods of little spread. We can document this trend by tabulating
the average degree of nodes recently selected into the virus' sample,
compared to the average degree of nodes not in the sample and to nodes
that have been in the sample for longer. We find the degree of nodes
recently selected is higher. The same is true for their out-degree,
the number of links a node in the sample has to nodes not in the
In a node that has recently been infected, a strain of virus with a
high transmission rate will have an advantage, on average, relative to a strain
with a low rate. The high rate strain can take advantage of the high
number of links out to spread farther. There is a tradeoff, however,
since the higher virulence associated with the high transmission rate
brings a higher mortality rate to the host person. Later in the same
person, there tends to be fewer if any links out to nodes not already
infected. At that stage a strain of virus with low transmission rate
is favored. With the longer expected survival time of the host, the
virus has a chance over time of new links being formed.
The advantage to the virus of having a high transmission rate early
for a short time and a low transmission rate later for a long time is
amplified by clustering of network links in time and space. As
infection spreads through a cluster a strain with higher transmission
rate, particularly higher early rate, will spread explosively through
the cluster, faster than other strains, and become locally more
predominant. Over time their are fewer links to uninfected nodes in
the cluster, as more of the nodes locally are infected. Over a longer
period of time the cluster breaks up through social drift. Survival
time of a host person is higher for a low virulence, which gives that
strain a better chance of being the lead strain in igniting the next
cluster, should it be encountered in the drift over time as old links
dissolve and new ones are formed.
The pattern of simulation results comparing strains adds further light
on this issue. Comparing strains with fixed early and chronic rates,
an optimal strain having a transmission probability per sexual contact of
about 0.008 during chronic stage and about 15 times that during early
stage emerges. Compared to a strain with the same chronic rate and an
early rate the same as that, that is, an early rate factor of 1, the
strain with low early rate reaches over time reaches an equilibrium
presence about ten percent lower. More noticeably, the low early rate
strain tends to take much longer to get off the ground, starting from
just one or a few cases. A strain with a very high early rate of 50
times or more tends to rise very fast and then crash to a lower
equilibrium because of the higher host mortality induced. Strains
with an early factor between about 5 and 35 perform about as well as
the optimal strain, in terms of equilibrium level. In each run a
single strain exists in a single population.
When we put two strains with fixed rates together in competition, the
optimal strain almost always nearly completely dominate over time over
either the one with low early rate or very high early rate.
Next we put in a random selection of strains and let them evolve
together. When a strain transmits to a new node, the new early
transmission rate is the value of the one in the transmitting node
plus a small random increment (uniform or normal), constrained to not
go below zero or above 1.0. The result, over time, is independent of
the initial distribution and tends to produce a variable distribution
of early rates in the population, with mean early rate factor between
15 and 20 and the bulk of the distribution between 5 and 40.
Contributing factors to the persistence of variability of early rate
in the population are the variability at transmission, producing
genetic drift, and the fact that at any given moment different nodes
are under different selective pressures, depending on the number of
open links around them.
The two-pronged strategy of HIV with a high early transmission rate
for a short period followed by a low transmission, low mortality rate
for a longer period, presents challenges for interventions to lower
the incidence and prevalence of HIV in a human population. The
highest transmission rate comes before the virus can be detected with
standard tests. Explosions into concentrated clustered tend to be
well established before responses can be readied. And in the survival
period that that averages around ten years without treatment only a
few of those cases need to make an ignition of a new explosive area
for the epidemic to persist.
§.§.§ Pair consensus intervention strategy of safer sex early in
relationship followed by HIV home testing
Individuals and pairs of individuals have the most direct agency in
forming and dissolving links, which creates and maintains the network
over time and potentially can change it. As described earlier we
think of this as $N_t$ designs, as many as there are individuals. The
seemingly insignificant day to day choices and actions of individuals
create the uneven temporal network topology in which the HIV epidemic
expands or shrinks locally and throughout the world.
In this section we look at a strategy that relies on the agency of
individuals and cooperating pairs. The idea is to counteract the
virus strategy with it's high rate of transmission in the early stage
of an infection. The problem is, the standard tests of for the
presence of HIV are not sensitive in the early part of an infection,
and the person having an infection in early stage is likely not to
know they are infected. Instead, we consider the strategy in which a
pair uses safer sex practices early in a relationship, followed by
each person testing the other. We consider the use of one of the new,
cheek swab home tests.
The strategy is as follows. The couple use safer sex practices during
the first k weeks of a relationship, followed by HIV tests. If both
tests are negative, then the restriction to safer sex practices is
lifted. If both tests are positive, then also the pair can abandon
the safer sex practices with each other. If one test is positive and
the other negative, the pair continues to use the safer sex
A safer sex practice is one that reduces probability of transmission
in a contact to a specified proportion $p_s$ of penetrative
intercourse. In the simulations on the heterosexual epidemic I
use the value $p_s = 0.10$. The literature is sparse but
appears that not just the use of a condom but alternatively a range of
sexual techniques collectively referred to as “outercourse” reduce
transmission probability to something like 10 percent of what it would
be. These include various methods of oral sex and hand-genital
contacts. We omit from the list fellatio involving ejaculation into a
partner's mouth, as one study estimated is transmission rate as 50
percent that of penetrative sex.
For the duration of the safer sex period from the start of a
relationship the simulations use 12 weeks, or 84 days. The expected
duration of early stage infection from the start of HIV infection in
the simulations is 75 days, or between 10 and 11 weeks. In this way
the infection has a high probability of being detectable when the
tests are given and the chance of being exposed to the high
transmission rate from a new partner is greatly reduced.
In simulations when everyone in a population is using this strategy it
is surprisingly effective in bringing down the epidemic even in dense
preferential attachment situations. Suppose only 50 percent, or 10
percent of the population uses this strategy. What is the effect on
them, what is the effect for others. The answer depends on the
pattern of adoption of the strategy. If individuals with the strategy
tend to assort to relationships with others using the same strategy
the benefit for them is great and others are relatively unaffected.
If the two types of behaviors mix randomly to benefit to the people
using it is dampened and others accrue benefit. We can this type of
exploration to the extreme and ask, what is the effect on one person
if she uses this strategy while others do what they will do anyway,
including dense, high risk social settings. To use this strategy
takes the cooperation of both partners in a relationship. If her
partner has other relationships also, her risk is affected if the
partner uses the strategy with just her or with their other partners
as well, at least while they are in a relationship with her.
The concept of a strategy of this type is that individuals and
cooperating pairs bring down the potential rates of transmission in
relationships, with particular focus on early rates and untestable
§.§.§ Two types of cures, one conferring immunity
At the time of writing there is no practical cure for HIV infection,
that is, no treatment that will clear the virus completely from the
body and leave the person in good health. There is, however,
plausible theory on approaches for developing cures for HIV. One
person has apparently been cleared of the virus through stem cell
transfusion giving him immune cells having a protective mutation but
for various reasons it is not considered practical to spread this
approach widely. A cure would be the the best thing that could happen
to an individual who is infected with HIV and in our simulations a
cure emerges as having the best network dynamics for bringing the
epidemic down through prevention of further spread, even in very
difficult network situations.
In the simulation two types of cures, both equally effective in curing
an individual, emerge as having markedly different network dynamics.
One type of cure clears the virus from the person but leaves him or
her susceptible to reinfection. The other type of cure, in addition
to clearing the virus, confers subsequent immunity to infection. In
between are cures that would confer some degree of immunity, which is
incorporated as a resistance factor between 0 and 1.
One research
approach uses antiretroviral drugs to deeply suppress reproduction of
HIV particles in the blood or other body fluids and seeks another
class of drugs to induce the virus in refuge as protovirus segments of
human DNA to express themselves and emerge to be in turn prevented
from further reproduction. A second research approach in seeking a
cure involves gene therapies to insert strengthening proteins or other
HIV resistant features in immune cells. There is no apparent reason
to anticipate that the first type of cure would confer immunity
whereas the second type could reasonably expected to confer at least some
degree of immunity following cure.
Each of the interventions studied in the simulation can be effective
in bringing down the epidemic in many network settings and can be
overwhelmed by others. The most difficult of the temporal networks in
the simulation are dense and highly clustered in terms of links, and
are represented in the simulations by preferential attachment
tendencies in the formations of links, giving a high degree of
clustering in space and in time. What happens with the cure that does
not confer immunity is that the high-degree, highly connected, or
high-change individual who is most likely to be infected and serve as a
conduit of transmission to others, once cured is likely to be
reinfected and again serve as a conduit as least over near time. The
cure of an individual by the second type of cure not only reduces
prevalence in the population by one but also makes the links of that
person unavailable or unconducive for further spread.
§ ACKNOWLEDGMENTS
This work is supported by the Natural Science and
Engineering Research Council of Canada.
|
1511.00025
|
Quantum Sciences Group, Army Research Laboratory, 2800 Powder Mill Rd., Adelphi, Maryland 20783, USA
Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
Quantum Sciences Group, Army Research Laboratory, 2800 Powder Mill Rd., Adelphi, Maryland 20783, USA
We present a simplified version of a repeater protocol in a cold neutral-atom ensemble with Rydberg excitations optimized for two-node entanglement generation and describe a protocol for quantum teleportation. Our proposal draws from previous proposals [Zhao, et al., Phys. Rev. A 81, 052329 (2010)] and [Han, et al. Phys. Rev. A 81, 052311 (2010)] who described efficient and robust protocols for long-distance entanglement with many nodes. Using realistic experimental values we predict an entanglement generation rate of $\sim25$ Hz and teleportation rate of $\sim$ 5 Hz. Our predicted rates match the current state of the art experiments for entanglement generation and teleportation between quantum memories. With improved efficiencies we predict entanglement generation and teleportation rates of $\sim$7.8 kHz and $\sim$3.6 kHz respectively, representing a two order of magnitude improvement over the currently realized values.
Cold-atom ensembles with Rydberg excitations are promising candidates for repeater nodes because collective effects in the ensemble can be used to deterministically generate a long-lived ground state memory which may be efficiently mapped onto a directionally emitted single photon.
§ INTRODUCTION
Quantum repeaters can be used to entangle remote quantum memories by interfering flying qubits that are in turn entangled with the memories<cit.>. Networks of quantum repeaters are capable of extending the distance of quantum communication beyond what is capable in purely photonic systems by dividing the communication channel into smaller segments with a quantum memory at each node<cit.>. In addition, a network of quantum memories could be enabled with a quantum register of multiple memories at each node for logical operations and error correction<cit.>. This type of quantum network could realize applications such as cluster state generation<cit.>, distributed quantum computation<cit.>, and entanglement enhanced measurement<cit.>. In order for the entanglement to be distributed to remote sites, it is desirable for the nodes of a network to include long-lived quantum memories entangled with photonic flying qubits for long-distance communication. Quantum teleportation<cit.> is a vital protocol to realize on such quantum networks because it allows the transmission of an unknown quantum state from one node to another while still adhering to the no-cloning theorem <cit.>.
In this paper we present a protocol for teleportation between quantum repeater nodes based on Rydberg excitations in neutral atom ensembles. We describe the protocol in detail and examine the performance of entanglement generation and teleportation protocols for two-nodes. The photon collection is enhanced by using collective effects in the ensemble for directional photon emission. The fidelity of each step in a many step protocol can limit the success rate of the protocol. The use of Rydberg blockade allows us to improve the fidelity of each step, particularly the memory generation step, over processes that rely solely on spontaneous emission. The entanglement generation protocol we use is a modified version of that proposed in the previous work of Zhao et al.<cit.>. The optimization for our two-node protocol minimizes the number of steps and ground states needed which in principle improves the probability of success, reduces the time needed for the protocol and improves the state fidelities. Our two-node optimization comes at the expense of the many-node scaling characteristics of the previously proposed protocols. We compare our protocol with those of <cit.> and <cit.> and find that a simplified protocol can be advantageous for two node protocols because it has less experimental overehead at each node and higher rates for small rapid networking of small numbers of nodes.
Using experimental values that have been achieved in similar systems, we predict that entanglement generation and teleportation rates of $\sim$25 Hz and $\sim$5 Hz are possible. If technology such as pulse shaping, three dimensional optical lattices, or the use of optical cavities is used, it is reasonable to predict that one could achieve experimental efficiencies that lead to entanglement generation rates as high as $\sim$7.8 kHz with corresponding teleportation rates $\sim$3.6 kHz. This would represent asignificant improvement over the highest currently achieved rates for two-node protocols with memory<cit.>.
Teleportation between matter nodes has been realized in ions<cit.>, neutral atoms<cit.> and most recently in NV centers<cit.>. These examples rely on spontaneous emission from an excited state to generate the memory state. Because of a combination of probabilistic memory generation and low photon collection efficiency, teleportation between quantum memories has generally had low rates, on the order of one every few minutes. Approaches to achieving Hz-level rates have included custom high numerical aperture collection lenses<cit.> or placing the quantum memory in an optical cavities <cit.>.
Photonic systems can also be used for quantum teleportation<cit.>, and secure quantum key distribution protocols<cit.>. Photonic systems could realize applications in quantum communication and computation with cluster states through one-way measurement-based computation<cit.>. However, purely photonic systems may be limited because of the difficulties associated with the distance limitation from exponential photon loss in an optical fiber and the incorporation of information processing of multiple qubits at each node. A quantum network enabled with quantum memories addresses both of these difficulties.
The paper is structured as follows. First, neutral atom based quantum repeaters are introduced in Section <ref>. The process for producing ground state memories entangled with directionally emitted photons is detailed in Section <ref>. In Section <ref> we discuss the proposed system for experimental realization. In Section <ref> we show a simplified version of the protocol for generating entangled flying qubits based on the protocols in Zhao et al. <cit.>. In Section <ref> we show how the entanglement can be generated between remote memory pairs. In Section <ref> we demonstrate a theoretical protocol for quantum teleportation within this framework. In Section <ref> we analyze the entanglement generation and teleportation rates for two node protocols in this system. Finally, in Section <ref> we analyze a model for the many node entanglement generation and teleportation rates.
§ NEUTRAL ATOM BASED QUANTUM REPEATERS
The Duan, Lukin, Cirac, and Zoller (DLCZ) proposal <cit.>theoretically described a realizable repeater protocol based on directional single-photon emission from a neutral atom ensemble that relies only on linear optics. The DLCZ protocol uses weak laser beams to probabilistically excite a single spin-wave in an ensemble. The spin-wave serves as the quantum memory and can be read out with a subsequent strong pulse. A phase matching condition, similar to that in four-wave mixing, provides a collective enhancement in the photon emission direction and ensures that the single photons can be efficiently captured. However, in order to reduce two-photon errors, the probability of exciting a single spin wave quantum memory and generating the heralding photon, or the `write' photon, must be kept to $\sim 10^{-3}$ or lower, leading to low rates of entanglement generation<cit.>
In contrast, for the case of neutral atoms, the Rydberg blockade mechanism offers a route to improve these rates by generating the quantum memory deterministically<cit.>. Rydberg excitations in ensembles can utilize collective enhancement from phase matching, similar to the DLCZ scheme, to ensure efficient collection of a single photon entangled with a quantum memory. This can in principle increase the rate of successfully generating a quantum memory as compared to the DLCZ protocol by three orders of magnitude.
Entanglement between a collective Rydberg excitation and a single photon was demonstrated experimentally in Li et al.<cit.>. The single photon was entangled with a quantum memory by using a partial readout of the Rydberg level. However, using the Rydberg level as the memory limits the lifetime to a few tens of $\mu$s.
To improve the memory lifetime, the Rydberg excitation can be shelved in a long-lived atomic ground state. Shelving single collective excitations into ground states via Rydberg states was experimentally demonstrated by <cit.> with lifetimes as long as a few ms <cit.>. To increase the coherence time of Zeeman state memories, it may be possible to adapt methods that have been used to increase the coherence times of ground state memories to several seconds<cit.>. Extraction of a single photon entangled with a long-lived ground-state quantum memory via the Rydberg blockade mechanism has not yet been realized experimentally.
The Rydberg blockade mechanism can also be used to perform two-qubit gates, which are a critical component of the entangling and teleportation protocols described below. A two-qubit gate using Rydberg blockade between two neutral atoms has been demonstrated experimentally<cit.>. In addition, Rydberg blockade between two ensembles has been demonstrated<cit.>. The ability to efficiently perform deterministic gates between local qubits can allow for advanced protocols such as error correction.
Zhao et al.<cit.> and Han et al.<cit.> developed protocols for quantum repeaters using cold atom ensembles with Rydberg excitations that have favorable scaling to long distances and many nodes. In these protocols, the memories are deterministically generated via Rydberg blockade. Multiple memories are stored by coherently driving single excitations to different Zeeman ground states. Gates are performed between the multiple memories using Rydberg blockade. Entanglement is generated between remote pairs of nodes through photon interference. In addition to the memory generation, Rydberg blockade is used for deterministic entanglement swapping through local Rydberg interactions rather than photon interference and detection, which is typically probabilistic and of low efficiency. A separate proposal includes coupling the atomic ensembles to optical cavities to make use of the high efficiency photon absorption in a cavity in order to generate remote entanglement without the need for photon detection <cit.>.
Because of their potentially high rates of communication and information processing capabilities, quantum repeater nodes based on neutral atom ensembles with Rydberg excitations have the potential to enable large-scale quantum networks. Our work aims to simplify these protocols for small numbers of nodes and flesh out the details in order to pave the way for initial experimental demonstrations.
§ SINGLE PHOTON STORAGE AND READOUT
Here we examine the process of writing a single quantum memory and efficiently mapping it onto a photon mode. Consider a simplified atomic structure as in Fig. <ref> (a) with two $^{87}$Rb ground states $\Ket{g}$ and $\Ket{u}$, an excited state $\Ket{e}$ which, for example, could be in the $P_{3/2}, F=2$ manifold, and a high lying Rydberg level $\Ket{R}$. The atoms are initially optically pumped into the $\Ket{g}$ state which serves as the reservoir state where most atoms will remain. We couple the ground states to $\Ket{R}$ with a two-photon processes through the intermediate excited state $\Ket{e}$, as shown in the the energy-level diagram in Fig. <ref>. For a two-photon transition, detuning from the intermediate state ensures minimal population is transferred to $\Ket{e}$ during the process and two-photon resonance ensures that the state is transferred with high fidelity.
Due to the large interactions between Rydberg states, the presence of one Rydberg excitation will shift the Rydberg energy levels of near-by atoms out of resonance with the excitation beams. This ensures that only one excitation occurs within a Rydberg radius, $r_b$. This radius depends on the target Rydberg states and the linewidth of the excitation laser. If the trapped atomic ensemble has a diameter smaller than $r_b$ and the excitation beams are significantly larger than the atomic ensemble, then the excitation beams interact with all atoms with equal strength and the ensemble can contain only one Rydberg excitation.
When a two-photon $\pi$-pulse is applied, i.e. $\vec{\bf{k_1}}$ and $\vec{\bf{ k_2}}$ in Fig. <ref> (a) we assume that each atom has an equal probability of being excited, as is discussed in Section <ref>. The resulting state is an equal superposition of one atom in the excited Rydberg state with the remaining atoms in the reservoir state $\Ket{g}$. The collective state of the ensemble is a $\Ket{W}$ state and has the form given in Eq. (<ref>).
The level scheme shows the reservoir state $\Ket{g}$ on the same hyperfine manifold as the memory state $\Ket{\bf u}$. Both $\Ket{g}$ and $\Ket{\bf u}$ are coupled with two-photon transitions to a Rydberg state $\Ket{\bf R}$ through an intermediate excited state $\Ket{\bf e}$. (a) shows the generation of a single memory state, and (b) shows the readout from the Rydberg level. Collective enhancement occurs when the spontaneously emitted photon $\vec{\bf{k_8}}$ brings the state back to the reservoir $\Ket{g}$.
\begin{equation}
\Ket{\bf{R}}=\frac{1}{\sqrt{N}}\sum_{j=1}^N e^{i\phi_j}\Ket{g...R_j...g}
\label{eq:super}
\end{equation}
Where $ N $ is the total number of atoms within $r_b$. The phase, $\phi_j$, is determined by the wave vectors of the excitation laser beam and the position of the atom, i.e. ${ \phi_j =\sum^{pulses}\vec{\bf{k}}\cdot \vec{\bf{r_j}}}$.
Because the excitation is shared across the atoms, the state is robust against atom loss and the effective Rabi frequency of the two-photon transition between $\Ket{g}$ and $\Ket{R}$ is enhanced by a factor of $\sqrt{N}$ compared to the single-atom Rabi frequency<cit.>.
Importantly, when generated with the Rydberg blockade mechanism there is no vacuum component in the produced $\Ket{W}$ state, and therefore it can be prepared deterministically. In addition, the two-photon component depends on the detuning from the Rydberg blockade shifted state, which can be made to lead to low two-photon errors<cit.>.
This is in contrast to a $\Ket{W}$ state generated by the DLCZ protocol which produces a state that is mostly in the vacuum state, that is, almost all of the atoms remain in $\Ket{g}$ with no photonic component. Because of this, the memory generation must be heralded with a success probability in each shot being generally $p \sim 10^{-3}$. The undesired two-photon component scales as $p^{2}$ which can limit attempts to increase the rate of memory generation.
In the remainder of the paper, we will use a simplified notation where a bold-face letter, such as in, $\Ket{\bf{x}}$, represents a collective excitation in state $\Ket{x}$ that is in the form:
\begin{equation}
\Ket{\bf{x}}=\frac{1}{\sqrt{N}}\sum_{j=1}^N e^{i\phi_j}\Ket{g,...x_j,...g}
\label{eq:super2}
\end{equation}
Where `$x$' is any of the singly excited states (e.g. $ u$, $e$, $R$, etc.).
In order to ensure that the quantum memories do not de-phase during a storage time, it is desirable to transfer the single excitation from the Rydberg level, which has a relatively fast dephasing time, into a long-lived memory ground state $\Ket{\bf{u}}$, as in Fig. <ref>. This is done by applying another two-photon $\pi$-pulse from $\Ket{\bf{R}}$ to $\Ket{\bf{u}}$ through $\Ket{\bf{e}}$, i.e. steps $\vec{\bf{ k_3}}$ and $\vec{\bf{k_4}}$ in Fig. <ref> (a). The pulse sequence to generate a single memory state is shown in Fig. <ref> (a). At this point, the atomic ensemble is in the memory state $\Ket{\bf{u}}$, (i.e. a collective excitation of the form in Eq. (<ref>)), with a phase ${\phi_j =(\vec{\bf{ k_1}}+\vec{\bf{k_2}}+\vec{\bf{k_3}}+\vec{\bf{k_4}})\cdot \vec{\bf{r_j}}}$. This process has recently been used to generate a Fock state of atoms by Ebert et al.<cit.>.
To read the memory out photonically, we first apply a single-atom two-photon $\pi$-pulse from $\Ket{\bf{u}}$ to $\Ket{\bf{R}}$, steps $\vec{ \bf{k_5}}$ and $\vec{ \bf{ k_6}}$ in Fig. <ref> (b). This is followed by applying strong blue light nearly resonant with the $\Ket{\bf{R}}$ to $\Ket{\bf{e}}$ transition, $\vec{\bf{k_7}}$. The state $\Ket{\bf{e}}$ quickly decays to the ground state. The amplitude of emission into a given spatial mode with associated wave vector $\vec{\bf{k_e}}$, is given by the condition:
\begin{equation}
A\propto \frac{1}{N}\left|\sum_{j=1}^Ne^{-i({\vec{\bf{ k_{tot}}}-\vec {\bf{k_e}}})\cdot {\vec{ \bf{r_j}}}}\right|^2
\label{eq:collective}
\end{equation}
Where $\vec{\bf{k_{tot}}} = \sum_{j=1}^{7}(\vec{\bf{k_j}})$. In general, the spontaneous emission will be into $4\pi$, but emission into an arbitrary $\vec{\bf{k}}$ will result in the amplitude in Eq. (<ref>) averaging to 1. In the particular case when the emitted photon ${\vec k_e}$ brings the excitation back to the reservoir state and ${\vec {\bf{k_e}}} = {\vec{ \bf{k_{tot}}}}$, the exponent in Eq. (<ref>) is equal to zero, and the amplitude averages to $N$ <cit.>. This amounts to an enhancement of the spontaneous emission into a particular spatial mode, determined by the geometry of the pulses culminating in $\vec{\bf{k_{tot}}}$. Emission into the phase matched direction constructively interferes, while emission in an arbitrary direction destructively interferes <cit.>. In the presence of additional ground states, $\Ket{\bf{e}}$ decays preferentially to the reservoir state with an enhancement factor of $N$ <cit.>.
The process depicted in Fig. <ref> of memory generation and photon retrieval can be viewed as an eight-wave mixing process analogous to viewing the DLCZ process as a coherent time-delayed four-wave mixing. This enhances the single-photon collection efficiency from the quantum memory.
Thus, the Rydberg blockade mechanism in an atomic ensemble can be used as a high-efficiency source of directionally emitted photons as was originally theoretically proposed by Lukin et al.<cit.> and Saffman et al.<cit.>. Collectively enhanced spontaneous emission of a single photon from a Rydberg media was demonstrated by Li et al.<cit.>.
Alternatively, one could read out the memory by exciting from $\ket{\bf{u}}$ directly to $\ket{\bf{e}}$ giving an effective six-wave mixing to extract a directional photon. However, in this case, the strong beam would be at nearly the same wavelength as the single-photon, whereas in the eight-wave mixing case, the strong de-excitation beam has a very large spectral difference from the single-photon (i.e. 480 nm vs. 780 nm in the case of $^{87}$Rb). Though filtering schemes have allowed for good readout for certain single-atom states <cit.>, our approach allows us the versatility to store multiple memories within one ensemble, hence necessitating more optical beams. The advantage for filtering the single-photon signal from the read pulse will likely outweigh the simplification of applying fewer beams, yet this remains a possibility to explore.
§ EXPERIMENTAL REALIZATION
The energy level diagram of $^{87}$Rb used for the entanglement generation protocol where the bold indicates the light coupling to collectively excited states. The red light (780nm) couples the ground states ($\ket{\bf{u}}$ and $\ket{\bf{d}}$) to the intermediate excited states ($\ket{\bf{e_u}}$ and $\ket{\bf{e_d}}$). The blue light (480 nm) couples the intermediate excited state to two different Rydberg levels ($\ket{\bf{R_u}}$ and $\ket{\bf{R_d}}$). Light emitted when a state decays from the excited states $\ket{\bf{e_u}}$ or $\ket{\bf{e_d}}$ to $\ket{g}$ will have orthogonally polarized circular polarization, $\sigma^+$ or $\sigma^-$ respectively. The states with subscripts t are used later in the teleportation protocol
We briefly discuss an experimental realization of these protocols using a laser cooled ensemble of $^{87}$Rb atoms. The relevant energy diagram of $^{87}$Rb is given Fig. <ref>. The degeneracy of the hyperfine levels is lifted by applying a magnetic field of $\sim$ 0.5 mT which splits the ground states by $\sim3.5$ MHz.
We use a Rydberg level of n $\sim$90 which has been shown to have a lifetime of around $30$ $\mu s$ and $\sim$10 MHz Rydberg blockade shift at a distance of 10 $\mu$m with excitation laser linewidths on the order of a few kHz <cit.>. Hence we use 10 $\mu$m as $r_b$. Two-photon Rabi frequencies to n$\sim$90 Rydberg levels on the order of 1 MHz have been demonstrated with inferred fidelities of nearly $0.9$ <cit.>. The Rabi-frequency must be large enough to insure that operations can take place faster than the dephasing of the Rydberg levels.
Multiple memories can be stored in the different Zeeman sublevels shown in Fig. <ref>. For the entanglement generation protocol, we will use two ground-states, $\ket{\bf{u}}$ and $\ket{\bf{d}}$ for the quantum memory and a reservoir state $\ket{g}$. If beneficial for a particular protocol, the assignment of memory states and reservoir state could be changed. The states $\ket{\bf{u_t}}$ and $\ket{\bf{d_t}}$ will be used in Section <ref>. The Zeeman-state coherence can be one limit to the memory lifetime. Maintaining control at the few percent level of a few mT magnetic field would still allow for $\sim 100$ $\mu s$ lifetime, which would be sufficient to link nodes at a distance of $\sim 20$ km.
The memory states $\ket{\bf{u}}$ and $\ket{\bf{d}}$ are coupled to independently addressable Rydberg levels, $\ket{\bf{R_u}}$ and $\ket{\bf{R_d}}$, through different intermediate excited states $\ket{\bf{e_u}}$ and $\ket{\bf{e_d}}$. Photons emitted by a state decaying from $\ket{\bf{e_u}}$ or $\ket{\bf{e_d}}$ to $\ket{g}$ will have orthogonal circular polarizations, $\sigma^+$ or $\sigma^-$, respectively. These polarizations can be mapped onto the desired polarizations $\ket{V}$ or $\ket{H}$ with quarter wave-plates.
The $^{87}$Rb atoms will be collected in a magneto-optical trap (MOT) and then loaded into a crossed optical dipole trap. If the optical trap has a diameter smaller than $r_b$, and the excitation beams have waists larger than $r_b$ we can ensure that the beams interact only with the atoms within one $r_b$, and that all atoms have the about same interaction strength with the excitation beams, see Fig. <ref>. We assume that the MOT temperatures is cold enough that it does not limit the memory lifetime. The memory lifetime can be increased by transferring the atoms to a far off-resonant optical lattice at the Rydberg magic wavelength <cit.> or by using dynamic decoupling techniques <cit.>. Even without the use of a magic wavelength lattice, memory times of a few ms have been demonstrated in similar systems<cit.>.
If quantum repeater nodes with Rydberg excitations are incorporated into a large scale fiber network, quantum frequency conversion would need to be used to overcome photon loss in the fiber. Frequency conversion of the rubidium signal in the near infrared (780 nm) to a telecom band ($\sim1324$ nm or $\sim1550$ nm) is promising as it can be done with a one-step conversion process. This could be implemented using the atoms as the non-linear device <cit.>, or preferably, a non-linear crystal waveguide converter <cit.>.
A full analysis of the potential errors and limiting factors is beyond the scope of this paper, but we list some of the more important ones here. Fluctuation of static electric and magnetic fields can interfere with the Rydberg or ground-state Zeeman levels. Atomic motion or collisions can lead to dephasing which can limit the single-photon collection efficiency and memory coherence time. Two-photon excitations or dark counts in single photon detectors can lead to erroneous heralding of entanglement generation. AC stark shifts from the trapping laser or excitation beams or off-resonant excitations to Rydberg levels can limit the state preparation fidelity. In a dense atomic sample resonances with molecular Rydberg states can cause de-phasing<cit.>. In order to mitigate this the atoms can be loaded into a 3D optical lattice to control the inter-atomic spacing, and the principal quantum number and two-photon detuning can be adjusted to avoid the resonances.
The atoms are trapped in an optical dipole trap with a waist $< r_b$. The excitation beams ($k_{e_u-R_u}$ and $k_{u-e_u}$), with a waist diameter $> r_b$, are overlapped on a dichroic beam splitter (DBS) and intersect the atoms perpendicularly to the trap. The excitation beams for the $\ket{d}$ states (not shown) can be overlapped with the excitation beams for the $\ket{\bf{u}}$ states using polarizing beam splitters. This produces a spherical interaction region (yellow) with a diameter $= r_b$ where a single excitation is produced. The single excitation is mapped onto a directionally emitted photon which can be converted into the desired polarization with a $\lambda/4$ wave-plate. A second DBS filters the blue excitation light out from the signal photon. The two outgoing photons can be overlapped in time with the use of a PBS and a delay line before being sent to a Bell state measurement
§ GENERATING A FLYING QUBIT ENTANGLED WITH MEMORY
Following the proposals of Zhao et al.<cit.> and Han et al. <cit.>, we describe how to prepare an entangled state suitable for quantum communication protocols. In short, we first produce two spin waves into different magnetic sublevels in the ensemble by applying the steps of Fig. <ref> (a) twice, produce entanglement between them via a Rydberg blockade gate, and then read the components of that state into orthogonally polarized photons by applying the steps of Fig. <ref> (b) twice. The state produced is comprised of two ground state memories entangled to photons of a flying polarization qubit. This structure, sometimes known as `dual-rail' <cit.>, does not require interferometric, i.e. on the order of the wavelength, stability along the optical path <cit.>. Rather it requires the path length to be stable to the level of the photon coherence length, which is much less stringent. The steps to produce this state are summarized in Table <ref> and are detailed in the text, where all states are atomic states as labeled in Fig. <ref> and the photonic states have as subscript $\gamma$. A graphical depiction is shown in Fig. <ref> where the steps correspond to those in Table <ref>.
A graphical depiction of the steps in Table <ref> using the state identified in Fig. <ref>. Optical pumping in step i) is followed by a series of $\pi$-pulses (solid lines) and $\pi/2$-pulses (dashed lines) in order to prepare a memory qubit entangled with a flying photonic qubit.
Entanglement preparation of flying qubit entangled with long-lived ground-state excitations, where $\pi_N$ identifies collectively enhanced rotations and $\pi$ identifies single-atom rotations.
Step Pulse Result
i. Optically pump to ($\Ket{g}$) $\Ket{g}$
ii. $\pi_{N}$($\Ket{g}$ to $\Ket{\bf{R_{d}}}$) $\Ket{\bf{R_{d}}}$
iii. $\pi$($\Ket{\bf{R_{d}}}$ to $\Ket{\bf{d}}$) $\Ket{\bf{d}}$
vi. $\pi_{N}$($\Ket{g}$ to $\Ket{\bf{R_{u}}}$) $\Ket{\bf{d}}\ket{\bf{ R_{u}}}$
v. $\pi/2$($\Ket{\bf{R_{u}}}$ to $\Ket{\bf{u}}$) $\Ket{\bf{d}}(\Ket{\bf{u}}+\Ket{\bf{R_{u}}})$
vi. $\pi$($\Ket{\bf{d}}$ to $\Ket{\bf{R_{d}}}$) $(\Ket{\bf{R_{d}}}\Ket{\bf{u}}+\Ket{\bf{d}}\Ket{\bf{R_{u}}})$
vii. readout ($\Ket{\bf{R_{d}}}$ to $\Ket{\bf{e_{d}}}$) $(\Ket{\sigma^-}_{\gamma}\ket{\bf{u}}+\Ket{\bf{d}}\ket{\bf{R_{u}}})$
viii. readout ($\Ket{\bf{R_{u}}}$ to $\Ket{\bf{e_{u}}}$) $(\Ket{\sigma^-}_{\gamma}\Ket{\bf{u}}+\Ket{\bf{d}}\ket{\sigma^+}_{\gamma})$
The ensemble state is initialized in step i. of Table <ref> by optically pumping all atoms to the reservoir state, $\ket{g}$. This can be done with $\pi$ polarized light if $\ket{g}$ is the $F=1, m_f =0$ state. The $N$ subscripts in Table <ref> refer to transitions that have an enhanced effective Rabi frequency, all other pulses are for single atoms with single atom Rabi frequencies. All transitions between Rydberg levels and ground states are two-photon transitions.
In step ii. of Table <ref> we apply a two-photon $\pi_N$-pulse with an enhanced Rabi frequency from $\ket{g}$ to a high lying (n $\sim$ 90) Rydberg level, to create the state $\ket{\bf{R_d}}$. We use the intermediate excited state $F=2, m_f =-1$ of the $P_{3/2}$ D2 line, i.e. the state $\ket{\bf{e_{d}}}$. Next we shelve the Rydberg excitation in one of the ground states, $F=1, m_f =-1$, to produce the state $\ket{\bf{d}}$, by applying a $\pi$ pulse from $\ket{\bf{R_d}}$ through $\ket{\bf{e_{d}}}$, step iii. of Table <ref>. Recall that here, and in the rest of the paper, all excited states are single excitation superpositions in the form of Eq. (<ref>).
Next we excite a second Rydberg excitation,$\ket{\bf{R_u}}$, step iv of Table <ref>, with a $\pi_{N}$ pulse from $\ket{g}$ which can be addressed independently from $\ket{\bf{R_d}}$ because of the frequency difference between $\ket{\bf{R_u}}$ and $\ket{\bf{R_d}}$. The state of the ensemble is now given by the product state of two ground state memories as shown at the end of step v. in Table <ref>. The notation, $\ket{\bf{\bf{R_u}}}\ket{\bf{d}} \equiv \ket{\bf{\bf{R_u}d}}$ represents a double sum of product states analogous to Eq. (<ref>) where each term in the sum is a product state of two different atoms in different states. In this way, multiple memories are stored within the same atomic ensemble <cit.>.
The Rydberg level $\ket{\bf{R_u}}$ is coupled to a second ground state memory $F=2, m_f =1$, $\ket{\bf{u}}$. The excitation to $\ket{\bf{u}}$ uses a different intermediate excited state, $F=2, m_f =1$ of the $P_{3/2}$ D2 line, $\ket{\bf{e_u}}$.
To produce an entangled state, in step v. of Table <ref> we apply a $\pi/2$ pulse from $\ket{\bf{R_{u}}}$ to $\ket{\bf{u}}$ producing the superposition, see Fig. <ref>:
\begin{equation}
\label{eq:psi1}
\end{equation}
This is followed by a $\pi$-pulse from $\ket{\bf{d}}$ to $\ket{\bf{R_{d}}}$, step vi. of Table <ref>. Though $\ket{\bf{R_u}}$ and $\Ket{\bf{R_d}}$ are different, they still experience strong interactions and Rydberg blockade one another. In the $\ket{\bf{u}}\ket{\bf{d}}$ component $\ket{\bf{d}}$ is transferred to $\ket{\bf{R_d}}$, whereas in the $\ket{\bf{R_u}}\ket{\bf{d}}$ component, blockade between the Rydberg levels $\Ket{\bf{R_u}}$ and $\ket{\bf{R_d}}$ shifts the $\Ket{\bf{R_d}}$ state out of resonance, and the component is unchanged. The resulting state at the end of step vi. in Table <ref> is given by:
\begin{equation}
\ket{\psi}=1/\sqrt{2}(\ket{\bf{u}}\ket{\bf{R_{d}}}+\ket{\bf{d}}\ket{\bf{R_{u}}})
\label{eq:psi}
\end{equation}
This is an entangled state between two Rydberg excitations and two ground state excitations in the same ensemble.
Next, a partial readout maps the components of the qubit into a photonic qubit. To map the excitation in $\ket{\bf{R_d}}$ to a photon, we apply a strong blue beam nearly resonant with the transition from $\ket{\bf{R_d}}$ to $\ket{\bf{e_d}}$, see step vii. of Table <ref> and Fig. <ref>. This intermediate excited state quickly decays to the reservoir state and preferentially scatters into the spatial mode set by the phase matching condition for the eight-wave mixing process as discussed previously in relation to Eq. (<ref>). This is followed by mapping the $\ket{\bf{R_u}}$ state onto a photon with a beam nearly resonant with the $\ket{\bf{R_u}}$ to $\ket{\bf{e_u}}$ transition, step viii. of Table <ref>. The two memories could potentially be read simultaneously, which would reduce the time it takes to perform the atomic protocol, though not the probability of successfully reading out the state, as discussed in Section <ref>.
Alternatively the qubits could be read sequentially, in which case the time-bin qubit can be mapped onto a polarization qubit by delaying the first with a delay line, such as a long optical fiber and then overlapping it with the second photon, as shown in Fig. <ref>. A fiber delay of a few hundred meters will be sufficient to overlap the two photons but will not significantly contribute to the distance the photon must travel in a fiber to entangle two nodes located several $\sim$ km apart.
The photons read from $\ket{\bf{u}}$ and $\ket{\bf{d}}$ have orthogonal circular polarization and we have arrived at the end of step ix. of Table <ref>. As shown in Fig. <ref>, using a $\lambda/4$ wave-plate, we rotate the $\ket{\sigma^-}_{\gamma}$ and $\ket{\sigma^+}_{\gamma}$ photons into horizontal and vertical polarization respectively, which we label $\ket{H}_{\gamma}$ and $\ket{V}_{\gamma}$ to obtain the state:
\begin{equation}
\ket{\psi}=1/\sqrt{2}(\ket{\bf{u}}\ket{H}_{\gamma}+\ket{\bf{d}}\ket{V}_{\gamma})
\label{eq:mem}
\end{equation}
Here we have a maximally entangled state between a flying photonic polarization qubit and long-lived ground state memories. Importantly, because the two photons are emitted with orthogonal polarizations and the single photon is emitted hundreds of nm detuned from the de-excitation beam it should be possible to implement this protocol with all beams on a single axis combined with dichroic beam splitters and polarizing beam splitters as shown in Fig. <ref>. The dichroic beam splitter downstream from the ensemble filters the de-excitation beam from the signal photon. Additional filters will likely be needed to fully attenuate the de-excitation beam from the single photon signal.
This is in contrast to the DLCZ schemes where off-axis collection of the single photon is extremely useful to aid in filtering out the de-excitation beam and the close spectral proximity of the signal to the pump can require spectral filtering <cit.>. Further, off-axis geometry can limit the memory lifetime <cit.>. The switch to an on-axis geometry greatly reduces the experimental alignment complexity and should aid in achieving a high memory read-out efficiency.
The state in Eq. <ref> is suitable for many quantum communication protocols including repeaters, as described in Section <ref>, or teleportation as described in Section <ref>.
Our protocol differs from the Zhao et al.<cit.> proposal by skipping several steps. We read the Rydberg states out directly, whereas they shelve the Rydberg excitations to two additional ground state levels for long-term memory and deterministic on-site entanglement swapping. However, in the case of two-node communication this is unnecessary and we can simply read the state out directly. This enables us to produce entangled flying qubits with fewer steps and use three ground-states instead of five. This reduces experimental complexity for two-node protocols at the expense of the many node scaling of the full protocol, as will be described in the following section.
§ GENERATING REMOTE ENTANGLEMENT
To generate entanglement between two different remote ensembles consider two systems, A and B each prepared in a state in the form of Eq. (<ref>). The resulting combined state is:
\begin{equation}
1/2(\ket{ \bf{u_{A}}}\Ket{H_{A}}_{\gamma} + \Ket{\bf{d_{A}}}\ket{ V_{A}}_{\gamma})\otimes (\ket{\bf{ u_{B}}}\Ket{H_{B}}_{\gamma} + \Ket{\bf{d_{B}}}\ket{ V_{B}}_{\gamma})
\label{eq:generation}
\end{equation}
Photons from the nodes A and B are input to the Bell state analyzer and are interfered on a PBS with axes orthogonal to the polarization of light. Then each arm is then sent through a $\lambda/2$ wave-plate which rotates the polarization by 45 degrees. The photons in each arm are then sent through another PBS. All four output ports are measured with single photon detectors, $D_1$-$D_4$.
The flying qubits from A and B are overlapped on a polarizing beam splitter (PBS) and then subject to a Bell state analyzer<cit.> such as shown in Fig. <ref>. The axes of the PBS are oriented in the H and V axis of the light. The two outputs of the PBS are sent through $\lambda/2$ wave-plates which rotate the polarization by 45 degrees and are then incident on a second PBS. All four output ports are measured with single photon detectors ($D_1 - D_4$). Particular pairs of two photon coincidence measurements will project the state onto a Bell state. In this setup, if coincidences counts between $D_1$ and $D_4$ or $D_2$ and $D_3$ are measured, we project the remaining wave-function onto the state $1/\sqrt{2}( \ket{\bf{u_{A}}}\ket{\bf{u_{B}}} + \ket{\bf{d_{A}}}\ket{\bf{d_{B}}} )$. However, if on the other hand we measure coincidence counts between $D_1$ and $D_3$ or $D_2$ and $D_4$, then we produce the state $1/\sqrt{2}( \ket{\bf{u_{A}}}\ket{\bf{u_{B}}} - \ket{\bf{d_{A}}}\ket{\bf{d_{B}}} )$. In the remaining half of the terms in the expansion of Eq. (<ref>), two photons are sent to one detector, and since we cannot discriminate photon number these counts are lost. Thus if coincidence counts between $D_1$ and $D_4$ or $D_2$ and $D_3$ are measured we produce the desired state. If coincidence counts between $D_1$ and $D_3$ or $D_2$ and $D_4$ are measured, then we perform a local unitary operation to transform the state into the desired state:
\begin{equation}
\ket{\psi}=1/\sqrt{2}( \ket{\bf{u_{A}}}\ket{\bf{u_{B}}} + \ket{\bf{d_{A}}}\ket{\bf{d_{B}}} )
\label{eq:bell}
\end{equation}
This state represents a maximally entangled state between two remote long-lived ground state memories and can be used as the base entanglement resource for further protocols.
To use this protocol in a repeater to distribute entanglement we take Eq. <ref> as the starting point and iterate the procedure in Table <ref>. This simple model captures several of the features of many-node networks. Fig. <ref> shows the sequence of entanglement distribution. First we prepare A and B in the state described by Eq. <ref>, as seen in step i. of Fig. <ref>. Next, system A and B are entangled via a Bell state measurement as described above and depicted in step ii. of Fig. <ref>. Then, we simultaneously map the qubit in B onto a photonic qubit and prepare ensemble C in the state $1/\sqrt{2}(\ket{ \bf{u_{C}}} \Ket{H_{C}}_{\gamma}+ \Ket{\bf{d_{C}}}\ket{ V_{C}}_{\gamma})$. This results in the composite state shown in Eq. <ref>.
\begin{equation}
1/2( \ket{\bf{u_{A}}}\ket{H_{B}}_{\gamma} + \ket{\bf{d_{A}}}\ket{V_{B}}_{\gamma} ) \otimes(\ket{ \bf{u_{C}}}\Ket{H_{C}}_{\gamma} + \Ket{\bf{d_{C}}}\ket{ V_{C}}_{\gamma})
\label{eq:bell2}
\end{equation}
Next, the flying qubits from system B and C are entangled via a Bell state measurement, step iii. in Fig. <ref>. The entanglement between A and B is then transferred to entanglement between A and C, and the result is the long-lived entangled state given in step iv and Eq. <ref>:
\begin{equation}
1/\sqrt{2}( \ket{\bf{u_{A}}}\ket{\bf{u_{C}}} + \ket{\bf{d_{A}}}\ket{\bf{d_{C}}} )
\label{eq:bell3}
\end{equation}
This process can be repeated to continue spreading the entanglement to further nodes as long as the memories in A or any node to which A is entangled do not decohere. This could require the use of high-speed optical switches to ensure that the emitted photons are directed towards the appropriate links.
Graphical depiction of entanglement distribution. The nodes are labeled A-D. A solid line represents photons being passed between two nodes. A dashed line represents entanglement established between two nodes. i) The ensembles A and B are prepared in the state described by Eq. <ref>. ii) A and B are entangled via a Bell state measurement producing the state in Eq. <ref>. iii) The qubits in B and C are mapped onto photons. iv) A Bell state measurement between the photons from B and C extends the entanglement from A to C. v) The qubits in C and D are mapped onto photons. vi) A Bell state measurement between the photons from C and D extends the entanglement from A to D.
Teleportation of a collective excitation from one ensemble to a remote ensemble
Pulse Result
i. initial $\alpha(\Ket{\bf{uuu}} + \Ket{\bf{udd}}) + \beta (\ket{\bf{duu}} + \Ket{\bf{ddd}})$
ii. $\pi$($\Ket{\bf{u_{t}}}$ to $\Ket{\bf{R_{d}}}$) $\alpha (\Ket{\bf{Ruu}} + \Ket{\bf{Rdd}}) + \beta (\Ket{\bf{duu}} + \Ket{\bf{ddd}})$
iii. $\pi/2$($\Ket{\bf{u_{A}}}$ to $\Ket{\bf{d_{A}}}$) $\alpha/\sqrt{2}(\Ket{\bf{R}(\bf{u}+\bf{d})\bf{u}} + \Ket{\bf{R}(\bf{u}-\bf{d})\bf{d}}) + \beta/\sqrt{2}(\Ket{\bf{d}(\bf{u}+\bf{d})\bf{u}} + \Ket{\bf{d}(\bf{u}-\bf{d})\bf{d}})$
iv. $2\pi$($\Ket{\bf{u_{A}}}$ to $\Ket{\bf{R_{u}}}$) $\alpha/\sqrt{2} (\Ket{\bf{R}(\bf{u}+\bf{d})\bf{u}} + \Ket{\bf{R}(\bf{u}-\bf{d})\bf{d}}) - \beta/\sqrt{2}(\Ket{\bf{d}(\bf{u}-\bf{d})\bf{u}} + \Ket{\bf{d}(\bf{u}+\bf{d})\bf{d}})$
v. $\pi/2$($\Ket{\bf{u_{A}}}$ to $\Ket{\bf{d_{A}}}$) $\alpha(\Ket{\bf{Ruu}} + \Ket{\bf{Rdd}}) - \beta(\Ket{\bf{ddu}} + \Ket{\bf{dud}})$
vi. $\pi$($\Ket{\bf{R_{d}}}$ to $\Ket{\bf{u_{t}}}$) $\alpha(\Ket{\bf{uuu}} + \Ket{\bf{udd}}) + \beta(\Ket{\bf{ddu}} + \Ket{\bf{dud}})$
We note that we could use this type of protocol to perform the nested entanglement generation and entanglement swapping architectures that are characteristic of repeater protocols<cit.>. For example, to entangle nodes A and D we could first produce the state:
\begin{equation}
1/2( \ket{\bf{u_{A}}}\ket{\bf{u_{B}}} + \ket{\bf{d_{A}}}\ket{\bf{d_{B}}})\otimes( \ket{\bf{u_{C}}}\ket{\bf{u_{D}}} + \ket{\bf{d_{C}}}\ket{\bf{d_{D}}})
\label{eq:bell4}
\end{equation}
Where we could wait until the entanglement between the pairs AB and CD are both successful. We could swap entanglement by reading out the photons in B and C and entangling them on a Bell state analyzer which would produce the state:
\begin{equation}
1/\sqrt{2}( \ket{\bf{u_{A}}}\ket{\bf{u_{D}}} + \ket{\bf{d_{A}}}\ket{\bf{d_{D}}})
\label{eq:bell5}
\end{equation}
This approach relies on photon detection for the entanglement swapping and does not take advantage of the deterministic entanglement swapping proposed by the earlier protocols. Deterministic entanglement swapping relies on the use of additional ensembles at each node such as the approach in Zhao et al. <cit.> which would require additional atom traps or makes more use of the multi-plexed quantum memory storage in the multiple ground states of an ensemble such as in Han et al. <cit.> which would require more addressing laser beams. However, since we are more concerned with comparing the performance of entanglement generation and teleportation for two-nodes, we take advantage of the reduction in experimental complexity and analyze the performance of the protocol using one ensemble and a minimum number of ground state levels. The experimental implementation we have described could be adapted to include the deterministic entanglement swapping as described by earlier proposals.
The errors that could arise in this protocol are primarily given by the two-photon errors and the detector dark counts. The two-photon error arises when two Rydberg excitations are produced in one ensemble at the same time. This is dependent on the Rydberg blockade detuning, laser linewidth, and off-resonant coupling strength. If two photons are produced, the additional photon could trigger a detector which would mistakenly herald the creation of an entangled state, and would thus produce an error. Similarly, a dark count would mistakenly herald the creation of a memory when none had actually been created. More detailed analysis of errors in these types of systems are given by Zhao et al. <cit.> and Han et al. <cit.>.
§ TELEPORTATION BETWEEN REMOTE ENSEMBLES
We wish to teleport an unknown quantum state from node A to node B. To do this we need two additional ground states which will store the state that will be teleported. We must also perform qubit rotations which could be done with a Raman transition between Zeeman sublevels<cit.>. We choose the $F=2, m_f=-1$ and $F=1, m_f=1$ ground states as the qubit encoding the state to be teleported, see Fig. <ref>. These states will be used to produce the states $\ket{\bf{u_t}}$ and $\ket{\bf{d_t}}$. This `target' qubit pair is only used at node A, not at node B.
The $\ket{\bf{u_t}}$ and $\ket{\bf{d_t}}$ states share the intermediate excited states and Rydberg levels associated with the $\ket{\bf{d}}$ and $\ket{\bf{u}}$ states respectively. Making this choice allows us to access all the necessary states while minimizing the number of Rydberg excitations, reducing experimental complexity. Because two-qubit gate operations only occur between the original pairs or the target pairs, re-using the Rydberg states for the target pairs will not compromise the protocol even though the initial qubit at node A and the target qubit are physically located in the same ensemble of atoms.
The two remote ensembles are initially prepared in the state in Eq. <ref>. We establish entanglement by using the steps described in Section <ref> and Section <ref>. This generates the entanglement resource shared between two remote ensembles. Next, we produce the target state by exciting a spin wave in ensemble A to the state $\ket{\bf{u_t}}$ analogously to how we produced the state $\ket{\bf{u}}$. This results in the state $1/\sqrt{2}\ket{\bf{u_t}}(\Ket{\bf{u_{A}u_{B}}} + \Ket{\bf{d_{A} d_{B}}})$. This is followed by an off-resonant two-photon Raman transition from $\ket{\bf{u_t}}$ to $\ket{\bf{d_t}}$ through an intermediate excited state $F=2, m_{F}=0$, $\ket{\bf{e_{g}}}$. The Raman pulse can be chosen to give any arbitrary rotation, resulting in the state $\alpha \ket{\bf{u_t}} + \beta \ket{\bf{d_t}}$. The energy levels used in the teleportation protocol are shown in Fig. <ref>, but the beams are not.
After generating entanglement between the two systems and producing the target state, the wave-function of the system is described by:
\begin{equation}
\ket{\psi}=1/\sqrt{2}(\alpha \Ket{\bf{u_t}} + \beta \Ket{\bf{d_t}})\otimes (\Ket{\bf{u_{A}u_{B}}} + \Ket{\bf{d_{A} d_{B}}})
\label{eq:six}
\end{equation}
The steps to perform teleportation between nodes A and B are summarized in Table <ref>. The notation is simplified by identifying the first, second, and third elements in a ket as belonging to the `target' qubit, the initial A qubit, and the B qubit respectively.
After the initial state, we transfer the components of the target state that are in $\ket{\bf{u_t}}$ to the Rydberg state $\ket{\bf{R_d}}$ with a two-photon $\pi$-pulse, step ii. in Table <ref>. This effectively blocks any transitions in the $\alpha$ component of the wave-function until the end of the protocol. Subsequent operations only affect components in the $\beta$ component of the wave-function.
Next, in step iii of Table <ref>, we perform a $\pi/2$ pulse between $\ket{\bf{u_A}}$ and $\ket{\bf{d_A}}$ with an off-resonant two-photon Raman transition. This transfers $\ket{\bf{u_A}}$ to $1/\sqrt{2}(\ket{\bf{u_A}} + \ket{\bf{d_A}})$ and $\ket{\bf{d_A}}$ to $1/\sqrt{2}(\ket{\bf{u_A}} - \ket{\bf{d_A}})$.
In step iv. of Table <ref> we apply a $2\pi$ pulse from $\ket{\bf{u_A}}$ to $\ket{\bf{R_u}}$. Again, since the $\alpha$ component of the state already contains a Rydberg excitation, any state transfer on these states is prohibited by Rydberg blockade. In the $\beta$ component, the state $\ket{\bf{u_A}}$ receives a $\pi$ phase shift, i.e. it acquires a negative sign.
Next, in step v. in Table <ref>, we perform another $\pi/2$ rotation between $\ket{\bf{u_A}}$ and $\ket{\bf{d_A}}$. This results in the A qubit on the $\beta$ component swapped with respect to the initial state. Finally, in step vi. of Table <ref>, the state $\ket{\bf{R_d}}$ is rotated back to $\ket{\bf{u_t}}$, which acquires a $\pi$ phase shift because it has accumulated a $2\pi$ rotation. The result is the final state in Table <ref> which has an overall $\pi$ phase shift removed. Table <ref> essentially performs a CNOT gate with the `target' qubit as the control and the A qubit as the target <cit.>
At the end of Table <ref>, we now rotate the `target' qubit using a $\pi/2$ Raman pulse. If we apply this to the result of Table <ref> and rearrange the terms, the total state of the system is now given by:
\begin{eqnarray}
\ket{\psi} = & 1/2\ket{\bf{uu}}(\alpha \ket{\bf{u}}+ \beta \ket{\bf{d}}) \nonumber \\
&+1/2\ket{\bf{du}}(\alpha \ket{\bf{u}}- \beta \ket{\bf{d}}) \nonumber \\
&+1/2\ket{\bf{ud}}(\alpha \ket{\bf{d}}+\beta \ket{\bf{u}}) \nonumber \\
&+1/2\ket{\bf{dd}}(\alpha \ket{\bf{d}}-\beta \ket{\bf{u}})
\label{eq:measure}
\end{eqnarray}
From this state it is clear that if the two qubits at A are measured, this projects the state of the B qubit onto one of the terms in Eq. (<ref>). Once the classical result of the measurement is sent from A to B, then B can perform the appropriate rotation on the state of the qubit in B to produce the initially desired target state $\alpha \ket{\bf{u}}+ \beta \ket{\bf{d}}$.
§ TWO NODE RATE ANALYSIS
To estimate the average time required for any protocol to be successful, we compute the sum of the time required for each step of the protocol, $t_i$, divided by the product of probabilities of success for each step, $p_i$:
\begin{equation}
T=\frac {\sum t_{i}}{\prod p_{i}}
\label{eq:seven}
\end{equation}
A quantum network with separated nodes has a round-trip time of light between the two nodes ($2d/c$), where $d$ is the optical distance between nodes and $c$ is the speed of light. One factor of $d/c$ accounts for the time of transmission of the flying qubit from one node to the second, and the other $d/c$ accounts for the time it takes the result of the measurement to be transmitted back to the first node as classical information. As the distance between nodes increases the total time of the protocol can become dominated by the round trip time of the light. For example, for a $\sim$10 km node separation, the time of flight, $d/c$, is 50 $\mu$s in an optical fiber. The round trip travel time of light sets an absolute maximum entanglement generation rate assuming there are no losses and that the atomic protocol is significantly faster than the light travel time. For 10 km the speed of light sets an absolute maximum rate of entanglement generation 10 kHz in an optical fiber.
However, for estimating protocol times for short distance light propagation, we must look in more detail at the time it takes to perform the atomic protocol. The transition from a ground state to a Rydberg state is repeated many times for the protocol. For this estimate we ignore factors of two for $\pi/2$ pulses or $\sqrt{N}$ for the collective enhancement and assume that all of these are given by the rate of a two-photon $\pi$-pulse determined by the two-photon Rabi frequency, $\Omega_{R}$. Since a relatively small proportion of the pulses are $\pi/2$ pulses or have an Rabi frequency enhanced by $\sqrt{N}$, this will not significantly change the result. In similar experimental arrangements to the proposed one, i.e. $n=90$ Rydberg levels excited in $^{87}$Rb, two-photon Rabi frequencies as high as 750 kHz were achieved<cit.>. In order to make calculations simple, we estimate the time of a single operation to be $t_o =1 \mu$s. A summary of the definitions of the parameters and subscripts used in Sections <ref> and <ref> is given in Table <ref>.
Interestingly, a potential source of additional time to perform the atomic protocol is the time it takes the photon to exit the cloud, which depends on the group velocity and ensemble size. The group velocity of light in the very similar experimental arrangement, i.e. an electromagnetically-induced-transparency beam configuration using Rydberg excitations in rubidium, has been shown to be in the range of 10 to 30 m/s <cit.>, which, though very slow for light, will leave the 10 $\mu$m blockade radius in 0.1 to 0.3 $\mu$s. For this estimate, we take advantage of the comparable time scales and assume the readout time is close enough to the operation time, $t_o =1 \mu$s. With the simplifications above, we estimate the total time of the protocol to generate entanglement between two nodes with the total number of operations required, $n_{G}$, and the distance between nodes:
\begin{equation}
t_G=n_G t_o + 2d/c
\label{eq:rate1}
\end{equation}
The total number of transitions to and from the Rydberg level and photon readout for the entanglement generation is $n_G=7$, as read off Table <ref>. Since the two ensembles can be prepared simultaneously, we do not need to include a factor of two in the atomic protocol time. If we assume the nodes are separated by a minimal distance, the time of the atomic protocol $n_G t_o$, sets the absolute maximum repetition rate of the experiment to be 140 kHz.
More critical for the average time estimate of Eq. <ref> are the success probabilities. The four-photon transitions through a Rydberg level and down to atomic ground states have been performed with a probability of $0.62$ <cit.>. Thus, we estimate the probability of success for a single transition to the Rydberg level to be $P_R=\sqrt{0.62}=0.79$. Given an atomic density, the photon collection efficiency is estimated by<cit.>:
\begin{equation}
\label{eq:rate}
\end{equation}
An atomic density of $n=5 \times 10^{11} cm^{-3}$ and $r_b$ of $10 \mu$m leads to $\sim 2000$ atoms within the blockade radius and an optical density (OD) per blockade radius of 3 which predicts a photon collection efficiency of $P_{\gamma} = 0.3$. This photon collection efficiency is typical in neutral atom ensemble memories without Rydberg excitations<cit.>. A photon collection efficiency 11 % has already been achieved for the first attempt of collecting photons from a collective Rydberg excitation in an ensemble<cit.> which had a comparable atom number within the blockade radius. Shelving a single Rydberg excitation into a long-lived ground state with the higher atom numbers needed for high photon collection efficiency has not yet been experimentally realized. In order to achieve reasonable fidelities, 3D optical lattices can be employed and different principal quantum numbers and detunings can be used to avoid de-phasing resonances. In addition, there is no fundamental reason that the collection efficiency from a memory produced via Rydberg excitations would be smaller than from other neutral atom memories. Finally, we have a probability of obtaining a useful Bell state from the Bell state analyzer of $P_{B}=1/2$.
Summary of parameters and subscripts used in Sections <ref> and <ref>
Parameter Description
$n$ Number of nodes
$n_G, n_T, n_S$ Number of steps in a protocol
$t_o, t_G, t_T$ Time for an single protocol
$P_\gamma, P_R,P_B,P_G,P_T$ Probability of success
$T_G,T_T$ Average time for a single protocol
$T_G[n],T_T[n],T_{T'}[n]$ Average time for an n step protocol
Subscript Description
$G$ Entanglement generation
$S$ Extending entanglement
$T$ Teleportation protocol
$T'$ Alternative protocol (see Fig. <ref>)
$o$ Single operation
$\gamma$ Photon collection
$R$ Rydberg excitation efficiency
$B$ Bell measurement
Thus, we estimate the average time to successfully generate remote entanglement between two nodes as:
\begin{equation}
T_G=\frac{(n_G-1) t_o + 2d/c}{(P_{R}^{n_{G}}P_{\gamma}^{2})^{2}P_{B}}=\frac{t_G}{P_G}
\label{eq:rate2}
\end{equation}
The factor $(P_{R}^{n_{G}}P_{\gamma}^{2})$ in the denominator is squared because of the fact that both ensembles must produce a flying qubit entangled with the memory. There is one Bell-state measurement. The factor of $(n_G-1)$ appears in the numerator because we have assumed simultaneous readout, whereas the denominator contains $n_G$ because simultaneous readout doesn't change the probability of successfully performing the operations. We label the denominator, $P_G$, as the probability of successfully generating entanglement and the numerator, $t_G$ as the total time for the atomic protocol and light travel time.
We use values that have been observed in experiments as identified above, $P_R =0.79$, $P_\gamma=0.3$, $t_o=1$ $\mu$s, and $P_B$. We also set $d=0$, which allows us to compare with other teleportation protocols over small distances. We calculate the average rate of entanglement generation using Eq. <ref> to be $1/T_G=25 $Hz. This is the same order of magnitude as the highest rates between matter qubits currently reported in ion entanglement at $\sim 5$ Hz<cit.>.
Similarly, we can estimate the total time it takes to successfully perform teleportation by using Eq. <ref>. We use entanglement generation as the first step in the teleportation protocol. Since the entanglement generation is heralded, if we use the average time to generate entanglement $T_G$ given in Eq. <ref>, we do not need to include the probability $P_G$ in the estimate for average teleportation time, as it is included in the estimate of $T_G$.
Thus, the estimate for the average time to successfully teleport a quantum state between two remote nodes is:
\begin{equation}
T_T=\frac{T_G+n_{T}t_o+ 2d/c}{P_{R}^{n_{T}}}=\frac{T_G+t_T}{P_T}
\label{eq:rate3}
\end{equation}
Where $n_T$ is the total number of operations used for the teleportation protocol, i.e. the steps in Table <ref>, as well as a pulse to project the final state onto the desired state given the result of the Bell-state measurement. For simplicity in this estimate we take $n_T \simeq n_G=7$. The time of the teleportation protocol is defined as $t_T=n_{T}t_o+ 2d/c$. The final state measurement can in principle be done with near unit efficiency with field selective ionization <cit.> and is not considered for this estimate. The total probability for the steps used in just the teleportation protocol, $P_T$ thus does not include any photon collection, which again, was included in the estimate of $T_G$.
Using the same values as above for the efficiency parameters, we estimate the rate of successful teleportation events predicted by Eq. <ref> to be $\sim$ 5 Hz. A $\sim$Hz rate is on the same order as the rate of teleportation achieved with a single-atom coupled to a high-finesse optical cavity <cit.> whereas typical realizations of teleportation between matter qubits have had rates on the order of one every few minutes<cit.>.
To improve the rate of entanglement generation, the transition efficiency to the collective states and the photon collection efficiency need to be improved. To see the effect of higher efficiencies on the protocol rates, we estimate the efficiencies that might be achieved with improved technology. One potential for improving the Rydberg transition probability is to use techniques such as the pulse shaping developed by Beterov et al.<cit.>. In addition, the atoms could instead be loaded into a three-dimensional optical lattice in order to fix the separation between pairs of atoms and eliminate their motion. For this estimate, we use an improved Rydberg transition efficiency of $P_R = 0.9$ which is consistent with the theoretical prediction in Ref. <cit.> for higher atom numbers. To dramatically improve the photon collection efficiency, the ensemble can be coupled to a high finesse optical cavity. Photon collection efficiency as high as 0.84 has been achieved in non-Rydberg ensemble based systems <cit.>. For this estimate we use a photon collection efficiency of $P_{\gamma}=0.80$. If these efficiencies can be achieved, these improved parameters would predict average rates of success for entanglement generation and teleportation given by Eq. <ref> and Eq. <ref> to be 7.8 kHz and 3.6 kHz respectively, representing a significant improvement over the current state of the art.
a) Plot of the rate of the entanglement generation protocol as a function of the number of steps in the protocol, $n_G$, for two sets of experimental parameters. The blue dashed line uses the conservativel efficiency estimates of $P_R = 0.79$ and $P_{\gamma}=0.3$ and the red solid line is the prediction using the optimistic efficiencies of $P_R = 0.9$ and $P_{\gamma}=0.8$. b) Plot of the rate of the entanglement generation protocol (dashed blue line) and teleportation protocol (solid red line) as functions of $P_R$ for a fixed $n_G = 7$ and $P_{\gamma}=0.8$.
The rate of entanglement generation, i.e. Eq. <ref>, as a function of $n_G$ is plotted for the two different parameter sets mentioned in Fig. <ref> (a). The rate of entanglement generation and teleportation for two-node protocols are compared using the optimistic photon collection efficiency $P_\gamma =0.8$, as a function of the Rydberg transition efficiency in Fig. <ref> (b).
To compare to the rate of the Zhao et al. protocol <cit.>, we note that for two nodes, the only difference is in the preparation of the flying qubit entangled with the quantum memory. In our case, $n_G =7$ whereas in Zhao et al., $n_G = 12$. As can be seen in Fig. <ref>, this results in our protocol having a factor of $~\sim 20$ higher rate of entanglement generation than the Zhao et al. protocol for the initial efficiency estimate $P_R = 0.79$, and a factor of 5 higher rate for the optimistic efficiency estimate $P_R = 0.9$.
§ MANY NODE RATE ANALYSIS
Next, we analyze the rates of these protocols in our model when extended to many equidistant nodes. Since the successful entanglement of two nodes is heralded by the detection of photons as described in Section <ref>, the average time it takes to entangle two nodes can be used with a unity probability of success because we assume the entanglement of two nodes is successful every time the detection of the two photon state is heralded. The total average time to produce entanglement in the $n_{th}$ step, where $n$ is defined as one less than the number of nodes (because entanglement generation and teleportation are not defined for less than two nodes), is given by:
\begin{eqnarray}
T_{G}[n]=\frac{T_{G}[n-1] + n_{S}t_o+ 2d/c}{(P_{R}^{n_{S}}P_{\gamma}^{2})^2P_{B}} \nonumber \\
=\frac{T_{G}[n-1]+ n_{S}t_o+ 2d/c}{P_{S}}
\label{eq:rate4}
\end{eqnarray}
Where $n_S$ is the number of additional atomic transitions required to prepare a subsequent flying qubit entangled with the quantum memory for the photonic entanglement swapping step described in Section <ref>. In general $n_S \neq n_G$, as the number of steps to read out a memory that is already created is less than the number of steps required to produce a memory and read it out, but for the sake of simplicity, we will assume that $n_S \simeq n_G$ so that the total probability of successfully extending the entanglement is equal to the probability of generating entanglement between two nodes, i.e. $P_S=P_G$, which should be a good estimate for our purposes.
If we set $P_R=1$ and use the average time to generate entanglement as the time for the first step, i.e. $T_{G}[1]=T_{G}$, we recover the logical solution of $T_G[n]=n T_G$. However, if instead we make the simplifying assumptions that the number of nodes is large and the probability of generating entanglement on a single shot is low, i.e. n $\gg 1$ and $P_G < 1$, then the solution is given by:
\begin{equation}
\label{eq:rate5}
\end{equation}
Because the entanglement swapping is not deterministic in this model, the protocols in Zhao et al. <cit.> and Han et al. <cit.>, which do have deterministic entanglement swapping, will outperform this one by a factor $ \mathcal{O}(P_G^n)$.
The time for entanglement generation with deterministic entanglement swapping can be calculated to be <cit.>:
\begin{equation}
\label{eq:rate5.5}
\end{equation}
Where k level of entanglement purification nesting such that $n = 2^k$, as before, $t_G$ is the time for the entanglement generation protocol, and $P_G$ is the probability of a successful entanglement event. Even for three nodes Eq. <ref> predicts an estimated improvement in the entanglement generation by a factor of 400 over Eq. <ref> for the initial efficiency estimates of $P_R = 0.79$ and $P_\gamma = 0.3$. For the improved efficiency estimates, $P_R = 0.90$ and $P_\gamma = 0.8$, the improvement using deterministic entanglement swapping is a factor of 100.
A dashed line represents establishing entanglement between two nodes and a double line represents teleporting a state between two nodes. a) A multi-node teleportation scheme (T) in which the entanglement is first generated between nodes and then the state is teleported. This followed by subsequently entangling nodes and teleporting the state until the final node is reached. b) A multi-node teleportation scheme (T') in which the entanglement is distributed from the first node to the last node by successive entangling operations. After the first and final node are entangled, the state is teleported.
Next we analyze the rate of multi-node teleportation in the two cases shown in Fig. <ref>. In the first case entanglement is generated between the first and second node. Then teleportation is performed on the target state to transfer it from the first to the second node. This is followed by subsequently entangling and teleporting the state down the chain until the target state is teleported to the final node, Fig. <ref>(a). The rate can be calculated by assuming we have successfully teleported the state from the first node to the $n-1$ node. Then we solve for the intermediate step of generating the shared Bell-pair between the $n-1$ and $n_{th}$ nodes with a finite probability $P_G$:
\begin{equation}
\widetilde{T_{T}}[n]=\frac{T_{T}[n-1]+t_G}{P_G}
\label{eq:rate6}
\end{equation}
Once this is successfully completed, we perform the teleportation protocol given in Table <ref>:
\begin{eqnarray}
T_{T}[n]=\frac{\widetilde{T_{T}}[n]+t_T}{P_T} \nonumber \\
\label{eq:rate7}
\end{eqnarray}
To solve this, we make the simplifying assumptions that the number of steps is very large, n $\gg 1$ and $P_T,P_G < 1$ so that $(P_TP_G)^n \ll 1$ and that the time it takes for teleportation from the first to second node is given by $T_T$ from Eq. <ref>. The solution for large $n$ simplifies to:
\begin{equation}
\label{eq:rate8}
\end{equation}
Next, we want to analyze the rate of the teleportation scheme shown in Fig. <ref>(b). For this scheme, the entanglement is generated from one node to the last node, followed by a single teleportation step. The time it takes to do this is given by Eq. <ref>. This is followed by a single teleportation step. If we take $T_G[n]$ as the first step, and assume that the time for the teleportation protocol is negligible compared to the time of the entanglement generation of all of the nodes, we simply have $T_{T'}[n]=T_G[n]/P_T$ or:
\begin{equation}
\label{eq:rate9}
\end{equation}
This is significantly faster than the rate of Protocol $T_T$, i.e. Eq. <ref>, by a factor of $(P_R^{n_T})^n$, and could in fact be improved with a nested entanglement quantum repeater protocol, which is not possible with protocol $T_T$.
If we use the optimistic efficiency estimates, i.e. $P_R = 0.9$ and $P_\gamma = 0.8$, then Eq. <ref> and Eq. <ref> predict an average time to teleport a state between three nodes, i.e. from node A to node C, of 145 ms for protocol $T_T$ and 34 ms for protocol $T_{T'}$. Protocol $T_{T'}$ predicts an average time of 90 s to teleport a state to the 6th node where protocol $T_{T}$ predicts an average time of around one hour.
However, protocol $T_{T'}$, Fig. <ref>(b) requires that the memory lifetime of the first node to be long enough for the entire protocol to be successful, whereas protocol $T_T$, Fig. <ref>(a) only requires a memory time long enough to teleport a state between neighboring nodes. In addition, protocol $T_{T}$ is more resource intensive, requiring the `target' qubit pair used in the teleportation protocol at each node, while the $ T_{T'}$ protocol only requires the `target' pair at the initial node.
§ CONCLUSIONS
Using a multi-mode Rydberg excitation scheme in an atomic ensemble, teleportation between long-lived memory states can achieve high rates. This system has also been shown to be compatible with a large-scale quantum network architecture. We have examined the performance of a quantum repeater node based on cold-atom ensembles with Rydberg excitations and theoretically described a teleportation protocol. We analyzed the rates of two-node entanglement generation and teleportation and found that the teleportation rates achievable on realistic systems could approach the kHz level, two orders of magnitude improvement over the current highest achieved rate. This two-node performance can be used as a metric and benchmark for incorporating Rydberg-based cold-atom ensemble quantum repeater nodes into a larger scale network. We also analyzed a model for many node protocols.
It could be possible to spatially multiplex a cold atom ensemble node by addressing several Rydberg radii of atoms along the length of the optical dipole trap or by multi-site trapping of an atomic ensemble on a chip with individual site addressing technology <cit.>. The technology for coherent control of Rydberg atoms on a chip, though challenging, is currently being pursued by several groups <cit.> where it might be possible to realize multi-Rydberg atom trapping on atom chips. These types of multi-plexing could realize larger quantum registers at each node and would enable temporal multiplexing to increase data transmission rates.
The presence of a single Rydberg excitation produces a large non-linearity in the ensemble which can affect other atoms in the ensemble or photons entering the ensemble. This can lead to non-linear effects at the single photon level such as single photon EIT<cit.>, single photon switches<cit.>, single photon transistors<cit.>, and effective photon-photon interactions<cit.> which have all been experimentally demonstrated. Rydberg excitations in cold atom ensembles promise a rich and viable path towards interesting applications in quantum communication and information.
Because of the potential high rates of entanglement generation and the potential for scalability, quantum repeaters based on neutral atom ensembles with Rydberg excitations are a promising route towards long-distance quantum network.
§ ACKNOWLEDGMENTS
We would like to thank Steve Rolston and Alexey Gorshkov for helpful discussions. NS is an Oak Ridge Associated Universities (ORAU) postdoctoral fellow.
Research was sponsored by the Army Research Laboratory. The views and
conclusions contained in this document are those of the Authors and should
not be interpreted as representing the official policies, either expressed
or implied, of the Army Research Laboratory or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation herein.
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1511.00007
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Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
37.30.+i, 42.50.-p, 05.30.Rt, 11.30.Qc, 67.85.Hj, 05.65.+b
Insights into complex phenomena in quantum matter can be gained from simulation experiments with ultracold atoms, especially in cases where theoretical characterization is challenging. However, these experiments are mostly limited to short-range collisional interactions. Recently observed perturbative effects of long-range interactions were too weak to reach novel quantum phases <cit.>.
Here we experimentally realize a bosonic lattice model with competing short- and infinite-range interactions, and observe the appearance of four distinct phases - a superfluid, a supersolid, a Mott insulator and a charge density wave. Our system is based on an atomic quantum gas trapped in an optical lattice inside a high finesse optical cavity. The strength of the short-ranged on-site interactions is controlled by means of the optical lattice depth. The infinite-range interaction potential is mediated by a vacuum mode of the cavity <cit.> and is independently controlled by tuning the cavity resonance. When probing the phase transition between the Mott insulator and the charge density wave in real-time, we discovered a behaviour characteristic of a first order phase transition. Our measurements have accessed a regime for quantum simulation of many-body systems where the physics is determined by the intricate competition between two different types of interactions and the zero point motion of the particles.
Experiments with cold atoms have contributed in many ways to elucidate fundamental behaviour of quantum matter <cit.>. An example is the realization of the Bose-Hubbard model, where the balance between the kinetic energy of particles moving in an optical lattice and the on-site collisional interactions drives a quantum phase transition from a superfluid to a Mott insulating phase <cit.>. Whilst collisions between atoms are naturally present in quantum gases and give rise to short-range interactions <cit.>, longer ranged interactions are more elusive. In order to get a handle on the latter, ultracold gases of particles with large magnetic or electric dipole moments <cit.>, atoms in Rydberg states <cit.>, or cavity-mediated interactions <cit.> have been studied. Indeed, already Hubbard models with additional nearest-neighbour interactions are predicted to show intriguing phases like charge and spin density waves, supersolids, topological phases or checkerboard and stripe phases <cit.>.
Illustrations of the experimental scheme realising a lattice model with on-site and infinite-range interactions.
A stack of 2D systems along the $y$-axis is loaded into a 2D optical lattice (red arrows). The cavity induces atom-atom interactions of infinite range. Illustration of the competing energy scales: tunnelling $t$, on-site interactions $U_{\mathrm{s}}$ and long-range interactions $U_{\mathrm{l}}$.
In our experiment, we achieve independent control over three energy scales by combining an optical lattice with cavity-mediated interactions, see Fig. 1. The underlying static lattices along all three directions are necessary to study the direct competition between short- and long-range interactions, as compared to the situation very recently investigated in <cit.>.
Aspects of this scenario, in which on-site interactions compete with infinite-range interactions, have been theoretically studied in the context of self-consistent extended Hubbard models and various phases have been predicted <cit.>.
The starting point is a Bose-Einstein condensate (BEC) of $4.2(4) \times 10^4$ $^{87}$Rb atoms which is prepared inside the ultrahigh-finesse optical cavity.
The optical lattice is formed by three mutually orthogonal standing waves. The lattice along the $y$-axis at wavelength $\lambda_y = 670\, \unit{nm}$ splits the BEC into a stack of about 60 weakly coupled two-dimensional (2D) layers. These 2D layers are then exposed to a square lattice in the $xz$-plane formed by one free space lattice and one intracavity optical standing wave, both at a wavelength of $\lambda = 785.3\, \unit{nm}$. They create periodic optical potentials of equal depths $V_{\mathrm{2D}}$ along both directions, which we will specify in units of the recoil energy $E_{\mathrm{R}}= h^2/2m\lambda^2$, where $m$ denotes the mass of $^{87}$Rb. In addition to the lattice potential, the atoms are exposed to an overall harmonic confinement, which results in a maximum density of 2.8 atoms per lattice site at the center of the trap. The standing wave along the $z$-axis fulfills a second role as it controls long-range interactions via off-resonant scattering into the optical resonator mode.
The photons are scattered off the trapped atoms and are delocalized within the cavity mode thereby mediating atom-atom interactions of infinite range (see Methods).
These infinite-range interactions create $\lambda$-periodic atomic density-density correlations on the underlying $\lambda/2$-periodic square lattice <cit.>. The correlations can lead to the breaking of a $\mathbb{Z}_2$-symmetry between the two checkerboard sublattices <cit.>, defined by either even or odd sites, resulting in the appearance of a self-consistent optical potential with alternating strength.
In a wide range of the parameter space, the system can be described by a lattice model with long-range interactions (see Methods and Extended Data Fig. 1), given by:
\begin{eqnarray}
\begin{split}
\hat{H} = - t \sum_{\langle e,o \rangle}\left(\hat{b}^{\dagger}_{e}\hat{b}_{o} + \mathrm{h}.\mathrm{c}.\right) + \frac{U_{\mathrm{s}}}{2} \sum_{i\in e,o} \hat{n}_{i}(\hat{n}_{i}-1) \\
- \frac{U_{\mathrm{l}}}{K}\left(\sum_{e}{\hat{n}_{e}} - \sum_{o}{\hat{n}_{o}}\right)^{2} - \sum_{i\in e,o}{\mu_i\hat{n}_{i}}.
\end{split}
\label{eq:BHHam}
\end{eqnarray}
Here $e$ and $o$ denote all even and odd lattice sites respectively, $\hat{b}_i$ $(\hat{b}^{\dagger}_i)$ are the bosonic annihilation (creation) operators at site $i$, $\hat{n}_i$ counts the number of atoms on site $i$ and $K$ is the total number of sites. The first term represents the tunnelling between neighbouring sites at rate $t$ and favours delocalization of the atoms within a 2D layer, supporting superfluidity. The second term describes the on-site interaction with strength $U_{\mathrm{s}}$ controlled via $V_{\mathrm{2D}}$. Its energy is minimized if the atomic wavefunctions are localized on individual lattice sites, with balanced populations on even and odd sites and vanishing spatial coherence. The infinite-range interactions are captured by the third term and favour, for positive $U_{\mathrm{l}}$, a particle imbalance between even and odd sites. This global atom-atom interaction strength $U_{\mathrm{l}}$ is proportional to $V_{\mathrm{2D}}$ and inversely proportional to the detuning $\Delta_{\mathrm{c}} = \omega_z - \omega_{\mathrm{c}}$, where $\omega_z$ is the frequency of the $z$-lattice beam and $\omega_{\mathrm{c}}$ is the cavity resonance frequency (see Methods). The last term represents the effective chemical potential $\mu_i = \mu - \epsilon_{i}$, incorporating the chemical potential $\mu$ and the external trapping potential $\epsilon_{i}$ on lattice site $i$. In the absence of long-range interactions, equation (1) reduces to the Bose-Hubbard model.
To explore the phase diagram of $\hat{H}$, the lattices along the $x$ and $z$-direction are simultaneously ramped up to a certain value $V_{\mathrm{2D}}$, keeping the total ramp time constant. This procedure is repeated for different relative strength of short- and long-range interactions ($U_{\mathrm{l}}$/$U_{\mathrm{s}}$), controlled via the detuning $\Delta_{\mathrm{c}}$.
In order to detect a superfluid-insulator phase transition, we probe the spatial coherence of the gas by turning off all confining potentials and taking absorption images of the atomic cloud after ballistic expansion.
Fig. 2a shows measured projected momentum distributions for four different $V_{\mathrm{2D}}$, together with extracted vertical line sums.
For small lattice depth $V_{\mathrm{2D}}$, spatial coherence can be observed, characterized by a narrow momentum distribution of the cloud and a large BEC fraction $f$, extracted from a bimodal fit to the distribution.
When increasing $V_{\mathrm{2D}}$, the momentum distributions broaden, indicating a drop of coherence and $f$ reduces.
We observe a kink in $f$ as a function of the interaction strength $U_{\mathrm{s}}/t$ (for details, see Methods and Extended Data Fig. 5), which we associate with the formation of an insulating phase in the cloud and a loss of superfluidity <cit.>. The extracted transition points are shown as white points in Fig. 2b. We confirmed that coherence between different lattice sites is restored when ramping down the 2D lattice potential again.
Characterization of the phases via spatial coherence (a, b) and via even-odd imbalance (c, d).
a, Absorption images in the xz-plane (upper panels) and the same signal integrated along the cavity axis (lower panels, red), taken after a ballistic expansion for lattice depths $V_{\mathrm{2D}}$ of $2 \,E_{\mathrm{R}}$ (I), $6.5 \,E_{\mathrm{R}}$ (II), $11\,E_{\mathrm{R}}$ (III) and $16 \,E_{\mathrm{R}}$ (IV) at $\Delta_{\mathrm{c}} /2 \pi =-22\, \unit{MHz}$. Black lines show fits with a bimodal distribution including higher momentum peaks. Due to the cavity mirrors, the field of view along the $x$-direction is restricted.
b, Extracted BEC fraction $f$ as a function of $V_{\mathrm{2D}}$ and $\Delta_{\mathrm{c}}$. White points mark the transition from a superfluid to an insulating phase and are obtained from a piecewise linear fit to the BEC fraction (see Extended Data Fig. 5). Error bars indicate fit uncertainties (see Methods)
c, Scattered photons $n_{\mathrm{ph}}$ of single repetitions as a function of $V_{\mathrm{2D}}$ for pump-cavity detunings $\Delta_{\mathrm{c}} /2 \pi$ of $-12\,\unit{MHz}$ (i), $-22\,\unit{MHz}$ (ii) and $-32\,\unit{MHz}$ (iii).
d, Imbalance $\Theta$ mapped as a function of $\Delta_{\mathrm{c}}$ and $V_{\mathrm{2D}}$. We assign the onset of a scattered cavity light field (black points) to the formation of a phase with even-odd imbalance. In the region indicated by the three dotted lines at values $\Delta_{\mathrm{c}}/2 \pi=\{-47, -49.5, -52\} \,\unit{MHz}$, the onset of the cavity light field showed a large variation. Error bars indicate the s.d. of the fit, an additional systematic error of $0.2 \,E_{\mathrm{R}}$ stems from the data analysis. The detection background is growing with decreasing $V_{\mathrm{2D}}$ and increasing detuning from cavity resonance (see Methods). Grey areas were not recorded.
An even-odd imbalance causes a $\lambda$-periodic density modulation that acts as a Bragg grating, off which photons from the $z$-lattice beam are scattered into the cavity mode and vice versa. The amplitude of the scattered light field adiabatically follows the atomic density distribution <cit.> and is continuously monitored using a heterodyne detection (see Methods).
Fig. 2c displays mean intracavity photon numbers $n_{\mathrm{ph}}$ measured as a function of $V_{\mathrm{2D}}$. The onset of a cavity field is clearly visible and is taken as the transition point to a phase with even-odd imbalance $\Theta$, marked with black points in Fig. 2d (for details, see Methods and Extended Data Fig. 5). The imbalance $\Theta$ can be quantified using (see Methods):
\begin{eqnarray}
\Theta = \left|\frac{\sum_{e}{\left\langle\hat{n}_{e}\right\rangle} - \sum_{o}{\left\langle\hat{n}_{o}\right\rangle}}{\sum_{e}{\left\langle\hat{n}_{e}\right\rangle} + \sum_{o}{\left\langle\hat{n}_{o}\right\rangle}}\right|\approx \frac{1}{N}\sqrt{n_{\mathrm{ph}}\, \frac{\Delta_{\mathrm{c}}^2}{\eta^2 }}.
\label{eq:Theta}
\end{eqnarray}
Here, $\eta$ is the two-photon Rabi frequency of the scattering process and $N$ is the total atom number.
Phase diagram.
The four phases are indicated by different colors: SF (red), SS (violet), CDW (blue), MI (yellow). Simplified density distributions are schematically illustrated for the homogeneous case with, on average, one atom per site. Data points (from Fig. 2b,d) show the experimentally obtained phase transition points recorded for increasing $V_{\mathrm{2D}}$: Black data points indicate the onset of an even-odd imbalance, white data points depict where spatial coherence is lost. Increasing the 2D lattice depth $V_{\mathrm{2D}}$ simultaneously increases short- and long-range interactions. The detuning $\Delta_{\mathrm{c}}$ changes only the strength of the long-range interactions. The slanted lines indicate the region where CDW and MI may coexist. At detuning $\Delta_{\mathrm{c}} / 2 \pi= +8\, \unit{MHz}$, $U_{\mathrm{l}}$ becomes negative and favours zero imbalance, thus only SF and MI phases appear. No data was taken at detunings indicated by the grey bar. A version of the phase diagram in Hamiltonian parameters is shown in Ext. Data Fig. 2.
To establish a phase diagram, we combine all determined transition points in Fig. 3.
We identify four phases that arise from the competition of the three energy scales: a superfluid (SF), a supersolid (SS), a Mott insulator (MI) and a charge density wave (CDW) phase.
Far away from cavity resonance, i.e. $\Delta_{\mathrm{c}}/2 \pi\lesssim -52\,\unit{MHz}$, $U_{\mathrm{l}}$ becomes small and the system undergoes, for large enough $V_{\mathrm{2D}}$, a transition from a superfluid to a Mott insulating phase. The latter is characterized by a loss of coherence, as well as the absence of an even-odd imbalance.
The observed SF to MI transition line is shifted to larger values of $V_{\mathrm{2D}}$ than theoretically expected for a homogeneous system <cit.>, which we attribute to the harmonic confinement of our 2D systems <cit.>.
Approaching cavity resonance increases $U_{\mathrm{l}}$. Above $\Delta_{\mathrm{c}}/2 \pi\approx -52\,\unit{MHz}$, this leads to the formation of a structured phase with even-odd imbalance, heralded by the onset of a light field scattered into the cavity.
Depending on the relative strength of tunnelling and short-range interactions, the structured phase can either be a SS, where superfluidity is supported, or a CDW phase, where spatial coherence is lost. The identification of the SS phase is further supported by the observation of additional interference peaks corresponding to a $\lambda$-periodic density modulation (see Extended Data Fig. 4) <cit.>.
The SF to SS phase boundary shifts to smaller $V_{\mathrm{2D}}$ when approaching the cavity resonance <cit.>. The transition line from a SS to a CDW follows the same trend. We attribute the loss of coherence in the CDW phase to a reduced nearest-neighbour tunnelling, which has its origin in an energy offset between even and odd lattice sites. This energy offset is a result of the optical potential created by the interference of the field scattered into the cavity with the field of the $z$-lattice, and is shown in Extended Data Fig. 6. Our experimental resolution currently does not allow us to assess the precise topology of the multicritical region.
For long-range interactions dominating over tunnelling and short-range interactions, we observe a maximum even-odd imbalance $\Theta$ above $0.9$, implying that mostly even or odd sites are occupied <cit.>. This imbalance is significantly lower between $-32\, \unit{MHz} \gtrsim \Delta_{\mathrm{c}}/2 \pi\gtrsim - 52\,\unit{MHz}$, see Fig. 2d. A possible explanation is the coexistence of CDW and MI phases <cit.>, which is supported by the external trapping potential, making it energetically costly for the system to place atoms away from the trap center. Contrary, a homogeneous system with non-integer filling would turn into a structured phase with even-odd imbalance for any finite long-range interaction (see Methods and Extended Data Fig. 3). Since the particles in the Mott insulating regions do not scatter into the cavity, the cavity field will rapidly vanish when the size of these regions increases. We conclude from the signal-to-noise ratio of our detection that below $\Delta_{\mathrm{c}}/2\pi = -52\,\unit{MHz}$ the technical noise does not allow us to detect an even-odd imbalance below $0.01$.
We now study the evolution between the predominantly insulating CDW and MI phases.
We initialize the system in the insulating region ($V_{\mathrm{2D}} = 14\, E_{\mathrm{R}}$) at a certain detuning $\Delta_{\mathrm{c}}$ and then continuously vary $U_{\mathrm{l}}$ by changing $\Delta_{\mathrm{c}}$, before returning to the initial value of $\Delta_{\mathrm{c}}$ (see Methods).
The cavity output field tracks the instantaneous even-odd imbalance $\Theta$ in real-time.
Fig. 4a shows the evolution of the imbalance when decreasing $\Delta_{\mathrm{c}}$ from an initial value in the CDW phase. The data shows a hysteretic behaviour with a lower imbalance on return. The imbalance evolution for a starting value of $\Delta_{\mathrm{c}}$ in the transition region between CDW and MI shows a similar behaviour when decreasing $\Delta_{\mathrm{c}}$ and the opposite behaviour when increasing $\Delta_{\mathrm{c}}$. In Fig. 4c, where we started in the MI phase, the imbalance remains low throughout the measurement. We measured the hysteretic behaviour to be insensitive to the ramp speed, see Extended Data Fig. 7.
Hysteretic behaviour of the CDW to MI transition.
Imbalance $\Theta$ recorded by varying $\Delta_{\mathrm{c}}$ at a rate of $0.67 \,\unit{MHz/ms}$, for fixed $V_{\mathrm{2D}} = 14\, E_{\mathrm{R}}$. The initial detunings $\Delta_{\mathrm{c}}/2 \pi$, indicated by stars, are $-32 \,\unit{MHz}$ (a), $-42 \,\unit{MHz}$ (b) and $-52\, \unit{MHz}$ (c). Arrows signify the ramp directions; dashed lines show the return to the starting point. Curves are rescaled to take atom loss into account and contain three to nine averages, binned at $400\,\unit{\mu s} $ (see Methods).
This hysteretic behaviour of the system points towards a first order phase transition between CDW and MI. When starting in a MI and increasing $U_{\mathrm{l}}/U_{\mathrm{s}}$ beyond a certain point, the CDW will become energetically favourable, but cannot be reached due to an energy barrier between the two phases. Further increasing $U_{\mathrm{l}}/U_{\mathrm{s}}$ lowers this energy barrier until the system is driven out of the metastable state. We suggest that this is activated in our inhomogeneous system by $\lambda$-periodic density-density correlations that are created in residual compressible regions, or superfluid shells, acting as impurities. In the opposite direction, moving from a CDW to a MI phase, the energy offset between even and odd lattice sites stabilizes the CDW phase beyond the point where the MI gets energetically favourable, which results in the observed hysteretic behaviour.
§ METHODS
§.§ Preparation of a BEC in 2D layers.
We produce a Bose-Einstein condensate (BEC) of $4.2(4) \times 10^4$ $^{87}$Rb atoms at a temperature of $42(2) \,\unit{nK}$ in the $|F,m_F\rangle = |1,-1\rangle$ hyperfine state where $F$ and $m_{F}$ are the total angular momentum and the corresponding magnetic quantum number. The quantization axis is defined by a magnetic field pointing along the $z$-direction.
The BEC is confined to the center of a TEM$_{00}$ mode of the cavity by an optical dipole trap at a wavelength of $852 \,\unit{nm}$, with trap frequencies of $\omega_{x,y,z}/2 \pi = \left[70.6(3),\,31.4(5),\,29.4(2)\right] \,\unit{Hz}$. Further details on the cavity setup can be found in <cit.>.
The trapped BEC is loaded into a blue-detuned optical lattice of wavelength $\lambda_y=670 \,\unit{nm}$ oriented along the $y$-direction. This is done by implementing a smooth amplitude ramp (S-ramp) in time $t$ which is of the form: $V(t) = V_0\left[3\left(t/t_0\right)^2 - 2\left(t/t_0\right)^3 \right]$, where $V_0$ is the final lattice depth and $t_0$ is the total duration of the ramp. The lattice depth is increased to a final value of $24.9(1)\,E^{670}_{\mathrm{R}}$ in $100 \,\unit{ms}$, where ${E^{670}_{\mathrm{R}}}={{h}^2}/{2{m}{\lambda^2_y}}$ is the atomic recoil energy with $m$ being the mass of a $^{87}$Rb atom. The trap frequencies are kept constant during the loading by increasing the dipole trap depth simultaneously with the blue-detuned lattice. In this way, the whole BEC is cut into roughly 60 2D layers with about 1300 atoms in the central layer.
§.§ Loading into the square lattice.
After the preparation of 2D layers, the BEC is exposed to a 2D optical lattice in the $xz$-plane at a wavelength of $\lambda=785.3\,\unit{nm}$. The lattice along the $z$-direction is formed by a free space retro-reflected standing wave laser field which is linearly polarized along the $y$-direction. The lattice along the $x$-direction is created by pumping the TEM$_{00}$ mode of the cavity with linear polarization along the $z$-direction. The effect of interference between the $x$- and $z$-lattices on atoms is minimized by introducing a frequency offset of at least $5\, \unit{MHz}$ between the two laser frequencies. Both lattices are ramped simultaneously within a fixed time of 50 $\unit{ms}$ to a variable lattice depth $V_{\rm{2D}}$ using again the S-ramp. The lattice potential seen by the atoms is of the form: $V(x,z) = V_{\mathrm{2D}}\left[\cos^2(kx) + \cos^2(kz)\right]$, where $k=2 \pi /\lambda$ is the wave number and $V_{\mathrm{2D}}$ is the depth of the lattice in units of the corresponding recoil energy $E_{\mathrm{R}}=h^2/2 m\lambda^2$.
§.§ Characterization of the optical lattices.
The lattice depths along the $y$- and $z$-direction are calibrated using Raman-Nath diffraction <cit.>, whereas the lattice depth along the $x$-direction is calibrated using amplitude modulation spectroscopy between the lowest and the first two excited Bloch bands <cit.>. We estimate the calibration uncertainties on all lattice depths to be smaller than $4\%$. The uncertainty in the intracavity optical lattice depth is enlarged to about $10\%$ by shifts of the cavity resonance frequency due to atomic redistribution during the $V_{\rm{2D}}$ ramp and residual drifts of the incoupling to the resonator.
The heating effect of the near-resonant $xz$-optical lattices on the BEC is characterized by ramping back down the lattices after reaching the insulating regime. We recover a BEC fraction larger than 0.45 and observe an atom loss of 5$-$10$\%$. Loading of lattices in the $xz$-plane also increases the overall confinement. The trap frequencies are $\omega'_{x,z}/2 \pi = \left[170,\,165\right]\,\unit{Hz}$ at a typical lattice depth of $10\,E_{\mathrm{R}}$.
§.§ Lattice model with long-range interactions.
The single-particle Hamiltonian $\hat{H}_{\rm{sp}}$, describing the dynamics of an atom strongly coupled to a single cavity mode and moving in a 2D layer in the presence of static optical lattices, is given as <cit.>:
\begin{eqnarray}
\begin{split}
\hat{H}_{\rm{sp}} &= \hat{H}_0 + {V}_{\mathrm{Trap}}(x,z) + \hbar \eta (\hat{a}^{\dagger} + \hat{a}) \cos (kx)\cos (kz) \\
& - \hbar \left(\Delta _{\mathrm{c}} - {U}_{0} \cos ^{2}(kx)\right)\hat{a}^{\dagger}\hat{a}.
\end{split}
\end{eqnarray}
$\hat{H}_0$ consists of the kinetic energy of the particle and the potential seen due to the optical lattices in the $xz$-plane:
\begin{equation}
\hat{H}_0 = \frac{\hat{p}^{2}_x}{2m} + \frac{\hat{p}^{2}_z}{2m} + V_{\mathrm{2D}} \left(\cos ^{2} (kx) + \cos ^{2} (kz) \right).
\end{equation}
$V_{\rm{Trap}}(\textit{x},\textit{z})$ incorporates the inhomogeneous confining potential seen by the atoms. $\hat{a}$ ($\hat{a}^{\dagger}$) annihilates (creates) a photon in the cavity mode. Scattering of the light field from the $z$-lattice into the cavity mode at a two-photon Rabi frequency $\eta$ creates a self-consistent checkerboard lattice for the atoms and is represented by the third term in $\hat{H}_{\rm{sp}}$. This term describes how the atomic motion self-consistently determines the occupation of the cavity field mode inducing infinite-range interactions between the atoms. The last term in $\hat{H}_{\rm{sp}}$ represents the cavity field in the rotating frame of the $z$-lattice with $\Delta_{\mathrm{c}} = \omega_z - \omega_{\mathrm{c}}$. The effect of the dispersive shift of the cavity resonance frequency is also included with $U_0$ being the maximum light shift per atom.
The many-body description of the system is obtained by introducing the bosonic field operator $\hat{\Psi}({\bf r})$ ($\hat{\Psi}^{\dagger}(\textbf{r})$) which annihilates (creates) a particle at position $\textbf{r} = (x,z)$ and satisfies bosonic commutation relations. In the framework of second quantization, the many-body Hamiltonian $\hat{{H}}^{\mathrm{2nd}}$ reads:
\begin{equation}
\begin{split}
\begin{aligned}
\hat{{H}}^{\mathrm{2nd}} &= \int \mathrm{d}\textbf{r}\,\hat{\Psi}^{\dagger} (\textbf{r}) \Big[\hat{H}_{\mathrm{sp}} - \mu \\ & \mathrel{\phantom{=\int}}\quad{} +g_{\rm{2D}}\hat{\Psi}^{\dagger}(\textbf{r})\hat{\Psi}(\textbf{r})\Big]\hat{\Psi} (\textbf{r}),
\end{aligned}
\end{split}
\end{equation}
where $\mu$ is the chemical potential and $g_{\rm{2D}}$ is the modified short-range interaction strength in a 2D layer <cit.>. We expand $\hat{\Psi}(x,z)$ in the basis of Wannier functions localized on different lattice sites which are obtained from the lowest Bloch band defined by $\hat{H}_{\rm{0}}$:
\begin{equation}
\hat{\Psi}=\sum_{\textbf{m}} W_{\textbf{m}}(x,z)\hat{b}_{\textbf{m}},
\end{equation}
where $\hat{b}_{\textbf{m}}$ ($\hat{b}^{\dagger}_{\textbf{m}}$) represent the annihilation (creation) operators of a single particle at site $\textbf{m}=(m_x,m_z)\frac{\lambda}{2}$ and $W_{\textbf{m}}(x,z)$ is the Wannier function localized on site $\textbf{m}$. A site is referred to as even (odd) if $m_x+m_z$ is even (odd). The Wannier functions localized on neighbouring lattice sites are related to each other by a translation of the lattice constant. Keeping interactions only up to the nearest neighbouring sites, we obtain the Bose-Hubbard model with additional terms <cit.>:
\begin{eqnarray}
\begin{split}
\hat{H}^{\mathrm{2nd}}_{\rm{Wan}} &=
- t \sum_{\langle e,o \rangle}\left(\hat{b}^{\dagger}_{e}\hat{b}_{o} + \mathrm{h}.\mathrm{c}.\right) + \frac{U_{\mathrm{s}}}{2} \sum_{i\in e,o} \hat{n}_{i}\left(\hat{n}_{i}-1\right) \\
&+ \hbar\eta M_0\left(\hat{a}^{\dagger} + \hat{a}\right)\left(\sum_{e}{\hat{n}_{e}} - \sum_{o}{\hat{n}_{o}}\right) \\
&- \hbar \left(\Delta_{\rm{c}}-\delta\right) \hat{a}^{\dagger}\hat{a} - \sum_{i\in e,o}{\mu_i\hat{n}_{i}},
\end{split}
\label{eq:semi_BHHam}
\end{eqnarray}
where $t$ and $U_{\mathrm{s}}$ represent tunnelling and contact interaction in the Bose-Hubbard model <cit.> and are defined as:
\begin{equation}
t = \int\int \mathrm{d}x \,\mathrm{d}z \,W_i^{*}(x,z) \,\hat{H}_0 \,W_i(x,z-\lambda/2)
\label{eq:t}
\end{equation}
\begin{equation}
U_{\rm{s}} = g_{\rm{2D}}\int\int \mathrm{d}x \, \mathrm{d}z\, |W_i(x,z)|^4,
\label{eq:Us}
\end{equation}
$\mu_i = \mu-\epsilon_i$ describes the local chemical potential at site $i$ incorporating the effect of $V_{\mathrm{Trap}}$ and $\hat{n}_{i} = \hat{b}^{\dagger}_i\hat{b}_i$ counts the number of particles on site $i$. Indices $e$ and $o$ refer to all even and odd lattice sites, respectively. $\delta = U_{\rm{0}}M_{\rm{1}}N$ is the dispersive shift of the cavity due to the BEC with $N$ being the total number of atoms. The two overlap integrals $M_{\rm{0}}$, $M_{\rm{1}}$ are defined as:
\begin{eqnarray}
M_{\rm{0}} &=& \int\int \mathrm{d}x \,\mathrm{d}z \,W_i^{*}(x,z) \cos{\left(kx\right)} \cos{\left(kz\right)} W_i(x,z) \notag \\
M_{\rm{1}} &=& \int\int \mathrm{d}x \, \mathrm{d}z\, W_i^{*}(x,z)\cos^2{\left(kx\right)} W_i(x,z) \notag .
\end{eqnarray}
A higher order correction to the tunnelling along the $x$-direction by the self-consistent cavity lattice is neglected. The cavity decay rate $\kappa$ is large compared to the atomic recoil frequency which allows us to adiabatically eliminate the cavity field <cit.>. Its steady state value is given by:
\begin{equation}
\hat{a} = \frac{ \eta M_{0} }{ \Delta_{c}-\delta + i \kappa }\left(\sum_{e}{\hat{n}_{e}} - \sum_{o}{\hat{n}_{o}}\right).
\label{eq:cavity_field}
\end{equation}
Hence the light leaking out of the cavity will be proportional to the imbalance of the number of atoms on the two kind of sites and it can herald the presence of a phase with broken $\mathbb{Z}_2$-symmetry of the underlying static lattice. Inserting equation (10) into equation (7), we recover equation (1) of the main text, with cavity-mediated long-range interaction strength $U_{\rm{l}}$ given by:
\begin{eqnarray}
\begin{split}
&U_{\rm{l}} = -K \hbar |\eta M_{0}|^{2} \frac{\Delta_{\rm{c}}-\delta}{(\Delta_{\rm{c}}-\delta)^{2} + \kappa^{2}} \\
&\hspace{-0.3cm}\stackrel{\text{$|\Delta_{\rm{c}}| \gg \kappa, |\delta |$}}{\approx} -K \hbar |M_{0}|^{2} \frac{\eta^2}{\Delta_{\rm{c}}} \propto \frac{V_{\mathrm{2D}}}{\Delta_{\rm{c}}}.
\end{split}
\label{eq:Ul}
\end{eqnarray}
To describe our stack of 2D layers within this theoretical framework, we assume that a system of many 2D layers can be combined to form one 2D layer with accordingly larger number of lattice sites, containing all atoms.
Validity of the theoretical model: In the derivation of equation (1), we assume the validity of the single-band approximation. To deduce the experimental parameter space where this assumption holds, we compare the strength of all Hamiltonian parameters with the excitation energy to the next higher Bloch band, as shown in Extended Data Fig. 1.
Validity of the single-band approximation. The energy scales of the Hamiltonian are plotted in units of the minimum gap $\Delta_{\rm{ex}}$ between the lowest and the first excited Bloch band. The single-band approximation is assumed to be valid if all the energy scales $U_{\rm{s}}$, $U_{\rm{l}}$ and $t$ are at least 5 times smaller than $\Delta_{\rm{ex}}$, i.e. if they lie below the black dashed line. This criterion is fulfilled for $\Delta_{\rm{c}}/2\pi < -18.3\,\unit{MHz}$ and $18\,E_{\rm{R}}>V_{\rm{2D}} > 3\,E_{\rm{R}}$. For detunings in the interval $-18.3 \,\unit{MHz} < \Delta_{\rm{c}}/2\pi < -10.9\,\unit{MHz}$, the approximation is only partially valid, depending on $V_{\rm{2D}}$. We use this information to illustrate the region of validity in Extended Data Fig. 2.
§.§ Phase diagram in terms of Hamiltonian parameters.
Phase diagram plotted as a function of Hamiltonian parameters $U_{\rm{s}}/t$ and $U_{\rm{l}}/t$. These are derived by converting experimental parameters from Fig. 3 into Hamiltonian parameters. The region of validity for this conversion lies to the right of the solid black line, grey areas were not recorded. The white data points indicate where spatial coherence is lost, and the black data points depict the onset of an even-odd imbalance. The white shaded regions around the data points represent the respective converted error bars. The dotted black lines show, as in Fig. 3, the region where the onset of the cavity light field showed a large variation.
We convert the phase diagram from Fig. 3 into Hamiltonian parameters (see Extended Data Fig. 2) using equations (8), (9) and (11), taking into account the effect of two nearly-degenerate polarization modes of the cavity in the definition of $U_{\rm{l}}$.
Starting in a SF phase and increasing $U_{\rm{l}}/t$ takes the system into a SS phase and eventually into a CDW phase. At the transition from the SF to SS phase, the system needs to overcome additional short-range interaction energy. As a result, an increasingly larger critical long-range interaction strength is required to enter the SS phase for increasing $U_{\rm{s}}/t$. A similar effect is seen for the transition from a SS to a CDW phase.
For negligible tunnelling, a direct transition from a MI to a CDW phase is found at a relative strength of $\frac{U_{\rm{l}}}{U_{\rm{s}}}= 0.66(4)$. In the absence of tunnelling and trapping potential the Hamiltonian (1) supports a stable MI only for commensurate filling. In this case, the phase boundary between MI and CDW lies at a relative strength of $\frac{U_{\rm{l}}}{U_{\rm{s}}} = 0.5$. Deviations from this value can be attributed to the presence of the trap, incommensurate filling and the non-local nature of the long-range interactions.
For negligible $U_{\mathrm{l}}$, the transition from SF to MI is observed at a relative strength of $\frac{U_{\rm{s}}}{t} = 28(4)$. The value is larger than the theoretically predicted value of $\frac{U_{\rm{s}}}{t} \approx 16$ for a homogeneous system with unity filling <cit.>, as discussed in the main text.
§.§ Effect of the trapping potential
The harmonic trapping potential experienced by the atoms has a stabilizing effect on the MI phase in the presence of long-range interactions. Extended Data Fig. 3 illustrates, for fixed atom number and zero tunnelling, the effect of the trapping potential in the presence of the self-consistent lattice potential. For any non-zero $U_{\rm{l}}$ and for non-integer filling, the homogeneous system will arrange in a structured phase with even-odd imbalance. However, the presence of a trapping potential can favour the coexistence of MI and CDW phases or of a MI phase alone, since the system has to pay additional energy for arranging atoms away from the trap center. This energy cost has to be compared with the gain in energy due to the formation of a CDW phase.
For larger fillings, we expect that the system develops a wedding cake like structure similar to experimental realizations of MI phases. In our system, the plateaus can also host partially modulated and fully modulated CDW phases. The presence of any CDW in the system will be signalled by a finite light field scattered into the cavity. We do not expect a qualitative change of the phase diagram depending on the steepness of the trap.
Influence of the trapping potential on the long-range interacting system. Shown are sketches of a 1D slice through a 2D layer, displaying the ground state configurations of 13 particles depending on the relative influence of trapping potential and long-range interaction. Panel (a) shows the situation for a homogeneous system with finite $U_{\rm{l}}$. Panels (b)-(d) show the state of the system for increasing $U_{\rm{l}}$, starting with small but finite $U_{\rm{l}}$.
§.§ Extraction of the BEC fraction.
We take an absorption image of the atomic distribution in the $xz$-plane after $15 \,\unit{ms}$ of ballistic expansion. The obtained momentum distribution is integrated over the cavity direction. We perform a bimodal fit to the resulting distribution, in which we distinguish two contributions. The first component represents coherent atoms diffracted by the lattice potential, captured by a Thomas-Fermi profile plus two Gaussian interference peaks at $\pm 2\hbar k$. The second component is a broad Gaussian distribution resulting from the incoherent addition of atomic signals from the insulating part of the cloud. The BEC fraction $f$ is finally extracted from the ratio $f=N_\mathrm{c}/N$, where $N_\mathrm{c}$ is the integrated atom number in the coherent part and $N$ is the total atom number.
$N$ is obtained from the mean total atom number of all experimental data at low lattice depths, $V_{\mathrm{2D}}\leq 2 \,E_{\mathrm{R}}$. For deeper lattices, the growing spatial extent of the incoherent background is affected by inhomogeneities in the imaging and by the cropped field of view.
To constrain the number of free fit parameters, we fix the position of the interference peaks with respect to the central peak. Furthermore, their widths are linearly correlated to the width of the central peak <cit.>, see Fig. 2a. We double count the interference peaks to correct for the non-visible peaks along the $x$-direction, where the field of view is cropped by the cavity mirrors. The contribution from the $\pm2 \hbar k$ peaks to the total atom number is on the order of a few percent at most. The checkerboard lattice in the supersolid phase leads to extra interference peaks, which lie outside the field of view (see Extended Data Fig. 4). Their contribution to the overall atom number is even lower than the one from the $\pm2 \hbar k$ peaks in most parts of the phase diagram and is therefore neglected.
Momentum distribution in the SS phase. Absorption image from a calibration measurement taken after a short ballistic expansion of $7\,\unit{ms}$ at a detuning of $\Delta_\mathrm{c}/2\pi=-23\,\unit{MHz}$ and a lattice depth $V_{\mathrm{2D}}$ 39% above the onset of an even-odd imbalance in the SS phase. We observe interference peaks at $p_z =\pm 2\hbar k$. Additional interference peaks resulting from the emerging checkerboard lattice appear at $(p_x,p_z) = (\pm hk, \pm hk)$. This observation indicates a SS phase. These additional momentum peaks lie outside the field of view for the longer ballistic expansion time of $15\,\unit{ms}$.
§.§ Extraction of the even-odd imbalance.
During each experimental sequence, the Bragg scattered light leaking out of the cavity is detected with a heterodyne setup <cit.> having a sensitivity of $0.67(1) \,\unit{V}^2$ per intracavity photon. The heterodyne detection is insensitive to the laser field creating the static lattice along the cavity axis by the choice of orthogonal polarizations and a minimum frequency difference of $5 \,\unit{MHz}$. Both phase and magnitude of the light field are recorded. To separate out the coherent part of the light field, we apply a low pass filter to the quadratures before taking the absolute square to obtain an intracavity photon number $n_\mathrm{ph}$.
It is mapped to an even-odd particle imbalance obtained from equation (10) with $n_{\mathrm{ph}} = \langle \hat{a}^{\dagger}\hat{a}\rangle$. We define the effective even-odd imbalance $\Theta$ under the assumption of completely localized atoms on either even or odd sites ($M_0 = 1$):
\begin{eqnarray}
\Theta = \left| \frac{\sum_{e}{\left\langle\hat{n}_{e}\right\rangle} - \sum_{o}{\left\langle\hat{n}_{o}\right\rangle}}{\sum_{e}{\left\langle\hat{n}_{e}\right\rangle} + \sum_{o}{\left\langle\hat{n}_{o}\right\rangle}}\right| = \frac{1}{N}\sqrt{n_{\mathrm{ph}}\, \frac{\Delta_{\mathrm{c}}^2}{\eta^2 }}\frac{1}{F(\Delta_{\mathrm{c}})},
\label{eq:Theta}
\end{eqnarray}
\begin{eqnarray}
\begin{split}
F&(\Delta_{\mathrm{c}})= \sqrt{\frac{\Delta_{\mathrm{c}}^2\cos^2(\alpha)}{(\Delta_{\mathrm{c}}-\delta-\frac{\delta_{\mathrm{B}}}{2})^2+\kappa^2} + \frac{\Delta_{\mathrm{c}}^2 \sin^2(\alpha)}{(\Delta_{\mathrm{c}}-\delta+\frac{\delta_{\mathrm{B}}}{2})^2+\kappa^2}}\notag\\
&\hspace{-0.12cm}\stackrel{\text{$|\Delta_{\rm{c}}| \gg \kappa, |\delta |,\delta_{\mathrm{B}}$}}{\approx} 1 \notag
\end{split}
\end{eqnarray}
describing the scattering into two linearly polarized $\mathrm{TEM}_{00}$ eigenmodes of the cavity, separated due to birefringence by $\delta_{\mathrm{B}} = 2\pi\times 2.2\,\unit{MHz}$ and oriented at an angle $\alpha=22^\circ$ with respect to the $y$- respectively $z$-axis.
The cavity decay rate $\kappa$ is $2\pi\times 1.25\,\unit{MHz}$ and the effective two-photon Rabi frequency $\eta$ for scattering into the two cavity modes is given by $\eta = 2\pi \times 2.7\sqrt{V_{\mathrm{2D}}/\hbar}\,\sqrt{\mathrm{Hz}}$. Close to cavity resonance, the polarization of the Bragg scattered cavity field rotates slightly due to the birefringence, which we include in the detection efficiency of the heterodyne detection. The maximum dispersive shift ${U}_{0}$ per atom of each of the two cavity modes is $-2\pi\times 45.9 \,\unit{Hz}$. The intracavity photon number is determined with a systematic uncertainty of $8\%$, leading to a relative uncertainty in $\Theta$ of less than $6\%$. The technical background level of the photodetection is converted into an imbalance background, which depends on $\Delta_{\rm{c}}$ and $V_{\mathrm{2D}}$. This causes the signal in the lower left corner of Fig. 2d. However, in the MI phase, we estimate from the s.d. of this background a resolution for $\Theta$ which is better than $1\%$.
§.§ Phase boundaries.
Coherence: We convert the 2D lattice depth $V_{\mathrm{2D}}$ to the corresponding ratio $U_{\mathrm{s}}/t$ of short-range interaction strength and nearest neighbour tunnelling by using the Wannier functions obtained from the lowest Bloch band of the applied static lattices. In this way, we obtain a BEC fraction $f$ as a function of $U_{\mathrm{s}}/t$ (see Extended Data Fig. 5), which we fit with a piecewise linear function. The first kink in the fit is associated with the transition point to an insulating phase <cit.>. By analyzing the stability of the fit with respect to initial parameters, we deduce an additional uncertainty on top of the s.d. and include it in the error bar displayed for each transition point in Fig. 2b and 3.
Even-odd imbalance: From each experimental repetition, we obtain a time trace of the light field scattered into the cavity. The maximum photon number $n_{\mathrm{ph,max}}$ at the end of the trace and the corresponding lattice depth $V_{\mathrm{2D}}$ are averaged in a time window of 10 ms (spanning 7.8% of $V_{\mathrm{2D}}$), resulting in one data point extracted per time trace. For each detuning, $n_{\mathrm{ph,max}}(V_{\mathrm{2D}})$ is fitted with a piecewise linear and power law function (see Extended Data Fig. 5) to determine the point where the intracavity light field starts building up. This method largely increases the signal to noise ratio while keeping systematic shifts of the onset point to below 0.2 $E_{\mathrm{R}}$. In the region of $-52\,\mathrm{MHz}\leq\Delta_\mathrm{c}/2\pi\leq-47\,\mathrm{MHz}$, the intracavity field becomes very small and fluctuates strongly from shot to shot (see Extended Data Fig. 5c). We therefore indicate a region for the transition to a phase with $\lambda$-periodic density modulation by dashed lines in Fig. 2d and 3. The starting point of these dashed lines indicate the earliest onset of an even-odd imbalance including the s.d. of the fit. Due to the fixed time of the lattice ramp, we cross the transition to an even-odd imbalanced phase non-adiabatically when ramping into deep lattices. The non-adiabaticity leads to a small shift of the onset point towards higher lattice depths, which can be seen in Fig. 2c. This behaviour was studied previously and explained with Kibble-Zurek theory <cit.>. The described method of discretizing the data is intrinsically less sensitive to this type of shift compared to fitting a single time trace to extract the transition point.
Determination of the phase boundaries. BEC fraction $f$ (averaged into 100 equally spaced bins) and maximum photon number $n_{\mathrm{ph,max}}$ (closed and open symbols, respectively) as a function of $U_{\mathrm{s}}/t$ for detunings $\Delta_\mathrm{c}/2\pi$ of $-12\, \unit{MHz}$ (a), $-22\, \unit{MHz}$ (b) and $-47\, \unit{MHz}$ (c). The red curve shows the result of a piecewise linear fit to $f$. We confirmed that the initial BEC fraction has no systematic dependence on $\Delta_{\mathrm{c}}$. The blue curve displays a power law fit to $n_{\mathrm{ph,max}}$.
§.§ Self-consistent checkerboard lattice.
The SS and CDW phases give rise to a light field inside the cavity due to the Bragg scattering of $z$-lattice photons. This cavity field is self-consistent as it depends on the strength of the $\lambda$-periodic density modulation in the atomic cloud and has a depth $V_{\mathrm{c}}=n_{\mathrm{ph}} \times 12.3 \times 10^{-3} \,E_{\mathrm{R}}$. Interference of this self-consistent $x$-lattice with the field of the $z$-lattice produces a checkerboard lattice potential of depth $V_{\mathrm{CB}} = 2\sqrt{V_{\mathrm{2D}} V_{\mathrm{c}}}$, displayed in Extended Data Fig. 6. The line of constant $V_{\mathrm{CB}}$ bends towards smaller values of $V_{\mathrm{2D}}$ when approaching cavity resonance, substantiating the assumption that this energy offset causes the observed behaviour of the SS to CDW boundary line.
When the energy offset between even and odd sites due to $V_{\mathrm{CB}}$ gets comparable to the tunnelling energy, the effective tunnelling strength between nearest-neighbours reduces and higher order tunnelling processes begin to play a significant role <cit.>.
Strength of self-consistent checkerboard (CB) lattice. The CB lattice depth extracted from the measured mean intracavity photon number $n_{\mathrm{ph}}$, as a function of the applied lattice depth $V_{\mathrm{2D}}$ and detuning $\Delta_{\mathrm{c}}$. The CB lattice depth gets comparable to the depth of the static lattices close to cavity resonance, but drops rapidly when moving away due to its detuning dependence. Exemplary equipotential lines at $0.05\,E_{\mathrm{R}}$, $1\,E_{\mathrm{R}}$ and $3\,E_{\mathrm{R}}$ are shown.
§.§ Hysteresis measurements.
We initialize the system in the insulating region at either $V_{\mathrm{2D}}= 14\,E_{\mathrm{R}}$ or $18\,E_{\mathrm{R}}$ using a $50\,\unit{ms}$ long S-shaped amplitude ramp. The detuning $\Delta_{\mathrm{c}}$ is then changed with an S-shaped frequency ramp at an average speed of $0.67 \,\unit{MHz/ms}$ reaching a different detuning value. After holding for $10\,\unit{ms}$, we scan back to the initial detuning. Residual atom loss continuously reduces the measured mean intracavity photon number $n_{\mathrm{ph}}$, which we take into account by rescaling the data before converting it into an imbalance $\Theta$. The scaling factor is extracted from reference measurements, where we hold at different $\Delta_{\mathrm{c}}$ for $50\,\unit{ms}$. We deduce a linear decrease in $n_{\mathrm{ph}}$ by $48(4)\%$ ($41(4)\%)$ for lattice depths of $V_{\mathrm{2D}}=14\,E_{\mathrm{R}}$ $(18 \,E_{\mathrm{R}}$). After rescaling the data, we observe a remaining relative drift of the imbalance level of $8(4)\%$ during the hold time.
Extended Data Fig. 7 shows detuning scans performed at $V_{\mathrm{2D}}=18\,E_{\mathrm{R}}$, where a similar hysteretic behaviour is observed as in Fig. 4 of the main text. To test the sensitivity of the hysteretic behaviour on the ramp speed, we slow down the frequency ramp by a factor of two and observe a comparable evolution of the even-odd imbalance, see Extended Data Fig. 7.
Sensitivity on the ramp speed. Hysteretic behaviour in the insulating regime, at $V_{\mathrm{2D}}=18\, E_{\mathrm{R}}$. The detuning $\Delta_{\mathrm{c}}$ is ramped at two speeds, $0.67 \, \unit{MHz/ms}$ (blue) and $0.33 \,\unit{MHz/ms}$ (orange). Lines result from an average of two to five measurements, using $400\,\unit{\mu s}$ time bins. Stars signify starting points, arrows show the scan direction and dashed lines indicate the return back to the starting point.
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Author Contributions
R.L., L.H., N.D. and M.L. took the data and analyzed them together with T.D. Contributions to the design of the experiment were made by R.M. All work was supervised by T.E. All authors contributed to discussions and the preparation of the manuscript.
Author Information
The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to T.D. ([email protected]).
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1511.00411
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Activity in neocortex exhibits a range of behaviors, from irregular to
temporally precise, and from weakly to strongly correlated. So far
there has been no single theoretical framework that could explain all
these behaviors, leaving open the possibility that they are a
signature of radically different mechanisms. Here,
we suggest that this is not the case. Instead, we show that a single
theory can account for a broad
spectrum of experimental observations,
including specifics such as the fine temporal details of subthreshold
cross-correlations. For the model underlying our theory, we need only
assume a small number of well-established properties common to all
local cortical networks. When these assumptions are combined with
realistically structured input, they produce exactly the repertoire of
behaviors that is observed experimentally, and lead to a number of
testable predictions.
Given the immense complexity and computational power of the mammalian
cortex, it is perhaps surprising that under a broad range of
conditions neurons are relatively stereotyped: spikes are irregular –
often near Poisson<cit.> – and weakly
correlated<cit.>, and membrane potentials exhibit
approximately Gaussian variability
<cit.>. This apparent randomness was
explained by van Vreeswijk and Sompolinsky, who showed
that high variability is a necessary consequence of the high yet sparse
connectivity, strong synaptic coupling, and relatively low firing
rates that are ubiquitous in cortex<cit.>. This
was an important advancement, as understanding the bulk of neuronal
activity is a prerequisite for understanding how networks carry out
computations, and van Vreeswijk and Sompolinsky's theory has developed
into the de-facto standard model of cortical dynamics.
Although this standard model generally provides a good description of
the behavior of neurons, there are a growing number of observations,
most of them based on intracellular recordings in vivo, that are
at odds with it: near-synchronous activity
<cit.>, precise relative timing between excitation
and inhibition<cit.>, non-Gaussian membrane potential
and rapid switching between
states that is mediated by both behavior<cit.>
and sensory stimuli<cit.>. These
observations would seem to
suggest that the standard model needs to be extended, if not replaced
altogether. Here we show that it does, indeed, need to be
extended, but in a way that does not require additional anatomical
or physiological assumptions. The standard model describes
behavior in a regime in which networks exhibit a stable
equilibrium at moderate firing rates. Our proposal is
to also allow networks to operate in a regime in which
the stable equilibrium shifts to zero or near zero firing rates.
This extended standard model, like the standard one, is a model of
randomly connected networks of excitatory and inhibitory neurons. We
investigate analytically, and through simulations,
the dynamics of these networks, and what we find
is that even unstructured, randomly connected networks
can exhibit the behavior reported in a wide
variety of studies
This does not imply that
networks in the brain are randomly connected. It does, though, imply
that to uncover the computational principles used by the brain,
it will be necessary to design experiments that
go beyond the dynamics
expected from randomly connected
excitatory-inhibitory networks.
We study the dynamics of a model for a generic local cortical network
with a radius of up to about 150 microns and containing on the order
of thousands of recurrently connected excitatory and inhibitory
neurons. One may think of this network as representing a layer within
a cortical column. We assume only a small set of well-established
properties: every neuron receives hundreds to thousands of inputs from
within its local neighborhood; synaptic coupling is strong (strong
enough that without inhibition the network would be
epileptic<cit.>), with an average EPSP size of
approximately 0.5 mV<cit.>; neurons are
sparsely connected; and, because inhibition is primarily local within
cortex<cit.>, external input to the local
network from the “rest of the brain” is assumed to be excitatory.
Given these properties, we determine the range of possible behaviors
of the network. To do that, we write down a relatively generic network
model, argue that the behavior of the network is determined largely by
the conductances, and then study their behavior. In the next two
we present the mathematical formulation of our model and our approach
to elucidating its dynamics; in the three sections after that we
present the results.
phvNetwork model
We consider a network of $N_E$ excitatory and $N_I$ inhibitory neurons
coupled via spike-driven conductance changes and exhibiting
essentially arbitrary single neuron dynamics. The network is described
by a set of equations for the membrane potentials. Using $V_i^\alpha$
to denote the membrane potential of neuron $i$ of type $\alpha$, where
$\alpha$ can be either $E$ (excitatory) or $I$ (inhibitory), the
equations are
\begin{equation} \label{neuron-dyn}
C_m \frac{dV_i^\alpha}{dt} =
I_i^\alpha(V_i^\alpha, \b c_i^\alpha) -
G_i^{\alpha E}
\big( V_i^\alpha - \mathcal{E}_E \big) -
G_i^{\alpha I}
\big( V_i^\alpha - \mathcal{E}_I \big) +
\, .
\end{equation}
Here $C_m$ is membrane capacitance, $I_i^\alpha(V_i^\alpha, \b
specifies the single neuron dynamics (including spike
generation), $\b c_i^\alpha$ is a set of channels for neuron $i$, each with
its own dynamics, $I_i^{\alpha,ext}(t)$ is external,
$\mathcal{E}_E$ and $\mathcal{E}_I$ are
reversal potentials, and
$G_i^{\alpha E}$ and $G_i^{\alpha I}$ are the total excitatory and
inhibitory conductances,
\begin{equation} \label{G_def}
G_i^{\alpha \beta}(t) = \sum_{j=1}^{N_\beta} W_{ij}^{\alpha \beta}
\end{equation}
where $W_{ij}^{\alpha \beta}$ are synaptic strengths and
the $g_j^\beta(t)$ are the individual conductance –
essentially, $g_j^\beta(t)$ exhibits a small
increase whenever neuron $j$ of type $\beta$ fires. Note that both
$I_i^\alpha$ and the conductance changes
can be chosen from a variety of standard models, making Eq. (<ref>)
extremely general – it can
display single neuron dynamics ranging from linear integrate and fire to
Hodgkin-Huxley, and the conductance changes can range from
simple functions of time (e.g., instantaneous rise and exponential
decay) to complex dynamics that includes failures and adaptation.
For our simulations, $I_i^\alpha$ corresponds to a quadratic integrate
and fire neuron and the conductance changes exhibit an instantaneous
rise whenever there's a spike, followed by an exponential decay; see
Methods for details, including network parameters.
To analyze the dynamics of the network described in
Eq. (<ref>), we focus on the conductances, $G_i^{\alpha
\beta}(t)$, associated with the recurrently generated spikes. That's
because if we knew the conductances, we would know the activity of the
neurons. Importantly, that activity is approximately independent of the
single neuron model we use, since for essentially all single
neuron models the effect of the conductances is the same: increasing
the excitatory conductance increases firing rates, increasing the
inhibitory conductance decreases firing rates, and increasing the
fluctuations of either of them increases irregularity.
Thus, our results apply to other single neuron models
besides the quadratic integrate and fire.
Our starting point for analyzing the conductances
is similar to that of other
mean field models of neuronal networks,
which is to divide the conductance seen
by each neuron into mean and fluctuating pieces,
\begin{equation} \label{G}
G_i^{\alpha \beta}(t) = G_{\alpha \beta}(t) + \delta G_i^{\alpha \beta}(t)
\end{equation}
where $G_{\alpha \beta}(t)$ is the population-averaged conductance
(the “shared” term) and the $\delta G_i^{\alpha \beta}(t)$ are the
neuron-specific offsets from – and fluctuations around – that
average (the “individual” term); see
Figs. <ref>a and b.
The vast majority of mean field models consider what is known as
the asynchronous regime, a regime in which the shared conductances,
the $G_{\alpha \beta}$, are
A key, and nontrivial, result that has come out of those models is
that the individual terms, the
$\delta G_i^{\alpha \beta}(t)$, are rapidly fluctuating, and the
fluctuations are weakly correlated across neurons
<cit.>; exactly what is
seen in Fig. <ref>c.
The properties of the individual and shared conductances
are consistent with theoretical predictions.
a̱. Excitatory-to-excitatory conductances, $G_i^{EE}(t)$,
onto five neurons (thin red lines), the shared excitatory-to-excitatory
conductance, $G_{EE}(t)$
(thick red line), approximated by averaging over
$20$ neurons, and the excitatory population
firing rate, $\nu_E(t)$ (black line).
As predicted,
the individual conductances,
$\delta G_i^{EE}(t) = G_i^{EE}(t) - G_{EE}(t)$,
show fast and strong fluctuations
around the shared conductance, and
the shared conductance is proportional to the excitatory population
firing rate.
ḇ. Analogous to panel a but for
conductances, $G_i^{EI}(t)$, onto the same five neurons
(thin blue lines), the shared inhibitory-to-excitatory conductance,
$G_{EI}(t)$ (thick blue line), and the
inhibitory population firing rate, $\nu_I(t)$ (black line).
c̱. Cross correlations between individual conductances,
$\delta G_i^{\alpha \beta}(t)$, onto two
randomly chosen neurons.
Red: cross correlation between two
excitatory-to-excitatory individual
conductances; purple: cross
correlation between excitatory-to-excitatory and
inhibitory-to-excitatory individual conductances (onto the same two
In agreement with the theory, the individual conductances
are essentially uncorrelated.
ḍ. Same as panel c, but for the total (shared plus individual)
conductances, $G_i^{\alpha \beta}(t)$.
absence of correlations in the individual conductances (panel
c) implies that the high correlations
in the total conductances (this panel) are entirely due to
the correlations in the shared
which in turn are due to the fluctuations in the population
rates (panels a and b).
e̱. Cross correlation of the membrane potential between the same two
neurons as in panels c and d. Because inhibition largely cancels excitation,
the membrane potentials are almost completely uncorrelated.
Network parameters are described in Methods.
What we do that is new is extend these results to the case of
time-varying shared conductances (the $G_{\alpha \beta}$
depend on time). This time dependence introduces strong
correlations in the input to single neurons (as shown, for example, in
Fig. <ref>d), and so leads to qualitatively
different behavior compared to the asynchronous regime.
To understand this behavior, we need a model for how
the shared conductances depend on time. For that we make use of two
observations. First, as we show in Methods, and as can be seen in
Figs. <ref>a and b, the shared conductances
are proportional to the average excitatory and inhibitory firing
rates, with the constant of proportionality given by the
connection strengths,
\begin{equation} \label{gqr}
G_{\alpha \beta} (t) \propto W_{\alpha \beta} \nu_\beta(t)
\end{equation}
where $W_{\alpha \beta}$ is the average synaptic strength made by a
neuron of type $\beta$ onto a neuron of type $\alpha$, assuming that a
connection is made,
and $\nu_\beta(t) \equiv N_\beta^{-1} \sum_j \nu_j^\beta(t)$ is the
average firing rate of population $\beta$.
Second, we use a highly successful phenomenological
model of the average firing rates, the Wilson and Cowan
\begin{align}
\tau_E \frac{d \nu_E}{dt} &= f_E
\big( J_{EE} \nu_E - J_{EI} \nu_I + I_E(t) \big) - \nu_E
\\
\tau_I \frac{d \nu_I}{dt} &= f_I
\big( J_{IE} \nu_E - J_{II} \nu_I + I_I(t) \big) - \nu_I
\end{align}
where $f_E$ and $f_I$ are approximately
sigmoidal gain functions,
$J_{\alpha \beta}$ is approximately proportional to $W_{\alpha \beta}$, and
$I_E(t)$ and
$I_I(t)$ represent external input.
While this model can't provide quantitative results, it can place
severe constraints on the firing rate dynamics, and thus on the
dynamics of the shared conductances. Consequently, it allows us to
categorize the expected range of network behaviors, and thus, via
Eq. (<ref>) and Eq. (<ref>), predict single-neuron sub-threshold
dynamics, including properties of correlations across neurons.
phvTwo distinct network states
To understand the constraints implied by the Wilson and Cowan model,
and, therefore, the possible range of network
behaviors, we apply phase
plane analysis. This analysis starts by constructing the excitatory
and inhibitory nullclines, which are curves in $\nu_E$-$\nu_I$ space
along which $d \nu_E/dt$ and $d \nu_I/dt$ are zero, respectively.
Then, given the nullclines, network dynamics can be inferred
relatively easily (see Fig. <ref> caption). For
essentially any reasonable shape of the gain functions, $f_E$ and
$f_I$, two qualitatively different regimes can be
identified <cit.>. One corresponds to the
“active state,” for which the nullclines intersect at nonzero rate
(Fig. <ref>a, b); the other to the “quiescent
state,” for which the nullclines intersect at near zero rate
(Fig. <ref>c).
Network models
typically consider only the active state
<cit.>. However, the
quiescent state is just as important. In fact, as we will show, it is key to
explaining, and reconciling, a diverse
set of in vivo cortical data.
In the next three sections we elaborate on this point, using as a
guide the nullclines given in Fig. <ref>.
We first briefly review – and extend – the properties of
the active state (Figs. <ref>a and b); we then describe
the properties of the quiescent state (Fig. <ref>c);
and, finally, we discuss switches between the two.
Nullclines and population-averaged firing rates, or “population
rates”, corresponding to Eq. (<ref>), in three different regimes.
Left panels: excitatory (red) and inhibitory (blue)
nullclines. Points along the excitatory nullcline represent firing rate
equilibria at fixed inhibition; above it the
excitatory firing rate decreases with time while below it the
excitatory firing rate increases (red arrows).
Correspondingly, points along the inhibitory nullcline represent
firing rate
equilibria at fixed excitation; to the left of it
the inhibitory firing rate decreases while to the right of it
the inhibitory firing rate increases (blue arrows). The
dashed black curves correspond to trajectories.
Right panels: population rates; red lines are excitatory population
rates and blue ones are inhibitory rates.
See reference latham_intrinsic_2000a for a detailed
description of how these nullclines are constructed.
a̱. The input is sufficiently strong that there is a robust, low
population rate equilibrium, and inhibition is sufficiently fast
that the equilibrium is
The trajectory that
spirals into the fixed point represents transient behavior; after long
times the population rates are constant, as shown in the right panel.
ḇ. Same as panel a, but with slower inhibition, which
destabilizes the equilibrium.
In this regime the population rates oscillate.
c̱. Sufficiently weak input that the only equilibrium is at
near zero
population rate. The network is still excitable, though, and brief,
strong input produces transient activity, as shown by the two
trajectories in the right panel.
phvThe active state
The active state is characterized by an equilibrium at nonzero population
rates. This equilibrium, however, may or may not be stable. Let us
first consider the stable case, for which
the network exhibits constant mean excitatory and
inhibitory population rates, as shown in
Fig. <ref>a. Because the shared conductances,
the $G_{\alpha \beta}$, are proportional to the population rates
(see Eq. (<ref>)), they too are constant, and so the
dynamics of single neurons are determined solely by the individual
conductances, the $\delta G_i^{\alpha \beta}$. As has been shown
(and as we show in Figs. <ref>a and b),
these conductances exhibit essentially stochastic,
fluctuations <cit.>.
This leads to approximately Gaussian distributed conductances,
and, consequently,
approximately Gaussian distributed
membrane potentials.
Dynamic balance <cit.> ensures that the
mean excitatory and inhibitory currents nearly
cancel each other, so that spiking is
caused by the stochastic fluctuations of the membrane
potentials. This results in irregular spike times that are very weakly
correlated across
neurons <cit.>.
Strongly-coupled spiking networks operating at
a stable equilibrium have become the de-facto standard model of
cortical network dynamics <cit.>, and the dynamic properties in this regime
are referred to as the “asynchronous state”.
Although a constant population rate equilibrium is a convenient
abstraction, in fact population rates are never constant. That's
because finite-size effects produce fluctuations, which
in turn lead to trajectories that spiral in a counter-clockwise
direction around the equilibrium.
The resulting time-depending population rates produce, via
Eq. (<ref>), time-dependent fluctuations in the shared
conductances, and thus strong correlations.
This would seem to
rule out operation in the asynchronous regime, which by
definition is characterized by essentially
uncorrelated membrane potentials and
action potentials. However, because of the
nearly tangential intersection of the excitatory and inhibitory
nullclines, the trajectories are elongated
(Fig. <ref>b),
and so the shared excitatory and inhibitory conductances closely track
each other. This close tracking causes the correlations to
mostly cancel, producing nearly uncorrelated membrane potentials, and,
consequently, irregular spike times that are also nearly uncorrelated.
These predictions, which are consistent with rigorous analytic results
for networks of binary neurons <cit.>,
illustrated in Fig. <ref> (see in
particular panel e, which shows a complete
absence of correlations in the membrane potential).
Thus, even though the excitatory and inhibitory
conductances are strongly correlated across neurons, their difference
(suitably weighted by the reversal potentials) is not, and
so even finite size networks can
exhibit asynchronous activity. This can be seen in our
network simulations (Fig. <ref>a and b),
in vivo recordings
(Fig. <ref>c and d), and
numerous other simulation studies
Simulations and in vivo data in the active state.
a̱. Conductance traces and histograms from
simulations. Conductances are approximately Gaussian distributed, and
inhibitory conductances have a larger mean and wider distribution than
excitatory ones. Network parameters,which are described in
Methods are the same as for Fig. <ref>.
ḇ. Membrane potential traces and histograms from
the same set of simulations as in panel a. Sub-threshold membrane potentials are
approximately Gaussian distributed, and
spike timing is irregular and asynchronous
across neurons.
c̱. Conductance traces and histogram in vivo, adapted from
Fig. 7b of reference rudolph_characterization_2005;
permission requested.
ḏ. Membrane potential traces and histograms in vivo,
adapted from Box 1a of reference
destexhe_high-conductance_2003, with
permission from Macmillan Publishers Ltd: Nat. Rev. Neurosci.,
copyright 2003.
The active state also applies when the population rates change slowly.
These slow changes can occur either because the external input is time
varying, or because the population rate equilibrium becomes unstable,
producing oscillations
<cit.>. In either case
the population rates become correlated and, therefore, so do the
excitatory and inhibitory conductances (see Eq. (<ref>)). However,
as in the case of fluctuations driven by finite size effects,
the shared excitatory and inhibitory conductances
again nearly cancel. Thus, almost all of the results for constant
input apply to time-varying input
and oscillations: excitatory and
inhibitory conductances inherit cross-correlations from the time-varying
population rates, but the cancellation of these correlations at the
level of the membrane potentials, in combination with the strongly
fluctuating conductance terms,
lead to irregular spiking
and approximately Gaussian-distributed membrane potentials.
Moreover, conditioned on population rates,
both the membrane potential and
spike times are approximately uncorrelated, and the network remains
effectively asynchronous.
How slowly do the changes in population rate need to be for the network to
stay asynchronous? It turns out that the external input can change on
a faster time scale than that of single-neuron dynamics; this is, in
fact, one of the hallmarks of strongly-coupled balanced
networks<cit.>: sudden,
strong increases in input can induce a sudden rise in excitation
before inhibition can catch up, causing many neurons to spike almost
simultaneously. When this happens, neurons can display increased
temporal precision in spike timing in response to sharp stimulus
onsets, which provides a robust, network-level explanation of
precise timing effects in cortex <cit.>.
Although the active state has provided a great deal of insight into
cortical dynamics, there is a growing body of experimental data that
is not consistent with it. As we will show in the next two sections,
the other nullcline regime – the one corresponding to the quiescent
state – is needed to provide a robust explanation of that data.
phvThe quiescent state
In the active state, the external input is strong enough for the
network to stay continuously active. What happens if we reduce its
strength? This will cause the excitatory nullcline to shift down and
the inhibitory nullcline to shift to the right. For small enough
external input – and thus large enough shifts – the equilibrium
vanishes. When this happens, a new, stable equilibrium appears at zero
or near zero population rate (Fig.<ref>c, which
shows an equilibrium at zero firing rate). This equilibrium
corresponds to a silent or nearly silent network, which is why we
refer to the corresponding nullcline regime as the quiescent state.
While a silent network may seem uninteresting, the strong
recurrent connections among the excitatory neurons means that such a
network is highly excitable. Consequently, excitatory
can result in nontrivial dynamics.
Two typical
population rate trajectories caused by brief input applied to a
silent network are shown in Fig. <ref>c, both in phase
space (left panel) and versus time (right panel).
Note that the trajectories are highly stereotyped: they all consist of
a sudden, rapid increase in
excitatory population rate, followed, with a small delay, by a rapid
increase in inhibitory rate, and then
a slightly slower – but still fast – drop in both
excitatory and inhibitory population rates. Thus, sufficiently strong
input applied to the
quiescent state results in short bursts of activity in which
inhibition peaks slightly later than excitation. These bursts differ
in amplitude, but vary little in shape.
Because population firing rates are related to conductances
(Eq. (<ref>)) and conductances drive neurons
(Eq. (<ref>)),
knowing how the population firing rates evolve over time
allows us to
make inferences about the behavior of individual
neurons. Consequently, Fig. <ref>c leads to the
following picture: between
bursts of activity
the synaptic drive is essentially zero and the neurons are at
rest; during a burst, first the shared excitatory conductance to each
neuron increases very rapidly, and then, a few milliseconds later, the shared
inhibitory conductance increases very rapidly; after they have peaked,
they decay back toward zero, with both decaying at about the same
To verify this picture, we analyzed the same network as
in our previous simulations (Fig. <ref>),
but with
input that consisted of brief pulses rather than sustained drive. As
predicted, the conductances in the simulations showed large, brief
excursions which were highly correlated across
neurons, excitation
led inhibition by a small amount, and there was on average no time lag
between the excitatory drives on pairs of neurons or between
the inhibitory drives (Fig. <ref>a).
Simulations and in vivo data during brief excursions from
the quiescent state.
a̱. Simulations.
The excursions were caused by brief, synchronous input applied to
the whole network. Data is shown for two neurons; for these neurons
the single neuron dynamics was linearized to prevent spiking, and
constant current was injected to adjust
the resting membrane potential (see
In panel a-i the resting membrane potential was set to -70 mV (the
inhibitory reversal potential) to isolate the excitatory drive. (Cell
2 was offset slightly to aid visibility.)
In panel a-ii the resting membrane potential was set to -40 mV
to (partially) isolate the inhibitory drive. Because isolation was
incomplete, there are brief upward deflections associated with the
initial surge of excitation.
Panel a-iii shows the membrane potential correlation between
cells 1 and 2 when they are both hyperpolarized to -70 mV
(H-H; black) and when they are both depolarized to -40 mV (D-D; magenta).
Panel a-iv shows the same thing, but when
cell 1 is depolarized and cell 2 is hyperpolarized (D-H,
orange trace) and when cell 1 is hyperpolarized and cell 2 is depolarized
(H-D, green trace). Because of our convention for computing the
correlation (see Methods),
the minimum at a positive lag for the orange trace and a negative lag
for the green trace both indicate that excitation leads inhibition.
Network parameters, which are described in Methods, are the same
as for Figs. <ref> and <ref>
except for the input.
ḇ. Same as panel a, except for experimental data.
Reproduced with permission from reference okun_instantaneous_2008.
In vivo experiments have, critically, verified that the
quiescent state exists in cortex, and that excitation of the quiescent state
leads to behavior consistent with our predictions. In
particular, DeWeese and Zador
observed brief membrane potential
excursions (which the authors called “bumps”) separated
by periods of silence, and they established that during the silent
periods there was essentially no synaptic
More detailed experiments involving paired recordings from nearby
neurons in somatosensory cortex of anesthetized rats
<cit.> yielded results that are identical to
our predictions of the
relative timing of excitation and inhibition across pairs of neurons.
In those experiments, excitatory and inhibitory drive to different
neurons occurred at about the same time, whereas excitation led
by several milliseconds, on average. A time lag between
excitation and inhibition was also seen in single neuron intracellular
recordings in
anesthetized rat auditory cortex<cit.>.
Although in the latter study the
authors could not directly compute cross-correlograms,
they could compute trial-averages of both excitatory and inhibitory
drive, and they reported that “inhibition and excitation occurred in
a precise and stereotyped temporal sequence,” with excitation leading
inhibition by a few milliseconds, again in
agreement with our predictions.
The absence of a time lag for both excitation and inhibition across
different neurons, and the short lag of inhibition behind excitation
during these brief events, is exactly what our theory predicts –
both qualitatively and quantitatively
(see Figs. <ref>a and b).
However, although in some experiments cortical
networks spend much of their time in the quiescent state, with only
brief periods of activity, more commonly they switch – often fairly
rapidly – between the quiescent and active states. We consider that
behavior next.
phvState switching
In the quiescent state, brief supra-threshold input produces short,
stereotyped bursts of activity. Longer supra-threshold input leads to
different dynamics: instead of exhibiting
brief stereotyped activity
bursts, the network switches to the active state; that is, the
nullclines switch to the topology shown in
Fig. <ref>a<cit.>.
The transition from the quiescent to the active state
has the same properties as the initial phase of the brief activity
bursts: it occurs synchronously in all neurons, and the spikes at the
onset are closely temporally aligned (in contrast to later
spikes). When the input falls below threshold, the network
becomes quiescent again; this also occurs synchronously on all
neurons, although with somewhat less precision than for the
quiescent-to-active transition.
Repeated alternations between super-threshold
and sub-threshold input is what we call state switching (see
Fig. <ref>).
Spike rasters and membrane potential during state switching.
a̱. The input to the network consists of a slowly oscillating
ḇ. Spike rasters. As predicted, spikes are highly synchronous
during the transition from the quiescent to active state, but quickly
c̱. Membrane potential of two randomly chosen neurons. In the active
state, fluctuations are large and, especially at the initial onset,
highly correlated. In the quiescent state, on the other hand, the
membrane potential is virtually flat.
Network parameters, which are described in Methods, are the same
as for Figs. <ref>, <ref> and <ref>
except for the input.
Note that state switching is very different from the bursts of
activity that occur when the quiescent state is activated by brief
input. For state switching, the nullcline topology alternates between
that shown in Figs. <ref>a and b and that shown in
Fig. <ref>c (discussed in detail in reference
latham_intrinsic_2000a). Therefore, termination of the active state
occurs either through a decrease in external input or a change in
single neuron properties (e.g., spike after-hyperpolarization, as
discussed in the next paragraph).
For brief bursts of activity, on the other hand, the
nullclines do not change, and activity is terminated by inhibition.
Although state switching can be driven by external input,
it can also occur
spontaneously, without changes in external input. This is because
modulation in single-neuron excitability has the same effect as
changing the strength of the external input. A typical example of such
a modulation is spike after-hyperpolarization,
which results in a slow
decrease in the effective strength of the input during active
and a slow increase during quiescent periods.
Therefore, even with constant
external input, there can be shifts in the
nullclines that produce sudden switches between active and
quiescent state dynamics. The resulting activity is commonly referred to as up
and down states. Such a mechanism, which was first described in the
context of networks for generating breathing <cit.>,
has been confirmed both theoretically
and experimentally
Internally driven up-down states are not the only example of state
switching. In fact, a central prediction of our theory is that any
local cortical network will undergo state switching
whenever the effective input switches between
sustained supra-threshold and sub-threshold levels. In the
supra-threshold regime, the network will have all of the properties of
the active state, including weak correlations in membrane potential
and spike times (at least after the initial transient).
However, correlations computed
in periods that are long compared to the switching time will be stronger – typically much stronger.
That's because the switches occur in all neurons nearly
synchronously. Consequently, the membrane potential on any one neuron
is highly predictive of the membrane potential on any other neuron
(i.e., high membrane potential on any one neuron predicts high membrane
potential on another, and similarly for low membrane potential.)
To illustrate the effect of state switching on correlations across
neurons, we performed simulations using the same randomly connected
spiking network as for our previous results (Figs 2 and 3).
Unlike in the
previous simulations, however, we switched between two input
regimes: (1) a regime in which the input consisted of slow
oscillations that periodically crossed the network-activating
threshold, termed “Sub/Suprathreshold”, and (2) a regime in which
the input had a rich temporal structure, and which kept the network
continuously active, termed “Suprathreshold”. The choice of the two
input regimes and their structure were motivated by
experiments<cit.>, as detailed below.
When we stimulated our network with the Sub/Suprathreshold input,
whenever the input crossed the threshold from below, a transition from
the quiescent to the active state occurred; when it crossed from
above, the opposite transition occurred.
As predicted by our
theory, in this regime
membrane potentials were highly correlated, as shown in the
Sub/Suprathreshold regions of Fig. <ref>a.
(Note that the only important
feature of the input in this regime is the repeated
crossing of the activation threshold;
the crossings do need not to be caused by oscillating
input, nor do need they need to be at regular intervals.)
In the Suprathreshold regime, the input consisted of repeated
temporal Gabor filters with a center frequency of 8 Hz; this was meant
to mimic whisking-like input (compare to Fig. 1i in reference
poulet_internal_2008). In this regime, despite the strong and
fast modulation of input amplitude, the input was sustained enough to
keep the network in the active state, yet it was weak enough that the
time-average of single-neuron firing was $\leq 1$ Hz; about the same firing rate
as during the Sub/Suprathreshold input.
Because in the Suprathreshold regime the network is
in the active state, our prediction
is that membrane potential correlations should be low, despite strong
fluctuations in the input. Indeed they are, as can be seen in the
central region in Fig. <ref>a.
Exactly such behavior was found in a recent study by Poulet and
Petersen <cit.> who, in an experimental tour de
force, recorded membrane potentials in awake animals in pairs of
neurons in mouse somatosensory cortex. Their finding was that the
degree of membrane potential correlations was tightly linked to the
behavioral state of the animal: during quiet resting, membrane
potentials of nearby neurons were highly correlated, whereas during
active whisking, correlations decreased markedly
(Fig. <ref>b).
Within our framework, this is expected if
the input to the sensory cortex
alternates between supra-threshold and sub-threshold during
quiet resting, but is sustained during active whisking.
As for the origin of the intermittent input
during quiet resting, our theory suggests a number of possibilities:
it could be due to invading activity from other
cortical or subcortical areas, an
operating regime that is close to the activation threshold together
with adaptation effects (analogous to spontaneous up-down state
switching, as described above), or a combination
of the two.
A key observation in the experiments by Poulet and
was that even after cutting the sensory nerves that relay whisker
sensation to somatosensory cortex, the neurons still switched between
high and low correlations whenever the behavior changed between
resting and whisking. Based on this observation, our theory predicts
that the rodent somatosensory cortex receives sustained input from
non-sensory areas whenever the animal is actively whisking.
Simulations and in vivo data during state switching.
a̱. Simulations.
In the region marked “Sub/Suprathreshold,” the input is
identical to that of Fig. <ref>: slowly
oscillating, with amplitude and offset chosen so that
near the peaks of the oscillations the network is in the
active state and the rest of the time it is in the quiescent state. In
the region marked “Suprathreshold,” on the other hand,
the input is sufficiently large
that the network is always in the active state.
In both regions the correlation
of the membrane potentials in two cells is
computed over 1 second windows and
averaged over the central 4 ms.
Network parameters, which are described in Methods, are the same
as for Figs. <ref> and
except for the input.
ḇ. Same as panel a, except for experimental
data (Supplementary Fig. 4 from reference
poulet_internal_2008). The green trace is whisker angle.
According to our theory, the absence of whisking (low angular
deflection) corresponds to input that frequently drops below the
activation threshold, while whisking (high angular deflection)
corresponds to input that stays continuously above the activation
threshold. This is reflected in the high correlations in the absence
of whisking and the weak
correlations during whisking.
Reproduced with permission by the author.
We have shown how the rich repertoire of cortical networks –
ranging from weakly correlated with irregular firing to highly
correlated and temporally precise, and including rapid switches between
different states – can be readily understood within a single
theoretical framework. The mechanistic model underlying this framework
is extremely general and its essential ingredients apply to any local
patch of cortex.
Our primary result is that cortical networks can operate in one of two
distinct regimes, which we termed active and quiescent.
In the active
regime, the network exhibits ongoing activity with population firing
rates that never drop to zero, so that the membrane potentials of all
neurons are depolarized away from their resting values. This regime
contains as a special case asynchronous activity, in
which spiking is irregular, membrane potential distributions are
Gaussian, and correlations are weak, as has been
studied previously
It also contains
oscillating activity<cit.>, and, more generally, activity that
fluctuates due to finite-size effects<cit.>,
or changes in magnitude due to
changes in input strength.
In the quiescent regime, on the other hand, the topology of the
nullclines is different from that in the active state, and the
network is silent or nearly silent.
It is, though, highly excitable and brief input
results in precisely-timed stereotyped dynamics both at
the level of population activity and single-neuron subthreshold
A key result of this work is that the regime in which the network
operates is determined primarily by the strength of the
external input:
sufficiently strong external input leads to the active state;
sufficiently weak input to the quiescent state.
Importantly, while changes in single-neuron properties can move
cortical networks from
one state to another,
no such changes
are required – changing the strength and duration of external
spiking input is sufficient.
In which of the two regimes do cortical networks operate in vivo? It appears that the
answer is both
<cit.>. However, a consistent observation is that
cortical networks do not remain completely silent for any extended period of
time. Consequently, networks that do operate in the quiescent state
tend to undergo frequent brief activations even in the absence of any
sensory stimulation.
Based on the above picture, we make a number of specific predictions,
and violation of any of them would invalidate our theory.
First, brief activations in the quiescent state are a
network effect shaped by the recurrent interactions, so
changes in excitatory and inhibitory drive occur
in all local neurons nearly
simultaneously. To test this prediction, one needs to record
intracellularly from at least two nearby neurons.
The only study we are aware of
that has performed such measurements across many pairs
of neurons was carried out in rodent somatosensory
cortex<cit.>, and, in line with our theory,
excitation and inhibition were synchronous across neurons.
We predict that this must be true for
any cortical network that operates in the quiescent state, including
auditory cortex,
for which brief activations of single neurons have been reported
previously<cit.>. Furthermore, our theory predicts that
switches between the quiescent and the active state (not just
brief, graded excursions) have to occur in
all local neurons nearly synchronously, which implies that membrane
potentials have to be highly correlated across neurons whenever
switches between up and down states are observed in any individual
neuron<cit.>. In essence,
observing a single neuron is sufficient to predict the timing and
general features of the subthreshold activities of all other nearby
Second, we predict that the excitatory and
inhibitory drives to individual neurons should rise and fall
together, with a short lag of inhibition behind excitation –
irrespective of network state. This should be true across neurons as well, so
for any pair of neurons in a local network, the cross-correlation
between excitatory drives and between inhibitory drives should always
peak at zero time lag, and the cross-correlation between excitatory and
inhibitory drives should exhibit a small time lag.
This should be a robust phenomenon that doesn't depend on either the wiring details nor on the
momentary amount of synaptic
Third, oscillations should be very common, and should change with
network state – a phenomenon that is seen frequently, and often cited
evidence that oscillations are involved in information processing
(reviewed in reference wang_neurophysiological_2010).
Our theory also explains more subtle effects,
such as why, during oscillations, bigger excursions in excitation are
followed by bigger excursions in inhibition after a short delay of a
few milliseconds<cit.>.
Besides making a specific set of predictions, our theory provides an
alternative explanation of state switching, which is that it can
be caused by simply modulating the strength or temporal features of
the external input. Typically, state switching – switching between
the active and
quiescent states – is attributed to
neuromodulators that change the membrane and synaptic
properties of neurons<cit.>.
This mechanism
is compatible with our framework because, as pointed out above,
changes in single neuron properties
can, by themselves,
cause transitions between the active and quiescent states.
However, if all state switches in cortex were to depend on
neuromodulation, then both the timing and location of the switches
would be difficult to control on either a fine temporal
or spatial scale. That's because neuromodulators are
released globally, rather than locally, and the timescale
for removing their effect is too long to permit rapid switches.
Neuromodulators are not, of course, the only candidate drivers of
state switches. A recent study reported that the level of tonic firing
in thalamus can control the state of its target area in
However, the mechanism was speculated to be specific to
thalamic input, and separate from the effect of sensory signals
transmitted by the thalamus. In contrast, our theory shows that any
changes in external input, from any area, can lead to state switching.
Another recent study explored the effect of
visual stimuli on
the strength and spatial spread of spike-triggered average LFP waves
in primary visual cortex (V1)<cit.>. While such
averaged LFP waves were strong without stimulation, they were
essentially absent during strong full-field stimulation – in
effect, the external input caused a state switch. The authors
proposed that this state switch occurred because
the external input modified the
effectiveness of lateral connectivity, perhaps due to a
modulation of the spatial spread of dendritic integration. In light of
our theory, however, this phenomenon can be explained by
noting that strong sensory input is likely
to move the network from a
quiescent state punctuated by brief periods of
activity to a continually active
state. As discussed above, spikes that occur during the brief
periods of activity are temporally precise and
spatially correlated. Consequently, they exhibit
long-range correlations with each other, and thus long-range
correlations with the LFP.
Spikes that occur during the active state, on the other
hand, are irregular and spatially uncorrelated, and thus can exhibit
only short range correlations with the LFP.
phvImplications for information processing
Perhaps the
most important conclusion we can draw from our analysis is that
observation of any of the phenomena predicted by our theory is not, in
and of itself, evidence for any deep computational principle – it is
evidence that one is recording from an area with all the properties
of a randomly connected network. In particular, one should not be
surprised by brief, highly synchronous bursts of activity,
precise timing
between the excitatory and inhibitory drives to individual or different neurons,
oscillations that change with network state<cit.>,
or correlations between
neurons<cit.>, and between
neurons and the local field
that change with network state.
This does not imply, of course, that the brain doesn't make use of
some – or all – of these phenomena. Indeed, Loebel
and colleagues<cit.>
showed theoretically that brief, highly
synchronized bursts of activity could provide a precise temporal code
for complex sounds;
there have been a large number of proposals for the computational role
of oscillations<cit.>; and the ability to
control the correlational structure among neurons simply by modifying
the input to an area (a key outcome of our model) could play a role in
gating information. However, because all of these phenomena occur
naturally in randomly connected networks, providing a direct link to
an underlying computational
principle becomes a nontrivial experimental task.
In sum, starting from a limited set of generic cortical network
properties, our theory
reconciles and unifies a large body of existing experimental data that
previously appeared either unrelated or suggested fundamental
differences between cortical areas. At the same time, it
makes a set of specific predictions about
possible firing patterns and correlational structures for
a range of input
regimes – predictions that should hold for any layer in any
cortical area.
To the best of our knowledge, the data in
all of the existing
studies are in excellent agreement with our theory.
However, not all of our predictions have been tested, nor have
parameters, layers and areas been thoroughly explored, in
large part because of the difficulty of recording intracellularly from
multiple cortical neurons in vivo.
If our theory does hold up, though,
the fact that a relatively simple framework can
explain a large variety of electrophysiological recordings
in vivo would indicate that, at least at the level of
population firing rates and subthreshold and spiking correlations
between neurons, cortical network dynamics is reasonably
straightforward to understand.
Finally, given that there is a great deal of structure in cortical
networks, even at levels that span just several layers within a single
column <cit.>, it might seem surprising that a model
based on randomly connected networks could explain such a large body
of in vivo data. However, it is important to note that the
observations we explain are essentially observations about collective
properties: excitation and
inhibition of individual neurons
(which are due to the summation of many
presynaptic neurons), membrane
potential and
single-neuron spike statistics, population-averaged firing rates,
correlations, and oscillations.
A solid understanding of the
universal properties of cortical dynamics, as we have attempted
to provide here, is a prerequisite for
uncovering those activity patterns that are a signature of
specific functional roles.
This work was supported by the Gatsby Charitable Foundation and and US
National Institute of Mental Health grant R01 MH62447. We would like
to thank Ken Harris and David Barrett for insightful comments on the
The network equations we simulated follow the form of those
given in
the main text, Eq. (<ref>); all that
remains to be specified is
the single neuron model, $I_i^\alpha(V_i, \b c_i)$
the equation for the synaptic conductances, the $g_j^\beta$, and an
expression for the external input, $I_i^{\alpha, ext}$.
For the single neuron model we
use a quadratic integrate and fire neuron
<cit.> with
parameters that are independent of neuron type, so that
$I^{\alpha}(V, \b c) \rightarrow I(V)$. Using $R_m$ to denote the membrane
resistance, $I(V)$ is given by
\begin{equation} \label{single-neuron-dyn}
I(V) =
\frac{(V - V_r)(V - V_{th})}{R_m(V_{th} - V_r)}
\, .
\end{equation}
For the synaptic conductances, $g_j^\beta(t)$, we
assume an instantaneous rise and exponential decay,
\begin{equation} \label{g-dyn}
\tau_s \frac{d g_j^\beta}{dt} = - g_j^\beta
+ g_0 \tau_s \sum_k \delta \big(t-t_j^k \big)
\, ,
\end{equation}
where $g_0$ is the conductance associated with a unitary event,
$t_j^k$ is the time of the $k^{\rm th}$ spike on
neuron $j$, and $\delta(t)$ is the
Dirac delta function. A spike is emitted when the voltage reaches
$+\infty$, at which point it is reset to $-\infty$.
For the external input we assumed a conductance-based coupling,
\begin{equation} \label{i_ext}
I_i^{\alpha, ext}(t) =
W_\alpha^{ext} G_i^{ext}(t) \big( V_i^\alpha - \mathcal{E}_E \big)
\, .
\end{equation}
The conductance associated with the external
input, $G_i^{ext}(t)$, varied from one simulation to another, and is
discussed below.
Connectivity was sparse: the connection probability between any two
neurons was 0.1, independent of neuron type. If there was a connection,
the normalized connection strengths, $W_{ij}^{\alpha \beta}$, was
equal to $W_{\alpha \beta}$ on average, with Gaussian noise around
that mean,
\begin{equation} \label{weights}
W_{ij}^{\alpha \beta} = \left\{ \begin{array}{ll}
W_{\alpha \beta}(1 + 0.1 \eta) & \ \ \hbox{probability } 0.1
\\
0 & \ \ \hbox{probability } 0.9
\end{array} \right.
\end{equation}
where $\eta$ is a zero mean, unit variance Gaussian random variable,
taken to be independent across weights.
The network parameters are given in Table I. Note that $R_m$ and $C_m$
were chosen so that the membrane time constant is 20 ms.
The connection strengths,
$W_{EE}$, $W_{EI}$, etc., given in that table were chosen to generate
the following PSP sizes:
\begin{equation} \label{psps}
\begin{array}{ll}
E \rightarrow E: \ \ 0.95 \hbox{ mV}, &
\ \ \ \ \ \ E \rightarrow I: \ \ 1.19 \hbox{ mV}
\\
I \rightarrow E \ : -1.96 \hbox{ mV}, &
\ \ \ \ \ \ \ I \rightarrow I: -1.96 \hbox{ mV} \, .
\end{array}
\end{equation}
To find the connection strengths that
generate the right PSP sizes, we
linearized the single neuron dynamics around rest ($V_i=V_r)$,
the PSP amplitude in response to a single presynaptic spike, and
adjusted the connection strength to achieve the values given in
Eq. (<ref>). The resulting expression for the average
weights in terms of PSP amplitudes is
\begin{equation} \label{v_to_w}
W_{\alpha \beta} =
\frac{V_{\alpha \beta}}{\mathcal{E}_\beta - V_r}
\frac{1}{R_m g_0}
\frac{\tau_m}{\tau_s} \exp \left[ \frac{\log
\tau_m/\tau_s}{\tau_m/\tau_s-1} \right]
\end{equation}
where $V_{\alpha \beta}$ is the PSP on a neuron of type $\alpha$ given
a presynaptic spike on a neuron of type $\beta$. This formula yields
the values for the weights given in Table I.
Table I. Network parameters.
number of excitatory neurons, $N_E$ 1600
number of inhibitory neurons, $N_I$ 400
connection probability 0.1
membrane resistance, $R_m$ 100 M$\Omega$
membrane capacitance, $C_m$ 200 pF
membrane time constant, $R_m C_m \equiv \tau_m$ 20 ms
synaptic time constant, $\tau_s$ 5 ms
unitary conductance, $g_0$ 0.928 nS
resting membrane potential, $V_r$ -65 mV
threshold, $V_{th}$ -50 mV
excitatory reversal potential, $\mathcal{E}_E$ 0 mV
inhibitory reversal potential, $\mathcal{E}_I$ -70 mV
$W_{EE}$ 1
$W_{IE}$ 1.253
$W_{EI}$ 26.82
$W_{II}$ 26.82
$W_E^{ext}$ 1
$W_I^{ext}$ 0.667
Although we report only simulations with $N = 2,000$ neurons, we
performed simulations with $N$ ranging from 1,000 to 12,000. For all
simulations, the connection probability was fixed at 0.1. The PSP size,
however, followed the $K^{-1/2}$ scaling suggested by van Vreeswijk
end Sompolinsky <cit.>, where $K$ is
the average number of connections per neuron; to achieve this, we
scaled the synaptic weights
relative to the values shown in Table I
by a factor of $(K_0/K)^{1/2}$ where
$K_0 = 200 \, (= \, 0.1 \times 2,000)$
is the number of connections/neuron when $N=2,000$, and $K=0.1N$.
It is also necessary to scale the external connection strengths, but
by $K^{1/2}$ instead of $K^{-1/2}$; we thus scaled $W_E^{ext}$
and $W_I^{ext}$ by a factor of $(K/K_0)^{1/2}$.
With this scaling, results were qualitatively the same for all network sizes.
Simulations were performed using a 4th-order Runge-Kutta
integration scheme with a time step of 0.2 ms. To avoid the
infinities associated with spike threshold and reset, we made the
change of variables $V_i = (V_{th}+V_r)/2 + (V_{th}-V_r) \tan(\theta_i/2)$,
and integrated $\theta_i$ rather than $V_i$.
This moved the
spike threshold and reset to $\theta_i=\pi$.
External input
The only difference among the simulations used to make
Figs. <ref> and
was the external
conductance, $G_i^{ext}(t)$. Although ultimately this conductance
comes from spikes in other areas, for simplicity we
characterized it as a time-dependent function filtered by
the synaptic time constant,
\begin{equation} \label{g-ext}
\tau_s \frac{d G_i^{ext}}{dt} = - G_i^{ext} + g_0 h_i^{ext}(t)
\, .
\end{equation}
The normalized drive, $h_i^{ext}(t)$,
was constant for Figs. <ref>
and <ref> (active state),
consisted of brief synchronous pulses for
Fig. <ref> (quiescent state),
was sinusoidal for Fig. <ref>,
and alternated between sinusoidal and
Gabor functions for Fig. <ref> (state switching).
More quantitatively, for
Figs. <ref> and <ref> we
\begin{equation} \label{g_active}
h_i^{ext}(t) = 5
\, .
\end{equation}
Because $h_i^{ext}(t)$ is constant, for this case $G_i^{ext}$ is also
constant (and independent of neuron), and equal to $5g_0$.
Given that the integral of the conductance
change per spike is $g_0 \tau_s$ (see Eq. (<ref>)), this
corresponds to the mean input produced by 40 neurons firing at
rate 25 Hz (40 $\times$ 25 Hz $\times$ 5 ms = 5).
For Fig. <ref> the input consisted of 40 ms pulses
arriving randomly at an average frequency of 8 Hz, with amplitudes
drawn from a uniform distribution. More specifically,
\begin{equation} \label{g_quiescent}
h_i^{ext}(t) =
\sum_k a_k \hat{T}\big(t-(\tau_k+\delta t_i) \big)
\end{equation}
where $\hat{T}(t)$ is a 40 ms top hat function ($\hat{T}(t) = 1$ if $0 < t
< 40$ ms and zero otherwise); the $\tau_k$, the average start times of
the pulses, were Poisson distributed with rate 8 Hz; the $a_k$, the
pulse amplitudes, were
drawn from a uniform distribution over
the range 0 to 5; and $\delta t_i$, which produces different
arrival times of the pulses at different neurons, was drawn from a
uniform distribution over the range $-2$ to $+2$ ms. Note that a
pulse can arrive before the previous one is finished; if this happens,
we didn't actually sum the pulses; instead, we used the amplitude of the
second pulse. We didn't indicate this in Eq. (<ref>) to
avoid overly complex notation.
In Fig. <ref>,
we report excitatory and inhibitory
drives. To determine these, we injected either positive or negative
constant current and
measured the membrane potential. (Although we could have
reported the excitatory and inhibitory conductances directly, we
instead report membrane potentials to make contact with experiments
To keep the cell from spiking during this procedure, we linearized the
single neuron dynamics and added a constant current, denoted $I_{clamp}$.
This had the effect of replacing the first term on the right hand
side of Eq. (<ref>)
by $-(V-V_r+I_{clamp}R_m)/R_m$.
To isolate excitatory drive, $I_{clamp}$ was chosen so that the
resting membrane potential shifted from -65 to -70 mV
($I_{clamp} R_m = -5$ mV);
to (partially) isolate inhibitory drive, $I_{clamp}$ was chosen so that the
resting membrane potential shifted to -40 mV ($I_{clamp} R_m = 25$ mV).
For Fig. <ref>, the input
was sinusoidal with a frequency of 1.5 Hz and it was offset to make it non-negative,
\begin{equation} \label{state_raster}
h_i^{ext}(t) = 1 + \sin (2 \pi \times 1.5 \, t)
\, .
\end{equation}
For Fig. <ref>, the input in the Sub/Suprathreshold
regime was the same as in Fig. <ref>.
In the Suprathreshold regime the input consisted of
a constant offset modulated by a series of Gabor
functions, denoted $f_G(t)$ and given by
\begin{equation} \label{gabor}
f_G(t) =
\cos (\omega_G t + \phi_G) \exp ( -t^2/2 \sigma_G^2 )
\end{equation}
where $\omega_G = 2 \pi \times 80$ Hz,
$\phi_G = 0.628$ radians (36
degrees) and $\sigma_G = 0.015$ s. The two regimes alternated,
\begin{equation} \label{g_switch}
h_i^{ext}(t) =
\left\{ \begin{array}{ll}
1 + \sin (2 \pi \times 1.5 \, t)
& \ \ 0 < t \le 4 \hbox{s and } 9 \hbox{s} < t \le 15 \hbox{s}
\\
1.6 + 2 \sum_n f_g(t_G + t - 2 n t_G)
& \ \ 4 \hbox{s} < t \le 9 \hbox{s}
\end{array} \right.
\end{equation}
where $t_G = 0.055 s$ and the $n$ are
chosen so that the Gabor functions fill
the region between 4 and 9 seconds. The phase offset, $\phi_G$, was included
to make the Gabor functions slightly asymmetric, as seen in
experiments <cit.>.
In Figs. <ref> and
<ref> we reported
correlation coefficients. For these we used the standard formula,
\begin{equation} \label{corr}
C(\tau) =
\frac{\hbox{Covar}[X_1(t+\tau),X_2(t)]}
{\left(\hbox{Var}[X_1] \hbox{Var}[X_2]\right)^{1/2}}
\end{equation}
where $X$
can be either conductance or membrane potential and the average is
over time.
Relationship between conductances and firing rates
Here we show that the shared conductance are indeed proportional to
the firing rates, as indicated in Eq. (<ref>). Our starting point
is an explicit expression for the shared conductances,
\begin{equation} \label{gqr_def}
G_{\alpha \beta} (t) \equiv
\frac{1}{N_\alpha} \sum_{i=1}^{N_\alpha} G_i^{\alpha \beta}(t) =
\frac{1}{N_\alpha} \sum_{i=1}^{N_\alpha} \sum_{j=1}^{N_\beta} W^{\alpha
\beta}_{ij} g_j^\beta(t)
\, .
\end{equation}
This follows from Eq. (<ref>) and
the fact that the shared conductances are, by
definition, the average conductance seen by all neurons of a given
The sum over $i$ in the rightmost term,
which acts only on $W_{ij}^{\alpha \beta}$, results
in a quantity that depends on $j$, but that dependence is smaller than
the mean by a factor of $1/K^{1/2}$.
Thus, because $K$ is large in cortex,
it is a good approximation to ignore the
$j$-dependence, and set $\sum_i W_{ij}^{\alpha \beta}$
to $K W_{\alpha \beta}$
where, as in the main text,
$W_{\alpha \beta}$ is the average synaptic strength made by a
neuron of type $\beta$ onto a neuron of type $\alpha$, assuming that a
connection is made (see Eq. (<ref>)).
With this replacement, Eq. (<ref>) becomes
\begin{equation} \label{gab1}
G_{\alpha \beta} (t) =
\frac{K W_{\alpha \beta}}{N_\alpha}
\, \sum_{j=1}^{N_\beta} g_j^\beta(t)
\, .
\end{equation}
To relate synaptic conductances,
$g_j^\beta(t)$, to firing rates, we note that the
$g_j^\beta(t)$ can be written as a sum over spike times,
\begin{equation} \label{gsingle1}
g_j^\beta(t) =
\sum_k f_j^\beta(t-t_j^k)
\end{equation}
where $f_j^\beta(t)$ is the conductance change associated with a
single presynaptic spike at time 0 and $t_j^k$ is the $k^{\rm th}$
spike on neuron $j$. (For our network simulations, $f_j^\beta(t) =
e^{-t/\tau_s}\Theta(t)$ where $\Theta(t)$ is the Heaviside step
function, but our analysis would hold for essentially any conductance
The synaptic conductances, $g_j^\beta(t)$,
can be broken into a trial-averaged piece and
fluctuations around that average. The first of these is just an
average over the probability of
spikes, which is determined by the firing rate; the second we denote
$\delta g_j^\beta(t)$. This gives
\begin{equation} \label{gsingle2}
g_j^\beta(t) =
\int d\tau \, \nu_j^\beta(\tau) f_j^\beta(t-\tau)
+ \delta g_j^\beta(t)
\, .
\end{equation}
If the fluctuating terms are independent, which we assume here, in the
limit of a large number of neurons they make a negligible contribution
to the sum over $j$ in Eq. (<ref>) (at least compared to the
We thus need to focus only on the first term in Eq. (<ref>).
Assuming the firing rates change slowly compared to the time course of the
conductance changes – which is reasonable, given that the latter
happen at a millisecond timescale – we may replace
$\nu_j^\beta(\tau)$ by $\nu_j^\beta(t)$. Then, ignoring the
fluctuating piece, $\delta g_j^\beta(t)$, we have
\begin{equation} \label{gsingle3}
g_j^\beta(t) \approx
\nu_j^\beta(t) \int d\tau \, f_j^\beta(t-\tau)
\equiv
\nu_j^\beta(t) F_j^\beta
\end{equation}
where $F_j^\beta$ is the time integral of a single conductance change.
For our model, $F_j^\beta = \tau_s$, but for just about any model
we can choose the weights so that $F_j^\beta$ is independent of $j$ and
$\beta$. Thus, $g_j^\beta(t)$ is approximately proportional to
$\nu_j^\beta(t)$, and we recover Eq. (<ref>).
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1511.00096
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Visual motion estimation is a computationally intensive, but important task for sighted animals. Replicating the robustness and efficiency of biological visual motion estimation in artificial systems would significantly enhance the capabilities of future robotic agents. 25 years ago, in this very journal, Carver Mead outlined his argument for replicating biological processing in silicon circuits. His vision served as the foundation for the field of neuromorphic engineering, which has experienced a rapid growth in interest over recent years as the ideas and technologies mature. Replicating biological visual sensing was one of the first tasks attempted in the neuromorphic field. In this paper we focus specifically on the task of visual motion estimation. We describe the task itself, present the progression of works from the early first attempts through to the modern day state-of-the-art, and provide an outlook for future directions in the field.
§ INTRODUCTION
Visual sensing is a computationally intensive, but crucial task for sighted animals <cit.>. Although motion estimation is just one aspect of visual sensing, its importance is easily understood when observing its wide range of uses in biology <cit.>, including depth perception, ego-motion estimation, collision avoidance and triggering escape reflexes, time-to-contact estimation (landing control), prey detection and identification, segmentation by motion, and visual odometry.
The ability to reliably estimate visual motion in artificial systems would find applications ranging from surveillance and tracking, to visual flight control <cit.>, to video compression, image stabilization, and even the computer mouse <cit.>. However, the most relevant application of bio-inspired visual motion estimation is for embedded sensing onboard robotic agents capable of moving through and interacting with their environment, since this is precisely the function for which biological visual motion systems have evolved. Key characteristics which distinguish this application from others are: the sensor must operate in real-time, the sensor must operate under egomotion, the sensor must be physically carried by the agent, and the sensor should provide information which is relevant for enabling the agent to interact meaningfully with its environment. In fact, Wolpert argues in a recent TED talk that control of motion is the primary purpose of the brain <cit.>.
There are significant differences between biological and artificial systems regarding how visual information is acquired and processed. State-of-the-art modern visual motion estimation methods still rely on capturing sequences of images (frames) in rapid succession, even though the majority of data in these images is redundant <cit.>. The problem of storing and transmitting this redundant information is partially overcome by using dedicated video compression ASICs, or in the case of standalone visual motion sensors, by computing on chip <cit.>. Nevertheless, these artificial approaches capture frames at pre-determined discrete time points regardless of the visual scene. On the other hand, biological retinae continuously capture data and perform a combination of compression and pre-processing in analog (using graded potentials) at the focal plane itself, with the visual scene largely driving when and where data is transmitted as spikes (voltage pulses) down the optic tract. Spikes are similar to digital pulses in artificial systems in that their signal amplitude can be restored and they are therefore particularly useful for communicating over longer distances, such as along the optical tract.
Processing also differs significantly between biological and artificial systems. Similarly to how artificial systems typically capture data at a constant rate, they must also compute at a constant rate to ensure all the captured data is processed, thus processing of visual information continues even if the scene is static. On the other hand, computation in biological systems is driven by the sparse captured data (spikes), in turn ensuring that neuron activation is sparse <cit.> (since neuron activation is driven by the sparse incoming data). This sparsity combined with the low power consumption of neurons which are not computing <cit.> results in significant energy savings. Modern biologically inspired sensors generate sparse data (events) in response to activity in the scene <cit.> and this data can be used to drive sparse computation on modern neural simulator platforms <cit.>. Together, these sensors and neural simulators allow both data capture and computation to scale with scene activity.
The architectures used for processing also differ. Typical artificial systems compute on a small number of parallel processors, each of which performs sequential operations in a precise repeatable manner, and operates at a timescale on the order of nanoseconds. On the other hand, biology relies on massively parallel processing using a very large number of imprecise computing elements (neurons), each of which operates on a timescale on the order of milliseconds <cit.>. However, parallelism in artificial systems is increasing, particularly for visual processing (GPUs), and emerging custom neural hardware platforms <cit.>. State of the art ASICs dedicated to visual motion estimation are also optimized to perform processing in as parallel a manner as possible.
Despite the imprecise nature of individual neurons, biological systems perform robustly and continue to do so even after the death of individual neurons. The same is not true of artificial systems, where a single fault can cause catastrophic failure of the entire system. Similar fault tolerance is highly desirable in artificial systems, especially as the number of transistors per device continues to increase, and as the size limits of silicon technology continue to be pushed. Even looking past silicon to nanotechnology, device yield continues to be a major challenge <cit.>.
Beyond handling minor faults, biological systems are also able to learn and adapt to changes in the visual system itself <cit.>, as well as to different environments through visual experience <cit.>, allowing them to operate effectively under a wide range visual of conditions. Such self-contained on-line learning and adaptation would also prove valuable for artificial systems, removing any need for manual tuning of parameters for operation in different environments.
Biology's robust processing, low power consumption, and ability to learn and adapt all present desirable characteristics for artificial systems and drive the field of bio-inspired sensing and computation.
In this paper we provide a brief introduction to the visual motion estimation problem and provide background on the methods used in traditional computer vision versus biological systems, before reviewing advances in bio-inspired visual motion estimation for artificial systems, presenting our own approach to the problem, and discussing future directions.
§ THE VISUAL MOTION ESTIMATION PROBLEM
When relative motion is present between an observer (eye or camera) and objects in a scene, the projections of these objects onto the image plane (retina or pixel sensor) will move. Visual motion estimation is the task of estimating how the projections of these objects move on the image plane.
Definition of axes used in (<ref>) for a pinhole camera approximation with unit focal length. The $z$ axis points out towards the scene perpendicular to the image plane, while the $x$ and $y$ axes are parallel to the image plane.
Assuming a pinhole camera approximation, with the image plane at unit focal length, visual motion can be described as a function of the image plane co-ordinates $(x,y)$, the relative rotation $(\omega_x, \omega_y, \omega_z)$ and translation $(T_x, T_y, T_z)$ between the camera and the object being viewed, and the depth of the object $(z)$ <cit.> as shown in Fig. <ref> and (<ref>) below.
\begin{equation}\label{eq:VisualMotion}
\begin{array}{l l l}
\frac{\delta x}{\delta t} &= &\frac{T_zx-T_x}{z} - \omega_y + \omega_zy + \omega_xxy - \omega_yx^2\\
\frac{\delta y}{\delta t} &= &\frac{T_zy-T_y}{z} + \omega_x - \omega_zx - \omega_yxy + \omega_xy^2\\
\end{array}
\end{equation}
Visual motion is constrained to lie in the image plane and therefore has no $z$-direction component. The first term in each equation describes the visual motion due to translation, which is depth dependent, while the remaining terms describe visual motion due to rotation, which is independent of depth. In other words, visual motion due to rotation does not depend on the structure of the scene, while visual motion due to translation does depend on scene structure. The rotations and translations in the equation above are for motion of the camera relative to the origin as depicted in Fig. <ref>.
The aperture problem illustrated with a triangle moving between an initial position (grey) and a final position (black), while viewed only through three apertures (blue circles). For the leftmost aperture, the image only varies in the horizontal direction, and therefore only the horizontal component of motion can be estimated. Similarly, for the middle aperture, only the vertical component of motion can be estimated. For the rightmost aperture, the viewed image varies along both the horizontal and vertical directions and therefore motion can be uniquely determined.
The relationship described in (<ref>) also shows that multiple different combinations of scene structure and relative motion can result in identical visual motion. Thus visual motion alone is not enough to infer relative motion or scene structure and additional information is required. For example, if the scene is static and the rotational motion of the sensor is known, then the translational motion direction can be determined, and a relationship between scene depth and camera translation speed can be obtained. Thus, measuring or even eliminating rotational motion allows additional valuable information to be derived from visual motion.
Visual motion can only be estimated in the presence of an intensity gradient. A shape of uniform colour moving against a background of identical colour will have no intensity gradient and will therefore not elicit a visual motion stimulus. More specifically, for motion to be detected, the intensity gradient must be non-zero in the direction of motion. This is an example of the aperture problem <cit.>, to which all visual systems are prone, and is illustrated in Fig. <ref>.
When considering only a small image region which has no intensity gradient in a particular direction, the magnitude of image velocity in that direction cannot be determined unless additional information is available. The component of motion in the direction of the maximum image gradient can be determined, and is known as the “normal flow", since it is perpendicular (normal) to the edge orientation. The larger the image region under consideration, the more likely it will contain gradients in different directions, helping to alleviate the aperture problem. A common approach is to simultaneously consider multiple neighbouring image regions and assume their motion to be either consistent <cit.> or smoothly varying <cit.>, thus providing the additional constraint required to uniquely determine motion.
§ APPROACHES TO VISUAL MOTION ESTIMATION
For the purpose of providing background for later sections, we introduce here the basic theory underlying each of the three main classes of visual motion estimation approaches: correlation methods, gradient methods, and frequency methods.
Underlying all three of these methods is the assumption of brightness constancy, known as the brightness constancy constraint, which states that the brightness of a point remains constant after moving a small distance on the image plane $[\Delta x, \Delta y]$, within a small period of time, $\Delta t$. Formally this can be written as:
\begin{equation}\label{eq:ImageConstancy1}
\begin{array}{l l l}
I(x,y,t) &\approx & I(x+\Delta x, y+\Delta y, t+ \Delta t)\\
\end{array}
\end{equation}
where $I(x,y,t)$ is the intensity of the point located at $(x,y)$ on the image plane at time $t$.
§.§ Correlation Methods
Correlation methods for motion estimation rely on detecting the same visual feature at different points in time as it moves across the image plane. Correlation is used to determine whether two feature signals detected at different points in time relate to the same or different features. The feature signals on which correlation is computed can take the form of continuous-time analog signals, discrete-time analog signals, discrete digital signals, or even single bit binary tokens indicating only the presence or absence of a feature (“token methods"). The change in feature location can be combined with the change in time between detections to determine the feature's motion. The simplest features to use are brightness patterns, or derivatives thereof, the appearance of which remains constant over small time periods as described in (<ref>).
Frequency representation of the motion described in (<ref>). The left plot (red) shows the relationship between time and location for a point moving with constant velocity. The right plot (blue) shows the velocity dependent relationship between spatial and temporal frequency for the same moving point.
Most state of the art commercially available motion estimation ASICs rely on correlation methods <cit.>. These devices capture frames at high frame rates and detect correlations in local pixel intensity patterns between images to determine motion. One very common approach is the Sum of Absolute Differences (SAD) block-matching algorithm <cit.>, which matches “blocks" of pixels between frames by computing the SAD for pixel intensities. This process is repeated for many different blocks and the estimates from all of these blocks are combined to determine motion.
§.§ Gradient Methods
Gradient methods rely on the Taylor series expansion of (<ref>), which for a first order expansion can be rearranged into the form:
\begin{equation}\label{eq:ImageConstancy2}
\begin{array}{l l l}
\frac{\delta I(x,y,t)}{\delta x}\frac{\delta x}{\delta t} + \frac{\delta I(x,y,t)}{\delta y}\frac{\delta y}{\delta t} +\frac{\delta I(x,y,t)}{\delta t} &=& 0\\
\end{array}
\end{equation}
where $\frac{\delta x}{\delta t}$ and $\frac{\delta y}{\delta t}$ are the visual motion values that must be estimated, while $\frac{\delta I(x,y,t)}{\delta x}$, $\frac{\delta I(x,y,t)}{\delta y}$ and $\frac{\delta I(x,y,t)}{\delta t}$ are intensity derivatives which can be obtained from captured frames.
Notice that if the intensity derivative in either spatial direction is zero, then motion in that direction is removed from the equation and cannot be estimated. This is the aperture problem discussed in Section <ref>. When these spatial derivatives are non-zero, but small, they are sensitive to noise and can still result in erroneous motion measurements. Even if accurate non-zero intensity derivatives are available, (<ref>) is a single equation with two unknowns and thus does not provide a unique solution.
To arrive at a unique solution additional constraints must be imposed, such as that the motion of all points in an image patch will be equal (as is used in the Lucas-Kanade algorithm <cit.>), or that motion varies smoothly across image locations (as is used in the Horn-Schunk algorithm <cit.>).
§.§ Frequency Methods
Frequency based methods rely on the observation that there is a relationship between temporal frequency, spatial frequency, and velocity <cit.>. For simplicity consider a point (Dirac Delta function <cit.>) moving in the $x$ direction. This point will trace out a line in the space-time plot on the left of Fig. <ref> with slope equal to the velocity. Taking the Fourier transform results in a line in frequency space with slope equal to the inverse of velocity, as shown on the right of Fig. <ref>.
\begin{equation}\label{eq:FrequencyMethod}
\begin{array}{l l l}
I(x,t) &=& \delta(x-v_x t)\\
v_x &=& \frac{\Delta x}{\Delta t}\\
I_{\omega}(\omega_x,\omega_t) &=& \delta(\omega_t+v_x\omega_x)\\
\frac{1}{v_x} &=& -\frac{\Delta\omega_x}{\Delta\omega_t}\\
\end{array}
\end{equation}
where $I(x,t)$ is the intensity at location $x$ at time $t$, $v_x$ is the velocity in the $x$ direction, $\delta$ is the Dirac Delta function, and $I_{\omega}(\omega_x,\omega_t)$ is the Fourier transform <cit.> of $I(x,t)$.
The case described above is an ideal case where the stimulus is a point (Dirac Delta function) and therefore has equal energy at all spatial frequencies. In the more general case of a stimulus with an arbitrary distribution of spatial frequency content, the energy will still be constrained to lie along the line shown on the right of Fig. <ref>.
Visual motion can be estimated by finding the slope of the line in Fig. <ref>, which can be achieved by tiling frequency space with spatiotemporal filters as shown in Fig. <ref> and combining their responses to find the location of the energy peak.
An example placement of four different quadrature pairs of spatiotemporal filters in frequency space. Each quadrature pair is indicated by a different color and is symmetric about the $\omega_t$ axis and the plane $\omega_t=0$ (shaded gray). The red and dark blue quadrature pairs are sensitive to different velocities in the $y$-direction. The green quadrature pair is sensitive to motion of a specific speed in the $x$-direction. The light blue quadrature pair is most sensitive to a particular velocity which has both $x$ and $y$ components.
§.§ Implementation
Typical artificial approaches to motion estimation rely on a frame-based camera to capture snapshots (frames) of the scene at fixed intervals. A processor is then used to apply one of the three methods described above to estimate visual motion. In computer vision typically correlation or gradient based methods are used, with frequency methods regarded as bio-inspired approach as will be discussed in Section <ref>.
In all three of the methods outlined above, increasing the frame rate improves the accuracy of the algorithm because the brightness constancy constraint relies on the assumption of a small time period between observations (frames).
For correlation methods, increased frame-rate also decreases the distance a feature can move between frames, thereby helping to restrict the search region for that feature in subsequent frames. For example, typical optical mouse algorithms operate at thousands of frames per second (albeit small frames), and the search can be in increments of fractions of a pixel. For gradient and frequency methods, increased frame-rate is equivalent to a higher temporal sampling rate, reducing aliasing and allowing for higher order digital filters to be used when estimating the temporal derivative or frequency.
However, increasing frame-rate also increases the computing power required to sustain real-time operation, since more frames must be processed within the same time period. The additional computing needs are typically met by using more powerful hardware, such as GPUs, FPGAs, and custom ASICs. Tight coupling of a frame-based sensor and ASIC is often used to reduce communication costs for embedded applications, such as for a stand alone motion estimation unit relying on a high frame rate, or in-camera video compression relying on block matching.
Some artificial approaches to motion estimation do not rely on frames, but instead process on continuous-time analog signals derived from CMOS photodiodes. Notable examples include <cit.>.
For all three of the approaches described, motion estimates within a local image region can be computed independently of motion estimates for other image regions. This opens up the possibility of simultaneously computing motion for different image regions in parallel. GPUs, FPGAs, and ASICs all take advantage of this.
The computational complexity of each approach varies. A major disadvantage of gradient methods is that they typically require the expensive computation of a matrix inverse (or pseudo-inverse) in order to find the solution which best satisfies (with the least square error) the brightness constancy (<ref>) and secondary constraints. However, gradient methods have the advantage of computing on instantaneous gradient values and therefore require very little memory (only enough to estimate temporal gradients).
Computing correlations for correlation methods is computationally simpler, but requires the system to have memory of previously observed features, whether by delaying feature signals or by explicitly storing them.
Digital implementations of frequency methods require even more memory because many time points are required to detect temporal frequencies, particularly if very low frequencies are present. The spatial and temporal frequency content of the scene is typically not known in advance, so frequency methods require a large number of spatiotemporal filters in order to accurately detect the frequency content of different motion stimuli. Implementing these filters digitally is costly both in terms of computation and memory.
The robustness of the approaches also varies. When computing the correlation between two signals, it is not always clear whether a large output has resulted from a strong correlation or from large input signals. Signals can be normalized before correlation to overcome this ambiguity. However, the typical signals on which correlation is computed arise from a combination of image motion and image spatial contrast and it is not always possible to disambiguate the effects of motion versus contrast in the final output.
Gradient methods rely on the ratio of the temporal gradient to the spatial gradient (<ref>) and are therefore very sensitive to noise, particularly when spatial gradient signals are weak.
Spatiotemporal frequency models are far more robust to noise, but as discussed above, the computational and memory requirements for estimating frequencies is far higher.
§ VISUAL MOTION ESTIMATION IN BIOLOGY
A comparison of the Hassenstein-Reichardt (left), Barlow-Levick (center), and Adelson-Bergen (right) models, similar to that in <cit.>. Spatial filters (dark blue) can either differ in location (as shown) or in phase. Two temporal filters (red) are used, with the second ($\omega_{t2}$) having a longer delay than the first ($\omega_{t1}$). The Reichardt model detects correlation between the signal at one location and the delayed signal from a neighbouring location. The correlation can be modelled as a multiplication (brown), or as a logical AND if using single bit signals. The Barlow-Levick model instead uses the signal to block response to motion in the null direction. This can be modelled as a summation (green) followed by half wave rectification (light blue) to prevent negative intermediate responses. If using 1-bit signals, the inhibition can be modelled as an AND gate with the inhibiting (delayed) input negated. The Adelson-Bergen model combines separable spatiotemporal filter responses incorporating a squaring (light blue) non-linearity to compute motion energy. The output of the Adelson-Bergen model is formally equivalent to that of the Hassenstein-Reichardt model <cit.>.
The animal kingdom is incredibly diverse and the visual systems of many creatures have evolved independently <cit.> (although they may share a common origin), resulting in variations in size, number, shape, location, wavelength sensitivity, and acuity of eyes across families <cit.>. Similarly to how eyes vary by family, so does the process of visual motion estimation. For the sake of discussion we will focus specifically on Drosophila (fruit flies) and macaque (monkeys), which are both well studied genera, but possess very different visual systems. Properties of the vision systems of Drosophila and macaque generalise to many other insects and mammals respectively. This section is intended only as a brief introduction. For more details, the reader is directed to neuroscience reviews covering Drosophila <cit.> and primate vision <cit.>.
Despite the differences between macaque and Drosophila vision systems, there are also many similarities. In both systems initial computation is performed at the focal plane by neurons which communicate using a combination of spiking and graded responses. These neurons respond to intensity changes, with responses to intensity increases (ON) and decreases (OFF) processed independently by parallel pathways <cit.>. In Drosophila, motion estimation can still be performed in the L1 (ON) pathway if the L2 (OFF) pathway is blocked, and vice versa.
In macaque, direction selective Starburst Amacrine Cells (SACs) <cit.>
can be found in the retina itself, and separate SACs are used to process ON and OFF responses in parallel. Although direction selective, these cells show very limited speed sensitivity, with true velocity sensitive neurons only found in higher visual areas.
In Drosophila, as with many other animals, a fast escape reflex triggered by visual motion aids in evading approaching predators. Low latency motion detection is critical for this task and is achieved by keeping the motion processing circuitry relatively simple and located close to the photoreceptors <cit.>. These motion processing circuits rely on correlation methods, which are fast, and allow for compact implementation which can be realised in the limited space available near the photoreceptors <cit.>. Furthermore, these circuits are tuned to detect stimuli characteristics indicative of an approaching predator, rather than to accurately measure a wide range of complex motion stimuli.
On the other hand, the macaque visual system is more concerned with accurately estimating motion for a wide range speeds and visual stimuli than detecting approaching predators. Accurate motion estimates help to achieve a deeper scene understanding, which can then be used for action planning. This link between motion estimation and scene understanding requires interaction between motion detectors and cortex, and motion sensitive neurons are therefore found in various cortical areas <cit.>.
Low latency detection is also desirable for macaque, but is not as important as when triggering escape reflexes, allowing the luxury of using more complex motion estimation methods and taking advantage of the computational resources available in cortex. The macaque visual system is therefore not restricted to using simple correlation methods. The presence of spatiotemporal frequency sensitive neurons in the Middle Temporal (MT) visual area, which plays an important role in motion estimation <cit.>, suggests that motion estimation in primates relies on frequency methods <cit.>.
In both macaque and Drosophila the visual motion estimation system is tightly coupled with the motor system. In macaque, the vestibular ocular reflex is important for visual perception <cit.>. Visual inputs can trigger saccades, and saccades can suppress visual responses <cit.>. In Drosophila, the visual system can trigger motor responses through the optomotor reflex, and flight control is heavily reliant on visual motion estimation <cit.>.
It is therefore important when considering biological vision systems to note that they do not exist in isolation. The visual system is part of an embodied system capable of moving through, and interacting with the environment. The visual and motor systems are tightly coupled and deficits in either system can affect the other, as documented in both primates <cit.> and Drosophila <cit.>. The optomotor response is so strong in many insects that motor outputs in response to visual stimuli can provide insight into the visual system .
It was through an investigation of the optomotor response of the beetle Clorophanus in the 1950s that Hassenstein and Reichardt arrived at their seminal model of the Elementary Motion Detector (EMD) <cit.>, shown in Fig. <ref>a.
The Hassenstein-Reichardt EMD computes the correlation between the signal of one photoreceptor and the time-delayed signal of a neighbouring photoreceptor. The delay is typically modelled as a low pass filter and correlation performed as multiplication. Strong correlation indicates the presence of motion in the preferred direction. A mirror symmetric circuit detects motion in the opposite direction, and the difference between the circuit outputs indicates motion direction.
The Hassenstein-Reichardt EMD does not provide a direct measure of the stimulus velocity. Instead, it provides an indication of how well the motion stimulus matches the EMD's preferred combination of speed and spatial frequency. Multiple combinations of speed and spatial frequency can result in the same EMD output magnitude, so speed cannot be uniquely determined. Even if the spatial frequency is known, there are speeds greater than and less than the EMD's preferred speed for which the response magnitude would be equal, so speed would still not be uniquely determined.
These observations led to many interesting predictions which were later verified. Evidence of the existence of the Hassenstein-Reichardt model has since been found in many other visual systems, including that of Drosophila.
By the 1960s Hubel and Weisel had isolated directionally selective units in the cat cortex <cit.>. Later similar responses were observed in the tectum of pigeons and frogs.
Barlow and Levick found such directional responses even earlier in the visual pathway of the rabbit, in the retina itself. Based on their recordings, they proposed what is now known as the Barlow-Levick model <cit.>
shown in Fig. <ref>b.
The Barlow-Levick model relies on inhibition instead of excitation as the underlying mechanism, with null direction motion inhibiting a motion unit's response. Although the presence of Hassenstein-Reichardt and Barlow-Levick motion models have been ruled out in macaque,
a similar mechanism relying on unbalanced inhibition is thought to underly directional selectivity of starburst amacrine cells in primate retina <cit.>.
As mentioned earlier, the presence of cells in MT sensitive to specific spatiotemporal frequencies indicates that frequency methods likely underly motion perception in macaque. In 1985, Adelson and Bergen <cit.> and Watson and Ahumada <cit.> proposed similar architectures for motion estimation based on spatiotemporal frequency filtering. The Adelson-Bergen motion energy model is shown in Fig. <ref>c. Each unit computes the energy at a specific spatiotemporal frequency, with the relationship between spatial and temporal frequency being indicative of speed as outlined in Section <ref>. Adelson and Bergen also showed how the opponent energy output by their model was equivalent to the output of the Hassenstein-Reichardt model.
The Adelson-Bergen model has since been used to explain many visual illusions <cit.> and Simoncelli and Heeger <cit.> have proposed a model describing how the spatiotemporal responses in MT may be computed in biology.
Simoncelli <cit.> also took the brightness constancy constraint (<ref>) relied upon by gradient methods and used it to develop a probabilistic Bayesian framework capable of explaining responses of MT neurons, although the framework does not describe how such responses are computed physiologically.
In the abovementioned models, individual motion units do not encode velocity directly, but rather are selective to specific spatiotemporal stimuli. To infer velocity, the responses of motion units must be combined as described in <cit.>.
§ REVIEW OF BIO-INSPIRED WORKS
Summary of bio-inspired VLSI visual motion estimation works
1|c|Author 1|c|Year 1|c|Process 2|c|Array 1|c|Motion 1|c|Method 1|c|Feature
Tanner <cit.> 1986 2$\mu$ 1D (1x16) Global Block-Matching Intensity variation
Tanner <cit.> 1986 1.5$\mu$ 2D (8x8) Global Gradient -
Franceschini <cit.> 1989 - 1D (1x100) Local Token Temporal edge
Andreou <cit.> 1991 2$\mu$ 1D (1x25) Global Reichardt ON-center OFF-surround
Etienne-Cummings <cit.> 1992 2$\mu$ 2D (5x5) Global Token (TI) Temporal edge of center surround
Horiuchi <cit.> 1992 2$\mu$ 1D (1x17) Local Token (TS) Temporal edge
Delbruck <cit.> 1993 2$\mu$ 2D (25x25) Local Reichardt Temporal contrast
Sarpeshkar <cit.> 1993 2$\mu$ NA NA NA Token Temporal edge of spatial contrast
Gottardi <cit.> 1995 2$\mu$ 1D (1x115) Global Block-Matching Intensity values
Kramer <cit.> 1995 2$\mu$ 1D (1x8) Local Token (FS) Temporal edge
Yakovleff <cit.> 1996 2$\mu$ 1D (1x61) Local Block-Matching Sign of spatiotemporal gradients
Arreguit <cit.> 1996 2$\mu$ 2D (9̃x9) Local Block-Matching Spatial edge
Etienne-Cummings <cit.> 1997 2$\mu$ 2D (9x9) Global Token Temporal edge of center surround
Moini <cit.> 1997 1.2$\mu$ 2x1D (2x64) Local Block-Matching Spatiotemporal templates
Harrison <cit.> 1998 2$\mu$ 2D (1x2) Local Reichardt Temporal contrast
Higgins <cit.> 1999 1.2$\mu$ 2D (14x13) Local Token (ITI, FS) Temporal edge
Indiveri <cit.> 1999 1.2$\mu$ 2D (8x8) Global Token (FS) Temporal edge
Jiang <cit.> 1999 0.6$\mu$ 2D (32x32) Global Token (ISI) Temporal edge of spatial contrast
Etienne-Cummings <cit.> 1999 2$\mu$ 2x1D (2x18) Global Adelson-Bergen Spatiotemporal energy of edge map
Barrows <cit.> 2000 1.2$\mu$ 2x1D (2x4) Global Token (FS) Spatial features
Liu <cit.> 2000 1.2$\mu$ 1D (1x37) Global Reichardt Temporal contrast
Pant <cit.> 2000 1.6$\mu$ 2D (13x6) Local Reichardt Temporal contrast
Higgins <cit.> 2000 1.2$\mu$ 2D (13x15) Local Token (FS) Temporal edge
Harrison <cit.> 2000 1.2$\mu$ 1D (1x22) Global Reichardt Temporal contrast
Higgins <cit.> 2002 1.2$\mu$ 2D (27x29) Local Token (ITI) Temporal edge
Yamada <cit.> 2003 1.5$\mu$ 2D (2x10) Local Token (FS) Spatial edge
Ozalevi <cit.> 2003 1.5$\mu$ 2D (6x6) Global Adelson-Bergen Spatiotemporal energy
Massie <cit.> 2003 0.5$\mu$ 12x1D (12x90) Yaw/Pitch/Roll Token (FS) Temporal edge of spatial contrast
Stocker <cit.> 2004 0.8$\mu$ 2D (30x30) Local Gradient -
Ozalevi <cit.> 2005 1.6$\mu$ 2D (6x7) Global Reichardt Temporal contrast
Harrison <cit.> 2005 0.5$\mu$ 2D (16x16) Global Reichardt Temporal contrast
Shoemaker <cit.> 2005 0.35$\mu$ 1D (1x7) Global Reichardt Temporal contrast
Mehta <cit.> 2006 0.5$\mu$ 2D (95x52) Local Gradient -
Moeckel <cit.> 2007 1.5$\mu$ 1D (1x24) Local Token (FS) Temporal contrast
Bartolozzi <cit.> 2011 0.6$\mu$ 1D (1x64) Local Token (FS) Temporal edge
Roubieu <cit.> 2013 - 1D (1x5) Global Token (ISI) Temporal contrast
In this section we discuss integrated real-time bio-inspired visual motion estimation works, which are compared in Table <ref>.
The discussion is divided into three subsections, one for each of the three methods described in Section <ref>, and within each subsection, works are described in chronological order.
§.§ Gradient Methods
In his seminal 1986 thesis, Tanner <cit.> presented two VLSI implementations of visual motion estimation. The first made use of the correlation method and will be discussed in the next subsection. Tanner's second implementation used a 2D analog VLSI gradient based approach relying a feedback loop to arrive at a minimum-error solution which simultaneously satisfies both the brightness constancy constraint (<ref>) and a local smoothness constraint. This gradient based approach was later adopted and extended in other works by Stocker <cit.> and Mehta <cit.>.
§.§ Correlation Methods
§.§.§ Block Matching
Tanner's second implementation uses a correlation method. It comprises a linear array of 16 pixels which is sampled and binarized in a single step to produce a binary image. These binary images are captured and their correlation to shifted versions of an initial image are computed (the shifts are 1 pixel left, no shift, and 1 pixel right). Digital pulses indicate when leftward or rightward displacement by a single pixel has occurred, at which point a new initial image is captured and the process is repeated. Velocity is encoded by the digital pulse rate at the output of the sensor.
Gottardi <cit.> would later present a similar approach using CCD pixels coupled with CMOS circuity, but capable of detecting motion of up to 5 pixels per image.
Yakovleff <cit.> presented a local matching approach which uses the sign of temporal gradients as the feature to be matched by digital circuits.
Arreguit <cit.> presented the first 2D array for block matching, which used spatial edges as features and was designed for use as a pointing device (computer mouse). Pixels computed motion locally, and local estimates were combined to estimate global motion.
§.§.§ Hassenstein-Reichardt and Barlow-Levick Models
At a high level, Tanner's implementation can be seen as sequentially implementing Reichardt detectors with increasing time delays until sufficient correlation is detected. In 1991 Andreou <cit.> reported an analog implementation of the Reichardt detector which instead outputs the correlation value itself. The sensor computes ON-center OFF-surround features in analog and uses an all-pass filter to implement the delay. Outputs from all the Reichardt detectors along a linear array are summed and output as a differential current.
Delbruck <cit.> later presented a 2D analog variation of the Reichardt detector. Temporal contrast was used as the input signal and was delayed using a multi-tap analog delay line capable of propagating a signal across multiple pixels. At each pixel the correlation is computed between the propagating signal and the local signal before being propagated to the next pixel. The output therefore incorporates signals from many pixels and increases if motion is sustained across multiple pixels. However, the magnitude of the output is highly dependent on contrast.
Harrison <cit.> presented another analog Reichardt implementation which also computes correlation based on temporal contrast, but exhibits decreased dependence on contrast magnitude. The detectors in the chip are either accessed individually, or combined using a linear summation to obtain a global response. Harrison also used the approach to generate an artificial optomotor response (torque signal) and compared it to measurement of Drosophila in experiments <cit.>.
Pant <cit.> presented an analog Reichardt implementation in which the responses of multiple Reichardt detectors are combined on-chip in a non-linear fashion. Pant showed how the output can be used to generate a torque signal to control the gaze direction of a robot during visual tracking, even though adjusting the gaze of the robot induces optical flow through egomotion (<ref>).
Liu <cit.> also presented an analog Reichardt implementation with non-linear on-chip integration of detector outputs, and showed that the frequency response of the chip is similar to the frequency response of Horizontal System (HS) neurons in Drosophila <cit.>.
Popular token methods. The left image (a) shows how tokens are typically generated by thresholding (light blue) the output of a temporal bandpass filter (red) to detect changes in pixel intensities. Spatial filtering (blue) is optionally performed as a first step. The Trigger and Inhibit (TI), Facilitate and Trigger (FT), Facilitate Trigger and Inhibit (FTI), and Facilitate and Sample (FS) methods are shown in (b-e) respectively for a rightward moving stimulus, which will trigger responses from left (first) to right (last) when moving across the array. Red arrows indicate preferred direction of motion in each case.
Shoemaker <cit.> presented an analog Reichardt implementation based on visual motion processing in the fly. The sensor makes use of non-linearities in the processing chain to reduce the dependence of the sensor output on scene contrast and spatial frequencies.
Harrison <cit.> later presented a time-to-contact sensor for collision avoidance based on detecting 2D looming motion fields with Reichardt detectors.
Although partially addressed in the latter of the works mentioned thus far, one of the shortcomings of the Reichardt detector is the dependence of the output on stimulus contrast. This can be overcome by using so-called “token" methods, where thresholding provides a 1 bit token to signal the presence or absence of a stimulus.
§.§.§ Token Methods
Horiuchi <cit.> developed a linear array which used detection of a sufficient temporal intensity derivative (edge) as a digital token. Tokens from neighbouring pixels propagate down a digital delay line in opposite directions until they cross. The location in the delay lines at which they cross indicates both speed and direction. The delay line between each pixel pair contributes a “vote" into a Winner-Take-All
circuit which outputs the location with the most votes.
Etienne-Cummings <cit.> developed a chip for 2D motion detection which uses the temporal derivative of a center-surround feature as the token. The chip encoded speed using the width of a pulse initiated by departure of a token from one pixel and terminated by arrival of the token at either neighbour. A voting scheme was used to determine direction.
Sarpeshkar <cit.> also took a token approach in which tokens trigger digital pulses. The pulse from one pixel would be delayed and correlated against its neighbour. The output provided a measure of how well the observed stimulus matched the circuit's optimal stimulus, but the response to non-optimal is ambiguous because both increasing and decreasing the stimulus speed causes the response to decrease. In the same paper, Sarpeshkar proposed a facilitate and trigger approach to overcome the speed ambiguity.
Similar approaches to that proposed by Sarpeshkar soon became popular. Fig. <ref> outlines some of these token methods. The Trigger and Inhibit (TI) mechanism (Fig. <ref>b) triggers a pulse when a token is detected at a pixel, and ends (inhibits) the pulse when the token is detected at the next pixel, thereby providing a pulse with width inversely proportional to speed. However, for motion in the null direction, the inhibition occurs before the trigger, and the pulse can continue indefinitely. The shortest of two pulses from directionally opposing circuits is typically assumed to be the correct one.
The Facilitate and Trigger (FT) method (Fig. <ref>c) only triggers a pulse if a facilitation signal generated by the previous pixel is present. The pulse ends when the facilitation signal ends, thereby limiting the maximum pulse width to the width of the facilitation pulse. For motion in the null direction, the trigger occurs before the facilitation signal and no pulse is generated. Unlike the TI case, the pulse from the FT approach is directly proportional to speed.
The Facilitate, Trigger, and Inhibit (FTI) method (Fig. <ref>d) combines the outputs of three pixels, using a facilitation signal for directional selectivity, followed by a Trigger and Inhibit mechanism to generate a pulse inversely proportional to the stimulus speed.
In the TI, FT, and FTI methods (Fig. <ref>b-d) the duration a pulse must be measured in order to infer speed. The Facilitate and Sample (FS) method (Fig. <ref>e) overcomes this by using a shaped facilitation pulse. Instead of triggering an output pulse, the facilitation pulse is sampled providing a value proportional to speed. For motion in the null direction, the facilitation signal will be zero when it is sampled.
Other methods have been proposed which take a similar approach, but rely on inhibition to suppress response in the null direction rather than facilitation to enable response in the preferred direction.
Kramer <cit.> presented the first FTI and the first FS token implementations, each using an 8 pixel linear array with temporal contrast edges as the token.
Etienne-Cummings <cit.> developed a foveated sensor for tracking and stabilization, consisting of a 19x17 pixel array for detecting onset and offset of spatial edges, with the middle 5x5 pixels replaced by a 9x9 array of smaller motion estimating pixels which output motion direction only, thereby realising a “bang-bang" output.
Higgins <cit.> later presented the first Inhibit, Trigger, and Inhibit (ITI) implementation as well as the first 2D FS implementation, both using temporal contrast edges for the token. His FS implementation also subtracted the samples of opposite direction circuits on-chip to provide a signed velocity. He later further developed the concept and extended it to larger array <cit.>. Jiang <cit.> meanwhile presented the first 2D FTI implementation, using temporal edges of spatial contrast as the token, while Yamada <cit.> demonstrated an FS token implementation using 1D arrays and reported on its possible application to traffic flow measurement and monitoring blind corners while driving.
As neuromorphic front end sensors for temporal contrast detection improved, Higgins <cit.> and Indiveri <cit.> both adopted multi-chip approaches to motion-estimation, relying on a stand-alone specialized front end sensor for temporal contrast detection, and a separate chip for the FS token algorithm implementation. The multi-chip approach carries the disadvantage of requiring additional power for off-chip communication between the front-end sensor and the motion computation chip. However, moving the in-pixel motion computation circuits to a separate chip reduces pixel size on the front-end chip, allowing a denser pixel array at the front-end.
Barrows <cit.> implemented a multi-chip token method which combined a microcontroller for postprocessing with a front end sensor for extracting programmable spatial features. Different spatial features were found to work well for different stimuli, and the combined use of multiple spatial features was used to improve performance. The chips were used to control the rudder of a Micro Aerial Vehicle (MAV) to help it avoid obstacles.
The multi-chip approach also allows signals to be remapped between the front-end sensor and back-end motion computation chip, allowing mapping from cartesian to polar co-ordinates, which can be useful for measuring looming motion fields as Higgins and Indiveri both demonstrated <cit.>. Furthermore, signals can easily be copied and routed to multiple motion processing chips. Higgins demonstrated such an approach for simultaneously computing motion in cartesian and polar co-ordinates using two motion processing chips in parallel. Higgins also demonstrated how two front-end sensors can feed into a single motion processing chip to compute motion only at a specific disparity (depth in $z$ direction) <cit.>.
Massie <cit.> presented a combination imager and motion estimation chip for roll, pitch, and yaw estimation. The chip consists of 12 linear 90 pixel arrays (2 for yaw, 2 for pitch, and 8 for roll) relying on the token based FS method. Integrated into the same chip was a 128x128 pixel variable acuity imager capable of providing maximum resolution on objects of interest, while conserving bandwidth by combining pixel responses from “uninteresting" regions.
Ozalevi <cit.> presented a multi-chip approach which used a separate front end sensor to generate temporal edge tokens, but a low-pass filter was used to convert these tokens back into analog signals which were processed by separate chips implementing Hassenstein-Reichardt and Barlow-Levick models. The low pass filter also serves to create the delays required by these models (see Fig. <ref>). Thus, an analog implementation of the Hassenstein-Reichardt and Barlow-Levick models is realised, but the intermediate “token" stage serves to normalize signal amplitude, largely removing the dependence on stimulus contrast.
Moeckel <cit.> presented a linear array relying on the FS token method with improved robustness to noise allowing the chip to extract motion over 2 decades of speeds.
Bartolozzi <cit.> recently presented a prototype linear array motion tracking chip which relies on the FS token method using temporal contrast edges as the token. Temporal derivatives are computed as part of the token generation process, but the chip also computes spatial derivatives in parallel. Both these derivatives as well as the motion estimates themselves are fed into a WTA circuit with programmable input weights, allowing the user to track the most salient feature in the array. The programmability of the WTA allows the most salient feature to be defined as a weighted summation the spatial contrast, temporal contrast, and motion features.
oubieu <cit.> presented a 23.3mm $\times$ 12.3mm sensor weighing under 1 gram (including optics). The sensor consists of 5 pairs of 1D motion sensors which use tokens to measure the time for a feature to travel between neighbouring locations, similar to the Trigger and Inhibit (TI) token method.
The authors <cit.> proposed an algorithm in which simple spiking neurons with pre-programmed synaptic delays can be combined with a silicon retina <cit.> to implement motion sensitive receptive fields. Similarly to Fig. <ref>a, where a point with motion in 1 spatial dimension traces out a line in a 2D space-time plot, an edge with motion in 2 spatial dimensions will trace out plane in a 3D space-time plot, with the slope of the plane encoding the local motion velocity. This can be seen as a Reichardt detector, where the inter-pixel delays uniquely describe a plane, although the algorithm still computes on temporal contrast tokens provided by the front end sensor. This approach is elaborated on in Section <ref>.
In the author's implementation each neuron is designed to detect the presence of a particular local space-time plane (i.e. a specific inter-pixel delay) but combines responses from 5x5 pixels per motion unit to simultaneously determine both the direction and speed of the normal flow. The responses of multiple motion units are then combined in a second layer of the neural network to attenuate errors due to the aperture problem. However, this algorithm has not yet been implemented on embedded hardware or in real-time.
At a similar time, Benosman <cit.> proposed an approach which also relies on detection of local planes in data provided by a silicon retina. Instead of using multiple receptive fields tuned to detect different motions (planes), the best fit for a single local plane is mathematically computed, with the normal of the computed fit indicating the normal flow locally. The algorithm runs in JAVA on a host computer.
§.§ Frequency Methods
Others have focused on Adelson-Bergen type models <cit.> relying on spatiotemporal filtering for motion detection. The first such implementation was reported by Etienne-Cummings <cit.>, using a front-end silicon retina to compute a binary map of spatial edges, thereby providing an input signal of normalized amplitude. Subsequent processing using a multi-chip reconfigurable neural processor
implements pairs of spatial and temporal filters to extract the oriented energy at a particular spatiotemporal frequency, thereby implementing an Adelson-Bergen motion unit. The oriented energy from multiple motion units of different frequencies are computed in parallel and their outputs are combined (as described in <cit.>) to obtain an estimate of image motion.
Ozalevi <cit.>, in an approach similar to his implementation of the Hassenstein-Reichardt and Barlow-Levick models, described a multi-chip Adelson-Bergen model. Although successful, this model only implemented a single motion energy unit per pixel, therefore indicating the presence of a preferred motion stimulus and direction without indicating speed.
Modern computing technologies allows for processing on a larger scale than ever before. The author <cit.> demonstrated how an FPGA can be used to implement and combine 720 Adelson-Bergen motion energy units per pixel in real-time for a 128x128 pixel array running at 30FPS.
§ A SPIKING NEURAL NETWORK FOR VISUAL MOTION ESTIMATION
As mentioned in the previous section, the authors have developed a spiking neural network architecture for visual motion estimation <cit.> which relies on synaptic delays to create motion sensitive receptive fields.
Discrete temporal contrast tokens from a separate front end sensor <cit.> are used as input spikes to the architecture and Leaky Integrate and Fire (LIF) neurons with a linear decay are used for computation. Such neurons are good at detecting temporal coincidence of their inputs, but motion signals are inherently spread over time, as modelled in (<ref>).
As stated in <cit.> and repeated here for convenience, a motion sensitive unit in the architecture relies on the assumption that if we consider a small enough spatial region, a moving edge can be approximated as being a straight edge moving with constant velocity. The equation below shows how a motion stimulus is modelled.
Construction of a 3$\times$3 pixel receptive field sensitive to motion of an edge parallel to the y-axis travelling in the positive x-direction with speed $v_x = 1/5$ pixels per millisecond. The location of the edge can be described by $x = v_xt$, where $x$ is measured in pixels and $t$ is measured in milliseconds. This equation describes a spatiotemporal plane (shown in gray). Red crosses are located where the plane crosses pixel locations and indicate which pixels are expected to respond when (blue circles indicate actual recorded data). The length of green arrows above each pixel location indicate the synaptic delay for the synapse connecting from that pixel, and the green plane indicates the time at which the neuron would respond to this stimulus.
\begin{equation}
\begin{array}{l l}
I(x,y,t) &= H(x-v_xt)\\
\frac{dI(x,y,t)}{dt} &= \delta(t-\frac{x}{v_x})\\
E(x,y,t) &= \delta(t-\frac{x}{v_x})III_1(x) III_1(y)\\
\end{array}\label{eq:plane}
\end{equation}
where $x$ and $y$ describe a location on the image plane. $I$ is intensity, $t$ is time (milliseconds), $H$ is the Heaviside step function, $v_x$ is the x-component of velocity in pixels per millisecond, $\delta$ is the Dirac delta function, $dI(x,y,t)/dt$ is the temporal derivative of image intensity, $E(x,y,t)$ is the sensor output, and $III_1$ is a sampling comb with period 1 pixel. Multiplying by the sampling combs converts the continuous space signal into a discrete space signal which only has values at integer pixel locations.
The multilayer architecture detecting the motion of a blue box. (a) shows a set of neurons tuned to detect different speeds and directions of motion. A full set of such neurons is present at every image location. (b) shows multiple image locations as they detect the motion of a stimulus (blue box moving upwards to the left). Black circles show locations of activated neurons, with arrows indicating which neuron (speed and direction) was activated. (c) shows a layer 2 neuron which determines the correct motion by combining layer 1 outputs to alleviate the aperture problem.
Fig. <ref> shows how a receptive field sensitive to a specific motion stimulus (in this case a speed of 1/5 pixels per millisecond in the x-direction) can be constructed. The underlying concept relies on using synaptic delays (green arrows) to convert a temporal sequence of spikes (red crosses) into a group of spikes coincident in time (green plane). The delayed spikes serve as input to a LIF neuron, which is good at detecting temporal coincidence of its inputs. In practice there will not be perfect temporal coincidence because the actual spikes received from the front end sensor (blue circles) will not perfectly match the spike times predicted by our model (red crosses).
Lowering the threshold voltage of the LIF neuron will cause it to still respond when its inputs are slightly spread in time, and the threshold value can be used to control how much time spreading can be tolerated before the neuron stops responding.
Our approach can be seen as a Reichardt detector covering multiple pixels, since it is effectively delaying the signal from neighbouring pixels while the LIF neuron detects multi-pixel correlations in the delayed spikes.
As with the Reichardt detector, multiple detectors (in our case neurons) are required in order to detect different speeds and directions of motion. Our architecture uses 8$\times$8 neurons per pixel location to detect all possible combinations of 8 different directions and 8 different speeds as shown in Fig. <ref>a. Directions vary from 0 to 315 degrees in steps of 45 degrees, while speeds vary from $\sqrt{2}/{50}$ to $\sqrt{2}^8/{50}$ pixels/ms by factors of $\sqrt{2}$.
The equations in (<ref>) are independent of motion parallel to the edge direction, presenting a form of the aperture problem where only motion perpendicular to an edge (the normal flow) can be detected. An edge moving in a direction perpendicular to its orientation at speed $s$ would look identical to the same edge moving in a direction 45 degrees to its orientation with speed $s\sqrt{2}$ (see dotted arrows in Figs. <ref> and <ref>) since in both cases the perpendicular component of motion is just $s$. This relationship gives rise to the $\sqrt{2}$ factor used between different speeds.
A key feature which sets this work apart from previous token and Reichardt works is the use of a second stage of processing to overcome the aperture problem. The second stage of processing is implemented by a second layer of neurons, with each neuron receiving inputs over a wider spatial region than first layer neurons (Fig <ref>c). A layer 2 neuron sensitive to speed $s$ and direction $d$ would incorporate inputs from layer 1 neurons sensitive to the same speed and direction, but also from layer 1 neurons sensitive to speed $s/\sqrt{2}$ and directions $d\pm45^o$. This multi-layer approach bears resemblance to gradient based methods such as Lucas-Kanade <cit.> and Horn-Schunck <cit.> which compute normal flow locally in a first step before incorporating the normal flow from other nearby locations to more accurately approximate the true optical flow.
Motion of a moving bus as detected by the network. The left pane shows temporal change tokens (black for decrease, white for increase, grey for no change). The middle pane shows outputs of Layer 1, which tend to be perpendicular to edges. The right pane shows the output of the Layer 2, which uses the data from Layer 1 to detect the actual flow. In each pane only 3.3ms worth of data is shown. Grayscale values are obtained using the exposure measurement function of the Asynchronous Time-based Image Sensor (ATIS) <cit.>.
Fig <ref> shows the system architecture. At each pixel location there are 8$\times8$ neurons sensitive to different speeds and directions of motion (Fig <ref>a), but mutual inhibition ensures only one neuron (shown in black) can respond to a stimulus (produce an output spike) at any time.
Fig <ref>b shows a stimulus covering 3$\times$3 pixel locations. Dark circles indicate locations where neurons are responding, with solid arrows indicating the speed and direction selectivity of the neuron responding at that location. Dotted arrows indicate other speeds and directions consistent with the aperture problem as discussed above. Fig <ref>c shows a layer 2 neuron determining the correct motion from the inputs it receives from the layer 1 neurons of Fig <ref>b.
Fig. <ref> shows actual outputs from each layer for a real world scene of a bus crossing a bridge. Layer 1 outputs tending to be perpendicular to edges, while layer 2 outputs more accurately describe the actual motion of the bus by incorporating data over a larger spatial region.
Fig. <ref> presents the output of the architecture for a controlled stimulus consisting of a spinning spiral. The figure shows how it can reliably detect different speeds and directions of motion. The top images show data accumulated over multiple rotations of the spiral. Colour is used to encode speed (left) and direction (right) according to the legend provided above the images. Speed varies as a function of distance to the axis of rotation, while direction varies with angle. Near the axis of rotation, the motion is slower than the slowest receptive field and therefore elicits no responses. The lower image shows the motion architecture's output accumulated over a period of 10ms. The colour of arrows helps to encode the motion direction, while their length encodes speed. The spiral stimulus used has been superimposed on the image as a yellow dashed line.
Output of the architecture when viewing a rotating spiral. Top images show speed (left) and direction (right) outputs accumulated over multiple rotations. The lower image shows 10ms of output data while the spiral is spinning. The motion vectors all point outwards from the center, creating a looming field.
An example real world scene (top) consisting of a moving car and a startled bird. The lower subimages show a cropped region around the bird during flight at different points in time. The time in milliseconds is indicated at the top left of each image. Notably, the motion of each wing is detected as well as the motion of the body.
Fig. <ref> shows another example of the architecture operating on real world data. The top part of the image shows a full scene captured with the ATIS. There are two moving objects in the scene, a rightward moving car in the lower part, and a bird in the top right. The rest of the images in Fig. <ref> show motion responses elicited by the flying bird. Inset numbers indicate the time (in milliseconds) at which each image was captured. Each image shows 3.3ms of motion output data. In the first frame (t = 0) rightward motion of the bird's body is detected, while separate motion is detected for each wing. In the second frame (t = 7ms) retraction of the left wing is detected, followed by retraction of the right wing at 37ms, as shown by the opposite motion estimates for each wing while the body continues motion upward to the right. 10ms later extension of the left wing is detected (red arrows at t = 47ms). By 90ms, the bird has returned to a similar pose to that seen at 0ms. At 117ms, both wings have been pulled in front of the body, causing motion upwards. After 150ms, the bird exits the scene.
The bird example is particularly tricky for conventional motion estimation techniques due to the bird significantly and rapidly changing its appearance while moving.
The model as presented here has not yet been integrated into a real-time implementation. However, the model is computationally efficient. Neurons are only updated when an input spike arrives, and each neuron update only requires 4 additions, 2 greater-than comparisons, and 1 multiplication. Sparsity of the incoming spikes from the ATIS front-end sensor keeps the required number of operations per second low and easily achievable with commercially available hardware.
The challenge in achieving real-time implementation does not lie in the computation rate, but rather in the implementation of synaptic delays. Incoming spikes must be delayed and stored, which drives up memory requirements. These memory requirements can be reduced by observing that only a small percentage of synapses are active at any one time, so memory can be shared between synapses. However, different synapses have different delays, so a simple First In First Out (FIFO) buffer will not work because the order in which spikes are placed in the buffer will not be the same as the order in which they must be read out.
§ FUTURE DIRECTIONS
In the previous sections we have summarized past and present works. In this section we outline the directions for future works.
Although great progress has been made, the organisation of computational circuits in artificial approaches still differs significantly from biology. In the retina, neurons are arranged in interconnected layers stacked on top of each other and lying above the photoreceptor layer. Similarly, in visual cortex neurons are arranged in 3D layers, with both short local connections, and longer range axonal connections between more distant regions of cortex.
In silicon, photoreceptors and computational circuits mimicking different layers of biological processing are restricted to lying side by side within the same plane, which limits both the photoreceptor size and spacing, which in turn affects the signal strength and spatial resolution respectively. When mapping a 3D biological structure onto 2D silicon, short vertical connections are often mapped to long lateral connections, increasing line capacitance, energy consumption, and occupying valuable space. This is overcome in many artificial implementations by only considering motion in one lateral direction, then stacking circuits in the other lateral direction instead of vertically <cit.>.
As 3D stacked silicon technology matures, it can be used to alleviate the wiring problem, allow for larger photodiode fill-factors, and achieve a more biologically realistic organisation of neural circuits. This compact 3D organisation is typical of insect vision and primate retina, however, the early stages of primate retina and visual cortex are located far from each other, and thus compact integration of cortical circuits and photoreceptors is not necessarily accurate to biology.
Some of the described works have relied on an approach in which a spiking “silicon retina" and neural processing are implemented in separate chips <cit.> (although 3D integration is useful for both chips). Implementing retinal and cortical processing as two different components provides advantages during system development. First, an improvement in either component can be achieved without re-fabricating the other, and second, data can be recorded as it is transmitted between the two, allowing for in depth off-line analysis which can provide insights into how neural algorithms for processing can be further improved.
The last decade has seen silicon retinae mature to the point where they are now commercially available and are used by many labs around the world and there are a number of works emerging (see other papers in this special issue) which argue for the superiority of these sensors for high speed visual tasks which must be executed in real-time on a limited power budget. Reconfigurable neural processing platforms are also rapidly maturing, spurred by the dramatic increase in interest and funding the neuromorphic field has experienced in the last few years.
It is an exciting time for the neuromorphic area, with major companies including Qualcomm, Samsung, Intel, and IBM coming on board and launching their own research projects in the field. There are also major projects in the US (the $200 million BRAIN initiative <cit.>) and Europe (the €1 billion Human Brain Project <cit.>) which are incorporating neuromorphic aspects. The US Defense Advanced Research Projects Agency (DARPA) has also been taking notice, funding projects such as the Unconventional Processing Of Signals For Intelligent Data Exploitation (UPSIDE) and Systems of Neuromorphic Adaptive Plastic Scalable Electronics (SYNAPSE) projects.
Modern CMOS technology is quite different from biological “wetware". CMOS typically operates at frequencies ranging from megahertz to gigahertz, while a general rule of thumb is that biological neurons do not operate at frequencies above 1kHz. This massive speed difference is not necessarily an advantage for silicon. In fact, slowing silicon neural circuits down to biologically realistic time-scales can prove quite challenging, and often requires extra design effort and cost to implement.
Biology leverages parallel processing, and the speed of CMOS can be useful when one wants to approximate multiple parallel units from biology using a single high speed sequential unit in silicon. However, this approach comes at a disproportionate power cost. Higher operating speeds require higher operating voltages, and power scales proportionally to voltage squared. It is therefore preferable to have many low speed, low voltage processors (like biology) than a few high speed, high voltage processors (like modern CMOS). Hence, biology provides a road map for the future, where the scaling of CMOS will allow the realization of ultra-low voltage (hence low-power) circuits performing massively parallel computation in very small and three dimensionally stacked dies. As technology moves in this direction, CMOS can learn about 3D connectivity, massively parallel computation, density of computational elements, and stochastic circuits from biology.
Despite technological improvements, wiring remains an issue in the connectivity which can currently be achieved between artificial neurons. Even though 3D integration can help, inter-neuron connectivity with present technologies is constrained to remain far sparser than in biology. The combined use of silicon circuits and carbon nanotube crossbar arrays has been proposed to improve physical connectivity, with memristor devices capable of learning proposed for use as synaptic connections between nanotubes <cit.>.
The development of new online learning algorithms and architectures, whether relying on memristor devices or conventional silicon, are likely to play an increasingly important role. Using learning through visual experience to help configure and organise a neural architecture can improve fault tolerance (and therefore device yield) and save man hours spent on manual configuration. This learning is especially important if copies of the same device are to adapt to operation under very different visual conditions, such as in urban versus forested environments, or onboard flying versus ground vehicles.
As mentioned in the Introduction, biological sensors are embodied, and have evolved in conjunction with motor systems. The interplay between motor and sensory systems can be useful for sensing. Examples of this include the peering behaviour used by many animals to induce motion parallax for depth perception <cit.>, the optomotor response in insects <cit.>, and the vestibular ocular reflex in humans <cit.>. Recent studies also suggest that micro-saccades during fixation play an important role in perception, particularly for object recognition in humans <cit.>. In Drosophila motion is found to also play a role in motion perception <cit.>.
An embodied biological sensor also serves a particular purpose, to provide information relevant to the agent for self-preservation and meaningful interaction with the environment. The value metric of biological motion estimates is therefore not directly assessed by how accurately motion is perceived, but rather by how motion estimates improve the effectiveness of the agent's behaviour (although to a degree, more accurate motion estimates will be more effective in affecting behaviour).
It is therefore important to keep in mind the intended use of the system being constructed. Inspiration from biology is useful, but at some stage the design must deviate from precise bio-mimicry. A micro-aerial vehicle may benefit from an artificial version of the Drosophila vision system, but for the vehicle to be of value to the operator, it will be expected to execute a goal oriented task rather than simply behave like Drosophila. Also, at some point making a system more biologically accurate will come at a performance cost rather than benefit due to the inherent differences between silicon circuits and biological neurons. It was Carver Mead who first developed the concept of imitating neural processing in silicon circuits by noting the similarities between the two <cit.>, but it was also Carver Mead who said “Listen to the technology; find out what it's telling you".
Nevertheless, we are still a long way from matching the power efficient performance of biology in artificial systems, so for the foreseeable future, continued research into bio-inspired visual motion estimation techniques will reap rewards for artificial systems.
[]Garrick Orchard
[]Ralph Etienne-Cummings
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1511.00499
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$^1$ School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, UK
$^2$ School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455, USA
[email protected]; [email protected]
The ground-state (GS) phase diagram of the frustrated spin-$s$
$J_{1}$–$J_{2}$–$J_{3}$ Heisenberg antiferromagnet on the
honeycomb lattice is studied using the coupled cluster method
implemented to high orders of approximation, for spin quantum
numbers $s=1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. The model has
antiferromagnetic (AFM) nearest-neighbour, next-nearest-neighbour and
next-next-nearest-neighbour exchange couplings (with strength
$J_{1}>0$, $J_{2}>0$ and $J_{3}>0$, respectively). We specifically
study the case $J_{3}=J_{2}=\kappa J_{1}$, in the range $0 \leq
\kappa \leq 1$ of the frustration parameter, which includes the
point of maximum classical ($s \rightarrow \infty$) frustration,
viz., the classical critical point at $\kappa_{{\rm
cl}}=\frac{1}{2}$, which separates the Néel phase for
$\kappa < \kappa_{{\rm cl}}$ and the collinear striped AFM phase for
$\kappa > \kappa_{{\rm cl}}$. Results are presented for the GS
energy, magnetic order parameter and plaquette valence-bond crystal
(PVBC) susceptibility. For all spins $s \geq \frac{3}{2}$ we find a
quantum phase diagram very similar to the classical one, with a
direct first-order transition between the two collinear AFM states
at a value $\kappa_{c}(s)$ which is slightly greater than
$\kappa_{{\rm cl}}$ [e.g., $\kappa_{c}(\frac{3}{2}) \approx
0.53(1)$] and which approaches it monotonically as $s \rightarrow
\infty$. By contrast, for the case $s=1$ the transition is split
into two such that the stable GS phases are one with Néel AFM
order for $\kappa < \kappa_{c_{1}} = 0.485(5)$ and one with striped
AFM order for $\kappa > \kappa_{c_{2}} = 0.528(5)$, just as in the
case $s=\frac{1}{2}$ (for which $\kappa_{c_{1}} \approx 0.47$ and
$\kappa_{c_{2}} \approx 0.60$). For both the $s=\frac{1}{2}$ and $s=1$ models
the transition at $\kappa_{c_{2}}$ appears to be of first-order
type, while that at $\kappa_{c_{1}}$ appears to be continuous.
However, whereas in the $s=\frac{1}{2}$ case the intermediate phase
appears to have PVBC order over the entire range $\kappa_{c_{1}} <
\kappa < \kappa_{c_{2}}$, in the $s=1$ case PVBC ordering either
exists only over a very small part of the region or, more likely, is
absent everywhere.
§ INTRODUCTION
Quantum spin-lattice models, in which the sites of a give regular
periodic lattice are all occupied by magnetic ions with spin quantum
number $s$, offer a rich arena for the study of exotic ground-state
(GS) phases that are not present in their classical ($s \rightarrow
\infty$) counterparts. Whereas interactions between the classical
spins give rise to magnetic ground states in which the spins are ordered such
that each individual spin is oriented in a specific direction, quantum
fluctuations can act either to diminish the corresponding magnetic
order parameter (viz., the average local onsite magnetization) or to
destroy it altogether. In the former case, where the long-range order
(LRO) is only partially reduced, such quasiclassical magnetically
ordered states spontaneously break both the SU(2) spin-rotation and
time-reversal symmetries.
By contrast, such intrinsically quantum-mechanical states as the
various valence-bond crystalline (VBC) phases, in which specific
combinations of the lattice spins combine into spin singlets, have
zero magnetic order and break neither of the SU(2) spin-rotation and
time-reversal symmetries, although they still break some lattice
symmetries. Yet other states exist in which, for example,
time-reversal symmetry is preserved, so that magnetic order is
definitely absent, but for which the SU(2) spin-rotation symmetry is
still broken. These are the so-called multipolar or spin-nematic
phases. Finally, of course, one also has the possibility of quantum
spin-liquid (QSL) phases that preserve all of the symmetries,
including the lattice symmetries.
For a given regular lattice in $d$ spatial dimensions, one is usually
interested in the interplay between frustration (which may be tuned,
for example, by varying the relative strengths of competing
interactions in the model Hamiltonian that separately tend to promote
different forms of magnetic LRO) and quantum fluctuations. In broad
terms quantum fluctuations are larger for lower values of both the
spatial dimensionality $d$ and the spin quantum number $s$. They are
typically also larger, for given spatial dimensionality $d$, for
lattices with smaller values of the coordination number $z$.
For the case of $d=1$ isotropic Heisenberg chain systems the
Mermin-Wagner theorem <cit.> excludes the possibility of GS
magnetic order even at zero temperature ($T=0$), since it is
impossible to break a continuous symmetry for any such system. The
Mermin-Wagner theorem similarly implies the absence of magnetic LRO in
any $d=2$ isotropic Heisenberg system at all nonzero temperatures ($T
> 0$). The behaviour and GS quantum phase structure of two-dimensional (2D)
spin-lattice models at $T=0$ has thus come to occupy a special role in
the study of quantum phase transitions.
Spin-lattice systems are said to be frustrated when constraints are
present that preclude the formation of a GS phase which satisfies all
of the (generally, pairwise) interactions among the spins.
Frustration is strongly associated with macroscopic degeneracy of the
GS phase, with the consequent existence of strong quantum fluctuations
among the states in the degenerate manifold. Either quantum or
thermal fluctuations can then, in such a situation, suppress magnetic
LRO, and the possibility of such exotic non-classical states as those
discussed above forming the stable GS phase under certain conditions
is heightened. The low coordination number, $z=3$, of the honeycomb
lattice further enhances the quantum fluctuations, and thereby makes
it a special 2D spin-lattice model candidate for the study of its
$T=0$ GS quantum phase diagram when dynamical frustration is introduced via
competing interactions.
In the present paper we study the so-called $J_{1}$–$J_{2}$–$J_{3}$
model on the honeycomb lattice with antiferromagnetic (AFM) Heisenberg
exchange interactions between pairs of nearest-neighbour (NN) spins
(of strength $J_{1}$), next-nearest-neighbour (NNN) spins (of strength
$J_{2}$), and next-next-nearest-neighbour (NNNN) spins (of strength
$J_{3}$). Even in the classical ($s \rightarrow \infty$) limit the
model has a rich phase diagram
<cit.>, as we discuss further in
Sec. <ref> below. In the case where all three bonds are
AFM in nature (i.e., $J_{i} > 0$; $i = 1,2,3$), the classical system
exhibits two collinear AFM phases, namely the so-called Néel and
striped phases, as well as a spiral phase. The three phases meet in a
triple point at $J_{3} = J_{2} = J_{1}/2$ (see, e.g., Refs.<cit.>). This is the point of
maximum classical frustration, where the classical GS phase has
macroscopic degeneracy. For the present study we consider the model
along the line $J_{3}=J_{2} \equiv \kappa J_{1}$ for the case
$J_{1}>0$, as a function of the frustration parameter $\kappa$ in the
range $0 \leq \kappa \leq 1$. The classical ($s \rightarrow \infty$)
version of the model thus has a single quantum phase transition in its
$T=0$ phase diagram at $\kappa_{{\rm cl}} = \frac{1}{2}$. For $\kappa
< \kappa_{{\rm cl}}$ the stable GS phase is the Néel AFM phase,
whereas for $\kappa > \kappa_{{\rm cl}}$ the stable GS phase is the
striped AFM phase. In fact, at $T=0$, there actually exists an
infinite family of non-coplanar states, all of which are degenerate in
energy with the striped state. However, it is asserted <cit.>
that both thermal and quantum fluctuations break this degeneracy in
favour of the collinear striped state, at least in the large-$s$ limit <cit.>.
Whereas the spin-$\frac{1}{2}$ $J_{1}$–$J_{2}$–$J_{3}$ model on the
honeycomb lattice, or particular cases of it (e.g., when $J_{3}=J_{2}$
or $J_{3}=0$), have been investigated by many authors with a variety
of theoretical tools (see, e.g., Refs.<cit.>),
there are far fewer studies of the model in the case $s>\frac{1}{2}$.
A particular exception is a very recent study
<cit.> of the $s=1$ $J_{1}$–$J_{2}$ model
(i.e., when $J_{3}=0$) on the honeycomb lattice, using the
density-matrix renormalization group (DMRG) method. Our specific aim
here is to extend earlier work using the coupled cluster method (CCM)
applied to the spin-$\frac{1}{2}$ version of the
$J_{1}$–$J_{2}$–$J_{3}$ model on the honeycomb lattice along the
line $J_{3}=J_{2}$
to cases $s>\frac{1}{2}$. In particular, we now compare results for
the case $s=\frac{1}{2}$ with those for
We note that by now there exist many experimental realizations of
frustrated honeycomb-lattice systems with AFM interactions. These
include such magnetic compounds as Na$_{3}$Cu$_{2}$SbO$_{6}$
<cit.>, InCu$_{2/3}$V$_{1/3}$O$_{3}$
<cit.>, $\beta$-Cu$_{2}$V$_{2}$O$_{7}$
<cit.>, and Cu$_{5}$SbO$_{6}$
<cit.>, in each of which the $s=\frac{1}{2}$
Cu$^{2+}$ ions are situated on the sites of weakly coupled
honeycomb-lattice layers. The iridates A$_{2}$IrO$_{3}$ (A $=$ Na, Li)
are also believed to be magnetically ordered Mott insulators in which
the Ir$^{4+}$ ions form effective $s=\frac{1}{2}$ moments arrayed on
weakly-coupled honeycomb-lattice layers. Other similar honeycomb
materials include, for example, the families of compounds
BaM$_{2}$(XO$_{4}$)$_{2}$ (M $=$ Co, Ni; X $=$ P, As)
<cit.> and Cu$_{3}$M$_{2}$SbO$_{6}$ (M $=$ Co, Ni)
<cit.>. In both of these families the magnetic
ions M$^{2+}$ are again disposed in weakly-coupled layers where they
occupy the sites of a honeycomb lattice. In both families the
Ni$^{2+}$ ions appear to take the high-spin value $s=1$, whereas the
Co$^{2+}$ ions appear to take the low-spin value $s=\frac{1}{2}$ in
the former family BaCo$_{2}$(XO$_{4}$)$_{2}$ and the high-spin value
$s=\frac{3}{2}$ in the latter compound Cu$_{3}$Co$_{2}$SbO$_{6}$. As a
last example of an $s=\frac{3}{2}$ honeycomb-lattice AFM material, we
also mention the layered compound Bi$_{3}$Mn$_{4}$O$_{12}$(NO$_{3}$)
<cit.> in
which the spin-$\frac{3}{2}$ Mn$^{4+}$ ions sit on the sites of
the honeycomb layers.
The remainder of the paper is organized as follows. In Sec.<ref> we discuss further the model itself, before we give a
brief description in Sec. <ref> of the CCM formalism that
we apply to it. The results are then presented in Sec.<ref>, and we conclude in Sec. <ref> with a
discussion and summary.
§ THE MODEL
The Hamiltonian of the $J_{1}$–$J_{2}$–$J_{3}$ model on the honeycomb lattice is given by
\begin{equation}
H = J_{1}\sum_{\langle i,j \rangle} \mathbf{s}_{i}\cdot\mathbf{s}_{j} + J_{2}\sum_{\langle\langle i,k \rangle\rangle} \mathbf{s}_{i}\cdot\mathbf{s}_{k} + J_{3}\sum_{\langle\langle\langle i,l \rangle\rangle\rangle} \mathbf{s}_{i}\cdot\mathbf{s}_{l}\,,
\label{Hamiltonian}
\end{equation}
where the sums over $\langle i,j \rangle$, $\langle \langle i,k \rangle
\rangle$ and $\langle \langle \langle i,l \rangle \rangle \rangle$ run
over all NN, NNN and NNNN bonds, respectively, on the lattice,
counting each pair of spins once and once only in each of the three
sums. Each site $i$ of the honeycomb lattice carries a spin-$s$
particle described by the SU(2) spin operator ${\bf s}_{i} \equiv
(s^{x}_{i}, s^{y}_{i}, s^{z}_{i})$, with ${\bf s}^{2}_{i} = s(s+1)$,
and, for the cases considered here,
$s=1,\,\frac{3}{2},\,2,\frac{5}{2}$. The lattice and the Heisenberg
exchange bonds are illustrated in Fig. <ref>(a).
The $J_{1}$–$J_{2}$–$J_{3}$ model on the honeycomb
lattice, showing (a) the bonds ($J_{1} = $ —–, $J_{2} = $ - - -,
$J_{3} = $ - $\cdot$ - ) and the Néel state, and (b) the
triangular Bravais lattice vectors $\mathbf{a}$ and $\mathbf{b}$
and one of three equivalent striped states. Sites on the two
triangular sublattices ${\cal A}$ and ${\cal B}$ are shown by
filled and empty circles respectively, and the spins are
represented by the (red) arrows on the lattice sites.
For the present study we are interested in the case where each of the
three types of bonds is AFM in nature (i.e., $J_{m} > 0$; $m=1,2,3$).
Without loss of generality we may put $J_{1} \equiv 1$ to set the
overall energy scale, and we will specifically consider the case where
$J_{3}=J_{2} \equiv \kappa J_{1}$, in the interesting window $0 \leq \kappa \leq 1$ of the frustration parameter $\kappa$.
The honeycomb lattice is bipartite, but comprises two triangular
Bravais sublattices ${\cal A}$ and ${\cal B}$. The basis vectors
$\mathbf{a}=\sqrt{3}d\hat{x}$ and
$\mathbf{b}=(-\sqrt{3}\hat{x}+3\hat{z})d/2$ are illustrated in Fig.<ref>(b), where the lattice is defined to lie in the
$xz$ plane as shown, and where $d$ is the lattice spacing (i.e., the
distance between NN sites). The unit cell $i$ at position vector
$\mathbf{R}_{i} = m_{i}\mathbf{a}+n_{i}\mathbf{b}$, where $m_{i},\,n_{i} \in
\mathbb{Z}$, now comprises the two sites at $\mathbf{R}_{i} \in {\cal
A}$ and $(\mathbf{R}_{i}+d\hat{z}) \in {\cal B}$. The reciprocal
lattice vectors corresponding to the real-space vectors $\mathbf{a}$
and $\mathbf{b}$ are thus
$\boldsymbol{\alpha}=2\pi(\sqrt{3}\hat{x}+\hat{z})/(3d)$ and
$\boldsymbol{\beta}=4\pi/(3d)\hat{z}$. The Wigner-Seitz unit cell and the
first Brillouin zone are thus the parallelograms formed, respectively,
by the pairs of vectors $(\mathbf{a}$, $\mathbf{b})$ and
$(\boldsymbol{\alpha}$, $\boldsymbol{\beta})$. Both may equivalently be taken
as being centred on a point of sixfold rotational symmetry in their
respective spaces. Thus, the Wigner-Seitz unit cell may be taken as
being bounded by the sides of a primitive hexagon of side length $d$
as in Fig. <ref>. In this case the first Brillouin
zone is also a hexagon, now of side length $4\pi/(3\sqrt{3}d)$, and which is
also rotated by 90$^{\circ}$ with respect to the Wigner-Seitz hexagon.
The classical ($s \rightarrow \infty$) version of the $J_{1}$–$J_{2}$–$J_{3}$ model of Eq. (<ref>) on the honeycomb lattice already itself displays a rich $T=0$ GS phase diagram (see, e.g., Refs. <cit.>). The generic stable GS phase is a coplanar spiral configuration of spins defined
by a wave vector Q, together with an angle $\theta$ that is the
relative orientation of the two spins in the same unit cell $i$ characterized by the lattice
vector $\mathbf{R}_{i}$. The two classical spins in unit
cell $i$ are given by
\begin{equation}
\mathbf{s}_{i,\rho}=-s[\cos(\mathbf{Q}\cdot\mathbf{R}_{i}+\theta_{\rho})\hat{z}_{s}+\sin(\mathbf{Q}\cdot\mathbf{R}_{i}+\theta_{\rho})\hat{x}_{s}]\,; \quad \rho={\cal A},\,{\cal B}\,, \label{eq_classical-spins}
\end{equation}
where $\hat{x}_{s}$ and $\hat{z}_{s}$ are two orthogonal unit vectors
that define the spin-space plane, as shown in Fig.<ref>. We choose the two angles $\theta_{\rho}$ such that
$\theta_{{\cal A}}=0$ and $\theta_{{\cal B}}=\theta$.
When all three bonds are AFM in nature (i.e., $J_{m}>0$; $m=1,2,3$), as considered here, it has been shown <cit.> that the
classical model has a $T=0$ GS phase diagram consisting of three
different phases. With reference to an origin at the centre of the hexagonal Wigner-Seitz cell, one may show that one value
of the spiral wave vector $\mathbf{Q}$ that minimizes the classical GS energy of the model is given by
\begin{equation}
\mathbf{Q}=\frac{2}{\sqrt{3}d}\cos^{-1}\left[\frac{(J_{1}-2J_{2})}{4(J_{2}-J_{3})}\right]\hat{x}\,, \label{eq_spiral-wave-factor}
\end{equation}
together with $\theta=\pi$.
Equation (<ref>) is clearly only valid when
\begin{equation}
-1 \leq \frac{J_{1}-2J_{2}}{4(J_{2}-J_{3})} \leq 1 \,. \label{eq_conditon_Q}
\end{equation}
If we define $x \equiv J_{2}/J_{1}$ and $y \equiv J_{3}/J_{1}$, Eq. (<ref>) is equivalent to the inequalities,
\begin{equation}
y \leq \frac{3}{2}x-\frac{1}{4}\,; \quad y \leq \frac{1}{2}x+\frac{1}{4}\,. \label{eq_xy}
\end{equation}
In the positive quadrant (i.e., $x \geq 0$, $y \geq 0$) of the $xy$
plane the classical model has the spiral phase described by the wave
vector $\mathbf{Q}$ of Eq. (<ref>) and $\theta=\pi$
as the stable GS phase in the region defined by Eq. (<ref>).
Everywhere on the boundary line $y = \frac{3}{2}x-\frac{1}{4}$ of the
spiral phase, $\mathbf{Q} = \mathbf{\Gamma} = (0,0)$, which simply
describes the Néel AFM phase shown in Fig.<ref>(a). Similarly, everywhere on the other boundary
line $y = \frac{1}{2}x+\frac{1}{4}$,
$\mathbf{Q}=2\pi/(\sqrt{3}d)\hat{x}$, which describes the collinear
striped AFM phase shown in Fig. <ref>(b). Both the
phase transitions between the spiral and Néel phases and between
the spiral and striped phases are clearly continuous ones. The two
phase boundaries meet at the tricritical point
$(x,y)=(\frac{1}{2},\frac{1}{2})$. Finally, one can also easily show
that there is a first-order phase transition between the two collinear
AFM phases along the line $x=\frac{1}{2}$, $y>\frac{1}{2}$. To
summarize, in the regime where $J_{1}>0$ and $x \geq 0$, $y \geq 0$,
the classical version of the honeycomb-lattice
$J_{1}$–$J_{2}$–$J_{3}$ model has three stable GS phases at $T=0$.
These are: (a) a spiral phase for $0<y<\frac{3}{2}x-\frac{1}{4}$,
$\frac{1}{6}<x<\frac{1}{2}$ and $0<y<\frac{1}{2}x+\frac{1}{4}$,
$x>\frac{1}{2}$; (b) a Néel AFM phase for $y>0$, $0<x<\frac{1}{6}$
and $y>\frac{3}{2}x-\frac{1}{4}$, $\frac{1}{6}<x<\frac{1}{2}$; and (c)
a striped collinear AFM phase for $y>\frac{1}{2}x+\frac{1}{4}$,
$x>\frac{1}{2}$. Clearly, along the line $y=x$ considered here, which
includes the tricritical point at ($\frac{1}{2},\frac{1}{2}$), there
are just two stable GS phases, namely the collinear Néel and
striped AFM phases.
It is worth noting that both the spiral and the striped states described
by the wave vector of Eq. (<ref>)
and its appropriate limiting form (and $\theta=\pi$) have two other equivalent states
rotated by $\pm \frac{2}{3}\pi$ in the honeycomb $xz$ plane. We also
note that in the limiting case when the $J_{2}$ bond dominates (i.e.,
when $x \rightarrow \infty$ for a fixed finite value of $y$), the
spiral pitch angle
$\phi=\cos^{-1}[\frac{1}{4}(J_{1}-2J_{2})/(J_{2}-J_{3})] \rightarrow
\frac{2}{3}\pi$. Clearly, in this limit, the classical model reduces
to two disconnected Heisenberg antiferromagnets (HAFs) on
interpenetrating triangular lattices, each with the 3-sublattice
Néel ordering of NN spins (on each triangular lattice) oriented
at an angle $\frac{2}{3}\pi$ to one another. Precisely in this limit
the wave vector $\mathbf{Q}$ of Eq. (<ref>)
becomes one of the six corners,
$\mathbf{K}^{(1)}=4\pi/(3\sqrt{3}d)\hat{x}$, of the hexagonal first
Brillouin zone. The two inequivalent corner vectors thus describe the
two distinct 3-sublattice Néel orderings for a classical
triangular-lattice HAF.
For spiral pitch angles in the range $\frac{2}{3}\pi < \phi
\leq \pi$ the wave vector $\mathbf{Q}$ of Eq.(<ref>) lies outside the first hexagonal Brillouin
zone. It can equivalently be mapped back inside this range of values,
when $\mathbf{Q}$ then moves continuously from a corner at position
$\mathbf{K}^{(3)}=2\pi(-\hat{x}+\sqrt{3}\hat{z})/(3\sqrt{3}d)$ along
an edge to its midpoint at $\mathbf{M}^{(2)}=2\pi/(3d)\hat{z}$. Thus,
the striped AFM state shown in Fig. <ref>(b) may
equivalently be described by the ordering wave vector
$\mathbf{Q}=\mathbf{M}^{(2)}$ (with the relative angle between the two
triangular sublattices ${\cal A}$ and ${\cal B}$ being $\theta=\pi$).
The two other equivalent striped states have wave vectors
corresponding to the other two inequivalent midpoints of the hexagonal
Brillouin zone edges, at
$\mathbf{M}^{(1)}=\pi(\sqrt{3}\hat{x}+\hat{z})/(3d)$ and
$\mathbf{M}^{(3)}=\pi(-\sqrt{3}\hat{x}+\hat{z})/(3d)$ (with
$\theta=0$ in these two cases).
While Eq. (<ref>) is generally sufficient to
describe the classical GS spin configuration
<cit.>, it relies on the assumption that the
GS order either is unique (up to a global rotation of all spins by the
same amount) or exhibits, at most, a discrete degeneracy (e.g., as
associated with rotations of the wave vector $\mathbf{Q}$ by $\pm
120^{\circ}$ about the $\hat{y}$ axis). Nevertheless, the assumption can be shown to be false
for special values of $\mathbf{Q}$
<cit.>, which include the
cases when $\mathbf{Q}$ equals either one half or one quarter of a
reciprocal lattice vector $\mathbf{G}$. This includes precisely the
case for the striped states for which the wave vectors
$\mathbf{Q}=\mathbf{M}^{(i)}$, $i=1,2,3$, equal one half of
corresponding reciprocal lattice vectors. In this case, it has been
shown <cit.> that the GS ordering now
spans a 2D manifold of non-planar spin configurations, all of which are
degenerate in energy with those of the striped states.
Classical spin-lattice systems that display such
an infinitely degenerate family (IDF) of GS phases in some region of
their $T=0$ phase space are well known to be prime candidates for the
emergence of novel quantum phases with no classical counterparts in the
corresponding quantum systems. Quantum fluctuations then often act to
lift this accidental GS degeneracy (either wholly or in part) by the
order by disorder mechanism
in favour of just one (or several) member(s) of the classical IDF. As
we noted previously in Sec. <ref>, the striped collinear state
is indeed energetically selected by quantum fluctuations
<cit.> in the present case in the large-$s$ limit
<cit.> where first-order linear spin-wave theory
(LSWT) becomes exact.
Of course, quantum fluctuations can also be expected in such cases of
macroscopic degeneracy of the classical GS phase, to destroy
completely the magnetic LRO, as we discussed in Sec. <ref>,
Clearly, this is most likely to occur for small values of $s$, when
the results of LSWT become less likely to remain valid and when
quantum fluctuations become larger. Since the specific case when
$J_{3}=J_{2}\equiv \kappa J_{1}$ includes the classical tricritical point at
$\kappa_{{\rm cl}} = \frac{1}{2}$, which is the point of
maximum classical frustration, we restrict further attention to this
potentially rich regime in the entire parameter space of the $J_{1}$–$J_{2}$–$J_{3}$ model.
Along this line, the classical ($s \rightarrow \infty$) model at $T=0$
undergoes a first-order transition from the Néel phase, which is
the stable GS phase for $\kappa < \kappa_{{\rm cl}}$, to the striped
phase which is the stable GS phase for $\kappa > \kappa_{{\rm cl}}$. From our above discussion it
is clear that the most promising regime for novel quantum states, with
no classical counterparts, to emerge is the region around $\kappa
\approx \frac{1}{2}$.
In an earlier paper <cit.> the
$J_{1}$–$J_{2}$–$J_{3}$ model with $J_{3}=J_{2}\equiv \kappa J_{1} >
0$ was studied for the case $s=\frac{1}{2}$ in the window $0 \leq
\kappa \leq 1$, using the CCM. It was found
<cit.> that the classical ($s \rightarrow \infty$)
transition at $\kappa_{{\rm cl}}=\frac{1}{2}$ is split in the
$s=\frac{1}{2}$ case into two transitions at $\kappa_{c_{1}} < \kappa_{{\rm
cl}}$ and $\kappa_{c_{2}} > \kappa_{{\rm cl}}$, with the Néel phase
surviving for $\kappa < \kappa_{c_{1}} \approx 0.47$ and the striped
phase for $\kappa > \kappa_{c_{2}} \approx 0.60$. A paramagnetic
phase, with no discernible magnetic LRO, was indeed found to exist in
the intermediate regime $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$.
CCM calculations were also performed to measure the susceptibilities of
the two AFM phases on either side of the paramagnetic regime against
the formation of plaquette valence-bond crystalline (PVBC) order
<cit.>. It was thereby concluded that the paramagnetic
state was most likely one with PVBC order over the entire intermediate
regime $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$. On the basis of
all the CCM calculations (i.e., for the GS energy per spin $E/N$, the
GS magnetic order parameter $M$, and the susceptibility $\chi_{p}$
against the formation of PVBC order, for the two AFM states on either
side of the intermediate regime), the accumulated evidence pointed
towards the quantum phase transition (QPT) at the quantum critical
point (QCP) $\kappa = \kappa_{c_{2}}$ being a first-order one, just as is
the classical phase transition at $\kappa = \kappa_{{\rm cl}}$. By
contrast, the QPT at $\kappa = \kappa_{c_{1}}$ appeared to be a
continuous one on the basis of the CCM results presented. Since the
quasiclassical Néel phase and the quantum PVBC phase break different
symmetries, however, the usual Landau-Ginzburg-Wilson scenario of continuous
phase transitions is inapplicable, and it was suggested that the CCM
results <cit.> provide strong evidence for the QPT
at $\kappa=\kappa_{c_{1}}$ being of the deconfined quantum critical
type <cit.>.
In view of the qualitative differences between the GS phase diagrams
of the above spin-$\frac{1}{2}$ and classical ($s \rightarrow \infty$)
versions of the model, it is obviously of considerable interest to
examine the model in the case where the spin quantum number $s >
\frac{1}{2}$. One of the great strengths of the CCM is that it is
relatively straightforward, both conceptually and computationally, to
examine a given spin-lattice model for different values of $s$, within
a unified and consistent hierarchy of approximations. Hence, we now
use the method to examine the $J_{1}$–$J_{2}$–$J_{3}$ model on the
honeycomb lattice, along the line $J_{3}=J_{2}\equiv \kappa J_{1} >
0$, with $0 \leq \kappa \leq 1$, for the cases
$s=1,\,\frac{3}{2},\,2\,,\frac{5}{2}$, in order to compare them both
with the extreme quantum limit ($s=\frac{1}{2}$) case and with the
classical ($s \rightarrow \infty$) case.
§ THE COUPLED CLUSTER METHOD
We briefly describe here the key features of the CCM, and refer the
interested reader to the extensive literature (and see, e.g., Refs.<cit.>
and references cited therein) for further details. To implement the
method in practice one first needs to choose a suitable normalized
model (or reference) state $|\Phi\rangle$, against which the
correlations present in the exact GS wave function can be
incorporated. The properties required of $|\Phi\rangle$ are described
more fully below, but in general terms it plays the role of a
generalized vacuum state. For the present study suitable choices for
the model state $|\Phi\rangle$ will turn out to be the two
quasiclassical AFM states (viz., the Néel and collinear striped
states) that form the stable GS phases of the classical version of the
model under consideration in their respective regimes of the $T=0$
phase diagram.
The exact GS ket- and bra-state wave functions, $|\Psi\rangle$ and
$\langle\tilde{\Psi}|$, respectively, are chosen to have the normalization
\begin{equation}
\langle\tilde{\Psi}|\Psi\rangle = \langle{\Phi}|\Psi\rangle =
\langle{\Phi}|\Phi\rangle \equiv 1\,. \label{norm_conditions}
\end{equation}
These exact states are now parametrized with respect to the model
state $|\Phi\rangle$ in the exponentiated forms,
\begin{equation}
|\Psi\rangle=e^{S}|\Phi\rangle\,; \quad \langle\tilde{\Psi}|=\langle\Phi|\tilde{S}e^{-S}\,, \label{exp_para}
\end{equation}
that are a characteristic
hallmark of the CCM. Although the correlation operator $\tilde{S}$ may formally be expressed in terms of its counterpart $S$ as
\begin{equation}
\langle\Phi|\tilde{S} = \frac{\langle\Phi|e^{S^{\dagger}}e^{S}}{\langle\Phi|e^{S^{\dagger}}e^{S}|\Phi\rangle}\,, \label{correlation-opererators-relationship}
\end{equation}
by using Hermiticity, the CCM chooses not to restrain this relationship between $|\Psi\rangle$ and $\langle\tilde{\Psi}|$ explicitly. Instead, the
two correlation operators $S$ and $\tilde{S}$ are
formally decomposed independently as
\begin{equation}
S=\sum_{I\neq 0}{\cal S}_{I}C^{+}_{I}\,; \quad \tilde{S}=1+\sum_{I\neq 0}\tilde{{\cal S}}_{I}C^{-}_{I}\,, \label{correlation_oper}
\end{equation}
where $C^{+}_{0}\equiv 1$ is defined to be the identity operator in the many-body Hilbert space, and
where the set index $I$ denotes a complete set of single-particle
configurations for all $N$ particles. What is required of $|\Phi\rangle$ and the set of (multiconfigurational) creation operators $\{C^{+}_{I}\}$ is that $|\Phi\rangle$ is a fiducial (or cyclic) vector with respect to these operators, i.e., as a generalized vacuum state. Explicitly we require that the set of states $\{C^{+}_{I}|\Phi\rangle\}$ form a complete basis for the ket-state Hilbert space, and that
\begin{equation}
\langle\Phi|C^{+}_{I} = 0 = C^{-}_{I}|\Phi\rangle\,, \quad \forall I
\neq 0\,, \label{creat-destruct-operators-relationship}
\end{equation}
where the destruction operators $C^{-}_{I} \equiv
(C^{+}_{I})^{\dagger}$. Lastly, and importantly, we require that all
members of the complete set of operators $\{C^{+}_{I}\}$ are mutually
The rather general CCM paramerizations of Eqs.(<ref>)–(<ref>)
have several immediate important consequences. While Hermiticity is
not made explicit, and while the exact correlation operators $S$ and
$\tilde{S}$ will certainly fulfill Eq.(<ref>), when approximations are made
(e.g., by truncating the sums over configuration $I$ in Eq.(<ref>), as is usually done in practice),
Hermiticity may be only approximately maintained. Against this loss,
however, come several advantages, which usually far outweigh it.
Firstly, the CCM parametrizations guarantee that the Goldstone linked
cluster theorem is exactly preserved, as we describe in more detail
below, even if truncations are made in Eq. (<ref>).
In turn, this feature guarantees size-extensivity at any such level of
truncation, so that the GS energy, for example, is always calculated as
an extensive variable. Thus, the CCM has the first advantage that we
may work from the outset in the thermodynamic limit ($N \rightarrow
\infty$), thereby obviating the need for any finite-size scaling, as
is required in most alternative methods. A second key feature of the
CCM, which is guaranteed by its exponentiated parametrizations, is
that it also exactly preserves the important Hellmann-Feynman theorem
at any level of truncation or approximate implementation.
Clearly, a knowledge of the CCM $c$-number correlation
coefficients $\{{\cal S}_{I}, \tilde{{\cal S}}_{I}\}$ completely suffices to determine the GS expectation value of any operator. They are now found by minimization of the GS energy expectation functional,
\begin{equation}
\bar{H}=\bar{H}({\cal S}_{I},{\tilde{\cal S}_{I}}) \equiv
\langle\Phi|\tilde{S}e^{-S}He^{S}|\Phi\rangle\,, \label{eq_GS_E_xpect_funct}
\end{equation}
from Eq. (<ref>),
with respect to each of the coefficients $\{{\cal S}_{I},{\tilde{\cal
S}}_{I}\,; \forall I \neq 0\}$ separately. Variation of $\bar{H}$ from Eq. (<ref>), with
respect to ${\tilde{\cal S}}_{I}$ from Eq. (<ref>), immediately yields
\begin{equation}
\langle\Phi|C^{-}_{I}e^{-S}He^{S}|\Phi\rangle = 0\,, \quad \forall I \neq 0\,, \label{ket_eq}
\end{equation}
which is a coupled set of non-linear equations for the
coefficients $\{{\cal S}_{I}\}$, with the same number of equations as
parameters. A similar variation of $\bar{H}$ from Eq.(<ref>), with respect to ${\cal S}_{I}$ from Eq. (<ref>) yields
\begin{equation}
\langle\Phi|\tilde{S}e^{-S}[H,C^{+}_{I}]e^{S}|\Phi\rangle=0\,, \quad \forall I \neq 0\,, \label{bra_eq}
\end{equation}
as a coupled set of linear equations for the coefficients $\{{\tilde{\cal S}}_{I}\}$,
again with the same number of equations as parameters, once the
coefficients $\{{\cal S}_{I}\}$ are used as input after Eq.(<ref>) has been solved for them.
The GS energy $E$, which is simply the value of $\bar{H}$ from Eq. (<ref>) at the minimum, may then be expressed as
\begin{equation}
E=\langle\Phi|e^{-S}He^{S}|\Phi\rangle=\langle\Phi|He^{S}|\Phi\rangle\,, \label{eq_GS_E}
\end{equation}
using Eqs. (<ref>) and (<ref>). By making use of Eq.(<ref>), we may rewrite the set of linear equations (<ref>) in the equivalent form,
\begin{equation}
\langle\Phi|\tilde{S}(e^{-S}He^{S}-E)C^{+}_{I}|\Phi\rangle=0\,, \quad \forall I \neq 0\,, \label{bra_eq_alternative}
\end{equation}
which is just a set of generalized linear eigenvalue equations for the
set of coefficients $\{\tilde{\cal S}_{I}\}$.
Up to this point in the CCM procedure and implementation we have made no approximations. However, clearly
Eqs. (<ref>) that determine the creation coefficients $\{{\cal S}_{I}\}$ are
intrinsically highly nonlinear in view of the exponential terms. Hence one may ask if we now need to make truncations to evaluate these terms. We note, though, that these always
appear in the equations to be solved in the combination $e^{-S}He^{S}$ of a similarity transformation of the Hamiltonian. This may itself be expanded as the well-known nested commutator sum,
\begin{equation}
e^{-S}He^{S} = \sum^{\infty}_{n=0}\frac{1}{n!}[H,S]_{n}\,, \label{eq_expon_nested_commutator}
\end{equation}
where $[H,S]_{n}$ is an $n$-fold nested commutator, defined
iteratively as
\begin{equation}
[H,S]_{n}=[[H,S]_{n-1},S]\,; \quad [H,S]_{0}=H\,.
\end{equation}
Another key feature of the CCM is that this otherwise infinite sum now
(usually) terminates exactly at some finite order, when used in the
equations to be solved, due to the facts that all of the terms in the
expansion of Eq. (<ref>) for $S$ commute with one
another and that $H$ itself is (usually, as here) of finite order in the
relevant single-particle operators. For example, if $H$ contains up
to $m$-body interactions, in its second-quantized form it contains
sums of terms involving products of up to $2m$ single-particle
(destruction and creation) operators, and the sum in Eq.(<ref>) will terminate exactly with the
term $n=2m$. In our present case where the Hamiltonian of Eq.(<ref>) is bilinear in the SU(2) spin operators, the sum
terminates at $n=2$. Finally, we also note here that the fact that
all of the operators in the set $\{C^{+}_{I}\}$ that comprise $S$ by
Eq. (<ref>) commute with one another, automatically
guarantees that all (nonzero) terms in the sum of Eq.(<ref>) are linked to the Hamiltonian.
Unlinked terms simply cannot appear, and hence the Goldstone theorem
and size-extensivity are satisfied, at any level of truncation.
Hence, for any implementation of the CCM, the only approximation
made in practice is to restrict the set of multiconfigurational
set-indices $\{I\}$ that are retained in the expansions of Eq.(<ref>) for the correlation operators
$\{S,\tilde{S}\}$ to some appropriate (finite or infinite) subset.
How this choice is made must clearly depend on the problem at hand and
on the particular choices that have been made for the model state
$|\Phi\rangle$ and the associated set of operators $\{C^{+}_{I}\}$. Let
us, therefore, now turn to how such choices are made for the present
model in particular and for quantum spin-lattice models in general.
The simplest choice of model state $|\Phi\rangle$ for a quantum
spin-lattice problem is a straightforward independent-spin product
state in which the spin projection (along some specified quantization axis) of the spin on each lattice site is
specified independently. The two quasiclassical collinear AFM states
shown in Figs. <ref>(a) and <ref>(b),
viz., the Néel and striped states, are examples. In order to
treat all such states in the same way it is very convenient to make a
passive rotation of each spin independently (i.e., by making a
suitable choice of local spin quantization axes on each site independently), so that
on every site the spin points downwards, say, in the negative $z_{s}$
direction, as in the spin-coordinate frame shown in Fig.<ref>. Such rotations are clearly just unitary
transformations that leave the basic SU(2) spin commutation relations
unchanged. In this way each lattice site $k$ is completely equivalent
to all others, and all such independent-spin product model states now
take the universal form
In this representation it is now clear that $|\Phi\rangle$ can indeed
be regarded as a fiducial vector with respect to a set of mutually
commuting creation operators $\{C^{+}_{I}\}$, which may now be chosen
as a product of single-spin raising operators, $s^{+}_{k} \equiv
s^{x}_{k}+is^{y}_{k}$, such that $C^{+}_{I} \rightarrow
s^{+}_{k_{1}}s^{+}_{k_{2}}\cdots s^{+}_{k_{n}};\; n=1,2,\cdots , 2sN$.
The corresponding set index $I$ thus becomes a set of lattice-site
indices, $I \rightarrow \{k_{1},k_{2},\cdots , k_{n};\; n=1,2,\cdots ,
2sN\}$, in which each site index may be repeated up to $2s$ times.
Once the local spin coordinates have been selected by the above procedure
(i.e., for the given model state $|\Phi\rangle$), one simply
re-expresses the Hamiltonian $H$ in terms of them.
We now turn to the choice of approximation scheme, which hence simply
involves a choice of which configurations $\{I\}$ to retain in the
decompositions of Eq. (<ref>) for the CCM
correlation operators $(S,\tilde{S})$. A powerful and rather general
such scheme, the so-called SUB$n$–$m$ scheme, retains the
configurations involving a maximum of $n$ spin-flips (where each
spin-flip requires the action of a spin-raising operator $s^{+}_{k}$
acting once) spanning a range of no more than $m$ contiguous sites on
the lattice. A set of lattice sites is defined to be contiguous if
every site in the set is the NN of at least one other in the set (in a
specified geometry). Clearly, as both indices become indefinitely
large, the approximation becomes exact. Different schemes can be
defined according to how each index approaches infinity.
For example, if we first let $m \rightarrow \infty$, we arrive at the
so-called SUB$n$ $\equiv$ SUB$n$–$\infty$ scheme, which is the
approximation scheme most commonly employed, more generally, for
systems defined in a spatial continuum, such as atoms and molecules in
quantum chemistry <cit.> or finite atomic nuclei or
nuclear matter in nuclear physics <cit.> (and see,
e.g., Refs. <cit.> for further
details). By contrast to continuum theories, for which the notion of
contiguity is not easily applicable, in lattice theories both indices
$n$ and $m$ may be kept finite. A very commonly used scheme is the
so-called LSUB$m$ scheme
<cit.>, defined to retain,
at the $m$th level of approximation, all spin clusters described by
multispin configurations in the index set $\{I\}$ defined over any
possible lattice animal (or polyomino) of size $m$ on the lattice.
Again, such a lattice animal is defined in the usual graph-theoretic
sense to be a configured set of contiguous (in the above sense) sites
on the lattice. Clearly, the LSUB$m$ scheme is equivalent to the
SUB$n$–$m$ scheme when $n=2sm$ for particles of spin quantum number
$s$, i.e., LSUB$m \equiv$ SUB$2sm$–$m$. The LSUB$m$ scheme was precisely the
truncation scheme used in our previous study of the present model for
the case $s=\frac{1}{2}$ <cit.>.
At a given $m$th level of LSUB$m$ approximation the number $N_{f}$ of
fundamental spin configurations that are distinct (under the symmetries
of the lattice and the specified model state), which are retained is
lowest for $s=\frac{1}{2}$ and rises sharply as $s$ is increased.
Since $N_{f}$ typically also increases rapidly (typically, faster than
exponentially) with the truncation index $m$, an alternative scheme
for use in cases $s > \frac{1}{2}$ is to be set $m=n$ and hence employ
the resulting SUB$n$–$n$ scheme. Clearly, the two schemes are
equivalent only for the case $s=\frac{1}{2}$, for which LSUB$m$
$\equiv$ SUB$m$–$m$. We note too that the numbers $N_{f}$ of
fundamental configurations at a given SUB$n$–$n$ level are still
higher for the cases $s>\frac{1}{2}$ considered here than for the case
$s=\frac{1}{2}$. Thus, whereas for the the present model we were able
to perform LSUB$m$ calculations with $m \leq 12$ previously for the
case $s=\frac{1}{2}$ <cit.>, we are now restricted
for the cases $s>\frac{1}{2}$ considered here to perform SUB$n$–$n$
calculations with $n \leq 10$, with similar amounts of supercomputer
resources available. Thus, for example, for the case $s=\frac{1}{2}$,
at the LSUB12 level of approximation we have $N_{f}=103,097 (250,891)$
using the Néel (striped) state as the CCM model state. By
comparison, at the SUB10–10 level of approximation we have
$N_{f}=219,521 (552,678)$ for the case $s=1$, and $N_{f}=538,570
(1,436,958)$ for the case $s=\frac{5}{2}$, in each case using the
Néel (striped) state as the CCM model state. Just as before
<cit.> we employ massively parallel computing
<cit.> both to derive (with computer algebra) and to solve
(and see, e.g., Ref.<cit.>) the respective
coupled sets of CCM equations (<ref>) and
Once the coefficients $\{{\cal S}_{I},{\tilde{{\cal S}}}_{I}\}$ retained in a given
SUB$n$–$n$ approximation have been calculated by solving Eqs.(<ref>) and (<ref>), we may calculate any GS
quantity at the same level of approximation. Thus, for example, the
GS energy $E$ may be calculated from Eq. (<ref>) in terms
of the ket-state coefficients $\{{\cal S}_{I}\}$ alone. Any other GS
quantity requires a knowledge also of the bra-state coefficients
$\{\tilde{{\cal S}}_{I}\}$. For example, we also calculate here the
magnetic order parameter $M$, which is defined to be the average
on-site GS magnetization,
\begin{equation}
M = -\frac{1}{N}\sum^{N}_{k=1}\langle\Phi|\tilde{S}
e^{-S}s^{z}_{k}e^{S}|\Phi\rangle\,, \label{M_eq}
\end{equation}
in terms of the local rotated spin-coordinate frames that we have
described above.
As a last step, and as essentially the only approximation made in the
whole CCM implementation, we need to extrapolate the raw SUB$n$–$n$
data points for $E$ and $M$ to the exact $n \rightarrow \infty$ limit.
Although no exact extrapolation rules are known, a great deal of experience
has by now been accumulated for doing so, from the many applications
of the technique to a wide variety of spin-lattice problems that have
been examined with the method. For the GS energy per spin, for
example, a very well tested and highly accurate extrapolation ansatz
(and see, e.g., Refs.<cit.>)
\begin{equation}
\frac{E(n)}{N} = a_{0}+a_{1}n^{-2}+a_{2}n^{-4}\,, \label{extrapo_E}
\end{equation}
while for the magnetic order parameter $M$ different schemes have been
used in different situations. Unsurprisingly, the GS expectation
values of other physical observables generally converge less rapidly
than the GS energy, i.e., with leading exponents less than two. More
specifically, the leading exponent for $M$ tends to depend on the
amount of frustration present, generally being smaller for the most
highly frustrated cases.
Thus, for unfrustrated models or for models with only moderate amounts of
frustration present, a scaling ansatz for $M(n)$ with leading power $1/n$ (rather than $1/n^{2}$ as for the GS energy),
\begin{equation}
M(n) = b_{0}+b_{1}n^{-1}+b_{2}n^{-2}\,, \label{M_extrapo_standard}
\end{equation}
has been found to work well in many cases (and see, e.g., Refs.<cit.>).
For systems that are either close to a QCP or for which the magnetic
order parameter $M$ for the phase under study is either zero or close
to zero, the extrapolation ansatz of Eq. (<ref>)
tends to overestimate the extrapolated value and hence to predict a
somewhat too large value for the critical strength of the frustrating
interaction that is driving the respective phase transition. In such
cases a great deal of evidence has now shown that a scaling ansatz
with leading power $1/n^{1/2}$ fits the SUB$n$–$n$ data much better.
Thus, as an alternative in those instances to Eq.(<ref>), a more appropriate scaling scheme (and see,
e.g., Refs.<cit.>)
\begin{equation}
M(n) = c_{0}+c_{1}n^{-1/2}+c_{2}n^{-3/2}\,. \label{M_extrapo_frustrated}
\end{equation}
Since the extrapolation schemes of Eqs.(<ref>)–(<ref>) contain three fitting
parameters, it is clearly preferable to use at least four SUB$n$–$n$
data points in each case. Furthermore since the lowest-order SUB2–2
approximants are less likely to conform well to the extrapolation
schemes, we prefer to perform fits using SUB$n$–$n$ data with with $n
\geq 4$.
§ RESULTS
We show in Fig. <ref>(a) our CCM results for the
GS energy per spin, $E/N$, of the spin-1 model at
various SUB$n$–$n$ levels of approximation with $n=\{4,6,8,10\}$,
using both the Néel and striped AFM states as separate choices of
the CCM model state.
CCM results for the GS energy per spin, $E/N$, for
the $J_{1}$–$J_{2}$–$J_{3}$ model on the honeycomb lattice (with $J_{1} \equiv 1, J_{3}=J_{2} \equiv \kappa J_{1} > 0$), as a function of $\kappa$,
using the Néel and striped states as the CCM model states. (a) The results for the $s=1$ model are shown using the SUB$n$–$n$
approximations with $n=\{4,6,8,10\}$, together with the corresponding
extrapolated SUB$\infty$–$\infty$ results obtained using equation (<ref>), with this data set. (b) We show extrapolated (SUB$\infty$–$\infty$) results for $E/(Ns^{2})$ as a function of $\kappa$, using equation
(<ref>) together with the data set $n=\{6,8,10,12\}$ for the case $s=\frac{1}{2}$, and the corresponding data sets with $n=\{4,6,8,10\}$ for the four cases $s=1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. We also show the corresponding classical ($s \to \infty$) result.
We observe very clearly that the results converge very rapidly with
increasing values of the truncation index $n$, and we also show the
extrapolated $n \rightarrow \infty$ results, $a_{0}$, from Eq.(<ref>). We also observe that each of the energy curves
based on a particular model state terminates at a critical value of
the frustration parameter $\kappa$ that depends on the SUB$n$–$n$
approximation used. Beyond those critical values no real solutions
can be found to the corresponding CCM equations (<ref>). Such
termination points of the CCM coupled equations are very common in
practice, and are well understood (see, e.g., Refs.<cit.>).
They are direct manifestations of the corresponding QCP in the system,
at which the respective form of magnetic LRO in the model state used
melts. As is usually the case, the CCM SUB$n$–$n$ solutions for a
given finite value of $n$ and for a given phase extend beyond the
actual SUB$\infty$–$\infty$ QCP, i.e., into the unphysical regime
beyond the termination point. The extent of the unphysical regime
diminishes (to zero) as the truncation order $n$ increases (to the
exact $n \rightarrow \infty$ limit).
In Fig. <ref>(b) we compare the corresponding
extrapolated curves for the scaled GS energy per spin, $E/(Ns^{2})$,
using both the Néel and striped AFM states separately as our
choice of CCM model state, for the five cases
$s=\frac{1}{2},\,1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. In each case the
extrapolation is performed with Eq. (<ref>). For the case
$s=\frac{1}{2}$ alone the input SUB$n$–$n$ data points are
$n=\{6,8,10,12\}$, while for each of the cases $s>\frac{1}{2}$ the
input set is $n=\{4,6,8,10\}$. We also show in Fig.<ref>(b) the corresponding classical ($s \rightarrow
\infty$) results, $E^{{\rm
N\acute{e}el}}_{{\rm cl}}/(Ns^{2})=\frac{3}{2}(-1+\kappa)$ and $E^{{\rm
striped}}_{{\rm cl}}/(Ns^{2})=\frac{1}{2}(1-5\kappa)$. We observe clear
preliminary evidence from Fig. <ref>(b) for an
intermediate phase (between the phases with Néel and striped
magnetic LRO) in the $s=1$ case, although with a range of stability in
the frustration parameter $\kappa$ now markedly less than in the
$s=\frac{1}{2}$ case. The preliminary evidence from the energy
results is also that there is no such intermediate phase present in
each of the cases $s > 1$. Lastly, Fig. <ref>(b) also
shows that, at least so far as the energy results are concerned, all
cases with $s \gtrsim 2$ are rather close to the classical limit.
In order to get more detailed evidence on the phase structures of the
model for various values of the spin quantum number $s$ we now turn to
the results for the GS magnetic order parameter, $M$, of Eq.(<ref>). Thus, firstly, in Fig. <ref>(a) we show
our CCM results for $M$ for the spin-1 model at various SUB$n$–$n$
levels of approximation with $n=\{4,6,8,10\}$, using both the Néel
and striped AFM states as separate choices for the model state.
CCM results for the GS magnetic order parameter, $M$, for
the $J_{1}$–$J_{2}$–$J_{3}$ model on the honeycomb lattice (with $J_{1} \equiv 1, J_{3}=J_{2} \equiv \kappa J_{1} > 0$), as a function of $\kappa$,
using the Néel and striped states as the CCM model states. (a) The results for the $s=1$ model are shown using the SUB$n$–$n$
approximations with $n=\{4,6,8,10\}$, together with the corresponding
extrapolated SUB$\infty$–$\infty$ results obtained using equation (<ref>), with this data set. (b) We show extrapolated (SUB$\infty$–$\infty$) results for $M/s$ as a function of $\kappa$, using equation
(<ref>) together with the data set $n=\{6,8,10,12\}$ for the case $s=\frac{1}{2}$, and the corresponding data sets with $n=\{4,6,8,10\}$ for the four cases $s=1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. We also show the corresponding classical ($s \to \infty$) result.
Hence, what is shown in Fig. <ref>(a) for $M$ is just
the precise analogue of what is shown in Fig. <ref>(a)
for $E/N$. It is clear that the SUB$n$–$n$ sequence of
approximations for $M$ converges more slowly than for $E/N$, just as
expected. We also show in Fig. <ref>(a) the
extrapolated results for the spin-1 model, where we have used the data
set shown, $n=\{4,6,8,10\}$, as input to the extrapolation scheme of
Eq. (<ref>). As was explained in Sec.<ref>, while the alternative scheme of Eq.(<ref>) is certainly more appropriate when the
frustration parameter $\kappa$ is zero or small, that of Eq.(<ref>) is certainly preferable for larger values
of $\kappa$ (e.g., in the striped phase) or when the order parameter
$M$ becomes small (i.e., near any QCPs).
The SUB$\infty$–$\infty$ extrapolation shown in Fig.<ref>(a) now clearly validates the earlier, more
qualitative, results from the GS energy, namely the existence of a GS
phase intermediate between the quasiclassical Néel and striped
collinear AFM states, just as in the spin-$\frac{1}{2}$ case. Once
again, Néel LRO exists over the range $0 \leq \kappa <
\kappa_{c_{1}}$, while striped LRO exists for $\kappa >
\kappa_{c_{2}}$, where $\kappa_{c_{1}} < \kappa_{{\rm cl}} =
\frac{1}{2}$ and $\kappa_{c_{2}} > \kappa_{{\rm cl}} = \frac{1}{2}$.
The values obtained for the two QCPs from the extrapolations using
Eq. (<ref>) with the data set $n=\{4,6,8,10\}$,
taken as the points where $M \rightarrow 0$, as shown in Fig.<ref>(a), are $\kappa_{c_{1}} \approx 0.486$ and
$\kappa_{c_{2}} \approx 0.527$. A more detailed analysis of the
errors associated with the fits, and by comparison with comparable
extrapolations using alternative data sets (e.g., $n=\{6,8,10\}$ and
$n=\{4,6,8\}$), yields our best estimates for the spin-1 model QCPs,
$\kappa_{c_{1}}=0.485(5)$ and $\kappa_{c_{2}}=0.528(5)$. These may be
compared with the corresponding values for the spin-$\frac{1}{2}$
model QCPs <cit.>, $\kappa_{c_{1}}=0.47$ and
In Fig. <ref>(b) we now compare the corresponding
extrapolated curves for the scaled magnetic order parameter, $M/s$,
using both the Néel and striped AFM states separately as CCM model
states, for the five cases
$s=\frac{1}{2},\,1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. In each case
shown the extrapolation has been performed with the ansatz of Eq.(<ref>), together with the data set
$n=\{6,8,10,12\}$ for the $s=\frac{1}{2}$ case and the sets
$n=\{4,6,8,10\}$ for each case with $s>\frac{1}{2}$. The results once
again validate our earlier, more qualitative, findings from the GS
energy results, that the intermediate phase is present only for the
two cases $s=\frac{1}{2},\,1$, with a direct transition from the
Néel to the striped phase in all cases $s > 1$, just as in the
classical $(s \rightarrow \infty)$ limit. For all cases $s > 1$ this
direct transition clearly occurs at values very close to the classical
value $\kappa_{{\rm cl}} = \frac{1}{2}$. The actual crossing points
of the order parameter curves shown in Fig. <ref>(b)
occur at values $\kappa_{m} \approx 0.517$ for $s=\frac{3}{2}$,
$\kappa_{m} \approx 0.508$ for $s=2$, and $\kappa_{m} \approx 0.505$
for $s=\frac{5}{2}$. What is apparent from Fig. <ref>(b)
is that the curves for the striped phase approach zero much more steeply
than for the Néel phase for all values of $s$. For the case
$s=\frac{1}{2}$ it was argued <cit.> that this was
a reflection of the transition at $\kappa_{c_{2}}$ being of
first-order type, while that at $\kappa_{c_{1}}$ is of continuous (and
hence probably of the deconfined) type. The difference in the shapes
of the curves near the crossing point is what leads to the direct
transition apparently being at values slightly larger than
$\kappa_{{\rm cl}} = \frac{1}{2}$ for finite values of $s > 1$.
Clearly the precise crossing points $\kappa_{m}$ of the magnetic order
curves for the Néel and striped phases for the cases $s>1$ depend
rather critically on the extrapolations, particularly those for the
striped phase, where the slope become large. In such cases more
precise values of the corresponding QCP for the direct transition
between the two quasiclassical phases can be expected to come from the
analogous crossing points, $\kappa_{e}$, of the extrapolated energy
curves. The respective values from Fig. <ref>(b) are
$\kappa_{e} \approx 0.544$ for $s=\frac{3}{2}$, $\kappa_{e} \approx
0.534$ for $s=2$, and $\kappa_{e} \approx 0.528$ for $s=\frac{5}{2}$.
It is reassuring that the respective pairs of values of $\kappa_{e}$
and $\kappa_{m}$ agree so well in each case, for what are essentially
quite independent results.
Before discussing how we can investigate the nature of the
intermediate phase for the present $s=1$ case within the CCM
framework, let us briefly comment on the case of the pure HAF on the
honeycomb lattice, with NN interactions only (i.e., the limiting
case $\kappa=0$ of the present model). In this case, the
extrapolation ansatz of Eq. (<ref>) becomes
applicable, rather than that of Eq. (<ref>)
shown in Fig. <ref>. We show in Table <ref> the
scaled values for the GS energy per spin and magnetic order
parameters, $E/(Ns^{2})$ and $M/s$, respectively, for our present
model calculations at the unfrustrated limiting value $\kappa=0$.
GS parameters of the HAF on the honeycomb lattice, with NN interactions only (of strength $J_{1}=1$), for various values of the spin quantum number $s$.
$s$ $E/(Ns^{2})$ $M/s$
$\frac{1}{2}$ -2.17866 0.5459
1 -1.83061 0.7412
$\frac{3}{2}$ -1.71721 0.8249
2 -1.66159 0.8689
$\frac{5}{2}$ -1.62862 0.8955
$\infty$ -1.5 1
The corresponding extrapolation schemes of Eqs. (<ref>) and
Eq. (<ref>) have been used in Table <ref>,
together with the input data sets with $n=\{6,8,10,12\}$ for
$s=\frac{1}{2}$, and with $n=\{4,6,8,10\}$ for $s>\frac{1}{2}$.
Another way to estimate the accuracy of our extrapolated CCM results
for the higher spin values is to use them to extract, for example, the
coefficients of the expansions of $E/(Ns^{2})$ and $M/s$ in inverse
powers of $s$, and compare them with the results of higher-order
spin-wave theory (SWT). For example, at second-order, we may fit our
results of Table <ref> to the forms,
\begin{equation}
\frac{E}{Ns^{2}} = -\frac{3}{2} + \frac{e_{1}}{s} + \frac{e_{2}}{s^{2}}\,, \label{honey-pure_E-fit_inversePower}
\end{equation}
\begin{equation}
\frac{M}{s} = 1 + \frac{m_{1}}{s} + \frac{m_{2}}{s^{2}}\,, \label{honey-pure_M-fit_inversePower}
\end{equation}
and then compare with the corresponding results of second-order SWT,
i.e., SWT(2). If we simply take our results from Table <ref>
for the two highest spin values calculated, viz., $s=2,\,\frac{5}{2}$,
and fit them to Eqs. (<ref>) and
(<ref>), we obtain values $e_{1} \approx
-0.31503$ and $e_{2} \approx -0.01630$ for the GS energy, and $m_{1}
\approx -0.2575$ and $m_{2} \approx -0.0095$ for the GS Néel
magnetic order parameter (i.e., the sublattice magnetization). The
corresponding (exact) SWT(2) results <cit.> are
$e_{1} = -0.31476$, $e_{2} = -0.01651$, $m_{1} = -0.2582$, and $m_{2}
= 0$. The agreement is rather striking.
We now turn finally to the question of what is the nature of the
intermediate phase in the case $s=1$. For the analogous
$s=\frac{1}{2}$ case it was shown <cit.> that the
intermediate paramagnetic phase likely had PVBC order. It is natural
now to consider this possibility for the $s=1$ case. To do so we now
calculate within the CCM framework the susceptibility, $\chi_{p}$,
which measures the response of the system to an applied external field
that promotes PVBC order. More generally, let us add an infinitesimal
field operator $F \equiv \delta\; \hat{O}$ to the Hamiltonian $H$ of Eq.(<ref>). We then calculate the perturbed energy per site,
$E(\delta)/N=e(\delta)$, for the perturbed Hamiltonian $H+F$, using
the same CCM procedure as above, and using the same previous model
states. The susceptibility of the system to the perturbed operator
$\hat{O}$ is then defined as usual to be
\begin{equation}
\chi_{F} \equiv -
\left. \frac{\partial^2e(\delta)}{\partial {\delta}^2} \right|_{\delta=0\,,}
\end{equation}
so that the energy,
\begin{equation}
e(\delta) = e(0) - \frac{1}{2}\chi_{F}\delta^{2}\,,
\end{equation}
is a maximum at $\delta=0$ for $\chi_{F}>0$. A clear signal of the
system becoming unstable against the perturbation $F$ is the finding
that $\chi_{F}$ diverges or, equivalently, that $\chi_{F}^{-1}$
becomes zero (and then possibly changes sign).
In our present case the perturbing operator $F$ is now chosen to
promote PVBC order, and it is illustrated in Fig.<ref>(a).
(a) The
fields $F=\delta\; \hat{O}$ for the plaquette susceptibility
$\chi_{p}$. Thick (red) and thin (black) lines correspond respectively
to strengthened and weakened NN exchange couplings, where $\hat{O} =
\sum_{\langle i,j \rangle} a_{ij}
\mathbf{s}_{i}\cdot\mathbf{s}_{j}$, and the sum runs over all NN
bonds, with $a_{ij}=+1$ and $-1$ for thick (red) and thin (black)
lines respectively. (b) CCM results for the inverse plaquette
susceptibility, $1/\chi_{p}$, for
the $J_{1}$–$J_{2}$–$J_{3}$ model on the honeycomb lattice (with $J_{1} \equiv 1, J_{3}=J_{2} \equiv \kappa J_{1} > 0$), as a function of $\kappa$,
using the Néel and striped states as the CCM model states. The results for the $s=1$ model are shown using the SUB$n$–$n$
approximations with $n=\{4,6,8,10\}$, together with the corresponding
extrapolated SUB$\infty$–$\infty$ results obtained using equation (<ref>), with this data set. (c) We show extrapolated (SUB$\infty$–$\infty$) results for $1/\chi_{p}$ as a function of $\kappa$, using equation
(<ref>) together with the data set $n=\{6,8,10,12\}$ for the case $s=\frac{1}{2}$, and the data set $n=\{4,6,8,10\}$ for the case $s=1$.
It clearly breaks the translational symmetry
of the system. The SUB$n$–$n$ estimates, $\chi_{p}(n)$, for the
resulting susceptibility of our system to the formation of PVBC order,
then need to be extrapolated to the exact ($n \rightarrow \infty$)
limit. Previous experience <cit.> has shown that
an appropriate extrapolation ansatz is
\begin{equation}
\chi_{p}^{-1}(n) = x_{0}+x_{1}n^{-2}+x_{2}n^{-4}\,. \label{X_extrapo}
\end{equation}
We show in Fig. <ref>(b) our CCM results for
$\chi_{p}^{-1}(n)$ at SUB$n$–$n$ levels of approximation with
$n=\{4,6,8,10\}$, using both the Néel and striped AFM states
separately as model states, in complete analogy to what is displayed
in Figs. <ref>(a) and <ref>(a) for the GS
energy per spin, $E/N$, and magnetic order parameter, $M$. In Fig.<ref>(b) we also show the extrapolated results ($x_{0}$)
obtained by inserting the set of raw results shown into Eq.(<ref>). Just as in the spin-$\frac{1}{2}$ case
<cit.>, the results for $\chi_{p}^{-1}(n)$
converge much faster for the Néel state than for the striped
state. For both states there are clear critical points at which
$\chi_{p}^{-1}$ vanishes. However, the shapes of the curves for
$\chi_{p}^{-1}$ near their respective critical points differ markedly,
just as in the spin-$\frac{1}{2}$ case. Thus, on the Néel side,
$\chi_{p}^{-1} \rightarrow 0$ with a slope that is small. By
contrast, on the striped side, $\chi_{p}^{-1} \rightarrow 0$ with a
very large slope (and which is probably compatible with being
infinite, within extrapolation errors). These differences reinforce
our earlier findings that the critical point at which Néel order
vanishes is likely to mark a continuous phase transition, while that
at which striped order vanishes is likely to mark a first-order
In Fig. <ref>(c) we compare the extrapolated CCM results
for $\chi_{p}^{-1}$ for the two cases ($s=\frac{1}{2},\,1$) for which
our findings indicate the existence in the model of a phase in its
$T=0$ GS phase diagram intermediate between the two quasiclassical
phases with magnetic LRO. Whereas in the spin-$\frac{1}{2}$ case
there exists a clear gap along the frustration parameter, $\kappa$,
axis between the two points at which $\chi_{p}^{-1}$ vanishes (one for
each quasiclassical phase), the gap in the spin-1 case is much less
marked. Indeed, its very existence is open to doubt, as we explain
below. For the spin-$\frac{1}{2}$ case <cit.> the
two values on Fig. <ref>(c) at which $\chi_{p}^{-1}
\rightarrow 0$ are $\kappa \approx 0.473$ and $\kappa \approx 0.586$.
These may be compared with the corresponding values on Fig.<ref>(b) at which $M \rightarrow 0$, which are
$\kappa_{c_{1}} \approx 0.466$ and $\kappa_{c_{2}} \approx 0.601$.
The very close agreement between the corresponding values was taken
<cit.> to be good evidence that the PVBC phase
occurs at (or is very close to) the transition points $\kappa_{c_{1}}$
and $\kappa_{c_{2}}$ where the quasiclassical magnetic LRO vanishes.
The fact that the slope of the $\chi_{p}^{-1}(\kappa)$ curve on the
Néel side is vanishingly small (within numerical errors) at the
point where $\chi_{p}^{-1} \rightarrow 0$, also provided strong
evidence that $\chi_{p}^{-1}$ vanishes over the entire intermediate
region in Fig. <ref>(c) between the points, where the
CCM calculations have been performed with the two classical model
states. All of this evidence pointed strongly to the stable GS phase
in the whole of the region $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$
being one with PVBC order, in the $s=\frac{1}{2}$ case. If any other
phase exists in part of this region, its region of stability is
clearly constrained by the CCM calculations <cit.>
to be a very small part of the intermediate region.
The corresponding situation for the $s=1$ case is now subtly
different, however. Thus, firstly, a close inspection of Fig.<ref>(b) (and, especially, the inset) shows that on the
Néel side the extrapolated SUB$\infty$–$\infty$
$\chi_{p}^{-1}(\kappa)$ curve has not quite reached zero at the
SUB10–10 termination point (or at least as far as we have managed to
perform numerical calculations, which, as we discussed above, become
increasingly difficult and computationally costly the closer one
approaches a termination point). A simple further extrapolation of
the curve, however, yields a value $\kappa \approx 0.530(2)$ at which
$\chi_{p}^{-1} \rightarrow 0$ on the Néel side. This is very close to the corresponding value in Fig.<ref>(b) of $\kappa \approx 0.535$ at which
$\chi_{p}^{-1} \rightarrow 0$ on the striped phase side.
Significantly, both of these values are indubitably greater than
$\kappa_{{\rm cl}} = \frac{1}{2}$. By contrast, the corresponding
values from Fig. <ref>(b) at which $M \rightarrow 0$ for
the spin-1 model are $\kappa_{c_{1}} \approx 0.486$ and
$\kappa_{c_{2}} \approx 0.527$. The most likely interpretation of our
$\chi_{p}$ results is hence that the CCM using both model states is
showing that $\chi_{p}^{-1}$ vanishes only at a single point, viz.,
the QCP $\kappa_{c_{2}}$, for the spin-1 model. This interpretation
is lent further weight by the observation that, unlike in the
spin-$\frac{1}{2}$ case, the slope of the $\chi_{p}^{-1}(\kappa)$
curve on the Néel side does not appear to be zero at the
(extrapolated) point at which it becomes zero.
In summary, if a PVBC-ordered phase is stable anywhere in the
intermediate regime $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$, our
findings are that it is confined only to a very narrow range close to
$\kappa_{c_{2}}$, and that it is definitely not the stable GS phase
over the whole interval. The more likely scenario is that the
intermediate regime is occupied by a paramagnetic phase (or more than
one such phase) with a form of order other than PVBC.
§ CONCLUSIONS
We have investigated higher-spin versions of a frustrated
$J_{1}$–$J_{2}$–$J_{3}$ HAF model on the honeycomb lattice, in the
specific case where $J_{1} > 0$, $J_{3}=J_{2} \equiv \kappa J_{1}>0$,
over the range $0 \leq \kappa \leq 1$ of the frustration parameter.
This includes the point of maximum frustration in the classical ($s
\rightarrow \infty$) limit, viz., the tricritical point at
$\kappa_{{\rm cl}} = \frac{1}{2}$, at which there is a direct
first-order transition (along the line $J_{3}=J_{2}$) between a Néel-ordered AFM phase for
$\kappa < \kappa_{{\rm cl}}$ and a collinear stripe-ordered AFM phase
for $\kappa > \kappa_{{\rm cl}}$. Whereas the spin-$\frac{1}{2}$
version of the model has been studied previously, higher-spin versions
have received no attention, so far as we are aware.
In particular, the CCM has been applied to the spin-$\frac{1}{2}$
model in an earlier study <cit.> that yielded
accurate results for its entire $T=0$ GS phase diagram. Since the method
generally provides values for the QCPs of a wide range of spin-lattice
systems, which are among the most accurate available by any
alternative methodology, we have now used the CCM to study spin-$s$
versions of the model with values of the spin quantum number
$s>\frac{1}{2}$. An aim has been to compare the $T=0$ GS phase
diagrams of the higher-spin models with the two extreme limits,
$s=\frac{1}{2}$ (where quantum effects should be greatest) and $s
\rightarrow \infty$ (where quantum effects vanish). In particular, it
has been shown <cit.> that the direct classical
transition at $\kappa_{{\rm cl}} = \frac{1}{2}$ is split by quantum
fluctuations in the spin-$\frac{1}{2}$ model into two separate
transitions at $\kappa_{c_{1}} < \kappa_{{\rm cl}}$ and
$\kappa_{c_{2}} > \kappa_{{\rm cl}}$, at the first of which Néel
AFM order of the type shown in Fig. <ref>(a) breaks
down, and at the second of which striped AFM order of the type shown
in Fig. <ref>(b) breaks down. Between the regimes
$\kappa < \kappa_{c_{1}}$ in which the stable GS phase has Néel
magnetic LRO and $\kappa > \kappa_{c_{2}}$ in which the stable GS
phase has striped magnetic LRO, there opens an intermediate
paramagnetic regime $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$.
Strong evidence was presented <cit.> that the
stable GS phase in this entire intermediate regime in the
spin-$\frac{1}{2}$ case is an intrinsically quantum-mechanical one
with PVBC order.
We have now performed analogous CCM calculations for spin-$s$ versions
of the same honeycomb lattice model for values
$s=1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. A primary finding is that an
intermediate phase also exists for the case $s=1$, but that for all
higher spins ($s \geq \frac{3}{2}$) the intermediate phase disappears
in favour of a direct transition between the two quasiclassical states
with magnetic LRO. For all finite values of the spin quantum number
$s \geq \frac{3}{2}$ the direct transition seems to occur at a value
$\kappa_{c}$ marginally higher than the classical value of 0.5 [e.g.,
for $s=\frac{3}{2}$, $\kappa_{c} \approx 0.53(1)$, and for
$s=\frac{5}{2}$, $\kappa_{c} \approx 0.52(1)$], with $\kappa_{c}$
tending monotonically to 0.5 as $s \rightarrow \infty$.
The range of the intermediate phase is smaller for the spin-1 model
($\kappa_{c_{1}} \approx 0.49$, $\kappa_{c_{2}} \approx 0.53$) than
for its spin-$\frac{1}{2}$ counterpart ($\kappa_{c_{1}} \approx
0.47$, $\kappa_{c_{2}} \approx 0.60$), as expected. Interestingly,
all of the evidence garnered here is that, unlike in the
spin-$\frac{1}{2}$ case the intermediate phase for the spin-1 model
does not have PVBC ordering. On the other hand, both the
spin-$\frac{1}{2}$ and spin-1 models seem to share that the transition
at $\kappa_{c_{2}}$ is a direct first-order one while that at
$\kappa_{c_{1}}$ is continuous.
On the basis of the present calculations, the nature and properties of
the intermediate phase in the spin-1 version of the model remain
open questions. It will be of considerable interest to study this phase
further, both by the CCM and by the use of alternative techniques. A
particularly promising such alternative technique in this respect is
the DMRG method, which has been used very recently
<cit.> in an analysis of the quantum
($T=0$) phase diagram of the spin-1 version of the related
$J_{1}$–$J_{2}$ Heisenberg model on the honeycomb lattice.
§ ACKNOWLEDGMENTS
We thank the University of Minnesota Supercomputing Institute for the
grant of supercomputing facilities. One of us (RFB) also gratefully
acknowledges the Leverhulme Trust for the award of an Emeritus
Fellowship (EM-2015-07).
§ REFERENCES
|
1511.00261
|
Binghamton University, Binghamton, NY 13902.
Ohio State University, Columbus, OH 43210.
Absence of Anosov diffeomorphisms]Aspherical products which do not support Anosov diffeomorphisms
$^\ast$A.G. was partially supported by NSF grant DMS-1266282. J.-F.L. was partially supported by NSF grant
MSC Primary 37D20; Secondary 55R10, 57R19, 37C25.
We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms.
§ INTRODUCTION
Let $M$ be a smooth closed $n$-dimensional Riemannian manifold.
Recall that a diffeomorphism $f$ is called Anosov if there
exist constants $\lambda \in (0,1)$ and $C>0$ along with a
$df$-invariant splitting $TM=E^s\oplus E^u$ of the tangent bundle of
$M$, such that for all $m \ge 0$
\begin{multline}
\label{def_anosov}
\qquad\|df^mv\|\le C\lambda^m\|v\|,\;v\in E^s,\; \\
\qquad\shoveleft{\|df^{-m}v\|\le C\lambda^{m}\|v\|,\;v\in E^u.
\hfill}
\end{multline}
The invariant distributions $E^s$ and $E^u$ are called the stable and unstable distributions.
If either fiber of $E^s$ or $E^u$ has dimension $k$ with $k\le\lfloor n/2\rfloor$ then $f$ is
called a codimension $k$ Anosov diffeomorphism. An Anosov diffeomorphism is called transitive if there exist a point whose orbit is dense in $M$.
All currently known examples of Anosov diffeomorphisms are conjugate to affine automorphisms of infranilmanifolds. It is a famous classification problem, which dates back to Anosov and Smale, to decide whether there are other manifolds which carry Anosov diffeomorphisms. In particular, Smale <cit.> asked whether manifolds which support Anosov diffeomorphisms must be covered by Euclidean spaces.
It is a more restrictive but still very interesting problem to classify Anosov diffeomorphisms on such manifolds. The goal
of our current work is to evince certain coarse geometric obstructions (presence of negative curvature) to the existence
of Anosov diffeomorphisms on manifolds which are covered by Euclidean spaces.
Let $N$ be a closed infranilmanifold and let $M$ be a smooth aspherical manifold whose fundamental group
$\Gamma=\pi_1(M)$ has the following three properties: (i) $\Gamma$ is Hopfian, (ii) $Out(\Gamma)$ is finite,
and (iii) the intersection of all maximal nilpotent subgroups of $\Gamma$ is trivial. Then $M\times N$ does not
support Anosov diffeomorphisms.
We will give a brief overview of known results in Section <ref>, and discuss background
material in Sections <ref>, <ref>, and <ref>.
The proof of the Main Theorem will be given in
Section <ref>. Finally, in Section <ref>, we discuss further open problems. We
also provide some concrete classes of manifolds satisfying our Main Theorem, for instance:
Let $N$ be any closed infranilmanifold, and
let $M_1, \ldots ,M_k$ be a collection of closed smooth aspherical manifolds of dimension $\geq 3$,
each of which satisfies one of the following properties:
* it has Gromov hyperbolic fundamental group; or
* it is an irreducible higher rank locally symmetric space with no local $\mathbb H^2$-factors or $\mathbb R$-factors.
Then the product $M_1\times \cdots M_k \times N$ does not support any Anosov diffeomorphisms.
Since closed negatively curved Riemannian manifolds are aspherical and have Gromov hyperbolic fundamental
group, the corollary shows that any product of such manifolds of dimension $\ge 3$ with a nilmanifold does not
support Anosov diffeomorphisms.
Acknowledgements. We would like to acknowledge extremely helpful communications with
Tom Farrell, Sheldon Newhouse and Federico Rodriguez Hertz. We also would like to thank Rafael Potrie and the anonymous referee for useful communications.
§ PREVIOUS RESULTS
In this section, we provide the reader with a brief overview of the current state of the classification problem.
A full answer to the classification question was achieved under certain additional assumptions:
* Franks and Newhouse <cit.> proved that codimension one Anosov diffeomorphisms can only exist on manifolds which are homeomorphic to tori.
* Franks and Manning <cit.> proved that Anosov diffeomorphisms on infranilmanifolds are conjugate to affine Anosov diffeomorphisms. In particular, Anosov diffeomorphisms on nilmanifolds are conjugate to hyperbolic automorphisms of nilmanifolds.
* Brin and Manning <cit.> showed that “sufficiently pinched" Anosov diffeomorphisms can only exist on infranilmanifolds.
In light of these results, one is interested in the classification of Anosov automorphisms of nilmanifolds. In general, such a classification seems to be hopeless. However it was achieved in low dimension (see <cit.>, corrections in <cit.> and references therein).
A number of classes of manifolds are known to not support any Anosov diffeomorphisms, or to not support
Anosov diffeomorphisms of a certain type:
* Hirsch <cit.> showed that certain manifolds with polycyclic fundamental group do not admit Anosov diffeomorphisms. In particular, he showed that mapping tori of hyperbolic toral automorphisms do not carry Anosov diffeomorphisms.
* Shiraiwa <cit.> noted that an Anosov diffeomorphism with orientable stable (or unstable) distribution cannot induce the identity map on homology in all dimensions. It follows, for example, that spheres, lens spaces and projective spaces do not admit Anosov diffeomorphisms.
* Ruelle and Sullivan <cit.> showed that if $M$ admits a codimension $k$ transitive Anosov diffeomorphism
$f$ with orientable invariant distributions then $H^k(M; \mathbb R)\neq0$. In fact, they show that there is a non-zero
cohomology class $\alpha\in H^k(M; \mathbb R)$ and positive $\lambda \neq 1$ with the property that
$f^*(\alpha)= \lambda \cdot \alpha$. Existence of such a class provides an obstruction to the existence of Anosov diffeomorphisms on the level of the cohomology ring.
* Gogolev and Rodriguez Hertz <cit.> have recently shown that certain products of spheres cannot
support Anosov diffeomorphisms.
* Finally, the most relevant reference with respect to the current paper is <cit.>, where Yano used bounded cohomology to show that negatively curved manifolds do not support Anosov diffeomorphisms, (cf. Section <ref> for an alternate proof).
§ PERIODIC POINTS OF ANOSOV DIFFEOMORPHISMS
Here we collect some well known facts. We refer the reader to <cit.> and <cit.> for further details on this material.
Recall that if $X$ is a closed manifold and $f\colon X\to X$ is a self-map with finitely many fixed points, then
the Lefschetz formula calculates the sum of indices of the fixed points — the Lefschetz number — as follows
\Lambda(f)\stackrel{\mathrm{def}}{=}\sum_{p\in Fix(f)}ind_f(p)=\sum_{k\ge 0}(-1)^k Tr(f_*|_{H_k(X; \R)}).
Now assume that $X$ is a closed oriented manifold and $f$ is an Anosov diffeomorphism with oriented unstable subbundle $E^u$,
and that $f$ preserves the orientation of the unstable subbundle. Then
$$ind_{f^m} (x)=(-1)^{\dim E^u}
for all $x\in Fix(f^m)$, $m\ge 1$. Hence the number of points fixed by $f^m$ can be calculated as follows:
\begin{equation}
\label{eq_lefschetz}
\left|Fix(f^m)\right|=\left|\Lambda(f^m)\right|
\end{equation}
On the other hand, $|Fix(f^m)|$ can be calculated from the Markov coding. In particular, for a transitive
Anosov diffeomorphism $f$ the following asymptotic formula holds
\begin{equation*}\label{form_asymp_transitive}
\end{equation*}
where $h_{top}(f)$ is the topological entropy of $f$.
For general, not necessarily transitive, Anosov diffeomorphism $f$ the formula takes the form
\begin{equation}\label{form_asymp_general}
\end{equation}
where $r$ is the number of transitive basic sets with entropy equal to $h_{top}(f)$.
Finally, we will need a formula due to Manning <cit.> for the Lefschetz number of an automorphism of a
nilmanifold in terms of its eigenvalues. Let $L\colon N\to N$ be an automorphism of a compact nilmanifold $N$. Then
\begin{equation}\label{eq_manning}
\Lambda(L^m)=\prod_{\lambda\in \text{Spec}(L)} (1-\lambda^m), \;\;\;m\ge 1,
\end{equation}
where the product is taken over all eigenvalues (counted with multiplicity) of the Lie algebra automorphism induced
by $L$.
§ ABSENCE OF ANOSOV DIFFEOMORPHISMS ON NEGATIVELY CURVED MANIFOLDS
The following Lemma is presumably well-known to experts, but does not seem to appear in the literature.[We thank the referee for observing that the argument based on the Lefschetz formula works for proving Lemma <ref>, cf. arXiv:1511.00261v1]
Let $M$ be a closed aspherical manifold with fundamental group $\Gamma:= \pi_1(M)$. If $\text{Out}(\Gamma)$ is
torsion, then $M$
does not support any Anosov diffeomorphism.
Let us assume $M$ supports an Anosov diffeomorphism. Lifting to a finite cover, we may assume that the
invariant distributions are oriented. For $M$ aspherical, there is an identification $H^*(M; \mathbb R) \cong
H^*(\Gamma; \mathbb R)$ between the cohomology of the manifold $M$ and the (group) cohomology of $\Gamma$.
Moreover, this identification is functorial, so one has a commutative diagram
\xymatrix{
H^*(M; \mathbb R) \ar[r]^{f^*} \ar[d]^{\simeq}& H^*(M; \mathbb R) \ar[d]^{\simeq} \\
H^*(\Gamma ; \mathbb R) \ar[r]^{(f_\sharp)^*} & H^*(\Gamma ; \mathbb R)
At the level of group cohomology, any inner automorphism induces the identity map on group cohomology.
Since $\text{Out}(\Gamma)$ is torsion, a finite power of $f_\sharp$ is inner, and hence a finite power $f_\sharp^k$ is the identity on cohomology.
Therefore the cohomology automorphisms $(f^{mk})^*$, $m\ge 1$ are the identity on $H^*(M)$. Hence, using
Poincaré Duality, we conclude that the Lefschetz numbers $\Lambda(f^{mk})$, $m\ge 1$ are uniformly bounded.
But this is in contradiction with the growth of periodic points, by (<ref>) and (<ref>).
There exist aspherical manifolds $M$ which do not admit Anosov diffeomorphisms for the same reason, but have infinite order elements in $\text{Out}(\pi_1M)$. Indeed, the manifolds $M$ obtained from a certain (twisted) double construction as in <cit.>,
<cit.>, have infinite order elements in $\text{Out}(\pi_1M)$. However, it is easy to see that (after passing to a finite power) all infinite order elements act trivially on cohomology. Hence, by the above argument, such $M$ also do not admit Anosov diffeomorphisms.
We note that Lemma <ref> already gives a substantial restriction on the possible closed aspherical
manifolds that can support an Anosov diffeomorphism. For instance, it follows from Mostow rigidity that the outer
automorphism group of a lattice $\Gamma$ in a simply-connected, connected, non-compact, real semisimple Lie group $G$
has finite outer automorphism provided $G$ has no local $SO(2,1)$ factors. This yields the immediate:
Let $M$ be a non-positively curved locally symmetric manifold, whose universal cover $\tilde M$ does not split
off an $\mathbb H^2$-factor or an $\R$-factor. Then $M$ cannot support any Anosov diffeomorphisms.
There are hyperbolic 3-manifolds $M$ which are known to support Anosov flows (see Goodman <cit.>). Thus,
the analogous result does not hold for Anosov flows or for partially hyperbolic diffeomorphisms. Note also, that by taking the product of the time-1 map of such a flow and an Anosov diffeomorphism of a nilmanifold $N$ we can obtain a partially hyperbolic diffeomorphism on the product $M\times N$ with one dimensional center distribution (cf. the Main Theorem).
As another application of this result we recover the result of Yano <cit.>.
Let $M$ be a closed negatively curved Riemannian manifold of dimension $n\geq 3$. Then $M$ does not support any Anosov diffeomorphism.
Since $M$ is negatively curved, $\Gamma:= \pi_1(M)$ is a torsion-free Gromov hyperbolic group. Moreover, the
boundary at infinity $\partial ^\infty \Gamma$ is homeomorphic to the boundary at infinity $\partial ^\infty \tilde M$
of the universal cover of $M$, which we know is a sphere $S^{n-1}$ of dimension $n-1 \geq 2$. If $M$ supports
an Anosov diffeomorphism, then Lemma <ref> tells us $Out(\Gamma)$ must be infinite. Work of Paulin <cit.>
and Bestvina and Feighn <cit.> then implies that $\Gamma$ splits over a cyclic subgroup. By Bowditch
<cit.>, this forces $\partial ^\infty \Gamma \cong S^{n-1}$ to have a local cut-point (i.e., an open
connected subset $U\subset S^{n-1}$ and a point $p\in U$ with the property that $U\setminus \{p\}$ is disconnected),
a contradiction since the boundary at infinity is a sphere of dimension $\geq 2$.
§ LOCALLY MAXIMAL HYPERBOLIC SETS
Let $f\colon M\to M$ be a diffeomorphism. Recall that an $f$-invariant closed set $K$ is called hyperbolic if the tangent
bundle over $K$ admits a $df$-invariant splitting $T_KM=E^s\oplus E^u$, where vectors in $E^s$ ($E^u$) decay
(grow) exponentially fast – in the sense of equation (<ref>). A hyperbolic set $K$ is called locally
maximal if there exists an open neighborhood $V$ of $K$ such that
\begin{equation}
\label{eq_loc_maximal}
\end{equation}
We note that $f$ is Anosov if and only if the entire manifold $M$ is a hyperbolic set. Thus, the notion of hyperbolicity is
intended to reflect some “Anosov-type” behavior on the set $K$.
A hyperbolic set $K$ is locally maximal if and only if it has a local product structure.
We refer to <cit.> for the definition of local product structure and the proof of this proposition.
Let $K$ be a locally maximal hyperbolic set for $f$ and let $\Omega\subset K$ be the set of non-wandering points
for the restriction $f|_K$. Then periodic points of $f|_K$ are dense in $\Omega$.
We will use the above proposition for the proof of the following result.
Let $K$ be an uncountable locally maximal hyperbolic set for $f$. Then the restriction $f|_K$ has infinitely many
periodic points.
Let $\Omega$ be the set of non-wandering points of $f|_K$. (Note that $\Omega$ may not be the same as the set of
non-wandering points of $f$ in $K$.) By Proposition <ref> periodic points of $f|_K$ are dense in
$\Omega$. Hence, if $\Omega$ is infinite, then Proposition <ref> follows.
Now assume that $\Omega$ consists of finitely many periodic orbits. By passing to a finite iterate of $f$ we can also
assume that $f(p)=p$ for all $p\in\Omega$. Denote by $W^{s,K}(p)$ and $W^{u,K}(p)$ the stable and unstable sets of
$p\in\Omega$, respectively; that is,
W^{s, K}(p)=W^s(p)\cap K,\;\; W^{u, K}(p)=W^u(p)\cap K,
where $W^s(p)$ and $W^u(p)$ are the stable and unstable manifolds of $p$, respectively. Then $K$ decomposes
as a disjoint union
\begin{equation*}
\end{equation*}
Indeed, if a point $x\in K$ does not belong to any stable set, then neither does its forward orbit $\{f^n(x);n\ge 1\}$. Let $q$ be an $\omega$-limit point for $\{f^n(x);n\ge 1\}$. But then $q$ is a non-wandering point for $f|_K$, which does not belong to $\Omega$, yielding a contradiction. Similarly,
\begin{equation*}
\end{equation*}
Therefore, each wandering point of $K$ is a heteroclinic (or homoclinic) point. However, the stable and unstable manifolds
of points in $\Omega$ intersect in at most countably many points. Hence $K$ is countable, which yields a contradiction.
§ PROOF OF THE MAIN THEOREM
Our proof proceeds by assuming that there exists an Anosov diffeomorphism $f\colon M\times N\to M\times N$.
We first will go through a series of reductions (R1)-(R4).
§.§ A reduction
Here we show that, by passing to finite iterates of $f$ and finite covers of $M\times N$ we can additionally
assume the following.
R1. $N$ is a nilmanifold;
R2. the stable and unstable subbundles $E^u$ and $E^s$ are oriented and $f$ preserves these orientations;
Let $N'$ be a closed nilmanifold cover of $f$. Then some iterate of $f$ lifts to $M\times N'$ because of the following
more general assertion.
Let $X$ be a closed manifold, let $\tilde X\to X$ be a finite cover of $X$ and let $f\colon X\to X$ be a diffeomorphism.
Then there exists $n\ge 0$ such that $f^n\colon X\to X$ lifts to a diffeomorphism $\widetilde{f^n}\colon\tilde X\to\tilde X$;
the diagram
\xymatrix{
\tilde X\ar[d]\ar^{\widetilde{f^n}}[r] & \tilde X\ar[d] \\
X\ar^{f^n}[r] & X
By the lifting criterion it suffices to show that some iterate of $f$ preserves the conjugacy class of the covering subgroup.
The covering subgroup is a finite index subgroup of the fundamental group of $X$. Because the fundamental group of
$X$ is finitely generated, there are only finitely many subgroups of given finite index (by a theorem of M. Hall).
Hence, by the pigeonhole principle, some iterate of $f$ fixes the conjugacy class of the covering subgroup.
Now, if $E^u$ is not orientable we can pass to the orienting double covering and to $f^2$, which ensures
that the unstable distribution
are orientable and $f$ preserves the orientation. If $E^s$ is still non-orientable then the same procedure can
be applied once again.
In fact, we might need to pass to a double cover (and to a finite iterate) yet once more in order to be able to keep the
product structure of the total space. Indeed, let $\Gamma=\pi_1M$ and $G=\pi_1N$. Let $H\subset \Gamma\times G$ be
the orienting double cover subgroup. Then $H$ has index two and we let
$H_\Gamma=H\cap (\Gamma\times \{id_G\})$ and $H_G=H\cap(\{id_\Gamma\}\times G)$. We identify $H_\Gamma$
and $H_G$ with subgroups of $\Gamma$ and $G$, respectively. It is easy to see that $H_\Gamma$ and $H_G$ are
either index one or index two subgroups. If one of these subgroups has index one, then $H=H_\Gamma\times H_G$
and the double cover has a product structure. Otherwise $H_\Gamma\times H_G$ is an index two subgroup of $H$.
In this case we pass to the 4-fold cover that corresponds to $H_\Gamma\times H_G$, which is clearly a product.
Because $H_\Gamma\times H_G\subset H$ this 4-fold cover is also orienting.
§.§ Further reduction: the induced automorphism
Again, by passing to a finite iterate if necessary, we can assume that $f$ has a fixed point $p$. Consider the
induced automorphism
f_\#\colon \pi_1(M\times N, p)\to\pi_1(M\times N, p)
Let $\Gamma=\pi_1M$ and $G=\pi_1N$. We identify $\pi_1(M\times N, p)$ and $\Gamma\times G$ in the obvious way.
The induced automorphism $f_\#$ has the following form
f_\#(\gamma, g)=(\alpha(\gamma), \rho(\gamma)L(g)),
where $\alpha\colon\Gamma\to\Gamma$ and $L\colon G\to G$ are automorphisms and $\rho\colon \Gamma\to\cZ(G)$
is a homomorphism into the center of $G$.
Recall that, by hypothesis, the group $\Gamma$ has the property that the intersection of all its maximal nilpotent
subgroups is trivial. This implies that, for the group $\Gamma\times G$, the intersection of all the maximal nilpotent
subgroups is precisely the subgroup $\{id_\Gamma\}\times G$. It follows that the subgroup $\{id_\Gamma\}\times G$
is a characteristic subgroup in the product, hence is invariant under $f_{\#}$. We denote by
$L = f_\#|_{\{id_\Gamma\}\times G} \in \text{Aut}(G)$ the induced automorphism of $G$.
Next we define $\alpha$ and $\rho$ via the following formula: for $\gamma \in \Gamma$, we set
$$f_\# \left((\gamma, id_G)\right) = \left(\alpha(\gamma), \rho(\gamma) \right),$$
where $\alpha: \Gamma \rightarrow \Gamma$, $\rho:\Gamma \rightarrow G$ are homomorphisms.
Let us verify the expression for $f_\#$ given in the Lemma.
If $(\gamma, g)\in \Gamma \times G$ is arbitrary, then we have:
\begin{align*}
f_\#((\gamma, g)) &= f_\# \big((\gamma, id_G) (id_\Gamma, g) \big)
= f_\# \big((\gamma, id_G)\big) f_\# \big((id_\Gamma, g) \big) \\
&= \left(\alpha(\gamma), \rho(\gamma) \right)(id_\Gamma, L(g)) = (\alpha(\gamma), \rho(\gamma)L(g)).
\end{align*}
Now $\alpha$ is clearly surjective, and $\Gamma$ is Hopfian, so $\alpha \in \text{Aut}(\Gamma)$. Finally, since
$f_\#$ is a homomorphism, the fact that $(\gamma, id_G)$ and $(id_\Gamma, g)$ always commute tells us that
$\rho(\gamma) L(g) = L(g) \rho(\gamma)$. This forces the homomorphism $\rho$ to have image in the center
$\cZ(G)$, and concludes the proof of the Lemma.
Because $\text{Out}(\Gamma)$ is finite, by passing to a further iterate of $f$, we can (and do) assume the following.
$\alpha\colon\Gamma\to\Gamma$ is an inner automorphism.
§.§ The model
Recall that, by work of Mal$'$cev <cit.>, a closed nilmanifold $N$ can be identified with the quotient space
$\tilde N/G$, where $\tilde N$ is a simply connected nilpotent Lie group and $G\subset \tilde N$ is a cocompact
lattice (which we identify with $\pi_1N$). Also $\cZ(G)=G\cap\cZ(\tilde N)$.
Now let $L\colon G\to G$ be an automorphism, then, again by work of Mal$'$cev, $L$ uniquely extends to an
automorphism of $\tilde N$, which we continue to denote by $L$.
Denote by $\tilde M$ the universal cover of $M$. The fundamental group $\Gamma=\pi_1M$ acts on $\tilde M$
cocompactly by deck transformations, and we identify $\Gamma$ with an orbit of a base-point in $\tilde M$.
Let $\rho\colon\Gamma\to\cZ(G)$ be a homomorphism. Then it extends to a smooth equivariant map
$\rho\colon\tilde M\to\cZ(\tilde N)$,
\forall \gamma\in\Gamma,\;\;\forall x\in \tilde M\;\; \rho(\gamma x)=\rho(\gamma)\rho(x).
Triangulate $\tilde M$ in an equivariant way so that $\Gamma\subset \tilde M$ belongs to the 0-skeleton.
Extend $\rho$ to the 0-skeleton equivariantly in an arbitrary way. Recall that the center $\cZ(\tilde N)$ is a
Euclidean space. Because $\cZ(\tilde N)$ is connected $\rho$ can be equivariantly extended to 1-skeleton.
And because $\cZ(\tilde N)$ is aspherical we can extend $\rho$ equivariantly to all skeleta by induction on
dimension. To finish the proof we approximate the resulting map by a smooth equivariant map. It is easy to
see (by using charts) that this can be done without changing the values on $\Gamma\subset \tilde M$.
Let $L\colon G\to G$ and $\rho \colon \Gamma\to\cZ(G)$ be given by Lemma <ref>.
Then $\rho$ extends to $\rho\colon\tilde M\to\cZ(\tilde N)$ by Lemma <ref>. It is straightforward
to check that the map
\tilde M\times \tilde N\ni (x,y)\mapsto (x, \rho(x)L(y))\in \tilde M\times \tilde N
descends to a (model) map $\bar f\colon M\times N\to M\times N$. We continue abusing notation and denote by
$L$ the induced automorphism $L\colon N\to N$ and by $\rho$ the induced map $\rho\colon M\to \cZ(\tilde N)/\cZ(G)$.
Then we can write
\bar f(x,y)=(x,\rho(x)L(y)).
By construction, the induced homomorphism $\bar f_\#\colon \pi_1(M\times N)\to\pi_1(M\times N)$ is given by
\bar f_\#(\gamma,g)=(\gamma,\rho(\gamma)L(g)).
Hence, by the discussion in the previous subsection, $f_\#$ and $\bar f_\#$ are conjugate. Therefore $f$ and
$\bar f$ are homotopic maps.
Note that because $\rho$ is smooth the model map $\bar f$ is, in fact, a diffeomorphism. However
this is not used in the sequel. We only use smoothness of $\rho$ for Lemma <ref>.
§.§ Hyperbolicity of the model
Consider a gradient vector field on $M$ with finitely many fixed points $q_1, q_2,\ldots ,q_k$, each of which is
hyperbolic. Denote by $i_1, i_2,\ldots ,i_k$ the dimensions of the unstable manifolds at $q_1, q_2,\ldots ,q_k$,
respectively. Recall that the classical Poincaré-Hopf theorem yields the following formula for the Euler characteristic:
\begin{equation}
\label{eq_euler}
\chi(M)=\sum_{j=1}^k ind(q_j)=\sum_{j=1}^k(-1)^{i_j}
\end{equation}
Denote by $\varphi$ the time-one map of the gradient flow and define $\bar{\bar f}\colon M\times N\to M\times N$
as follows
\bar{\bar f}(x,y)=(\varphi(x),\rho(x)L(y))
By a direct calculation one can see that
(\bar{\bar f}\,)^m(x,y)=(\varphi^m(x), \rho_m(x)L^m(y)),
where $\rho_m\colon M\to \cZ(\tilde N)/\cZ(G)$ has an explicit expression in terms of $\rho$ and $L$.
Note that, for each of the maps $(\bar{\bar f}\,)^m$, the only possible fixed points must lie on the fibers
$\{q_i\} \times N$.
We now proceed to modify the maps $(\bar{\bar f}\,)^m$, in order to control the fixed points on each of the fibers.
For each $m\ge 1$ and each $j=1,\ldots ,k$ consider a small perturbation $\widetilde{L^m_j}$ of
$\rho_m(q_j)L^m\colon N\to N$ such that $\widetilde{L^m_j}$ has finitely many fixed points, each of which is
hyperbolic. Now consider a small perturbation of $(\bar{\bar f}\,)^m$ of the form
\widetilde{f^m}(x,y)=(\varphi^m(x),\widetilde{L^m_x}(y))
such that $\widetilde{L^m_{q_j}}=\widetilde{L^m_j}$ for $j=1,\ldots ,k$. Note that we have homotopies
$\widetilde{L^m_j}\simeq \rho_m(q_j)L^m\simeq L^m$ and hence
\begin{equation}
\label{eq_fiber_lefschetz}
\Lambda(\widetilde{L^m_j})=\Lambda(L^m)
\end{equation}
Also note that by construction $\widetilde{f^m}$ has finitely many fixed points each of which has the form $(q_j,y)$,
$j=1,\ldots ,k$. Moreover,
\begin{equation}
\label{eq_index_formula}
ind_{\widetilde{f^m}}(q_j,y)=(-1)^{i_j}ind_{\widetilde{f^m}|_{\{q_j\}\times N}}(y)=(-1)^{i_j}ind_{\widetilde{L^m_j}}(y)
\end{equation}
From the homotopies $f\simeq \bar f \simeq \bar{\bar f}$ and $(\bar{\bar f}\,)^m\simeq \widetilde{f^m}$ we can carry
out the following calculation of the Lefschetz number.
\begin{align*}
\Lambda(f^m)=\Lambda(\widetilde{f^m})&=\sum_{(q_j,y)\in Fix(\widetilde{f^m})}ind_{\widetilde{f^m}}(q_j,y)
\stackrel{(\ref{eq_index_formula})}{=}\sum_{j=1}^k\left[(-1)^{i_j}\sum_{y\in Fix(\widetilde{L^m_j})}ind_{\widetilde{L^m_j}}(y)\right]\\
\stackrel{(\ref{eq_euler})}{=}\chi(M)\Lambda(L^m)
\end{align*}
Note that if the Euler characteristic $\chi(M)$ vanishes (for instance, if $M$ is odd-dimensional) then we immediately obtain a contradiction with (<ref>): the Lefschetz number simultaneously must grow exponentially fast, and must equal zero. This already completes the proof of the Main Theorem in the special case where $\chi(M)=0$. The rest of the paper deals with the case where $\chi(M)\neq 0$.
Now if $\chi(M)\neq 0$, by combining the above formula with (<ref>), (<ref>) and (<ref>) we obtain the following equality for all $m\ge 1$.
\frac{r}{\chi(M)}e^{mh_{top}(f)}+o(e^{mh_{top}(f)})=\prod_{\lambda \in spec(L)}|1-\lambda^m|
Modulo the coefficient $r/\chi(M)$ this equality is the same as equation (2) in <cit.>. Therefore,
the argument of Manning <cit.> yields the following result.
The eigenvalues of $L$ lie off the unit circle, $L\colon N\to N$ is an Anosov automorphism.
§.§ Further reduction: a global change of coordinates
The restriction of the map
\tilde N\ni y\mapsto L(y)y^{-1}\in \tilde N
to the center of $\tilde N$ is a homomorphism. By Lemma <ref>, the restriction of $L$ to $\cZ(\tilde N)$
is hyperbolic, and hence, the above homomorphism is, in fact, an automorphism. Note that, since $\cZ(\tilde N)$
is a Euclidean space, the restriction of $L$ to $\cZ(\tilde N)$ can be represented by a matrix (also denoted $L$). The
map $y\mapsto L(y)y^{-1}$ on $\cZ(\tilde N)$ is then represented by the matrix $L-id$.
Denote by $(L-id)^{-1}$ the inverse of the restriction of this automorphism to $\cZ(G)\subset\cZ(\tilde N)$.
Define a homomorphism $\theta\colon\Gamma\to\cZ(G)$ by
\theta(\gamma)=(L-id)^{-1}\rho(\gamma).
And define an automorphism $h_\#\colon \Gamma\times G\to\Gamma\times G$ by
\begin{equation}
\label{eq_h_sharp}
h_\#(\gamma,g)=(\gamma, \theta(\gamma)g).
\end{equation}
It is straightforward to check that
f^{new}_\#\stackrel{\mathrm{def}}{=}h_\#\circ f_\#\circ h_\#^{-1}
is given by
There exists a self diffeomorphism $h\colon M\times N\to M\times N$ such that the induced automorphism
$h_\#\colon \Gamma\times G\to\Gamma\times G$ is given by (<ref>).
– see Farrell and Jones <cit.>.
Recall that, by Lemma <ref>, the homomorphism $\rho\colon \Gamma\to\cZ(G)$
extends to a smooth equivariant map $\rho\colon\tilde M\to \cZ(G)$. Hence, by letting
\theta(x)=(L-id)^{-1}\rho(x)
we extend $\theta\colon\Gamma\to\cZ(G)$ to a smooth equivariant map, which in turn descends to a map
$\theta\colon M\to\cZ(G)$. Then the formula
defines the posited diffeomorphism.
Now let
f^{new}=h\circ f\circ h^{-1}.
Clearly $f^{new}$ is also an Anosov diffeomorphism. Thus, by replacing $f$ with $f^{new}$ if necessary, we can
assume the following.
The induced automorphism $f_\#\colon \pi_1(M\times N, p)\to\pi_1(M\times N, p)$ has the following form
f_\#(\gamma, g)=(\alpha(\gamma), L(g)).
§.§ A locally maximal hyperbolic set $K$ by applying Franks' theorem
Consider the diagram
\xymatrix{
\Gamma\times G\ar[d]\ar^{f_\#}[r] & \Gamma\times G \ar[d] \\
G\ar^L[r] & G
By the reduction in the previous subsection this diagram commutes. Also, recall that $L\colon N\to N$ is an
Anosov automorphism, and hence, is a $\pi_1$-diffeomorphism in the sense of Franks. Then Franks' thesis <cit.>
yields a semi-conjugacy $h\colon M\times N\to N$, which induces $h_\#\colon (\gamma,g)\mapsto g$ and makes the
\xymatrix{
M\times N\ar_h[d]\ar^{f}[r] & M\times N \ar_h[d] \\
N\ar^L[r] & N
(Recall that $p$ is a fixed point of $f$.) Clearly $K$ is an $f$-invariant closed set.
§.§ $K$ contains infinitely many periodic points
The set $K$ is a locally maximal hyperbolic set. (Hence has local product structure by Proposition
The set $K$ is hyperbolic because $f$ is Anosov. We check that $K$ is locally maximal by using the
definition (<ref>).
The point $h(p)$ is a locally maximal hyperbolic set for $L$,
where $U$ is a sufficiently small neighborhood of $h(p)$. Let $V=h^{-1}(U)$, then
\begin{align*}
&=\bigcap_{m\in\Z} f^k(h^{-1}(U))=\bigcap_{m\in\Z} f^k(V).\\
\end{align*}
establishing the Lemma.
We will now exploit the fact that the dimension of $M$ is positive.
The set $K$ is uncountable.
For each $x\in M$ consider the restriction $h|_{\{x\}\times N}\colon N\to N$. This restriction induces an
isomorphism of fundamental groups. The isomorphism of fundamental groups induces an isomorphism
on top-degree (group) cohomology. Since $N$ is aspherical, this implies the map induces an isomorphism between
the top-degree cohomology of the spaces. Hence the map has degree one, so must be onto. We conclude that
$h^{-1}(h(p))\cap (\{x\}\times N)\neq\varnothing$, for each $x\in M$. Since $\dim(M)>0$, this shows $K$ is uncountable.
We conclude that Proposition <ref> applies to $K$ and yields the following lemma.
There are infinitely many periodic points in the invariant set $K$.
§.§ Lifting the dynamics to $M\times\tilde N$
Let $\bar\pi\colon\tilde N\to N$ be the universal covering. By taking the product with the identity map $id_M$ we
obtain the covering
\pi\colon M\times\tilde N\to M\times N.
Choose a base point $\tilde p\in M\times\tilde N$ such that $\pi(\tilde p)=p$. Then, using reduction R4, we see that
$f\colon M\times N\to M\times N$ uniquely lifts to $\tilde f\colon M\times \tilde N\to M\times\tilde N$ with
$\tilde f(\tilde p)=\tilde p$. We denote by $d$ the distance induced by the (lifted) Riemannian metric on $M\times\tilde N$.
The diffeomorphism $\tilde f$ is homotopic to the model diffeomorphism $id_M\times\tilde L\colon (x,y)\mapsto (x, \tilde Ly)$ via a homotopy $H$. Moreover,
$d(H_t, id_M\times\tilde L)$ is uniformly bounded for all $t\in[0,1]$.
By reduction R4 the diffeomorphism $f\colon M\times N\to M\times N$ and the model map $(x,y)\mapsto (x, Ly)$
induce the same outer automorphism of $\pi_1(M\times N)$ and hence are homotopic <cit.>.
This homotopy lifts to the posited homotopy $H$ on $M\times \tilde N$.
The distance $d(H_t, id_M\times\tilde L)$ is bounded because $H$ is a lift of a homotopy on a compact manifold $M\times N$.
We also lift the semi-conjugacy $h\colon M\times N\to N$ constructed in Subsection <ref> to a
semi-conjugacy $\tilde h\colon M\times \tilde N\to\tilde N$ and the automorphism $L\colon N\to N$ to an
automorphism $\tilde L\colon\tilde N\to\tilde N$ so that $\tilde L(\tilde h(\tilde p))=\tilde h(\tilde p)$. Then we
have the following commutative diagram.
\xymatrix{
*[r]{\;(M\times\tilde N,\tilde p)\;} \ar@<4ex>_\pi[d]\ar@<0.1ex>[rr]^-{\tilde h} \ar@(dl,ul)[]^{\tilde f}
& & *[l]{\;(\tilde N, \tilde h(\tilde p))\,} \ar@(dr,ur)[]_{\tilde L} \ar@<-5.3ex>^{\bar\pi}[d]\\
*[r]{\;(M\times N, p)\;} \ar@<0.1ex>[rr]^-{ h} \ar@(dl,ul)[]^{ f} && *[l]{\;( N, h( p))\,} \ar@(dr,ur)[]_{ L}
We now define the set $\tilde K:=\tilde h^{-1}(\tilde h(\tilde p))$. Clearly $\tilde K$ is $\tilde f$-invariant.
The restriction $\pi|_{\tilde K}\colon\tilde K\to K$ is a homeomorphism.
Indeed, to see that $\pi|_{\tilde K}$ is a bijection notice that
\pi^{-1}(K)=\pi^{-1}\big(h^{-1}(h(p))\big)=\tilde h^{-1}\big(\bar\pi^{-1}(h(p))\big)
=\tilde h^{-1}\Big(\bigcup_{g\in G} g.\tilde h(\tilde p)\Big)\\
=\bigcup_{g\in G} g.\tilde K,
where the union over the group of deck transformations is disjoint. Hence each $g.\tilde K$ projects homeomorphically
onto $K$.
This lemma together with the preceding commutative diagram provides a certain understanding of the dynamics of
$\tilde f$ which we summarize below.
Let $\tilde f\colon M\times\tilde N\to M\times\tilde N$ and $\tilde K$ be as above. We equip $M\times\tilde N$
with a Riemannian metric lifted from $M\times N$. Then
* $\tilde f$ is an Anosov diffeomorphism with infinitely many periodic points each of which belongs to the
$\tilde f$-invariant compact set $\tilde K$;
* $d_{C^0}(\tilde f, id_M\times \tilde L)<const$;
* any point $x\notin\tilde K$ escapes to infinity either in positive or in negative time.
§.§ Fiberwise compactification
Our goal now is to change the diffeomorphism $\tilde f$ so that the set
of periodic points stays the same, and the new map $\hat f$ (which is not necessarily a diffeomorphism) extends
to the fiberwise compactification $M\times \S^k$, where $k=\dim N$.
Let $\tilde f\colon M\times\tilde N\to M\times\tilde N$ and $\tilde K$ be as above. Then there exists a map
$\hat f\colon M\times\tilde N\to M\times\tilde N$ which has the following properties:
* $\hat f$ coincides with $\tilde f$ on a large compact set $B_1$ which contains $\tilde K$;
* $\hat f$ has infinitely many periodic points each of which belongs to $\tilde K$;
* Outside of a large compact set $B_2\supset B_1$
\hat f(x,y)=(x, \tilde L(y)).
We first choose a pair of large nested compact sets $D_i \subset \tilde N$, $i=1,2$, chosen so that each
$D_i$ is homeomorphic to $\DD^k$, and such that the intermediate region $D_2 \setminus Int(D_1)$
is homeomorphic to $\S^{k-1} \times [0,1]$. We fix an identification $D_2 \setminus Int(D_1)\rightarrow \S^{k-1} \times [0,1]$, giving us a
coordinate system on the region $D_2\setminus Int(D_1)$.
The sets $B_i$ will be given by $M\times D_i$.
Recall that $H_t: M\times \tilde N \rightarrow M\times \tilde N$
is the proper homotopy obtained by lifting the homotopy $f\simeq Id\times L$ to the cover $M\times \tilde N$
(parametrized so that $H_0 = Id_M\times \tilde L$ and $H_1 = \tilde f$).
Now we define the map $\hat f$ via the following formula:
\hat f(x,y)=
\begin{cases} \tilde f(x,y) & \mbox{if } (x,y)\in B_1 \\
H_{1-t}(x, y) & \mbox{if } (x,y)\in B_2 \setminus Int(B_1), y\in \S^{k-1}\times \{t\} \\
(x, \tilde L(y)) & \mbox{if } (x,y)\notin B_2
\end{cases}
Note that on the slice $\S^{k-1}\times \{0\} = \partial B_1$, we have that $H_1 = \tilde f$, while on the slice
$\S^{k-1}\times \{1\} = \partial B_2$, we have $H_0 = Id_M\times \tilde L$. So the above map is indeed continuous.
Properties P1 and P3 are clear from the construction. To verify P2 note that
\begin{equation}
\label{eq_hat_f}
d(\hat f, id_M\times\tilde L)<R
\end{equation}
where the constant $R$ is independent of the choices which we made while constructing $\hat f$, that is, $R$ is independent of the choice of $B_1$, $B_2$ and the
coordinate system on $B_2\backslash B_1$. Indeed, this is clear from the estimate on the homotopy $H$ from Lemma <ref>.
Therefore, estimate (<ref>) guarantees that for sufficiently large $B_1$ any point $(x,y)\notin B_1$ will escape to infinity in positive or negative time under
$\hat f$, because it escapes to infinity under $id_M\times\tilde L$. It remains to notice that any point $(x,y)\in B_1\backslash\tilde K$ escapes $B_1$ in positive or
negative time by the last statement in Proposition <ref> (because $\hat f|_{B_1}=\tilde f|_{B_1}$).
Consider the space $M\times \S^k$ (where $k=\dim (N)$), obtained by taking the one point compactification of each
$\tilde N$-fiber in $M\times \tilde N$.
The collection of compactifying points for each of the fibers give a copy $M_\infty$ of $M$, which we call
the fiber at infinity.
Since the map $\hat f$ restricted to each fiber $\{p\}\times \tilde N$ coincides with the map $\tilde L$ (outside of a compact set), we
can extend the map on each fiber to the one-point compactification. This gives us a new map,
denoted $\hat f$, from $M\times \S^k$ to $M\times \S^k$. Note that the new map $\hat f$ restricts to
the identity on $M_\infty$. We
further homotope $\hat f$ in a small neighborhood of $M_\infty$ (keeping the $\tilde N$ coordinates constant)
so that the restriction $\hat f|_{M_\infty}$ becomes a time-one map of a gradient flow. Then
the following Proposition is immediate from P2, the discussion in Section <ref> and the Poincaré-Hopf formula (<ref>).
Let $\hat f\colon M\times\S^k\to M\times\S^k$ be the map constructed above. Then the Lefschetz numbers can
be calculated as follows
\Lambda(\hat f^m)=\sum_{p\in Fix(\hat f^m)}ind_{\hat f^m}(p)=(-1)^{s}\chi(M)+(-1)^{\dim E^u}\left|Fix(\hat f^m|_{\tilde K})\right|,
where $s$ is the dimension of the stable subspace of $L$. In particular, as $m$ tends to infinity, $\Lambda(\hat f^m)$ is unbounded. (cf. Section <ref>.)
For each $x\notin\tilde K$ $\tilde f^m(x)\to\infty$ as $m\to\infty$.
this lemma is for counting periodic point on the dynamics on the double
Recall that $\tilde L$ is hyperbolic. Hence for each $y\neq \tilde h(\tilde p)$ we have $\tilde L^m(y)\to\infty$ as
$m\to\infty$. But if $x\notin\tilde K$ then $\tilde h(x)\neq \tilde h(\tilde p)$. Hence, ?because $\tilde h$ is
quasi-isometry?, $\tilde f^m(x)\to\infty$ as $m\to\infty$.
§.§ Fiberwise compactification
Next, we perturb the map $\tilde f$ in order to control the behavior of $\tilde f$ on large scales.
Let $g: M \times \tilde N \rightarrow M \times \tilde N$ be an arbitrary map.
Then $g$ can be homotoped to a map $\bar g$ which sends $\tilde N$-fibers to $\tilde N$-fibers, i.e. $\bar g$ is
of the form
$$\bar g ( x, y) = \left( \phi(x), \psi(x,y) \right)$$
where $\phi: M \rightarrow M$ and $\psi: M \times \tilde N \rightarrow N$ are continuous maps.
We have a basepoint $p=(x_0, y_0) \in M\times \tilde N$, and image basepoint $q:=g(p)=(x_1, y_1) \in M\times \tilde N$.
Given a $y\in \tilde N$, we denote by $M_y:= M \times \{y\}$, and by $g_y: M \rightarrow M$ the composite
\xymatrix{
M \ar[rr]^{Id \times \{y\}} & & M_y \ar[r]^(.4){g}& M\times \tilde N \ar[r]^(.6){p} & M
where $p$ denotes the projection onto $M$. We need to modify $g$ so that all of the maps $g_y$ coincide. Note that the
maps $g_y$ vary continuously with respect to the parameter $y\in \tilde N$, and hence all induce the same map on the level
of $\pi_1$. Choosing a path in $\tilde N$ that connects $y$ to the basepoint $y_0$, we obtain a homotopy
$H_t\colon M\times I \rightarrow M$ of maps from $H_1\equiv g_y$ to $H_0 \equiv g_{y_0}$. Fixing lifts
$\tilde x_0, \tilde x_1 \in \tilde M$ of the basepoints $x_0, x_1 \in M$, and noting that $g_{y_0}(x_0) = x_1$, we can lift the
homotopy $H$ to a homotopy $\tilde H: \tilde M \times I \rightarrow \tilde M$ with the property that $\tilde H_0(\tilde x_0)
= \tilde x_1$. Note that this homotopy $\tilde H_t$ depends on the choice of path connecting $y$ to $y_0$, but the time
one map $\tilde H_1$ which lifts $g_y$ is independent of this choice of path (since the path is chosen in the simply connected
parameter space $\tilde N$). We thus obtain a canonical family of lifts $\tilde g_y : \tilde M \rightarrow \tilde M$
of the maps $g_y$. Moreover, these lifts vary continuously with respect tothe parameter $y\in \tilde N$, and each of these
is equivariant with respect to the deck transformation action of $\pi_1(M)$ on the $\tilde M$. Since $\tilde M$ supports a
negatively curved Riemannian metric, we can use a geodesic homotopy to deform each of the maps $\tilde g_y$ to $\tilde g_{y_0}$.
These homotopies are continuous with respect to the parameter $y\in \tilde M$, and are also equivariant with respect to the
$\pi_1(M)$ action, hence descend to a (continuously varying) family of homotopies from the individual $g_y$ to $g_{y_0}$.
Applying each of these homotopies to the first coordinate of the map $g$ gives us the desired map $\bar g$.
We now use the homotopy from Lemma <ref> to modify the map $\tilde f$. Specifically, we have the set
$K\subset M\times \tilde N$ in which we have good control of the dynamics. We look at the projection of $K$ onto the $\tilde N$
factor, take a $K'\subset \tilde N$ whose interior contains this projection, and a smooth function $\rho: \tilde N \rightarrow [0,1]$
which is identically zero on $K$ and identically one outside of $K'$. We now use the function $\rho$ as the time parameter in
the homotopy, in order to form the map $\hat f$ which has the following properties:
* $\hat f$ coincides with the original map $\tilde f$ on the set $K$,
* outside of $M\times K'$, $\hat f$ sends $\tilde N$-fibers to $\tilde N$-fibers, and
* $\hat f$ is homotopic to $\tilde f$, and hence homotopic to the identity map on $M\times \tilde N$.
Observe that the homotopy we use only changes the $M$-coordinate of the map $\tilde f$, hence we always have
$d(\hat f (p) , \tilde f(p)) < \text{Diam}(M)$ (recall that $M$ is compact). It follows that the maps $\tilde f$ and $\hat f$ are
at bounded distance apart.
§.§ Lefschetz numbers on $M\times \S^k$
We have now built a self-map $\hat f: M\times \S^k \rightarrow M\times \S^k$. As explained in the last section,
from the construction of $\hat f$, we have
a good understanding of the periodic points. On the other hand, we can easily compute the Lefschetz numbers for the map
$\hat f$ and its powers.
The map $\hat f$ is homotopic to the identity map on $M\times \S^k$.
Reversing the small homotopy near $M_\infty$, we can assume that $\hat f$ coincides with $Id_M\times \tilde L$
on the complement of $B_2$ (and is the identity on $M_\infty$). Next we define a homotopy $\bar H$ on the
space $M\times \tilde N \subset M\times \S^k$ as follows:
\bar H_s(x,y)=
\begin{cases} H_s(x,y), \;\;\mbox{if}\;\;\; (x,y)\in B_1 \\
H_{s(1-t)}(x, y), \;\;\mbox{if}\;\;\; (x,y)\in B_2 \setminus Int(B_1), y\in \S^{k-1}\times \{t\} \\
H_0(x, y), \;\;\mbox{if}\;\;\; (x,y)\notin B_2
\end{cases}
Let us observe that $\bar H_0 = H_0 = Id_M\times \tilde L$, while $\bar H_1 = \hat f$. Moreover this
is a proper homotopy, whose support is entirely contained in the compact set $B_2$. So it extends to the fiberwise
one-point compactification, giving us a homotopy from the self-map $\hat f$ to the map
consisting of the fiberwise one-point compactification of $Id_M\times \tilde L$. Finally, we note that $\tilde L$ is properly
homotopic to $Id_{\tilde N}$. Composing with these (fiber-wise) homotopies, we obtain the desired homotopy, completing
the proof of the Lemma.
We choose a CW-structure on $M$, and the standard CW-structure on $S^k$ (consisting of a single $0$-cell at the
point $\infty$,
and a single $k$-cell). From these, we obtain the product CW-structure on $M\times S^k$, where each cell is the
product of a cell $e_1$ in $M$ and a cell $e_2$ in $S^k$. We want to show that the map $\phi$ can be homotoped to
the identity map. We will denote by $M_\infty:= M\times \{\infty\}$.
First, by a small homotopy, we may arrange for the map $\psi$ to be transverse to the point $\infty$, and to
still satisfy $\psi^{-1}(\infty)=\{\infty\}$. Next, we can homotope $\psi$ near $\infty$ to coincide with the linear map
$d\psi$. Performing this small homotopy on each $U_1$-fiber of the neighborhood $M\times U_1$, we see that
locally near $M_\infty$, the map $\phi$ is given on the neighborhood $M\times U_1$ by $(x, y)\mapsto (x, Ay)$
where $A\in GL_k^+(\mathbb R)$ is a fixed orientation-preserving matrix. Since $GL_k^+(\mathbb R)$ is
connected, up to performing a further fiber-wise homotopy, we may assume that $\phi$ on the neighborhood
$M\times \{U_1\}$ is given by the identity map.
By hypothesis, if $e_2$ is the $0$-cell $\infty \in S^k$, then the map $\phi$ is the identity on all the cells of the form
$e_1\times e_2 \subset M_\infty$. So it is sufficient to consider what $\phi$ does on the cells $e_1\times e_2$,
where $e_1$ is an arbitrary cell in $M$, and $e_2$ is the $k$-dimensional cell in $S^k$. We will construct the
homotopy by induction on the dimension of $e_1$.
If $e_1$ is a $0$-cell in $M$, then $e_1 = \{q\}$ for some $q\in M$. Let us analyze the $\phi$-image
of the fiber $\{q\}\times S^k$. By hypothesis, $\phi$ fixes a small neigborhood of the point $\left(q,\infty \right)$,
and $\phi$ maps the complement of the point $(q,\infty)$ into the subset
$\left(M\times S^k \right) \setminus M_\infty = M\times \mathbb R^k$.
We now have two maps from the $e_1 \times e_2$ cell
$\cong \mathbb D^{k}$ into $M\times \mathbb R^k$: the restriction of $\phi$,
and the identity map. These two maps coincide in a small neighborhood of the boundary, so concatenate to
give a map from a sphere $S^{k}$ into $M\times \mathbb R^k$. But $\pi_k(M\times \mathbb R^k) = \pi_k(M) = 0$
since $M$ is aspherical and $k\geq 2$. So this
concatenation bounds a disk, which provides us with a homotopy from the restriction of $\phi$ to the identity. This shows
that the $\phi$-image of the fiber $\{q\} \times S^k$ is homotopic to the original fiber $\{q\}\times S^k$.
Now we are ready for the inductive step. If $M^{(r)}$ denotes the $r$-skeleton of $M$, we can now assume that the
restriction of $\phi$ to the product $M^{(r)}\times S^k$ has already been homotoped to the identity. Take an $(r+1)$-cell
$e_1$ in $M$, and look at $\phi$ restricted to the $(r+k+1)$-dimensional cell $e:=e_1\times e_2$ in $M\times S^k$
(recall $e_2$ is the single $k$-dimensional cell in $S^k$). We want to construct a
homotopy from $\phi|_e$ to the identity map $Id_{e}$, and by induction, we already have a homotopy from
$\phi|_{\partial e}$ to the identity on $\partial e$. This homotopy can be viewed as a map from
$\partial e \times I \cong S^{r+k}\times \mathbb [0,1]$ into $M\times \mathbb R^k$. The map $\phi|_e$ and $Id|_e$ are
a pair of maps from $\mathbb D^{r+k+1}$ into $M\times \mathbb R^k$, which coincide, along the boundary
$\partial D^{r+k+1}=S^{r+k}$ with the homotopy at times $\{0\}$ and $\{1\}$. Thus these three maps glue together to give
a map from $S^{r+k+1}$ into the space $M\times \mathbb R^k$. Since $r+ k+ 1 \geq 2$, and $M$ is aspherical,
we have that $\pi_{r+k+1}(M\times \mathbb R^k)$ vanishes. This means the map from $S^{r+k+1}$ extends to a map
from $\mathbb D^{r+k+2} \cong \mathbb D^{r+k+1}\times [0,1]$. This extension can be viewed as a homotopy from
$\phi|_e$ to the identity map. This completes the inductive step, and the proof of the lemma.
As an immediate consequence, we obtain the following.
For all $m\geq 0$, we have that $\Lambda (\hat f ^m) = \Lambda(\text{Id}|_{M\times \S^k}) = \chi(M)\chi(\S^k)$ is a constant,
independent of $m$.
But this contradicts the computation of the Lefschetz number obtained in Proposition <ref>,
completing the proof of the Main Theorem.
§ APPLICATIONS AND CONCLUDING REMARKS
§.§ Concrete examples
We now prove Corollary <ref>. We first consider the case of a single factor.
Let $M^n$ ($n\geq 3$) be either (a) an aspherical manifold with Gromov hyperbolic
fundamental group, or (b) a higher rank locally symmetric space with no local $\mathbb R$ or $\mathbb H^2$-factors.
We need to verify that the fundamental group $\Gamma$ of $M$ satisfies conditions (i)-(iii) in the statement of our
Main Theorem.
Property (i) states that $\Gamma$ is Hopfian. For torsion-free Gromov hyperbolic groups, this is a deep result of
Sela <cit.>. For higher rank locally symmetric spaces, the fundamental group is linear (i.e. has a
faithful finite-dimensional complex representation). But Mal'cev <cit.>
showed that finitely generated linear groups are Hopfian.
Property (ii) states that $\text{Out}(\Gamma)$ is finite. For lattices in higher-rank semisimple Lie groups, this is a
direct consequence of Mostow's rigidity theorem: $\text{Out}(\Gamma)$ coincides with the compact Lie group
$\text{Isom}(M)$. But the latter must be $0$-dimensional (hence finite), for otherwise the action of a $1$-parameter subgroup
would give a local $\mathbb R$-factor in $M$. In the case where $\Gamma$ is Gromov hyperbolic, the work of
Paulin <cit.> and Bestvina and Feighn <cit.> reduce the problem to deciding whether the boundary at infinity
$\partial ^\infty \Gamma$ has any local cutpoints (see the discussion in the proof of Corollary <ref>).
But Bestvina has shown that $\partial ^\infty \Gamma$ is a homology manifold of dimension $n-1\geq 2$
Thus we can find arbitrarily small open neighborhoods $U$ around any given point $p\in \partial ^\infty \Gamma$,
with the property that $H_1(U, U\setminus \{p\})\cong H_0(U, U \setminus \{p\}) \cong 0$. The homology long exact
sequence for the pair $(U, U\setminus \{p\})$ then shows that $U$ and $U\setminus \{p\}$ have the same number of
connected components, and hence $p$ cannot be a local cutpoint.
Finally, property (iii) concerns the intersection of maximal nilpotent subgroups of $\Gamma$. When $\Gamma$ is
torsion-free Gromov hyperbolic, these are precisely the (maximal) cyclic subgroups of $\Gamma$.
Since $M$ has dimension
$>1$, $\Gamma$ is not virtually cyclic, so must contain a free subgroup. In particular, we have a pair of cyclic
subgroups that intersect trivially, verifying (iii). In the case where $M$ is an irreducible higher rank locally
symmetric space, the intersection of all the maximal nilpotent subgroups forms
a normal subgroup of $\Gamma=\pi_1(M)$. From Margulis' normal subgroup theorem
(see <cit.>, or <cit.>)
this intersection is either finite, or
finite index. Since the intersection is nilpotent, it cannot be finite index in $\Gamma$, hence it must be finite. Since
$\Gamma$ is torsion-free, it follows that it is trivial.
Now we pass to the case where $M= M_1\times \cdots \times M_k$ is a product of manifolds of the above type.
We again need to verify that the fundamental group $\Gamma = \Gamma_1 \times \cdots \times \Gamma_k$ of
such a manifold verifies the algebraic conditions (i)-(iii) from our Main Theorem.
For (iii) we observe that a product of nilpotent groups is nilpotent, and the homomorphic image of
a nilpotent group is nilpotent. It follows that the maximal nilpotent subgroups
in $\Gamma$ are precisely the products of the maximal nilpotent subgroups in the individual $\Gamma_i$. From
this, one can see that the intersection of the maximal nilpotent subgroups in $\Gamma$ coincides with the product
of the intersection of the maximal nilpotents in the individual $\Gamma_i$. But we saw earlier that the later are all
trivial, which shows property (iii).
Properties (i) and (ii) are harder to establish. As a first step, let us analyze
surjective homomorphisms $\phi: \Gamma \rightarrow \Gamma$. By an abuse of notation,
we will also denote by $\Gamma_i$ the subgroup of $\Gamma$ consisting of all tuples $(g_1, \ldots, g_k)$
with $g_j=id$ when $j\neq i$. Restricting the homomorphism $\phi$ to the component $\Gamma_i$, we can
express $\phi|_{\Gamma_i} = (\phi_{i1}, \ldots ,\phi_{ik})$, where each $\phi_{ij}: \Gamma_i \rightarrow \Gamma_j$
is a homomorphism (not necessarily surjective). The homomorphism $\phi$ is then completely determined by the
collection of homomorphisms $\phi_{ij}$ (where $1\leq i, j \leq k$), since we have the formula:
$$\phi\big( (g_1, \ldots , g_k)\big) = \Big( \prod_{i=1}^k\phi_{i1}(g_i), \prod_{i=1}^k\phi_{i2}(g_i), \cdots ,
\prod_{i=1}^k\phi_{ik}(g_i)\Big).$$
Note that, in the expression above, the products are unambiguous: since the subgroups $\Gamma_i$ inside
$\Gamma$ all commute, their $\phi$-images also commute, so the order in which the product is taken does not
matter. We now define the image subgroups $\Lambda_{ij}:= \phi_{ij}(\Gamma_i) \leq \Gamma_j$.
Since $\Gamma_i$ is normal inside
$\Gamma$, and the morphism $\phi$ is surjective, each of the image subgroups $\Lambda_{ij}$ are normal inside
the corresponding $\Gamma_j$.
We will now try to understand the individual $\Lambda_{st}$. If $\Gamma_t$ is an irreducible lattice in a higher
rank semisimple Lie group, then from Margulis' normal subgroup theorem, it follows that each $\Lambda_{rt}$ is either
finite (whence trivial) or finite index in $\Gamma_t$. We claim exactly one of these is finite index (and must hence have
index one). Indeed, if we have two distinct $\Lambda_{rt}$ and $\Lambda_{st}$
($r\neq s$) both of which are finite index, then their intersection $\Lambda_{rt}\cap \Lambda_{st}$ has finite
index in $\Gamma_t$. Since $\Gamma_r, \Gamma_s$ commute in $\Gamma$, their images commute inside
$\Gamma_t$, and thus this intersection must be an abelian subgroup. So one obtains that $\Gamma_t$ must be
virtually abelian, a contradiction.
Next let us assume that $\Gamma_t$ is torsion-free Gromov hyperbolic. Again, since the image subgroups $\Lambda_{rt}$
and $\Lambda_{st}$ commute inside $\Gamma_t$, we have that $\Lambda_{rt}$ is actually a subgroup of the centralizer
$\text{Cent}_{\Gamma_t}(\Lambda_{st})$. We will need the following:
Let $H$ be a subgroup in a torsion-free Gromov hyperbolic group $G$,
* if the subgroup $H$ is trivial, the centralizer is the whole group $G$.
* if the subgroup $H$ is (infinite) cyclic, then every element in the centralizer generates a cyclic subgroup
which intersects $H$ non-trivially.
* if $H$ is not cyclic, then the centralizer is trivial.
Statement (1) is obvious. If $H$ contains an element $x$ of infinite order, then any element $y\in \text{Cent}_G(H)$ in the
centralizer must
commute with $x$. Since $G$ cannot contain any $\mathbb Z^2$-subgroups, this implies that the torsion-free abelian subgroup
$\langle x,y \rangle$ generated by the two elements is isomorphic to $\mathbb Z$. If $H$ is infinite cyclic, this immediately
gives (2).
Next, if $H$ is not infinite cyclic, then it contains two elements $a,b$ which do not jointly
lie inside a $\mathbb Z$ subgroup (recall that a finitely generated locally cyclic group is cyclic). From the Tits' alternative,
there exists a $k>0$ with the property that $\langle a^k, b^k\rangle$ is a free group (see <cit.>).
Now if $z$ is an arbitrary non-trivial element in the centralizer of $H$, then from the discussion above,
$\langle z\rangle \cap \langle a^k\rangle$ and $\langle z\rangle \cap \langle b^k \rangle$ both have finite index
inside $\langle z \rangle$, so $\langle a^k \rangle \cap \langle b^k\rangle \cong \mathbb Z$ inside the free
group $\langle a^k, b^k\rangle$, a contradiction.
This establishes statement (3).
We now finish our analysis of the possible $\Lambda_{rt}$ when $\Gamma_t$ is Gromov hyperbolic. We first
note that none of the $\Lambda_{rt}$ can be infinite cyclic. Indeed, if $\Lambda_{rt}$ is infinite cyclic, then from
the discussion above, we obtain that all the $\Lambda_{st}$ ($s\neq r$) lie inside the centralizer of
$\Lambda_{rt}$. Hence, the subgroup spanned by them also lies inside the centralizer. But by
surjectivity of the map $\phi$, this means that the entire (finitely generated) group $\Gamma_t$ coincides
with the centralizer. Property (2) in Lemma <ref> would then imply that $\Gamma_t$ is a torsion-free $2$-ended group (i.e.
$\partial ^\infty \Gamma_t$ consists of just two points), so must be isomorphic
to $\mathbb Z$. This is impossible, as $\Gamma_t$
is the fundamental group of an aspherical manifold of dimension $>1$.
Having showed that none of the $\Lambda_{rt}$ are infinite cyclic, we now note that at least one of
the $\Lambda_{rt}$ must be non-cyclic. For otherwise all the $\Lambda_{rt}$would be finite cyclic, and hence trivial,
contradicting the surjectivity of $\phi$. So assume $\Lambda_{rt}$ is not cyclic. Then from statement (3) in Lemma <ref>,
one sees that all the remaining $\Lambda_{st}$ ($s\neq t$) must be trivial. Finally, surjectivity of $\phi$ then forces that
$\Lambda_{rt} = \Gamma_t$. We summarize the discussion so far in the following:
Fact: For each $j$, there is a unique $1\leq \tau(j) \leq k$ with the property that
$$\Lambda_{ij} = \begin{cases}
\Gamma_j & i=\tau(j) \\
1 & i\neq \tau(j) \\
\end{cases}$$
Since we only have finitely many factors in $\Gamma$, surjectivity of $\phi$ implies that $\tau$ is a bijection,
i.e. defines an element in the symmetric group $\text{Sym}(k)$. We denote by $\sigma := \tau ^{-1}$ the inverse
Let us first show the Hopfian property (i). From the discussion above, to any surjective homomorphism $\phi: \Gamma
\rightarrow \Gamma$, we have associated a permutation $\sigma\in \text{Sym}(k)$, which encodes how the surjective
homomorphism permutes the direct factors in $\Gamma$. It follows that a high enough power $\phi^r:\Gamma\rightarrow
\Gamma$ is a surjective homomorphism which preserves each factor, and hence induces a surjection from each
factor to itself. As we know each factor is Hopfian, it follows that $\phi^r$ is injective, and hence $\phi$ must also be
injective. Thus $\phi$ is an isomorphism, and we conclude that $\Gamma$ is Hopfian, verifying (i).
For (ii), we note that the above discussion gives us a well-defined homomorphism $\text{Aut}(\Gamma)\rightarrow
\text{Sym}(k)$. Since an inner automorphism preserves each factor, this descends to a homomorphism
$\text{Out}(\Gamma) \rightarrow \text{Sym}(k)$. The image of this homomorphism lies in a finite group, while the
kernel can easily be seen to be $\text{Out}(\Gamma_1) \times \cdots \times \text{Out}(\Gamma_k)$. Since each
of the $\text{Out}(\Gamma_i)$ are finite, we obtain that $\text{Out}(\Gamma)$ is also finite, establishing (ii). This
concludes the proof of Corollary <ref>.
The obstruction mentioned in Lemma <ref> does not apply to the examples in our
Corollary <ref>. Indeed, the outer automorphism group of $\pi_1(M\times N)$ contains the outer
automorphism group of $\pi_1(N)$. If $N$ is an infranilmanifold supporting an Anosov diffeomorphism, then the
later has infinite order, and hence so does $\text{Out}(\pi_1(M\times N))$.
§.§ Open problems
There are a number of interesting questions we encountered on this project.
Concerning Smale's question, it seems like the following special case is still open:
Let $M$ be a compact non-positively curved locally symmetric manifold and let $f\colon M\to M$ be an Anosov diffeomorphism. Then a finite cover of $M$ is a torus.
The crucial difficulty in resolving this problem seems to be the following:
Show that the product of two surfaces at least one of which has genus $\ge 2$ does not support an Anosov diffeomorphism.
Finally, regarding Lemma <ref>, it seems quite hard to produce high-dimensional aspherical manifolds
whose fundamental group has infinite outer automorphism group. While several classes of aspherical manifolds
are known to have fundamental groups with infinite outer automorphism group (see e.g. <cit.>, <cit.>, <cit.>,
<cit.>), they all appear to contain $\mathbb Z^2$-subgroups.
In particular, we do not know the answer to the following:
Let $M$ be an aspherical manifold of dimension $\geq 3$, and assume $\Gamma:= \pi_1(M)$ contains no
$\mathbb Z^2$-subgroup. Does it follow that $\text{Out}(\Gamma)$ is finite?
This question has a positive answer for 3-dimensional manifolds.
Indeed, one notes that such a manifold $M$ is aspherical and atoroidal. Hence $M$ is prime, and has trivial JSJ
decomposition. From geometrization, it follows that $M$ must be modeled on one of the eight $3$-dimensional
geometries: $\mathbb H^3$, $\mathbb E^3$, $\mathbb S^3$, $\mathbb S^2\times \mathbb E^1$,
$\mathbb H^2\times \mathbb E^1$, $\widetilde{PSL}(2,\mathbb R)$, $NIL$, or $SOL$. Since $M$ is aspherical,
we can rule out the two model geometries $\mathbb S^3$ and $\mathbb S^2\times \mathbb E^1$.
And since $\pi_1(M)$ has no $\mathbb Z^2$-subgroup, we can rule out the five model geometries $\mathbb E^3$,
$\mathbb H^2\times \mathbb E^1$, $\widetilde{PSL}(2,\mathbb R)$, $NIL$, and $SOL$. Thus $M$ supports a
hyperbolic metric, and finiteness of $\text{Out}(\Gamma)$ follows from Mostow rigidity.
A famous question of Gromov asks whether CAT(0) groups with no $\mathbb Z^2$-subgroup are Gromov hyperbolic.
An affirmative answer to Gromov's question would then imply, by a standard argument (see Corollary <ref>
and the discussion at the beginning of Section <ref>) an affirmative answer to our question as well.
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[Y83]Y K. Yano, There are no transitive Anosov diffeomorphisms on negatively curved manifolds.
Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, p. 445.
[Z84]zimmer R. J. Zimmer, Ergodic Theory and Semisimple Groups,
Monographs in Mathematics 81, Birkhaüser, 1984.
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1511.00425
|
Preconditioned ADMM with nonlinear operator constraint
Benning et al.
University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
({mb941, cbs31, tjmv3}@cam.ac.uk) New York University, Center for Advanced Imaging, Innovation and Research, New York 4th Floor 660 First Avenue New York, NY 10016, USA
Preconditioned ADMM with nonlinear operator constraint
Martin Benning
We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).
§ INTRODUCTION
Non-smooth regularisation methods are popular tools in image processing. They allow to promote sparsity of inverse problem solutions with respect to specific representations; they allow to implicitly restrict the null-space of the forward operator while guaranteeing noise suppression at the same time. The most prominent representatives of this class are total variation regularisation <cit.> and $\ell^1$-norm regularisation as in the broader context of compressed sensing <cit.>.
In order to solve convex, non-smooth regularisation methods with linear operator constraints computationally, first-order operator splitting methods have gained increasing interest over the last decade, see <cit.> to name just a few. Despite some recent extensions to certain types of non-convex problems <cit.> there has to our knowledge only been made little progress for nonlinear operators constraints <cit.>.
In this paper we are particularly interested in minimising non-smooth, convex functionals with nonlinear operator constraints. This model covers many interesting applications; one particular application that we are going to address is the joint reconstruction of the spin-proton-density and coil sensitivity maps in parallel MRI <cit.>.
The paper is structured as follows: we will introduce the generic problem formulation, then address its numerical minimisation via a generalised ADMM method with linearised operator constraints. Subsequently we will show connections to the recently proposed NL-PDHGM method (indicating a local convergence result of the proposed algorithm) and conclude with the joint spin-proton-density and coil sensitivity map estimation as a numerical example.
§ PROBLEM FORMULATION
We consider the following generic constrained minimisation problem:
\begin{align}
(\hat{u}, \hat{v}) &= \argmin_{u, v} \left\{ H(u) + J(v) \ \text{subject to} \ F(u, v) = c\right\} \, \text{.}\label{eq:nonlinprobform}
\end{align}
Here $H$ and $J$ denote proper, convex and lower semi-continuous functionals, $F$ is a nonlinear operator and $c$ a given function. Note that for nonlinear operators of the form $F(u, v) = G(u) - v$ and $c = 0$ problem (<ref>) can be written as
\begin{align}
\hat{u} &= \argmin_{u} \left\{ H(u) + J(G(u)) \right\} \text{.}\label{eq:nonlinvarprob}
\end{align}
In the following we want to propose a strategy for solving (<ref>) that is based on simultaneous linearisation of the nonlinear operator constraint and the solution of an inexact ADMM problem.
§ ALTERNATING DIRECTION METHOD OF MULTIPLIERS
We solve (<ref>) by alternating optimisation of the augmented Lagrange function
\begin{align}
\mathcal{L}_\delta(u, v; \mu) = H(u) + J(v) + \langle \mu, F(u, v) - c \rangle + \frac{\delta}{2} \| F(u, v) - c\|_2^2 \, \text{.}\label{eq:auglag}
\end{align}
Alternating minimisation of (<ref>) in
$u$, $v$ and subsequent maximisation of $\mu$ via a step of gradient ascent yields this nonlinear version of ADMM <cit.>:
\begin{align}
u^{k + 1} &\in \argmin_u \left\{ \frac{\delta}{2} \| F(u, v^k) - c\|_2^2 + \langle \mu^k, F(u, v^k) \rangle + H(u) \right\} \, \text{,} \label{eq:admmup1}\\
v^{k + 1} &\in \argmin_v \left\{ \frac{\delta}{2} \| F(u^{k + 1}, v) - c\|_2^2 + \langle \mu^k, F(u^{k + 1}, v) \rangle + J(v) \right\} \, \text{,} \label{eq:admmvp1} \\
\mu^{k + 1} &= \mu^k + \delta \left( F(u^{k + 1}, v^{k + 1}) - c \right) \, \text{.} \label{eq:admmupmu}
\end{align}
Not having to deal with nonlinear subproblems, we replace $F(u^{k + 1}, v^k)$ and $F(u^{k + 1}, v^{k + 1})$ by their Taylor linearisations around $u^k$ and $v^k$, which yields $F(u, v^k) \approx F(u^k, v^k) + \partial_u F(u^k, v^k)\left( u - u^k \right)$ and $F(u^{k + 1}, v) \approx F(u^{k + 1}, v^k) + \partial_v F(u^{k + 1}, v^k)\left( v - v^k \right)$, respectively. The updates (<ref>) and (<ref>) modify to
\begin{align}
u^{k + 1} &\in \argmin_u \left\{ \frac{\delta}{2} \left\| A^k u - c_1^k\right\|_2^2 + \langle \mu^k, A^k u \rangle + H(u) \right\} \, \text{,} \label{eq:admmup2}\\
v^{k + 1} &\in \argmin_v \left\{ \frac{\delta}{2} \left\| B^k v - c_2^k \right\|_2^2 + \langle \mu^k, B^k v \rangle + J(v) \right\} \, \text{,} \label{eq:admmvp2}
\end{align}
with $A^k := \partial_u F(u^k, v^k)$, $B^k := \partial_v F(u^{k + 1}, v^k)$, $c_1^k := c + A^k u^k - F(u^k, v^k)$ and $c_2^k := c + B^k v^k - F(u^{k + 1}, v^k)$. Note that the updates (<ref>) and (<ref>) are still implicit, regardless of $H$ and $J$. In the following, we want to modify the updates such that they become simple proximity operations.
§ PRECONDITIONED ADMM
Based on <cit.>, we modify (<ref>) and (<ref>) by adding the surrogate terms $\| u^{k + 1} - u^k \|_{Q^k_1}^2 / 2$ and $\| v^{k + 1} - v^k \|_{Q^k_2}^2 / 2$, with $\| w \|_Q := \sqrt{\langle Qw, w\rangle}$ (note that if $Q$ is chosen to be positive definite, $\| \cdot \|_Q$ becomes a norm). We then obtain
\begin{align*}
u^{k + 1} &\in \argmin_u \left\{ \frac{\delta}{2} \left\| A^k u - c_1^k\right\|_2^2 + \langle \mu^k, A^k u \rangle + H(u) + \frac{1}{2}\| u - u^k \|_{Q_1^k}^2 \right\} \, \text{,} \\
v^{k + 1} &\in \argmin_v \left\{ \frac{\delta}{2} \left\| B^k v - c_2^k \right\|_2^2 + \langle \mu^k, B^k v \rangle + J(v) + \frac{1}{2}\| v - v^k \|_{Q_2^k}^2 \right\} \, \text{.}
\end{align*}
If we choose $Q_1^k := \tau_1^k I - \delta A^k {}^* A^k$ with $\tau_1^k \delta < 1/\| A^k \|^2$ and $Q_2^k := \tau_2^k I - \delta B^k {}^* B^k$ with $\tau_2^k \delta < 1/\| B^k \|^2$ and if we define $\overline{\mu}^k := 2\mu^k - \mu^{k - 1}$ we obtain
\begin{align}
u^{k + 1} &= \left( I + \tau_1^k \partial H \right)^{-1} \left( u^k - \tau_1^k A^k{}^*\overline{\mu}^k \right)\, \text{,} \label{eq:admmup3}\\
v^{k + 1} &= \left( I + \tau_2^k \partial J \right)^{-1} \left( v^k - \tau_2^k B^k{}^*\left( \mu^k + \delta \left(F(u^{k + 1}, v^k) - c \right) \right) \right) \, \text{,} \label{eq:admmvp3}
\end{align}
with $(I + \alpha \partial E)^{-1}(w)$ denoting the proximity or resolvent operator
\begin{align*}
(I + \alpha \partial E)^{-1}(w) := \argmin_u \left\{ \frac{1}{2} \|u - w\|_2^2 + \alpha E(u) \right\} \, \text{.}
\end{align*}
The entire proposed algorithm with updates (<ref>), (<ref>) and (<ref>) reads as
Preconditioned ADMM with nonlinear operator constraint
Parameters: $H, \ J, \ F, \ c$
Initialization: $u^0$, $v^0$, $\mu^0$, $\delta$
$\overline{\mu}^0 = \mu^0$
convergence criterion is not met
$A^k = \partial_u F(u^k, v^k)$
Set $\tau_1^k$ such that $\tau_1^k \delta < 1/\| A^k \|^2$
$u^{k + 1} = \left( I + \tau_1^k \partial H \right)^{-1} \left( u^k - \tau_1^k A^k{}^*\overline{\mu}^k \right)$
$B^k = \partial_v F(u^{k + 1}, v^k)$
Set $\tau_2^k$ such that $\tau_2^k \delta < 1/\| B^k \|^2$
$v^{k + 1} = \left( I + \tau_2^k \partial J \right)^{-1} \left( v^k - \tau_2^k B^k{}^*\left( \mu^k + \delta \left(F(u^{k + 1}, v^k) - c \right) \right) \right)$
$\mu^{k + 1} = \mu^k + \delta \left( F(u^{k + 1}, v^{k + 1}) - c \right)$
$\overline{\mu}^{k + 1} = 2\mu^{k + 1} - \mu^k$
$u^{k}$, $v^k$, $\mu^k$, $\overline{\mu}^k$
§ CONNECTION TO NL-PDHGM
In the following we want to show how the algorithm simplifies in case the nonlinear operator constraint is only nonlinear in one variable, which is sufficient for problems of the form (<ref>). Without loss of generality we consider constraints of the form $F(u, v) = G(u) - v$, where $G$ represents a nonlinear operator in $u$. Then we have $A^k = \jac G(u^k)$ (with $\jac G(u^k)$ denoting the Jacobi matrix of $G$ at $u^k$), $B^k = -I$ and if we further choose $\tau_2^k = 1/\delta$ for all $k$, update (<ref>) reads
\begin{align*}
v^{k + 1} = \left( I + \frac{1}{\delta} \partial J \right)^{-1} \left( G(u^{k + 1}) + \frac{1}{\delta}\mu^k \right) \, \text{.}
\end{align*}
Applying Moreau's identity <cit.> $b = \left(I + \frac{1}{\delta} \partial J\right)^{-1}(b) + \frac{1}{\delta}(I + \delta \partial J^*)^{-1}(\delta b)$ yields
\begin{align*}
\mu^{k + 1} = \left( I + \delta \partial J^{*} \right)^{-1}\left( \mu^k + \delta G(u^{k + 1}) \right) \, \text{.}
\end{align*}
If we further change the order of the updates, starting with the update for $\mu$, the whole algorithm reads
\begin{align*}
\mu^{k + 1} &= \left( I + \delta \partial J^{*} \right)^{-1}\left( \mu^k + \delta G(u^k) \right) \, \text{,}\\
\overline{\mu}^{k + 1} &= 2\mu^{k + 1} - \mu^k \, \text{,}\\
u^{k + 1} &= \left( I + \tau_1^k \partial H \right)^{-1} \left( u^k - \tau_1^k \jac G(u^k)^* \overline{\mu}^{k + 1} \right)\, \text{.}
\end{align*}
Note that this algorithm is almost the same as NL-PDHGM proposed in <cit.> for $\theta = 1$, except that the extrapolation step is carried out on the dual variable $\mu$ instead of the primal variable $u$. In the following we want to briefly sketch how to prove convergence for this algorithm in analogy to <cit.>. We define
\begin{align*}
N(\mu^{k + 1}, u^{k + 1}) &:= \left( \begin{array}{c}
\partial J^*(\mu^{k + 1}) - \nabla G(u^k) u^{k + 1} - c^k\\
\partial H(u^{k + 1}) + \jac G(u^k)^* \mu^{k + 1}
\end{array} \right) \, \text{,}\\
L^k &:= \left( \begin{array}{cc}
\frac{1}{\delta} I & \jac G(u^k)\\
\jac G(u^k)^* & \frac{1}{\tau_1^k} I
\end{array}\right) \, \text{,}
\end{align*}
with $c^k := G(u^k) - \jac G(u^k)u^k$.
Now the algorithm is: find $(\mu^{k+1}, u^{k+1})$ such that
\[
N(\mu^{k + 1}, u^{k + 1}) + L^k(\mu^{k + 1}-\mu^k, u^{k + 1}-u^k) \ni 0.
\]
If we exchange the order of $\mu$ and $u$ here, i.e., reorder the rows of $N$, and the rows and columns of $L^k$, we obtain almost the “linearised” NL-PDHGM of <cit.>. The difference is that the sign of $\jmc G$ in $L^k$ is inverted. The only points in <cit.> where the exact structure of $L^k$ ($M_{x^k}$ therein) is used, are Lemma 3.1, Lemma 3.6 and Lemma 3.10. The first two go through exactly as before with the negated structure. Reproducing Lemma 3.10 demands bounding actual step lengths $\norm{u^k-u^{k+1}}$ and $\norm{\mu^k-\mu^{k+1}}$ from below, near a solution for arbitrary $\epsilon>0$. A proof would go beyond the page limit of this proceeding. Let us just point out that this can be done, implying that the convergence results of <cit.> apply for this algorithm as well. This means that under somewhat technical regularity conditions, which for TV type problems amount to Huber regularisation, local convergence in a neighbourhood of the true solution can be guaranteed.
§ JOINT ESTIMATION OF THE SPIN-PROTON DENSITY AND COIL SENSITIVITIES IN PARALLEL MRI
We want to demonstrate the numerical capabilities of the algorithm by applying it to the nonlinear problem of joint estimation of the spin-proton density and the coil sensitivities in parallel MRI. The discrete problem of joint reconstruction from sub-sampled k-space data on a rectangular grid reads
\begin{align*}
\left(\begin{array}{c}
\hat{u}\\
\hat{c_1}\\
\vdots\\
\hat{c_2}
\end{array}\right)
\in \argmin_{\mathbf{v}=(u, c_1, \ldots, c_n)} \left\{ \frac{1}{2} \sum_{j = 1}^n \| S\mathcal{F}(G(\mathbf v))_j - f_j \|_2^2 + \alpha_0 R_0(u) + \sum_{j = 1}^n \alpha_j R_j(c_j) \right\} \, \text{,}
\end{align*}
where $\mathcal{F}$ is the 2D discrete Fourier transform, $f_j$ are the k-space measurements for each of the $n$ coils, $S$ is the sub-sampling operator and $R_j$ denote appropriate regularisation functionals. The nonlinear operator $G$ maps the unknown spin-proton density $u$ and the different coil sensitivities $c_j$ as follows <cit.>:
\begin{align}
G(u, c_1, \ldots, c_n) = (u c_1, u c_2, \ldots, u c_n)^T \, \text{.}\label{eq:coilsensop}
\end{align}
In order to compensate for sub-sampling artefacts in sub-sampled MRI it is common practice to use total variation as a regulariser <cit.>. Coil sensitivities are assumed to be smooth, cf. Figure <ref>, motivating a reconstruction model similar to the one proposed in <cit.>. We therefore choose the discrete isotropic total variation, $R_0(u) = \|\nabla u \|_{2, 1}$, and the smooth 2-norm of the discretised gradient, i.e. $R_j(c_j) := \| \nabla c_j \|_{2, 2}$, for all $j > 0$, following the notation in <cit.>. We further introduce regularisation parameters $\lambda_j$ in front of the data fidelities and rescale all regularisation parameters such that $\alpha_0 + \sum_{j = 1}^n \lambda_j + \sum_{j = 1}^n \alpha_j = 1$. In order to realise this model via Algorithm <ref> we consider the following operator splitting strategy. We define $F(u_0, \ldots, u_n, v_0, \ldots, v_{2n})$ as
\begin{align*}
F(u_0, \ldots, u_n, v_1, \ldots, v_n) := \left( \begin{array}{c}
G(u_0, \ldots, u_n)\\ \begin{array}{cccc} \nabla u_0 & 0 & \cdots & 0\\ 0 & \nabla u_1 & \ddots & \vdots\\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \nabla u_n\end{array}
\end{array}\right) - \left( \begin{array}{c}
v_0 \\ \vdots \\ v_n \\ \vdots \\ v_{2n}
\end{array}\right) \, \text{,}
\end{align*}
set $H(u_0, \ldots, u_n) \equiv 0$, and $J(v_0, \ldots, v_{2n}) = \sum_{j = 0}^{2n} J_j(v_j)$ with $J_j(v_j) := \frac{\lambda_j}{2}\| S\mathcal{F} v_j - f_j \|_2^2$ for $j \in \{0, \ldots, n - 1\}$, $J_n(v_n) = \alpha_0 \| v_n \|_{2, 1}$ and $J_j(v_j) = \alpha_{j - n} \| v_j \|_{2, 2} $ for $j \in \{ n + 1, \ldots, 2n\}$. Note that with these choices of functions, all the resolvent operations can be carried out easily. In particular, we obtain
\begin{align*}
(I + \tau_1^k \partial H)^{-1}(w) &= w \, \text{,}\\
(I + \tau_2^k \partial J_j)^{-1}(w) &= \mathcal{F}^{-1} \left( \frac{\mathcal{F}w_j + \tau_2^k \lambda_j S^T f_j}{1 + \tau_2^k \lambda_j \text{diag}(S^T S) } \right) \ \text{for} \ j \in \{0, \ldots, n - 1\} \, \text{,}\\
(I + \tau_2^k \partial J_n)^{-1}(w) &= \frac{w_n}{\| w_n \|_2}\max\left( \| w_n \|_2 - \alpha_0 \tau_2^k, 0 \right) \, \text{,}\\
(I + \tau_2^k \partial J_j)^{-1}(w) &= \frac{w_n}{\| w_n \|_{2, 2}}\max\left( \| w_n \|_{2, 2} - \alpha_j \tau_2^k, 0 \right) \ \text{for} \ j \in \{n + 1, \ldots, 2n\} \, \text{.}
\end{align*}
Moreover, as $B_k = -I$ (and thus, $\| B^k \| = 1$) for all $k$, we can simply eliminate $\tau_2^k$ by replacing it with $1/\delta$, similar to
Section <ref>.
[Brain phantom]
[25% sub-sampling]
[1st coil]
[2nd coil]
[3rd coil]
[4th coil]
[5th coil]
[6th coil]
[7th coil]
[8th coil]
Figure <ref> shows the brain phantom as described in Section <ref>. Figure <ref> - <ref> show visualisations of the measured coil sensitivities of a water bottle. Figure <ref> shows the simulated, spiral-shaped sub-sampling scheme used to sub-sample the k-space data.
[Reconstruction $u$]
[1st coil]
[2nd coil]
[3rd coil]
[4th coil]
[5th coil]
[6th coil]
[7th coil]
[8th coil]
Reconstructions for noise with low noise level $\sigma = 0.05$. Despite the sub-sampling, features of the brain phantom are very well preserved. In addition, the coil sensitivities seem to correspond well to the original ones, despite a slight loss of contrast. Note that coil sensitivities remain the initial value where the signal is zero.
§.§ Experimental setup
We now want to discuss the experimental setup. We want to reconstruct the synthetic brain phantom in Figure <ref> from sub-sampled k-space measurements. The numerical phantom is based on the design in <cit.> with a matrix size of $190 \times 190$. It consists of several different tissue types like cerebrospinal fluid (CSF), gray matter (GM), white matter (WM) and cortical bone. Each pixel is assigned a set of MR tissue properties: Relaxation times $\text{T}_1(x,y)$ and $\text{T}_2(x,y)$ and spin density $\rho(x,y)$. These parameters were also selected according to <cit.>. The MR signal $s(x,y)$ in each pixel was then calculated by using the signal equation of a fluid attenuation inversion recovery (FLAIR) sequence <cit.>:
\begin{align*}
s(x,y) = \rho(x,y)(1-2 ~e^{-\text{TI}/\text{T}_1(x,y)})(1 - e^{-\text{TR}/\text{T}_1(x,y)}) ~e^{-\text{TE}/\text{T}_2(x,y)}.
\end{align*}
The sequence parameters were selected: TR = 10000 ms, TE = 90 ms. TI was set to 1781 ms to achieve signal nulling of CSF ($\text{T}_1^\text{csf} \log(2)$ with $\text{T}_1^\text{csf} = 2569 \text{ms}$).
In order to generate artificial k-space measurements for each coil, we proceed as follows. First, we produce 8 images of the brain phantom multiplied by the measured coil sensitivity maps shown in Figure <ref> - <ref>. The coil sensitivity maps were generated from the measurements of a water bottle with an 8-channel head coil array. Then we produce artificial k-space data by applying the 2D discrete Fourier-transform to each of those individual images. Subsequently, we sub-sample only approx. 25% of each of the k-space datasets via the spiral shown in Figure <ref>. Finally, we add Gaußian noise with standard deviation $\sigma$ to the sub-sampled data.
[Reconstruction $u$]
[1st coil]
[2nd coil]
[3rd coil]
[4th coil]
[5th coil]
[6th coil]
[7th coil]
[8th coil]
Reconstructions for noise with high noise level $\sigma = 0.95$. Due to the large amount of noise, higher regularisation parameters are necessary. As a consequence, fine structures are smoothed out and in contrast to the case of little noise, compensation of sub-sampling artefacts is less successful.
§.§ Computations
For the actual computations we use two noisy versions $f_j$ of the simulated k-space data; one with small noise ($\sigma = 0.05$) and one with a high amount of noise ($\sigma = 0.95$). As stopping criterion we simply choose a fixed number of iterations; for both the low noise level as well as the high noise level dataset we have fixed the number of iterations to 1500. The initial values used for the algorithm are $u_j^0 = \textbf{1}$ with $\textbf{1} \in \R^{l \times 1}$ being the constant one-vector, for all $j \in \{0, \ldots, n\}$. All other initial variables ($v^0$, $\mu^0$, $\overline{\mu}^0$) are set to zero.
§.§.§ Low noise level
We have computed reconstructions from the noisy data with noise level $\sigma = 0.05$ via Algorithm <ref>, with regularisation parameters set to $\lambda_j = 0.0621$, $\alpha_0 = 0.062$ and $\alpha_j = 0.9317$ for $j \in \{1, \ldots, n\}$. We have further created a naïve reconstruction by averaging the individual inverse Fourier-transformed images obtained from zero-filling the k-space data. The modulus images of the results are visualised in Figure <ref>. The results are visualised in Figure <ref>. The PSNR values for the averaged zero-filled reconstruction is 10.2185, whereas the PSNR of the reconstruction with the proposed method is 24.5572.
§.§.§ High noise level
We proceeded as in the previous section, but for noisy data with noise level $\sigma = 0.95$. The regularisation parameters were set to $\lambda_j = 0.0149$, $\alpha_0 = 0.0135$ and $\alpha_j = 0.9716$ for $j \in \{1, \ldots, n\}$. The modulus images of the results are visualised in Figure <ref>. The PSNR values for the averaged zero-filled reconstruction is 9.9621, whereas the PSNR of the reconstruction with the proposed method is 16.672.
§ CONCLUSIONS & OUTLOOK
We have presented a novel algorithm that allows to compute minimisers of a sum of convex functionals with nonlinear operator constraint. We have shown the connection to the recently proposed NL-PDHGM algorithm which implies local convergence results in analogy to those derived in <cit.>. Subsequently we have demonstrated the computational capabilities of the algorithm by applying it to a nonlinear joint reconstruction problem in parallel MRI.
For future work, the convergence of the algorithm in the general setting has to be verified, and possible extensions to guarantee global convergence have to be studied. Generalisation of stopping criteria such as a linearised primal-dual gap will also be of interest as well. With respect to the presented parallel MRI application, exact conditions for the convergence (like the exact norm of the bounds) have to be verified. The impact of the algorithm- as well as the regularisation-parameters on the reconstruction has to be analysed, and rigorous study with synthetic and real data would also be desirable. Moreover, future research focus will be on alternative regularisation functions, e.g. based on spherical harmonics motivated by <cit.>. Last but not least, other applications that can be modelled via (<ref>) should be considered in future research.
§.§.§ Acknowledgments
MB, CS and TV acknowledge EPSRC grant EP/M00483X/1. FK ackowledges National Institutes of Health grant NIH P41 EB017183.
|
1511.00293
|
An ordering between the quantum states emerging from a single mode gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e. diagonal with decreasing eigenvalues) produces an output which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others.
§ INTRODUCTION
The minimum von Neumann entropy at the output of a quantum communication channel can be crucial for the determination of its classical communication capacity <cit.>.
Most communication schemes encode the information into pulses of electromagnetic radiation, that travels through metal wires, optical fibers or free space and is unavoidably affected by attenuation and noise.
Gauge-covariant quantum Gaussian channels <cit.> provide a faithful model for these effects, and are characterized by the property of preserving the thermal states of electromagnetic radiation.
It has been recently proved <cit.> that the output entropy of any gauge-covariant Gaussian quantum channel is minimized when the input state is the vacuum.
This result has permitted the determination of the classical information capacity of this class of channels <cit.>.
However, it is not sufficient to determine the capacity region of the quantum Gaussian broadcast channel <cit.> and the triple trade-off region of the quantum-limited Gaussian attenuator <cit.>.
Indeed, solving these problems would require to prove that Gaussian thermal input states minimize the output von Neumann entropy of a quantum-limited Gaussian attenuator among all the states with a given entropy.
This still unproven result would follow from a stronger conjecture, the Entropy Photon-number Inequality (EPnI) <cit.>, stating that Gaussian states minimize the output von Neumann entropy of a beamsplitter among all the couples of input states, each one with a given entropy.
So far, it has been possible to prove only a weaker version of the EPnI, the quantum Entropy Power Inequality <cit.>, that provides a lower bound to the output entropy of a beamsplitter, but is never saturated.
Actually, Ref.'s <cit.> do not only prove that the vacuum minimizes the output entropy of any gauge-covariant quantum Gaussian channel.
They also prove that the output generated by the vacuum majorizes the output generated by any other state, i.e. applying a convex combination of unitary operators to the former, we can obtain any of the latter states.
This paper goes in the same direction, and proves a generalization of this result valid for any one-mode gauge-covariant quantum Gaussian channel.
Our result states that the output generated by any quantum state is majorized by the output generated by the state with the same spectrum diagonal in the Fock basis and with decreasing eigenvalues, i.e. by the state which is passive <cit.> with respect to the number operator (see <cit.> for the use of passive states in the context of quantum thermodynamics).
This can be understood as follows: among all the states with a given spectrum, the one diagonal in the Fock basis with decreasing eigenvalues produces the less noisy output.
Since all the states with the same spectrum have the same von Neumann entropy, our result implies that the input state with given entropy minimizing the output entropy is certainly diagonal in the Fock basis, and then reduces the minimum output entropy quantum problem to a problem on discrete classical probability distributions.
Thanks to the classification of one-mode Gaussian channels in terms of unitary equivalence <cit.>, we extend the result to the channels that are not gauge-covariant with the exception of the singular cases $A_2)$ and $B_1)$, for which we show that an optimal basis does not exist.
We also point out that the classical channel acting on discrete probability distributions associated to the restriction of the quantum-limited attenuator to states diagonal in the Fock basis coincides with the channel already known in the probability literature under the name of thinning.
First introduced by Rényi <cit.> as a discrete analogue of the rescaling of a continuous random variable, the thinning has been recently involved in discrete versions of the central limit theorem <cit.>
and of the Entropy Power Inequality <cit.>.
In particular, the Restricted Thinned Entropy Power Inequality <cit.> states that the Poissonian probability distribution minimizes the output Shannon entropy of the thinning among all the ultra log-concave input probability distributions with a given Shannon entropy.
The techniques of this proof could be useful to prove that Gaussian thermal states minimize the output von Neumann entropy of a quantum-limited attenuator among all the input states diagonal in the Fock basis with a given von Neumann entropy (but without the ultra log-concavity constraint).
Then, thanks to the main result of this paper it would automatically follow that Gaussian thermal states minimize the output von Neumann entropy of a quantum-limited attenuator among all the input states with a given von Neumann entropy, not necessarily diagonal in the Fock basis.
The paper is organized as follows.
In Section <ref> we introduce the Gaussian quantum channels, and in Section <ref> the majorization.
The Fock rearrangement is defined in Section <ref>, while Section <ref> defines the notion of Fock optimality and proves some of its properties.
The main theorem is proved in Section <ref>, and the case of a generic not gauge-covariant Gaussian channel is treated in Section <ref>.
Finally, Section <ref> links our result to the thinning operation.
§ BASIC DEFINITIONS
In this section we recall some basic definitions and facts on Gaussian quantum channels.
The interested reader can find more details in the books <cit.>.
The trace norm of an operator $\hat{X}$ is
\begin{equation}
\left\|\hat{X}\right\|_1:=\mathrm{Tr}\sqrt{\hat{X}^\dag\hat{X}}\;.
\end{equation}
If $\left\|\hat{X}\right\|_1$ is finite, we say that $\hat{X}$ is a trace-class operator.
A quantum operation is a linear completely positive map on trace-class operators continuous in the trace norm.
A trace-preserving quantum operation is a quantum channel.
We consider the Hilbert space $\mathcal{H}$ of the harmonic oscillator, i.e. the irreducible representation of the canonical commutation relation
\begin{equation}\label{CCR}
\left[\hat{a},\;\hat{a}^\dag\right]=\hat{\mathbb{I}}\;.
\end{equation}
$\mathcal{H}$ has a countable orthonormal basis
\begin{equation}
\{|n\rangle\}_{n\in\mathbb{N}}\;,\qquad \langle m|n\rangle=\delta_{mn}
\end{equation}
called the Fock basis, on which the ladder operators act as
\begin{eqnarray}\label{acta}
\hat{a}\;|n\rangle &=& \sqrt{n}\;|n-1\rangle\\
\hat{a}^\dag\;|n\rangle &=& \sqrt{n+1}\;|n+1\rangle\;.\label{actadag}
\end{eqnarray}
We can define a number operator
\begin{equation}
\hat{N}=\hat{a}^\dag\hat{a}\;,
\end{equation}
\begin{equation}
\hat{N}\;|n\rangle=n\;|n\rangle\;.
\end{equation}
The Hilbert-Schmidt norm of an operator $\hat{X}$ is
\begin{equation}
\left\|\hat{X}\right\|_2^2:=\mathrm{Tr}\left[\hat{X}^\dag\;\hat{X}\right]\;.
\end{equation}
(Hilbert-Schmidt dual)
Let $\Phi$ be a linear map acting on trace class operators and continuous in the trace norm.
Its Hilbert-Schmidt dual $\Phi^\dag$ is the map on bounded operators continuous in the operator norm defined by
\begin{equation}
\mathrm{Tr}\left[\hat{Y}\;\Phi\left(\hat{X}\right)\right]=\mathrm{Tr}\left[\Phi^\dag\left(\hat{Y}\right)\;\hat{X}\right]
\end{equation}
for any trace-class operator $\hat{X}$ and any bounded operator $\hat{Y}$.
The characteristic function of a trace-class operator $\hat{X}$ is
\begin{equation}
\chi_{\hat{X}}(z):=\mathrm{Tr}\left[e^{z\;\hat{a}^\dag-\bar{z}\;\hat{a}}\;\hat{X}\right]\;,\qquad z\in\mathbb{C}\;,
\end{equation}
where $\bar{z}$ denotes the complex conjugate.
It is possible to prove that any trace-class operator is completely determined by its characteristic function.
The characteristic function provides an isometry between the Hilbert-Schmidt product and the scalar product in $L^2(\mathbb{C})$, i.e. for any two trace-class operators $\hat{X}$ and $\hat{Y}$,
\begin{equation}\label{Trint}
\mathrm{Tr}\left[\hat{X}^\dag\;\hat{Y}\right]=\int_{\mathbb{C}}\overline{\chi_{\hat{X}}(z)}\;\chi_{\hat{Y}}(z)\;\frac{d^2z}{\pi}\;.
\end{equation}
See e.g. Theorem 5.3.3 of <cit.>.
A gauge-covariant quantum Gaussian channel with parameters $\lambda\geq0$ and $N\geq0$ can be defined by its action on the characteristic function: for any trace-class operator $\hat{X}$,
\begin{equation}\label{chiPhi}
\chi_{\Phi\left(\hat{X}\right)}(z)=e^{-|\lambda-1|\left(N+\frac{1}{2}\right)|z|^2}\;\chi_{\hat{X}}\left(\sqrt{\lambda}\;z\right)\;.
\end{equation}
The channel is called quantum-limited if $N=0$.
If $0\leq\lambda\leq1$, it is a quantum-limited attenuator, while if $\lambda\geq1$ it is a quantum-limited amplifier.
Any gauge-covariant quantum Gaussian channel is continuous also in the Hilbert-Schmidt norm.
Easily follows from its action on the characteristic function (<ref>) and the isometry (<ref>).
Any gauge-covariant quantum Gaussian channel can be written as a quantum-limited amplifier composed with a quantum-limited attenuator.
See <cit.>.
The Hilbert-Schmidt dual of the quantum-limited attenuator of parameter $0<\lambda\leq1$ is $1/\lambda$ times the quantum-limited amplifier of parameter $\lambda'=1/\lambda\geq1$, hence its restriction to trace-class operators is continuous in the trace-norm.
Easily follows from the action of the quantum-limited attenuator and amplifier on the characteristic function (<ref>) and formula (<ref>); see also <cit.>.
The quantum-limited attenuator of parameter $0\leq\lambda\leq1$ admits the explicit representation
\begin{equation}\label{kraus}
\Phi_\lambda\left(\hat{X}\right)=\sum_{l=0}^\infty\frac{(1-\lambda)^l}{l!}\;\lambda^\frac{\hat{N}}{2}\;\hat{a}^l\;\hat{X}\;\left(\hat{a}^\dag\right)^l\;\lambda^\frac{\hat{N}}{2}
\end{equation}
for any trace-class operator $\hat{X}$.
Then, if $\hat{X}$ is diagonal in the Fock basis, $\Phi_\lambda\left(\hat{X}\right)$ is diagonal in the same basis for any $0\leq\lambda\leq1$ also.
The channel $\Phi_\lambda$ admits the Kraus decomposition (see Eq. (4.5) of <cit.>)
\begin{equation}
\Phi_\lambda\left(\hat{X}\right)=\sum_{l=0}^\infty\hat{B}_l\;\hat{X}\;\hat{B}_l^\dag\;,
\end{equation}
\begin{equation}
\hat{B}_l=\sum_{m=0}^\infty\sqrt{\binom{m+l}{l}}\;(1-\lambda)^\frac{l}{2}\;\lambda^\frac{m}{2}\;|m\rangle\langle m+l|\;,\qquad l\in\mathbb{N}\;.
\end{equation}
Using (<ref>), we have
\begin{equation}
\hat{a}^l=\sum_{m=0}^\infty\sqrt{l!\;\binom{m+l}{l}}\;|m\rangle\langle m+l|\;,
\end{equation}
and the claim easily follows.
The quantum-limited attenuator of parameter $\lambda=e^{-t}$ with $t\geq0$ can be written as the exponential of a Lindbladian $\mathcal{L}$, i.e. $\Phi_\lambda=e^{t\mathcal{L}}$, where
\begin{equation}\label{lindblad}
\mathcal{L}\left(\hat{X}\right)=\hat{a}\;\hat{X}\;\hat{a}^\dag-\frac{1}{2}\hat{a}^\dag\hat{a}\;\hat{X}-\frac{1}{2}\hat{X}\;\hat{a}^\dag\hat{a}
\end{equation}
for any trace-class operator $\hat{X}$.
Putting $\lambda=e^{-t}$ into (<ref>) and differentiating with respect to $t$ we have for any trace-class operator $\hat{X}$
\begin{equation}
\frac{d}{dt}\Phi_{\lambda}\left(\hat{X}\right)=\mathcal{L}\left(\Phi_{\lambda}\left(\hat{X}\right)\right)\;,
\end{equation}
where $\mathcal{L}$ is the Lindbladian given by (<ref>).
\begin{equation}\label{Xdiag}
\hat{X}=\sum_{k=0}^\infty x_k\;|\psi_k\rangle\langle\psi_k|\;,\quad\langle\psi_k|\psi_l\rangle=\delta_{kl}\;,\quad x_0\geq x_1\geq\ldots
\end{equation}
be a self-adjoint Hilbert-Schmidt operator.
Then, the projectors
\begin{equation}\label{Pin}
\hat{\Pi}_n=\sum_{k=0}^n |\psi_k\rangle\langle\psi_k|
\end{equation}
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}_n\;\hat{X}\right]=\sum_{k=0}^n x_k\;.
\end{equation}
Easily follows from an explicit computation.
Let $\hat{X}$ be a positive Hilbert-Schmidt operator with eigenvalues $\{x_k\}_{k\in\mathbb{N}}$ in decreasing order, i.e. $x_0\geq x_1\geq\ldots\;$,
and let $\hat{P}$ be a projector of rank $n+1$.
\begin{equation}\label{TrPiX}
\mathrm{Tr}\left[\hat{P}\;\hat{X}\right]\leq\sum_{k=0}^n x_k\;.
\end{equation}
(See also <cit.>).
Let us diagonalize $\hat{X}$ as in (<ref>).
The proof proceeds by induction on $n$.
Let $\hat{P}$ have rank one.
\begin{equation}
\hat{X}\leq x_0\;\hat{\mathbb{I}}\;,
\end{equation}
we have
\begin{equation}
\mathrm{Tr}\left[\hat{P}\;\hat{X}\right]\leq x_0\;.
\end{equation}
Suppose now that (<ref>) holds for any rank-$n$ projector.
Let $\hat{P}$ be a projector of rank $n+1$.
Its support then certainly contains a vector $|\psi\rangle$ orthogonal to the support of $\hat{\Pi}_{n-1}$, that has rank $n$.
We can choose $|\psi\rangle$ normalized (i.e. $\langle\psi|\psi\rangle=1$), and define the rank-$n$ projector
\begin{equation}
\hat{Q}=\hat{P}-|\psi\rangle\langle\psi|\;.
\end{equation}
By the induction hypothesis on $\hat{Q}$,
\begin{equation}\label{ineqQpsi}
\mathrm{Tr}\left[\hat{P}\,\hat{X}\right]=\mathrm{Tr}\left[\hat{Q}\,\hat{X}\right]+\langle\psi|\hat{X}|\psi\rangle\leq\sum_{k=0}^{n-1}x_k+\langle\psi|\hat{X}|\psi\rangle\;.
\end{equation}
Since $|\psi\rangle$ is in the support of $\hat{\mathbb{I}}-\hat{\Pi}_{n-1}$, and
\begin{equation}
\left(\hat{\mathbb{I}}-\hat{\Pi}_{n-1}\right)\hat{X}\left(\hat{\mathbb{I}}-\hat{\Pi}_{n-1}\right)\leq x_n\;\hat{\mathbb{I}}\;,
\end{equation}
we have
\begin{equation}\label{ineqpsi}
\langle\psi|\hat{X}|\psi\rangle\leq x_n\;,
\end{equation}
and this concludes the proof.
Let $\hat{X}$ and $\hat{Y}$ be positive Hilbert-Schmidt operators with eigenvalues in decreasing order $\{x_n\}_{n\in\mathbb{N}}$ and $\{y_n\}_{n\in\mathbb{N}}$, respectively.
\begin{equation}
\sum_{n=0}^\infty (x_n-y_n)^2\leq\left\|\hat{X}-\hat{Y}\right\|_2^2\;.
\end{equation}
We have
\begin{equation}\label{TrXYr}
\left\|\hat{X}-\hat{Y}\right\|_2^2-\sum_{n=0}^\infty (x_n-y_n)^2=2\sum_{n=0}^\infty x_ny_n-2\mathrm{Tr}\left[\hat{X}\hat{Y}\right]\geq0\,.
\end{equation}
To prove the inequality in (<ref>), let us diagonalize $\hat{X}$ as in (<ref>).
We then also have
\begin{equation}
\hat{X}=\sum_{n=0}^\infty\left(x_n-x_{n+1}\right)\hat{\Pi}_n\;,
\end{equation}
\begin{equation}
\hat{\Pi}_n=\sum_{k=0}^n|\psi_k\rangle\langle\psi_k|\;.
\end{equation}
We then have
\begin{eqnarray}
\mathrm{Tr}\left[\hat{X}\;\hat{Y}\right] &=& \sum_{n=0}^\infty\left(x_n-x_{n+1}\right)\mathrm{Tr}\left[\hat{\Pi}_n\;\hat{Y}\right]\nonumber\\
&\leq& \sum_{n=0}^\infty\left(x_n-x_{n+1}\right)\sum_{k=0}^n y_k\nonumber\\
&=& \sum_{n=0}^\infty x_n\;y_n\;,
\end{eqnarray}
where we have used Ky Fan's Maximum Principle (Lemma <ref>) and rearranged the sum (see also the Supplemental Material of <cit.>).
§ MAJORIZATION
We recall here the definition of majorization.
The interested reader can find more details in the dedicated book <cit.>, that however deals only with the finite-dimensional case.
Let $x$ and $y$ be decreasing summable sequences of positive numbers.
We say that $x$ weakly sub-majorizes $y$, or $x\succ_w y$, iff
\begin{equation}
\sum_{i=0}^n x_i\geq\sum_{i=0}^n y_i\quad\forall\;n\in\mathbb{N}\;.
\end{equation}
If they have also the same sum, we say that $x$ majorizes $y$, or $x\succ y$.
Let $\hat{X}$ and $\hat{Y}$ be positive trace-class operators with eigenvalues in decreasing order $\{x_n\}_{n\in\mathbb{N}}$ and $\{y_n\}_{n\in\mathbb{N}}$, respectively.
We say that $\hat{X}$ weakly sub-majorizes $\hat{Y}$, or $\hat{X}\succ_w\hat{Y}$, iff $x\succ_w y$.
We say that $\hat{X}$ majorizes $\hat{Y}$, or $\hat{X}\succ\hat{Y}$, if they have also the same trace.
From an operational point of view, majorization can also be defined with:
Given two positive operators $\hat{X}$ and $\hat{Y}$ with the same finite trace, the following conditions are equivalent:
* $\hat{X}\succ\hat{Y}$;
* For any continuous nonnegative convex function $f:[0,\infty)\to\mathbb{R}$ with $f(0)=0\,$,
\begin{equation}\label{Trf}
\mathrm{Tr}\;f\left(\hat{X}\right)\geq\mathrm{Tr}\;f\left(\hat{Y}\right)\;;
\end{equation}
* For any continuous nonnegative concave function $g:[0,\infty)\to\mathbb{R}$ with $g(0)=0\,$,
\begin{equation}\label{Trg}
\mathrm{Tr}\;g\left(\hat{X}\right)\leq\mathrm{Tr}\;g\left(\hat{Y}\right)\;;
\end{equation}
* $\hat{Y}$ can be obtained applying to $\hat{X}$ a convex combination of unitary operators, i.e. there exists a probability measure $\mu$ on unitary operators such that
\begin{equation}
\hat{Y}=\int\hat{U}\,\hat{X}\,\hat{U}^\dag\;d\mu\left(\hat{U}\right)\;.
\end{equation}
See Theorems 5, 6 and 7 of <cit.>.
Notice that Ref. <cit.> uses the opposite definition of the symbol “$\succ$” with respect to most literature (and to Ref. <cit.>), i.e. there $\hat{X}\succ\hat{Y}$ means that $\hat{X}$ is majorized by $\hat{Y}$.
If $\hat{X}$ and $\hat{Y}$ are quantum states (i.e. their trace is one), (<ref>) implies that the von Neumann entropy of $\hat{X}$ is lower than the von Neumann entropy of $\hat{Y}$, while (<ref>) implies the same for all the Rényi entropies <cit.>.
§ FOCK REARRANGEMENT
In order to state our main theorem, we need to define
Let $\hat{X}$ be a positive trace-class operator with eigenvalues $\{x_n\}_{n\in\mathbb{N}}$ in decreasing order.
We define its Fock rearrangement as
\begin{equation}
\hat{X}^\downarrow:=\sum_{n=0}^\infty x_n\;|n\rangle\langle n|\;.
\end{equation}
If $\hat{X}$ coincides with its own Fock rearrangement, i.e. $\hat{X}=\hat{X}^\downarrow$, we say that it is passive <cit.> with respect to the Hamiltonian $\hat{N}$.
For simplicity, in the following we will always assume $\hat{N}$ to be the reference Hamiltonian, and an operator with $\hat{X}=\hat{X}^\downarrow$ will be called simply passive.
The Fock rearrangement of any projector $\hat{\Pi}_n$ of rank $n+1$ is the projector onto the first $n+1$ Fock states:
\begin{equation}\label{Pin*}
\hat{\Pi}_n^\downarrow=\sum_{i=0}^n|i\rangle\langle i|\;.
\end{equation}
We define the notion of passive-preserving quantum operation, that will be useful in the following.
We say that a quantum operation $\Phi$ is passive-preserving if $\Phi\left(\hat{X}\right)$ is passive for any passive positive trace-class operator $\hat{X}$.
We will also need these lemmata:
For any self-adjoint trace-class operator $\hat{X}$,
\begin{equation}
\lim_{N\to\infty}\left\|\hat{\Pi}_N^\downarrow\;\hat{X}\;\hat{\Pi}_N^\downarrow-\hat{X}\right\|_2=0\;,
\end{equation}
where the $\hat{\Pi}_N^\downarrow$ are the projectors onto the first $N+1$ Fock states defined in (<ref>).
We have
\begin{eqnarray}\label{PXP}
\left\|\hat{\Pi}_N^\downarrow\hat{X}\hat{\Pi}_N^\downarrow-\hat{X}\right\|_2^2 &=& \mathrm{Tr}\left[\hat{X}\left(\hat{\mathbb{I}}+\hat{\Pi}_N^\downarrow\right)\hat{X}\left(\hat{\mathbb{I}}-\hat{\Pi}_N^\downarrow\right)\right]\nonumber\\ &\leq&2\;\mathrm{Tr}\left[\hat{X}^2\left(\hat{\mathbb{I}}-\hat{\Pi}_N^\downarrow\right)\right]\nonumber\\
&=&2\sum_{n=N+1}^\infty\langle n|\hat{X}^2|n\rangle\;,
\end{eqnarray}
where we have used that
\begin{equation}
\hat{\mathbb{I}}+\hat{\Pi}_N^\downarrow\leq 2\;\hat{\mathbb{I}}\;.
\end{equation}
Since $\hat{X}$ is trace-class, it is also Hilbert-Schmidt, the sum in (<ref>) converges, and its tail tends to zero for $N\to\infty$.
A positive trace-class operator $\hat{X}$ is passive iff for any finite-rank projector $\hat{P}$
\begin{equation}\label{PP*X}
\mathrm{Tr}\left[\hat{P}\;\hat{X}\right]\leq\mathrm{Tr}\left[\hat{P}^\downarrow\;\hat{X}\right]\;.
\end{equation}
First, suppose that $\hat{X}$ is passive with eigenvalues $\{x_n\}_{n\in\mathbb{N}}$ in decreasing order, and let $\hat{P}$ have rank $n+1$.
Then, by Lemma <ref>
\begin{equation}
\mathrm{Tr}\left[\hat{P}\;\hat{X}\right]\leq\sum_{i=0}^n x_i=\mathrm{Tr}\left[\hat{P}^\downarrow\;\hat{X}\right]\;.
\end{equation}
Suppose now that (<ref>) holds for any finite-rank projector.
Let us diagonalize $\hat{X}$ as in (<ref>).
Putting into (<ref>) the projectors $\hat{\Pi}_n$ defined in (<ref>),
\begin{equation}
\sum_{i=0}^n x_i=\mathrm{Tr}\left[\hat{\Pi}_n\;\hat{X}\right]\leq\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\hat{X}\right]\leq\sum_{i=0}^n x_i\;,
\end{equation}
where we have again used Lemma <ref>.
It follows that for any $n\in\mathbb{N}$
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\hat{X}\right]=\sum_{i=0}^n x_i\;,
\end{equation}
\begin{equation}
\langle n|\hat{X}|n\rangle=x_n\;.
\end{equation}
It is then easy to prove by induction on $n$ that
\begin{equation}
\hat{X}=\sum_{n=0}^\infty x_n\;|n\rangle\langle n|\;,
\end{equation}
i.e. $\hat{X}$ is passive.
Let $\left\{\hat{X}_n\right\}_{n\in\mathbb{N}}$ be a sequence of positive trace-class operators with $\hat{X}_n$ passive for any $n\in\mathbb{N}$.
Then also $\sum_{n=0}^\infty\hat{X}_n$ is passive, provided that its trace is finite.
Follows easily from the definition of Fock rearrangement.
Let $\Phi$ be a quantum operation.
Suppose that $\Phi\left(\hat{\Pi}\right)$ is passive for any passive finite-rank projector $\hat{\Pi}$.
Then, $\Phi$ is passive-preserving.
Choose a passive operator
\begin{equation}
\hat{X}=\sum_{n=0}^\infty x_n\,|n\rangle\langle n|\;,
\end{equation}
with $\{x_n\}_{n\in\mathbb{N}}$ positive and decreasing.
We then also have
\begin{equation}
\hat{X}=\sum_{n=0}^\infty z_n\;\hat{\Pi}_n^\downarrow\;,
\end{equation}
where the $\hat{\Pi}_n^\downarrow$ are defined in (<ref>), and
\begin{equation}
\end{equation}
Since by hypothesis $\Phi\left(\hat{\Pi}_n^\downarrow\right)$ is passive for any $n\in\mathbb{N}$, according to Lemma <ref> also
\begin{equation}
\Phi\left(\hat{X}\right)=\sum_{n=0}^\infty z_n\;\Phi\left(\hat{\Pi}_n^\downarrow\right)
\end{equation}
is passive.
Let $\hat{X}$ and $\hat{Y}$ be positive trace-class operators.
* Suppose that for any finite-rank projector $\hat{\Pi}$
\begin{equation}\label{majproj}
\mathrm{Tr}\left[\hat{\Pi}\,\hat{X}\right]\leq\mathrm{Tr}\left[\hat{\Pi}^\downarrow\,\hat{Y}\right]\;.
\end{equation}
Then $\hat{X}\prec_w\hat{Y}$.
* Let $\hat{Y}$ be passive, and suppose that $\hat{X}\prec_w\hat{Y}$.
Then (<ref>) holds for any finite-rank projector $\hat{\Pi}$.
Let $\{x_n\}_{n\in\mathbb{N}}$ and $\{y_n\}_{n\in\mathbb{N}}$ be the eigenvalues in decreasing order of $\hat{X}$ and $\hat{Y}$, respectively, and let us diagonalize $\hat{X}$ as in (<ref>).
* Suppose first that (<ref>) holds for any finite-rank projector $\hat{\Pi}$.
For any $n\in\mathbb{N}$ we have
\begin{equation}\label{eqtr}
\sum_{i=0}^n x_i=\mathrm{Tr}\left[\hat{\Pi}_n\,\hat{X}\right]\leq\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\,\hat{Y}\right]\leq\sum_{i=0}^n y_i\;,
\end{equation}
where the $\hat{\Pi}_n$ are defined in (<ref>) and we have used Lemma <ref>.
Then $x\prec_w y$, and $\hat{X}\prec_w\hat{Y}$.
* Suppose now that $\hat{X}\prec_w\hat{Y}$ and $\hat{Y}=\hat{Y}^\downarrow$.
Then, for any $n\in\mathbb{N}$ and any projector $\hat{\Pi}$ of rank $n+1$,
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}\,\hat{X}\right]\leq \sum_{i=0}^n x_i\leq\sum_{i=0}^n y_i=\mathrm{Tr}\left[\hat{\Pi}^\downarrow\,\hat{Y}\right]\;,
\end{equation}
where we have used Lemma <ref> again.
Let $\hat{Y}$ and $\hat{Z}$ be positive trace-class operators with $\hat{Y}\prec_w\hat{Z}=\hat{Z}^\downarrow$.
Then, for any positive trace-class operator $\hat{X}$,
\begin{equation}
\mathrm{Tr}\left[\hat{X}\;\hat{Y}\right]\leq\mathrm{Tr}\left[\hat{X}^\downarrow\;\hat{Z}\right]\;.
\end{equation}
Let us diagonalize $\hat{X}$ as in (<ref>).
Then, it can be rewritten as
\begin{equation}\label{XPi}
\hat{X}=\sum_{n=0}^\infty d_n\,\hat{\Pi}_n\;,
\end{equation}
where the projectors $\hat{\Pi}_n$ are as in (<ref>) and
\begin{equation}
\end{equation}
The Fock rearrangement of $\hat{X}$ is
\begin{equation}\label{X*Pi}
\hat{X}^\downarrow=\sum_{n=0}^\infty d_n\,\hat{\Pi}_n^\downarrow\;.
\end{equation}
We then have from Lemma <ref>
\begin{eqnarray}
\mathrm{Tr}\left[\hat{X}\;\hat{Y}\right] &=& \sum_{n=0}^\infty d_n\;\mathrm{Tr}\left[\hat{\Pi}_n\;\hat{Y}\right]\leq\sum_{n=0}^\infty d_n\;\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\hat{Z}\right]\nonumber\\
\end{eqnarray}
Let $\left\{\hat{X}_n\right\}_{n\in\mathbb{N}}$ and $\left\{\hat{Y}_n\right\}_{n\in\mathbb{N}}$ be two sequences of positive trace-class operators, with $\hat{Y}_n=\hat{Y}_n^\downarrow$ and $\hat{X}_n\prec_w\hat{Y}_n$ for any $n\in\mathbb{N}$.
\begin{equation}
\sum_{n=0}^\infty\hat{X}_n\prec_w\sum_{n=0}^\infty\hat{Y}_n\;,
\end{equation}
provided that both sides have finite traces.
Let $\hat{P}$ be a finite-rank projector.
Since $\hat{X}_n\prec_w\hat{Y}_n$ and $Y_n=Y_n^\downarrow$, by the second part of Lemma <ref>
\begin{equation}
\mathrm{Tr}\left[\hat{P}\;\hat{X}_n\right]\leq\mathrm{Tr}\left[\hat{P}^\downarrow\;\hat{Y}_n\right]\qquad\forall\;n\in\mathbb{N}\;.
\end{equation}
\begin{equation}
\mathrm{Tr}\left[\hat{P}\;\sum_{n=0}^\infty\hat{X}_n\right]\leq\mathrm{Tr}\left[\hat{P}^\downarrow\;\sum_{n=0}^\infty\hat{Y}_n\right]\;,
\end{equation}
and the submajorization follows from the first part of Lemma <ref>.
The Fock rearrangement is continuous in the Hilbert-Schmidt norm.
Let $\hat{X}$ and $\hat{Y}$ be trace-class operators, with eigenvalues in decreasing order $\{x_n\}_{n\in\mathbb{N}}$ and $\{y_n\}_{n\in\mathbb{N}}$, respectively.
We then have
\begin{equation}
\left\|\hat{X}^\downarrow-\hat{Y}^\downarrow\right\|_2^2=\sum_{n=0}^\infty(x_n-y_n)^2\leq\left\|\hat{X}-\hat{Y}\right\|_2^2\;,
\end{equation}
where we have used Lemma <ref>.
§ FOCK-OPTIMAL QUANTUM OPERATIONS
We will prove that any gauge-covariant Gaussian quantum channel satisfies this property:
We say that a quantum operation $\Phi$ is Fock-optimal if for any positive trace-class operator $\hat{X}$
\begin{equation}\label{conjectureeq}
\Phi\left(\hat{X}\right)\prec_w\Phi\left(\hat{X}^\downarrow\right)\;,
\end{equation}
i.e. Fock-rearranging the input always makes the output less noisy, or among all the quantum states with a given spectrum, the passive one generates the least noisy output.
If $\Phi$ is trace-preserving, weak sub-majorization in (<ref>) can be equivalently replaced by majorization.
We can now state the main result of the paper:
Any one-mode gauge-covariant Gaussian quantum channel is passive-preserving and Fock-optimal.
See Section <ref>.
Any linear combination with positive coefficients of gauge-covariant quantum Gaussian channels is Fock-optimal.
Follows from Theorem <ref> and Lemma <ref>.
In the remainder of this section, we prove some general properties of Fock-optimality that will be needed in the main proof.
Let $\Phi$ be a passive-preserving quantum operation.
If for any finite-rank projector $\hat{P}$
\begin{equation}\label{hypproj}
\Phi\left(\hat{P}\right)\prec_w\Phi\left(\hat{P}^\downarrow\right)\;,
\end{equation}
then $\Phi$ is Fock-optimal.
Let $\hat{X}$ be a positive trace-class operator as in (<ref>), with Fock rearrangement as in (<ref>).
Since $\Phi$ is passive-preserving, for any $n\in\mathbb{N}$
\begin{equation}
\Phi\left(\hat{\Pi}_n\right)\prec_w\Phi\left(\hat{\Pi}_n^\downarrow\right)=\Phi\left(\hat{\Pi}_n^\downarrow\right)^\downarrow\;.
\end{equation}
Then we can apply Lemma <ref> to
\begin{equation}
\Phi\left(\hat{X}\right)=\sum_{n=0}^\infty d_n\;\Phi\left(\hat{\Pi}_n\right)\prec_w\sum_{n=0}^\infty d_n\;\Phi\left(\hat{\Pi}_n^\downarrow\right)=\Phi\left(\hat{X}^\downarrow\right)\;,
\end{equation}
and the claim follows.
A quantum operation $\Phi$ is passive-preserving and Fock-optimal iff
\begin{equation}\label{conjPQeq}
\mathrm{Tr}\left[\hat{Q}\;\Phi\left(\hat{P}\right)\right]\leq\mathrm{Tr}\left[\hat{Q}^\downarrow\;\Phi\left(\hat{P}^\downarrow\right)\right]
\end{equation}
for any two finite-rank projectors $\hat{Q}$ and $\hat{P}$.
Suppose first that $\Phi$ is passive-preserving and Fock-optimal, and let $\hat{P}$ and $\hat{Q}$ be finite-rank projectors.
\begin{equation}
\Phi\left(\hat{P}\right)\prec_w\Phi\left(\hat{P}^\downarrow\right)=\Phi\left(\hat{P}^\downarrow\right)^\downarrow\;,
\end{equation}
and (<ref>) follows from Lemma <ref>.
Suppose now that (<ref>) holds for any finite-rank projectors $\hat{P}$ and $\hat{Q}$.
Choosing $\hat{P}$ passive, we get
\begin{equation}
\mathrm{Tr}\left[\hat{Q}\;\Phi\left(\hat{P}\right)\right]\leq\mathrm{Tr}\left[\hat{Q}^\downarrow\;\Phi\left(\hat{P}\right)\right]\;,
\end{equation}
and from Lemma <ref> also $\Phi\left(\hat{P}\right)$ is passive, so from Lemma <ref> $\Phi$ is passive-preserving.
Choosing now a generic $\hat{P}$, by Lemma <ref>
\begin{equation}
\Phi\left(\hat{P}\right)\prec_w\Phi\left(\hat{P}^\downarrow\right)\;,
\end{equation}
and from Lemma <ref> $\Phi$ is also Fock-optimal.
We can now prove the two fundamental properties of Fock-optimality:
Let $\Phi$ be a quantum operation with the restriction of its Hilbert-Schmidt dual $\Phi^\dag$ to trace-class operators continuous in the trace norm.
Then, $\Phi$ is passive-preserving and Fock-optimal iff $\Phi^\dag$ is passive-preserving and Fock-optimal.
Condition (<ref>) can be rewritten as
\begin{equation}\label{PQdag}
\mathrm{Tr}\left[\Phi^\dag\left(\hat{Q}\right)\hat{P}\right]\leq\mathrm{Tr}\left[\Phi^\dag\left(\hat{Q}^\downarrow\right)\hat{P}^\downarrow\right]\;,
\end{equation}
and is therefore symmetric for $\Phi$ and $\Phi^\dag$.
Let $\Phi_1$ and $\Phi_2$ be passive-preserving and Fock-optimal quantum operations with the restriction of $\Phi_2^\dag$ to trace-class operators continuous in the trace norm.
Then, their composition $\Phi_2\circ\Phi_1$ is also passive-preserving and Fock-optimal.
Let $\hat{P}$ and $\hat{Q}$ be finite-rank projectors.
Since $\Phi_2$ is Fock-optimal and passive-preserving,
\begin{equation}
\Phi_2\left(\Phi_1\left(\hat{P}\right)\right)\prec_w\Phi_2\left(\Phi_1\left(\hat{P}\right)^\downarrow\right)=\Phi_2\left(\Phi_1\left(\hat{P}\right)^\downarrow\right)^\downarrow\;,
\end{equation}
and by Lemma <ref>
\begin{align}\label{eqphi12}
\mathrm{Tr}\left[\hat{Q}\;\Phi_2\left(\Phi_1\left(\hat{P}\right)\right)\right] &\leq \mathrm{Tr}\left[\hat{Q}^\downarrow\;\Phi_2\left(\Phi_1\left(\hat{P}\right)^\downarrow\right)\right]\nonumber\\
&= \mathrm{Tr}\left[\Phi_2^\dag\left(\hat{Q}^\downarrow\right)\Phi_1\left(\hat{P}\right)^\downarrow\right]\;.
\end{align}
Since $\Phi_1$ is Fock-optimal and passive-preserving,
\begin{equation}
\Phi_1\left(\hat{P}\right)^\downarrow\prec_w\Phi_1\left(\hat{P}^\downarrow\right)=\Phi_1\left(\hat{P}^\downarrow\right)^\downarrow\;.
\end{equation}
From Theorem <ref> also $\Phi_2^\dag$ is passive-preserving, and $\Phi_2^\dag\left(\hat{Q}^\downarrow\right)$ is passive.
Lemma <ref> implies then
\begin{align}\label{eqphi3}
\mathrm{Tr}\left[\Phi_2^\dag\left(\hat{Q}^\downarrow\right)\Phi_1\left(\hat{P}\right)^\downarrow\right] &\leq \mathrm{Tr}\left[\Phi_2^\dag\left(\hat{Q}^\downarrow\right)\Phi_1\left(\hat{P}^\downarrow\right)\right]\nonumber\\
&= \mathrm{Tr}\left[\hat{Q}^\downarrow\;\Phi_2\left(\Phi_1\left(\hat{P}^\downarrow\right)\right)\right]\;,
\end{align}
and the claim follows from Lemma <ref> combining (<ref>) with (<ref>).
Let $\Phi$ be a quantum operation continuous in the Hilbert-Schmidt norm.
Suppose that for any $N\in\mathbb{N}$ its restriction to the span of the first $N+1$ Fock states is passive-preserving and Fock-optimal, i.e. for any positive operator $\hat{X}$ supported on the span of the first $N+1$ Fock states
\begin{equation}
\Phi\left(\hat{X}\right)\prec_w\Phi\left(\hat{X}^\downarrow\right)=\Phi\left(\hat{X}^\downarrow\right)^\downarrow\;.
\end{equation}
Then, $\Phi$ is passive-preserving and Fock-optimal.
Let $\hat{P}$ and $\hat{Q}$ be two generic finite-rank projectors.
Since the restriction of $\Phi$ to the support of $\hat{\Pi}_N^\downarrow$ is Fock-optimal and passive-preserving,
\begin{align}
\Phi\left(\hat{\Pi}_N^\downarrow\;\hat{P}\;\hat{\Pi}_N^\downarrow\right) &\prec_w \Phi\left(\left(\hat{\Pi}_N^\downarrow\;\hat{P}\;\hat{\Pi}_N^\downarrow\right)^\downarrow\right)\nonumber\\ &=\left(\Phi\left(\left(\hat{\Pi}_N^\downarrow\;\hat{P}\;\hat{\Pi}_N^\downarrow\right)^\downarrow\right)\right)^\downarrow\;.
\end{align}
Then, from Lemma <ref>
\begin{equation}\label{TrPQN}
\mathrm{Tr}\left[\hat{Q}\;\Phi\left(\hat{\Pi}_N^\downarrow\;\hat{P}\;\hat{\Pi}_N^\downarrow\right)\right] \leq \mathrm{Tr}\left[\hat{Q}^\downarrow\;\Phi\left(\left(\hat{\Pi}_N^\downarrow\;\hat{P}\;\hat{\Pi}_N^\downarrow\right)^\downarrow\right)\right]\;.
\end{equation}
From Lemma <ref>,
\begin{equation}
\left\|\hat{\Pi}_N^\downarrow\;\hat{P}\;\hat{\Pi}_N^\downarrow-\hat{P}\right\|_2\to0\qquad\text{for}\;N\to\infty\;,
\end{equation}
and since $\Phi$, the Fock rearrangement (see Lemma <ref>) and the Hilbert-Schmidt product are continuous in the Hilbert-Schmidt norm, we can take the limit $N\to\infty$ in (<ref>) and get
\begin{equation}
\mathrm{Tr}\left[\hat{Q}\;\Phi\left(\hat{P}\right)\right] \leq \mathrm{Tr}\left[\hat{Q}^\downarrow\;\Phi\left(\hat{P}^\downarrow\right)\right]\;.
\end{equation}
The claim now follows from Lemma <ref>.
Let $\Phi_1$ and $\Phi_2$ be Fock-optimal and passive-preserving quantum operations.
Then, also $\Phi_1+\Phi_2$ is Fock-optimal and passive-preserving.
Easily follows from Lemma <ref>.
§ PROOF OF THE MAIN THEOREM
First, we can reduce the problem to the quantum-limited attenuator:
If the quantum-limited attenuator is passive-preserving and Fock-optimal, the property extends to any gauge-covariant quantum Gaussian channel.
From Lemma <ref>, any quantum gauge-covariant Gaussian channel can be obtained composing a quantum-limited attenuator with a quantum-limited amplifier.
From Lemma <ref>, the Hilbert-Schmidt dual of a quantum-limited amplifier is proportional to a quantum-limited attenuator, and from Lemma <ref> also the amplifier is passive-preserving and Fock-optimal.
Finally, the claim follows from Theorem <ref>.
By Lemma <ref>, we can restrict to quantum states $\hat{\rho}$ supported on the span of the first $N+1$ Fock states.
Let now
\begin{equation}
\hat{\rho}(t)=e^{t\mathcal{L}}\left(\hat{\rho}\right)\;,
\end{equation}
where $\mathcal{L}$ is the generator of the quantum-limited attenuator defined in (<ref>).
From the explicit representation (<ref>), it is easy to see that $\hat{\rho}(t)$ remains supported on the span of the first $N+1$ Fock states for any $t\geq0$.
In finite dimension, the quantum states with non-degenerate spectrum are dense in the set of all quantum states.
Besides, the spectrum is a continuous function of the operator, and any linear map is continuous.
Then, without loss of generality we can suppose that $\hat{\rho}$ has non-degenerate spectrum.
\begin{equation}
\end{equation}
be the vectors of the eigenvalues of $\hat{\rho}(t)$ in decreasing order, and let
\begin{equation}
s_n(t)=\sum_{i=0}^n p_i(t)\;,\qquad n=0,\ldots,\,N\;,
\end{equation}
their partial sums, that we similarly collect into the vector $s(t)$.
Let instead
\begin{equation}\label{pndt}
p_n^\downarrow(t)=\langle n|e^{t\mathcal{L}}\left(\hat{\rho}^\downarrow\right)|n\rangle\;,\qquad n=0,\,\ldots,\,N
\end{equation}
be the eigenvalues of $e^{t\mathcal{L}}\left(\hat{\rho}^\downarrow\right)$ (recall that it is diagonal in the Fock basis for any $t\geq0$), and
\begin{equation}
s_n^\downarrow(t)=\sum_{i=0}^n p_i^\downarrow(t)\;,\qquad n=0,\,\ldots,\,N\;,
\end{equation}
their partial sums.
Notice that $p(0)=p^\downarrow(0)$ and then $s(0)=s^\downarrow(0)$.
Combining (<ref>) with the expression for the Lindbladian (<ref>), with the help of (<ref>) and (<ref>) it is easy to see that the eigenvalues $p_n^\downarrow(t)$ satisfy
\begin{equation}
\frac{d}{dt}p_n^\downarrow(t)=\left(n+1\right)p_{n+1}^\downarrow(t)-n\,p_n^\downarrow(t)\;,
\end{equation}
\begin{equation}
\frac{d}{dt}s_n^\downarrow(t)=(n+1)\left(s^\downarrow_{n+1}(t)-s^\downarrow_n(t)\right)
\end{equation}
for their partial sums.
The proof of Theorem <ref> is a consequence of:
The spectrum of $\hat{\rho}(t)$ can be degenerate at most in isolated points.
$s(t)$ is continuous in $t$, and for any $t\geq0$ such that $\hat{\rho}(t)$ has non-degenerate spectrum it satisfies
\begin{equation}\label{sdot}
\frac{d}{dt}s_n(t)\leq(n+1)(s_{n+1}(t)-s_n(t))\;,\qquad n=0,\,\ldots,\,N-1\;.
\end{equation}
If $s(t)$ is continuous in $t$ and satisfies (<ref>), then
\begin{equation}
s_n(t)\leq s_n^\downarrow(t)
\end{equation}
for any $t\geq0$ and $n=0,\,\ldots,\,N$.
Lemma <ref> implies that the quantum-limited attenuator is passive-preserving.
Indeed, let us choose $\hat{\rho}$ passive.
Since $e^{t\mathcal{L}}\left(\hat{\rho}\right)$ is diagonal in the Fock basis, $s_n^\downarrow(t)$ is the sum of the eigenvalues corresponding to the first $n+1$ Fock states $|0\rangle,\;\ldots,\;|n\rangle$.
Since $s_n(t)$ is the sum of the $n+1$ greatest eigenvalues, $s_n^\downarrow(t)\leq s_n(t)$.
However, Lemma <ref> implies $s_n(t)=s_n^\downarrow(t)$ for $n=0,\,\ldots,\,N$.
Thus $p_n(t)=p_n^\downarrow(t)$, so the operator $e^{t\mathcal{L}}\left(\hat{\rho}\right)$ is passive for any $t$, and the channel $e^{t\mathcal{L}}$ is passive-preserving.
Then from the definition of majorization and Lemma <ref> again,
\begin{equation}
e^{t\mathcal{L}}\left(\hat{\rho}\right)\prec_w e^{t\mathcal{L}}\left(\hat{\rho}^\downarrow\right)
\end{equation}
for any $\hat{\rho}$, and the quantum-limited attenuator is also Fock-optimal.
§.§ Proof of Lemma <ref>
The matrix elements of the operator $e^{t\mathcal{L}}\left(\hat{\rho}\right)$ are analytic functions of $t$.
The spectrum of $\hat{\rho}(t)$ is degenerate iff the function
\begin{equation}
\phi(t)=\prod_{i\neq j}\left(p_i(t)-p_j(t)\right)
\end{equation}
This function is a symmetric polynomial in the eigenvalues of $\hat{\rho}(t)=e^{t\mathcal{L}}\left(\hat{\rho}\right)$.
Then, for the Fundamental Theorem of Symmetric Polynomials (see e.g Theorem 3 in Chapter 7 of <cit.>), $\phi(t)$ can be written as a polynomial in the elementary symmetric polynomials in the eigenvalues of $\hat{\rho}(t)$.
However, these polynomials coincide with the coefficients of the characteristic polynomial of $\hat{\rho}(t)$, that are in turn polynomials in its matrix elements.
It follows that $\phi(t)$ can be written as a polynomial in the matrix elements of the operator $\hat{\rho}(t)$.
Since each of these matrix element is an analytic function of $t$, also $\phi(t)$ is analytic.
Since by hypothesis the spectrum of $\hat{\rho}(0)$ is non-degenerate, $\phi$ cannot be identically zero, and its zeroes are isolated points.
§.§ Proof of Lemma <ref>
The matrix elements of the operator $e^{t\mathcal{L}}\left(\hat{\rho}\right)$ are analytic (and hence continuous and differentiable) functions of $t$.
Then for Weyl's Perturbation Theorem $p(t)$ is continuous in $t$, and also $s(t)$ is continuous (see e.g. Corollary III.2.6 and the discussion at the beginning of Chapter VI of <cit.>).
Let $\hat{\rho}(t_0)$ have non-degenerate spectrum.
Then, $\hat{\rho}(t)$ has non-degenerate spectrum for any $t$ in a suitable neighbourhood of $t_0$.
In this neighbourhood, we can diagonalize $\hat{\rho}(t)$ with
\begin{equation}
\hat{\rho}(t)=\sum_{n=0}^N p_n(t) |\psi_n(t)\rangle\langle\psi_n(t)|\;,
\end{equation}
where the eigenvalues in decreasing order $p_n(t)$ are differentiable functions of $t$ (see Theorem 6.3.12 of <cit.>),
\begin{equation}
\frac{d}{dt}p_n(t)=\langle\psi_n(t)|\mathcal{L}\left(\hat{\rho}(t)\right)|\psi_n(t)\rangle\;.
\end{equation}
We then have
\begin{equation}
\frac{d}{dt}s_n(t)=\mathrm{Tr}\left[\hat{\Pi}_n(t)\;\mathcal{L}\left(\hat{\rho}(t)\right)\right]\;,
\end{equation}
\begin{equation}
\hat{\Pi}_n(t)=\sum_{i=0}^n|\psi_i(t)\rangle\langle\psi_i(t)|\;.
\end{equation}
We can write
\begin{equation}
\hat{\rho}(t)=\sum_{n=0}^N d_n(t)\;\hat{\Pi}_n(t)\;,
\end{equation}
\begin{equation}
\end{equation}
so that
\begin{equation}
\frac{d}{dt}s_n(t)=\sum_{k=0}^N d_k(t)\;\mathrm{Tr}\left[\hat{\Pi}_n(t)\;\mathcal{L}\left(\hat{\Pi}_k(t)\right)\right]\;.
\end{equation}
With the explicit expression (<ref>) for $\mathcal{L}$, it is easy to prove that
\begin{equation}
\sum_{k=0}^N d_k(t)\;\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\mathcal{L}\left(\hat{\Pi}_k^\downarrow\right)\right]=(n+1)(s_{n+1}(t)-s_n(t))\;,
\end{equation}
so it would be sufficient to show that
\begin{equation}\label{PL}
\mathrm{Tr}\left[\hat{\Pi}_n(t)\;\mathcal{L}\left(\hat{\Pi}_k(t)\right)\right]\overset{?}{\leq} \mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\mathcal{L}\left(\hat{\Pi}_k^\downarrow\right)\right]\;.
\end{equation}
We write explicitly the left-hand side of (<ref>):
\begin{equation}\label{PLext}
\mathrm{Tr}\left[\hat{\Pi}_n(t)\;\hat{a}\;\hat{\Pi}_k(t)\;\hat{a}^\dag-\hat{\Pi}_n(t)\;\hat{\Pi}_k(t)\;\hat{a}^\dag\hat{a}\right]\;,
\end{equation}
where we have used that $\hat{\Pi}_n(t)$ and $\hat{\Pi}_k(t)$ commute.
* Suppose $n\geq k$.
\begin{equation}
\hat{\Pi}_n(t)\;\hat{\Pi}_k(t)=\hat{\Pi}_k(t)\;.
\end{equation}
Using that
\begin{equation}
\hat{\Pi}_n(t)\leq\hat{\mathbb{I}}
\end{equation}
in the first term of (<ref>), we get
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}_n(t)\;\hat{a}\;\hat{\Pi}_k(t)\;\hat{a}^\dag-\hat{\Pi}_n(t)\;\hat{\Pi}_k(t)\;\hat{a}^\dag\hat{a}\right]\leq0\;.
\end{equation}
On the other hand, since the support of $\hat{a}\,\hat{\Pi}_k^\downarrow\,\hat{a}^\dag$ is contained in the support of $\hat{\Pi}_{k-1}^\downarrow$, and hence in the one of $\hat{\Pi}_n^\downarrow$, we have also
\begin{equation}
\hat{\Pi}_n^\downarrow\;\hat{a}\;\hat{\Pi}_k^\downarrow\;\hat{a}^\dag=\hat{a}\;\hat{\Pi}_k^\downarrow\;\hat{a}^\dag\;,
\end{equation}
so that
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\hat{a}\;\hat{\Pi}_k^\downarrow\;\hat{a}^\dag-\hat{\Pi}_n^\downarrow\;\hat{\Pi}_k^\downarrow\;\hat{a}^\dag\hat{a}\right]=0\;.
\end{equation}
* Suppose now that $k\geq n+1$.
\begin{equation}
\hat{\Pi}_n(t)\;\hat{\Pi}_k(t)=\hat{\Pi}_n(t)\;.
\end{equation}
Using that
\begin{equation}
\hat{\Pi}_k(t)\leq\hat{\mathbb{I}}
\end{equation}
in the first term of (<ref>), together with the commutation relation (<ref>), we get
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}_n(t)\;\hat{a}\;\hat{\Pi}_k(t)\;\hat{a}^\dag-\hat{\Pi}_n(t)\;\hat{\Pi}_k(t)\;\hat{a}^\dag\hat{a}\right]\leq n+1\;.
\end{equation}
On the other hand, since the support of $\hat{a}^\dag\,\hat{\Pi}_n^\downarrow\,\hat{a}$ is contained in the support of $\hat{\Pi}_{n+1}^\downarrow$ and hence in the one of $\hat{\Pi}_k^\downarrow$, we have also
\begin{equation}
\hat{\Pi}_k^\downarrow\;\hat{a}^\dag\;\hat{\Pi}_n^\downarrow\;\hat{a}=\hat{a}^\dag\;\hat{\Pi}_n^\downarrow\;\hat{a}\;,
\end{equation}
so that
\begin{equation}
\mathrm{Tr}\left[\hat{\Pi}_n^\downarrow\;\hat{a}\;\hat{\Pi}_k^\downarrow\;\hat{a}^\dag-\hat{\Pi}_n^\downarrow\;\hat{\Pi}_k^\downarrow\;\hat{a}^\dag\hat{a}\right]=n+1\;.
\end{equation}
§.§ Proof of Lemma <ref>
Since the quantum-limited attenuator is trace-preserving, we have
\begin{equation}
\end{equation}
We will use induction on $n$ in the reverse order: suppose to have proved
\begin{equation}
s_{n+1}(t)\leq s_{n+1}^\downarrow(t)\;.
\end{equation}
We then have from (<ref>)
\begin{equation}
\frac{d}{dt}s_n(t)\leq(n+1)\left(s_{n+1}^\downarrow(t)-s_n(t)\right)\;,
\end{equation}
\begin{equation}
\frac{d}{dt}s_n^\downarrow(t)=(n+1)\left(s_{n+1}^\downarrow(t)-s_n^\downarrow(t)\right)\;.
\end{equation}
\begin{equation}
\end{equation}
we have $f_n(0)=0$, and
\begin{equation}
\frac{d}{dt}f_n(t)\geq-(n+1)f_n(t)\;.
\end{equation}
This can be rewritten as
\begin{equation}
\end{equation}
and implies
\begin{equation}
\end{equation}
§ GENERIC ONE-MODE GAUSSIAN CHANNELS
In this section we extend Theorem <ref> to any one-mode quantum Gaussian channel.
We say that two quantum channels $\Phi$ and $\Psi$ are equivalent if there are a unitary operator $\hat{U}$ and a unitary or anti-unitary $\hat{V}$ such that
\begin{equation}\label{Psi}
\Psi\left(\hat{X}\right)=\hat{V}\;\Phi\left(\hat{U}\;\hat{X}\;\hat{U}^\dag\right)\;\hat{V}^\dag
\end{equation}
for any trace-class operator $\hat{X}$.
Clearly, a channel equivalent to a Fock-optimal channel is also Fock-optimal with a suitable redefinition of the Fock rearrangement:
Let $\Phi$ be a Fock-optimal quantum channel, and $\Psi$ be as in (<ref>).
Then, for any positive trace-class operator $\hat{X}$,
\begin{equation}
\Psi\left(\hat{X}\right)\prec_w\Psi\left(\hat{U}^\dag\left(\hat{U}\;\hat{X}\;\hat{U}^\dag\right)^\downarrow\hat{U}\right)\;.
\end{equation}
The problem of analyzing any Gaussian quantum channel from the point of view of majorization is then reduced to the equivalence classes.
§.§ Quadratures and squeezing
In order to present such classes, we will need some more definitions.
The quadratures
\begin{eqnarray}
\hat{Q} &=& \frac{\hat{a}+\hat{a}^\dag}{\sqrt{2}}\\
\hat{P} &=& \frac{\hat{a}-\hat{a}^\dag}{i\sqrt{2}}
\end{eqnarray}
satisfy the canonical commutation relation
\begin{equation}
\left[\hat{Q},\;\hat{P}\right]=i\;\hat{\mathbb{I}}\;.
\end{equation}
In this section, and only in this section, $\hat{Q}$ and $\hat{P}$ will denote the above quadratures, and not generic projectors.
We can define a continuous basis of not normalizable vectors $\left\{|q\rangle\right\}_{q\in\mathbb{R}}$ with
\begin{eqnarray}
\hat{Q}|q\rangle &=& q|q\rangle\;,\\
\langle q|q'\rangle &=& \delta(q-q')\;,\\
\int_{\mathbb{R}}|q\rangle\langle q|\;dq &=& \hat{\mathbb{I}}\;,\\
e^{-iq\hat{P}}|q'\rangle &=& |q'+q\rangle\;,\qquad q,\,q'\in\mathbb{R}\;.
\end{eqnarray}
For any $\kappa>0$ we define the squeezing unitary operator <cit.> $\hat{S}_\kappa$ with
\begin{equation}
\hat{S}_\kappa |q\rangle=\sqrt{\kappa}\;|\kappa q\rangle
\end{equation}
for any $q\in\mathbb{R}$.
It satisfies also
\begin{equation}
\hat{S}_\kappa^\dag\;\hat{P}\;\hat{S}_\kappa = \frac{1}{\kappa}\;\hat{P}\;.
\end{equation}
§.§ Classification theorem
Then, the following classification theorem holds <cit.>:
Any one-mode quantum Gaussian channel is equivalent to one of the following:
* a gauge-covariant Gaussian channel as in Definition <ref> (cases $A_1)$, $B_2)$, $C)$ and $D)$ of <cit.>);
* a measure-reprepare channel $\Phi$ of the form
\begin{equation}\label{class2}
\Phi\left(\hat{X}\right)=\int_{\mathbb{R}}\langle q|\hat{X}|q\rangle\;e^{-iq\hat{P}}\;\hat{\rho}_0\;e^{iq\hat{P}}\;dq
\end{equation}
for any trace-class operator $\hat{X}$, where $\rho_0$ is a given Gaussian state (case $A_2)$ of <cit.>);
* a random unitary channel $\Phi_\sigma$ of the form
\begin{equation}\label{Phieta}
\Phi_\sigma\left(\hat{X}\right)=\int_{\mathbb{R}}e^{-iq\hat{P}}\;\hat{X}\;e^{iq\hat{P}}\;\frac{e^{-\frac{q^2}{2\sigma}}}{\sqrt{2\pi\sigma}}\;dq
\end{equation}
for any trace-class operator $\hat{X}$, with $\sigma>0$ (case $B_1)$ of <cit.>).
From Lemma <ref>, with a suitable redefinition of Fock rearrangement all the channels of the first class are Fock-optimal.
On the contrary, for both the second and the third classes the optimal basis would be an infinitely squeezed version of the Fock basis:
§.§ Class 2
We will show that the channel (<ref>) does not have optimal inputs.
Let $\hat{\omega}$ be a generic quantum state.
Since $\Phi$ applies a random displacement to the state $\hat{\rho}_0$,
\begin{equation}\label{PhiX0}
\Phi\left(\hat{\omega}\right)\prec\hat{\rho}_0\;.
\end{equation}
Moreover, $\Phi\left(\hat{\omega}\right)$ and $\hat{\rho}_0$ cannot have the same spectrum unless the probability distribution $\langle q|\hat{\omega}|q\rangle$ is a Dirac delta, but this is never the case for any quantum state $\hat{\omega}$, so the majorization in (<ref>) is always strict.
Besides, in the limit of infinite squeezing the output tends to $\hat{\rho}_0$ in trace norm:
\begin{align}
&=\left\|\int_{\mathbb{R}}\langle q|\hat{\omega}|q\rangle\left(e^{-i\kappa q\hat{P}}\;\hat{\rho}_0\;e^{i\kappa q\hat{P}}-\hat{\rho}_0\right)dq\right\|_1\nonumber\\
&\leq\int_{\mathbb{R}}\langle q|\hat{\omega}|q\rangle\left\|e^{-i\kappa q\hat{P}}\;\hat{\rho}_0\;e^{i\kappa q\hat{P}}-\hat{\rho}_0\right\|_1dq\;,
\end{align}
and the last integral tends to zero for $\kappa\to0$ since the integrand is dominated by the integrable function $2\langle q|\hat{\omega}|q\rangle$, and tends to zero pointwise.
It follows that the majorization relation
\begin{equation}
\Phi\left(\hat{S}_\kappa\;\hat{\omega}\;\hat{S}_\kappa\right)\prec\Phi\left(\hat{\omega}\right)
\end{equation}
will surely not hold for some positive $\kappa$ in a neighbourhood of $0$, and $\hat{\omega}$ is not an optimal input for $\Phi$.
§.§ Class 3
For the channel (<ref>), squeezing the input always makes the output strictly less noisy.
Indeed, it is easy to show that for any positive $\sigma$ and $\sigma'$
\begin{equation}
\Phi_\sigma\circ\Phi_{\sigma'}=\Phi_{\sigma+\sigma'}\;.
\end{equation}
Then, for any $\kappa>1$ and any positive trace-class $\hat{X}$
\begin{align}
\hat{S}_\kappa\;\Phi_\sigma\left(\hat{X}\right)\;\hat{S}_\kappa^\dag &= \Phi_{\kappa^2\sigma}\left(\hat{S}_\kappa\;\hat{X}\;\hat{S}_\kappa^\dag\right)\nonumber\\
\end{align}
hence, recalling that $\Phi$ applies a random displacement,
\begin{equation}
\Phi_\sigma\left(\hat{X}\right)\prec \Phi_{\sigma}\left(\hat{S}_\kappa\;\hat{X}\;\hat{S}_\kappa^\dag\right)\;.
\end{equation}
§ THE THINNING
The thinning <cit.> is the map acting on classical probability distributions on the set of natural numbers that is the discrete analogue of the continuous rescaling operation on positive real numbers.
In this Section we show that the thinning coincides with the restriction of the Gaussian quantum-limited attenuator to quantum states diagonal in the Fock basis, and we hence extend Theorem <ref> to the discrete classical setting.
The $\ell^1$ norm of a sequence $\{x_n\}_{n\in\mathbb{N}}$ is
\begin{equation}
\|x\|_1=\sum_{n=0}^\infty |x_n|\;.
\end{equation}
We say that $x$ is summable if $\|x\|_1<\infty$.
A discrete classical channel is a linear positive map on summable sequences that is continuous in the $\ell^1$ norm and preserves the sum, i.e. for any summable sequence $x$
\begin{equation}
\sum_{n=0}^\infty\left[\Phi(x)\right]_n=\sum_{n=0}^\infty x_n\;.
\end{equation}
The definitions of passive-preserving and Fock-optimal channels can be easily extended to the discrete classical case:
Given a summable sequence of positive numbers $\{x_n\}_{n\in\mathbb{N}}$, we denote with $x^\downarrow$ its decreasing rearrangement.
We say that a discrete classical channel $\Phi$ is passive-preserving if for any decreasing summable sequence $x$ of positive numbers $\Phi(x)$ is still decreasing.
We say that a discrete classical channel $\Phi$ is Fock-optimal if for any summable sequence $x$ of positive numbers
\begin{equation}\label{optimalcl}
\Phi(x)\prec\Phi\left(x^\downarrow\right)\;.
\end{equation}
Let us now introduce the thinning.
Let $N$ be a random variable with values in $\mathbb{N}$.
The thinning with parameter $0\leq\lambda\leq1$ is defined as
\begin{equation}
T_\lambda(N)=\sum_{i=1}^N B_i\;,
\end{equation}
where the $\{B_n\}_{n\in\mathbb{N}^+}$ are independent Bernoulli variables with parameter $\lambda$, i.e. each $B_i$ is one with probability $\lambda$, and zero with probability $1-\lambda$.
From a physical point of view, the thinning can be understood as follows:
consider a beam-splitter of transmissivity $\lambda$, where each incoming photon has probability $\lambda$ of being transmitted, and $1-\lambda$ of being reflected, and suppose that what happens to a photon is independent from what happens to the other ones.
Let $N$ be the random variable associated to the number of incoming photons, and $\{p_n\}_{n\in\mathbb{N}}$ its probability distribution, i.e. $p_n$ is the probability that $N=n$ (i.e. that $n$ photons are sent).
Then, $T_\lambda(p)$ is the probability distribution of the number of transmitted photons.
It is easy to show that
\begin{equation}\label{Tn}
\left[T_\lambda(p)\right]_n=\sum_{k=0}^\infty r_{n|k}\;p_k\;,
\end{equation}
where the transition probabilities $r_{n|k}$ are given by
\begin{equation}\label{rnk}
\end{equation}
and vanish for $k<n$.
The map (<ref>) can be uniquely extended by linearity to the set of summable sequences:
\begin{equation}\label{Tne}
\left[T_\lambda(x)\right]_n=\sum_{k=0}^\infty r_{n|k}\;x_k\;,\qquad \|x\|_1<\infty\;.
\end{equation}
The map $T_\lambda$ defined in (<ref>) is continuous in the $\ell^1$ norm and sum-preserving.
For any summable sequence $x$ we have
\begin{equation}
\sum_{n=0}^\infty\left|T_\lambda(x)\right|_n\leq\sum_{n=0}^\infty\sum_{k=0}^\infty r_{n|k}\;|x_k|=\sum_{k=0}^\infty|x_k|\;,
\end{equation}
where we have used that for any $k\in\mathbb{N}$
\begin{equation}
\sum_{n=0}^\infty r_{n|k}=1\;.
\end{equation}
Then, $T_\lambda$ is continuous in the $\ell^1$ norm.
An analogous proof shows that $T_\lambda$ is sum-preserving.
Let $\Phi_\lambda$ and $T_\lambda$ be the quantum-limited attenuator and the thinning of parameter $0\leq\lambda\leq1$, respectively.
Then for any summable sequence $x$
\begin{equation}
\Phi_\lambda\left(\sum_{n=0}^\infty x_n\;|n\rangle\langle n|\right)=\sum_{n=0}^\infty \left[T_\lambda(x)\right]_n\;|n\rangle\langle n|\;.
\end{equation}
Easily follows from the representation (<ref>), (<ref>) and (<ref>).
As easy consequence of Theorem <ref> and Theorem <ref>, we have
The thinning is passive-preserving and Fock-optimal.
§ CONCLUSIONS
We have proved that for any one-mode gauge-covariant bosonic Gaussian channel, the output generated by any state diagonal in the Fock basis and with decreasing eigenvalues majorizes the output generated by any other input state with the same spectrum.
Then, the input state with a given entropy minimizing the output entropy is certainly diagonal in the Fock basis and has decreasing eigenvalues.
The non-commutative quantum constrained minimum output entropy problem is hence reduced to a problem in classical discrete probability, that for the quantum-limited attenuator involves the thinning channel, and whose proof could exploit the techniques of the proof of the Restricted Thinned Entropy Power Inequality <cit.>.
Exploiting unitary equivalence we also extend our results to one-mode trace-preserving bosonic Gaussian channel which are not gauge-covariant, with the notable exceptions of those special maps admitting normal forms $A_2)$ and $B_1)$ <cit.> for which we show that no general majorization ordering is possible.
§ ACKNOWLEDGMENT
The Authors thank Andrea Mari, Luigi Ambrosio, Seth Lloyd and Alexander S. Holevo for comments and fruitful discussions.
GdP thanks G. Toscani and G. Savaré for the ospitality and the useful discussions in Pavia.
[]Giacomo De Palma
was born in Lanciano (CH), Italy, on March 15, 1990.
He received the B.S. degree in Physics and the M.S. degree in Physics from the University of Pisa, Pisa (PI), Italy, in 2011 and 2013, respectively. He also received the “Diploma di Licenza” in Physics from Scuola Normale Superiore, Pisa (PI), Italy, in 2014.
He is getting the Ph.D. degree in Physics at Scuola Normale Superiore in 2016.
His research interests include quantum information, quantum statistical mechanics and quantum thermodynamics.
He is author of nine scientific papers published in peer-reviewed journals.
[]Dario Trevisan
was born in the Province of Venice, Italy. He received the B.S. degree in mathematics from the University of Pisa, Pisa, Italy, in 2009, the M.S. degree in mathematics from the University of Pisa, in 2011, and the Ph.D. degree in mathematics from the Scuola Normale Superiore, Pisa, Italy, in 2014.
He is currently Assistant Professor at the University of Pisa.
Dr. Trevisan is a member of the GNAMPA group of the Italian National Institute for Higher Mathematics (INdAM).
[]Vittorio Giovannetti
was born in Castelnuovo di Garfagnana (LU) Italy, on April 1, 1970. He received the M.S. degree in Physics from the University of Pisa and PhD degree in theoretical Physics from the University of Perugia.
He is currently Associate Professor at the Scuola Normale Superiore of Pisa.
|
1511.00236
|
$^{1}$ Institut für Theoretische Physik II - Soft Matter
Heinrich-Heine-Universität Düsseldorf
Building 25.32
Room O2.56
Universitätsstrasse 1
D-40225 Düsseldorf, Germany
$^{2}$Departamento de Física, Facultad de Ciencias, Universidad Nacional
Autónoma de México,
Ciudad Universitaria, México D.F. 04510, Mexico
[email protected], [email protected]
We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of
circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly efficient as the radius of the ostacles tends to $0$, the so-called Boltzmann–Grad limit.
The 2D algorithm applies continued fractions to obtain the exact disc with which a particle will collide at each step, instead of using periodic boundary conditions as in the classical algorithm. The 3D version incorporates the 2D algorithm by projecting to the three coordinate planes.
As an application, we calculate distributions of free path lengths close to the Boltzmann–Grad limit for certain Lorentz gases.
We also show how the algorithms may be applied to deal with general crystal lattices.
§ INTRODUCTION
Lorentz gases are simple physical systems that present deterministic chaos <cit.>, and are a popular model in statistical mechanics and nonlinear dynamics. This model consists of point particles that move freely until they encounter obstacles, often spheres, where they undergo elastic collisions.
These systems can have different configurations of obstacles, e.g., random arrangements <cit.> or quasiperiodic structure <cit.>. However, due to its simplicity, the periodic case has been most widely studied; see, e.g., <cit.>. In this case, the model is equivalent to a Sinai billiard <cit.>. Many of the results obtained theoretically for these gases are in the limit where obstacles are very small, i.e., the so-called Boltzmann–Grad limit <cit.>. There are still many interesting open questions in this area
The standard simulation method for periodic Lorentz gases is to reduce to a single cell with periodic boundary conditions, and, in the simplest case, an obstacle in the centre of the cell
<cit.>. However, this requires that the program check in each cell whether the particle collides with the obstacle in the cell, or if it will move to the next cell.
If the obstacle is large, it is quite likely that the particle will collide each time it crosses into a new cell. However, for very small obstacles, this method becomes very inefficient.
Instead, we would like to just find the coordinates of the next obstacle with which the particle will collide, given its initial position and velocity.
This turns out to be closely related to the best rational approximant to an irrational number, and can be solved using the continued-fraction algorithm.
Continued fractions have often been used to provide information about the free path distribution of the periodic Lorentz gas in the Boltzmann–Grad limit <cit.>.
An algorithm along these lines was previously developed: see comments in <cit.>; however, it was never published
[T. Geisel, private communication].
Caglioti and Golse developed a method to encode the trajectories of particles using the continued fraction algorithm and the so-called 3-length theorem <cit.>.
However, Golse's algorithm works only if the particle leaves the surface of a disk. This restriction prevents the algorithm from being used in other geometries, such as two incommensurate overlapping arrays of square lattices, or with different shapes of obstacles; such systems may produce a number of surprising effects <cit.>.
On the other hand, due to the construction of Golse's algorithm, it is not possible to use it in higher dimensions, which is “a notoriously more difficult problem” <cit.>.
Recent advances on multidimensional continued-fraction algorithms may provide a possible future direction <cit.>, although here we have opted for a different approach for higher-dimensional systems.
In this paper, we develop an efficient algorithm to find a collision with a 2D square lattice of discs starting from an arbitrary initial condition. We then use that 2D algorithm as part of an efficient algorithm for a 3D simple cubic lattice by projecting onto coordinate planes. Finally, we show how obstacles arranged on arbitrary (periodic) crystal lattices may be treated.
§ CLASSICAL ALGORITHM FOR THE PERIODIC LORENTZ GAS
We begin by recalling the classical algorithm for a Lorentz gas on a $d$-dimensional (hyper)-cubic lattice, where each cell contains a single spherical obstacle of radius $r$. The simplest method is to locate the centre of the obstacle at the centre of a cubic cell $[-\frac{1}{2}, \frac{1}{2})^d$, and to track which cell $\vec{n} \in \mathbb{Z}^{d}$ the particle is in using periodic boundary conditions: when a particle hits a cell boundary, its position is reset to the opposite boundary and the cell counter $\vec{n}$ is updated accordingly; see Figure <ref>.
Reducing the dynamics in a periodic lattice to a single cell with periodic boundary conditions.
In each cell, the classical algorithm is as follows.
For a particle with initial position $\vec{x}$ and velocity $\vec{v}$, a collision occurs with the disc with centre at $\vec{c}$ and radius $r$ at a time $t^{\ast}$ if
\begin{equation}
\| \vec{x} + \vec{v} t^{\ast} - \vec{c} \| = r.
\end{equation}
This gives a quadratic equation for the collision time, and hence
\begin{equation}
t^{\ast} = -B - \sqrt{B^2 - C}
\label{eq:collision_time}
\end{equation}
\begin{equation}
B= \frac{(\vec{x} -\vec{c}) \cdot \vec{v}}{v^2}; \qquad
C= \frac{(\vec{x} - \vec{c})^2 -r^2}{v^2},
\end{equation}
provided that the condition $B^2 - C \ge 0$ is satisfied. If this happens, then the collision position is $\vec{x} + \vec{v} t^{\ast}$. If the condition is not satisfied, then the trajectory misses the disc.
If no collision with the obstacle occurs, i.e. when $B^{2} - C < 0$, the velocity is conserved and the particle will hit one of the cell boundaries. To determine which boundary will be hit, we find intersection times of the trajectory with each cell boundary (lines 2D or planes in 3D), given by
$$t_{i, \pm} = \frac{\pm \frac{1}{2} - x_i}{v_i},$$
where $i$ runs from 1 to the number of dimensions (2 or 3) and the sign corresponds to the two opposite faces in direction $i$. The least positive time then gives the collision time with the boundary. Depending on which boundary was hit, we move to the new unit cell and repeat the process: if $t_{i,\pm}$ is the minimum time, then the positive (resp. negative) $i$th boundary is hit, and the $i$th component of the cell is updated to $n_{i}' = n_{i} \pm 1$.
§ EFFICIENT 2D ALGORITHM
The classical algorithm is efficient for large radii $r$, but very inefficient once $r$ is small, since a trajectory will cross many cells before encountering a disc.
In this section, we develop an algorithm to simulate the periodic Lorentz gas on a 2D square lattice, based on the use of continued fractions, whose
goal is to calculate efficiently the first disc hit by a particle, even for very small values of the radius $r$. Without loss of generality, we will use the lattice formed by the integer coordinates in the 2D plane.
We wish to calculate the minimal time $t^{\ast}>0$ such that a collision occurs with some disc centred at $\vec{c} = (q,p)$, with $q$ and $p$ integers, by “jumping” straight to the correct disc; see Figure <ref>.
Cells covered by the classical algorithm, compared to the few steps required by the efficient algorithm.
§.§ Continued fraction algorithm: approximation of an irrational number by a rational
In this section we recall the continued fraction algorithm and some properties of continued fractions; see, e.g.,
<cit.> for proofs. The geometrical interpretation has been suggested before by many other authors; see, for example, <cit.>.
A continued fraction is obtained via an iterative process, representing a number $\alpha$ as the sum of its integer part, $a_0$, and the reciprocal of another number, $\alpha_1:=\alpha-a_0$, then writing $\alpha_1$ as the sum of its integer part, $a_1$, and the reciprocal of $\alpha_2:=\alpha_1-a_1$, and so on. This gives the continued fraction representation of $\alpha$:
\begin{equation*}
\alpha = a_0 + \frac{1}{\displaystyle a_1
+ \frac{1}{\displaystyle a_2
+ \frac{1}{\displaystyle a_3 + \dots}}}
\end{equation*}
This iteration produces a sequence of integers $\lfloor \alpha \rfloor=a_0$, $\lfloor \alpha_1 \rfloor=a_1$, $\lfloor \alpha_2 \rfloor=a_2$, etc.
We define inductively two sequences of integers $\{ p_n\}$ and $\{ q_n\}$ as follows:
\begin{eqnarray}
p_{-2} = 0; \quad p_{-1} = 1; \quad p_i &=a_i p_{i-1}+p_{i-2};
\label{eq:sucesion1}
\\
q_{-2} = 1, \quad q_{-1} = 0, \quad q_i &=a_i q_{i-1}+q_{i-2}.
\label{eq:sucesion2}
\end{eqnarray}
With this sequence we can approximate any irrational number $\alpha$ using the Hurwitz theorem:
For any irrational number, $\alpha$, all the relative prime integers $p_n$, $q_n$ of the sequences defined in equations (<ref>) and (<ref>) satisfy
\begin{equation}
|\alpha- \frac{p_n}{q_n}|\leq \frac{1}{{q_n}^2}.
\end{equation}
§.§ Collision with a disc
The classical algorithm finds the intersection between a line, corresponding to the trajectory of
the particle, and a circle, corresponding to the circumference of the disc, by solving the quadratic equation (<ref>) for $t^{\ast}$. A first improvement follows from observing that we may instead look for the intersection of the trajectory with another line, as follows. In the following, we take $v_1>0$ and $v_2>0$; by symmetry of the system, we can always rotate or reflect it such that these conditions are satisfied.
We write the equation of the particle's trajectory as $y=\alpha x+b$, with slope $\alpha = v_{2} / v_{1}$, and look for its intersection with the vertical line $x=q$ passing through the disc at $(q,p)$. As shown in figure <ref>, if $|\alpha q+b-p| < \delta := r/v_1$, then a collision with the disc $(q,p)$ will occur. Due to the periodic boundary conditions, we can redefine $b := \{|\alpha q+b-p|\}$ where $\{\cdot\}$ denotes the fractional part. Thus, $0 < b < 1$, and we need only solve $b<\delta$.
We do not need to apply periodic boundary conditions at every step; rather, we only need to check
\begin{equation}
|\{ \alpha q_n \}+b -1|< \delta,
\label{eq:master}
\end{equation}
where $\{ \cdot \}$ denotes the fractional part, and $q_n=q_{n-1}+1$, where $q_1$ is the $x$-coordinate of the closest obstacle to the particle at $t=0$. Then, the first $q_n$ that satisfies this inequality will be $q$. To calculate $p$, we use that either $p=\lfloor \alpha q +b\rfloor$ or $p=\lfloor \alpha q +b\rfloor+1$.
Relation between the intersection of a line and a circle with integer coordinates and the intersection of the line $x = q$.
Now, to simplify the algorithm further, consider the integer coordinates $(q_n, p_n)$ such that
\begin{equation}
|\alpha q_n -p_n + b|< \delta,
\label{eq:1}
\end{equation}
and for any pair of numbers $(i,j)$ such that $i<q_n$, then $|\alpha i -j+ b|> \delta$, $q=q_n$, and $p=p_n$.
But $|\alpha q_i - p_i + b|$ are the distances between the integer coordinates $(q_i, p_i)$ and the point $( q_i ,\alpha q_i + b)$. Thus, we would like a sequence such that
\begin{equation}
|\alpha q_i - p_i + b|<|\alpha q_{i-1} - p_{i-1} + b|
\label{eq:iteration}
\end{equation}
for every integer $i>1$. Also, the first pair of integer coordinates $q_0$ and $p_0$ should be $(0, 0)$ or $(0, 1)$, minimizing $| \alpha q_0 - p_0 + b |$, that is
\begin{eqnarray}
%|\alpha q_1 -p_1 + b|< f(b) = \begin{cases} b &\mbox{if } b < 1/2 \\
%1-b & \mbox{if } b > 1/2 \end{cases} .
|\alpha q_1 -p_1 + b|< f(b) =&
\left\{
\begin{array}{@{}l@{\quad}l}
b, & \mbox{if } b < \frac{1}{2} \\[\jot]
1 - b, & \mbox{if } b > \frac{1}{2}.
\end{array}
\right.
\label{eq:prima}
\end{eqnarray}
Note that if $b < 1/2$, we have
$p_n= \lfloor \alpha q_n +b \rfloor= \lfloor \alpha q_n \rfloor $, if $b+\alpha q_n-\lfloor \alpha q_n \rfloor < 1$, and
$p_{n} = \lfloor \alpha q_n \rfloor+1$, if $b+\alpha q_n-\lfloor \alpha q_n \rfloor > 1$.
Whereas if $b>1/2$, we have $p_n= \lfloor \alpha q_n +b \rfloor+1= \lfloor \alpha q_n \rfloor+1$, if $b+\alpha q_n-\lfloor \alpha q_n \rfloor < 1$ and
$ p_{n} = \lfloor \alpha q_n \rfloor+2$, if $b+\alpha q_n-\lfloor \alpha q_n \rfloor > 1$. Substituting these four cases in the two cases of equation (<ref>), we obtain that indeed $p_1= \lfloor \alpha q_1 \rfloor+1$. Iterating the inequality (<ref>) we obtain
\begin{equation}
p_n= \lfloor \alpha q_n \rfloor+1.
\label{eq:hn}
\end{equation}
Combining the inequality (<ref>) with equation (<ref>), we obtain again equation (<ref>).
Thus, we have reduced the solution from two linear equations and one quadratic to one linear equation.
Furthermore, now we do not check in every periodic cell, because if $\alpha >1$,
for every $q_n$ we advance $(\lfloor q_{n}\alpha \rfloor -\lfloor q_{n-1}\alpha \rfloor)$ cells.
And we do not need to apply periodic boundary conditions until we reach the obstacle.
§.§ The Diophantine inequality: $|\alpha p - q|\leq \delta$
Now, a better algorithm should find a way to find the set of $q_i$, such that inequality (<ref>) holds for every $i$, and there is no integer $q$ such that $q_i<q<q_{i-1}$ for some $i$ and
$|\{ \alpha q_i \}+b -1|<|\{ \alpha q \}+b -1| <|\{ \alpha q_{i-1} \}+b -1|$.
In order to do this, we can use the continued fraction algorithm to obtain solutions to the inequality $|\alpha q - p|\leq \delta$. This algorithm already gives a sequence of $(q_n,p_n)$ such that $|\alpha q_i - p_i|<|\alpha q_{i-1} - p_{i-1}|$ if $q_{i-1} <q_i$. In addition, the convergents of the continued fractions provide best approximants and hence the smallest solution of the inequality (<ref>).
So, if we turn our inequality (<ref>) into this other inequality, we will find our algorithm just by using the continued fraction algorithm.
Indeed, using equation (<ref>) and the inequality (<ref>), we obtain $|\{ \alpha q_1 \} -1|< 2b$ if $b < 1/2$ or $< 2(1-b)$ if $b > 1/2$,
which is almost the continued fraction inequality, except that $p_1$ is always equal to $\lfloor \alpha q_1 \rfloor+1$.
\begin{eqnarray}
| \alpha q - p | <
\left\{
\begin{array}{@{}l@{\quad}l}
& 2b, \quad \mbox{if } b < \frac{1}{2} \\
& 2(1-b), \quad \mbox{if } b > \frac{1}{2}.
\end{array}
\right.
\label{eq:master2}
\end{eqnarray}
Now we can apply the continued fraction algorithm to obtain $p_1$ and $q_1$ of inequality (<ref>). If ${\alpha q_1+b} < \delta$ or $1-{\alpha q_1+b} < \delta$, then we have found the center of the obstacle at $(p_1,q_1)$, with $p_1=\lfloor \alpha q_1 \rfloor+1$; otherwise, we have not found it, but we know that if the center of collision is at $(p,q)$ then
$p\geq p_1$, and $q \geq q_1$. Hence, we can just use $(q_1,p_1)$ even if they do not satisfy inequality (<ref>). Redefining $b_i$ as $b_0=b$, $b_i=\{\alpha q_i+b\}$, we can calculate a succession of $(p_i,q_i)$. If $b_n<\delta$ the algorithm stops, and the collision will take place with the obstacle centred at the coordinates $(q_n,p_n)$.
Otherwise, if $b_n = b$, then the particle has a rational slope equal to that of a channel, and so is travelling along and parallel to that channel, and hence will never undergo another collision with an obstacle. In this case, the algorithm throws an exception.
§.§ Complete 2D algorithm
We now have the necessary tools to implement the algorithm. Pseudo-code for the complete efficient 2D algorithm is given in the Appendix; source code for our implementation, written in the Julia programming language
may be found in the supplementary information.
The functions described above work only for velocities in the first octant, i.e. such that $0 < v_{2} < v_{1}$. If the velocity does not satisfy this condition, we use the symmetry of the system, applying rotations and reflections and then, after obtaining the
coordinates of the collision, use the inverse transformations to return to the original system; see the Appendix for details.
Finally, to calculate the exact collision point, we use the classical algorithm to obtain the intersection between a line and a circle, and from there the resulting post-collision velocity.
§ EFFICIENT 3D ALGORITHM
We now develop an efficient algorithm for calculating the next collision with a sphere in 3D on a simple cubic lattice, which again is designed to be efficient for a small radius $r$. The algorithm works by projecting the geometry onto the 2D coordinate planes and then using the above efficient 2D algorithm in each plane, as follows.
Suppose we project a particle trajectory in a 3D lattice onto one of the $x$–$y$, $x$–$z$ or
$y$–$z$ planes. We will obtain a periodic square lattice with a 2D trajectory. This trajectory is not a trajectory of the 2D Lorentz gas, however – it may pass through certain discs as if they were not there, and may have non-elastic reflections with other discs. Furthermore, the speed varies.
However, we will use this to apply the 2D algorithm for the projections in each plane, in order to obtain coordinates of a disc in each of the three planes at which the first collision in that plane is predicted to occur.
We now check whether the obstacle coordinates in these projections correspond to the same 3D obstacle, i.e. if the coordinates coincide pairwise. If not, then we have not found a true collision in 3D.
We move the particle to the cell containing the obstacle that is furthest away, i.e., has the maximum collision time in its respective plane, and continue.
If the obstacle coordinates do coincide pairwise, then this algorithm predicts that there is a collision. However, this may not be true, due to the geometry, as follows.
Calculating a collision with a disc in one of the planes $x$–$y$, $x$–$z$ or $y$–$z$ is equivalent to calculating a collision in space with a cylinder orthogonal to that plane. Joining these coordinates together means calculating a collision with the intersection of three orthogonal cylinders with the same radius.
Figure <ref> shows such an intersection of three cylinders, called a tricylinder or Steinmetz solid <cit.>, together with a sphere of the same radius. The sphere is contained inside the intersection of the cylinders, and has a smaller volume.
A sphere of radius $r$ embedded into the intersection of three orthogonal cylinders of the same radius. The volume inside the intersection but outside the sphere is the region where the 3D algorithm predicts false collisions.
Thus the algorithm may predict a false collision – with the tricylinder – even though the particle does not collide with the sphere. To control this, we check if the particle really does collide with this obstacle by using the classical algorithm; if so, then we have found a true collision, and if not, we move the particle to the next cell and continue applying the algorithm.
Numerically, we find the probability of a false collision to be around $0.17$.
Pseudo-code for the efficient 3D algorithm is given in the Appendix.
§ NUMERICAL MEASUREMENTS
§.§ Execution time
In order to test the efficiency of our algorithms, we measure the average execution time of the function that finds the first collision, starting from an initial point near the origin,
as a function of obstacle radius, for both the classical and efficient algorithms, in 2D and 3D.
Mean execution time to find the first collision in the 2D square Lorentz gas, for the classical (dotted curve) and efficient (solid curve) algorithms. The straight lines show power-law fits.
Mean execution time to find the first collision in the 3D simple cubic Lorentz gas, for the classical (dotted curve) and efficient (solid curve) algorithms. The straight lines show power-law fits.
Figures <ref> and <ref> show the results for the 2D and 3D algorithms, respectively. We performed power-law fits for the execution time as a function of obstacle radius. For the 2D case, we find an exponent of $-1.01$ for the classical algorithm and $-0.20$ for the efficient algorithm. For the 3D case, the exponents are $-2.25$ and $-1.20$ for classical and efficient, respectively. As we can see, our algorithms are increasingly more efficient for $r < 0.01$.
Similarly, we calculated the execution time per cell as a function of the obstacle radius, for both the 2D and 3D efficient algorithms, with comparison to the corresponding classical algorithms; see Figures <ref> and <ref>. Since the classical algorithms use periodic boundary conditions, the time per cell is basically constant, independent of the obstacle radius.
For small radii, we again observe the efficiency of the new algorithms.
Mean execution time per cell to find the first collision in a 2D square Lorentz gas, for the classical (dashed curve) and efficient (solid curve) algorithms, as a function of disc radius,
Mean execution time per cell to find the first collision in the 3D cubic Lorentz gas, for the classical (dashed curve) and efficient (solid curve) algorithms, as a function of sphere radius, $r$.
§.§ Asymptotic complexity of the classical and efficient algorithms
The scaling of the complexity of the classical algorithm as $1/r$ may be explained as follows.
The distance that a particle travels before it collides with an obstacle, i.e. the free path length, is a function of obstacle size: the smaller the obstacles, the longer the free paths.
In periodic Lorentz gases, there is a simple formula for the mean free path between consecutive collisions, $\tau$, that arises from geometrical considerations <cit.>: it is, up to a dimension-dependent constant, the ratio of the volume of the available space outside the obstacles to the surface area of the obstacles. For the square 2D Lorentz gas with discs of radius $r$, we have
\begin{equation}
\tau(r) = \frac{1 - \pi r^{2}}{2r},
\end{equation}
with asymptotics $r^{-1}$ for small $r$. Since the classical algorithm must cross this distance at speed $1$, it takes time proportional to $1/r$, as we find numerically. In applying this algorithm, approximately $1/r$ quadratic equations and four times as many linear equations must be solved.
For the simple cubic Lorentz gas in 3D with spheres of radius $r$, we have
\begin{equation}
\tau(r) = \frac{1 - \frac{4}{3} \pi r^{3}}{\pi r^2},
\end{equation}
with asymptotics $r^{-2}$, which is not far from our numerical results.
On the other hand, the efficient algorithm checks only one quadratic equation, and around
$2r^{-1/2}$ linear equations, giving an upper bound of $r^{-1/2}$ for the complexity. (This calculation uses Hurwitz's theorem.) Numerically, it turns out to be significantly more efficient than that.
§.§ Free flight distribution
As an example application of our algorithm, we measure the distribution of free flight lengths for the first collision for certain systems studied by Marklof and Strömbergsson <cit.>. They studied $N$ incommensurable, overlapping periodic Lorentz gases in the Boltzmann–Grad limit, $r \to 0$, and proved that the asymptotic decay of the probability density for free flights in that system is $\sim \ell^{-N-2}$. It follows that the asymptotic density of the first free flight should be
$\rho(\ell) \sim \ell^{-N-1}$.
Figure <ref> shows our numerical results for this distribution in the case of two and three overlapping lattices, compared to the asymptotic decay given by the rigorous result of <cit.>.
To obtain this plot, we fixed the radius as $r=10^{-4}$ and calculated free flights for a given initial condition for a 2D lattice, and for the same lattice rotated by $\pi/5$ and $\pi/7$, respectively. The first free flight for each lattice is calculated separately, and the minimum of those results is then taken to give the first free flight for the superposition of either two or three incommensurable lattices.
The distributions obtained numerically do indeed follow the power laws predicted. Naturally, it becomes increasingly difficult to obtain the asymptotic behaviour of the densities as the number of lattices increases.
Probability density of the first free flight for two and three incommensurable, overlapping periodic Lorentz gases with angles $\pi/5$ and $\pi/7$; a total of $10^{8}$ initial conditions was used. The results for a single lattice are shown for comparison. The dashed lines and labels show the theoretical asymptotics.
§ EXTENSION TO GENERAL PERIODIC LATTICES
So far, we have restricted attention to spherical obstacles on simple cubic lattices. In this section, we will show how to deal with arbitrary periodic crystal lattices. Such lattices consist of a basis (finite collection) of different spheres (atoms), in unit cells of a Bravais lattice; see, e.g., <cit.>.
This may be considered as the superposition of distinct Bravais lattices, one for each of the distinct atoms in the basis. Thus the efficient algorithm may be used separately for each such lattice, and then we take the minimum time to determine the next collision. In this way, we can now restrict attention to simulating a Bravais lattice with a single atom per unit cell. For simplicity we will describe the method in 2D; the 3D case is similar.
A Bravais lattice in 2D is the set of points given by linear combinations of the form $a_{1} \vec{u}_{1} + a_{2} \vec{u}_{2}$ of vectors $\vec{u}_{i}$ defining the directions of the lattice, where the $a_{i}$ are integers. We pass from the square lattice to the oblique lattice by applying the transformation matrix
$\mathsf{M}_\mathrm{so}$, defined such that its columns are the vectors $\vec{u}_{i}$:
\begin{equation}
\mathsf{M}_\mathrm{so} := (\vec{u}_1 | \vec{u}_2).
\end{equation}
To transform back from the Bravais lattice to the square lattice, we apply the inverse transformation
$\mathsf{M}_\mathrm{os} := \mathsf{M}_\mathrm{so}^{-1}$.
Starting from circular obstacles of radius $r$ in the Bravais lattice and applying $\mathsf{M}_\mathrm{os}$ gives one obstacle per unit cell at integer coordinates in the square lattice. However, this stretches the shape of the resulting obstacles into ellipses, as follows from the singular-value decomposition (SVD) of $\mathsf{M}_\mathrm{os}$; see, e.g., <cit.>.
The semi-major axis of the resulting ellipses is $r' = r \sigma_1$, where $\sigma_1$ is the first singular value of $\mathsf{M}_\mathrm{os}$.
We circumscribe the resulting ellipse by a circular obstacle of radius $r'$, giving a standard square periodic Lorentz gas, suitable for analysis using the corresponding efficient 2D algorithm; see Figure <ref>.
Starting from a given initial condition $\vec{x}_{0}$, $\vec{v}_0$ in the Bravais lattice we wish to simulate, we transform these to
$\vec{x}'_{0} := \mathsf{M}_\mathrm{os} \cdot \vec{x}_{0}$ and
$\vec{v}'_{0} := \mathsf{M}_\mathrm{os} \cdot \vec{v}_{0}$
in the square lattice.
We then apply the efficient algorithm in the square lattice to obtain a proposed disc or sphere with integer coordinates $\vec{n}$. These coordinates are mapped to the oblique lattice, giving a proposed disc or sphere with coordinates $\vec{n}' := \mathsf{M}_\mathrm{so} \cdot \vec{n}$. We must check, however, if this is a true collision with the obstacle at $\vec{n}'$ using the classical algorithm, since the proposed collision with a disc in the square lattice may not actually hit the true elliptical obstacle there. If it is not a true collision, then we move to the next cell and continue; if it is a true collision, we calculate the new post-collision velocity.
Provided the transformation $\mathsf{M}_\mathrm{so}$ does not stretch the obstacles too much, and the radius is small, this algorithm will still be very efficient.
Effect of applying the transformation $\mathsf{M}_\mathrm{os}$ to an oblique lattice of discs (left). The result is a square lattice of ellipses; circumscribed circles are also shown (right).
Finally, non-spherical obstacles may be dealt with in a similar way, using a circumscribed circular or spherical obstacle. In this way, we may simulate completely general crystal lattice structures.
§ CONCLUSIONS
We have introduced efficient algorithms to simulate periodic Lorentz gases in two and three dimensions, that work particularly well when the obstacles are small. We have compared the efficiency of these algorithms with the standard ones, showing that the relative efficiency indeed increases very fast in 2D and fast in 3D, and we have shown a sample application to calculate free flight distributions near the Boltzmann–Grad limit.
We have also shown how to extend our methods to arbitrary crystal lattices.
The extension of the 3D algorithm to higher dimensions and applications are in progress.
§ ACKNOWLEDGEMENTS
We thank Michael Schmiedeberg for useful comments and discussions about the algorithms, and the anonymous referees for their insightful remarks.
ASK received support from the DFG within the Emmy Noether program (grant Schm 2657/2).
NK is the recipient of a DGAPA-UNAM postdoctoral fellowship.
DPS acknowledges financial support from CONACYT grant CB-101246 and DGAPA-UNAM PAPIIT grants IN116212 and IN117214.
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1511.00252
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§ INTRODUCTION
The light-quark baryons, namely, the nonstrangeness $N^*$ and $\Delta^*$ baryons
and the $\Lambda^*$ and $\Sigma^*$ hyperons with strangeness $S=-1$,
provide rich information about QCD in the nonperturbative domain.
A variety of hadron models, such as constituent quark models <cit.> and
models based on Dyson-Schwinger equations <cit.>,
have been proposed to calculate the mass spectrum and form factors of light-quark baryons
and to clarify the role of the confinement and chiral symmetry breaking of QCD
in understanding of the properties of light-quark baryons.
Also, the real energy spectrum of QCD
under the (anti)periodic boundary condition
has been computed recently for the light-quark baryon sector
within the lattice QCD framework (see, e.g., Refs. <cit.>).
A critical nature of excited baryons is that they are unstable against
the strong interaction and exist only as resonance states in hadron reactions.
Poles of scattering amplitudes in the complex-energy plane
are identified as resonance states,
and the use of a multichannel reaction framework
is necessary for properly extracting information on
such resonance states from the reaction data.
In fact, a number of analysis groups including us have performed
comprehensive analyses of the $\pi N$ and $\gamma N$ reaction data
by making use of sophisticated multichannel approaches, such as
the on-shell $K$-matrix approaches (e.g., Refs. <cit.>) and
the dynamical-model approaches (e.g., Refs. <cit.>),
and they have successfully extracted the parameters
(complex pole masses and residues, etc.) associated
with $N^*$ and $\Delta^*$ resonances defined by poles of scattering amplitudes.
Among those studies, the ones with dynamical-model approaches
have further revealed the crucial role of (multichannel) reaction dynamics
in understanding the mass spectrum, structure, and dynamical origin of
baryon resonances (see, e.g., Refs. <cit.>).
Similar studies based on a dynamical-model approach have also performed recently
by us for the $\Lambda^*$ and $\Sigma^*$ sector <cit.>.
In this contribution, we give an overview of our recent efforts on the spectroscopy of
light-quark baryons, which is based on the so-called ANL-Osaka dynamical coupled-channels
(DCC) approach.
§ ANL-OSAKA DCC MODEL
The basic formula of our DCC approach is the coupled-channels integral equations obeyed by
the partial-wave amplitudes for $a \to b$ reactions <cit.>
(here we explain our approach by taking the $N^*$ and $\Delta^*$ sector as an example):
\begin{equation}
T^{(J^P I)}_{b,a} (p_b,p_a;W) =
V^{(J^PI)}_{b,a} (p_b,p_a;W)
+\sum_c \int_C dqq^2 V^{(J^PI)}_{b,c} (p_b,q;W) G_c(q;W) T^{(J^PI)}_{c,a} (p_c,p_a;W).
\label{lseq}
\end{equation}
Here, $p_a$ is the magnitude of the relative momentum for the channel $a$
in the center-of-mass frame;
$W$ is the total scattering energy; the superscripts $(J^PI)$ represent the total angular momentum
$J$, parity $P$, and isospin $I$ of the partial wave; and
the subscripts ($a,b,c$) represent the considered reaction channels (the indices associated
with the total spin and orbital angular momentum of the channels are suppressed).
For the $N^*$ and $\Delta^*$ sector, we have taken into account
the eight channels, $\gamma^{(*)}N$, $\pi N$, $\eta N$, $K\Lambda$, $K\Sigma$, $\pi\Delta$, $\rho N$, and $\sigma N$, where the last three are the quasi two-body channels
that subsequently decay into the three-body $\pi\pi N$ channel.
Meson-Baryon Green's functions $G_c(q;W)$.
For the quasi two-body channels, $\pi\Delta$, $\rho N$, and $\sigma N$,
The three-body $\pi \pi N$ cuts are produced in the intermediate processes as
indicated with the red lines.
The figure is from Ref. <cit.>.
Transition potentials $V_{b,a}^{(J^PI)}(p_b,p_a;W)$.
The three-body $\pi \pi N$ cuts are produced in the $Z$-potentials,
as indicated with the red lines.
The figure is from Ref. <cit.>.
The diagrammatic representation of the Green's functions $G_c(q;W)$ and
the transition potentials $V^{(J^PI)}_{b,a} (p_b,p_a;W)$ are
presented in Figs. <ref> and <ref>, respectively.
Here, the Green's functions for the quasi two-body channels (the right two diagrams
in Fig. <ref>) and the $Z$-potentials (the middle diagrams in Fig. <ref>)
produce the three-body $\pi \pi N$ cut in the intermediate processes,
and the implementation of both contributions is necessary for maintaining the three-body unitarity.
The s-channel processes mediated by the bare $N^*$ and $\Delta^*$ states
are also included in our DCC model.
Those bare states couple to the reaction channels through the reaction processes,
and then become resonance states.
Furthermore, the iterative processes of the exchange potentials
can also produce resonance poles dynamically.
Our model contains both possibilities in a consistent way.
By solving the coupled-channels integral equation (<ref>),
we can sum up all possible transition processes
between the considered reaction channels, and this ensures the multichannel two-body
as well as three-body unitarity for the resulting amplitudes.
Furthermore, off-shell rescattering effects are also taken into account explicitly
through the momentum integral in Eq. (<ref>), which are usually neglected
in the on-shell approaches.
To extract resonance parameters from the scattering amplitudes given by Eq. (<ref>),
one needs to make an analytic continuation of the amplitudes to the (lower half of)
complex energy plane.
This can be accomplished by appropriately changing the path of momentum integral $C$ in
Eq. (<ref>).
See Refs. <cit.> for the details of the analytic continuation
method employed for our analysis.
The DCC model for the $\Lambda^*$ and $\Sigma^*$ sector with strangeness $S=-1$
can be constructed in the same way as the $N^*$ and $\Delta^*$ sector by replacing
the reaction channels with $\bar K N$, $\pi \Sigma$, $\pi\Lambda$, $\eta \Lambda$,
$\pi\Sigma^*$, and $\bar K^* N$, where the last two are the quasi two-body channels
for the three-body $\pi\pi\Sigma$ and $\pi\bar K N$ channels, respectively,
and by modifying the Green's functions and transition potentials appropriately.
§ $N^*$ AND $\DELTA^*$ SPECTROSCOPY THROUGH COMPREHENSIVE ANALYSIS OF
$\PI N$, $\GAMMA N$, AND $E N$ REACTIONS
Differential cross sections for $\pi^- p \to K^0 \Sigma^0$.
Red solid curves are the preliminary results obtained from the current ongoing analysis,
while blue dashed curves are from the latest published analysis <cit.>.
The numbers shown in each panel are the corresponding total scattering energy $W$ in MeV.
See Ref. <cit.> for references of the data.
(Left) Differential cross sections for $\gamma p \to \pi^0 p$.
(Right) Differential cross sections for $\gamma p \to K^+ \Lambda$.
The numbers shown in each panel are the corresponding total scattering energy $W$ in MeV.
See Ref. <cit.> for references of the data.
Our latest published model <cit.> for the $N^*$ and $\Delta^*$ sector
was constructed by performing a comprehensive analysis of unpolarized differential cross sections
and polarization observables for the $\pi N \to \pi N, \eta N, K \Lambda, K \Sigma$ and
$\gamma p \to \pi N, \eta N, K \Lambda, K \Sigma$ reactions.
The constructed model covers the energy range from the threshold
up to $W = 2.3$ GeV for the $\pi N$ scattering and up to $W = 2.1$ GeV for the other reactions.
A couple of results of our fits are presented in Figs. <ref> and <ref>.
We have been updating our reaction model since the last publication <cit.>,
and in the figures the blue dashed curves represent the published results,
while the red solid curves represent the current updated ones.
Some improvements are actually seen in several kinematical regions,
particularly at low energies of $\pi^- p \to K^0 \Sigma^0$ (Fig. <ref>)
and at forward angles of pion and kaon photoproductions (Fig. <ref>).
Mass spectra for $N^*$ and $\Delta^*$ resonances.
Real parts of the resonance pole masses $M_R$ are plotted.
Also, only the resonances with $0 < -{\rm Im}(M_R) < 0.2$ GeV are presented.
The results are from (Red) ANL-Osaka (ours) <cit.>,
(Blue) Jülich <cit.>, and (Green) Bonn-Gatchina <cit.>.
The spectrum of four- and three-star resonances rated by PDG <cit.>
is also presented with the red and blue filled squares, respectively,
of which the length in the longitudinal direction represents
the range of the real parts of the resonance pole masses assigned by PDG.
In Fig. <ref>, the mass spectra for the $N^*$ and $\Delta^*$
resonances extracted by multichannel analysis groups are presented.
Our spectrum <cit.> shown in red are compared with
the ones extracted by the Jülich group in 2013 (blue) <cit.>
and the Bonn-Gatchina group in 2012 (green) <cit.>,
and also with the four- and three-star resonances assigned by PDG <cit.>.
The results show that the existence and mass values of low-lying resonances
have been well determined for most of the spin-parity states.
Thus establishing the spectrum of high-mass resonances
will be a next important task in the $N^*$ and $\Delta^*$ spectroscopy.
Here it is noted that the Jülich group has updated their mass spectrum recently, and it can be found in Ref. <cit.>.
The high-mass $N^*$ and $\Delta^*$ resonances are expected to couple strongly to the three-body $\pi\pi N$ channel.
This can be seen from the partial decay widths evaluated within an earlier version of our 8-channel DCC analysis
(see Fig. 6 of Ref. <cit.>),
which actually shows that the high-mass resonances decay dominantly to the $\pi\pi N$ channel.
It is worth mentioning that the second $P_{33}$ resonance, $\Delta(1600)3/2^+$
in the notation of PDG <cit.>, also has a large partial decay width
to the $\pi\pi N$ channel.
The double-pion production data are therefore
a key essential to establishing high-mass resonances
as well as the Roper-like state of the $\Delta$ baryon.
However, so far essentially no differential cross section data
that can be used for the detailed partial-wave analysis
were available for the $\pi N \to \pi \pi N$ reactions at high energies,
and this was a problem for the $N^*$ and $\Delta^*$ spectroscopy.
But now the situation is being improved by HADES <cit.> and J-PARC <cit.>.
In particular, with the J-PARC E45 experiment <cit.>, it is expected that
the world data of the $\pi N\to \pi\pi N$ reactions is increased by a factor of 100 or more.
Another important task in the $N^*$ and $\Delta^*$ spectroscopy is
to determine electromagnetic transition form factors
between the nucleon and the $N^*$ or $\Delta^*$ resonance.
By studying the form factors, we could see how the transition
between the effective degrees of freedom describing baryons
occurs with changes in $Q^2$.
To determine the $Q^2$ dependence of the form factors, one needs
to analyze meson electroproduction reactions.
So far, we have made such analyses for the data for $p(e,e'\pi)N$
up to $Q^2 = 1.5$ GeV$^2$ within our previous 6-channel DCC model <cit.>,
and, more recently, up to $Q^2 = 3$ GeV$^2$ within our latest 8-channel DCC model <cit.>.
Structure functions $\sigma_T + \epsilon \sigma_L$ at several $Q^2$ and $W$ values..
The left (right) three columns are for $ep \to e'\pi^0p$ ($ep \to e'\pi^+n$).
The structure function data are from Refs. <cit.>.
In Fig. <ref>, a couple of results of our recent analysis for the $p(e,e'\pi)N$ data
performed in Ref. <cit.> are presented.
Here we have used the structure functions as the data to analyze <cit.>,
rather than the original five-fold differential cross sections.
The results capture an overall shape of the structure functions data,
but it is still not sufficient for the purpose of $N^*$ form factor studies.
Here it is noted that the analysis in Ref. <cit.> is dedicated for studying the neutrino
reactions, and therefore our model parameters are not fine-tuned
for the purpose of studying $N^*$ and $\Delta^*$ transition form factors.
More elaborated analysis of single pion electroproductions is ongoing for the $Q^2$
region up to 6 GeV$^2$, and the results will be presented elsewhere.
The $Q^2$ dependence of the $M1$ form factor, $G^*_M(Q^2)$, for
the $\gamma^* p \to \Delta(1232)3/2^+$ transition
evaluated at the pole position of $\Delta(1232)3/2^+$.
Filled (open) circles are the real parts of $G^*_M(Q^2)$ obtained from
our 8-channel <cit.> (6-channel <cit.>) model,
while filled (open) triangles are the corresponding imaginary parts.
The diamonds are extracted by experimental groups using the Breit-Wigner
parametrization <cit.>.
$G_D$ is a dipole factor, $G_D = [1+(Q^2/\Lambda^2)]^{-2}$ with $\Lambda = 0.71$ GeV.
In Fig. <ref>, the extracted $M1$ transition form factors
between the nucleon and $\Delta(1232)3/2^+$ resonance are presented.
Here it is noted that in our analyses the transition form factors are
evaluated at the pole positions of the resonances, and thus they inevitably become
complex because of the fact that resonances are decaying particles.
This is in contrast to the form factors extracted by experiment groups,
where the phenomenological Breit-Wigner parametrizations are used
and the extracted values are real.
We find that for the $\Delta(1232)3/2^+$ case, the imaginary parts of the form factors
are small and the Breit-Wigner results seem close to the real parts of the form factors
defined by poles.
However, for the higher resonances, the imaginary parts can be comparable
with the real parts, and in such cases the correspondence between the form factors
defined by poles and by the Breit-Wigner parametrizations becomes unclear.
The clarification of those differences requires further investigations.
Recently, we have also made an analysis of the data for the single pion
photoproduction off the “neutron” target (Fig. <ref>).
Analysis of both proton- and “neutron”-target photoproductions is
necessary for decomposing the electromagnetic currents into the isoscalar and isovector currents
and determining the electromagnetic interactions of the $N^*$ resonances that have isospin $1/2$.
It is noted that such isospin currents are also
necessary for studying neutrino-induced reactions (see e.g., Refs. <cit.>).
Currently we use the “neutron”-target data extracted by other analysis groups from
the deuteron-target reactions.
However, in the future we need to analyze the deuteron-reaction data directly and
extract the $\gamma n \to N^*$ helicity amplitudes in a fully consistent way in our approach.
Differential cross sections (left) and photon asymmetries (right)
for $\gamma n \to \pi^- p$.
The numbers shown in each panel are the corresponding total scattering energy $W$ in MeV.
The data are taken from Ref. <cit.>.
§ $\LAMBDA^*$ AND $\SIGMA^*$ SPECTROSCOPY THROUGH
COMPREHENSIVE ANALYSIS OF $K^-P$ REACTIONS
The $Y^*$ ($= \Lambda^*, \Sigma^*$) resonances are much less understood than the $N^*$ and $\Delta^*$ resonances.
This can be seen, for example, from the fact that for the $Y^*$ resonances
only the so-called Breit-Wigner masses and widths had been listed by PDG before 2012 <cit.>.
This was a rather unsatisfactory situation <cit.> because the Breit-Wigner parameters are nothing more than
“approximation” of the resonance parameters defined by poles of scattering amplitudes
in the complex energy plane <cit.>,
where the latter has a clear physical meaning: the resonance states defined by poles are associated with
the exact (complex) energy eigenstates of the full Hamiltonian of the system under the purely outgoing
boundary condition (see, e.g., Refs. <cit.>).
In this situation, we have recently made a comprehensive partial-wave analysis of
the available $K^- p$ reaction data within our DCC approach <cit.>.
This was accomplished by developing a DCC model for strangeness $S=-1$ sector, which takes into account
couplings between the two-body $\bar K N$, $\pi\Sigma$, $\pi\Lambda$, $\eta\Lambda$, and $K\Xi$ channels and
the three-body $\pi \pi \Lambda$ and $\pi \bar K N$ channels that have resonant components of
$\pi \Sigma^*$ and $\bar K^* N$, respectively.
The model parameters are then determined by fitting to all available data of
$K^- p \to \bar K N,\pi\Sigma,\pi\Lambda,\eta\Lambda,K\Xi$ reactions from the threshold up to $W=2.1$ GeV.
Our analysis includes the data of both unpolarized and polarized observables, and this results in fitting more than 17,000 data points.
From this analysis, we have successfully determined the partial-wave amplitudes not only for $S$ wave but also $P$, $D$ and $F$ waves,
and also extracted the $Y^*$ mass spectrum defined by poles of scattering amplitudes.
The full details of the analysis and the extracted $Y^*$ resonance parameters can be found in Refs. <cit.>, and in the following
we will present a highlight of them.
Extracted mass spectra of $\Lambda^*$ resonances.
Here only the resonances of which complex pole mass has a value satisfying
$m_{\bar K} + m_N \leq {\rm Re}(M_R) \leq 2.1$ GeV and $0\leq-{\rm Im}(M_R) \leq 0.2$ GeV, are presented [$m_{\bar K}$ ($m_N$) is the antikaon (nucleon) mass].
The mass spectra extracted from our two analyses,
Model A (red) and Model B (blue) constructed in Ref. <cit.>,
are compared with the one from the KSU analysis <cit.> (green).
The Breit-Wigner masses and widths of the four- and three-star resonances rated by PDG <cit.> (black) are also presented.
The well-determined resonances are enclosed with the orange dashed circles.
In Fig. <ref>, we compare the mass spectra of $\Lambda^*$ resonances extracted from our analysis <cit.>
and the analysis by the Kent State University (KSU) group <cit.>.
In our analysis, we found two distinct solutions that have quite different values for our model parameters, yet both give similar quality of
the fits to the $K^-p$ reaction data included in our analysis.
We call them Model A and Model B, and
their resulting mass spectra are presented in red and blue, respectively.
In the same figure, the spectrum of four- and three-star
resonances assigned by PDG is also presented.
However, it is noted that the mass spectra of our two models
and the KSU analysis are the ones given as poles of
scattering amplitudes, while the PDG values are of the Breit-Wigner masses and widths.
We see that the spectra extracted from our two models and the KSU analysis
show an excellent agreement for several resonances,
but in overall, they are still fluctuating between the three analyses.
For example, a $J^P=3/2^+$ $\Lambda$ resonance with a mass
${\rm Re}(M_R)\sim 1.86$ GeV
is found in Model A and the KSU analysis, while not in Model B
(panel for $J^P=3/2^+$ spectra of Fig. <ref>).
If this resonance corresponds to the four-star $\Lambda(1890)3/2^+$ of PDG,
then this may be one example showing that a four-star resonance rated by PDG
using the Breit-Wigner parameters is not confirmed by the analyses
in which the resonance parameters are extracted at pole positions.
As already discussed and emphasized in Refs. <cit.>,
this kind of analysis dependence would originate
from the fact that the existing $K^-p$ reaction data are not sufficient
to eliminate such dependence on the extracted mass spectrum.
(Left) Total cross section for $K^- p \to \eta \Lambda$.
Red solid and blue dashed curves are from Model A and Model B <cit.>, respectively.
(Right) Total cross section for $\pi^- p \to \eta n$, where the curve is from our published DCC model for $\pi N$ and $\gamma N$ reactions <cit.>.
We can see from the lower left-most panel of Fig. <ref>
(spectra for $J^P=1/2^-$ $\Lambda$ resonances) that
all of the three analyses find a narrow $J^P=1/2^-$ $\Lambda$ resonance located close to
the $\eta \Lambda$ threshold, $W \sim 1.67$ GeV: $M_R = 1669^{+3}_{-8} -i (9^{+9}_{-1})$ MeV
for Model A, $M_R = 1667^{+1}_{-2} -i (12^{+3}_{-1})$ MeV for Model B, and
$M_R = 1667 -i 13$ MeV for the KSU analysis.
This resonance is known as $\Lambda(1670)1/2^-$ and found to be responsible
for the sharp peak in the $K^- p \to \eta \Lambda$ total cross section near the threshold (left panel of Fig. <ref>).
This behavior looks similar to $N(1535)1/2^-$ in
the $\pi N \to \eta N$ reaction (right panel of Fig. <ref>),
where the contribution from $N(1535)1/2^-$ dominates the peak of the $\pi N \to \eta N$ total cross section near the threshold.
It is also interesting to see that Model B has another very narrow resonance
with $J^P =3/2^+$ and $M_R = 1671^{+2}_{-8}-i(5^{+11}_{-2})$ MeV
(see the panel for $J^P=3/2^+$ spectra in Fig. <ref>),
which has almost the same ${\rm Re}(M_R)$ value as $\Lambda(1670)1/2^-$.
However, currently this resonance is found only in Model B.
Actually, in Model A the peak of the $K^- p \to \eta \Lambda$ total cross section near the threshold
is completely dominated by $\Lambda(1670)1/2^-$ [Fig. <ref>(a)], while in Model B, about 40% of the magnitude
of the peak is turned out to come from this narrow $P$-wave $J^P=3/2^+$ $\Lambda$ resonance [Fig. <ref>(b)].
Since both models reproduce the total cross section well, it is hard to judge whether
this new $P$-wave $\Lambda$ resonance should exist or not, as far as looking at the total cross section only.
However, we can get a deeper insight by looking at differential cross sections.
The lower panels of Fig. <ref> show the differential cross section of $K^- p \to \eta \Lambda$ at 1672 MeV,
which corresponds to the peak energy of the total cross section near the threshold.
We see that the differential cross section data show a clear concave-up angular dependence,
which cannot be described by the $S$-wave amplitudes.
In fact, we find that Model A, for which the total cross section is dominated by the $S$ wave,
does not reproduce the angular dependence well.
On the other hand, in Model B, the new $P$-wave $J^P=3/2^+$ $\Lambda$ resonance is responsible for the reproduction of the data,
suggesting that this angular dependence of the data seems to favor this new resonance.
Total cross section near the threshold (upper panels) and differential cross section at $W = 1672$ MeV (lower panels)
for $K^- p \to \eta \Lambda$
Left panels (right panels) are the results from Model A (Model B).
Solid curves are the full results, while the dashed curves are the results for which
the contribution from the $P_{03}$ partial wave is turned off.
The dashed curves are almost dominated by the $S_{01}$ partial wave that contains $\Lambda(1670)1/2^-$.
For Model B, the difference between the solid and dashed curves is due to
the new narrow $P$-wave $J^P = 3/2^+$ $\Lambda$ resonance.
§ SUMMARY AND PROSPECTS
We have performed comprehensive partial-wave analysis for the data of various meson production reactions off the nucleon
within the ANL-Osaka DCC approach.
We then have successfully extracted the resonance parameters associated with
the light-quark baryons ($N^*$, $\Delta^*$, $\Lambda^*$, $\Sigma^*$), which are defined
by poles of scattering amplitudes in the complex energy plane.
We may say that a recent progress on the light-quark baryon spectroscopy
triggered by multichannel analysis groups is quite remarkable.
However, a visible analysis dependence still exists in the extracted resonance parameters.
To eliminate such dependence, one would need not only to make further improvements of the analysis methods
of each analysis group, but also to have more extensive and accurate data of meson production reactions
including the polarization observables (see, e.g., Ref. <cit.>).
Regarding this, there were a lot of contributions on the experimental activities at
the electron, photon, and hadron beam facilities to this NSTAR2015 workshop, and
a variety of new or planned experiments were reported.
With the help of these experiments, we would be able to make further
progress towards understanding nonperturbative nature of the low energy QCD.
Finally, the framework of our DCC approach itself is quite general, and it has been applied not only to
the light-quark baryon spectroscopy, but also to the neutrino-induced reactions <cit.> associated with
the neutrino-oscillation experiments in the multi-GeV region and the meson spectroscopy <cit.>.
We plan to put more efforts into these directions, too.
The author would like to thank T.-S. H. Lee, S. X. Nakamura, and T. Sato for their collaborations.
This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 25800149.
The author also acknowledges the support of the HPCI Strategic Program (Field 5
“The Origin of Matter and the Universe”) of Ministry of Education, Culture, Sports, Science
and Technology (MEXT) of Japan.
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This is in contrast to the $N^*$ and $\Delta^*$ cases,
for which the resonance parameters defined by poles have also been
extensively studied by a number of multichnal analysis groups, and
both the pole and Breit-Wigner results have been listed by PDG for three decades.
This “approximation” often does not work well for the resonances that have a large inteference with backgrounds and/or
are not well isolated from the other resonance poles and the singularities (branch points etc.)
in the complex energy plane.
For example, the mass of the Roper resonance is often referred to as $\sim$1440 MeV.
However, it is the value obtained within the Breit-Wigner parametrization.
In the pole definition, (the real part of) the mass of the Roper resonance is $\sim$1370 MeV, i.e., $\sim 70$ MeV lower than
the Breit-Wigner mass.
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1511.00331
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Construction of multi-default models with full viability[a working version]
Shiqi Song
Laboratoire Analyse et Probabilités
Université d'Evry Val D'Essonne, France
We have the following idea why a financial market is not complete. In fact, initially any market is fair, smooth and complete, just like the Black-Scholes model. Only over the time, successive default events and crisis deteriorate the market condition. The market persists, but becomes unpredictable and incomplete. A natural way to model this evolution is to begin with a fair information flow $\mathbb{F}$, and then to expand $\mathbb{F}$ successively with random times $\tau_{1},\ldots, \tau_{n}$ (multi-default time model), where the random times $\tau_{1},\ldots, \tau_{n}$ are chosen so that the market with expanded information flow remains viable.
Notice that there does not exist many classes of random times with which successive expansion can be made with viability. One thinks naturally the class of honest times. But it is not a good idea, because honest times in general destroy the market viability (cf. <cit.>). One can take the class of initial times introduced in <cit.> to do the successive expansion as in <cit.>. Such an expansion may preserve the viability because of Jacod's criterion. But in this paper, we try to do the successive expansion with a new class of random times, namely the random times of $\natural$-models introduced in <cit.>.
§ NOTATIONS AND VOCABULARY
We employ the vocabulary of stochastic calculus as defined in <cit.> with the following specifications.
Probability space and random variables
A stochastic basis $(\Omega, \mathcal{A},\mathbb{P},\mathbb{F})$ is a quadruplet, where $(\Omega, \mathcal{A},\mathbb{P})$ is a probability space and $\mathbb{F}$ is a filtration of sub-$\sigma$-algebras of $\mathcal{A}$, satisfying the usual conditions.
The relationships involving random elements are always in the almost sure sense. For a random variable $X$ and a $\sigma$-algebra $\mathcal{F}$, the expression $X\in\mathcal{F}$ means that $X$ is $\mathcal{F}$-measurable. The notation $\mathbf{L}^p(\mathbb{P},\mathcal{F})$ denotes the space of $p$-times $\mathbb{P}$-integrable $\mathcal{F}$-measurable random variables.
An element $v$ in an Euclidean space $\mathbb{R}^d$ ($d\in\mathbb{N}^*$) is considered as a vertical vector. We denote its transposition by $\transp v$. The components of $v$ will be denoted by $v_h, 1\leq h\leq d$. The vectors are always defined as vertical vectors. Let $u=(u_{1},\ldots,u_{d})$ and $v=(v_{1},\ldots,v_{d})$ be two vectors. We denote by $\transp u v$ the inner product $u_{1}v_{1}+\ldots+u_{d}v_{d}$. We denote by $u\vdots v$ the vector composed of $u_{i}v_{i}, 1\leq i\leq d$.
The processes
The jump process of a càdlàg process $X$ is denoted by $\Delta X$, whilst the jump at time $t\geq 0$ is denoted by $\Delta_tX$. By definition, $\Delta_0X=0$ for any càdlàg process $X$. When we call a process $A$ a process having finite variation, we assume automatically that $A$ is càdlàg. We denote then by $\mathsf{d}A$ the (signed) random measure that $A$ generates.
We deal with finite family of real processes $X=(X_h)_{1\leq h\leq d}$. It will be considered as $d$-dimensional vertical vector valued process. The value of a component $X_h$ at time $t\geq 0$ will be denoted by $X_{h,t}$. When $X$ is a semimartingale, we denote by $[X,\transp X]$ the $d\times d$-dimensional matrix valued process whose components are $[X_i,X_j]$ for $1\leq i,j\leq d$.
The projections
With respect to a filtration $\mathbb{F}$, the notation ${^{\mathbb{F}\cdot p}}\bullet$ denotes the predictable projection, and the notation $\bullet^{\mathbb{F}\cdot p}$ denotes the predictable dual projection.
The martingales and the semimartingales
Fix a probability $\mathbb{P}$ and a filtration $\mathbb{F}$. For any $(\mathbb{P},\mathbb{F})$ special semimartingale $X$, we can decompose $X$ in the form (see <cit.>) :
\dcb
\dce
where $X^m$ is the martingale part of $X$ and $X^v$ is the drift part of $X$, $X^c$ is the continuous martingale part, $X^{da}$ is the part of compensated sum of accessible jumps, $X^{di}$ is the part of compensated sum of totally inaccessible jumps. We recall that this decomposition of $X$ depends on the reference probability and the reference filtration. We recall that every part of the decomposition of $X$, except $X_0$, is assumed null at $t=0$.
The stochastic integrals
In this paper we employ the notion of stochastic integral only about the predictable processes. The stochastic integral are defined as 0 at $t=0$. We use a point $\centerdot$ to indicate the integrator process in a stochastic integral. For example, the stochastic integral of a real predictable process ${H}$ with respect to a real semimartingale $Y$ is denoted by ${H}\centerdot Y$, while the expression $\transp{K}(\centerdot[X,\transp X]){K}$ denotes the process
\int_0^t \sum_{i=1}^k\sum_{j=1}^k{K}_{i,s}{K}_{j,s} \mathsf{d}[X_i,X_j]_s,\ t\geq 0,
where ${K}$ is a $k$-dimensional predictable process and $X$ is a $k$-dimensional semimartingale. The expression $\transp{K}(\centerdot[X,\transp X]){K}$ respects the matrix product rule. The value at $t\geq 0$ of a stochastic integral will be denoted, for example, by $\transp{K}(\centerdot[X,\transp X]){K}_t$.
The notion of the stochastic integral with respect to a multi-dimensional local martingale $X$ follows <cit.>. We say that a (multi-dimensional) $\mathbb{F}$ predictable process is integrable with respect to $X$ under the probability $\mathbb{P}$ in the filtration $\mathbb{F}$, if the non decreasing process $\sqrt{\transp{H}(\centerdot[X,\transp X]){H}}$ is $(\mathbb{P},\mathbb{F})$ locally integrable. For such an integrable process ${H}$, the stochastic integral $\transp{H}\centerdot X$ is well-defined and the bracket process of $\transp{H}\centerdot X$ can be computed using <cit.>. Note that two different predictable processes may produce the same stochastic integral with respect to $X$. In this case, we say that they are in the same equivalent class (related to $X$).
The notion of multi-dimensional stochastic integral is extended to semimartingales. We refer to <cit.> for details.
§ FULL VIABILITY
§.§ Definition
A financial market is modeled by a triplet $(\mathbb{P},\mathbb{F},S)$ of a probability measure $\mathbb{P}$ on a measurable space $(\Omega,\mathcal{A})$, of an information flow $\mathbb{F}=(\mathcal{F}_t)_{t\in\mathbb{R}_+}$ (a filtration of sub-$\sigma$-algebra in $\mathcal{A}$), and of an $\mathbb{F}$ asset process $S$ (a multi-dimensional $\mathbb{F}$ special semimartingale with strictly positive components). The notion of viability has been defined in <cit.> for a general economy. This notion is then used more specifically to signify that the utility maximization problems have solutions in <cit.>. The viability is closely linked to the absences of arbitrage opportunity (of some kind) as explained in <cit.> so that the word sometimes is employed to signify no-arbitrage condition. In this paper, this notion will be involved in a setting of information flow expansion. Let $\mathbb{G}=(\mathcal{G}_t)_{t\geq 0}$ be a second filtration. We say that $\mathbb{G}$ is an expansion (or an enlargement) of the filtration $\mathbb{F}$, if $\mathcal{F}_t\subset\mathcal{G}_t$, and then, we write $\mathbb{F}\subset \mathbb{G}$.
Let $T>0$ be an $\mathbb{F}$ stopping time. We call a strictly positive $\mathbb{F}$ adapted real process $Y$ with $Y_0=1$, a local martingale deflator on the time horizon $[0,T]$ for a (multi-dimensional) $(\mathbb{P},\mathbb{F})$ special semimartingale $S$, if the processes $Y$ and $Y S$ are $(\mathbb{P},\mathbb{F})$ local martingales on $[0,T]$. The same notion can be defined for the filtration $\mathbb{G}$.
We recall that the existence of local martingale deflators is equivalent to the no-arbitrage conditions and (cf. <cit.>). We know that, when the no-arbitrage condition is satisfied, the market is viable, and vice versa (cf. <cit.>). For this reason, we introduce the following definition.
(Full viability on $[0,T]$ for the expansion $\mathbb{F}\subset \mathbb{G}$) Let $T$ be a $\mathbb{G}$ stopping time. We say that the expansion $\mathbb{F}\subset \mathbb{G}$ is fully viable on $[0,T]$ under $\mathbb{P}$, if, for any $\mathbb{F}$ asset process $S$ possessing a $(\mathbb{P},\mathbb{F})$ deflator, the process $S$ possesses a deflator in the expanded market environment $(\mathbb{P},\mathbb{G})$ on the time horizon $[0,T]$.
The full viability implies Hypothesis $(H')$ on $[0,T]$.
§.§ Enlargements of filtrations and Hypothesis$(H')$
Consider the two filtrations $\mathbb{F}=(\mathcal{F}_t)_{t\geq 0}$ and $\mathbb{G}=(\mathcal{G}_t)_{t\geq 0}$ such that $\mathcal{F}_t\subset\mathcal{G}_t$. Let $T$ be a $\mathbb{G}$ stopping time. We introduce the Hypothesis$(H')$ (cf. <cit.>):
(Hypothesis$(H')$ on the time horizon $[0,T]$) We say that Hypothesis$(H')$ holds for the expansion pair $\mathbb{F}\subset \mathbb{G}$ on the time horizon $[0,T]$ under the probability $\mathbb{P}$, if all $(\mathbb{P},\mathbb{F})$ local martingale is a $(\mathbb{P},\mathbb{G})$ semimartingale on $[0,T]$.
Whenever Hypothesis$(H')$ holds, the associated drift operator can be defined (cf. <cit.>).
Suppose hypothesis$(H')$ on $[0,T]$. Then there exists a linear map $\Gamma$ from the space of all $(\mathbb{P},\mathbb{F})$ local martingales into the space of càdlàg $\mathbb{G}$-predictable processes on $[0,T]$, with finite variation and null at the origin, such that, for any $(\mathbb{P},\mathbb{F})$ local martingale $X$, $\widetilde{X}:=X-\Gamma(X)$ is a $(\mathbb{P},\mathbb{G})$ local martingale on $[0,T]$. Moreover, if $X$ is a $(\mathbb{P},\mathbb{F})$ local martingale and $H$ is an $\mathbb{F}$ predictable process and $X$-integrable (in $(\mathbb{P},\mathbb{F})$), then $H$ is $\Gamma(X)$-integrable and $\Gamma(H\centerdot X)=H\centerdot \Gamma(X)$ on $[0,T]$. The operator $\Gamma$ will be called the drift operator.
§.§ Drift multiplier assumption
In this paper we work especially with the drift operators having the following property.
(Drift multiplier assumption) We say that the drift multiplier assumption holds for the expansion $\mathbb{F}\subset \mathbb{G}$ on $[0,T]$ under $\mathbb{P}$, if
* Hypothesis$(H')$ is satisfied for the expansion $\mathbb{F}\subset \mathbb{G}$ on the time horizon $[0,T]$ with a drift operator $\Gamma$;
* there exist $N=(N_1,\ldots,N_\mathsf{n})$ an $\mathsf{n}$-dimensional $(\mathbb{P},\mathbb{F})$ local martingale, and ${\varphi}$ an $\mathsf{n}$ dimensional $\mathbb{G}$ predictable process such that, for any $(\mathbb{P},\mathbb{F})$ local martingale $X$, $[N,X]^{\mathbb{F}\cdot p}$ exists, ${\varphi}$ is $[N,X]^{\mathbb{F}\cdot p}$-integrable, and
\Gamma(X)=\transp{\varphi}\centerdot [N,X]^{\mathbb{F}\cdot p}
on the time horizon $[0,T]$. $N$ will be called the martingale factor and $\varphi$ will be called the integrant factor of the drift operator.
An immediate consequence of the drift multiplier assumption is the following lemmas.
For any $\mathbb{F}$ adapted cŕdlŕg process $A$ with $(\mathbb{P},\mathbb{F})$ locally integrable variation, we have
A^{\mathbb{G}\cdot p}=A^{\mathbb{F}\cdot p}+\Gamma(A-A^{\mathbb{F}\cdot p})=A^{\mathbb{F}\cdot p}+\transp\overline{\varphi}\centerdot[N,A-A^{\mathbb{F}\cdot p}]^{\mathbb{F}\cdot p}
on $[0,T]$. In particular, for $R$ a $\mathbb{F}$ stopping time either $(\mathbb{P},\mathbb{F})$ predictable or $(\mathbb{P},\mathbb{F})$ totally inaccessible, for $\xi\in\mathbb{L}^1(\mathbb{P},\mathcal{F}_{R})$,
(\xi\ind_{[R,\infty)})^{\mathbb{G}\cdot p}
(\xi\ind_{[R,\infty)})^{\mathbb{F}\cdot p}+\transp\overline{\varphi}\centerdot(\Delta_{R}N\xi\ind_{[R,\infty)})^{\mathbb{F}\cdot p}
on $[0,T]$.
This lemma is proved in <cit.>.
§.§ Martingale representation property
The full viability will be studied under the martingale representation property. On the stochastic basis $(\Omega,\mathcal{A},\mathbb{P},\mathbb{F})$, consider a $d$-dimensional stochastic process $W$. We say that $W$ has the martingale representation property in the filtration $\mathbb{F}$ under the probability $\mathbb{P}$, if $W$ is a $(\mathbb{P},\mathbb{F})$ local martingale, and if all $(\mathbb{P},\mathbb{F})$ local martingale is a stochastic integral with respect to $W$. We say that the martingale representation property holds in the filtration $\mathbb{F}$ under the probability $\mathbb{P}$, if there exists a local martingale $W$ which possesses the martingale representation property. In this case we call $W$ a representation process.
Supppose that $W$ has the martingale representation property in $(\mathbb{P},\mathbb{F})$. Then, the process $W$ satisfies the finite $\mathbb{F}$ predictable constraint condition. More precisely, there exist a finite number $\mathsf{n}$ of $d$-dimensional $\mathbb{F}$ predictable processes $\alpha_h, 1\leq h\leq \mathsf{n}$, such that
\Delta W
\sum_{h=1}^{\mathsf{n}}\alpha_h\ind_{\{\Delta W=\alpha_h\}}.
A very useful consequence of the finite predictable constraint condition is the following.
If the martingale representation property holds in $\mathbb{F}$ under $\mathbb{P}$, there exists always a locally bounded representation process, which has pathwisely orthogonal components outside of a predictable thin set.
The above two theorems are proved in <cit.>.
§.§ Results on the full viability
Here is an equivalence condition defined for expansion pair $\mathbb{F}\subset \mathbb{G}$.
For any $\mathbb{F}$ predictable stopping time $R$, for any positive random variable $\xi\in\mathcal{F}_R$, we have $\{\mathbb{E}[\xi|\mathcal{G}_{R-}]>0, R\leq T, R<\infty\}=\{\mathbb{E}[\xi|\mathcal{F}_{R-}]>0, R\leq T, R<\infty\}$.
Clearly, if the random variable $\xi$ is already in $\mathcal{F}_{R-}$ (or if $\mathcal{F}_{R-}=\mathcal{F}_{R}$), the above set equality holds. Hence, a sufficient condition for Condition <ref> to be satisfied is that the filtration $\mathbb{F}$ is quasi-left-continuous (cf. <cit.>).
The following theorem is proved in <cit.>. Let $T$ be a $\mathbb{G}$ stopping time.
Suppose that $(\mathbb{P},\mathbb{F})$ satisfies the martingale representation property. Then, $\mathbb{G}$ is fully viable on $[0,T]$, if and only if the drift multiplier assumption <ref> (with the factors $N$ and $\varphi$) and Condition <ref> are satisfied such that
\begin{equation}\label{fn-sur-fn}
\dcb
\transp{\varphi}(\centerdot[N^c,\transp N^c]){\varphi}\ \mbox{ is a finite process on $[0,T]$ and }\\
\\
\sqrt{\sum_{0<s\leq t\wedge T}\left(\frac{\transp {\varphi}_{s}\Delta_{s} N}{1+\transp{\varphi}_{s}\Delta_{s} N}
\right)^2},\ t\in\mathbb{R}_+,\
\mbox{ is $(\mathbb{P},\mathbb{G})$ locally integrable.}
\dce
\end{equation}
§ RECURSIVE CONSTRUCTION OF MULTI-DEFAULT TIME MODEL
In the previous sections, we have introduced in Definition <ref> the notion of full viability of a general expansion pair $\mathbb{F}\subset \mathbb{G}$ and a result in Theorem <ref> to ensure its validity. In present, we will consider more specifical expansions of $\mathbb{F}$ and study accordingly the full viability issue.
Fix a stochastic basis $(\Omega,\mathcal{A},\mathbb{P},\mathbb{F})$. Consider $n\in\mathbb{N}^*$ random times $\tau_1,\ldots,\tau_n$ defined on $(\Omega,\mathcal{A})$. We define $\mathbb{G}^{:0}=\mathbb{F}$, and, for $1\leq k\leq n$, define $\mathbb{G}^{:k}$ to be the progressive enlargement of the filtration $\mathbb{G}^{:(k-1)}$ with $\tau_k$ (cf. <cit.>). The filtration $\mathbb{G}^{:n}$ is an $n$-default times model.
§.§ Full viability transmission
We consider the full viability for the expansion pair $\mathbb{F}\subset \mathbb{G}^{:n}$. The following theorem shows that the full viability can pass harmonically through the recursive construction of the multi-default time models.
For any $1\leq k\leq n$, let $T_{k}$ be a $\mathbb{G}^{:k}$ stopping time, with the property $T_{k}\leq T_{k-1}$ ($T_{0}=\infty$). Then, if the full viability holds for the expansion pair $\mathbb{G}^{:(k-1)} \subset \mathbb{G}^{:k}$ on the horizon $[0,T_{k}]$, $1\leq k\leq n$, the full viability holds for the expansion pair $\mathbb{F} \subset \mathbb{G}^{:n}$ on the horizon $[0,T_{n}]$.
Proof. The theorem is true for $n=1$. Suppose that the theorem is proved for $n=n_{0}, 1\leq n_{0}<n$. Let $S$ be an asset process in $\mathbb{F}$ possessing a deflator. Then, $S$ be an asset process in $\mathbb{G}^{:n_{0}}$ possessing also a deflator on the horizon $[0,T_{n_{0}}]$. As, by assumption, the full viability holds for the expansion pair $\mathbb{G}^{:n_{0}} \subset \mathbb{G}^{:n_{0}+1}$ on the horizon $[0,T_{n_{0}+1}] \subset [0,T_{n_{0}}]$, $S$ is again an asset process in $\mathbb{G}^{:n_{0}+1}$ possessing a deflator on the horizon $[0,T_{n_{0}+1}]$.
§.§ Applicabiliy of Theorem <ref> in recursive construction
So, what is really essential is to establish the full viability for every expansion pair $\mathbb{G}^{:(k-1)}\subset \mathbb{G}^{:k}$. We will do this with Theorem <ref>. This necessitates beforehand to ensure that every expansion pair $\mathbb{G}^{:(k-1)}\subset \mathbb{G}^{:k}$ satisfies the two conditions in section <ref> and the martingale representation property. For this, we introduce three other conditions, which are easier to check when constructing multi-default time models. We consider the case of infinite horizon.
For any $1\leq k\leq n$, Hypothesis$(H')$ holds for the filtration expansion pair $\mathbb{G}^{:(k-1)}\subset\mathbb{G}^{:k}$ on the time horizon $[0,\infty]$ with a drift operator $\Gamma^{(k-1)|k}$.
The drift multiplier assumption holds for the expansion pair $\mathbb{G}^{:(k-1)}\subset\mathbb{G}^{:k}$ with the martingale factor $M^{(k-1)|k}$ (a $\mathbb{G}^{:(k-1)}$ local martingale) and the integrant factor $\overline{\psi}^{(k-1)|k}$ (a $\mathbb{G}^{:k}$ predictable process).
For any $1\leq k\leq n$, $\tau_k$ does not intercept the $\mathbb{G}^{:(k-1)}$ stopping times.
For any $1\leq k\leq n$, the $s\!\mathcal{H}$ measure covering condition on $(0,\infty)$ (cf. <cit.>) is satisfied for the expansion pair $\mathbb{G}^{:(k-1)}\subset \mathbb{G}^{:k}$.
By Assumption <ref>, the drift multiplier assumption for the expansion pair $\mathbb{G}^{:(k-1)}\subset \mathbb{G}^{:k}$ is ensured. To be able to apply Theorem <ref>, it remains to establish the martingale representation property in every $\mathbb{G}^{:(k-1)}$ and also Assumption <ref>.
Suppose the assumptions <ref> and <ref> and <ref>. Then, Hypothesis $(H')$ holds for the expansion pair $\mathbb{F}\subset \mathbb{G}^{:k}$. If $\mathbb{F}$ is quasi-left-continuous and satisfies the martingale representation property, for all $1\leq k\leq n$, $\mathbb{G}^{:k}$ also is quasi-left-continuous and satisfies the martingale representation property.
Proof. Consider the case where $k=1$. Hypothesis $(H')$ is satisfied for the expansion pair $\mathbb{G}^{:0}\subset \mathbb{G}^{:1}$ by Assumption <ref>.
The martingale representation property in $\mathbb{G}^{:1}$ is the consequence of Assumptions <ref> and <ref>, according to <cit.>.
Note that, because of Assumption <ref>, $\tau_1$ is $\mathbb{G}^{:1}$ totally inaccessible. By the drift multiplier Assumption <ref> and the quasi-left-continuity in $\mathbb{G}^{:0}$, $\Gamma^{0|1}$ generates always continuous processes. This means, by Lemma <ref>, that the $\mathbb{G}^{:0}$ totally inaccessible stopping times have continuous compensators in $\mathbb{G}^{:1}$ so that they remain to be $\mathbb{G}^{:1}$ totally inaccessible stopping times.
By the quasi-left-continuity, a $\mathbb{G}^{:0}$ local martingale can jump only at a $\mathbb{G}^{:0}$ totally inaccessible time. By <cit.>, the representation process in $\mathbb{G}^{:1}$ is linked with that in $\mathbb{G}^{:0}$. Hence, the representation process in $\mathbb{G}^{:1}$ jumps only at $\mathbb{G}^{:1}$ totally inaccessible times. This implies that the filtration $\mathbb{G}^{:1}$ also is quasi-left-continuous.
The theorem is proved for $k=1$. By induction, the theorem can be proved for all $1\leq k\leq n$.
§.§ Discussion on the martingale factor
Suppose the assumptions <ref> and <ref> and <ref>. Suppose that $\mathbb{F}$ is quasi-left-continuous and satisfies the martingale representation property. Then, there exist an $\mathbb{F}$ local martingale $N^{(k-1)|k}$ and a $\mathbb{G}^{:k}$ predictable process $\overline{\varphi}^{(k-1)|k}$ such that, for any $\mathbb{F}$ local martingale $X$, $[N^{(k-1)|k},X]^{\mathbb{G}^{:(k-1)}\cdot p}$ exists, $\overline{\varphi}^{(k-1)|k}$ is $[N^{(k-1)|k},X]^{\mathbb{G}^{:(k-1)}\cdot p}$ integrable, and
\begin{equation}\label{(k-1)N}
\Gamma^{(k-1)|k}(\widetilde{X}^{:(k-1)})=\transp\overline{\varphi}^{(k-1)|k}\centerdot [N^{(k-1)|k},X]^{\mathbb{G}^{:(k-1)}\cdot p},
\end{equation}
where $\widetilde{X}^{:(k-1)}$ denotes the martingale part of $X$ in $\mathbb{G}^{:(k-1)}$.
Proof. Note that the initial formula is
\Gamma^{(k-1)|k}(\widetilde{X}^{:(k-1)})=\transp\overline{\psi}^{(k-1)|k}\centerdot [M^{(k-1)|k},\widetilde{X}^{:(k-1)}]^{\mathbb{G}^{:(k-1)}\cdot p},
for a $\mathbb{G}^{:(k-1)}$ local martingale $M^{(k-1)|k}$. To prove the theorem, we only need to establish
[M^{(k-1)|k},\widetilde{X}^{:(k-1)}]^{\mathbb{G}^{:(k-1)}\cdot p}<\!\!< [N^{(k-1)|k},X]^{\mathbb{G}^{:(k-1)}\cdot p},
for some $\mathbb{F}$ local martingale $N^{(k-1)|k}$. Let $W$ be a locally bounded representation process for the martingale representation property in $\mathbb{F}$ (cf. Theorem <ref>). By repeated applications of <cit.>, the representation process in $\mathbb{G}^{:(k-1)}$ is linked with $W$. With this link, because $X$ is an $\mathbb{F}$ local martingale, because of the continuity of the drift of $X$ in $\mathbb{G}^{:(k-1)}$ so that $\widetilde{X}^{:(k-1)}$ has jumps only at $\mathbb{F}$ stopping times, we can find a $\mathbb{G}^{:(k-1)}$ predictable process $H$ such that
[M^{(k-1)|k},\widetilde{X}^{:(k-1)}]^{\mathbb{G}^{:(k-1)}\cdot p}
\transp H\centerdot [\widetilde{W}^{:(k-1)},\widetilde{X}^{:(k-1)}]^{\mathbb{G}^{:(k-1)}\cdot p}
\transp H\centerdot [W,X]^{\mathbb{G}^{:(k-1)}\cdot p},
where the last equality results from the continuity of the drifts of $W$ and of $X$ in $\mathbb{G}^{:(k-1)}$.
§.§ Recursive construction of the factors in the drift operators
We can say more about the drift operator of the expansion pair $\mathbb{F}\subset \mathbb{G}^{:k}$ with the formula (<ref>).
Suppose the assumptions <ref> and <ref> and <ref>. Suppose that $\mathbb{F}$ is quasi-left-continuous and satisfies the martingale representation property. Then, Hypothesis $(H')$ is satisfied for the expansion pair $\mathbb{F}\subset \mathbb{G}^{:k}$, $1\leq k\leq n$, and the corresponding drift operator $\Gamma^{:k}$ satisfies the drift multiplier assumption whose factors $N^{:k},\overline{\varphi}^{:k}$ can be computed recursively by
\dcb
^\top N^{:0}=0, \ ^\top \overline{\varphi}^{:0}=0,\\
^\top N^{:k}=(^\top N^{:(k-1)},\ ^\top N^{(k-1)|k}) \ \mbox{(forming a vector of one more dimension)},\\
^\top \overline{\varphi}^{:k} =(^\top\overline{\varphi}^{:(k-1)},\ \transp(1+\transp\overline{\varphi}^{:(k-1)} \gamma^{(k-1)|k} )\vdots\overline{\varphi}^{(k-1)|k}),\\
\dce
where $\gamma^{(k-1)|k}=(\gamma^{(k-1)|k}_{i})_{1\leq i\leq d^{(k-1)|k}}$ is family of vectors which are determined by the relation $(\Delta N^{:(k-1)}{\centerdot}[N^{(k-1)|k}_i,\transp W])^{\mathbb{F}\cdot p} =\gamma^{(k-1)|k}_{i}{\centerdot}[N^{(k-1)|k}_i,\transp W]^{\mathbb{F}\cdot p}$, and $(1+\transp\overline{\varphi}^{:(k-1)} \gamma^{(k-1)|k} )$ is the vertical vector of components $(1+\transp\overline{\varphi}^{:(k-1)} \gamma^{(k-1)|k}_{i} )$, $1\leq i\leq d^{(k-1)|k}$ (the dimension of $N^{(k-1)|k}$).
Note that $\gamma^{(k-1)|k}$ exists by <cit.>.
Note that $\Gamma^{:1}=\Gamma^{0|1}$ so that the drift multiplier assumption is satisfied for $k=1$.
Suppose the induction assumption that the drift multiplier assumption is satisfied from $\mathbb{F}$ to $\mathbb{G}^{:(k-1)}$ with drift operator
\Gamma^{:(k-1)}(X)
\transp\overline{\varphi}^{:(k-1)}{\centerdot}[N^{:(k-1)},X]^{\mathbb{F}\cdot p}.
Let us prove it for $k$.
Let $X$ be a $\mathbb{F}$ local martingale. Write the representation $X=\transp G\centerdot W$. The bracket process $[N^{(k-1)|k},X]^{\mathbb{F}\cdot p}$ is continuous, because of the quasi-left-continuity. By Lemma <ref>, for any component $N^{(k-1)|k}_i$,
\dcb
[N^{(k-1)|k}_i,X]^{\mathbb{G}^{:(k-1)}\cdot p}\\
[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}+\transp\overline{\varphi}^{:(k-1)}\centerdot[N^{:(k-1)},[N^{(k-1)|k}_i,X]-[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}]^{\mathbb{F}\cdot p}\\
[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}+\transp\overline{\varphi}^{:(k-1)}\centerdot[N^{:(k-1)},[N^{(k-1)|k}_i,X]]^{\mathbb{F}\cdot p}\\
&&\mbox{ because of the continuity $[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}$}, \\
[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}+\transp\overline{\varphi}^{:(k-1)}\centerdot(\Delta N^{:(k-1)}{\centerdot}[N^{(k-1)|k}_i,X])^{\mathbb{F}\cdot p}\\
[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}+\transp\overline{\varphi}^{:(k-1)}\centerdot(\Delta N^{:(k-1)}{\centerdot}[N^{(k-1)|k}_i,\transp W])^{\mathbb{F}\cdot p} G\\
[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}+\transp\overline{\varphi}^{:(k-1)}\gamma^{(k-1)|k}_{i}\centerdot [N^{(k-1)|k}_i,\transp W]^{\mathbb{F}\cdot p} G\\
[N^{(k-1)|k}_i,X]^{\mathbb{F}\cdot p}+\transp\overline{\varphi}^{:(k-1)}\gamma^{(k-1)|k}_{i}\centerdot [N^{(k-1)|k}_i, X]^{\mathbb{F}\cdot p} \\
(1+\transp\overline{\varphi}^{:(k-1)}\gamma^{(k-1)|k}_{i})\centerdot [N^{(k-1)|k}_i, X]^{\mathbb{F}\cdot p}.
\dce
Let $\widetilde{X}^{:(k-1)}$ denote the martingale part of $X$ in $\mathbb{G}^{:(k-1)}$. We can write
\dcb
\transp\overline{\varphi}^{(k-1)|k}{\centerdot}[N^{(k-1)|k},\widetilde{X}^{:(k-1)}]^{\mathbb{G}^{:(k-1)}\cdot p}\\
\transp\overline{\varphi}^{(k-1)|k}{\centerdot}[N^{(k-1)|k},X]^{\mathbb{G}^{:(k-1)}\cdot p}\ \mbox{ because $[N^{:(k-1)},X]^{\mathbb{F}\cdot p}$ is continuous,}\\
\transp(1+\transp\overline{\varphi}^{:(k-1)} \gamma^{(k-1)|k} )\vdots\overline{\varphi}^{(k-1)|k}\centerdot[N^{(k-1)|k},X]^{\mathbb{F}\cdot p}.
\dce
Notice that $\widetilde{X}^{:k}=X-\Gamma^{:k}(X)$ and $\widetilde{X}^{:(k-1)}=X-\Gamma^{:(k-1)}(X)$ and also
\dcb
\widetilde{X}^{:(k-1)} - \Gamma^{(k-1)|k}(\widetilde{X}^{:(k-1)}) \\
X-\Gamma^{:(k-1)}(X) - \Gamma^{(k-1)|k}(\widetilde{X}^{:(k-1)}) \\
X-\transp\overline{\varphi}^{:(k-1)}{\centerdot}[N^{:(k-1)},X]^{\mathbb{F}\cdot p} - \transp(1+\transp\overline{\varphi}^{:(k-1)} \gamma^{(k-1)|k} )\vdots\overline{\varphi}^{(k-1)|k}\centerdot[N^{(k-1)|k},X]^{\mathbb{F}\cdot p}.
\dce
The theorem is now proved with the formulas
\dcb
^\top N^{:k}=(^\top N^{:(k-1)},\ ^\top N^{(k-1)|k}),\\
^\top \overline{\varphi}^{:k} =(^\top\overline{\varphi}^{:(k-1)},\ \transp(1+\transp\overline{\varphi}^{:(k-1)} \gamma^{(k-1)|k} )\vdots\overline{\varphi}^{(k-1)|k}).\ \ok
\dce
§ MULTI-DEFAULT TIME $\NATURAL$-MODEL WITH FULL VIABILITY
We continue to work on the stochastic basis $(\Omega,\mathcal{A},\mathbb{P},\mathbb{F})$. We want to construct a multi-default time model $\mathbb{G}^{:n}$ which satisfies the full viability. We want to do so with Theorem <ref> and Theorem <ref>. It is therefore necessary to construct beforehand a multi-default time model $\mathbb{G}^{:n}$, to which the two theorems are applicable.
In the last section, Theorem <ref> shows that, under Assumptions <ref> and <ref> and <ref>, if the filtration $\mathbb{F}$ satisfies
the quasi-left-continuuity and
the martingale representation property,
the filtrations $\mathbb{G}^{:k}$ preserve the same two properties, which make Theorem <ref> and Theorem <ref> applicable. Hence, what we have to do is to construct a multi-default time model satisfying Assumptions <ref> and <ref> and <ref>.
§.§ The $\natural$-model
Can we have random times satisfying the above requirements ? We may mention honest times. However, <cit.> has shown that a honest time model typically does not satisfy the no-arbitrage property on a horizon beyond the default time $\tau$. In contrast, the $\natural$-model introduced in <cit.> is perfect to meet the requirements.
Consider a stochastic basis $(\Omega,\mathcal{A},\mathbb{F},\mathbb{P})$. Let $L$ be a positive $\mathbb{F}$ locally bounded local martingale and $\Lambda$ be a non decreasing $\mathbb{F}$ adapted continuous process with $\Lambda_{0}=0$. We suppose that the non negative process $Z:=Le^{-\Lambda}$ satisfies $Z>0, 1-Z>0$ on $(0,\infty)$. Let $\mathbf{Y}$ be a multi-dimensional $\mathbb{F}$ locally bounded local martingale. We adopt the notion of smooth Markovian $\natural$-pair $(\mathbf{F}, \mathbf{Y})$ ($\mathbf{F}$ being a smooth Markovian coefficient) in <cit.>. According to <cit.>, we have
Under the above conditions, there exists a random time $\tau$ defined on (an isomorphic extension of) the probability space $(\Omega,\mathcal{A},\mathbb{P})$ such that
* $Z$ is the Azema supermartingale of $\tau$ with respect to $\mathbb{F}$,
* the random time $\tau$ does not intercept the $\mathbb{F}$ stopping times,
* Hypothesis $(H')$ holds for the expansion pair $\mathbb{F}\subset\mathbb{G}$,
* the $s\!\mathcal{H}$ measure covering condition holds on $(0,\infty)$.
Moreover, the drift operator takes the form
\begin{equation}\label{dies}
\Gamma(X)
\left(\ind_{(0,\tau]}\frac{1}{Z_-}
\ind_{(\tau,\infty)}\frac{1}{1-Z_{-}}
\right)\centerdot[M,X]^{\mathbb{F}-p}
\ind_{(\tau,\infty)}{\mathtt{p}}(\tau)\centerdot[\mathbf{Y},X]^{\mathbb{F}-p},
\end{equation}
where $M$ is the $\mathbb{F}$ martingale part of $Z$ ($M=e^{-\Lambda}\centerdot L$), $\mathtt{p}$ is a bounded $\mathbb{F}$ predictable process with parameter (coming form the derivative of $\mathbf{F}$). The pair $(\mathbb{F},\tau)$ will be called a $\natural$-model based on $(Z,\mathbf{F},\mathbf{Y})$.
Note that the $s\!\mathcal{H}$ measure covering condition on $(0,\infty)$ is proved in <cit.>. Note also that $L$ is assumed $\mathbb{F}$ locally bounded. As a corollary, we have the next results.
§.§ A sufficient condition for the full viability
Applying the previous theorem step by step in the construction of multi-default time models, we obtain the following corollary.
Let, for $1\leq k\leq n$, $Z^{(k-1)|k}=L^{:(k-1)}e^{-\Lambda^{(k-1)|k}}$ be the Azema supermartingale of $\tau_{k}$ with respect to $\mathbb{G}^{:(k-1)}$. Suppose that $Z^{(k-1)|k}>0, 1-Z^{(k-1)|k}>0$, $L^{(k-1)|k}$ is locally bounded and $\Lambda^{(k-1)|k}$ is continuous. Suppose that $(\mathbb{G}^{:(k-1)},\tau_{k})$ is a smooth Markovian $\natural$-model based on $(Z^{(k-1)|k},\mathbf{F}^{(k-1)|k},\mathbf{Y}^{(k-1)|k})$ ($\mathbf{Y}^{(k-1)|k}$ being $\mathbb{G}^{:(k-1)}$ locally bounded). Then, Assumptions <ref>, <ref> and <ref> are satisfied for the expansion pair $\mathbb{G}^{:k-1}\subset\mathbb{G}^{:k}$.
Corollary <ref> provides the conditions to apply Theorem <ref> and Theorem <ref>. We can now conclude.
Suppose the same condition as in Corollary <ref>. Suppose that
$\frac{1}{Z^{(k-1)|k}}\ind_{(0,\tau_{k}]}$ and $\frac{1}{1-Z^{(k-1)|k}}\ind_{(\tau_{k},\infty)}$ are $\mathbb{G}^{:k}$ locally bounded processes and
$\mathbf{Y}^{(k-1)|k}$ is continuous.
Then, the full viability holds for the expansion pair $\mathbb{F} \subset \mathbb{G}^{:n}$, whenever $\mathbb{F}$ is quasi-left continuous and satisfies the martingale representation property.
Notice that a multi-default time model may also be constructed with initial times (cf. <cit.>). But in general, a density process will not be as easy as $Z^{(k-1)|k}$ to be tackled.
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1511.00454
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We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our
construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.
§ INTRODUCTION
§.§ Background.
Spectral triples, a central concept of noncommutative geometry, provide an analytical language for geometric objects. A prototype is given by the triple $(C^1(\mathcal{M}), L^2(\mathcal{M},\mathcal{S}), \Dirac)$ which is a spectral triple on the algebra $C(\mathcal{M})$ of continuous functions on $\mathcal{M}$, where $\mathcal{M}$ is a compact Riemannian manifold equipped with a spin$^C$ (or spin) structure, $C^1(\mathcal{M})$ a dense “smooth" subalgebra of $C(\mathcal{M})$ and $\Dirac$ is the corresponding Dirac operator acting on $L^2(\mathcal{M},\mathcal{S})$.
Connes <cit.>, <cit.> introduced spectral triples as a potential means of describing the homology and index theoretic aspects in the more general language of (locally) compact topological spaces, as well as to develop a theory of cyclic cohomology mimicking the de-Rham cohomology theory of manifolds. Further, Connes shows that geometric information about a Riemannian manifold $\mathcal{M}$, such as the geodesic distance and dimension, can all be recovered from the Dirac triple on $C(\mathcal{M})$.
Spectral triples are motivated by Kasparov theory and can be regarded as “Dirac-type" or elliptic operators on general C$^*$-algebras (usually assumed separable). In particular a spectral triple defines a $K$-homology class. Spectral triples with good properties can be used to encode geometric information on a C$^*$-algebra. Besides the link between summability and dimension which is well understood in the commutative case, we mention two examples of current areas of research.
The first is Connes' reconstruction programme, the aim of which is to find conditions or axioms under which a spectral triple on a commutative C$^*$-algebra can provide the spectrum of the algebra with the structure of a manifold. Several reconstruction theorems have been suggested in what has become a very prominent area of research (see for example <cit.>, <cit.>). Besides the noncommutative tori, there do not seem to be many examples of noncommutative C$^*$-algebras at present for which this sort of analysis can be extended to.
The second one is the idea to regard spectral triples as noncommutative (quantum) metric spaces, beginning with Connes' observation <cit.> that the Dirac triple on a Riemannian spin$^C$ manifold $\mathcal{M}$ recovers the geodesic distance between two points on the manifold. In fact Connes' expression for the geodesic distance extends immediately to a metric on the space of probability measures on $\mathcal{M}$. In more recent and general language, a spectral triple on a C$^*$-algebra determines a Lipschitz seminorm on the self-adjoint part of the smooth subalgebra, an analogue of the classical notion of Lipschitz continuous functions. In e.g. <cit.>, <cit.> and <cit.> Rieffel studies Lipschitz seminorms of this kind extensively. Under mild conditions such a seminorm defines a metric on the state space of the algebra by a formula analogous to the manifold case. However, in general, Lipschitz seminorms and corresponding metrics may be quite arbitrary. A natural condition one would expect this metric to satisfy is that it induces the weak-$*$-topology on the state space and Rieffel makes this the defining condition of his notion of a quantum metric space. Rieffel found a very useful characterisation of this metric condition for unital C$^*$-algebras (<cit.>, cf. Prop.<ref> below for the statement). We will refer to this condition as Rieffel's metric condition. Latrémolière later extended much of this work to non-unital C$^*$-algebras in <cit.> and <cit.>. A C$^*$-algebra equipped with a spectral triple satisfying this metric condition is sometimes called a spectral metric space.
Despite the longevity of spectral triples as a subject of study, general methods of constructing spectral triples on C$^*$-algebras are not well understood, much less still those satisfying the metric condition. There have been successful constructions of so-called spectral metric
spaces on certain noncommutative C*-algebras, such as approximately finite dimensional algebras (<cit.>), group C*-algebras of discrete hyperbolic groups (<cit.>) and algebras arising as $q$-deformations of the function algebras of simply connected simple compact Lie groups (<cit.>).
Building on previous authors' works, we are particularly interested in `building block' constructions i.e. constructing new spectral triples from old ones, which is also in the sprit of permanence properties. This point of view has been used by various authors to attempt to construct spectral triples on crossed products of C*-algebras by certain discrete groups (<cit.>, <cit.>). More specifically, the authors of those two references study C$^*$-dynamical systems $(A,G,\alpha)$ in which the algebra $A$ is equipped with the structure of a spectral triple with good metric properties and consider under what conditions it is possible to write down a spectral triple on $A \semidir_{r, \alpha} G$ using a natural implementation of the external product in Kasparov theory. It turns out that a necessary and sufficient condition is the requirement that the action of $G$ essentially implements an isometric action on the underlying spectral metric space. This is satisfied for a variety of group actions and, via this construction, the authors in collaboration with A. Skalski and S. White (<cit.>) were able to write down spectral triples with good metric properties on both the irrational rotation algebras and the Bunce-Deddens algebras and some of their generalisations.
Spectral triples define Baaj-Julg cycles, the unbounded analogue of a Kasparov bimodule in KK-theory (<cit.>). This perspective is increasingly being examined by various authors to write down spectral triples on C$^*$-algebras by means of an unbounded version of Kasparov's internal product, which is defined for C$^*$-algebras $A$, $D$, $B$ and $p, q \in \{0,1\}$ as a map $\otimes_B: KK^p(A,D) \times KK^q(D,B) \to KK^{p+q}(A,B)$. There are a couple of important recent developments in this area: Gabriel and Grensing (<cit.>) consider the possibility of writing down spectral triples on certain Cuntz-Pimsner algebras, generalising the setting of ordinary crossed products by $\Z$ but with the same property that the triples they construct represent the image of a given triple under the boundary map in the resulting six-term exact sequence. They succeed in implementing these techniques to construct a variety of spectral triples on certain quantum Heisenberg manifolds. Goffeng and Mesland (<cit.>) investigate how the Kasparov product can, under suggested modifications, be used to write down spectral triples on Cuntz-Krieger algebras, beginning with the spectral triple on the underlying subshift space. It is anticipated therefore that there will be a considerable interest in the interplay between spectral triples and the Kasparov product in the near future.
In this paper we construct spectral triples on extensions of C*-algebras out of given ones on the ideal and the quotient algebra. We are, however, primarily concerned with those which satisfy Rieffel's metric condition, thus implementing the structure of a quantum metric space on the extension, beginning with related structures on both the quotient and ideal. Techniques in Kasparov theory will be important to us too, but certain technical difficulties will prevent us from being able to give a full description of the resulting triples in terms of their representatives in K-homology. We remark that the ideas in this paper are closely linked to those of Christensen and Ivan (<cit.>) and are to some extent a generalisation of their results.
§.§ Outline of the paper.
We assume throughout the paper that all C$^*$-algebras and Hilbert spaces are separable.
Given a C$^*$-algebra $E$ and an essential ideal $I \subs E$, and given spectral triples on both $I$ and $E/I$, is there any way of constructing a spectral triple on $E$ out of the given spectral triples? In this paper we will be looking at the situation in which the quotient is a unital C$^*$-algebra $A$ and the ideal is the tensor product of a unital C$^*$-algebra by the algebra of compact operators, that is, we consider extensions of the form,
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & \mathcal{K} \otimes B \ar[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] & 0}.
\end{eqnarray}
This is a generalisation of the situation considered by Christensen and Ivan <cit.>, who looked at short exact sequences of
the form,
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] & 0}.
\label{ext-k}
\end{eqnarray}
They exploited the fact that a certain class of C$^*$-extensions by compacts (those which are semisplit) can be spatially represented over a Hilbert space: as outlined in Section 2.7 of <cit.>, we can regard $E$ as a subalgebra of the bounded operators on an infinite dimensional Hilbert space $H$ generated by compacts on $PH$ and the operators $\{ P\pi_A(a)P \in B(H): a \in A \}$, where $\pi_A: A \to B(H)$ is a faithful representation and $P \in B(H)$ is an orthogonal projection with infinite dimensional range. The algebra acts degenerately, only on the subspace $PH$.
There is a certain generalisation of this picture for semisplit extensions by general stable ideals of the form (<ref>) which is due to Kasparov (<cit.>, see also <cit.>). For such extensions, $E$ can always be regarded as a subalgebra of $\mathcal{L}_B(\ell_2(B)) = \mathcal{L}_B$, the C$^*$-algebra of bounded $B$-linear and adjointable operators on the Hilbert module $\ell_2(B)$. In fact, using semisplitness, there is a representation $\pi : A \to \mathcal{L}_B(\ell_2(B) \oplus \ell_2(B)) \cong \mathcal{L}_B(\ell_2(B)) $ and a projection $P \in \mathcal{L}_B$ such that $E$ is generated by $P\pi (A) P $ and $P (\mathcal{K} \otimes B) P= P (\mathcal{K}_B)P$ (cf. Section <ref> for more details). However, to construct spectral triples on $E$ we need a representation on a Hilbert space, not Hilbert module. Our given spectral triples come with concrete representations $\pi_A : A \to B(H_A)$ and $\pi_B : B \to B(H_B)$ on Hilbert spaces. It seems reasonable to study those extensions which act naturally on the tensor product $H_A \otimes H_B$, possibly degenerately i.e. only on a subspace of this tensor product. More precisely, we consider representations of the form
\pi: E \to B(H_A \otimes H_B), \;\;\; \;\; \pi(\mathcal{K} \otimes B) = \mathcal{K}(H_0)\otimes \pi_B(B)
where $H_0$ is an infinite dimensional subspace of $H$, and $\pi (k \otimes b) = \phi (k) \otimes \pi_B(b)$ with $ \phi : \mathcal{K} \to \mathcal{K} (H_0)$ an isomorphism. Not all extensions can be brought into this form. In Section <ref> we show that it is possible if the Busby invariant satisfies a certain factorisation property. In order to describe it in somewhat more detail recall that a short exact sequence of C$^*$-algebras,
\begin{eqnarray*}
\xymatrix{ 0 \ar[r] & \mathcal{K} \otimes B \ar[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] & 0}
\end{eqnarray*}
is characterised by a $^*$-homomorphism $\psi: A \to \mathcal{Q}_B$, the Busby invariant, where
$\mathcal{Q}_B := \mathcal{L}_B / \mathcal{K}_B$ is sometimes called the generalised Calkin algebra with respect to the C$^*$-algebra $B$. Since $\mathcal{L}_B \cong \mathcal{M} (\mathcal{K} \otimes B)$ there is an embedding of the ordinary Calkin algebra $\mathcal{Q}= \mathcal{M}(\mathcal{K})/\mathcal{K}$ into $\mathcal{Q}_B$ and the condition which characterises the extensions we consider is that there exists a semisplit extension of $A$ by $\mathcal{K}$ of the type (<ref>) with Busby invariant $\psi_0: A \to \mathcal{Q}$ such that $\psi$ factors through $\psi_0$ and the natural inclusion of $\mathcal{Q}$ into $\mathcal{Q}_B= \mathcal{M}(\mathcal{K} \otimes B)/\mathcal{K\otimes B}$. In KK-theoretic language, we need the class of $\psi$ in $KK^1(A,B)= \text{Ext}(A,B)^{-1}$ to factor into the class of an extension
$\psi_0$ in $K^1(A)=\text{Ext}^{-1}(A)$ and the $K_0(B)$-class of $1_B \in B$, i.e.
[\psi] = [\psi_0] \otimes [1_B].
In this situation we can view the algebra $E$ as a concrete subalgebra of $B(H_A \otimes H_B)$ generated by
elements of the form $PaP \otimes 1_B$ and $k \otimes b$, where $P \in B(H_A)$ is an orthogonal infinite dimensional projection, $a \in A$, $b \in B$ and $k$ is a compact operator in $PH$ (cf. Corollary <ref>). Throughout the paper we will assume that our extension is of this form. To avoid technicalities we will assume that $PaP \cap \mathcal{K} = \{0\}$ which is true for essential extensions and can always be arranged by replacing $\pi_A$ by an infinite ampliation $\pi_A^{\infty}$.
Starting from a pair of spectral triples $(\Alg, H_A, \Dirac_A)$ on $A$ and $(\Balg, H_B, \Dirac_B)$ on $B$ (cf. Section <ref> for notation), Kasparov theory can be used to write down a spectral triple on $A \otimes B$ whose representative in K-homology is the external Kasparov product of the representatives of $(\Alg, H_A, \Dirac_A)$ and $(\Balg, H_B, \Dirac_B)$. When the spectral triple on $A$ is odd and the spectral triple on $B$ is even, i.e. there is a direct sum representation $\pi_B^+ \oplus \pi_B^-$ and
$\Dirac_B$, acting on $H_B \otimes \C^2$, decomposes as the matrix,
\begin{bmatrix} 0 & \Dirac_B^+ \\ \Dirac_B^- & 0 \end{bmatrix},
then the spectral triple can be defined on the spatial tensor product $A \otimes B$ acting on the Hilbert space $H_A
\otimes H_B \otimes \C^2$ via the representation $(\pi_A \otimes \pi_B^+) \oplus (\pi_A \otimes \pi_B^-)$ with the
Dirac operator,
\begin{bmatrix} \Dirac_A \otimes 1 & 1 \otimes \Dirac_B^+ \\ 1 \otimes \Dirac_B^- & -\Dirac_A \otimes 1
\end{bmatrix},
(which can be interpreted as sum two graded tensor products), whereas the product of two ungraded triples is represented by the matrix
\begin{bmatrix} 0 & \Dirac_A \otimes 1 - i \otimes \Dirac_B\\
\Dirac_A \otimes 1 + i \otimes \Dirac_B & 0
\end{bmatrix},
acting on $H_A \otimes \C^2$ (see for example <cit.> pp. 433-434).
We mention these formulae since they serve as an inspiration for the Dirac operator we are going to write down for the extension. In fact our operator will be a combination of these two formulae which makes it difficult to interpret our construction in K-homological terms.
Returning to our set-up, our assumptions imply that we can, omitting representations, write down an isomorphism,
\begin{eqnarray*}
E \cong \mathcal{K}(P H_A) \otimes B + PAP \otimes \C I_B,
\end{eqnarray*}
where $[P, a]$ is a compact operator on $H_A$.
$E$ can be regarded as a concrete subalgebra of $B(H_A \otimes H_B)$ acting degenerately (effectively only on $PH_A \otimes H_B$). The corresponding representation is denoted by $\pi$. There is another representation $\pi_{\sigma}: E \to B(H_A \otimes H_B)$ given as the composition of the quotient map $\sigma: E \to A$ and the natural representation $\pi_A \otimes 1$ of $A$ on $B(H_A \otimes H_B)$. $\pi_{\sigma}$ is non-degenerate but not faithful. The information coming from both representations is essential to writing down a
spectral triple on $E$ which encodes the metric behaviour of both the ideal and quotient parts of the extension.
We will use this information, the presence of Dirac operators $\Dirac_A$ on $H_A$ and $\Dirac_B$ on $H_B$ together with the
aforementioned ideas in Kasparov theory to build spectral triples on $E$. The representation of this triple will be a suitable combination of the two representations of $E$.
In order to build a spectral triple we need the further requirement that $P$ commutes with $\Dirac_A$
which then decomposes into the direct sum $\Dirac_A = \Dirac_A^p \oplus \Dirac_A^q$, where $\Dirac_A^p = P
\Dirac_A P$, $\Dirac_A^q = Q \Dirac_A Q$ and $Q = 1-P$. Next we require
\begin{eqnarray*}
[P, \pi_A(a)] \in \mathcal{C}(H_A) \;\;\; \forall a \in \Alg,
\end{eqnarray*}
where $\mathcal{C}(H_A)$ is the dense $^*$-subalgebra of elements $x \in \mathcal{K}(H_A)$ such that $x \textup{dom}(\Dirac_A) \subseteq \textup{dom}(\Dirac_A)$ and both $x \Dirac_A$ and $\Dirac_A x$ extend to bounded operators ($P$-regularity cf. Defs.<ref>, <ref>). One may think of $\mathcal{C}(H_A)$ as the dense $^*$-subalgebra of `differentiable compacts', hence the notation used.
We summarise our assumptions on the extension and spectral triples in the following definition which is consistent with the article <cit.>.
Let $\pi_A : A \to B(H_A)$ and $\pi_B : B \to B(H_B)$ be faithful representations, where $A, B$ are separable unital C$^*$-algebras and $H_A ,H_B$ separable Hilbert spaces.
The extension
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & \mathcal{K} \otimes B \ar[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] & 0}
\label{extension}
\end{eqnarray}
is said to be of Toeplitz type if there exists an infinite dimensional projection $P \in B(H_A)$ such that
$$[P,\pi_A(a)] \in \mathcal{K}(H_A),$$
$$E \cong \mathcal{K}(P H_A) \otimes \pi_B(B) + P\pi_A(A)P \otimes \C I_B$$
$$ \mathcal{K}(P H_A) \otimes \pi_B(B) \cap P\pi_A(A)P \otimes \C I_B= \{0\}.$$
$(\pi_A, \pi_B , P)$ is then referred to as a Toeplitz triple for the extension.
If moreover $(\Alg, H_A ,\Dirac_A)$ is a spectral triple such that $\Dirac_A$ and $P$ commute and $[P, \pi_A(a)] \in \mathcal{C}(H_A)$ for all $a \in \Alg$, then the quadruple $(\Alg, H_A, \Dirac_A, P)$ (or just $P$) is said to be of Toeplitz type.
A Toeplitz type quadruple is said to be $P$-injective if $\ker(\Dirac_A^p \cap PH_A) = \{0\}$.
When $P$ coincides with the orthogonal projection into the closed span of the positive eigenspace for $\Dirac_A$, then the smoothness assumption turns out to be equivalent to saying that not only $[\Dirac_A, \pi_A(a)]$ but also $[|\Dirac_A|, \pi_A(a)]$ is a bounded operator for each $a \in \Alg$, which is related to the concept of regularity for spectral triples in Riemannian geometry due to Connes and Moscovici (<cit.>).
We state here the two main results of our paper asserting the existence of spectral triples with good summability properties on Toeplitz-type extensions under the assumption that $(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type
(Theorem <ref>) and that Rieffel's metric condition is preserved under the mild extra assumption that $\Dirac_A$ is $P$-injective (Theorem <ref>).
Before we can do so we need to introduce further notation. Let $\Pi_1, \Pi_2 : E \to B(H_A \otimes H_B \otimes \C^2)$ be the representations given by
\begin{eqnarray}
\Pi_1 = \pi_{\sigma} \oplus \pi_{\sigma} \textup{ and } \Pi_2 = \pi \oplus \pi_{\sigma}
\end{eqnarray}
and consider operators $\Dirac_{1}, \Dirac_{2}, \Dirac_{3}$ on $H_A \otimes H_B \otimes \C^2$ given by
\begin{eqnarray}
\Dirac_{1} =
\begin{bmatrix} \Dirac_A \otimes 1 & 1 \otimes \Dirac_B \\
1 \otimes \Dirac_B & -\Dirac_A \otimes 1
\end{bmatrix},
\end{eqnarray}
\begin{eqnarray}
\Dirac_{2} =
\begin{bmatrix} \Dirac_A^q \otimes 1 & \Dirac_A^p \otimes 1 \\
\Dirac_A^p \otimes 1 & -\Dirac_A^q \otimes 1
\end{bmatrix}
\begin{bmatrix} \Dirac_A^q & \Dirac_A^p \\
\Dirac_A^p & -\Dirac_A^q
\end{bmatrix}
\otimes I
\end{eqnarray}
\begin{eqnarray}
\Dirac_{3} =
\begin{bmatrix} 1 \otimes \Dirac_B & 0 \\
0 & 1 \otimes \Dirac_B
\end{bmatrix}
I \otimes
\begin{bmatrix} \Dirac_B & 0 \\
0 & \Dirac_B
\end{bmatrix}
\end{eqnarray}
Let $A$ and $B$ be unital C$^*$-algebras and suppose that $E$ arises as the short exact sequence
(<ref>) and that there exist spectral triples $(\Alg,H_A,\Dirac_A)$ on $A$ and
$(\Balg,H_B,\Dirac_B)$ on $B$, represented via $\pi_A$ and $\pi_B$ respectively, and an orthogonal projection $P \in B(H_A)$ such that $(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type. Let
\begin{eqnarray*}
\Pi = \Pi_1\oplus \Pi_2 \oplus \Pi_{2}, \;\;H = (H_A \otimes
H_B \otimes \C^2)^3, \textup{ and}
\end{eqnarray*}
\begin{eqnarray*}
\Dirac = \begin{bmatrix} \Dirac_{1} & 0 & 0 \\[1ex]
0 & 0 & \Dirac_2 - i \Dirac_{3}\\[1ex]
0 & \Dirac_2 +
i \Dirac_{3} &0 \end{bmatrix}.
\end{eqnarray*}
Then $(\mathcal{E}, H, \Dirac)$, represented
via $\Pi$, defines a spectral triple on $E$.
Moreover, the spectral dimension of this spectral triple is computed by
the identity
\begin{eqnarray*}
s_0(\mathcal{E}, H, \Dirac) = s_0(\Alg,H_A,\Dirac_A) + s_0(\Balg,H_B,\Dirac_B).
\end{eqnarray*}
The Dirac operator $\Dirac$ of the spectral triple defines a Lipschitz seminorm $L_{\Dirac}$ which in turn defines in good cases a metric on the state space of the C$^*$-algebra. We address the question of whether our spectral triples satisfy Rieffel's metric condition, which is a necessary and sufficient condition for the metric on the state space to induce the weak-$*$-topology (cf. Prop.<ref> and the definition thereafter). In this case the spectral triple together with the Lipschitz seminorm is called a spectral metric space. We show that under our natural assumptions this is always the case.
Let $A$ and $B$ be unital C$^*$-algebras and suppose $E$ arises as the short exact sequence
(<ref>). Suppose further that there exists spectral triples $(\Alg,H_A,\Dirac_A)$ on $A$ and
$(\Balg,H_B,\Dirac_B)$ on $B$, represented via $\pi_A$ and $\pi_B$ respectively, and an orthogonal projection $P \in B(H_A)$ such that
$(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type and $P$-injective. If the spectral triples $(\Alg,H_A,\Dirac_A)$ and
$(\Balg,H_B,\Dirac_B)$ satisfy Rieffel's metric condition then so does the spectral triple $(\mathcal{E}, H, \Dirac)$ so that $(\mathcal{E}, L_{\Dirac})$ is a spectral metric space.
We go on to show that there are numerous examples of C$^*$-algebra extensions which can be given the structure of
a spectral metric space. Our main focus is the single-parameter noncommutative (quantum) spheres $C(S_q^n)$ for $n
\geq 2$, which can be iteratively defined as C$^*$-algebra extensions of smaller noncommutative spheres. We shall
specifically study the cases $n = 2$ (the equatorial Podleś spheres) and $n = 3$ (the quantum $\textup{SU}_q(2)$ group) and merely comment on how these two examples can be used to study their higher dimensional counterparts.
The noncommutative spheres have garnered a lot of attention in the literature as examples of noncommutative
manifolds and many spectral triples have been suggested (e.g. <cit.>, <cit.>, <cit.>), though most of these from a very different perspective to ours, for example by looking at the representation theory of the ordinary $\textup{SU}(2)$ group and focusing on those triples which behave equivariantly with respect to the group co-action.
We remark that Chakraborty and Pal also considered the question of finding Lip-metrics starting from given ones on the ideal and quotient for
extensions of the same type as ours <cit.>. However, our goal is to construct spectral triples, rather than compact quantum metric spaces. Our spectral triples give rise to Lip-metrics with properties similar to theirs but we have existence results for Dirac operators and our constructions seem to be fairly different.
Acknowledgement: We are very much indebt to the anonymous referees and would like to thank them for many very helpful comments which improved the paper considerably.
§ A REVIEW OF SPECTRAL TRIPLES AND QUANTUM METRIC SPACES
§.§ Spectral Triples.
We begin with a short exposition of spectral triples.
For more information and context, we recommend the articles <cit.>, <cit.> and <cit.> which provide an excellent exposition of the theory and motivation behind spectral triples. We remind the reader that $C^*$-algebras and Hilbert spaces are assumed to be separable throughout this article.
Let $A$ be a separable C$^*$-algebra. A spectral triple $(\Alg,\Hil,\Dirac)$
on $A$ is given by a faithful $^*$-representation $\pi: A
\mapsto B(\Hil)$ on a Hilbert space $\Hil$, a dense $^*$-subalgebra $\Alg \subseteq A$ and a linear densely defined
unbounded self-adjoint operator $\Dirac$ on
$\Hil$ such that
* $\pi(\Alg) \textup{dom} (\Dirac) \subseteq \textup{dom} (\Dirac)$ and $[\Dirac, \pi(a)]: \textup{dom}(\Dirac)
\to \Hil$ extends to a bounded operator for each $a \in \Alg$ and
* $\pi(a)(I + \Dirac^2)^{-1}$ is a compact operator for each $a \in A$.
Unlike the majority of definitions provided in the literature, we do not make the assumption that the C$^*$-algebra $A$ is unital, or indeed that the representation over $\Hil$ is nondegenerate. Using faithfulness on the other hand we may identify $A$ with $\pi(A)$ and therefore omit the representation from notation, in particular writing $a\xi$ for $\pi(a)\xi$.
For a spectral triple on $A$, it is sometimes convenient to study the maximal Lipschitz subalgebra,
$C^1(A)$ which comprises those elements $a \in A$ such that $\pi(a) (\textup{dom}(\Dirac)) = a( \textup{dom}(\Dirac)) \subseteq \textup{dom}(\Dirac)$, the operator $[\Dirac, \pi(a)]: \textup{dom} (\Dirac) \to \Hil$ is closable and $\delta_{\Dirac}(a) := \textup{cl}[\Dirac, \pi(a)]$ is a bounded operator in $B(\Hil)$. It is an analogue of the algebra of Lipschitz continuous functions on a Riemannian spin$^C$ manifold.
It is not immediately obvious, as one might think, that $C^1(A)$ is a $^*$-algebra. This follows
from the fact that if any two elements $a$ and
$b$ leave the domain of $\Dirac$ invariant then so does $ab$ and the operator $[\Dirac,ab] = [\Dirac,a]b +
a[\Dirac,b]$, defined on $\textup{dom}(\Dirac)$, extends
to a bounded operator in $B(\Hil)$. It is less clear that $C^1(A)$ is closed under involution. We are grateful to Christensen for pointing out the following way to show this. In <cit.>, he shows that the above
condition can be replaced by requiring the sesquilinear form $S([\Dirac,a])$, defined on $\textup{dom}(\Dirac) \times \textup{dom}(\Dirac)$ as
\begin{eqnarray*} S([\Dirac,a])(\xi,\eta) :=
\ip{a\xi}{\Dirac \eta} - \ip{a\Dirac\xi}{\eta}, \;\xi,\eta \in \textup{dom}(\Dirac),
\end{eqnarray*}
to be defined and bounded. The equality $S([\Dirac,a^*])(\xi,\eta) = -S([\Dirac,a](\eta,\xi))^*$ ensures that
$C^1(A)$ is closed under involution. It is well known that $C^1(A)$ becomes an operator algebra when equipped with
the norm $\|a\|_1 := \|\pi(a)\| + \|[\Dirac,\pi(a)]\|$ and viewed as a concrete subalgebra of the bounded
operators on the first Sobolev space, $\Hil_1 := \textup{dom}(\Dirac)$, of $\Hil$, the latter being a Hilbert space with
respect to the inner product $\ip{\eta_1}{\eta_2}_1 := \ip{\eta_1}{\eta_2} + \ip{\Dirac \eta_1}{\Dirac \eta_2}$.
Depending on the context, it can be useful to think of $C^1(A)$ as either a dense $^*$-subalgebra of $A$ or as a
Banach algebra in its own right.
Recall that a spectral triple $(\Alg, \Hil, \Dirac)$ on a unital C$^*$-algebra is called
$p$-summable, (sometimes $(p,\infty)$-summability), where $p \in (0,\infty)$, if
$(I + \Dirac^2)^{-p/2} \in B(\Hil)$ lies in the Dixmier class which is strictly larger than the trace class.
The spectral dimension of $(\Alg,\Hil,\Dirac)$, defined on a unital C$^*$-algebra $A$, is
given by
\begin{eqnarray*}
s_0(\Alg,\Hil,\Dirac) = \inf\{ p \in (0,\infty) : \; (\Alg,\Hil,\Dirac) \textup{ is } p-\textup{summable} \}
\end{eqnarray*}
It can be shown that
\begin{eqnarray*}
s_0(\Alg,\Hil,\Dirac) =\inf\{ p \in (0,\infty): \;\textup{Tr}(I + \Dirac^2)^{-p/2} < \infty\}.
\end{eqnarray*}
Here $\textup{Tr}$ is the usual unbounded trace on $B(H)$.
We will often write $s_0(\Dirac)$ instead of $s_0(\Alg,\Hil,\Dirac)$ and employ this notation also for the summability of an unbounded essentially self-adjoint operator.
Summability can also be defined for spectral triples of non-unital C$^*$-algebras, as advocated by Rennie in <cit.>.
Because of the relationship between spectral triples and Fredholm modules in K-homology, spectral triples are
often distinguished into odd and
even varieties:
A spectral triple on $A$ is called graded or even if there exists an operator $\gamma \in B(\Hil)$ such that
$\gamma^2 = \textup{id}$, $\gamma \pi(a) = \pi(a) \gamma$ for each $a \in A$ and $\gamma \Dirac = -
\Dirac \gamma$. Otherwise it will be called
ungraded or odd.
Stated in a different way, an even spectral triple is one which can be formally represented via a direct sum representation over an orthogonal direct sum of Hilbert spaces $\Hil = \Hil^+ \oplus \Hil^-$ over which $\pi$ and
$\Dirac$ decompose as
\begin{eqnarray*}
\pi = \begin{bmatrix} \pi_0^+ & 0 \\ 0 & \pi_0^- \end{bmatrix},
\;\;\;\;\;
\Dirac =
\begin{bmatrix} 0 & \Dirac^- \\
\Dirac^+ & 0 \end{bmatrix},
\;\;\;\;\;
\gamma =
\begin{bmatrix} 1 & 0 \\
0 & -1 \end{bmatrix}.
\end{eqnarray*}
§.§ Compact quantum metric spaces.
One of the most interesting aspects of spectral triples in differential geometry is the possibility to recover the metric information of the manifold from the spectral triple. Connes <cit.> generalises this observation by showing that a spectral triple $(\Alg, \Hil,\Dirac)$ on a
C$^*$-algebra $A$ defines an extended metric (i.e. allowing the metric to take the value $\infty$)
$d_C: S(A) \times S(A) \to [0,\infty]$ on
the state space $S(A)$ of $A$, by the formula
\begin{eqnarray*}
d_C(\omega_1, \omega_2) := \sup\{|\omega_1(a) - \omega_2(a)|: \;a = a^* \in \Alg,
\;\|[\Dirac,\pi(a)]\| \leq 1\}
\end{eqnarray*}
Connes' metric $d_C$ in general depends on the algebra $\Alg$, so it is often better to write $d_{\Alg}$ to
this dependence. The motivating example is prescribed by the Dirac triple on a connected spin$^c$ manifold
$\mathcal{M}$ defined on the dense subalgebra of
“C$^{\infty}$-functions" for which $\|[\Dirac, f]\| = \|df\|$. The restriction of Connes' metric to the point
evaluation measures $d_C(p_x, p_y)$ then coincides
with the geodesic metric $d_{\gamma}(x,y)$ along $\mathcal{M}$.
In <cit.>, <cit.> and <cit.>, Rieffel considered the more general setting of Lipschitz seminorms, which can be viewed as a generalisation of
metric spaces, or Lipschitz functions, to order-unit
spaces. The theory is based on the observation of Kantorovich and Rubinstein, who demonstrated that a metric on a
compact topological space can be extended
naturally to the set of probability measures on that space. Recent work by Latrémolière in <cit.> and <cit.> has extended much
of this work to the setting of non-unital
In the context of this paper, a Lipschitz seminorm on a separable C$^*$-algebra $A$ is a seminorm $L:
\Alg \to \R^+$ defined on a dense subalgebra
$\Alg$ which is closed under involution with the property $L(a^*) = L(a)$ for each $a \in \Alg$ and also $L(1) =
0$ whenever $\Alg$ is unital. We say that a Lipschitz seminorm $L$ is
nondegenerate if the set $\{a \in \Alg: \;L(a) = 0\}$ is trivial or contains only multiples of the
identity when $A$ is unital. As pointed out in <cit.>, nondegeneracy of $L$ is independent of the choice of the dense subalgebra $\Alg$, but it should be stressed that many of the properties of the Lipschitz seminorm will depend on $\Alg$.
A Lipschitz seminorm on $A$ determines an extended metric $d_{\Alg,L}$ on $S(A)$ (occasionally written
$d_{\Alg}$, or $d_L$) in a way which provides a noncommutative analogue of the Monge-Kantorovich distance when $A$
is commutative. The metric is given by
\begin{eqnarray}
\label{NoncKan} d_{\Alg,L}(\omega_1,\omega_2) := \sup \{|\omega_1(a) - \omega_2(a)|: \;a \in \Alg, \;L(a) \leq
\end{eqnarray}
Conversely a metric $d$ on $S(A)$ defines a nondegenerate seminorm $L_d$ on $A$ via
\begin{eqnarray}
\label{Recov} L_d(a) := \sup \bigg{ \{ } \frac{|\mu(a) -
\nu(a)|}{d(\mu,\nu)}: \; \mu,\nu \in S(A),\;\mu \neq \nu \bigg{ \} }.
\end{eqnarray}
If $L$ is a Lipschitz seminorm on $A$, so is $L_{d_{\Alg,L}}$. When the
Lipschitz seminorm $L$ is lower semicontinuous, so that the set $\{a \in \Alg: \; L(a) \leq r\}$ is
closed in $A$ for any and hence all $r > 0$, then $L =
L_{d_{\Alg,L}}$. We shall further call $L$ closed if $L$ is lower semicontinuous and $\Alg =
\textup{dom}(L_{d_{\Alg,L}})$. Hence, starting from a lower
semicontinuous seminorm, the above procedure can be used to extend $L$ to a closed seminorm. All these
observations are well known in the case when $\Alg$ is
unital and the procedure of replacing $\Alg$ with its unitisation $\overline{\Alg} = \Alg \oplus \C I$ and
introducing the new seminorm $\overline{L}(a,\lambda) :=
L(a)$ can easily be used to generalise these results to the non-unital case.
Let $(\Alg, \Hil, \Dirac)$ be a spectral triple over a C$^*$-algebra $A$ with faithful representation
$\pi: A \to B(\Hil)$ such that $[\Dirac,\pi(a)] = 0 \iff a \in \C I_A$. Then $L_{\Dirac}(a) := \|[\Dirac, \pi(a)]\|$, defines a
lower semicontinuous Lipschitz seminorm on $\Alg$,
which is closed if and only if $\Alg = C^1(A)$. If the representation $\pi$ is nondegenerate and the spectral
triple comes with a cyclic vector $\xi$ for $(A,\pi)$
such that $\ker{\Dirac} = \C \xi$ then Connes' extended metric on $S(A)$ is a metric.
Rieffel addresses the question of whether a metric induced by a Lipschitz seminorm on a unital separable C$^*$-algebra (or order-unit space) has finite diameter and whether it induces the weak-$*$-topology of $S(A)$ which is a compact metrisable Hausdorff space in this situation.
To state his result we introduce some notation: for a given Lipschitz seminorm $L$ on $\Alg$, define
B_L(\Alg) = \{ a \in \Alg : L(a) \leq 1\}, \;\;\;\;\;
\widetilde{B}_L(\Alg) := \{\tilde{a} \in \Alg / \C I : L(a) \leq 1\}$$
(note that $L$ passes to the quotient $\Alg / \C I $) and
$$B_{1,L}(\Alg) := \{ a \in B_L(\Alg) : \|a\| \leq 1\}=B_L(\Alg) \cap \overline{\textup{B}}_A,$$
where $\overline{\textup{B}}_A$ is the closed unit ball in $A$.
(<cit.> 1.8 and 1.9, <cit.>) Let $A$ be a unital C$^*$-algebra, equipped with a nondegenerate Lipschitz seminorm $L$ on a dense $*$-subalgebra $\Alg$ of $A$. Then:
* Equation (<ref>) determines a metric $d_{L,\Alg}$ of finite diameter if and only if $\widetilde{B}_L(\Alg)
\subseteq A / \C I$ is
norm-bounded, and further diam$(\widetilde{B}_L(\Alg), \| \cdot \|_{A / \C I}) \leq r$ if and only if
diam$(S(A),d_L) \leq 2r$, for each $r > 0$.
* $d_{\Alg,L}$ metrises the weak-$*$-topology of $S(A)$ if and only if the following conditions are satisfied.
* $d_{\Alg,L}$ has finite diameter.
* $B_{1,L}(\Alg) \subseteq A$ is totally bounded in norm.
We will refer to the conditions 2.(a) and 2.(b) in Prop.<ref> as Rieffel's metric conditions or just metric conditions.
The situation when $A$ is non-unital is rather more complicated, but Latrémolière (<cit.>) shows that, provided one works only with Lipschitz seminorms which give a metric on
$S(A)$ with finite diameter, things are not too complicated. For this case he provides conditions similar to the ones in the preceeding Proposition <ref>
which characterise those seminorms that induce the weak-$*$-topology on $S(A)$.
Let $A$ be a separable C$^*$-algebra equipped with a Lipschitz seminorm $L$ on a suitable
dense subalgebra $\Alg$ with the property that $d_{\Alg,L}$ determines a metric of finite diameter inducing the weak-$*$-topology of $S(A)$. Then the pair $(\Alg, L)$ is called a
quantum metric space (or compact quantum metric space when $\Alg$ is unital).
Thus $(\Alg, L)$ with $\Alg$ unital will be compact quantum metric space if and only if $(\Alg, d_{\Alg,L})$ satisfies Rieffel's metric conditions.
The final definition is motivated by a similar definition in <cit.> which we will follow in this paper.
Let $(\Alg, H, \Dirac)$ be a spectral triple with corresponding Lipschitz seminorm $L_{\Dirac}$. If $(\Alg, L_{\Dirac})$ is a quantum metric space, then $(\Alg, H, \Dirac)$ (or $(\Alg, L_{\Dirac})$) is called a spectral metric space.
§ EXTENSIONS AND KASPAROV'S KK-THEORY.
In this section we recall and develop some background in KK-theory related to extensions. Further information can be found in Kasparov's seminal paper <cit.> and in <cit.>, <cit.>.
For a separable C$^*$-algebra $A$, we write $\ell_2(A)$ to mean the Hilbert bimodule $A$ consisting of sequences of the form $(a_n)_{n \in \N}$ such that $\sum_n a_n^* a_n$ converges in norm, equipped with the inner product $\ip{(a_n)}{(b_n)} := \sum_n a_n^* b_n$. We write
$\mathcal{L}_A$ to mean the set of adjointable right $A$-linear operators on $\ell_2(A)$, which becomes a C$^*$-algebra when equipped with the operator norm. We denote by $\mathcal{K}_A$ the C$^*$-subalgebra of $\mathcal{L}_A$ consisting of the closed linear span of operators of the form
$\theta_{x,y}(z) = x\ip{y}{z}, \;x,y,z \in
\ell_2(A)$. Then $\mathcal{K}_A$ is an ideal in $\mathcal{L}_A$ and is isomorphic to the spatial tensor product, $\mathcal{K} \otimes A$, of $A$ by the compact operators on a separable infinite dimensional Hilbert space. The algebra $\mathcal{L}_A$ is isomorphic to $\mathcal{M}(\mathcal{K} \otimes A)$.
We denote the quotient $\mathcal{L}_A
/ \mathcal{K}_A$ by $ \mathcal{Q}_A$ and will also use the notation $\mathcal{L}$ for $B(\ell^2)$ and $\mathcal{Q}$ for the quotient $B(\ell^2)/\mathcal{K}$.
§.§ Background and set-up
The extensions in this article are unital short exact sequences of separable C$^*$-algebras of the form,
\begin{eqnarray}
\label{ExtStableIdeal} \xymatrix{ 0 \ar[r] & \mathcal{K} \otimes B \ar[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] &
0}. \label{TopEx}
\end{eqnarray}
Recall that this means $\iota$ is an injective $^*$-homomorphism and regarded as an inclusion map, $\sigma$ is a
surjective $^*$-homomorphism and $\im(\iota) = \ker (\sigma)$. We will always assume that the C$^*$-algebras $A$ and $B$
are unital and that $\mathcal{K} \otimes B$ is the stabilisation of $B$ by compact operators on a
separable, infinite dimensional Hilbert space. Additionally we will always require the ideal $\mathcal{K} \otimes
B$ to be essential, i.e. it has non-zero intersection with any other ideal $I \subseteq E$.
The Busby invariant of (<ref>) is a $^*$-homomorphism $\psi: A \to \mathcal{L}_B
/ \mathcal{K}_B =: \mathcal{Q}_B$. The $^*$-homomorphism $\psi$ can be regarded as a characteristic of the extension itself, since the
extension can be recovered from $\psi$, up to isomorphism, as the pullback C$^*$-algebra,
\begin{eqnarray}
E \cong \mathcal{L}_B \oplus_{(q_B, \psi)} A := \{(x,a) \in \mathcal{L}_B \oplus A: \;q_B(x) = \psi(a)\} \label{pull-back}
\end{eqnarray}
(here, $q_B: \mathcal{L}_B \to \mathcal{Q}_B$ is the quotient map). The assumptions above imply that $\psi$ and consequently the map $\pi: E \to \mathcal{L}_B, \;\pi(x,a) = x$ is injective. The maps fit together in the commuting diagram
\begin{eqnarray}\label{commuting diagram}
\xymatrix{ 0 \ar[r] & \mathcal{K}_B \ar@{^{(}->}[r]^{\iota} \ar[d]^{\pi|_{\overline{B}}} & E \ar[r]^{\sigma}
\ar[d]^{\pi} & A \ar[r] \ar[d]^{\psi}
& 0\\ 0 \ar[r] & \mathcal{K}_B \ar@{^{(}->}[r]^{\iota_B} & \mathcal{L}_B \ar[r]^{q_B} & \mathcal{Q}_B \ar[r] &
\end{eqnarray}
We do not consider all such extensions, but restrict our attention to the situation in which the Busby invariant $\psi$ admits a unital completely positive lift, i.e., there is a unital completely positive map $s: A \to \mathcal{L}_B$ such that $q_B \circ s = \psi$. This is equivalent to the existence of a ucp lift of $\sigma$. Such extensions are called semisplit.
A well known application of the Kasparov-Stinespring Theorem shows that, in this setting, there is a faithful representation $\rho: A \to M_2(\mathcal{L}_B) \cong \mathcal{L}_B$ and an orthogonal projection $P \in M_2(\mathcal{L}_B) \cong \mathcal{L}_B$ such that $[P, \rho(a)] \in M_2(\mathcal{K}_B)$ and $\rho_{11}(a) = s(a)$
for each $a \in A$, where,
\begin{eqnarray*}
\rho = \begin{pmatrix} \rho_{11} & \rho_{12} \\ \rho_{21} & \rho_{22} \end{pmatrix}; \;\;P = \begin{pmatrix} 1 & 0
\\ 0
& 0 \end{pmatrix}.
\end{eqnarray*}
We call $(\rho, P)$ a Stinespring dilation of $s: A \to \mathcal{L}_B \cong \mathcal{M}(\mathcal{K} \otimes B)$.
The existence of such a map is not automatic, unless $A$ is a nuclear C$^*$-algebra in which case the existence of a completely positive lift follows from the Choi-Effros lifting theorem.
The Stinespring dilation $(P, \rho)$ can be used to define a Kasparov cycle $\psi^*$ which is the element of $KK^1(A,B)$
represented by the triple $(\ell_2 (B) \oplus \ell_2(B), \rho, 2P-1)=(\ell_2 (B) , \rho, 2P-1) $. A well known result of Kasparov (<cit.>) says that there is a six-term
exact sequence in both K-theory and K-homology. In the case of K-homology the sequence has the form
\begin{eqnarray}
\label{SixTerm} \xymatrix{K^0(A) \ar[r]^{\sigma^*} & K^0(E) \ar[r]^{\iota^*} & K^0(B) \ar[d]^{\delta_0^*} \\
\ar[u]^{\delta_1^*} & \ar[l]^{\iota^*} K^1(E) & \ar[l]^{\sigma^*} K^1(A), }
\end{eqnarray}
where the boundary maps are defined by taking the internal Kasparov product with $\psi^*$.
§.§ Toeplitz type extensions and KK-theory
In this section we discuss a characterisation of Toeplitz type extensions showing that they form a large class. Moreover, we will introduce the representations of the extension algebra which are relevant in order to define our spectral triple on the extension algebra.
In what follows, we will assume $B$ is unital and we shall let $j: \mathcal{L} \to \mathcal{L}_B$,
$\bar{\jmath}: \mathcal{Q} \to \mathcal{Q}_B$, $q: \mathcal{L} \to \mathcal{Q}$ and $q_B: \mathcal{L}_B
\mapsto \mathcal{Q}_B$ be the natural maps, so that $q_B \circ j = \bar{\jmath} \circ q$. The main result of this section is contained in the following:
Given an extension (<ref>), where $A$ and $B$ are separable C$^*$-algebras and $\psi: A \to \mathcal{Q}_B$ is the Busby invariant of this extension, the following are equivalent.
* There is a $^*$-homomorphism $\psi_0: A \to \mathcal{Q}$ such that $\bar{\jmath} \circ \psi_0 = \psi$.
* There is a C$^*$-algebra $E_0$, an injective $^*$-homomorphism $r: E_0 \to E$, an injective $^*$-homomorphism $\pi_0: E_0 \to \mathcal{L}$ and a surjective $^*$-homomorphism $\sigma_0: E_0 \to A$ such that the following diagrams commute:
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & \mathcal{K} \ar@{^{(}->}[r]^{\iota_0} \ar[d]^{r|_{\mathcal{K}}} & E_0 \ar[r]^{\sigma_0}
\ar[d]^{r} & A \ar[r] \ar@{=}[d] &
0\\ 0 \ar[r] & \mathcal{K}_B \ar@{^{(}->}[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] & 0}
\label{character1}
\end{eqnarray}
\begin{eqnarray}
\xymatrix{ E_0 \ar[r]^{\pi_0} \ar[d]^{r} &
\mathcal{L} \ar[d]^{j} \\ E \ar[r]^{\pi} & \mathcal{L}_B.}
\label{ch2}
\end{eqnarray}
$(1) \implies (2)$: Starting from a
homomorphism $\psi_0: A \to \mathcal{Q}$ as above, we can define $E_0$ as the pullback C$^*$-algebra
\begin{eqnarray*}
E_0 := B(\ell^2) \oplus_{(q, \psi_0)} A :=
\{(x,a) \in B(\ell^2) \oplus A : q(x) = \psi_0(a)\}.
\end{eqnarray*}
There are natural maps $\pi_0: E_0 \to B(\ell_2)$ and $\sigma_0: E_0 \to A$. Similarly, as pointed out in (<ref>) $E$ is given as a pullback $E= \mathcal{L}_B \oplus_{(q_B,\psi)} A$, so that we also have natural maps $\pi : E \to \mathcal{L}_B$ and $\sigma : E \to A$. Notice that $\ker(\sigma_0) = \ker (q \circ \pi_0)$ is isomorphic to the algebra of compact operators. The map $r: E_0 \to E$ can be defined by $r((x,a)) := (j \circ \pi_0(x), a)$, which is injective and thus establishes the first part of the proof.
$(2) \implies (1)$: Let $\psi_0$ be the map which is given by
\begin{eqnarray*}
\psi_0(\sigma_0(e)) = q(\pi_0(e))).
\end{eqnarray*}
This map is well defined: if $e_1, e_2 \in E_0$ are such that $\sigma_0(e_1) = \sigma_0(e_2)$ then $e_1 - e_2$ is a compact operator, so that $q(\pi_0(e_1 - e_2)))$ vanishes. For any $e \in
E_0$, we have
\begin{eqnarray*}
(\bar{\jmath} \circ \psi_0 \circ \sigma_0)(e) = (\bar{\jmath} \circ q \circ \pi_0)(e) && \textup{(by definition of $\psi_0$)} \\
= (q_B \circ j \circ \pi_0)(e) && \textup{(since $q_B \circ j = \bar{\jmath} \circ q$)} \\
= (q_B \circ \pi \circ r)(e) && \textup{(from diagram (\ref{ch2}))} \\
= (\psi \circ \sigma \circ r)(e) && \textup{(from diagram (\ref{commuting diagram}))} \\
= (\psi \circ \sigma_0)(e)&& \textup{(from diagram (\ref{character1})}.
\end{eqnarray*}
Since $\sigma_0$ is surjective, $\bar{\jmath} \circ \psi_0 = \psi$, completing the proof.
It is clear that in the setting of the last Proposition <ref>, the map $\psi_0$ is injective if and only if $\psi$ is injective, so that we can assume the extension corresponding to the top row of (<ref>) is essential. If $s: A \to
\mathcal{L}$ is a completely positive lift for $\psi_0$ then $j \circ s$ is a completely positive lift for $\psi$.
To apply this to our Toeplitz type extensions recall (Def.<ref>) that an extension (<ref>) is of Toeplitz type if $\pi_A: A \to B(H_A)$ and $\pi_B: B \to B(H_B)$ are faithful representations with $[P, \pi_A(a)] \in \mathcal{K}(H_A)$ for each $a \in A$, $P \pi_A(a)P \; \cap \;\mathcal{K}(P H_A) = \{0\}$,
and $E$ is isomorphic to the subalgebra of $B(H_A \otimes H_B)$ generated by $\mathcal{K}(PH_A) \otimes \pi_B(B)$ and $P\pi_A(A)P \otimes \C I_B$. Thus, omitting the representations, the extension is of the form
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & \mathcal{K} \otimes B \ar[r]^{} & \mathcal{K}(PH_A) \otimes B + PAP \otimes \C I_B \ar[r]^{} & A \ar[r] & 0.}
\label{E}
\end{eqnarray}
$(\pi_A, \pi_B, P)$ is called the corresponding Toeplitz triple. From this extension we obtain the extension
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & \mathcal{K}(PH_A) \ar[r]^{} & E_0 \ar[r] & A \ar[r] & 0}.
\label{E_0}
\end{eqnarray}
of $A$ by $\mathcal{K}$,
where $E_0= P\pi_A(A) P + \mathcal{K} (PH_A)$. There is a natural inclusion map $r: E_0 \hookrightarrow E$. Moreover, $E_0$ embedds naturally into $\mathcal{L}\cong B(H_A)$ defining a degenerate (i.e. non-unital) but faithful representation $\pi_0 : E_0 \to \mathcal{L}$.
Similarly, there is a degenerate but faithful representation $\pi : E \to B(H_A \otimes H_B)$ given by its very definition as a subalgebra. Since $H_A$ is separable and infinite dimensional we have
$\mathcal{L}_B \cong \mathcal{M}(\mathcal{K} \otimes B) \cong \mathcal{M}(\mathcal{K}(H_A) \otimes \pi_B (B)) \subseteq B(H_A \otimes H_B)$. We can therefore think of $\pi$ as a representation
$\pi : E \to \mathcal{L}_B$. There is a natural inclusion $r:E_0 \to E$ (using that $B$ is unital) such that the diagrams in
Prop.<ref>.(2) commute.
Note that the Busby invariants for the extension $E$ and $E_0$ are given by
\psi(a)= q_B((P\otimes I ) (\pi_A(a) \otimes I) (P \otimes I))
\psi_0 (a) = q(P \pi_A(a) P),
which implies that $\bar{\jmath} \circ \psi_0 = \psi$. Thus starting from a Toeplitz type extension $E$ of $A$ by $\mathcal{K} \otimes B$ we found an extension $E_0$ of $A$ by $\mathcal{K}$ satisfying the conditions (1) and, hence, (2) of Prop.<ref>.
We mention that with this interpretation $s(a):= P \pi_A(a)P \otimes 1$ can be regarded as a completely positive map $s : A \to \mathcal{M}(\mathcal{K}(PH_A) \otimes B)$ such that $q \circ s : A \to \mathcal{Q} (\mathcal{K} (P H_A) \otimes B)$ is the Busby invariant of the extension. Hence $s$ is a cp-lift of the extension and $(P \otimes 1, \pi_A \otimes 1)$ can be regarded as Stinespring dilation of the semisplit extension (<ref>).
Let $E$ be a Toeplitz type extension (<ref>) with corresponding extension (<ref>). Then, with $\pi, \pi_0, r, \psi, \psi_0$ as defined above, condition (1) and (2) of Prop.<ref> are satisfied.
Conversely, given an essential semisplit extension
\begin{eqnarray*}
\xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r]^{\iota} & E_0 \ar[r] & A \ar[r] & 0}.
\end{eqnarray*}
there exists faithful representations $\pi_A : A \to B(H_A)$ and $\pi_B : B \to B(H_B)$ and $P \in B(H_A)$ an infinite dimensional projection such that $E_0 \cong P\pi_A(A) P + \mathcal{K} (PH_A)$ and if we define $E$ by $E= \mathcal{K}(PH_A) \otimes B + PAP \otimes \C I_B$ and $\pi, \pi_0, r, \psi, \psi_0$ as before then (1) and (2) of Prop.<ref> are satisfied.
The first part follows from the discussion preceeding the Corollary. For the second part it is known (<cit.>, 2.7.10) that the required representation $\pi_A $ and projections can be found for every semisplit extension of the form $0 \to \mathcal{K} \to E_0 \to A \to 0$ (it is given by the Stinespring dilation we decribed.) Once we have that we can use any faithful representation $\pi_B : B \to B(H_B)$ and define $E$ and $\pi, \pi_0, r, \psi, \psi_0$ as before satisfying the required identities.
When a Toeplitz triple exists, we have our $^*$-homomorphism $\pi : E \to B(H_A \otimes H_B)$ given on the generators via
\begin{eqnarray*}
\pi(k \otimes b)(\eta \otimes \nu) &:= &k\eta \otimes \pi_B(b)\nu, \\
\pi(PaP \otimes I_B)(\eta \otimes \nu) &:=&
P\pi_A(a)P \eta \otimes \nu,
\end{eqnarray*}
where $a \in A$ and $k \in \mathcal{K}(PH_A)$ which is faithful but degenerate (i.e. not unital). We also have another representation $\pi_{\sigma}: E \to B(H_A \otimes H_B)$ given by
\pi_{\sigma} = \pi_A \circ \sigma \otimes 1,
where $\sigma : E \to A$ is the quotient map in the extension, given by
\sigma(k \otimes b) := 0, \textup{ and } \sigma(PaP \otimes I_B) := a,
where $a \in A$, $b \in B$ and $k \in \mathcal{K}(PH_A)$.
The representation $\pi_{\sigma}$ is non-degenerate (unital) but not faithful.
§ CONSTRUCTION OF THE SPECTRAL TRIPLE.
We will now begin to describe the steps needed to construct a spectral triple on an extension
(<ref>) of Toeplitz type with Toeplitz triple $(\pi_A,\pi_B,P)$ introduced earlier in Def.<ref>.
§.§ Smoothness criteria.
In this section we want to discuss the smoothness conditions in Def.<ref> which we need for our main result. As stated in Def.<ref> given a Toeplitz type extension
\begin{eqnarray*}
\xymatrix{ 0 \ar[r] & \mathcal{K} \otimes B \ar[r]^{} & \mathcal{K}(PH_A) \otimes B + PAP \otimes \C I_B \ar[r]^{} & A \ar[r] & 0}
\end{eqnarray*}
and a spectral triple $(\Alg, H_A, \Dirac_A)$ on $A$ we say that the quadruple $(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type if
* $P$ and $\Dirac_A$ commute,
* $[P, \pi_A (\Alg)] \subseteq \mathcal{C}(H_A) $,
where $\mathcal{C}(H_A)$ was discussed in the introduction and is formally defined below.
Condition (1) means that $P$ should leave the domain of $\Dirac_A$ invariant and
commute with each of the spectral projections of
$\Dirac_A$, so that the operator $[\Dirac_A, P]: \textup{dom}(\Dirac_A) \to H_A$ vanishes. Thus we can decompose $\textup{dom}(\Dirac_A)$ as an orthogonal direct sum of vector spaces,
\begin{eqnarray*}
(\textup{dom}(\Dirac_A) \cap PH_A ) \;\;\oplus \;\; (\textup{dom}(\Dirac_A) \cap (1-P)H_A),
\end{eqnarray*}
which are dense in $PH_A$ and $(1-P)H_A$ respectively. $\Dirac_A$ decomposes as a diagonal block matrix $\Dirac_A^p|_{P H_A} \oplus
\Dirac_A^q|_{(1-P) H_A}$ with respect to this decomposition, where $\Dirac_A^p := P \Dirac_A$ and $\Dirac_A^q := (1-P) \Dirac_A$.
To discuss the second condition we formally introduce the notion of differentiability for compact operators in the next definition.
We define the dense subalgebra of $\Dirac_A$-differentiable compacts, $\mathcal{C}(H_A) \subseteq \mathcal{K}(H_A)$, to be the algebra of all compact operators $y \in \mathcal{K}(H_A)$ such that,
* $y (\textup{dom}(\Dirac_A)) \subseteq \textup{dom}(\Dirac_A)$,
* the operators $y\Dirac_A: \textup{dom}(\Dirac_A) \to H_A$ and $\Dirac_A y: \textup{dom}(\Dirac_A) \to H_A$ are closable,
* the closures, $\textup{cl}(y \Dirac_A)$, $\textup{cl}(\Dirac_A y)$ respectively, are bounded operators.
Our motivation for choosing the term $\Dirac_A$-differentiable compacts is based on the following observation: the same information as was given above can be used to write down an even spectral triple on the algebra of compact operators. It is given by the triple
\begin{eqnarray*}
\begin{pmatrix} \mathcal{C}(H_A ), & \textup{id} \oplus 0, & \Dirac := \begin{bmatrix} 0 & \Dirac_A
\\[1ex] \Dirac_A & 0 \end{bmatrix} \end{pmatrix}.
\end{eqnarray*}
We note that $\mathcal{C}(H_A)$ can be viewed as a Banach $^*$-algebra when equipped with the norm $\|y\|_1
:= \|y\| + \max\{\|\Dirac_A y\| ,\|y \Dirac_A\|\}$. Thus $\mathcal{C}(H_A)$ plays the role of the `differentiable' elements with respect to this choice of spectral triple.
Let $(\Alg, H_A, \Dirac_A)$ be a spectral triple on $A$ and let $P \in B(H_A)$ be an orthogonal projection commuting with $\Dirac_A$. The three following conditions are equivalent:
* $[P, \pi_A(a)] \in \mathcal{C}(H_A)$ for each $a \in \Alg$,
* $[\Dirac_A^p, \pi_A(a)]$ and $[\Dirac_A^q, \pi_A(a)]$ extend to bounded operators in $B(H_A)$ for each $a \in \Alg$,
* $[(2P-1)\Dirac_A, \pi_A(a)]$ extends to a bounded operator in $B(H_A)$ for each $a \in \Alg$.
If (1) holds, then the operators
\begin{eqnarray*}
&& [\Dirac_A^p, \pi_A(a)] = P[\Dirac_A, \pi_A(a)] + [P,\pi_A(a)]\Dirac_A, \\ && [\Dirac_A^q, \pi_A(a)] =
(1-P)[\Dirac_A, \pi_A(a)] - [P,\pi_A(a)]\Dirac_A,
\end{eqnarray*}
viewed as operators on $\textup{dom}(\Dirac_A)$, are bounded for each $a \in \Alg$. We want to regard each of these operators as bounded operators in $B(H_A)$. To this end, we remark that the operator $[\Dirac_A^p, \pi_A(a)]$ is closable, since $P[\Dirac_A, \pi_A(a)]$ and $[P,\pi_A(a)]\Dirac_A$ are closable. Writing cl$(T)$ to denote the closure of $T$, and remarking that $\textup{cl}(P[\Dirac_A, \pi_A(a)]) = P \textup{cl}([\Dirac_A, \pi_A(a)])$, we conclude that
\begin{eqnarray}
&& \textup{cl}([\Dirac_A^p,
\pi_A(a)]) = P \textup{cl}([\Dirac_A, \pi_A(a)]) + \textup{cl}([P,\pi_A(a)]\Dirac_A) \\ && \textup{cl}([\Dirac_A^q, \pi_A(a)]) = (1-P)
\textup{cl}([\Dirac_A, \pi_A(a)]) - \textup{cl}([P,\pi_A(a)]\Dirac_A)
\\ && \textup{cl}([{\Dirac_A}, \pi_A(a)]) = \textup{cl}([\Dirac_A^p, \pi_A(a)]) + \textup{cl}([\Dirac_A^q, \pi_A(a)]), \label{third}
\end{eqnarray}
where the third identity follows from the first two. Thus (2) holds.
That (2) implies (3) is immediate from the equation $[(2P - 1)\Dirac_A, \pi_A(a)] = [\Dirac_A^p, \pi_A(a)] - [\Dirac_A^q, \pi_A(a)]$.
Finally, if (3) holds then we recover the identity
\begin{eqnarray*}
[(2P-1)\Dirac_A, \pi_A(a)] = (2P-1)[\Dirac_A, \pi_A(a)] + 2[P,\pi_A(a)]\Dirac_A
\end{eqnarray*}
for each $a \in \Alg$. Arguments similar to the first part of the proof now show that the operator $[P,\pi_A(a)]\Dirac_A$ is closable and extends to a bounded operator in $B(H_A)$, so that (1) holds.
Let us add the following comments on the three equivalent conditions in Proposition <ref>.
The first condition can be compared to a smoothness criterion proposed by Wang in <cit.> whilst the second was studied by Christensen and Ivan in <cit.>. In the special situation in which $P$ is the orthogonal projection onto the span of the positive eigenspaces of $\Dirac_A$ the third condition can be rephrased as requiring that the commutator $[|\Dirac_A|, a]$ is bounded for all $a \in \Alg$. This condition is the first part of Connes and Moscovici's regularity (called smoothness by some authors) which requires for all $a \in \Alg$ that $[\Dirac_A, a]$ and $\delta(a):=[|\Dirac_A|, a]$ are bounded but, moreover, that $a$ and $[\Dirac_A, a]$ both lie in $\bigcap_{n=1}^{\infty} \textup{dom}(\delta^n)$ (<cit.>).
The first condition of Proposition <ref> has already been mentioned in the introduction as a smoothness assumption. We formalise this in the following definition.
Let $(\Alg, H_A, \Dirac_A)$ be a spectral triple on $A$ and let $(\pi_A , \pi_B , P)$ be a Toeplitz triple such that $P$ commutes with $\Dirac_A$. The spectral triple $(\Alg, H_A, \Dirac_A)$ is said to be $P$-regular if the equivalent conditions (1), (2) and (3) of Proposition <ref> hold. (Recall (Def.<ref>) that in this case we say that the quadruple $(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type.)
§.§ The spectral triple on $E$.
In this section we define the Dirac operator on the extension algebra and establish some of its basic properties. We suppose that we have C$^*$-algebras $A$, $B$, $E$, a short exact
sequence (<ref>), spectral triples $(\Alg, H_A, \Dirac_A)$ on $A$ and $(\Balg, H_B, \Dirac_B)$ on
$B$ represented via $\pi_A$ and $\pi_B$ respectively and an orthogonal projection $P \in B(H_A)$ such that
$(\pi_A, \pi_B, P)$ is a Toeplitz triple and the quadruple $(\Alg,H_A,\Dirac_A, P)$ is of Toeplitz type (cf. Defs.<ref> and <ref>).
As pointed out in the introduction, the Dirac operator for the extension algebra $E$ will be a combination of two formulae for Kasparov products.
Recall the definition of the following representations
$$\Pi_1, \Pi_2 : E \longrightarrow B(H_A \otimes H_B \otimes \C^2)$$
given by
\begin{eqnarray}
\Pi_1 = \pi_{\sigma} \oplus \pi_{\sigma} \textup{ and } \Pi_2 = \pi \oplus \pi_{\sigma}
\end{eqnarray}
and consider unbounded operators $\Dirac_{1}, \Dirac_{2}, \Dirac_{3}$ on $H_A \otimes H_B \otimes \C^2$ given by
\begin{eqnarray}
\Dirac_{1} =
\begin{bmatrix} \Dirac_A \otimes 1 & 1 \otimes \Dirac_B \\
1 \otimes \Dirac_B & -\Dirac_A \otimes 1
\end{bmatrix},
\label{Dirac1}
\end{eqnarray}
\begin{eqnarray}
\Dirac_{2} =
\begin{bmatrix} \Dirac_A^q \otimes 1 & \Dirac_A^p \otimes 1 \\
\Dirac_A^p \otimes 1 & -\Dirac_A^q \otimes 1
\end{bmatrix}
\begin{bmatrix} \Dirac_A^q & \Dirac_A^p \\
\Dirac_A^p & -\Dirac_A^q
\end{bmatrix}
\otimes I
=: \bar{\Dirac}_2 \otimes I
\label{Dirac2}
\end{eqnarray}
\begin{eqnarray}
\Dirac_{3} =
\begin{bmatrix} 1 \otimes \Dirac_B & 0 \\
0 & 1 \otimes \Dirac_B
\end{bmatrix}
I \otimes
\begin{bmatrix} \Dirac_B & 0 \\
0 & \Dirac_B
\end{bmatrix} = : I \otimes \bar{\Dirac}_3,
\label{Dirac3}
\end{eqnarray}
and finally
\begin{eqnarray}
\Dirac_I:=
\begin{bmatrix} 0 & \Dirac_2 - i \Dirac_3 \\
\Dirac_2 + i \Dirac_3 & 0
\end{bmatrix},
\label{DiracI}
\end{eqnarray}
an unbounded operator on $H_A \otimes H_B \otimes \C^4$.
To begin with, the $\Dirac_i$ are defined on $\mathbb{D}:= \textup{dom}(\Dirac_A) \odot \textup{dom}(\Dirac_B) \otimes \C^2$ and $\Dirac_I$ on $\mathbb{D} \oplus \mathbb{D}$.
To show that all these operators are essentially self-adjoint we only need to show that each of them possesses a complete orthonormal basis of eigenvectors. Indeed note that if $T: H \to H$ is an unbounded linear operator on a complex separable Hilbert space $H$ with a complete set of orthonormal eigenvectors $(\xi_n) \subseteq H$ and corresponding sequence of real eigenvalues $(\lambda_n)$ so that $T\xi_n = \lambda_n \xi_n$ and thus $\textup{lin} \{\xi_n : n\in \N\} \subseteq \textup{dom}(T)$, then it is easy to see that $T \subseteq T^* = T^{**}=\textup{cl}(T)$ so that $T$ is essentially self-adjoint. Such an operator will be called diagonalisable (with real eigenvalues).
Not all of the operators defined in (<ref>) - (<ref>) have compact resolvent but $\Dirac_1$ and $\Dirac_I$ do have which can be shown by finding their eigenvalues. This also allows to prove summability results.
Let $\Dirac_i$, $i=1,2,3$ and $\Dirac_I$ be as above. Then
* $\Dirac_i: \mathbb{D} \to H_A \otimes H_B \otimes \C^2$, $i=1,2,3$ and $\Dirac_I : \mathbb{D} \oplus \mathbb{D} \to (H_A \otimes H_B \otimes \C^2)^2$
are essentially self-adjoint.
* $\Dirac_1$ and $\Dirac_I$ have compact resolvent (that is, $(I+\Dirac_1^2)^{-1/2} \in \mathcal{K}(H_A \otimes H_B \otimes \C^2)$ and $(I+\Dirac_I^2)^{-1/2} \in \mathcal{K}((H_A \otimes H_B \otimes \C^2)^2)$).
* If $\Dirac_A$ and $\Dirac_B$ are finitely summable then so are $\Dirac_1$ and $\Dirac_I$. Specifically, if $\textup{Tr} (I + \Dirac_A^2)^{-r/2} < \infty$ and $\textup{Tr}(I + \Dirac_B^2)^{-s/2} < \infty$ then $\textup{Tr}(I + \Dirac_1^2)^{-(r+s)/2} < \infty$ and $\textup{Tr}(I + \Dirac_I^2)^{-(r+s)/2} < \infty$. Hence if $(\Alg,H_A,\Dirac_A)$ and
$(\Balg,H_B,\Dirac_B)$ are respectively $r$-summable and $s$-summable then both $\Dirac_1$ and $\Dirac_I$ are
(1) Note that if $\lambda ,\mu \in \R $ then the eigenvalues of the self-adjoint matrices
\begin{bmatrix} \lambda & \mu \\[1ex] \mu & -\lambda \end{bmatrix}
are $\pm \sqrt{\lambda^2+\mu^2}$, whereas $
\begin{bmatrix} \lambda & 0\\[1ex] 0 & -\lambda \end{bmatrix}
$ and
\begin{bmatrix} 0 & \lambda \\[1ex] \lambda & 0 \end{bmatrix}
both have eigenvalues $\pm \lambda$. Now $\Dirac_A$ and $\Dirac_B$ are diagonalizable with orthonormal bases of eigenvectors $(\xi_m) \subseteq H_A$ of $\Dirac_A$ and $(\eta_n)\subseteq H_B$ of $\Dirac_B$ such that there exists $S \subseteq \N$ with $(\xi_m)_{m \in S}$ is an orthonormal basis of $PH_A$, with real eigenvalue sequences $(\lambda_m)$ and $(\mu_n)$ (i.e. $\Dirac_A \xi_m = \lambda_m \xi_m$ and $\Dirac_B \eta_n = \mu_n \eta_n$) and $|\lambda_m | \to \infty$ and $|\mu_n| \to \infty$. For fixed $m_0$ the subspace $V_{m_0} = \C \xi_{m_0} \otimes \overline{\textup{lin}}(\eta_n) \otimes \C^2$ is $\Dirac_1$-invariant and $\Dirac_1|_{V_{m_0}}$ is given by the matrix
\begin{bmatrix} \lambda_{m_0} I & \textup{Diag}(\mu_n) \\[1ex] \textup{Diag}(\mu_n) & -\lambda_{m_0} I \end{bmatrix}\cong \bigoplus_n \begin{bmatrix} \lambda_{m_0} & \mu_n \\[1ex] \mu_n & -\lambda_{m_0} \end{bmatrix},
and those matrices have eigenvalues $\pm \sqrt{\lambda_{m_0}^2+\mu_n^2}$. It follows that $\Dirac_1$ is diagonalisable with eigenvalues $\pm \sqrt{\lambda_m^2+\mu_n^2}$.
$\bar{\Dirac}_2$ has eigenvalues $\pm\lambda_m$, so its eigenvalue sequence $(\lambda_m')$ is given by $\lambda_1,-\lambda_1,\lambda_2, -\lambda_2, \ldots$ with corresponding eigenvectors $e_1,e_2,e_3,\ldots$.
$\bar{\Dirac}_3$ has the same eigenvalues as $\Dirac_B$ with doubled multiplicity so its eigenvalue sequence $(\mu_n')$ is $\mu_1,\mu_1,\mu_2,\mu_2, \ldots$ with corresponding eigenvectors $f_1,f_2, f_3, \ldots$.
$\Dirac_I$ restricted to the subspace $V_{m,n} = \C(e_m \otimes f_n) \otimes \C^2$ has the matrix representation
\begin{bmatrix} 0 & \lambda_m' - i\mu_n' \\[1ex]\lambda_m' + i\mu_n' & 0 \end{bmatrix},
and this matrix is diagonalisable with eigenvalues $\pm \sqrt{\lambda_m'^2 + \mu_n'^2}$.
Therefore all operators $\Dirac_i$, $i=1,2,3$ and $\Dirac_I$ are unbounded and essentially self-adjoint.
(2) By assumption $|\lambda_m|, |\mu_n| \to \infty$ and therefore also $|\lambda_m'|, |\mu_n'| \to \infty$ as $m,n \to \infty$. Since we have shown in the proof of (1) that the eigenvalues of $\Dirac_1$ and $\Dirac_I$ are given by $\pm \sqrt{\lambda_m^2 + \mu_n^2}$ and $\pm \sqrt{\lambda_m'^2 + \mu_n'^2}$ respectively it is easy to see that the sequences of absolute values of them tend to infinity. Thus $\Dirac_1$ and $\Dirac_I$ have compact resolvent.
(3) Finally assuming $\textup{Tr} (I + \Dirac_A^2)^{-r/2} < \infty$ and $\textup{Tr}(I + \Dirac_B^2)^{-s/2} < \infty$ means $\sum_m (1 + \lambda_m^2)^{-r/2} < \infty$ and
$\sum_n (1 + \mu_n^2)^{-s/2} < \infty$. As indicated in <cit.> in a similar context, the inequality
$$(x+y-1)^{-(\alpha+\beta)} \leq x^{-\alpha}y^{-\beta},$$
valid for $x,y >1$ and $\alpha,\beta>0$ then implies that $\sum_{m ,n}(1 + \lambda_m^2+ \mu_n^2)^{-(r+s)/2} < \infty$.
This shows that $\Dirac_1$ is $(r+s)$-summable. Since the eigenvalues of $\Dirac_I$ are given by $\pm \sqrt{\lambda_m'^2 + \mu_n'^2}$ and $(\lambda_m'^2)$ and $(\mu_n'^2)$ are just the sequences $(\lambda_m^2)$ and $(\mu_n^2)$ with each term repeated once it follows that also $\Dirac_I$ is $(r+s)$-summable.
Next we need to show boundedness of commutators with our operators $\Dirac_1$ and $\Dirac_I$. In order to do so let us point out the following elementary identity for commutators of matrices.
\begin{eqnarray}
\left[
\begin{bmatrix} a_{11} & a_{12} \\[1ex] a_{21} & a_{22} \end{bmatrix}, \begin{bmatrix} a& 0 \\[1ex] 0 & b \end{bmatrix} \right] = \begin{bmatrix} [a_{11},a] & a_{12}b - a a_{12} \\[1ex] a_{21}a - ba_{21} & [a_{22},b] \end{bmatrix}.
\label{diagcom}
\end{eqnarray}
Let $e$ be in the dense $^*$-subalgebra $\mathcal{E}$ of $E$ generated by elementary tensors $k \otimes b \in \mathcal{C}(P H_A) \odot
\Balg$ and $\{PaP \otimes I_B: \;a \in
\Alg\}$. Then the operators $[\Dirac_1,\Pi_1(e)]$
$[\Dirac_I,\Pi_2(e) \oplus \Pi_2(e)]$
are bounded.
Let $e= x + P\pi_A(a)P\otimes 1$ be an element in $\mathcal{E}$, where $x \in \mathcal{C}(P H_A) \odot \Balg$. Then direct calculations using (<ref>) reveal that
\begin{eqnarray*}
[\Dirac_1, \Pi_1(e)] = \begin{bmatrix} [\Dirac_A, \pi_A(a)] \otimes 1 & 0 \\[1ex] 0 & -[\Dirac_A, \pi_A(a)]
\otimes 1 \end{bmatrix},
\end{eqnarray*}
which is bounded. Next we determine the following commutators, again using (<ref>). (We will omit the $A$ and $B$ indices of $\Dirac_A$ and $\Dirac_B$ as well as the representation $\pi_A$.)
\begin{eqnarray*}
[\Dirac_2, \Pi_2(e)]
\left[
\begin{bmatrix} \Dirac^q \otimes 1 & \Dirac^p \otimes 1 \\
\Dirac^p \otimes 1 &
-\Dirac^q \otimes 1
\end{bmatrix}, \begin{bmatrix}
\pi(e) & 0 \\
0 & \pi_{\sigma}(e)
\end{bmatrix}
\right] \\
\begin{bmatrix} [\Dirac^q \otimes 1, \pi(e) ] & (\Dirac^p \otimes 1 ) \pi_{\sigma} (e) - \pi(e) (\Dirac^p \otimes 1) \\
(\Dirac^p \otimes 1) \pi (e) - \pi_{\sigma} (e) (\Dirac^p \otimes 1) &
-[ \Dirac^q \otimes 1 , \pi_{\sigma} (e) ]
\end{bmatrix} \\
\begin{bmatrix} 0 & \Dirac^p a\otimes 1 - PaP\Dirac^p \otimes 1 - x(\Dirac^p \otimes 1) \\
(\Dirac^pPaP \otimes 1 + (\Dirac^p \otimes1)x - a\Dirac^p \otimes 1 &
-[ \Dirac^q,a] \otimes 1
\end{bmatrix} \\
\begin{bmatrix} 0 & P[\Dirac^p, a] \otimes 1 - x(\Dirac^p \otimes 1) \\
[\Dirac^p,a] P \otimes 1 + (\Dirac^p \otimes1)x &
-[ \Dirac^q,a] \otimes 1
\end{bmatrix}, \\
\end{eqnarray*}
where we have used that $\Dirac^q \otimes 1 \perp \pi(e)$. Next
\begin{eqnarray*}
[ \Dirac_3 , \Pi_2 (e) ]
\left[
\begin{bmatrix} 1 \otimes \Dirac_B & 0 \\
0 &
1 \otimes \Dirac_B
\end{bmatrix},
\begin{bmatrix}
\pi(e) & 0 \\
0 & \pi_{\sigma}(e)
\end{bmatrix}
\right] \\
\begin{bmatrix} [1 \otimes \Dirac_B, \pi(e) ] & 0 \\
0 &
[1 \otimes \Dirac_B , \pi_{\sigma} (e) ]
\end{bmatrix} \\
\begin{bmatrix} [1 \otimes \Dirac_B, x ] & 0 \\
0 & 0
\end{bmatrix}. \\
\end{eqnarray*}
Using these identities we obtain
\begin{eqnarray*}
[ \Dirac_I , \Pi_2 (e) \oplus \Pi_2(e) ]
\left[
\begin{bmatrix} 0 & \Dirac_2 - i \Dirac_3 \\
\Dirac_2 + i \Dirac_3 & 0
\end{bmatrix},
\begin{bmatrix}
\Pi_2(e) & 0 \\
0 & \Pi_2(e)
\end{bmatrix}
\right] \\
\begin{bmatrix} 0 & [\Dirac_2 - i \Dirac_3, \Pi_2(e)] \\
[\Dirac_2 + i \Dirac_3,\Pi_2(e)] & 0
\end{bmatrix}
\end{eqnarray*}
\begin{eqnarray*}
[\Dirac_2 - i \Dirac_3, \Pi_2(e)]
\begin{bmatrix} - i [1 \otimes \Dirac,x] & P[\Dirac^p, a] \otimes 1 - x(\Dirac^p \otimes 1) \\
[\Dirac^p,a] P \otimes 1 + (\Dirac^p \otimes1)x &
-[ \Dirac^q,a] \otimes 1
\end{bmatrix}
\end{eqnarray*}
\begin{eqnarray*}
[\Dirac_2 + i \Dirac_3, \Pi_2(e)]
\begin{bmatrix}
i [1 \otimes \Dirac,x] & P[\Dirac^p, a] \otimes 1 - x(\Dirac^p \otimes 1) \\
[\Dirac^p,a] P \otimes 1 + (\Dirac^p \otimes1)x & -[ \Dirac^q,a] \otimes 1
\end{bmatrix} .
\end{eqnarray*}
The claim is now evident since all entries in all operator matrices of the commutators are indeed bounded.
We are now ready to state and prove the first of our main results.
Let $A$ and $B$ be unital C$^*$-algebras and suppose that $E$ arises as the short exact sequence
(<ref>) and that there exist spectral triples $(\Alg,H_A,\Dirac_A)$ on $A$ and
$(\Balg,H_B,\Dirac_B)$ on $B$, represented via $\pi_A$ and $\pi_B$ respectively, and an orthogonal projection $P \in B(H_A)$ such that $(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type. Let
\begin{eqnarray*}
\Pi = \Pi_1\oplus \Pi_2 \oplus \Pi_{2}, \;\;H = (H_A \otimes
H_B \otimes \C^2)^3, \textup{ and}
\end{eqnarray*}
\begin{eqnarray*}
\Dirac = \begin{bmatrix} \Dirac_{1} & 0 & 0 \\[1ex]
0 & 0 & \Dirac_2 - i \Dirac_{3}\\[1ex]
0 & \Dirac_2 +
i \Dirac_{3} &0 \end{bmatrix}.
\end{eqnarray*}
Then $(\mathcal{E}, H, \Dirac)$, represented
via $\Pi$, defines a spectral triple on $E$.
Moreover, the spectral dimension of this spectral triple is computed by
the identity
\begin{eqnarray*}
s_0(\mathcal{E}, H, \Dirac) = s_0(\Alg,H_A,\Dirac_A) + s_0(\Balg,H_B,\Dirac_B).
\end{eqnarray*}
Note first that the representation $\Pi$ is faithful and that $\Dirac = \Dirac_1 \oplus \Dirac_I$.
Since $\Dirac_1$ and $\Dirac_I$ have compact resolvent by Lemma <ref>.(2) so has $\Dirac$.
By Lemma <ref> the commutators $[\Dirac, \Pi (e)]= [\Dirac_1 , \Pi_1 (e)] \oplus [\Dirac_I , \Pi_2 (e) ]$ are indeed bounded for every $ e \in \mathcal{E}$.
The summability claim finally follows
since $s_0(\Dirac) = s_0 (\Dirac_1 \oplus \Dirac_I) = \max( s_0(\Dirac_1) , s_0(\Dirac_I)) $ and we have shown in Lemma <ref>.(3) that $s_0(\Dirac_1) = s_0(\Dirac_I)= s_0(\Alg,H_A,\Dirac_A) + s_0(\Balg,H_B,\Dirac_B)$.
§.§ The algebra $C^1(E)$
Theorem <ref> only provides the existence of a spectral triple for the dense subalgebra $\mathcal{E}$ of $E$. Given the Dirac operator we defined, it is natural to ask how large we can allow the dense subalgebra to be. More specifically, we ask: what is the largest `smooth' subalgebra of $E$ in which the construction in Theorem <ref> defines a spectral triple?
There seems to be a natural such algebra, the maximal Lipschitz algebra $C^1(E)$ associated to our Dirac operator mentioned after Def.<ref>. It is also an extension fitting into the short exact sequence (<ref>).
We think of $E$ as being represented via $\pi$ on $H_A \otimes H_B$ so that $E \subseteq \mathcal{K}(PH_A) \otimes B + PAP \otimes 1$, where the sum is algebraically direct.
Then $C^1(E)$ is the $*$-subalgebra of $E$ comprising all $e \in E$ such that
* $\Pi(e) \textup{dom}(\Dirac) \subseteq \textup{dom}(\Dirac)$,
* $[\Dirac, \Pi(e)]: \textup{dom}(\Dirac) \to H$ is closable and bounded.
Writing $e=x + PaP \otimes I$ uniquely, where $x \in \mathcal{K}(PH_A) \otimes B$ and $a \in A$ the formulas for $[\Dirac_1,\Pi_1(e)]$ and $[\Dirac_I,\Pi_2(e) \oplus \Pi_2(e)]$ in the proof of Lemma <ref> show that $e \in C^1(E)$ iff the following conditions are satisfied.
* $\pi_A(a) (\textup{dom}(\Dirac_A)) \subseteq \textup{dom}(\Dirac_A)$ and $[\Dirac_A, \pi_A(a)]$ is closable and bounded.
* $\pi (x) (\textup{dom}(1 \otimes \Dirac_B)) \subseteq \textup{dom}(1 \otimes \Dirac_B)$
and $(1 \otimes \Dirac_B) x , x (1 \otimes \Dirac_B) $ are bounded.
* $\pi_A(a) (\textup{dom}(\Dirac_A^p)) \subseteq \textup{dom}(\Dirac_A)$ and $\pi_A(a) (\textup{dom}(\Dirac_A^q)) \subseteq \textup{dom}(\Dirac_A)$ and
$[\Dirac_A^p, \pi_A(a)] $, $[\Dirac_A^q, \pi_A(a)]$ are closable and bounded.
We now define $C^1(\mathcal{K}_B) \subseteq C^1(E)$ to be the dense $^*$-subalgebra of $\mathcal{K}_B$ consisting of all $x \in \mathcal{K}_B$ such that
$\pi (x) (\textup{dom}(1 \otimes \Dirac_B)) \subseteq \textup{dom}(1 \otimes \Dirac_B)$
and $(1 \otimes \Dirac_B) \pi(x) , \pi(x) (1 \otimes \Dirac_B) $ are bounded. (More easily we could define $C^1(\mathcal{K}_B)= \{ x \in C^1(E) : x \in \mathcal{K}_B \}$.)
Finally, let $C^{1,P}(A)$ be the $^*$-subalgebra of $A$ consisting of all $a \in A$ such that
* $\pi_A(a) (\textup{dom}(\Dirac_A)) \subseteq \textup{dom}(\Dirac_A)$ and $[\Dirac_A, \pi_A(a)]$ is closable and bounded.
* $\pi_A(a) (\textup{dom}(\Dirac_A^p)) \subseteq \textup{dom}(\Dirac_A)$ and $\pi_A(a) (\textup{dom}(\Dirac_A^q)) \subseteq \textup{dom}(\Dirac_A)$ and
$[\Dirac_A^p, \pi_A(a)] $, $[\Dirac_A^q, \pi_A(a)]$ are closable and bounded.
Our definitions imply that we obtain the following short exact sequence
\begin{eqnarray}
\xymatrix{ 0 \ar[r] & C^1(\mathcal{K}_B) \ar[r]^{\iota_1} & C^1(E) \ar[r]^{\sigma_1} & C^{1,P}(A) \ar[r] & 0},
\label{smoothext}
\end{eqnarray}
where $\iota_1$ and $\sigma_1$ are the natural inclusion and quotient map respectively.
§ THE METRIC CONDITION FOR EXTENSIONS.
We are interested in the construction of spectral metric spaces and, as such, we turn now to the question of
whether the spectral triple on $E$ satisfies Rieffel's metric condition (cf. Prop.<ref>), giving $E$ the structure of a spectral metric space. There is an abundance of Lipschitz seminorms on each of $A$, $B$ and $E$ which we could study, depending on the choice of smooth subalgebras. In this section we will focus on the situation in which the smooth subalgebras (cf. Definition <ref>) are $\Alg = C^{1,P}(A)$ and $\Balg = C^1(B)$ and show that it is possible to construct a Lipschitz seminorm on $C^1(E)$ coming from a spectral triple with the desired properties. Our results can be adjusted to fit the setting of dense subalgebras possibly smaller than $C^{1,P}(A)$ or $C^1(B)$.
To this end, we assume that the spectral triples $(C^{1,P}(A),H_A,\Dirac_A)$ and $(C^1(B),H_B,\Dirac_B)$ on $A$ and $B$ respectively, together with Lipschitz seminorms $L_{\Dirac_A}$ on $C^1(A)$ and $L_{\Dirac_B}$ on $C^1(\mathcal{K}_B)$, give $A$ and $B$ the structure of spectral metric spaces. According to Rieffel's criterium (Proposition <ref>), this means that the spectral triples $(\Alg,H_A,\Dirac_A)$ and $(\Balg,H_B,\Dirac_B)$ are
nondegenerate, that the spaces
\begin{eqnarray*}
\widetilde{\mathcal{U}}_A &:=& \{\tilde{a} \in C^{1,P}(A) / \C I_A: \;\|[\Dirac_A, \pi_A(\tilde{a})]\| \leq 1\}, \\
\widetilde{\mathcal{U}}_B &:=& \{\tilde{b} \in C^1(B) / \C I_B: \;\|[\Dirac_B, \pi_B(\tilde{b})]\| \leq 1\}
\end{eqnarray*}
are bounded subsets of $A / \C I_A$ and $B / \C I_B$ respectively and that the sets
\begin{eqnarray*}
\mathcal{U}_{A,1} &:=& \{a \in C^{1,P}(A): \;\|a\| \leq 1, \;\|[\Dirac_A, \pi_A(a)]\| \leq 1\}, \\
\mathcal{U}_{B,1} &:=& \{b \in C^1(B): \;\|b\| \leq 1, \;\|[\Dirac_B, \pi_B(b)]\| \leq 1\}
\end{eqnarray*}
are norm totally bounded. From Rem.<ref> and the standing assumption of essentialness of our extension we conclude that for every element $\tilde{e} \in C^1(E) / \C I_E $ there is a unique $x \in C^1(\mathcal{K}_B)$ and $\tilde{a} \in C^{1,P}(A) / \C I_A$ such that $\tilde{e} = (x + PaP \otimes I)^{\sim}$. In this sense we have:
The equality $C^1(E) / \C I_E = C^1(\mathcal{K}_B) + P (C^{1,P}(A) / \C
I_A) P \otimes \C I_B$ holds, where the sum is direct.
Notice that $I_E = P I_A P \otimes I_B= P \otimes I_B$.
We now introduce the following spaces, where $X$ and $Y$ play the role of subscripts and do not refer to other objects.
\begin{eqnarray*}
\mathcal{U}_X &:=& \{x \in C^1(\mathcal{K}_B): \;\|[\Dirac_I, \Pi_2(x)\oplus \Pi_2(x)]\| \leq 1\}, \\
\widetilde{\mathcal{U}}_Y &:=& \{P\tilde{a}P \otimes I_B \in P (C^{1,P}(A) / \C
I_A) P \otimes \C I_B: \;\|[\Dirac_A, \pi_A(\tilde{a})]\|
\leq 1\}, \\
\mathcal{U}_{Y,1} &:=& \{PaP \otimes I_B \in P C^{1,P}(A)P \otimes \C I_B: \; \| a \| \leq 1,\;\;\|[\Dirac_A, \pi_A(a)]\| \leq
\end{eqnarray*}
Recall from Def.<ref> that a Toeplitz type quadruple $(\Alg, H_A, \Dirac_A, P)$ is called $P$-injective if $\ker ( \Dirac_A^p ) \cap P H_A = \{0\}$ (i.e. the operator $\Dirac_A^p|_{P H_A}$ has an inverse in
It will be necessary for us to impose this condition throughout this section.
We remark that necessarily $\ker(\Dirac_A^p \cap P H_A)$ is finite rank, so that if $P$-injectivity
fails then we can merely replace $P$ with $P - P_{\ker (\Dirac_A)}$, where $ P_{\ker (\Dirac_A)}$ is the orthogonal projection onto $ \ker (\Dirac_A)$. This procedure does not affect any
other aspects of the extension. Using Lemma <ref> we have the following observation.
One checks that the expressions for $[\Dirac_1, \Pi_1(e)]$ and $[\Dirac_I, \Pi_2(e) \oplus \Pi_2(e)]$ in the proof of Lemma <ref> define bounded operators for all $e \in C^1(E) \supseteq \mathcal{E}$.
Indeed, for any $e = x + PaP \otimes I_B \in C^1(E)$ such that $x \in C^1(\mathcal{K}_B)$ and $a \in C^{1,P}(A)$ we have
\begin{eqnarray}
[\Dirac_1, \Pi_1(e)] = \begin{bmatrix} [\Dirac_A, \pi_A(a)] \otimes 1 & 0 \\[1ex] 0 & -[\Dirac_A, \pi_A(a)]
\otimes 1 \end{bmatrix},
\label{1com}
\end{eqnarray}
\begin{eqnarray}
[ \Dirac_I , \Pi_2 (e) \oplus \Pi_2(e) ]
\begin{bmatrix} 0 & [\Dirac_2 - i \Dirac_3, \Pi_2(e)] \\
[\Dirac_2 + i \Dirac_3,\Pi_2(e)] & 0
\end{bmatrix}
\label{2com}
\end{eqnarray}
\begin{eqnarray}
[\Dirac_2 \pm i \Dirac_3, \Pi_2(e)]
\begin{bmatrix}
\pm i [1 \otimes \Dirac,x] & P[\Dirac^p, a] \otimes 1 - x(\Dirac^p \otimes 1) \\
[\Dirac^p,a] P \otimes 1 + (\Dirac^p \otimes1)x & -[ \Dirac^q,a] \otimes 1
\end{bmatrix}.
\label{3com}
\end{eqnarray}
The spectral triple $(\mathcal{E}, \Pi, \Dirac)$ on $E$ in Theorem <ref> determines a seminorm $L= L_{\Dirac}$
on $C^1(E)$ given by $L(e) = \max\{\|[\Dirac_1, \Pi_1(e)]\|,\|[\Dirac_I, \Pi_2(e) \oplus \Pi_2(e) ]\|\}$. Our first objective is to show that $L$ is a nondegenerate Lipschitz seminorm:
Let $e \in C^1(E)$, then $L(e) = 0$ iff there exists $\lambda \in \C$ with $e= \lambda I$.
The proof consists in showing $C^1(E) \cap L^{-1}(\{0\}) = \C I_E$. To this end, let $e = x + PaP \otimes I_B$, where $x \in C^1(\mathcal{K}_B)$ and $a \in C^{1,P}(A)$. If $L(e) = 0$ then $[\Dirac_1, \Pi_1(e)] = 0$ so that $[\Dirac_A, \pi_A(a)] = 0$ from eq. (<ref>). Since $\Dirac_A$ implements a nondegenerate spectral triple on $A$, necessarily $a = \lambda I_A$ for some $\lambda \in \C$, so that we can write $e = x + \lambda I_E$.
Moreover, we have $[\Dirac_I, \Pi_2(x) \oplus \Pi_2(x)] = 0$, so by (<ref>) and (<ref>) this means that $(\Dirac_A^p \otimes 1)\pi(x)= (\Dirac_A^p \otimes 1)x = 0$. By $P$-injectivity, $x = 0$, completing the proof.
Let $\widetilde{\mathcal{U}}_E$ and $\mathcal{U}_{E,1}$ be the subsets of $E / \C I_E$ and $E$ respectively defined by
\begin{eqnarray*}
\widetilde{\mathcal{U}}_E &:=& \{\tilde{e} \in C^1(E) / \C I_E:\;L(\tilde{e}) \leq 1\},\\
\mathcal{U}_{E,1} &:=& \{e \in C^1(E): \|e\| \leq 1, \;L(e) \leq 1\}.
\end{eqnarray*}
\widetilde{\mathcal{U}}_E \subseteq 7\mathcal{U}_X + \widetilde{\mathcal{U}}_Y:= \{ x+ PaP \otimes I/\C P \otimes I : \| [\Dirac_I, \Pi_2(x) \oplus \Pi_2(x)] \| \leq 7, \| [\Dirac_A , \pi_A(a) ] \| \leq 1 \}
$$\mathcal{U}_{E,1} \subseteq
7\mathcal{U}_{X} + \mathcal{U}_{Y,1}= \{ x+ PaP \otimes \C I : \| [\Dirac_I, \Pi_2(x) \oplus \Pi_2(x) ] \| \leq 7, \| [\Dirac_A , \pi_A(a) ] \| \leq 1 \}.$$
To show the first inclusion,
let $e \in C^1(E)$ be such that $L(e) \leq 1$ and write $e = x + PaP \otimes I_B$ for unique $x \in \mathcal{K}_B$ and $a \in C^{1,P}(A)$. We need to show that $\| [ \Dirac_A , \pi_A(a)]\| \leq 1$ and $\| [\Dirac_I , \Pi_2(x) \oplus \Pi_2(x) ]\| \leq 7$.
Now $L(e) \leq 1$ is equivalent to $\|[\Dirac_1, \Pi_1(e)]\| \leq 1$ and $\|[\Dirac_I, \Pi_2(e) \oplus \Pi_2(e)]\| \leq 1$. This means that the norms of all entries in the matrix expressions in Rem.<ref> are bounded by 1 from which we obtain the following inequalities:
* $\|[\Dirac_A, \pi_A(a)]\|= \|[\Dirac_A , a] \| \leq 1 $,
* $\| [ 1 \otimes \Dirac_B , x ]\| \leq 1$,
* $\| P [ \Dirac_A^p , a] \otimes 1 + x (D_A^p \otimes 1) \| \leq 1 $,
* $\| [D_A^p ,a] P \otimes 1 + (D^p \otimes 1)x \| \leq 1$,
* $\| [D_A^q , a ] \| \leq 1$.
Since $[ \Dirac_A , a ]= [ \Dirac_A^p , a]+[ \Dirac_A^q , a]$, we must have $\|[\Dirac_A^p,
a]\| \leq 2$, using (i) and (v).
Then (iii) and (iv) imply
\begin{eqnarray}
\| (\Dirac^p \otimes 1)x \|, \|x(\Dirac^p \otimes 1)\| \leq 3.
\label{xdp}
\end{eqnarray}
Now (ii) and (<ref>) imply
\begin{eqnarray*}
\|[\Dirac_2 \pm i \Dirac_3, \Pi_2(x)]\|
\left\|
\begin{bmatrix}
\pm i [1 \otimes \Dirac,x] & - x(\Dirac^p \otimes 1) \\
(\Dirac^p \otimes1)x & 0
\end{bmatrix}
\right\| \\
&\leq &
\| [1 \otimes \Dirac_B , x] \| + \| x (\Dirac^p \otimes 1 ) \| + \| (\Dirac^p \otimes 1) x \| \\
&\leq &
\end{eqnarray*}
This shows that $x \in 7 \: \mathcal{U}_X$ and the result follows.
The second inclusion can be shown in a similar way.
The next Lemma is immediate from our definitions.
Let $\sigma: E \to A$ be the quotient map with induced map $\tilde{\sigma} : E / \C I \to A / \C I$. Then the maps
\begin{eqnarray*}
\sigma|_{\mathcal{U}_{Y,1}}: (\mathcal{U}_{Y,1}, \|\cdot\|_E) \rightarrow (\mathcal{U}_{A,1}, \|\cdot\|_A), \;\;\;\;\tilde{\sigma}|_{\widetilde{\mathcal{U}}_Y}:
(\widetilde{\mathcal{U}}_Y, \|\cdot\|_{E / \C I_E}) \rightarrow (\widetilde{\mathcal{U}}_A, \|\cdot\|_{A / \C I_A})
\end{eqnarray*}
are isometric bijections. Therefore, since $(C^{1,P}(A), H_A , \Dirac_A)$ satisfies Rieffel's metric condition, $\widetilde{\mathcal{U}}_Y \subseteq E / \C I_E$ is bounded and $\mathcal{U}_{Y,1} \subseteq E$ is totally bounded.
The uniform norm estimates in the next result is of key importance to establish Rieffel's metric condition. It uses a norm estimate from the proof of Lemma <ref>.
Let $Y := ({\Dirac_A^p|_{P H_A}})^{-1} \in \mathcal{K}(PH_A)$, let
$\{P_k\}_{k \in \N}$ be the spectral projections of $Y$ and write $Q_n := \sum_{k =
1}^n P_k$. Then for each $\epsilon > 0$ there exists an $N \in \N$ such that, for each $ x \in \mathcal{U}_X$ and for all $n \geq N$,
\begin{eqnarray*}
\|x - (Q_n \otimes 1) x (Q_n \otimes 1) \| \leq \epsilon.
\end{eqnarray*}
Moreover, for each $x \in \mathcal{U}_X$ and for each $n \in \N$, $\|x_n\|\leq 3 \|Y\|$, where $x_n := (Q_n \otimes 1) x (Q_n \otimes 1)$.
Since $Y$ is a compact operator, it quickly follows that for each
$\epsilon > 0$ there exists an $N \in \N$ such that $\|Y - YQ_n\| \leq \frac{\epsilon}{6}$ and $\|Y - Q_n Y\| \leq
\frac{\epsilon}{6}$ whenever $n \geq N$. For $x \in \mathcal{U}_X$, using $P \Dirac_A Y = P$, we obtain
\begin{eqnarray*}
\|(Q_n \otimes 1) x (Q_n \otimes 1)\|
\|(Q_n \otimes 1) x (P \Dirac_A Y \otimes 1 ) (Q_n \otimes 1)\| \\
\|(Q_n \otimes 1) x (P \Dirac_A \otimes 1 )( Y \otimes 1) (Q_n \otimes 1)\| \\
\|Q_n \| \| x (P \Dirac_A \otimes 1 ) \| \| Y \| \|Q_n \| \\
& = &
\|x(P \Dirac_A \otimes 1)\| \|Y\| \leq 3\|Y\|,
\end{eqnarray*}
where the last inequality follows from (<ref>) in the proof of Lemma <ref>.
This proves the second statement. To prove the first statement note that for all $x \in \mathcal{U}_X$ and $ n \geq N$
\begin{eqnarray*}
\|x - x (Q_n \otimes 1)\| & \leq & \|x (P \Dirac_A \otimes 1)( Y \otimes 1) - x (P \Dirac_A \otimes 1 ) (Y Q_n \otimes 1)\| \\
& \leq & \|x(P \Dirac_A \otimes 1)\| \|Y \otimes 1 - Y Q_n \otimes 1 \| \\
& = & \|x(P \Dirac_A \otimes 1)\| \|Y - Y Q_n \| \\
& \leq & \frac{\epsilon}{2},
\end{eqnarray*}
and similarly $\|x - (Q_n \otimes 1) x\| \leq \frac{\epsilon}{2}$, so that
\begin{eqnarray*}
\|x - (Q_n \otimes 1) x (Q_n \otimes 1) x\|
\|x - x (Q_n \otimes 1)\| + \|x (Q_n \otimes 1) - (Q_n \otimes 1) x (Q_n \otimes 1)\| \\
&\leq & \epsilon.
\end{eqnarray*}
We can now prove our second main result.
Let $A$ and $B$ be unital C$^*$-algebras and suppose $E$ arises as the short exact sequence
(<ref>). Suppose further that there exists spectral triples $(\Alg,H_A,\Dirac_A)$ on $A$ and
$(\Balg,H_B,\Dirac_B)$ on $B$, represented via $\pi_A$ and $\pi_B$ respectively, and an orthogonal projection $P \in B(H_A)$ such that
$(\Alg, H_A, \Dirac_A, P)$ is of Toeplitz type and $P$-injective. If the spectral triples $(\Alg,H_A,\Dirac_A)$ and
$(\Balg,H_B,\Dirac_B)$ satisfy Rieffel's metric condition then so does the spectral triple $(\mathcal{E}, H, \Dirac)$ so that $(E, L_{\Dirac})$ is a spectral metric space.
According to Rieffel's criteria (Prop.<ref>) we need to show that $\widetilde{\mathcal{U}}_E$ is bounded and $\mathcal{U}_{E,1}$ is totally bounded.
By Lemma <ref> we know that $\widetilde{\mathcal{U}}_Y$ and $ \mathcal{U}_{Y,1}$ are bounded, respectively totally bounded. By Lemma <ref> we know that $\widetilde{\mathcal{U}}_E \subseteq 7\mathcal{U}_X + \widetilde{\mathcal{U}}_Y$ and $\mathcal{U}_{E,1} \subseteq 7\mathcal{U}_{X} + \mathcal{U}_{Y,1}$. So we have only to show that the set $\mathcal{U}_{X} \subseteq \mathcal{K}_B$ is totally norm bounded. Using Lemma <ref>, it will suffice for us to show that the sets
(Q_n \otimes 1) \mathcal{U}_X (Q_n \otimes 1)
are totally bounded for each $n \in \N$. Since we may regard $(Q_n \otimes 1) \mathcal{U}_X (Q_n \otimes 1)$ as a subset of $M_{m_n}(B)$, where $m_n = \textup{dim}(Q_n)$, any given element in this set can be expressed in the form
\begin{eqnarray*}
x_n = \sum_{i,j = 1}^{m_n} \pi_B(b_{i,j}) \otimes (| e_j \rangle \langle e_i|) ,
\end{eqnarray*}
where $b_{i,j} \in B$ and $\{e_i\}_{i = 1}^{m_n}$ is an orthonormal basis for the finite dimensional
Hilbert space $Q_n H_A$. We shall denote the corresponding projections in $B(Q_n H_A \otimes H_B)$ by $\{p_i\}_{i =
1}^{m_n}$. Since these commute with
$1 \otimes \Dirac_B$, we have that for $x \in \mathcal{U}_X$ and $n \in \N$,
\begin{eqnarray*}
\|\pi_B(b_{i,j})\| = \|p_j x_n p_i\| \leq \|x_n\| \leq 3 \|Y\|,
\end{eqnarray*}
\begin{eqnarray*}
\|[\Dirac_B, \pi_B(b_{i,j})]\| = \|[1 \otimes \Dirac_B, p_j x_n p_i]\| = \|p_j [1 \otimes \Dirac_B, x] p_i\| \leq
\end{eqnarray*}
since $\|[1 \otimes \Dirac_B, x] \| \leq 1$ from $\| \Dirac_I , \Pi_2(x) \oplus\Pi_2(x)]\| \leq 1$. These estimates tell us that the sets $Q_n \mathcal{U}_X Q_n$ are contained in the sets
\begin{eqnarray*}
S_n &:=& \Big\{ \sum_{i,j = 1}^{m_n}
\pi_B(b_{i,j}) \otimes (| e_j \rangle \langle e_i|): \;\;\; b_{i,j} \in \mathcal{B}, \;\|b_{i,j}\| \leq 3\|Y\|, \;\|[\Dirac_B,
\pi_B(b_{i,j})]\| \leq 1 \Big\} \\
& \subs & \Big\{ \sum_{i,j = 1}^{m_n}
\pi_B(b_{i,j}) \otimes (| e_j \rangle \langle e_i|): \;\;\; b_{i,j} \in 3\|Y\| \mathcal{U}_{B,1} \Big\}.
\end{eqnarray*}
Now we recall our assumption that the spectral triple on $B$ satisfies Rieffel's metric condition, so
that $\mathcal{U}_{B,1}$ is totally bounded and consequently the sets $S_n$ are totally bounded as well. This
concludes the proof of the Theorem.
§ EXAMPLES
§.§ Split extensions.
Recall that an extension (<ref>) is split when it is semisplit and the
splitting map $s: A \to \mathcal{L_B}$ can be chosen to be a $*$-homomorphism (rather than merely a completely positive map). If such an extension admits a Toeplitz representation, as in Definition <ref>, then $P$ is the identity in $B(\Hil_A)$, and we can restrict our attention to representations of this type. This significantly reduces the technicalities associated with the construction of spectral triples on such extensions. Our construction in Theorem <ref> reads in this case as follows:
Let $A$ and $B$ be unital C$^*$-algebras, endowed with spectral triples $(\Alg,H_A,\Dirac_A)$ and
$(\Balg,H_B,\Dirac_B)$ respectively. Let $E \cong \mathcal{K}(H_A) \otimes B + A \otimes I_B$ be a unital split extension of $A$ by the stabilisation of $B$. Then $(\mathcal{E}, H, \Dirac)$, represented via $\Pi$, defines a spectral triple on $E$. Here,
\begin{eqnarray*}
\Pi = \pi_{\sigma} \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma}, \;\;H = H_A \otimes
H_B \otimes \C^6,
\end{eqnarray*}
\begin{eqnarray*}
\Dirac = \begin{bmatrix} \Dirac_A \otimes 1 & 1 \otimes \Dirac_B & 0 & 0 &0 &0 \\[1ex]
1 \otimes \Dirac_B & -\Dirac_A \otimes 1 & 0 & 0 &0&0 \\[1ex]
0 & 0 & 0 & 0 & -i \otimes \Dirac_B & \Dirac_A \otimes 1 \\[1ex]
0 & 0 & 0 & 0 & \Dirac_A \otimes 1 & -i \otimes \Dirac_B \\[1ex]
0 & 0 & i \otimes \Dirac_B & \Dirac_A \otimes 1 & 0 & 0 \\[1ex]
0 & 0 & \Dirac_A \otimes 1 & i \otimes \Dirac_B & 0 & 0 \\[1ex]
\end{bmatrix}.
\end{eqnarray*}
If $\Dirac_A$ is invertible and the spectral triples $(\Alg,\Hil_A,\Dirac_A)$ and $(\Balg,\Hil_B,\Dirac_B)$
satisfy Rieffel's metric condition, so does the spectral triple on $E$.
In this case of split extensions other constructions are possible. For instance we can use the following representation and Dirac operator
\begin{eqnarray*}
\Pi = \pi_{\sigma} \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma}, \;\;H = H_A \otimes
H_B \otimes \C^4,
\end{eqnarray*}
\begin{eqnarray*}
\Dirac = \begin{bmatrix} \Dirac_A \otimes 1 & 1
\otimes \Dirac_B & 0 & 0 \\[1ex] 1 \otimes \Dirac_B & -\Dirac_A \otimes 1 & 0 & 0 \\[1ex] 0 & 0 & 1 \otimes \Dirac_B & \Dirac_A \otimes 1 \\[1ex] 0 & 0 & \Dirac_A \otimes 1 & - 1 \otimes \Dirac_B \end{bmatrix},
\end{eqnarray*}
which seems more natural from the point of view of K-homology.
§.§ Extensions by compacts.
An extension by compacts is a short exact sequence of the form,
\begin{eqnarray}
\label{ExtCompact} \xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r]^{\iota} & E \ar[r]^{\sigma} & A \ar[r] & 0}.
\end{eqnarray}
which we have mentioned before. From our point of view, these extensions correspond to the instance $B = \C$, the continuous functions on a single point. The canonical spectral triple on this space is the 'one-point' triple $(\C,\C,0)$. A second re-statement of Theorem <ref> is as follows:
Let $A$ be a unital C$^*$-algebra, endowed with a spectral triple $(\Alg,H_A,\Dirac_A)$. Let $E \cong PAP + \mathcal{K}(PH_A)$ be a unital extension of $A$ by compact operators such that $[P,a]$ is a compact operator for each $a \in A$, $PAP \cap \mathcal{K}(P H_A) = \{0\}$ and the quadruple $(\Alg,H_A,\Dirac_A, P)$ is of Toeplitz type. Then $(\mathcal{E}, H, \Dirac)$, represented via $\Pi$, defines a spectral triple on $E$. Here,
\begin{eqnarray*}
\Pi = \pi_{\sigma} \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma}, \;\;H = H_A \otimes \C^6,
\end{eqnarray*}
\begin{eqnarray*}
\Dirac = \begin{bmatrix} \Dirac_A & 0 & 0 & 0 & 0 &0 \\[1ex]
0 & -\Dirac_A & 0 & 0 &0&0 \\[1ex]
0 & 0 & 0 & 0 & \Dirac_A^q & \Dirac_A^p \\[1ex]
0 & 0 & 0 & 0 & \Dirac_A^p & \Dirac_A^q \\[1ex]
0 & 0 & \Dirac_A^q & \Dirac_A^p & 0 & 0 \\[1ex]
0 & 0 & \Dirac_A^p & \Dirac_A^q & 0 & 0 \\[1ex]
\end{bmatrix}.
\end{eqnarray*}
The spectral dimension of this triple is the same as the spectral dimension of $(\Alg, H_A, \Dirac_A)$. Moreover, if $\Dirac_A^p$ is invertible and the spectral triple $(\Alg,\Hil_A,\Dirac_A)$ satisfies the Rieffel metric condition then so does the spectral triple on $E$.
It is worth comparing our spectral triples with those considered by Christensen and Ivan <cit.>. They make the same assumptions that we do, but the difference is that their triple acts on the Hilbert space $P H_A \oplus P H_A \oplus Q H_A$, rather than the larger Hilbert space $H_A \otimes \C^6$. Their Dirac operator, like ours, is designed to obtain a spectral triple with good metric properties. In the spirit of Rieffel-Gromov-Hausdorff theory, Christensen-Ivan
introduce extra parameters $\alpha, \beta \in (0,1)$ which can be used to study the effects of "recovering" metric data on either the quotient algebra or the compacts itself, coming from the extension.
§.§ Noncommutative spheres.
The quantum group $SU_q(2)$ was introduced by Woronowicz as a 1-parameter deformation of the
ordinary $SU(2)$ group <cit.>. When one considers
the isomorphism $SU(2) \cong S^3$ of topological Lie groups, we can identify its C$^*$-algebra with a 1-parameter deformation of the continuous functions on the 3-sphere, $C(S_q^3)$, for each $q \in [0,1]$. It can be formally defined as the universal C$^*$-algebra for generators $\alpha$ and $\beta$ subject to the
\begin{eqnarray*}
\alpha^* \alpha + \beta^*\beta = I, \;\;\;\; \alpha \alpha^* + q^2\beta \beta^* = I,
\end{eqnarray*}
\begin{eqnarray*}
\alpha \beta = q \beta \alpha, \;\;\;\; \alpha \beta^* = q \beta^*\alpha,
\;\;\;\;\beta^*\beta = \beta \beta^*.
\end{eqnarray*}
Woronowicz shows that the C$^*$-algebras $C(S_q^3)$ are all isomorphic for $q \in [0,1)$. For $q \in (0,1)$,
there is an alternative description of $C(S_q^3)$ as a symplectic foliation (see <cit.>, <cit.>, <cit.>): write $H := \ell_2(\N_0) \otimes
\ell_2(\Z)$ and let $S$ and $T$ be respectively the unilateral shift on $\ell_2(\N_0)$ and the bilateral shift on $\ell_2(\Z)$, i.e $S e_k := e_{k+1}$ for each $k \geq 0$ and $T e_k := e_{k+1}$ for each $k \in \Z$. Let $N_q \in \mathcal{K}(\ell_2(\N_0))$ be defined by $N_q e_k := q^k e_k$. There exists a representation of $C(S_q^3)$ over $H$ defined by:
\begin{eqnarray*}
\pi(\alpha) := S^* \sqrt{1 - N_q^2} \otimes I, \;\;\; \pi(\beta) &:=& N_q \otimes T^*.
\end{eqnarray*}
and this representation is faithful. By considering the map $\sigma: C(S_q^3) \to C(\T)$ sending $\beta$ to $0$ and $\alpha$ to the generator $T^*$ of $C(\T)$, we soon obtain a short exact sequence,
\begin{eqnarray*}
0 \;\rightarrow \; \mathcal{K} \otimes C(\T) \;\rightarrow \; C(S_q^3) \;\rightarrow \; C(\T)
\;\rightarrow \; 0.
\end{eqnarray*}
We obtain an isomorphism,
\begin{eqnarray*}
C(S_q^3) \cong P C(\T) P \otimes \C I + \mathcal{K}(\ell_2(\N_0)) \otimes C(\T),
\end{eqnarray*}
where $P \in B(\ell_2(\Z))$ is the usual Toeplitz projection, with the property that $[P,x]$ is a
compact operator for each $x \in C(\T)$ and $PxP \otimes I \in C(S_q^3)$ for each $x \in C(\T)$. Note that we can write
\pi(\alpha) = -P T^* P(1 - \sqrt{1 - N_q^2}) \otimes I + P T^* P \otimes I \textup{ whilst } \pi(\beta) \in \mathcal{K} \otimes C(\T).
Because the algebra $C(S_q^3)$ has the requisite Toeplitz form, the construction in Theorem <ref> defines a spectral triple on $C(S_q^3)$ and it further provides $C(S_q^3)$ with the structure of a spectral metric space. For the latter, a slight perturbation of one of the Dirac operators is needed in this construction to ensure $P$-injectivity. In what follows, $\pi$ denotes the natural non-unital inclusion of $C(S_q^3)$ in $B(\ell_2(\Z) \otimes \ell_2(\Z))$, whilst $\pi_{\sigma}: C(S_q^3) \to B(\ell_2(\Z) \otimes \ell_2(\Z))$ is the map defined on the generators by $\pi_{\sigma}(\alpha) := T^* \otimes 1$, $\pi_{\sigma}(\beta) = 0$.
Let $(\Alg, \ell_2(\Z), M_{\ell})$, $M_{\ell} e_n = n e_n$ be the usual spectral triple on $C^1(\T)$, where $\Alg \subseteq C(\T)$ is any dense $^*$-subalgebra of $C(\T)$ such that $(\Alg, \ell_2(\Z), \Dirac)$ is a triple satisfying
\begin{eqnarray*}
[M_{\ell}, f] \in B(\ell_2(\Z)), \;\;[|M_{\ell}|, f] \in B(\ell_2(\Z)), \;\;f \in \Alg
\end{eqnarray*}
(e.g. $\Alg = C^1(\T)$).
Then, for each $\lambda \in \R$ and for each $q \in (0,1)$, there is a spectral triple $(\mathcal{E}, (\ell_2(\Z) \otimes \ell_2(\Z)) \otimes \C^6, \Dirac_{\lambda})$ on $C(S_q^3)$, represented via $\pi_{\sigma} \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma}$
and where
\begin{eqnarray*}
\Dirac_{\lambda} =
\begin{bmatrix} M_{\ell,\lambda} \otimes 1 & 1
\otimes M_{\ell} & 0 \\[1ex]
1 \otimes M_{\ell} & - M_{\ell,\lambda} \otimes 1 & 0\\[1ex]
0 & 0 & \Dirac_I \\[1ex]
\end{bmatrix},
\end{eqnarray*}
\begin{eqnarray*}
\Dirac_I=
\begin{bmatrix}
0 & 0 & M_{\ell,\lambda}^q \otimes 1 - i \otimes M_{\ell} & M_{\ell,\lambda}^p \otimes 1 \\[1ex]
0 & 0 & M_{\ell,\lambda}^p \otimes 1 & - M_{\ell,\lambda}^q \otimes 1 - i \otimes M_{\ell} \\[1ex]
M_{\ell,\lambda}^q \otimes 1 + i \otimes M_{\ell} & M_{\ell,\lambda}^p \otimes 1 & 0 & 0 \\[1ex]
M_{\ell,\lambda}^p \otimes 1 & - M_{\ell,\lambda}^q \otimes 1 + i \otimes M_{\ell} & 0 & 0 \\[1ex]
\end{bmatrix}.
\end{eqnarray*}
(Here, $M_{\ell,\lambda} := (M_{\ell} + \lambda I)$.) This spectral triple has spectral dimension 2. Moreover, for each $\lambda > 0$, the spectral triple implements the structure of a quantum metric space on $C(S_q^3)$.
There are numerous other constructions of spectral triples on the algebra $C(SU_q(2))$ in the literature, mostly with different spectral dimensions and no information about Rieffel's metric condition. The precise relation between those and our construction is unclear. The first spectral triples on $C(SU_q(2))$ were constructed by Chakraborty and Pal in <cit.> and <cit.>, whose focus was very different to ours. The named authors show that any spectral triple on $C(SU_q(2))$ which is of a certain natural form and which is equivariant for the quantum group co-action of $SU_q(2)$ must have spectral dimension at least 3, which is in contrast to our spectral triple of dimension 2. In <cit.> the same authors construct spectral triples on $C(SU_q(2))$ using an altogether different approach, focusing on those triples which are equivariant for the action of $\T^2$ on $C(SU_q(2))$, which might be closer to our spectral triple. The construction in <cit.> was used and further developed by Connes <cit.>. A different construction of a $3^+$-summable spectral triple on $C(SU_q(2))$ was developped in <cit.> using the classical Dirac operator. In another paper <cit.> the same authors give a construction of this triple via an extension using the cosphere bundle defined in <cit.> which appears somewhat similar to our construction.
The Podleś spheres were introduced as a family of quantum homogeneous spaces for the action of the
quantum $SU(2)$ group <cit.>. Probably the most widely studied algebraically
non-trivial examples are the so-called
equatorial Podleś spheres. They can be defined for each $q \in (0,1)$ as the universal C$^*$-algebra,
$C(S_q^2)$, for generators $\alpha$ and $\beta$, subject to the relations,
\begin{eqnarray*}
\beta^{^*} = \beta, \;\;\beta \alpha = q \alpha \beta, \;\;\alpha^* \alpha + \beta^2 = I, \;\; q^4 \alpha
\alpha^* + \beta^2 = q^4.
\end{eqnarray*}
Using the same notation as in Example <ref>, we can write down a representation of $C(S_q^2)$ over $H := \ell_2(\N) \otimes \C^2$ defined by:
\begin{eqnarray*}
\pi(\alpha) := T \sqrt{1 - N_q^4} \otimes \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \;\;\;
\pi(\beta) &:=& N_q^2 \otimes \begin{bmatrix} 1 & 0
\\ 0 & -1 \end{bmatrix},
\end{eqnarray*}
and this representation is faithful. By considering the map $\sigma: C(S_q^2) \to C(\T)$ sending $\beta$ to $0$ and $\alpha$ to $T \in C(\T)$, we soon obtain a short exact sequence,
\begin{eqnarray*}
0 \;\rightarrow \; \mathcal{K} \otimes \C^2 \;\rightarrow \; C(S_q^2) \;\rightarrow \; C(\T)
\;\rightarrow \; 0.
\end{eqnarray*}
We obtain an isomorphism,
\begin{eqnarray*}
C(S_q^2) \cong P C(\T) P \otimes \C I + \mathcal{K}(\ell_2(\N_0)) \otimes \C^2,
\end{eqnarray*}
where $P \in B(\ell_2(\Z))$ is again the usual Toeplitz projection, so that again $[P,x]$ is
compact for each $x \in \C(\T)$ and now $PxP \otimes 1 \in C(S_q^2)$ for each $x \in C(\T)$. As before we can write
\pi(\alpha) = -P T P(1 - \sqrt{1 - N_q^4}) \otimes I + P T P \otimes I, \textup{ and } \pi(\beta) \in \mathcal{K} \otimes \C^2.
As in Example <ref>, we can formulate the existence of spectral triples for the algebras $C(S_q^2)$ as follows: first, on $B$ we introduce the two-point triple, which turns $B$ into a spectral metric space
\begin{eqnarray*}
\left(\C^2, \C^2, \gamma := \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \right).
\end{eqnarray*}
Let $\pi$ denote the natural non-unital inclusion of $C(S_q^2)$ in $B(\ell_2(\Z) \otimes \C^2)$, whilst $\pi_{\sigma}: C(S_q^2) \to B(\ell_2(\Z) \otimes \C^2)$ is the map defined on the generators by $\pi_{\sigma}(\alpha) := T \otimes I_2$, $\pi_{\sigma}(\beta) = 0$.
Let $(\Alg, \ell_2(\Z), M_{\ell})$, $M_{\ell} e_n = n e_n$ be the usual spectral triple on $C^1(\T)$, where $\Alg \subseteq C(\T)$ is any dense $^*$-subalgebra of $C(\T)$ such that $(\Alg, \ell_2(\Z), \Dirac)$ satisfies
\begin{eqnarray*}
[M_{\ell}, f] \in B(\ell_2(\Z)), \;\;[|M_{\ell}|, f] \in B(\ell_2(\Z)), \;\;f \in \Alg
\end{eqnarray*}
(e.g. $\Alg = C^1(\T)$).
Then, for each $\lambda \in \R$ and for each $q \in (0,1)$, there is a spectral triple $(\mathcal{E}, (\ell_2(\Z) \otimes \C^2) \otimes \C^6, \Dirac_{\lambda})$ on $C(S_q^2)$, represented via $\pi_{\sigma} \oplus \pi_{\sigma} \oplus \pi \oplus \pi_{\sigma}$
and where
\begin{eqnarray*}
\Dirac_{\lambda} =
\begin{bmatrix}
M_{\ell,\lambda} \otimes 1 & 1 \otimes \gamma & 0 \\[1ex]
1 \otimes \gamma & - M_{\ell,\lambda} \otimes 1 & 0\\[1ex]
0 & 0 & \Dirac_I \\[1ex]
\end{bmatrix},
\end{eqnarray*}
\begin{eqnarray*}
\Dirac_I=
\begin{bmatrix}
0 & 0 & M_{\ell,\lambda}^q \otimes 1 - i \otimes \gamma & M_{\ell,\lambda}^p \otimes 1 \\[1ex]
0 & 0 & M_{\ell,\lambda}^p \otimes 1 & - M_{\ell,\lambda}^q \otimes 1 - i \otimes \gamma \\[1ex]
M_{\ell,\lambda}^q \otimes 1 + i \otimes \gamma & M_{\ell,\lambda}^p \otimes 1 & 0 & 0 \\[1ex]
M_{\ell,\lambda}^p \otimes 1 & - M_{\ell,\lambda}^q \otimes 1 + i \otimes \gamma & 0 & 0 \\[1ex]
\end{bmatrix}.
\end{eqnarray*}
(Here, $M_{\ell,\lambda} := (M_{\ell} + \lambda I)$.) This spectral triple has spectral dimension 1. Moreover, for each $\lambda > 0$, the spectral triple implements the structure of a quantum metric space on $C(S_q^2)$.
A spectral triple on $C(S_q^2)$ of dimension 2 has been constructed previously in <cit.>, again with no information about the metric condition. Also here the connection to our construction is unclear and left to future research. The relation seems even less clear than in the previous Example <ref> since the construction in <cit.> does not use any extensions.
The noncommutative $n$-spheres for higher dimensions can be defined inductively on $n$. The spheres of odd
dimension arise as short exact sequences of the form
\begin{eqnarray*}
0 \rightarrow \mathcal{K} \otimes C(S^1) \rightarrow C(S_q^{2n + 1}) \rightarrow C(S_q^{2n - 1})
\rightarrow 0, \;\;n \geq 1.
\end{eqnarray*}
and the spheres of even dimension as short exact sequences of the form
\begin{eqnarray*}
0 \rightarrow \mathcal{K}
\otimes \C^2 \rightarrow C(S_q^{2n}) \rightarrow
C(S_q^{2n - 1}) \rightarrow 0, \;\;n \geq 1.
\end{eqnarray*}
We suspect that the same process that was used to construct spectral metric spaces on $C(S_q^2)$ and $C(S_q^3)$ can, via this procedure, lead to the construction of spectral metric spaces for 1-parameter quantum spheres of any integer dimension. We can then relate these to similar constructions in the literature, e.g. <cit.>, <cit.>.
§ OUTLOOK.
In addition to the questions mentioned at the end of Example <ref> and <ref>
we briefly raise a number of questions related to this article which seem interesting.
1. The spectral triple we construct on the extension (<ref>) behaves well with respect to summability and induces metrics on the state space. This was our main goal. However, the following question still remains:
What is the KK-theoretical meaning of the spectral triple we construct on the extension?
2. Our construction of spectral triples is restricted to a special class of extensions (Toeplitz type extensions) but is applicable to several concrete examples as demonstrated in the last section. As discussed, there are similarities between a general semisplit extension by a stable ideal the Toeplitz type extensions we consider.
Can the construction of the spectral triple in Thm.<ref> be generalised to extensions which are not necessarily of Toeplitz type or only of Toeplitz type in a generalised sense?
3. Rieffel proposed a notion of distance between compact quantum metric spaces, modelled on the Gromov-Hausdorff
distance (<cit.>). It has since been used in a
number of questions relating to C$^*$-algebras endowed with seminorms. Some of the results are quite surprising:
Rieffel (<cit.>) shows how the common observation in quantum physics that `matrices converge to the 2-sphere' can be
illustrated quite well using Rieffel-Gromov-Hausdorff
There are various perspectives that we could take with respect to convergence for extensions in this chapter, especially for algebras arising as $q$-deformations.
One is to try to mimic the convergence studied by Christensen and Ivan in their approach. They construct
a two-parameter family of spectral
triples $(\mathcal{T}, H_A, \Dirac_{\alpha,\beta})$ for extensions of the form
\begin{eqnarray*} 0 \rightarrow
\mathcal{K} \rightarrow \mathcal{T} \rightarrow A
\rightarrow 0, \end{eqnarray*}
and for $\alpha, \beta > 0$, for which the quantum metric spaces converge to those on $A$ and $\mathcal{K}$ as $\alpha \to 0$ and $\beta \to 0$. However, for example in the case of the Podleś spheres this turns out not to be sufficient to study the Gromov-Hausdorff convergence aspects of varying the parameter $q$. The following two questions seem interesting, though we point out that the situation addressed in those questions is quite different from the matrix algebra convergence in <cit.> since the parameter $q$ does not change the algebras of the Podles spheres. We remark that classically $q$ can be regarded as a label for Poisson structures on $S^2$ (<cit.>).
Suppose that $(q_n)_{n \in \N} \subseteq (0,1)$ is a sequence converging to $q \in (0,1)$ and let
$(\Alg(C(S_{q_n}^2)),L)$ be one of the compact quantum metrics on
the Podleś sphere $C(S_{q_n}^2)$ for $n \in \N$ as defined in Thm.<ref>. Is it true that
$(\Alg(C(S_{q_n}^2)),L)$ converges to $(\Alg(C(S_{q}^2)),L)$ for
Rieffel-Gromov-Hausdorff convergence?
Suppose now that $(q_n)_{n \in \N} \subseteq
(0,1)$ converges to $1$. Let $(C^1(S^2),
L_{\Dirac})$ be the usual Lipschitz seminorm on the algebra $C(S^2) \cong C(S_1^2)$ for which the restriction of the metric to $S^2$ is the geodesic metric. Is it true that $(\Alg(C(S_{q_n}^2)),L)$ converges to $(C^1(S^2)),L)$, or any equivalent Lipschitz pair on the
two-sphere, for Rieffel-Gromov-Hausdorff convergence?
|
1511.00556
| |
1511.00075
|
We prove that there is no -algorithm that can approximate the
dominating set problem with any constant ratio, unless $\FPT= \W 1$. Our
hardness reduction is built on the second author's recent $\W 1$-hardness
proof of the biclique problem <cit.>. This yields, among other
things, a proof without the PCP machinery that the classical dominating
set problem has no polynomial time constant approximation under the
exponential time hypothesis.
c a
b ıi
ȷj u
§ INTRODUCTION
The dominating set problem, or equivalently the set cover problem, was among
the first problems proved to be -hard <cit.>. Moreover, it has been
long known that the greedy algorithm achieves an approximation ratio
$\approx \ln n$ <cit.>. And after a sequence
of papers (e.g. <cit.>), this is
proved to be best possible. In particular, Raz and Safra <cit.>
showed that the dominating set problem cannot be approximated with ratio
$c\cdot \log n$ for some constance $c\in \mathbb N$ unless $\P=
\NP$ <cit.>. Under a stronger assumption $\NP\not\subseteq
\DTIME\left(n^{O(\log\log n)}\right)$ Feige proved that no approximation
within $(1- \varepsilon)\ln n$ is feasible <cit.>. Finally Dinur and
Steuer established the same lower bound assuming only $\P\ne
\NP$ <cit.>. However, it is important to note that the
approximation ratio $\ln n$ is measured in terms of the size of an input
graph $G$, instead of $\ds(G)$, i.e., the size of its minimum dominating
set. As a matter of fact, the standard examples for showing the $\Theta(\log
n)$ greedy lower bound have constant-size dominating sets. Thus, the size of
the greedy solutions cannot be bounded by any function of $\ds(G)$. So the
question arises whether there is an approximation algorithm $\mathbb A$ that
always outputs a dominating set whose size can be bounded by
$\rho(\ds(G))\cdot \ds(G)$, where the function $\rho:\mathbb N\to\mathbb N$
is known as the approximation ratio of $\mathbb A$. The constructions
in <cit.> indeed show that we can rule out $\rho(x)\le \ln
x$. To the best of our knowledge, it is not known whether this bound is
tight. For instance, it is still conceivable that there is a polynomial time
algorithm that always outputs a dominating set of size at most
Other than looking for approximate solutions, parameterized
complexity <cit.> approaches
the dominating set problem from a different perspective. With the
expectation that in practice we are mostly interested in graphs with
relatively small dominating sets, algorithms of running time
$2^{\ds(G)}\cdot |G|^{O(1)}$ can still be considered efficient.
Unfortunately, it turns out that the parameterized dominating set problem is
complete for the second level of the so-called W-hierarchy <cit.>,
and thus fixed-parameter intractable unless $\FPT= \W 2$. So one natural
follow-up question is whether the problem can be approximated in fpt-time.
More precisely, we aim for an algorithm with running time $f(\gamma(G))\cdot
|G|^{O(1)}$ which always outputs a dominating set of size at most
$\rho(\ds(G))\cdot \ds(G)$. Here, $f:\mathbb N\to \mathbb N$ is an arbitrary
computable function. The study of parameterized approximability was
initiated in <cit.>. Compared to the classical
polynomial time approximation, the area is still in its very early stage
with few known positive and even less negative results.
§.§ Our results
We prove that any constant-approximation of the parameterized dominating set
problem is 1-hard.
For any constant $c\in \mathbb N$ there is no -algorithm $\mathbb
A$ such that on every input graph $G$ the algorithm $\mathbb A$ outputs a
dominating set of size at most $c\cdot \ds(G)$, unless $\FPT= \W 1$ (which
implies that the exponential time hypothesis () fails).
In the above statement, clearly we can replace “-algorithm” by
“polynomial time algorithm,” thereby obtaining the classical
constant-inapproximability of the dominating set problem. But let us mention
that our result is not comparable to the classical version, even if we
restrict ourselves to polynomial time tractability. The assumption $\FPT\ne
\W 1$ or is apparently much stronger than $\P\ne \NP$, and in fact
implies $\NP\not\subseteq \DTIME\left(n^{O(\log\log n)}\right)$ used
in aforementioned Feige's result. But on the other hand, our lower bound
applies even in case that we know in advance that a given graph has no large
dominating set.
Let $\beta:\mathbb N\to \mathbb N$ be a nondecreasing and unbounded
computable function. Consider the following promise problem.
[6.5]$\mds_\beta$A graph $G=(V,E)$ with $\ds(G)\le
\beta(|V|)$A dominating set $D$ of $G$$|D|$$\min$
Then there is no polynomial time constant approximation algorithm for
$\mds_{\beta}$, unless $\FPT= \W 1$.
The proof of Theorem <ref> is crucially built on a recent result
of the second author <cit.> which shows that the parameterized
biclique problem is $\W 1$-hard. We exploit the gap created in its hardness
reduction (see Section <ref> for more details). In the known
proofs of the classical inapproximability of the dominating set problem, one
always needs the PCP theorem in order to have such a gap, which makes those
proofs highly non-elementary. More importantly, it can be verified that
reductions based on the PCP theorem produce instances with optimal solutions
of relatively large size, e.g., a graph $G=(V,E)$ with $\ds(G)\ge
|V|^{\Theta(1)}$. This is inevitable, since otherwise we might be able to
solve every -hard problem in subexponential time. As an example, if it is
possible to reduce an -hard problem to the approximation of
$\textsc{Min-Dominating-Set}_\beta$ for $\beta(n)= \log \log \log n$, then
by brute-force searching for a minimum dominating set, we are able to solve
the problem in time $n^{O(\log\log\log n)}$. It implies $\NP\subseteq
\DTIME\left(n^{O(\log\log\log n)}\right)$. Because of this,
Corollary <ref>, and hence also Theorem <ref>, is
unlikely provable following the traditional approach.
Using a result of Chen et.al. <cit.> the lower bound in
Theorem <ref> can be further sharpened.
Assume holds.
Then there is no -algorithm which on every input graph $G$
outputs a dominating set of size at most $\sqrt[4+\varepsilon]{\log
(\ds(G))} \cdot \ds(G)$ for every $0<\varepsilon<1$.
§.§ Related work
The existing literature on the dominating set problem is vast. The most
relevant to our work is the classical approximation upper and lower bounds
as explained in the beginning. But as far as the parameterized setting is
concerned, what was known is rather limited.
Downey et. al proved that there is no additive approximation of the
the parameterized dominating set problem <cit.>. In the same paper,
they also showed that the independent dominating set problem has no approximation with any approximation ratio. Recall that an independent
dominating set is a dominating set which is an independent set at the same
time. With this additional requirement, the problem is no longer
monotone, i.e., a superset of a solution is not necessarily a
solution. Thus it is unclear how to reduce the independent dominating set
problem to the dominating set problem by an approximation-preserving
In <cit.> it is proved under that
there is no $c\sqrt{\log \ds(G)}$-approximation algorithm for the dominating
set problem[The papers actually address the set cover problem,
which is equivalent to the dominating set problem as mentioned in the
beginning.] with running time
$2^{O(\ds(G)^{(\log \ds(G))^d})}|G|^{O(1)}$, where $c$ and $d$ are some
appropriate constants. With the additional Projection Game Conjecture
due to <cit.> and some of its further strengthening, the authors of
<cit.> are able to even rule out
$\ds(G)^c$-approximation algorithms with running time almost doubly
exponential in terms of $\ds(G)$.
Clearly, these lower bounds are against far better approximation ratio than
those of Theorem <ref> and Theorem <ref>, while the drawback is that the dependence of the running time on $\ds(G)$ is not an arbitrary computable function.
The dominating set problem can be understood as a special case of the
weighted satisfiability problem of -formulas, in which all literals are
positive. The weighted satisfiability problems for various fragments of
propositional logic formulas, or more generally circuits, play very
important roles in parameterized complexity. In particular, they are
complete for the W-classes. In <cit.> it is shown that they have no
approximation of any possible ratio, again by using the
non-monotoncity of the problems. Marx strengthened this result significantly
in <cit.> by proving that the weighted satisfiability problem is not
approximable for circuits of depth 4 without negation gates, unless
$\FPT = \W 2$. Our result can be viewed as an attempt to improve Marx's
result to depth-2 circuits, although at the moment we are only able to rule
out approximations with constant ratio.
§.§ Organization of the paper
We fix our notations in
Section <ref>. In the same section we also explain the result
in <cit.> key to our proof. To help readability, we first prove that
the dominating set problem is not approximable with ratio smaller than
$3/2$ in Section <ref>. In the case of the clique problem, once we
have inapproximability for a particular constant ratio, it can be easily
improved to any constant by gap-amplification via graph products. But
dominating sets for general graph products are notoriously
hard to understand (see e.g. <cit.>). So to prove
Theorem <ref>, Section <ref> presents a modified
reduction which contains a tailor-made graph product.
Section <ref> discusses some consequences of our results. We
conclude in Section <ref>.
§ PRELIMINARIES
We assume familiarity with basic combinatorial optimizations and
parameterized complexity, so we only introduce those notions and notations
central to our purpose. The reader is referred to the standard textbooks
(e.g., <cit.> and <cit.>) for further
$\mathbb N$ and $\mathbb N^+$ denote the sets of natural numbers (that is,
nonnegative integers) and positive integers, respectively. For every $n\in
\mathbb N$ we let $[n]:= \{1, \ldots, n\}$. $\mathbb R$ is the set of real
numbers, and $\mathbb R_{\ge 1}:= \big\{r\in \mathbb R\bigmid r\ge 1\big\}$.
For a function $f: A\to B$ we can extend it to sets and vectors by defining
$f(S):= \{f(x)\mid x\in S\}$ and $f(\v):=\big(f(v_1), f(v_2), \cdots,
f(v_k)\big)$, where $S\subseteq A$ and $\v = (v_1, v_2, \cdots, v_k)\in A^k$
for some $k\in \mathbb{N}^+$.
Graphs $G= (V,E)$ are always simple, i.e., undirected and without loops and
multiple edges. Here, $V$ is the vertex set and $E$ the edge set,
respectively. The size of $G$ is $|G|:= |V|+|E|$. A subset
$D\subseteq V$ is a dominating set of $G$, if for every $v\in V$
either $v\in D$ or there exists a $u\in D$ with $\{u,v\}\in E$. In the
second case, we might say that $v$ is dominated by $u$, and this can be
easily generalized to $v$ dominated by a set of vertices. The
domination number $\ds(G)$ of $G$ is the size of a smallest
dominating set. The classical minimum dominating set problem is to
find such a dominating set:
[5.5]A graph $G=(V,E)$A dominating set $D$ of
The decision version of $\mds$ has an additional input $k\in \mathbb N$.
Thereby, we ask for a dominating set of size at most $k$ instead of
$\ds(G)$. But it is well known that two versions can be reduced to each
other in polynomial time. In parameterized complexity, we view the input $k$
as the parameter and thus obtain the standard parameterization of
[8.7]A graph $G$ and $k\in \mathbb N$$k$Decide whether $G$
has a dominating set of size at most $k$.
As mentioned in the Introduction, is complete for the parameterized
complexity class $\W 2$, the second level of the W-hierarchy. We will need
another important parameterized problem, the parameterized clique
[8.7]A graph $G$ and $k\in \mathbb N$$k$Decide whether
$G$ has a clique of size at most $k$.
which is complete for $\W 1$. Recall that a subset $S\subseteq V$ is a
clique in $G= (V,E)$, if for every $u,v \in S$ we have either $u=v$
or $\{u,v\}\in E$.
Those W-classes are defined by weighted satisfiability problems for
propositional formulas and circuits. As they will be used only in
Section <ref>, we postpone their definition until then.
§.§ Parameterized approximability
We follow the general framework
of <cit.>. However, to lessen the notational burden we restrict
our attention to the approximation of the dominating set problem.
Let $\rho:\mathbb N\to \mathbb R_{\ge 1}$. An algorithm $\mathbb A$ is a
parameterized approximation algorithm for with approximation
ratio $\rho$ if for every graph $G$ and $k\in \mathbb N$ with $\ds(G)\le k$
the algorithm $\mathbb A$ computes a dominating set $D$ of $G$ such that
\[
|D|\le \rho(k)\cdot k.
\]
If the running time of $\mathbb A$ is bounded by $f(k)\cdot |G|^{O(1)}$
where $f:\mathbb N\to \mathbb N$ is computable, then $\mathbb A$ is an approximation algorithm.
One might also define parameterized approximation directly for by
taking $\ds(G)$ as the parameter. The next result shows that essentially
this leads to the same notion.
Let $\rho:\mathbb N\to \mathbb R_{\ge 1}$ be a function such that
$\rho(k)\cdot k$ is nondecreasing. Then the following are equivalent.
* has an approximation algorithm with approximation ratio
* There exists a computable function $g:\mathbb N\to \mathbb N$ and an
algorithm $A$ that on every graph $G$ computes a dominating set $D$ of $G$ with $|D|\le
\rho(\ds(G))\cdot \ds(G)$ in time $g(\ds(G))\cdot |G|^{O(1)}$.
§.§ The Color-Coding
For every $n,k\in \mathbb N$ there is a family $\Lambda_{n,k}$ of polynomial
time computable functions from $[n]$ to $[k]$ such that for every
$k$-element subset $X$ of $[n]$, there is an $h\in \Lambda_{n,k}$ such that
$h$ is injective on $X$. Moreover, $\Lambda_{n,k}$ can be computed in time
$2^{O(k)}\cdot n^{O(1)}$.
§.§ The 1-hardness reduction of the parameterized biclique
Our starting point is the following theorem
proved in <cit.> which states that, on input a bipartite graph, it is
$\Wone$-hard to distinguish whether there exist $k$ vertices with large
number of common neighbors or every $k$-vertex set has small number of
common neighbors.
There is a polynomial time algorithm $\mathbb A$ such that for every graph
$G$ with $n$ vertices and $k\in \mathbb N$ with $\ceil{n^{\frac{6}{k+6}}}>
(k+6)!$ and $6\mid k+1$ the algorithm $\mathbb A$ constructs a bipartite
graph $H=(A \dotcup B, E)$ satisfying:
* if $G$ contains a clique of size $k$, i.e., $K_k\subseteq G$, then
there are $s$ vertices in $A$ with at least $\ceil
{n^{\frac{6}{k+1}}}$ common neighbors in $B$;
* otherwise $K_k\nsubseteq G$, every $s$ vertices in $A$ have at most
$(k+1)!$ common neighbors in $B$,
where $s=\binom{k}{2}$.
In our reductions from $\pclique$ to $\pds$, we use the following procedure
to ensure that the instance $(G,k)$ of $\pclique$ satisfies $6\mid k+1$.
Preprocessing. On input a graph $G$ and $k\in \mathbb
N^+$, if $6$ does not divide $k+1$, let $k'$ be the minimum integer such
that $k'\ge k$ and $6\mid k'+1$. We construct a new graph $G'$ by adding a
clique with $k'-k$ vertices into $G$ and making every vertex of this clique
adjacent to other vertices in $G$. It is easy to see that $k'\le k+5$, and
$G$ contains a $k$-clique if and only if $G'$ contains a $k'$-clique. Then
we proceed with $G\gets G'$ and $k\gets k'$.
§ THE CASE $\RHO< 3/2$
As the first illustration of how to use the gap created in
Theorem <ref>, we show in this section that $\pds$ cannot
be fpt approximated within ratio $< 3/2$. This serves as a stepping stone to
the general constant-inapproximability of the problem.
Let $\rho< 3/2$. Then there is no approximation of the parameterized
dominating set problem achieving ratio $\rho$ unless $\FPT=\W 1$.
We fix some $\varepsilon, \delta\in \mathbb R$ with $0< \varepsilon<
1$, $0< \delta< 1/2$, and
\begin{equation}\label{eq:rhoepsdelta}
\frac{3/2- \delta}{1+ \varepsilon} > \rho.
\end{equation}
Let $G$ be a graph with $n$ vertices and $k\in \mathbb N$ a parameter. We
set $s:= \binom{k}{2}$,
\begin{eqnarray*}
d:= \ceil{\frac{s}{\varepsilon}}^{2s},
& \text{and} &
t:= \ceil{\left(\frac{1}{2}- \delta\right)\cdot d^{1-1/2s}}.
\end{eqnarray*}
As a consequence, when $k$ and $n$ are sufficiently large, we have
\begin{equation}\label{eq:at}
s t< \varepsilon d,
\quad \left(\frac{1}{2}- \delta\right)\cdot \frac{d}{t}\le \sqrt[2s]{d},
\quad (k+1)!< 2 \delta \sqrt{d}-1,
\quad \text{and} \quad d\le \lceil n^{\frac{6}{k+1}}\rceil.
\end{equation}
By Theorem <ref> (and the preprocessing) we can compute in
-time a bipartite graph $H_0=(A_0 \dotcup B_0, E_0)$ such that:
- if $K_k\subseteq G$, then there are $s$ vertices in $A_0$ with
$d$ common neighbors in $B_0$;
- if $K_k\nsubseteq G$, then every $s$ vertices in $A_0$ have at
most $(k+1)!$ common neighbors in $B_0$.
Then using the color-coding in Lemma <ref>, again in -time, we
construct two function families $\Lambda_A:= \Lambda_{|A_0|, s}$ and
$\Lambda_B:= \Lambda_{|B_0|, d}$ such that
- for every $s$-element subset $X\subseteq A_0$ there is an $h\in
\Lambda_A$ with $h(X)= [s]$;
- for every $d$-element subset $Y\subseteq B_0$ there is an $h\in
\Lambda_B$ with $h(Y)= [d]$.
Define the bipartite graph $H= \big(A(H) \dotcup B(H), E(H)\big)$
\begin{align*}
A(H) &:= A_0\times \Lambda_A\times \Lambda_B,
\qquad B(H):= B_0\times \Lambda_A\times \Lambda_B\\
E(H) &:= \Big\{\big\{(u,h_1,h_2), (v, h_1,h_2)\big\}
\Bigmid \text{$u\in A_0$, $v\in B_0$, $h_1\in \Lambda_A$, $h_2\in \Lambda_B$,
and $\{u,v\}\in E_0$ }\Big\}.
\end{align*}
Moreover, define two colorings $\alpha: A(H)\to [s]$ and $\beta: B(H)\to
[d]$ by
\begin{eqnarray*}
\alpha(u,h_1,h_2):= h_1(u) & \text{and} & \beta(v,h_1,h_2):= h_2(v).
\end{eqnarray*}
It is straightforward to verify that
(H1) if $K_k\subseteq G$, then there are $s$ vertices of distinct
$\alpha$-colors in $A(H)$ with $d$ common neighbors of distinct
$\beta$-colors in $B(H)$;
(H2) if $K_k\nsubseteq G$, then every $s$ vertices in $A(H)$ have
at most $(k+1)!$ common neighbors in $B(H)$.
Now from $H$, $\alpha$, and $\beta$ we construct a new graph $G'=
\big(V(G'), E(G')\big)$ as
follows. First, its vertex set is defined by
\[
V(G'):= B(H) \dotcup \big\{x_i, y_i \bigmid i\in [d]\big\}
\dotcup C \dotcup W,
\]
\begin{eqnarray*}
C:= A(H) \times [t] & \text{and} &
W:= \left\{w_{b,j,i} \Bigmid b\in B(H), i\in [t], j\in [s]\right\}.
\end{eqnarray*}
Moreover, $G'$ contains the following types of edges.
(E1) $\{b,b'\}\in E(G')$ with $b,b'\in B(H)$, $b\ne b'$, and
$\beta(b)= \beta(b')$ (i.e., all vertices in $B(H)$ with the same
color under $\beta$ form a clique in $G'$).
(E2) Let $b\in B(H)$ and $c:= \beta(b)$. Then $\{x_c, b\}, \{y_c,
b\}\in E(G')$.
(E3) Let $b,b'\in B(H)$ with $\beta(b)= \beta(b')$ and $b\ne b'$.
Then $\big\{w_{b,j,i}, b'\big\}\in E(G')$ for every $i\in [t]$ and
$j\in [s]$.
(E4) $\big\{(a,i), w_{b,j,i}\big\}\in E(G')$ for every $\{a,b\}\in
E(H)$, $j= \alpha(a)$ and $i\in [t]$.
(E5) Let $a, a'\in A(H)$ with $a\ne a'$ and $i\in [t]$. Then
$\big\{(a,i), (a',i)\big\}\in E(G')$.
To ease presentation, for every $c\in [d]$ we set
\[
B_c:= \big\{b\in B(H)\bigmid \beta(b)= c\big\} \cup \{x_c, y_c\}.
\]
Claim 1. If $D$ is a dominating set of $G'$, then $D\cap
B_c\ne \emptyset$ for every $c\in [d]$.
Proof of the claim. We observe that every $x_c$ is only
adjacent to vertices in $B_c$. $\dashv$
Claim 2. If $G$ contains a $k$-clique, then
$\ds(G')< (1+\varepsilon) d$.
Proof of the claim. By (H1) the bipartite graph $H$ has a
$K_{s,d}$ biclique $K$ with $\alpha(A(H)\cap K)= [s]$ and $\beta(B(H)\cap
K)=[d]$. It is then easy to verify that
\[
\big(B(H)\cap K\big) \dotcup \big((A(H)\cap K)\times [t]\big)
\]
is a dominating set of $G'$, whose size is $d+ s\cdot t< (1+\varepsilon) d$
by (<ref>). $\dashv$
Claim 3. If $G$ contains no $k$-clique, then
every $s$-vertex set of $A(H)$ has at most $(k+1)!< 2 \delta \sqrt{d}-1$
common neighbors in $B(H)$.
Claim 4. If $G$ contains no $k$-clique, then
\[
\ds(G')> \left(\frac{3}{2}- \delta\right)\cdot d.
\]
Proof of the claim. Let $D$ be a dominating set of $G'$.
By Claim 1 we have $D\cap B_c\ne \emptyset$ for every $c\in [d]$. Define
\[
e:= \Big|\big\{c\in [d] \bigmid |D\cap B_c|\ge 2\big\}\Big|.
\]
If $e> (1/2- \delta) \cdot d$ then $|D|> d+ e> (3/2- \delta)\cdot d$ and we
are done.
So let us consider $e\le (1/2- \delta) \cdot d$ and without loss of
generality $|D\cap B_c| = 1$ for every $c\le (1/2+ \delta) \cdot d$. Fix
such a $c$ and assume $D\cap B_c = \{b_c\}$. Recall $x_c, y_c\in V(G')$ are
not adjacent to any vertex outside $B_c$, and there is no edge between them,
thus $b_c\in B_c\setminus \{x_c, y_c\}= \big\{b\in B(H)\bigmid \alpha(b)=
c\big\}$. Let
\[
W_1:= \left\{w_{b_c, j,i} \Bigmid \text{$i\in [t]$, $j\in [s]$,
and $c\le (1/2+ \delta) \cdot d$}\right\}\subseteq W.
\]
(E3) implies that every $w_{b_c, j,i}\in W_1$ is not dominated by any vertex
in $D\cap \bigcup_{c\in [d]} B_c$. Therefore, it has to be dominated by or
included in $D\cap (C\cup W)$.
If $|D\cap W_1|> (1/2- \delta)\cdot d $, then again we are done. So suppose
$|D\cap W_1|\le (1/2- \delta)\cdot d$. Without loss of generality let
\[
W_2:= \left\{w_{b_c, j,i} \Bigmid \text{$i\in [t]$, $j\in [s]$,
and $c\le 2\delta d$}\right\}\subseteq W_1
\]
and assume $W_2 \cap D= \emptyset$. Thus $W_2$ has to be dominated by $D\cap
C$. For later purpose, let
\[
Y:= \big\{b_c\bigmid c\le 2\delta d\big\}.
\]
Obviously, $|Y|\ge 2\delta d -1$.
Again we only need to consider the case $|D\cap C|\le (1/2- \delta)\cdot d$.
Recall $C= A(H)\times [t]$. Thus there is an $i \in [t]$ such that
\[
\Big|D\cap \big(A(H)\times \{i\}\big) \Big|
\le \left(\frac{1}{2}- \delta\right)\cdot \frac{d}{t}.
\]
Let $X:= \big\{a\in A(H)\bigmid (a,i)\in D\big\}$, and in particular,
$|X|\le (1/2-\delta)\cdot d/ t$. Since $W_2$ is dominated by $D\cap C$, we
have for all $b\in Y$ and $j\in [s]$ there exists $a\in X$ such that
$\big\{(a, i), w_{b,j,i}\big\}\in E(G')$, which means that $\{a, b\}\in
E(H)$ and $\alpha(a)= j$. It follows that in the graph $H$ every vertex of
$Y$ has at least $s$ neighbors in $X$. Recall that $(1/2- \delta)\cdot d/
t\le \sqrt[2s]{d}$ by (<ref>). There are at most $\sqrt{d}$ different
types of $s$-vertex sets in $X$, i.e.,
\[
\left|\binom{X}{s}\right| \le \binom{(1/2-\delta)\cdot d/ t}{s}
\le \left(\sqrt[2s]{d}\right)^{s} = \sqrt{d}.
\]
By the pigeonhole principle, there exists an $s$-vertex set of $X\subseteq
A(H)$ having at least $|Y|/ \sqrt{d}\ge 2\delta \sqrt{d}-1$ common
neighbors in $Y\subseteq B(H)$, which contradicts Claim 3. $\dashv$.
Claim 2 and Claim 4 indeed imply that there is an fpt-reduction from the
clique problem to the dominating set problem which creates a gap great than
\[
\frac{3/2 - \delta}{1+\varepsilon}.
\]
So if there is a $\rho$-approximation of the dominating set problem,
by (<ref>) we can decide the clique problem in fpt time.
§ THE CONSTANT-INAPPROXIMBILITY OF
Theorem <ref> is a fairly direct consequence of the following
There is an algorithm $\mathbb A$ such that on input a graph $G$, $k\ge 3$,
and $c\in \mathbb N$ the algorithm $\mathbb A$ computes a graph $G_c$ such
(i) if $K_k\subseteq G$, then $\ds(G_c)< 1.1\cdot d^c$;
(ii) if $K_k\nsubseteq G$, then $\ds(G_c)> c\cdot d^c/3$,
where $d= \left(30\cdot c^2\cdot (k+1)^2\right)^{4\cdot k^3+ 3c}$. Moreover
the running time of $\mathbb A$ is bounded by $f(k,c)\cdot |G|^{O(c)}$ for a
computable function $f: \mathbb N\times \mathbb N\to \mathbb N$.
Suppose for some $\epsilon> 0$ there is an -algorithm $\mathbb{A}(G)$
which outputs a dominating set for $G$ of size at most
$\sqrt[4+\varepsilon]{\log (\ds(G))} \cdot \ds(G)$. Of course we can further
assume that $\varepsilon< 1$. Then on input a graph $G$ and $k\in\mathbb N$,
\begin{eqnarray*}
c := \ceil{k^{1-\epsilon/5}}= o(k)
& \text{and} &
d:= \left(30\cdot c^2\cdot (k+1)^2\right)^{4\cdot k^3+3c}.
\end{eqnarray*}
We have
\[
\sqrt[4+\varepsilon]{\log(1.1\cdot d^c)}
= O\left(\sqrt[4+\varepsilon]{c\cdot k^3\cdot \log k}\right)
= o\left(k^{\frac{4}{4+\varepsilon}}\right)
= o(c).
\]
By Theorem <ref>, we can construct a graph $G_c$ with
properties (i) and (ii) in time
\[
f(k,c)\cdot |G|^{O(c)} = h(k)\cdot |G|^{o(k)}
\]
for an appropriate computable function $h:\mathbb N\to \mathbb N$. Thus, $G$
contains a clique of size $k$ if and only if $\mathbb A(G_c)$ returns a
dominating set of size at most
\[
1.1\cdot d^c\cdot \sqrt[4+\varepsilon]{\log(1.1\cdot d^c)}
= o\left(c\cdot d^c \right)< \frac{c\cdot d^c}{3},
\]
where the inequality holds for sufficiently large $k$ (and hence
sufficiently large $c\cdot d^c$).
Therefore we can determine whether $G$ contains a $k$-clique in time
$g(k)\cdot |G|^{o(k)}$ for some computable $g:\mathbb N\to \mathbb N$. This
contradicts a result in Chen et.al. <cit.> under .
§.§ Proof of Theorem <ref>
We start by showing a variant of Theorem <ref>.
Let $\Delta\in \mathbb N^+$ be a constant and $d: \mathbb N^+\to \mathbb
N^+$ a computable function. Then there is an -algorithm that on input a
graph $G$ and a parameter $k\in \mathbb N$ with $6 \mid k+1$ constructs a
bipartite graph $H= \big(A(H) \dotcup B(H), E(H)\big)$ together with two
\begin{eqnarray*}
\alpha: A(H)\to [\Delta s] & \text{and} & \beta: B(H)\to [d(k)]
\end{eqnarray*}
such that:
(H1) if $K_k\subseteq G$, then there are $\Delta s$ vertices of
distinct $\alpha$-colors in $A(H)$ with $d(k)$ common neighbors of
distinct $\beta$-colors in $B(H)$;
(H2) if $K_k\nsubseteq G$, then every $\Delta(s-1)+ 1$ vertices in
$A(H)$ have at most $(k+1)!$ common neighbors in $B(H)$,
where $s=\binom{k}{2}$.
Let $G$ be a graph with $n$ vertices and $k\in \mathbb N$. Assume
without loss of generality
\begin{eqnarray*}
\ceil{n^{\frac{6}{k+6}}}> (k+6)!
& \text{and} &
\ceil {n^{\frac{6}{k+1}}} \ge d(k).
\end{eqnarray*}
By Theorem <ref> we can construct in polynomial time a
bipartite graph $H_0= (A_0 \dotcup B_0, E_0)$ such that for $s:=
\binom{k}{2}$:
* if $K_k\subseteq G$, then there are $s$ vertices in $A_0$ with at
least $d(k)$ common neighbors in $B_0$;
* if $K_k\nsubseteq G$, then every $s$ vertices in $A_0$ have at most
$(k+1)!$ common neighbors in $B_0$.
\begin{align*}
A_1 := A_0\times [\Delta], \ B_1:= B_0, \ \text{and}\
E_1 := \big\{\{(u,i), v\} \bigmid \text{$(u,i)\in A_0\times [\Delta]$,
$v\in B_0$, and $\{u,v\}\in E_0$}\big\}.
\end{align*}
It is easy to verify that in the bipartite graph $(A_1\dotcup B_1, E_1)$
* if $K_k\subseteq G$, then there are $\Delta s$ vertices in $A_1$
with at least $d(k)$ common neighbors in $B_2$;
* if $K_k\nsubseteq G$, then every $\Delta (s-1)+1$ vertices in $A_1$
have at most $(k+1)!$ common neighbors in $B_1$.
Applying Lemma <ref> on
\begin{eqnarray*}
\big(n \gets |A_1|, k\gets \Delta s\big)
& \text{and} &
\big(n \gets |B_1|, k\gets d(k)\big)
\end{eqnarray*}
we obtain two function families $\Lambda_A:= \Lambda_{|A_1|, \Delta s}$ and
$\Lambda_B:= \Lambda_{|B_1|, d(k)}$ with the stated properties. Finally the
desired bipartite graph $H$ is defined by $\Big((A_1\times \Lambda_A\times
\Lambda_B) \dotcup (B_1\times \Lambda_A\times \Lambda_B), E)\Big)$
\[
E:= \Big\{\big\{(u,h_1,h_2), (v, h_1, h_2)\big\} \Bigmid
\text{$u\in A_1$, $v\in B_1$, $h_1\in \Lambda_A$, $h_2\in \Lambda_B$,
and $\{u,v\}\in E_1$}\Big\}
\]
and the colorings
\[
\alpha(u,h_1,h_2):= h_1(u)
\quad \text{and} \quad
\beta(v,h_1, h_2):= h_2(v). \benda
\]
Setting the parameters. Let $\Delta:= 2$. Recall that
$k\ge 3$, $s=\binom{k}{2}\ge 3$, and $c\in \mathbb N^+$. We first
\[
d:= d(k):= \left(30\cdot c^2\cdot (k+1)^2\right)^{4\cdot k^3+3c}.
\]
It is easy to check that:
(i) $d^{\frac{1}{2}-\frac{1}{2s}}> c\cdot s^c \ \Big(\!= c \cdot
\binom{k}{2}^c\Big)$.
(ii) $d> \big(3(k+1)!\big)^{2s}$.
(iii) $d> \left(10\Delta s\cdot c^2\right)^{2\Delta s}$.
Then let
\begin{equation}\label{eq:t}
t:= c \cdot d^{c-\frac{1}{2\Delta s}}.\footnotemark
\end{equation}
Here, we assume $d^{c-\frac{1}{2\Delta s}}$ is an integer.
Otherwise, let $d\gets d^{2\Delta s}$ which maintains (i)– (iii). From
(ii), (iii), and (<ref>) we conclude
\begin{equation}\label{eq:dt}
\Delta s c t< 0.1\cdot d^c,
\quad \frac{c\cdot d^c}{3t}\le \sqrt[2\Delta s]{d},
\quad \text{and}\ (k+1)!< \frac{\sqrt[2 s]{d}}{3}.
\end{equation}
Moreover by (i) and $\Delta= 2$ we have
\begin{equation}\label{eq:deltas}
c\cdot d^c+ c\Delta^c s^c d^{c-\frac{1}{2}+\frac{1}{2s}}< 2\Delta^cd^c.
\end{equation}
Construction of $G_c$. We invoke Theorem <ref>
to obtain $H=(A \dotcup B, E)$, $\alpha$, and $\beta$. Then we construct a
new graph $G_c= \big(V(G_c), E(G_c)\big)$ as follows. First, the vertex set
of $G_c$ is given by
\[
V(G_c):= \bigcup_{\i\in [d]^c} V_{\i}\dotcup C\dotcup W,
\]
\begin{equation*}
V_{\i}:= \big\{\v\in B^{c}\bigmid \beta(\v)= \i \big\}
\quad \text{for every $\i\in [d]^c$},
\end{equation*}
\begin{eqnarray*}
C:= A \times [c]\times [t], & \text{and} &
W:= \Big\{w_{\v,\j,i}\Bigmid \text{$\v\in V_{\i}$ for some $\i\in [d]^{c}$,
$\j\in [\Delta s]^c$ and $i\in [t]$}\Big\}.
\end{eqnarray*}
Moreover, $G_c$ contains the following types of edges.
(E1) For each $\i\in [d]^{c}$, $V_{\i}$ forms a clique.
(E2) Let $\i\in [d]^{c}$ and $\v, \v'\in V_{\i}$. If for all
$\ell\in [c]$ we have $\v(\ell)\ne \v'(\ell)$ then $\left\{w_{\v, \j,
i}, \v'\right\}\in E(G_c)$ for every $i\in [t]$ and $\j\in[\Delta
(E3) Let $i\in [t]$. Then $\big\{(u,\ell,i), w_{\v, \j, i}\big\}\in
E(G_c)$ if $\{u, \v(\ell)\}\in E$ and
$\j(\ell)= \alpha(u)$.
(E4) Let $u, u'\in A(H)$ with $u\ne u'$, $\ell\in[c]$, and $i\in
[t]$. Then $\big\{(u,\ell,i), (u',\ell,i)\big\}\in E(G_c)$.
Theorem <ref> then follows from the completeness and the
soundness of this reduction.
If $G$ contains $k$-clique, then $\ds(G_c)< 1.1
If $G$ contains no $k$-clique then $\ds(G_c)>c\cdot d^c/3$.
We first show the easier completeness.
By (H1) in Theorem <ref>, if $G$ contains a subgraph isomorphic
to $K_k$, then the bipartite graph $H$ has a $K_{\Delta s,d}$-subgraph $K$
such that $\alpha(A\cap K)= [\Delta s]$ and $\beta(B\cap K)= [d]$. Let
\[
D:= (B\cap K)^c\dotcup \big((A\cap K)\times[c]\times [t]\big).
\]
Obviously, $|D|= d^c+ \Delta sct< 1.1\cdot d^c$ by (<ref>). And (E1)
and (E4) imply that $D$ dominates every vertex in $C$ and every vertex in
$V_{\i}$ for all $\i\in [d]^c$.
To see that $D$ also dominates $W$, let $w_{\v,\j,i}$ be a vertex in $W$.
First consider the case where $\v(\ell)\notin B\cap K$ for all $\ell\in[c]$.
Since $\beta\big((B\cap K)^c\big)= [d]^c$, there exists a vertex $\v'\in
(B\cap K)^c$ with $\beta(\v')= \beta(\v)$ and $\v(\ell)\neq\v'(\ell)$ for
all $\ell\in[c]$. Then $w_{\v, \j, i}$ is dominated by $\v'$ because of
Otherwise assume $\v(\ell)\in B\cap K$ for some $\ell\in[c]$, then $A\cap
K\subseteq N^H(\v(\ell))= \big\{u\in A\bigmid \{u, \v(\ell)\}\in
There exists a vertex $u\in A\cap K$ such that $\alpha(u)= \j(\ell)$ and
$\big\{\v(\ell),u\big\}\in E$. By (E3), $w_{\v,\j,
i}$ is adjacent to $(u,\ell,i)$.
§.§ Soundness
Suppose $c, \Delta, t\in\mathbb{N}^+$ and $\Delta< t$. Let $V\subseteq
[t]^c$. If there exists a function $\theta: V\to [c]$ such that for all
$i\in [c]$ we have
\begin{equation}\label{eq:V}
\Big|\big\{\v(i)\bigmid \text{$\v\in V$ and $\theta(\v)= i$}\big\}\Big|
\le t- \Delta,
\end{equation}
then $|V|\le t^c- \Delta^c$.
When $c=1$, we have $|V|\le t- \Delta$ by (<ref>). Suppose the lemma
holds for $c\le n$ and consider $c= n+1$. Given $V\subseteq [t]^{n+1}$ and
$\theta$, let
\[
C_{n+1}:= \big\{\v(n+1)\bigmid \text{$\v\in V$ and $\theta(\v)= n+1$} \big\}.
\]
By (<ref>), $|C_{n+1}|\le t- \Delta$. If $|C_{n+1}|< t- \Delta$, we add
$\big(t-\Delta-|C_{n+1}|\big)$ arbitrary integers from $[t]\setminus
C_{n+1}$ to $C_{n+1}$. So we have $|C_{n+1}|= t- \Delta$. Let $A:=
\big\{\v\in V\bigmid \v(n+1)\in C_{n+1}\big\}$ and $B:= V\setminus A$. It
follows that
\begin{equation}\label{eq:A}
|A|\le (t- \Delta)t^{c-1},
\end{equation}
$\Big|\big\{\v(n+1)\bigmid \v\in B\big\}\Big|\le \Delta$, and
$\theta(\v)\in[c-1]$ for $\v\in B$. Let
\[
V':= \big\{(v_1,v_2,\cdots,v_n)\bigmid
\exists v_{n+1}\in [t], (v_1,v_2,\cdots,v_n,v_{n+1})\in B\big\}.
\]
We define a function $\theta': V'\to [c-1]$ as follows. For all $\v'\in V'$,
choose $\v\in B$ with the minimum $\v(c)$ such that for all $i\in[c-1]$ it
holds $\v'(i)= \v(i)$. By the definition of $V'$, such a $\v$ must exist,
and we let $θ'('̌):= θ()̌$. By~(\ref{eq:V}),
$|{'̌(i)$\v'\in V'$ and $\theta'(\v')=
i$}|≤t- Δ$ for all $i∈[c-1]$. Applying the induction
hypothesis, we get $|V'|≤t^c-1- Δ^c-1$. Obviously,
\begin{equation}\label{eq:B}
|B|\le \Delta|V'|\le \Delta t^{c-1}- \Delta^c.
\end{equation}
From~\eqref{eq:A} and~\eqref{eq:B}, we deduce that $|V|= |A|+ |B|≤(t-Δ)t^c-1+ Δt^c-1- Δ^c ≤t^c- Δ^c$. \proofend
\end{proof}
%Let $H=(X\;\dot\cup\; Y;E)$ be a bipartite graph, $p,q\in\mathbb{N}^+$. If
%every vertex in $Y$ has at least $p$ neighbors in $X$ and
%$|Y|>\binom{|X|}{p}q$, there must exists $p$ vertices in $X$ with $q$ common
%neighbors in $Y$.
We are now ready to prove the soundness of our reduction.
\begin{proof}[of Lemma~\ref{lem:soundness}]
Let $D$ be a dominating set of $G_c$.
%By Claim~1 we can assume that $D\cap V_c\ne \emptyset$ for every $c\in [d]$.
\[
a:= \Big|\big\{\i\in [d]^c \bigmid |D\cap V_{\i}|\ge c+1 \big\}\Big|.
\]
If $a> d^c/3$, then $|D|\ge (c+1)a> c\cdot d^c/3$ and we are done.
\medskip
So let us consider $a\le d^c/3$. Thus, the set
\[
I:= \big\{\i\in [d]^c \bigmid |D\cap V_{\i}|\le c \big\}
\]
has size $|I|\ge 2d^c/3$. Let $\i\in I$ and assume that $D\cap V_{\i}=
\big\{\v_1, \v_2, \ldots, \v_{c'}\big\}$ for some $c'\le c$. We define a
$\v_\i\in V_\i$ as follows. If $c'=0$, we choose an arbitrary $\v_\i\in
V_\i$.\footnote{Since the coloring $\beta$ is obtained by the color-coding
used in the proof of Theorem~\ref{thm:colgap}, for every $b\in[d]$ it holds
that $\{v\in B\mid \beta(v)=b\}\neq\emptyset$, hence $V_\i\neq
\emptyset$.} Otherwise, let
%Clearly we can
%find a tuple $\v \in B^c$ with
% $\v(i)= \v_i(i)$ for all $i\in[c']$.
\[
\v_\i(\ell):=
\begin{cases}
\v_\ell(\ell) & \text{for all $\ell\in[c']$}; \\
\v_1(\ell) & \text{for all $c'<\ell\le c$}. \\
\end{cases}
\]
Obviously, $\beta(\v_\i)= \i$.
\smallskip
(E2) implies that for every $\j\in [\Delta s]^c$ and every $i\in [t]$, the
vertex $w_{\v_{\i},\j,i}$ is not dominated by $D\cap V_{\i}$. Observe that
$w_{\v_{\i},\j,i}$ cannot be dominated by other $D\cap V_{\i'}$ with $\i'\ne
\i$ either, by (E2) and (E3). Therefore every vertex
in the set
\[
W_1:= \big\{w_{\v_{\i}, \j,i}
\bigmid \text{$\i\in I$, $\j\in [\Delta s]^c$, and $i\in [t]$} \big\}
\]
is \emph{not} dominated by $D\cap \bigcup_{\i\in [d]^c} V_{\i}$. As a
consequence, $W_1$ has to be dominated by or included in $D\cap (C\cup W)$.
\medskip
If $|D\cap W_1|> c\cdot d^c/3$, then again we are done. So suppose $|D\cap
W_1|\le c\cdot d^c/3$ and let $W_2:= W_1\setminus D$. It follows that $W_2$
has to be dominated by $D\cap C$. Once again we only need to consider the
case $|D\cap C|\le c\cdot d^c/3$, and hence there is an $i' \in [t]$ such
\begin{equation}\label{eq:DCsize}
\Big|D\cap \big(A\times[c]\times \{i'\}\big) \Big|
\le \frac{c\cdot d^c}{3 t}.
\end{equation}
Then we define
\[
Z:= \big\{w_{\v,\j, i}\in W_2\bigmid i=i'\big\}
= \big\{w_{\v_{\i},\j, i'}\bigmid \text{$\i\in I$,
$\j\in[\Delta s]^c$, and $w_{\v_{\i},\j, i'}\notin D$}\big\}.
\]
So $Z$ has to be dominated by $D\cap C$, and in particular those vertices of
the form $(u,\ell,i')\in D\cap C$. Moreover,
\begin{equation}\label{eq:Z}
|Z|\ge \Delta^cs^c|I|- |D\cap W_1|\ge \Delta^cs^c |I|- c\cdot d^c/3.
\end{equation}
Our next step is to upper bound $|Z|$. To that end, let
%\yrand{$u\in A(H)\mapsto u\in A\times \ldots$}
\[
X:= \big\{u\in A\bigmid \text{$(u,\ell,i')\in D$ for some $\ell\in [c]$}\big\}.
\]
Thus $Z$ is dominated by those vertices $(u,\ell,i')$ with $u\in X$. And
\[
|X|\le \frac{c\cdot d^c}{3 t}.
\]
%Y=\left\{v\in B\bigmid \text{$\exists u_1,u_2,\cdots,u_{\Delta(s-1)+1}\in X$
%s.t. for all distinct $i,j\in[\Delta(s-1)+1], \beta(u_i)\ne\beta(u_j)$ and
%$\{u_i,v\}\in E(H)$}\right\}
\[
Y:= \Big\{v\in B\Bigmid \big|N^H(v)\cap X\big|> \Delta(s-1)\Big\}.
\]
%It follows that in the graph $H$ every vertex of $Y$ has at least
%$\Delta(s-1)+1$ neighbors in $X$.
Recall that $c\cdot d^c/(3t)\le \sqrt[2\Delta s]{d}$ by~\eqref{eq:dt}. Hence
$X$ has at most $\sqrt{d}$ different subsets of size $\Delta(s-1)+ 1$, i.e.,
\[
\left|\binom{X}{\Delta(s-1)+1}\right|\le |X|^{\Delta(s-1)+1}\le |X|^{\Delta s}
\le \sqrt{d}.
\]
We should have
\begin{equation}\label{eq:Y}
|Y|\le \sqrt{d}\cdot (k+1)!\le \frac{d^{\frac{1}{2}+\frac{1}{2s}}}{3},
\end{equation}
where the second inequality is by~\eqref{eq:dt}.
%\yrand{Added:where the second inequality is by~\eqref{eq:dt}.}
Otherwise, by the pigeonhole
principle, there exists a $(\Delta(s-1)+1)$-vertex set of $X\subseteq A(H)$
having at least $|Y|/\sqrt{d}> (k+1)!$
common neighbors in $Y\subseteq B(H)$. However, if $G$ contains no
$k$-clique, then by (H2) every $\big(\Delta(s-1)+1\big)$-vertex set of
$A(H)$ has at most $(k+1)!$ common neighbors in $B(H)$, and we obtain a
%Let $Z:= \big\{w_{\v,\j, i}\in W_2\bigmid i=i'\big\}$. We have
%|Z|\ge|\mathcal I|\cdot \Delta^cs^c-|D\cap W_1|\ge \Delta^cs^c|\mathcal I|-cd^c/3
\medskip
\begin{align*}
Z_1 := & \big\{w_{\v, \j, i'}\in Z\bigmid \text{there exists an $\ell\in[c]$ with $\v(\ell)\in Y$}\big\}
\quad\Big(\!\!\subseteq Z\Big) \\
= & \big\{w_{\v_{\i}, \j, i'}\bigmid \text{$\i\in I$, $\j\in [\Delta s]^c$, $w_{\v_{\i}, \j, i'}\notin D$,
and there exists an $\ell\in[c]$ with $\v_\i(\ell)\in Y$}\big\} \\
% = & \big\{w_{\v_{\i}, \j, i'}\bigmid \text{$\i\in I$, $\j\in [\Delta s]^c$, $w_{\v_{\i}, \j, i'}\notin D$,
% and there exists an $\ell\in[c]$ with $\i(\ell)\in \beta(Y)$}\big\} \\
\text{and}\qquad
Z_2 := & Z\setminus Z_1
= \big\{w_{\v_{\i}, \j, i'}\bigmid \text{$\i\in I$, $\j\in [\Delta s]^c$, $w_{\v_{\i}, \j, i'}\notin D$,
and $\v_{\i}(\ell)\notin Y$ for all $\ell\in [c]$}\big\}.
\end{align*}%\brand{Change $\in Y$ to $\notin Y$ in the definition of $Z_2$.}
Moreover, let $I_1:=\{\i\in I\mid \text{there exists a $w_{\v_\i, \j, i'}\in
Z_1$}\}$. From the definition, we can deduce that
\[
\text{for all $\i\in I_1$ there exists an $\ell\in [c]$ such that $\i(\ell)\in \beta(Y)$}.
%\forall \i\in I_1,\exists \ell\in[c],\i(\ell)\in\beta(Y).
\]
Then $|I_1|\le c|Y|d^{c-1}$ and hence
\[
|Z_1|\le|I_1|\Delta^cs^c\le c |Y| d^{c-1} \Delta^c s^c.
\]
To estimate $|Z_2|$, let us fix an $\i\in I$ and thus fix the tuple
$\v_{\i}\in B^c$, and consider the set
\[
J_{\i}:= \big\{\;\j\in[\Delta s]^c\bigmid w_{\v_{\i},\j, i'}\in Z_2 \big\}.
\]
Recall that $Z$ is dominated by those vertices $(u, \ell, i')$ with $u\in
X$, so for every $\j\in J_\i$ the vertex $w_{\v_i, \j, i'}$ is adjacent to
some $(u, \ell, i')$ in the dominating set $D$ with $u\in X$. Moreover, for
every $\ell\in [c]$, in the original graph $H$ the vertex $\v_{\i}(\ell)\in
B$ has at most $\Delta(s-1)$ neighbors in $X$, by the fact that
$\v_{\i}(\ell)\notin Y$ and our definition of the set $Y$.
Define a function $\theta: J_{\i}\to [\Delta s]$ such that for each\; $\j\in
J_{\i}$, if $w_{\v_{\i}, \j, i'}$ is adjacent to a vertex $(u, \ell, i')\in
D$ with $u\in X$, then $\theta(\j)= \ell$. As argued above, such a $(u,
\ell, i')$ must exist, and if there are more than one such, choose an
arbitrary one.
Let \;$\j\in J_{\i}$ and $\ell:= \theta(\j)$. By (E3), in the graph $H$ the
vertex $\v_{\i}(\ell)$ is adjacent to some vertex $u\in X$ with $\alpha(u)=
\j(\ell)$. It follows that for each $\ell\in[c]$ we have
\[
\Big|\big\{\;\j(\ell)\bigmid \text{$\j\in J_{\i}$ and $\theta(\j)= \ell$}\big\}\Big|
\le \Big|\big\{\alpha(u)\bigmid \text{$u\in X$ adjacent to $\v_{\i}(\ell)$}\big\}\Big|
\le \Delta(s-1).
\]
Applying Lemma~\ref{lem:productgap}, we obtain
\begin{equation*}%\label{eq:J}
\big|J_{\i}\big|\le \Delta^cs^c- \Delta^c.
\end{equation*}
\[
\big|Z_2\big|= \sum_{\i\in I} \big|J_{\i}\big|
\le |I|(\Delta^cs^c-\Delta^c).
\]
%From (\ref{eq:J}), we can deduce that $|Z_2|\le |\mathcal
By~\eqref{eq:Z} and the definition of $Z_1$ and $Z_2$, we should have
\[
\Delta^cs^c|I|- c\cdot d^c/3\le |Z|=|Z_1|+|Z_2|
\le c |Y| d^{c-1} \Delta^c s^c + |I|(\Delta^cs^c-\Delta^c).
\]
That is,
\[
c\cdot d^c/3 + c |Y| d^{c-1} \Delta^c s^c\ge \Delta^c |I|\ge 2\Delta^cd^c/3.
\]
Combined with~\eqref{eq:Y}, we have
\[
c\cdot d^c +c \Delta^c s^cd^{c-\frac{1}{2}+\frac{1}{2s}}\ge 2\Delta^cd^c,
\]
which contradicts the equation~\eqref{eq:deltas}. \proofend
\end{proof}
\section{Some Consequences}\label{sec:consq}
%Let $\beta:\mathbb N\to \mathbb N$ be a computable function. Consider the
%following promise problem.
%\noptprob[6.5]{$\textsc{Min-Dominating-Set}_\beta$}{A graph $G=(V,E)$ with
%$\ds(G)\le \beta(|V|)$}{A dominating set $D$ of $G$}{$|D|$}{$\min$}
%Then there is no polynomial time constant approximation algorithm for
%$\textsc{Min-Dominating-Set}_{\beta}$, unless $\FPT= \W 1$.
\begin{proof}[of Corollary~\ref{cor:main1}]
Let $c\in \mathbb N^+$, and assume that $\mathbb A$ is a polynomial time
algorithm which on input a graph $G= (V,E)$ with $\ds(G)\le \beta(|V|)$
outputs a dominating set $D$ with $|D|\le c\cdot \ds(G)$. Without loss of
generality, we further assume that given $0\le k\le n$ it can be tested in
time $n^{O(1)}$ whether $k> c\cdot \beta(n)$.
Now let $G$ be an arbitrary graph. We first simulate $\mathbb A$ on $G$, and
there are three possible outcomes of $\mathbb A$.
\begin{itemize}
\item $\mathbb A$ does not output a dominating set. Then we know $\ds(G)
> \beta(|V|)$. So in time
\[
2^{O(|V|)} \le 2^{O(\beta^{-1}(\ds(G)))}
\]
we can exhaustively search for a minimum dominating set $D$ of $G$.
\item $\mathbb A$ outputs a dominating set $D_0$ with $|D_0|> c\cdot
\beta(|V|)$. We claim that again $\ds(G)> \beta(|V|)$. Otherwise, the
algorithm $\mathbb A$ would have behaved correctly with
\[
|D_0|\le c\cdot \ds(G)\le c\cdot \beta(|V|).
\]
So we do the same brute-force search as above.
\item $\mathbb A$ outputs a dominating set $D_0$ with $|D_0|\le c\cdot
\beta(|V|)$. If $|D_0|> c\cdot \ds(G)$, then
\[
c\cdot \beta(|V|) \ge |D_0|> c\cdot \ds(G), \quad \text{i.e.},
\ \beta(|V|)> \ds(G),
\]
which contradicts our assumption for $\mathbb A$. Hence, $|D_0|\le
c\cdot \ds(G)$ and we can output $D:= D_0$.
\end{itemize}
To summarize, we can compute a dominating set $D$ with $|D|\le c\cdot
\ds(G)$ in time $f(\ds(G))\cdot |G|^{O(1)}$ for some computable $f:\mathbb
N\to \mathbb N$. This is a contradiction to Theorem~\ref{thm:main1}.
\proofend
\end{proof}
\medskip
Now we come to the approximability of the monotone circuit satisfiability
\noptprob[6]{$\mcs$}{A monotone circuit $C$}{A satisfying assignment $S$ of
$C$}{The weight of $|S|$}{$\min$}
Recall that a Boolean circuit $C$ is \emph{monotone} if it contains no
negation gates; and the \emph{weight} of an assignment is the number of
inputs assigned to $1$.
As mentioned in the Introduction, Marx showed~\cite{mar13} that \mcs\ has no
fpt approximation with any ratio $ρ$ for circuits of depth 4, unless
$= 2$.
\begin{cor}\label{cor:wsat}
Assume $\FPT\ne \W 1$. Then \mcs\ has no constant fpt approximation for
circuits of depth 2.
\end{cor}
\begin{proof}
This is an immediate consequence of Theorem~\ref{thm:main1} and the
following well-known approximation-preserving reduction from \mcs\ to \mds.
Let $G= (V,E)$ be a graph. We define a circuit
\[
C(G)= \bigwedge_{v\in V} \bigvee_{\{u,v\}\in E} X_u.
\]
There is a one-one correspondence between a dominating set in $G$ of size
$k$ and a satisfying assignment of $C(G)$ of weight $k$. \proofend
\end{proof}
\begin{rem}
Of course the constant ratio in Corollary~\ref{cor:wsat} can be improved
according to Theorem~\ref{thm:main2}.
\end{rem}
\section{Conclusions}\label{sec:con}
We have shown that $$ has \emph{no} fpt approximation with any constant
ratio, and in fact with a ratio slightly super-constant. The immediate
question is whether the problem has fpt approximation with \emph{some} ratio
$ρ:ℕ→ℕ$, e.g., $ρ(k)= 2^2^k$. We tend to believe
that it is not the case.
Our proof does not rely on the deep PCP theorem, instead it exploits the gap
created in the \W 1-hardness proof of the parameterized biclique problem
in~\cite{lin15}. In the same paper, the second author has already proved
some inapproximability result which was shown by the PCP theorem before.
Except for the derandomization using algebraic geometry in~\cite{lin15} the
proofs are mostly elementary. Of course we are working under some stronger
assumptions, i.e., \ETH\ and $1$. It remains to be seen whether
we can take full advantage of such assumptions to prove lower bounds
matching those classical ones or even improve them as in
\medskip
\subsection*{Acknowledgement}
We thank Edouard Bonnet for pointing out a mistake in an earlier version of the paper.
\bibliographystyle{plain}
\bibliography{appds}
\end{document}
\section*{Appendix A}
The following theorem was proved in \cite{lin15}.
\begin{theo}\label{thm:gapreduction1}
We can construct a bipartite graph $H=(A \dotcup B, E)$ in $n^{O(1)}$-time
on input an $n$-vertex graph $G$ and a positive integer $k$ with $\lceil
n^{\frac{6}{k+6}}\rceil>(k+6)!$ and $6\mid k+1$ such that:
\begin{enumerate}
\item if $K_k\subseteq G$ then there are $s$ vertices in $A$ with at
least $\lceil n^{\frac{6}{k+1}}\rceil$ common neighbors in $B$;
\item if $K_k\nsubseteq G$ then every $s$ vertices in $A$ have at most
$(k+1)!$ common neighbors in $B$,
\end{enumerate}
where $s=\binom{k}{2}$.
\end{theo}
To obtain Theorem~\ref{thm:colgap}, we need some preparation.
\begin{defn}\label{def:universal}
Let $n, k\in \mathbb N$ and $U\subseteq \{0,1\}^n$. Then $U$ is
\emph{$(n,k)$-universal} if for every $1\le i_1< i_2< \cdots <i_k\le n$,
\[
\bigmid \{(\v(i_1),\v(i_2),\cdots,\v(i_k))| \v\in U\}\bigmid<2^k.
\]
%projection of $U$ on the coordinates $i_1, \ldots, i_k$ is the whole binary
\end{defn}
\begin{lem}\label{lem:universal}
Let $k,n\in \mathbb N$ with $k2^k< \sqrt{n}$. Then in polynomial time we can
construct an $(n,k)$-universal set $U$ with $|U|= n$.
\end{lem}
Lemma~\ref{lem:universal} can be deduced from Theorem 10.20 and Proposition
10.19 in \cite{jukna2011extremal}.
\begin{cor}\label{cor:universal}
Let $r,n\in \mathbb N$ with
\[
(r\cdot \log r) 2^{r\cdot \log r}< \sqrt{n\cdot \log r}.
\]
Then in time polynomial in $n\log r$ we can construct a set $\mathcal C$ of
functions $[n] \to [r]$ of size $n\cdot \log r$ such that for every $1\le
i_1< i_2< \cdots <i_r\le n$ and every $c: [r] \to [r]$ there is a function
$\mathbf c: [n]\to [r]$ in $\mathcal C$ with
\[
\mathbf c(i_j)= c(j)
\]
for all $j\in [r]$.
\end{cor}
\proof We use Lemma~\ref{lem:universal} with parameters
\begin{eqnarray*}
k\gets r\cdot \log r & \text{and} & n \gets n\cdot \log r
\end{eqnarray*}
to construct a $(n·logr, r·logr)$-universal set $𝒞⊆{0,1}^n·logr$. It is easy to see that every string
$𝐜∈{0,1}^n·logr$ can be understood as a function
$𝐜: [n]→[r]$ by dividing $𝐜$ into $n$ blocks, each of
length $logr$. The desired property of $𝒞$ as a set of functions
follows directly from the $(n·logr, r·logr)$-universality.
\proofend
Now we are ready to present the proof of Theorem~\ref{thm:colgap} based on
Theorem~\ref{thm:gapreduction} and Corollary~\ref{cor:universal}.
\begin{proof}[of Theorem~\ref{thm:colgap}]
Let $G$ be an $n$-vertex graph, $k\in\mathbb{N}^+$ with $6\mid k+1$ and
$g(k):=d(k)\log d(k)+\delta s\log(\delta s)$. Without lost of generality, we
can assume that
$n>\max\{g(k)^2,[(k+6)!]^{\frac{k+6}{6}},d(k)^{\frac{k+1}{6}}\}$. First, we
invoke Theorem~\ref{thm:gapreduction} to obtain a bipartite graph
$H=(A\;\dot\cup\;B,E)$. Then we construct a new bipartite graph $H_\delta$
from $H$ as follows.
\begin{itemize}
\item $A(H_\delta):=A\times[\delta]$, $B(H_\delta):=B$;
\item $E(H_\delta):=\{\{(u,i),v\}\mid i\in[\delta],u\in A,v\in
B,\{u,v\}\in E\}$.
\end{itemize}
Intuitively, we replace every vertex $v$ in $A$ with its $\delta$ copies and
make each copy adjacent to every vertex in $N^H(v)$. It follows that
$H_\delta$ satisfies the following properties:
\begin{itemize}
\item if $K_k\subseteq G$ then there are $\delta s$ vertices in
$A(H_\delta)$ with at least $d(k)$ common neighbors;
\item if $K_k\nsubseteq G$ then every $\delta(s-1)+1$ vertices in
$A(H_\delta)$ have at most $(k+1)!$ common neighbors.
\end{itemize}
Let $n_\delta:=|V(H_\delta)|$. Since $n_\delta\ge n>g(k)^2$, we have
$\sqrt{n_\delta \log(\delta s)} >\delta s\log(\delta s)$. We apply
Corollary~\ref{cor:universal} with parameters
\[
r\gets \delta s, \ n\gets n_\delta
\]
to obtain a set $\mathcal A$ of functions $\left[n_\delta\right]\to
\left[\delta s \right]$. Similarly, we have $\sqrt{n_\delta\log(d(k))}
>d(k)\log(d(k))$. Applying Corollary~\ref{cor:universal} with parameters
\[
r\gets d(k), \ n\gets n_\delta,
\]
we get a set $\mathcal B$ of functions $\left[n_\delta\right]\to \left[d(k)
\right]$. As $|V(H_\delta)|=n_\delta$, we can assume
$V(H_\delta)=[n_\delta]$. The bipartite graph $H_\delta$ together with
$\mathcal A$ and $\mathcal B$ satisfy the following property:
\begin{itemize}
\item[($\star$)] For every $X\in\binom{A}{\delta s}$ and
$Y\in\binom{B}{d}$, there exist $\a\in\mathcal A$ and $\b\in\mathcal
B$ such that $\a(X)=[\delta s]$ and $\b(Y)=[d]$.
\end{itemize}
Then we construct our target bipartite graph $H_{\mathcal A,\mathcal B}$
together with two colorings $\alpha$ and $\beta$ as follows.
\begin{itemize}
\item $A(H_{\mathcal A,\mathcal B}):=A(H_\delta)\times \mathcal A\times
\mathcal B$;
\item $B(H_{\mathcal A,\mathcal B}):=B(H_\delta)\times \mathcal A\times
\mathcal B$;
\item $E(H_{\mathcal A,\mathcal B}):=\{\{(v,\a,\b),(u,\a,\b)\}\bigmid
\{v,u\}\in E(H_\delta), \a\in \mathcal A,\b\in\mathcal B\}$;
\item for every $x=(v,\a,\b)\in A(H_{\mathcal A,\mathcal B})$,
\item for every $y=(u,\a,\b)\in B(H_{\mathcal A,\mathcal B})$,
\end{itemize}
Suppose $K_k\subseteq G$. Then there exist $X\in\binom{A(H_\delta)}{\delta
s}$ and $Y\in\binom{B(H_\delta)}{d(k)}$ such that $X\cup Y$ induces a
$K_{\delta s,d(k)}$-subgraph in $H_\delta$. By ($\star$), there exist
$\a\in\mathcal A$ and $\b\in\mathcal B$ such that $\a(X)=[\delta s]$ and
$\b(Y)=[d(k)]$. According to the definition, we have
$\alpha(X\times\{\a\}\times\{\b\})=[\delta s]$,
$\beta(Y\times\{\a\}\times\{\b\})=[d(k)]$ and every vertex in
$X\times\{\a\}\times\{\b\}$ is adjacent to all vertices in
$Y\times\{\a\}\times\{\b\}$. Thus $H_{\mathcal A,\mathcal B}$, $\alpha$ and
$\beta$ satisfy the property (H1) in Theorem~\ref{thm:colgap}.
On the other hand, suppose $K_k\nsubseteq G$. Then every $\delta(s-1)+1$
vertices in $A(H_\delta)$ have at most $(k+1)!$ common neighbors in
$B(H_\delta)$. We note that for any connected graph $K$, if $H_{\mathcal
A,\mathcal B}$ contains a subgraph isomorphic to $K$, then so does
$H_\delta$. It follows that $H_{\mathcal A,\mathcal B}$ also satisfies
\end{proof}
|
1511.00360
|
Prosody affects the naturalness and intelligibility of speech. However, automatic prosody prediction from text for Chinese speech synthesis is still a great challenge and the traditional conditional random fields (CRF) based method always heavily relies on feature engineering. In this paper, we propose to use neural networks to predict prosodic boundary labels directly from Chinese characters without any feature engineering. Experimental results show that stacking feed-forward and bidirectional long short-term memory (BLSTM) recurrent network layers achieves superior performance over the CRF-based method. The embedding features learned from raw text further enhance the performance.
automatic prosody prediction, speech synthesis, neural network, BLSTM, embedding features
§ INTRODUCTION
Prosody refers to the rhythm, stress and intonation of speech, including variations in duration, loudness and pitch. It is well known that speech prosody plays an important perceptual role in human speech communication <cit.>. Specifically, perception of prosodic boundaries is essential for listeners. In Chinese speech synthesis systems, typical prosody boundary labels consist of prosodic word (PW), prosodic phrase (PPH) and intonational phrase (IPH), which construct a three-layer prosody structure tree <cit.>, as shown in Fig. <ref>. The leaf nodes of tree structure are lexical words that can be derived from a lexical-based word segmentation module. Whether the prosody labels are properly predicted will directly affect the naturalness and intelligibility of the synthesized speech.
Previous studies have investigated a great number of features, their relevance to prosody generation in speech production and various prosodic modeling methods. Some syntactic cues like part-of-speech (POS), syllable identity, syllable stress and their contextual counterparts are commonly used for prosody boundary prediction <cit.>. Many statistical methods have been investigated to model speech prosody, including classification and regression tree <cit.>, hidden Markov model <cit.>, maximum entropy model <cit.> and conditional random fields (CRF) <cit.>. To our knowledge, the best reported results were achieved with CRF due to its ability of relaxing strong model independence assumption and solving the label bias problem <cit.>.
Despite years of research, it is still a great challenge to predict correct prosodic labels from unrestricted text for a text-to-speech (TTS) system. Obviously, there are two major drawbacks of the CRF-based prosody prediction in Chinese speech synthesis. First, it heavily relies on the performances of Chinese word segmentation (CWS) and POS tagging <cit.>. Second, the particle size and the inevitable segmentation errors in CWS have negative effects on the subsequent prosodic boundary prediction task. Moreover, the choice of effective features, from a broad set of feature templates, is critical to the success of such systems <cit.>. Much of the effort goes into feature engineering, which is notoriously labor-intensive, mainly based on the experience of an annotator.
Three-layer prosody structure tree in Chinese.
Recently, deep neural networks (DNN) have been increasingly investigated in order to minimize the effort of feature engineering in sequential labeling tasks. Zheng et al. <cit.> applied neural networks to CWS and POS tagging and proposed a perceptron-style algorithm to speed up the training process with negligible loss in performance. Pei et al. <cit.> proposed a max-margin tensor neural network for CWS to model interactions between tags and context characters by exploiting tag embeddings and tensor-based transformation. These researches have proved that DNN is able to achieve similar or even superior performance over CRF-based method with minimal feature engineering in sequential labeling tasks. Therefore, it is promising to apply DNN architectures to automatic prosody prediction. However, we notice that the neural networks used in previous researches are feed-forward structures that keep the assumption of sample independence and provide only limited context modeling ability by operating on a fixed-size window of input samples. Instead, bidirectional recurrent neural networks (BRNN) are able to incorporate contextual information from both past and future inputs <cit.>. Specifically, BRNN with long short-term memory (LSTM) cells, namely BLSTM-RNN, has become a popular model <cit.>.
In this paper, we address the prosodic boundary prediction problem using neural networks. There are three main contributions. (1) We propose a neural network approach to predict prosody labels directly from Chinese characters without any feature engineering. (2) We show that superior performance is achieved by stacking feed-forward and bidirectional long short-term memory (BLSTM) recurrent layers. (3) We leverage a large raw text corpus to obtain useful character embedding features. Both objective and subjective evaluations show that the proposed architecture achieves superior performance over the CRF-based method and the embedding features further enhance the performance.
§ THE PROPOSED APPROACH
Just like CWS and POS tagging, automatic prosody prediction can be treated as a sequential labeling task that assigns boundary labels to characters of an input sentence. In order to make the prediction models less dependent on the feature engineering, we choose to use a variant of the neural network architecture proposed by <cit.> for probabilistic language model. This architecture was subsequently used for CWS and POS tagging <cit.>. As shown in Fig. <ref>, the architecture takes raw text as input and maps each Chinese character into a basic feature vector. The following layers are two types of neural networks, FFNN and BLSTM-RNN, used to discover multiple levels of feature representations from the basic feature vectors. The output layer is a graph over which tag inference is achieved by the Viterbi algorithm.
The neural network architecture for prosodic boundary prediction. In tag inference, B, NB and O denote boundary, non-boundary and others (e.g., punctuation), respectively.
§.§ Feature Vectors
The characters fed into network are transformed into feature vectors by a mapping operation. Typically, a character dictionary $D$ of size $|D|$ is extracted from the training set and unknown characters are mapped to a special symbol that is not used elsewhere. Each Chinese character can be typically represented by a one-hot vector, the size of which is $|D|$, and all dimensions are marked as $0$ except the location of the character in $D$, which is marked as $1$. However, the one-hot representation, with high dimensions, fails to model the semantic similarity between the ideographic characters. In contrast, the distributed representation or embedding feature, in form of a low dimensional
continuous-valued vector learned using neural networks from raw text in a fully unsupervised way, is assumed to carry important syntactic and semantic information <cit.> <cit.>. Recently, Mansur et al. <cit.> have shown superior performance in Chinese word segmentation by the use of embedding features based on a neural language model <cit.>. Besides <cit.>, Mikolov et al. <cit.> proposed a faster skip-gram model called word2vec[https://code.google.com/p/word2vec/]. As our preliminary experiments do not show much performance difference among various embedding features, we simply choose word2vec in this study because it can be trained much faster.
§.§ Network Structures and Training
Two types of neural networks are investigated in this paper: FFNN and BLSTM-RNN. FFNN, trained with a back-propagation learning algorithm <cit.>, is widely used in many practical applications. In a typical FFNN, every unit in a layer is connected with all the units in the previous layer, which takes in the output of the previous layer and computes a new set of non-linear activations for next layer. However, the assumption of sample independence brings in only limited context modeling ability.
Researchers have proposed RNN to solve the limitation of FFNN. However, conventional RNN is only able to make use of previous context information. This is not accurate in modeling speech prosody that is highly related with both past and future contexts. Instead, bidirectional RNN can access both the preceding and succeeding input contexts with two separate hidden layers, which are then fed to the same output layer. The activation function $\mathcal{H}$ of RNN is usually a sigmoid or hyperbolic tangent function, which often causes the gradient vanishing problem that prevents RNN from modeling the long-span relations in sequence features. An LSTM architecture, which uses purpose-built memory cells to store information, can overcome this problem and model longer contexts. Fig. <ref> illustrates a single LSTM memory cell. For LSTM, $\mathcal{H}$ is implemented by the following functions:
\begin{align*}
& i_t = \sigma(W_{xi}x_t + W_{hi}h_{t-1} + W_{ci}c_{t-1} + b_i) \\
\end{align*}
\begin{align*}
& f_t = \sigma(W_{xf}x_t + W_{hf}h_{t-1} + W_{cf}c_{t-1} + b_f) \\
& c_t = f_tc_{t-1} + i_ttanh(W_{xc}x_t + W_{hc}h_{t-1} + b_c) \\
& o_t = \sigma(W_{xo}x_t + W_{ho}h_{t-1} + W_{co}c_t + b_o) \\
& h_t = o_ttanh(c_t)
\end{align*}
where $x = (x_1, x_2, ... x_t ..., x_T)$ is the input feature sequence, $\sigma$ is the logistic function, and $i$, $f$, $o$ and $c$ are the input gate, forget gate, output gate and cell memory, respectively. $W$ is the weight matrix and the subscript indicates it is the matrix between two different gates.
BLSTM-RNN is a combination of LSTM and BRNN. Deep bidirectional LSTM-RNN can be established by stacking multiple BLSTM-RNN hidden layers on top of each other. The output sequence of one layer is used as the input sequence of the next layer. The hidden state sequences, $h^n$, consist of forward and backward sequences $\overrightarrow{h}^n$ and $\overleftarrow{h}^n$, iteratively computed from $n = 1$ to $N$ and $t = 1$ to $T$ as follows:
\begin{align*}
& \overrightarrow{h}_t^n = \mathcal{H}(W_{\overrightarrow{h}^{n-1}\overrightarrow{h}^n}\overrightarrow{h}_t^{n-1} + W_{\overrightarrow{h}^n\overrightarrow{h}^n}\overrightarrow{h}_{t - 1}^n + b_{\overrightarrow{h}}^n), \\
& \overleftarrow{h}_t^n = \mathcal{H}(W_{\overleftarrow{h}^{n-1}\overleftarrow{h}^n}\overleftarrow{h}_t^{n-1} + W_{\overleftarrow{h}^n\overleftarrow{h}^n}\overleftarrow{h}_{t - 1}^n + b_{\overleftarrow{h}}^n), \\
& y_t = W_{\overrightarrow{h}^Ny}\overrightarrow{h}_t^N + W_{\overleftarrow{h}^Ny}\overleftarrow{h}_t^N + b_y.
\end{align*}
where $y = (y_1, y_2, ... y_t ..., y_T)$ is the output prosodic boundary sequence.
In our study, the feed-forward layers are trained with typical backpropagation (BP) algorithm and the back-propagation through time (BPTT) method is used for training of BLSTM layers. BPTT is applied to both forward and backward hidden nodes and back-propagates layer by layer. The weight gradients are computed over the entire utterance <cit.>. The neural networks can be trained effectively in a layer-wised training manner, which makes it convenient to stack different types of neural network layers on top of each other to form a deep architecture. The deep architecture is able to build up progressively higher level representations of the input data, which is a crucial factor of the recent success of hybrid systems <cit.>.
Long short-term memory cell.
§.§ Tag Inference
To model the tag dependency and infer the tag sequence globally, given a set of tags $G=\{B, NB, O\}$, a transition score $S_{ab}$ is introduced for jumping from tag $a \in G$ to tag $b \in G$. For the input character sequence of a sentence $c_{[1:T]}$ with a tag sequence $tag_{[1:T]}$, a sentence-level score is then given by the sum of transition and network scores <cit.>:
\begin{align*}
& l(c_{[1:T]},tag_{[1:T]},\theta) = \sum_{t=1}^{T}(S_{tag_{t-1}tag_t} + f_\theta(tag_t|c_t))
\end{align*}
where $f_\theta(tag_t|c_t))$ indicates the score output for $tag_t$ at the $t$-th character by the networks. Given a sentence $c_{[1:T]}$, we can find the best tag path $tag^*_{[1:T]}$ by maximizing the sentence score:
\begin{align*}
& tag^*_{[1:T]} = \arg\max_{\forall l_{[1:T]}} l(c_{[1:T]}, tag_{[1:T]}, \theta).
\end{align*}
The Viterbi algorithm can be used for tag inference. The description above shows that it is easy to stack feature vectors, neural networks and tag inference together. Thus, the proposed architecture can be trained in a layer-wised fashion.
§ EXPERIMENTS
Totally 48210 sentences randomly selected from People's Daily were used in our experiments. Prosodic boundaries (PW, PPH and IPH) were labelled by professional annotators with corresponding speech and labeling consistency is ensured. Word segmentation and POS tagging were carried out by a front-end preprocessing tool. The accuracy of word segmentation is 97% and the accuracy of POS tagging is 96%. The corpus was partitioned into three parts: a training set with 43390 utterances, a validation set with 2410 utterances for parameter tuning and a testing set with another 2410 utterances. A character dictionary $D$ of size 4030 was extracted from the training set. A large set of raw texts was also collected from People's Daily for unsupervised embedding feature learning. All texts were preprocessed with text normalization.
In the experiments, PW, PPH and IPH were predicted separately. That is to say, three separate neural network models were trained independently for PW, PPH and IPH using the CURRENNT toolkit <cit.>. Each character in a sentence was assigned to one of the following three boundary tags: B for a prosodic boundary, NB for a non-boundary, and O for other symbols such as punctuation. Precision (P), recall (R) and F-score (F) were calculated as standard objective evaluation criteria.
A CRF-based prosodic boundary prediction approach was used as baseline and boundary prediction (B, NB and O) was operated at word level. Atomic features in the CRF approach include word identity, POS tags, the length of word and the predicted tag from the previous boundary level. A linear statistical model was applied to optimize the feature templates. Parameters grid search was adopted to achieve the best performance of the CRF model. The CRF++ toolkit[http://taku910.github.io/crfpp/] was used for the CRF-based prosodic boundary prediction. The baseline results are shown in Table <ref>.
Boundary P (%) R (%) F (%)
PW 95.34 96.73 96.03
PPH 83.41 83.68 83.06
IPH 84.85 73.39 78.71
The results of CRF-based prosody prediction.
Topology c B, BB, BBB, BBBB
FFB, FBF, BFF, FBB, BFB, BBF
Num of nodes 32, 64, 128, 256
Different network configurations in the experiments.
We investigated the performance of neural network architecture with different topologies, as described in Table <ref>, where F and B denote a feed-forward layer and a BLSTM layer, respectively. The number of the nodes were kept the same for all hidden layers in every tested network architecture. Specifically, the network input is an $M$-dimensional feature vector, where $M$=4030 for the PW prediction and $M$=4031 for the PPH and IPH prediction [The predicted tag from the previous level was used as a feature.]. The network output corresponds to the three boundary tags (B, NB and O). All networks were trained with a momentum of 0.9, a learning rate of 1e-3 for PW and 1e-4 for PPH and IPH. BPTT was performed using stochastic gradient descent (SGD) with 32 parallel sentences. The training stops if no lower error on the validation set can be achieved within the last 10 epochs. The best performances for different prosodic boundary levels are shown in Table <ref>. We interestingly discover that the best performances at different levels are all obtained with a topology of FBB. When we compare Table 3 with the CRF-baseline Table 1, we find that the proposed neural network approach achieves competitive performance at the PW level and significant improvements at the PPH and IPH levels.
We also studied the effectiveness of the character embedding features. Different sizes of unsupervised training data (400M, 800M, 1200M, 1600M and 2000M text) and embedding feature sizes (100, 200, 300 and 400) were tested. The best network architectures, as shown in Table 3, were used in the experiments. Please note that the dimension of feature vector is greatly reduced as compared with the one-hot representation. The results shown in Table <ref> indicate that the embedding features can further improve the performance of automatic prosodic boundary prediction.
Boundary P (%) R (%) F (%) TP / Num of nodes
PW 96.02 96.69 96.35 FBB / 32
PPH 82.50 86.75 84.57 FBB / 128
IPH 84.06 79.33 81.63 FBB / 64
The best performance of each level and the corresponding network topology (TP).
Boundary P (%) R (%) F (%) Embedding feature size
PW 96.27 96.91 96.59 300
PPH 82.89 87.13 84.96 400
IPH 84.81 79.88 82.27 100
The results of neural network architecture with embedding features and the corresponding feature size.
We further conducted an A/B preference test on the naturalness of the synthesized speech. A set of 100 sentences were randomly selected from the test set and the prosodic boundary labels were achieved by:
(1) CRF-based model in Table <ref>;
(2) NN with one-hot representation in Table <ref>;
(3) NN with embedding features in Table <ref>.
We carried out two sessions of comparative evaluations: (1) vs (2) and (2) vs (3). A set of 20 sentence pairs of each session was randomly selected from the 100 pairs with different prosody prediction results and speech was generated through a typical HMM-based TTS system. A group of 10 subjects were asked to choose which one was better in terms of the naturalness of synthesis speech. The percentage preference is shown in Figure <ref>. We can clearly see that the NN architecture with one-hot representation can achieve better naturalness of synthesized speech as compared with CRF, while the use of embedding features further improves the natrualness.
The percentage preference of A/B test.
§ CONCLUSION AND FUTURE WORK
In this paper, we propose to use neural network architectures to predict prosodic boundary labels directly from Chinese characters without feature engineering. We show that superior performance is achieved by stacking feed-forward and bidirectional long short-term memory (BLSTM) recurrent layers. We obtain useful character embedding features from raw text. Both objective and subjective evaluations show that the proposed neural network approch achieves superior performance over the CRF-based approach and the use of embedding features can further boost the performance. For future work, it is promising to predict PW, PPH and IPH labels in a unified neural network and n-gram character embedding features can be further investigated.
§ ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China (61175018 and 61571363).
|
1511.00242
|
]Lianhua Zhu
]Peng Wang
]Zhaoli Guocor1
[cor1]Corresponding author
State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan, 430074, China
The general characteristics based off-lattice Boltzmann scheme (BKG) proposed by Bardow et al., <cit.> and the discrete unified gas kinetic scheme (DUGKS) <cit.> are two methods that successfully overcome the time step restriction by the collision time, which is commonly seen in many other kinetic schemes.
Basically, the BKG scheme is a time splitting scheme, while the DUGKS is an un-split finite volume scheme.
In this work, we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations.
It is found that the effects of collision term are considered in the reconstructions of the cell-interface distribution function in both schemes,
which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time.
The difference between the two schemes lies in the treatment of the integral of the collision term,
in which the Bardow's scheme uses the rectangular rule while the DUGKS uses the trapezoidal rule.
The performance of the two schemes, i.e., accuracy, stability, and efficiency are then compared by simulating several two dimensional flows,
including the unsteady Taylor-Green vortex flow, the steady lid-driven cavity flow, and the laminar boundary layer problem.
It is observed that, the DUGKS can give more accurate results than the BKG scheme.
Furthermore, the numerical stability of the BKG scheme decreases as the Courant-Friedrichs-Lewy (CFL) number approaches to 1,
while the stability of DUGKS is not affected by the CFL number apparently as long as $\text{CFL}<1$.
It is also observed that the BKG scheme is about one time faster than the DUGKS scheme with the same computational mesh and time step.
§ INTRODUCTION
The lattice Boltzmann method (LBM) has become a popular numerical tool for flow simulations. It solves the discrete velocity Boltzmann equation (DVBE) with sophistically chosen discrete velocity set. With the coupled discretization of velocity space and spatial space, the numerical treatment of convection term reduces to a very simple streaming processes, which provides the benefits of low numerical dissipation, easy implementation, and high parallel computing efficiency. Another advantage of LBM is that, the simplified collision term is computed implicitly while implemented explicitly, which allows for a large time step even though the collision term causess stiffness at a small relaxation time. This advantage makes the LBM a potential solver for high Reynolds number flows.
However, the coupled discretization of velocity and spatial spaces limits the LBM to the use of uniform Cartesian meshes, which prohibits its applications for practical engineering problems. Some efforts have been made to extend the standard LBM to non-regular (non-uniform, unstructured) meshes, and a number of so called off-lattice Boltzmann (OLB) methods have been developed by solving the DVBE using certain finite-difference, finite-volume, or finite-element schemes <cit.>. These OLB schemes differ from each other in the temporal and spatial discretizations. However, a straightforward implementation of the CFD techniques usually leads to the loss of the advantages of standard LBM, especially the low dissipation property and stability at large time step. For example, in many of the schemes <cit.>, the time step is limited by the relaxation time to get an accurate solution, even as the collision term is computed implicitly <cit.>. This drawback makes these OLB schemes very computational expensive when simulating high Reynolds number flows.
An alternative way to construct OLB schemes is to use the time-splitting strategy in solving the DVBE <cit.>, in which the DVBE is decomposed into a collision sub-equation and a followed pure advection sub-equation. The collision sub-equation is fully local and is discretized directly, leading to a collision step the same as the standard LBM; The collisionless advection subequation is then solved with certain numerical schemes on uniform or non-uniform meshes <cit.>, leading to a general streaming step. Specifically, the scheme proposed by Bardow et al. (denoted by BKG), which combines the variable transformation technique for the collision term and the Lax-Wendroff scheme for the streaming step, overcomes the time step restriction by the relaxation time. It was demonstrated that accurate and stable solutions can be obtained even as the time step is much larger than the relaxation time <cit.>.
The above OLB schemes are developed in the LBM framework, and are limited to continuum flows. Recently, a finite volume kinetic approach using general mesh, i.e., discrete unified gas kinetic scheme (DUGKS), was proposed for all Knudsen number flows <cit.>. In the DUGKS the numerical flux is constructed based on the governing equation i.e., the DVBE itself, instead of using interpolations. With such a treatment, the time step is not restricted by the relaxation time, and its superior accuracy and stability for high Reynolds continuum flows have been demonstrated <cit.>.
Since both the BKG and the DUGKS methods overcome the time step restriction from different approaches, it is still not clear the performance difference between them,
so in this work we will present a comparative study of these two kinetic schemes for continuum flows, even the DUGKS is not limited to such flows. We will also investigate the link between the two schemes by comparing them in the same finite volume framework.
The remaining part of this paper is organized as follows. Sec. 2 will introduce the DUGKS and BKG methods and discuss their relation, Sec. 3 will present the comparison results, and a conclusion is given in Sec. 4.
§ NUMERICAL FORMULATION
§.§ Discrete Velocity Boltzmann-BGK equation
The governing equation for the OLB schemes and DUGKS method is the Boltzmann equation with the Bhatnagar-Gross-Krook collision operator <cit.>,
\begin{equation}\label{BGK}
\frac{\partial f}{\partial t}+{\bm \xi}\cdot \bm \nabla f=\Omega(f)\equiv\frac{f^{eq}-f}{\tau},
\end{equation}
where $f\equiv f(\bm{x},\bm{\xi},t)$ is the distribution function (DF) with particle velocity $\bm{\xi}$ at position $\bm{x}$ and time $t$,
$\tau$ is relaxation time due to particle collisions, and $f^{eq}$ is the Maxwellian equilibrium distribution function.
In this article, we consider the isothermal two-dimensional-nine-velocities (D2Q9) lattice model.
The corresponding DVBE is
\begin{equation}\label{BGK_discrete}
\frac{\partial f_\alpha}{\partial t}+{\bm \xi_\alpha}\cdot\bm \nabla f_\alpha=\Omega(f_\alpha)\equiv\frac{f_\alpha^{eq}-f_\alpha}{\tau},
\end{equation}
where $f_\alpha \equiv f(\bm{x},\bm{\xi}_\alpha, t)$ and $f_\alpha^{eq} \equiv f^{eq}(\bm{x}, \bm{\xi}_\alpha, t)$ are the DF with discrete velocity $\bm{\xi}_\alpha$ and the
corresponding discrete equilibrium DF respectively.
The D2Q9 discrete velocity set $\bm{\xi}_\alpha$ is defined as
\begin{equation}
\bm{\xi}_\alpha=
\begin{cases}
(0,0) & \text{for} ~\alpha=0,\\
\sqrt{3RT} \left(\cos[(\alpha-1)\pi/2], \sin[(\alpha-1)\pi/2]\right) & \text{for} ~ \alpha=1,2,3,4,\\
\sqrt{3RT} \left(\cos[(2\alpha-9)\pi/4], \sin[(2\alpha-9)\pi/4]\right)\sqrt{2} & \text{for} ~ \alpha=5,6,7,8,
\end{cases}
\end{equation}
where $R$ is the gas constant and $T$ is the constant temperature.
Under the low Mach number condition, the discrete equilibrium DF
can be approximated by its Taylor expansion around zero particle velocity up to second order,
\begin{equation}\label{equilibrium}
f^{eq}_\alpha=w_\alpha\rho\left[1+\frac{\bm{\xi}_\alpha\cdot\bm{u}}{c_s^2}+ \frac{(\bm{\xi}_\alpha \cdot\bm{u})^2}{2c_s^4}-\frac{\mid\bm{u}\mid^2}{2c_s^2} \right],
\end{equation}
where $c_s = \sqrt{RT}$ is the lattice sound speed and the weights $w_\alpha$ are
\begin{equation}
\begin{cases}
4/9 & \text{for} ~ \alpha=0,\\
1/9 & \text{for} ~ \alpha=1,2,3,4,\\
1/36 & \text{for} ~ \alpha=5,6,7,8.
\end{cases}
\end{equation}
The fluid density $\rho$ and velocity $\bm u$ are the moments of the DF,
\begin{equation}\label{eq:macro}
\rho = \sum_\alpha f_\alpha, \quad \rho\bm u = \sum_\alpha\bm \xi_\alpha f_\alpha.
\end{equation}
The shear viscosity of the fluid is related to the relaxation time by
\begin{equation}
\nu = \tau RT,
\label{nu_tau}
\end{equation}
which can be deduced from Chapman-Enskog analysis <cit.>.
The conservation property of the collision term is maintained at its discrete velocity counterpart, i.e.,
\begin{equation}\label{eq:convervation}
\sum_\alpha \Omega(f_\alpha) = 0, \quad \sum_\alpha \bm \xi_\alpha \Omega(f_\alpha) = 0.
\end{equation}
§.§ Discrete unified gas kinetic scheme
The DUGKS employs a cell centered finite volume (FV) discretization of the DVBE <cit.>. The computational domain is firstly divided into small control volumes $V_k$.
For a clear illustration of the formulas, we denote the volume averaged DF with discrete velocity $\xi_\alpha$ in control volume $V_k$ at time level $t^n$ by $f^n_{\alpha,k}$, i.e.,
\begin{equation}\label{eq:dugks_fv}
f^n_{\alpha,k} = \frac{1}{|V_k|}\int_{V_k} f_\alpha(\bm x, t^n) dV.
\end{equation}
Then integrating Eq. (<ref>) from time $t^n$ to time $t^{n+1}$ and applying the Gauss theorem we can get
\begin{equation}\label{eq_raw}
f^{n+1}_{\alpha,k} - f^{n}_{\alpha,k} = \frac{\Delta t}{|V_k|}\mathcal{F}_{\alpha,k,\text{dugks}} + \frac{\Delta t}{2}\left[ \Omega(f^n_{\alpha,k}) + \Omega(f^{n+1}_{\alpha,k})\right ],
\end{equation}
where $\mathcal{F}_{\alpha,k,\text{dugks}}$ is the numerical flux that flows into the control volume from its faces, and $\Delta t= t^{n+1} - t^n$ is the time step size.
Note that trapezoidal rule is used for the collision term.
This implicit treatment of the collision term is crucial for its stability when the time step is much larger than the relaxation time.
This implicitness can be removed in the actual implementation using the following variable transformation technique, which is also adopted by the standard LBM,
\begin{equation}
\tilde f = f - \Delta t/2 \Omega(f), \quad \tilde f^+ = f + \Delta t/2 \Omega(f).
\end{equation}
Equation (<ref>) can then be rewritten in an explicit formulation,
\begin{equation}\label{eq_update}
\tilde f^{n+1}_{\alpha,k} = \tilde f^{n,+}_{\alpha,k} + \frac{\Delta t}{|V_k|}\mathcal{F}_{\alpha,k,\text{dugks}}.
\end{equation}
In the implementation, we track the evolution of $\tilde f_{\alpha,k}$ instead of the original DF.
Due to the conservation property of the collision term, the macroscopic variables can be calculated by the transformed DF,
\begin{equation}\label{eq:macro_var}
\rho = \sum_\alpha \tilde f_\alpha, \quad \rho\bm u = \sum_\alpha\bm \xi_\alpha \tilde f_\alpha.
\end{equation}
The key merit of DUGKS lies in its treatment of the advection term, i.e., the way to construct the numerical flux $\mathcal{F}_{\alpha,k,\text{dugks}}$.
In DUGKS, the middle point rule is used for the integration of the flux,
\begin{equation}
\mathcal{F}_{\alpha,k,\text{dugks}} = \int_{\partial V_k} ( \bm \xi_\alpha \cdot \bm n)f^{n+1/2}_{\alpha,\text{dugks}}dS.
\label{eq_flux}
\end{equation}
The integration over the faces is computed by the $f^{n+1/2}_\alpha$ at the centers of the faces,
which are computed using the characteristic solution of the kinetic equation (<ref>).
Supposed the center of a face is $\bm x_b$,
then integrating Eq. (<ref>) along the characteristic line in a half time step $h=\Delta t/2$ from $(t^n, \bm x_b - h\bm \xi_\alpha )$ to $(t^{n+1/2}, \bm x_b)$,
and applying the trapezoidal rule, we can get
\begin{equation}
f^{n+1/2}_{\alpha,\text{dugks}}(\bm x_b ) - f^n_\alpha(\bm x_b- h\bm\xi_\alpha) = \frac{h}{2}\left[ \Omega (f^n_\alpha(\bm x_b) ) + \Omega \left(f^{n+1/2}_\alpha(\bm x_b - h \bm \xi_\alpha ) \right) \right ].
\label{eq_charraw}
\end{equation}
Again the implicitness can be eliminated by introducing another two variable transformations,
\begin{equation}
\bar f = f - h/2 \Omega(f), \quad \bar f^+ = f + h/2 \Omega(f),
\label{eq_trans_2}
\end{equation}
and we can reformulate Eq. (<ref>) in an explicit form,
\begin{equation}
\bar f^{n+1/2}_\alpha(\bm x_b ) = \bar f^{n,+}_\alpha(\bm x_b- h\bm\xi_\alpha).
\label{eq_char}
\end{equation}
For smooth flow, $\bar f^{n,+}_\alpha(\bm x_b- h\bm\xi)$ can be interpolated linearly from its neighboring cell centers.
After getting $\bar f^{n+1/2}_\alpha(\bm x_b )$, the original DF $f^{n+1/2}_\alpha(\bm x_b)$ can be transformed back with the help of Eq. (<ref>).
The macroscopic fluid variable $\rho^{n+1/2}(\bm x_b)$ and $\bm u^{n+1/2}(\bm x_b)$ used by the collision term in Eq. (<ref>) are calculated from,
\begin{equation}
\left. \rho^{n+1/2}\right|_{\bm x_b} = \sum_\alpha \bar f_\alpha^{n+1/2}(\bm x_b), \quad
\left.(\rho\bm u)^{n+1/2}\right|_{\bm x_b} = \sum_\alpha\bm \xi_\alpha \bar f_\alpha^{n+1/2}(\bm x_b).
\label{}
\end{equation}
To insure the interpolation is stable, the time step is limited by the CFL condition,
\begin{equation}
\Delta t= \eta \frac{\Delta x}{|\bm \xi|_{\text{max}}} = \eta \frac{\Delta x}{\sqrt{6}c_s},
\label{eq:cfl}
\end{equation}
where $0<\eta <1$ is the CFL number and $\Delta x$ measures the size of the cell.
§.§ The BKG scheme
Unlike the DUGKS in which the collision and particle-transport are treated simultaneously,
the BKG scheme is a splitting method for Eq. (<ref>), which treats the convection term and collision term sequentially, i.e.,
\begin{align}\label{eq:split_a}
\frac{\partial f_\alpha}{\partial t}=\Omega(f_\alpha),\\
\label{eq:split_b}
\frac{\partial \tilde f_\alpha}{\partial t}+{\bm \xi_\alpha}\cdot\bm \nabla \tilde f_\alpha=0.
\end{align}
In the collision step, the collision term is integrated using trapezoidal rule <cit.>,
\begin{equation}
f^{*}_\alpha- f_\alpha^n = \frac{\Delta t}{2}\left [ \Omega(f^*_\alpha) + \Omega(f^n_\alpha) \right ].
\label{eq:splic_c}
\end{equation}
Using the same notation as used in the DUGKS, Eq. (<ref>) can be rewritten in an explicit formulation,
\begin{equation}
\tilde f^{*}_\alpha = \tilde f^{+,n}_\alpha,
\label{eq:bkg_collision}
\end{equation}
with $\tilde f^*_\alpha\equiv f^*_\alpha - (\Delta t/2)\Omega(f^*_\alpha) $.
It is noted that this treatment is identical to that of the standard LBM.
Then Eq. (<ref>) is solved with the Lax-Wendroff scheme with the initial value $\tilde f^*$ or $\tilde f^{+,n}$,
\begin{equation}
\tilde f^{n+1}_\alpha = \tilde f^{+,n}_\alpha - \Delta t \xi_{\alpha i} \frac{\partial \tilde f^{+,n}_\alpha}{\partial x_i}
+ \frac{\Delta t^2}{2}\xi_{\alpha i}\xi_{\alpha j} \frac{\partial^2 \tilde f^{+,n}_\alpha}{\partial x_i \partial x_j},
\label{eq:bkgfd}
\end{equation}
where the subscripts $i,j = 1,2$ denote the spatial indices. Equation (<ref>) forms the evolution of the BKG scheme. In the original works <cit.>, either finite element (FE) or finite difference (FD) is employed to discretize the spatial gradients in Eq. (<ref>). In Ref. [14], the central finite-difference scheme on a uniform mesh is used, i.e, the first and second order spatial derivatives are computed as <cit.>,
\begin{align}
\left. \frac{\partial \tilde f^{+,n}_\alpha}{\partial x_1} \right|_{l,m} &= \frac{\tilde f^{+,n}_{\alpha,l+1,m} - \tilde f^{+,n}_{\alpha,l-1,m}}{2\Delta x_1},\\
\left. \frac{\partial \tilde f^{+,n}_\alpha}{\partial x_2} \right|_{l,m} &= \frac{\tilde f^{+,n}_{\alpha,l,m+1} - \tilde f^{+,n}_{\alpha,l,m-1}}{2\Delta x_2},\\
\left. \frac{\partial^2 \tilde f^{+,n}_\alpha}{\partial x^2_1} \right|_{l,m} &= \frac{\tilde f^{+,n}_{\alpha,l+1,m} + \tilde f^{+,n}_{\alpha,l-1,m} -2 \tilde f^{+,n}_{\alpha,l,m}}{\Delta x_1^2}, \\
\left. \frac{\partial^2 \tilde f^{+,n}_\alpha}{\partial x^2_2} \right|_{l,m} &= \frac{\tilde f^{+,n}_{\alpha,l,m+1} + \tilde f^{+,n}_{\alpha,l,m-1} -2 \tilde f^{+,n}_{\alpha,l,m}}{\Delta x_2^2},\\
\left. \frac{\partial^2 \tilde f^{+,n}_\alpha}{\partial x_1 \partial x_2} \right|_{l,m} &=
\frac{1}{4\Delta x_1 \Delta x_2}
[\tilde f^{+,n}_ {\alpha,l+1,m+1}
- \tilde f^{+,n}_{\alpha,l-1,m+1}
- \tilde f^{+,n}_{\alpha,l+1,m-1}
+ \tilde f^{+,n}_{\alpha,l-1,m-1}],
\end{align}
where the computational stencil for each node is illustrated in Fig. <ref>.
It is noted that if we use the one dimensional Lax-Wendroff scheme to solve Eq. (<ref>) in each discrete velocity direction on a uniform Cartesian grid,
this characteristic based scheme reduces to the Lax-Wendroff LBE scheme developed in <cit.>.
Illustration of the finite volume discretization and the interpolation scheme.
The Lax-Wendroff scheme Eq. (<ref>) can also be expressed as
\begin{align}
\begin{split}\label{eq:xx1}
\tilde f^{n+1}_\alpha
&= \tilde f^{+,n}_\alpha - \Delta t \xi_{\alpha i} \frac{\partial}{\partial x_i}\left[\tilde f^{+,n}_\alpha -\frac{\Delta t}{2}\xi_{\alpha j} \frac{\partial \tilde f^{+,n}_\alpha}{ \partial x_j}\right] \\
&\approx \tilde f^{+,n}_\alpha - \Delta t \xi_{\alpha i} \frac{\partial}{\partial x_i}\left(\tilde f^{+,n}_\alpha(\bm{x}-\bm{\xi_\alpha}h)\right)\\
&\equiv \tilde f^{+,n}_\alpha - \Delta t \xi_{\alpha i} \frac{\partial}{\partial x_i}\left(f^{n+1/2}_\alpha(\bm{x})\right).
\end{split}
\end{align}
This means that the BKG scheme can be reformulated as a FV scheme,
\begin{equation}\label{eq:lwfv}
\tilde f^{n+1}_{\alpha,k} = \tilde f^{+,n}_{\alpha,k} + \frac{\Delta t}{|V_k|}\mathcal{F}_{\alpha,k,\text{bkg}},
\end{equation}
\begin{equation}
\mathcal{F}_{\alpha,k,\text{bkg}} \equiv \int_{\partial V_k} (\bm \xi_\alpha \cdot \bm n) f^{n+1/2}_{\alpha,\text{bkg}} dS,
\label{eq:G}
\end{equation}
\begin{equation}\label{eq:bkg_fb}
f^{n+1/2}_{\alpha, \text{bkg}}(\bm x_b) = \tilde f^{+,n}_{\alpha}(\bm x_b - h \bm \xi_\alpha)=f^{n}_{\alpha}(\bm x_b - h\bm\xi_\alpha) + h \Omega\left(f^n_{\alpha}(\bm x_b - h\bm\xi_\alpha)\right).
\end{equation}
To be more specific, we rewrite Eq. (<ref>) in the finite-volume form as
\begin{equation}\label{eq:bkg_fv}
\tilde f^{n+1}_{\alpha,l,m} = \tilde f^{+,n}_{\alpha,l,m} - \frac{\xi_{\alpha 1} \Delta t}{\Delta x_1} \left[ f^{n+1/2}_{\alpha,l+1/2,m} - f^{n+1/2}_{\alpha,l-1/2,m}\right]
- \frac{\xi_{\alpha 2} \Delta t}{\Delta x_2} \left[ f^{n+1/2}_{\alpha,l,m+1/2} - f^{n+1/2}_{\alpha,l,m-1/2}\right],
\end{equation}
where $f^{n+1/2}_{\alpha,l\pm 1/2,m}$ and $f^{n+1/2}_{\alpha,l,m \pm 1/2}$ are the DFs at the face-centers of cell $(l,m)$ at the half time step, which depend on interpolation schemes.
If the distribution function is assumed to be a linear piece-wise polynomial, we can obtain the distribution functions at the cell interfaces, e.g.,
\begin{align}
\begin{split}\label{eq:f_left}
f^{n+1/2}_{\alpha,l-1/2,m} &= \frac{1}{2}[\tilde f^{+,n}_{\alpha,l,m} + \tilde f^{+,n}_{\alpha,l-1,m}] -
\frac{\xi_{\alpha 1} \Delta t}{2\Delta x_1} [ \tilde f^{+,n}_{\alpha,l,m} - \tilde f^{+,n}_{\alpha,l-1,m} ] \\
&- \frac{\xi_{\alpha 2} \Delta t}{8\Delta x_2} [ \tilde f^{+,n}_{\alpha,l-1,m+1} + \tilde f^{+,n}_{\alpha,l,m+1}
-\tilde f^{+,n}_{\alpha,l-1,m-1} - \tilde f^{+,n}_{\alpha,l,m-1} ]\\
\tilde f^{+,n}_{\alpha,l-1/2,m} - \frac{\xi_{\alpha1}\Delta t}{\Delta x_1}\frac{\partial}{\partial x_1}\tilde f^{+,n}_{\alpha,l-1/2,m} - \frac{\xi_{\alpha2}\Delta t}{\Delta x_2}\frac{\partial}{\partial x_2}\tilde f^{+,n}_{\alpha,l-1/2,m}.
\end{split}
\end{align}
One can immediately check the equivalence of Eqs. (<ref>) and (<ref>), after calculating the rest flux terms in a similar way like Eq. (<ref>),
and inserting Eq. (<ref>) to Eq. (<ref>).
§.§ Comparison of the numerical fluxes in the DUGKS and BKG scheme
We now analyse the differences between the DUGKS and the BKG scheme in finite-volume formulation.
This is achieved by analyzing accuracy of the reconstructed distribution function at the cell interface center.
Firstly, it is noted that the exact solution of the DVBE at the cell interface center is
\begin{equation}\label{eq:ana}
f_{\alpha,\text{exact}}^{n+1/2}(\bm x_b) = f^n_{\alpha}(\bm x_b - h\bm\xi_\alpha) + \int_{0}^{h}\Omega\left(f_\alpha(\bm x_b - h\bm \xi_\alpha + s\bm \xi_\alpha, t^n+s)\right)ds.
\end{equation}
We can immediately find that if we approximate the integration of the collision term in Eq. (<ref>) explicitly,
i.e., assuming $\Omega(\bm x_b - h\bm \xi_\alpha + s\bm \xi_\alpha,s)=\Omega(\bm x_b-h\bm\xi_\alpha,t^n)$, we get Eq. (<ref>),
which is the reconstructed distribution function in the BKG scheme.
On the other hand, if we apply trapezoidal rule to the quadrature, we get Eq. (<ref>), i.e., the reconstructed cell-interface distribution functions in the DUGKS.
So, in both the DUGKS and the BKG methods, the flux is determined from the local characteristic solution of the DVBE,
and the convection and collision effects are considered simultaneously.
We also indicate that, for those FV/FD schemes that use the simple central difference <cit.> or upwind <cit.> schemes of traditional CFD methods,
the corresponding cell-interface distribution functions are
\begin{equation}
f^{n+1/2}_{\alpha,\text{central}}(\bm x_b) = f^{n}_{\alpha}(\bm x_b)
\label{eq:cfd1}
\end{equation}
\begin{equation}
f^{n+1/2}_{\alpha,\text{upwind}}(\bm x_b) = f^{n}_{\alpha}(\bm x_b - h\bm\xi_\alpha),
\label{eq:cfd2}
\end{equation}
where the collision effect is totally ignored.
The effects of the different treatments of the integration of the collision term in Eq. (<ref>)
can be analyzed by using the Chapman-Enskog expansion method <cit.>.
By approximating the distribution function by its first-order Chapman-Enskog solution,
\begin{equation}\label{eq:ce}
f_{\alpha }=f_{\alpha}^{(0)}+\tau f_{\alpha }^{(1)}+O({{\tau }^{2}}),
\end{equation}
with $f_{\alpha }^{(0)}=f_{\alpha }^{eq}$, we have
\begin{multline}\label{eq:ana_ce}
f_{\alpha,\text{exact}}^{n+1/2}(\bm x_b) = f^{(0),n}_{\alpha}(\bm x_b - h\bm \xi_\alpha ) + \tau f^{(1),n}(\bm x_b - h\bm \xi_\alpha)
- \int_0^h {f^{(1)}_\alpha}({\bm x_b} - h\bm\xi_\alpha + \bm \xi_\alpha s,s) ds + O(\tau^2).
\end{multline}
Therefore, for continuous flows where $\tau\ll h$,
the distribution function at the cell interface reconstructed in either the BKG or the DUGKS method is a second-order approximation of the exact one.
Furthermore, with the trapezoidal rule for the collision term, the DUGKS is expected to be more accurate than the BKG method which uses a lower order explicit rule.
In regions with large velocity gradients, where the collision effect or $f^{(1)}$ is important, the BKG scheme may yield significant error.
More importantly, as the DUGKS employs an implicit treatment of the collision term in the evaluation of numerical flux, it is expected to be more stable than the BKG scheme.
Finally, we shall remark that if the distribution functions at cell interfaces are obtained by direct interpolations,
i.e., neglecting the integral of the collision term in Eq. (<ref>), the leading error of the approximation is $-hf_\alpha^{(1),n}$.
As the integral of $f^{(1),n}$ on right hand side of Eq. (<ref>) contributes to the diffusive flux,
the lack of the collision term is equivalent to introduce a numerical viscosity proportional to $\Delta t$ <cit.>.
This explains why many other FV based lattice Boltzmann schemes have to keep the time step much smaller than the collision time to obtain accurate results.
§.§ No-slip boundary condition
In this subsection, we briefly mention the implementation of no-slip boundary condition for the BKG scheme and DUGKS methods,
The basic idea here is to mimic the half-way bounce-back rule of the standard LBM <cit.> by reversing the DFs at boundary faces at middle time steps.
Fig. <ref> illustrates a boundary face located at a no slip wall with velocity $\bm U_\text{w}$.
Implementation of no-slip boundary condition
Both of the incoming and outgoing DFs at the boundary face at middle time steps have to be provided to update the cell-centered DFs.
We denote the incoming and outgoing DFs by $\hat{\phi}_\text{w,in}^{n+1/2}$ and $\hat{\phi}_\text{w,out}^{n+1/2}$ respectively,
where $\hat{\phi}_\text{w}^{n+1/2}$ stands for $\bar{f}_\text{w}^{n+1/2}$ in the DUGKS and $f_\text{w}^{n+1/2}$ in BKG scheme.
The ghost cell method is used to facilitate the implementation of the no-slip boundary condition.
An extra layer of cells (ghost cells) are allocated outside of the wall.
The unknown DFs $\phi^n$ in the ghost cells are extrapolated linearly from the cell centers of the neighboring inner cells.
Here, $\phi^n$ stands for $\bar{f}^{+,n}$ in the DUGKS and $\tilde{f}^{+,n}$ in the BKG scheme.
Then we can compute the $\hat{\phi}_\text{w,out}^{n+1/2}$ normally.
After that, the incoming DFs are calculated in the same way as the half-way bounce-back rule in the standard LBM <cit.>,
\begin{equation}
\hat{\phi}_{\text{w},\bar\alpha}^{n+1/2}= \hat{f}_{\text{w}, \alpha}^{n+1/2} - 2w_\alpha \frac{\bm \xi_\alpha \cdot \bm U_\text{w}}{c_s^2},
\end{equation}
where $\alpha$ stands for an outgoing DF direction and $\bar \alpha$ is its reverse direction.
§ NUMERICAL TESTS
In this section, we compare the DUGKS and the BKG scheme in terms of accuracy, stability and computational efficiency by simulating several two dimensional flows.
The first one is the unsteady Taylor-Green vortex flow which is free from boundary effect, and exhibits an analytical solution exists for this problem,
the second test case is the lid-driven cavity flow, which is used to evaluate the accuracy and stability,
and the last one is the laminar boundary layer flow problem, which is used to verify the dissipation property of the the DUGKS and the BKG methods.
In all of our simulation, $c_s$ is set to be $1/\sqrt{3}$ and the CFL number is set to be 0.5 unless stated otherwise.
§.§ Taylor-Green vortex flow
This problem is a two dimensional unsteady incompressible flow in a square domain with periodical condition in both directions.
The analytical solution is given by
\begin{align}
u(x,y,t) = & - U_0\cos(2\pi x)\sin(2\pi y)\exp(-8\pi^2 \nu t),\\
v(x,y,t) = & U_0\sin(2\pi x)\cos(2\pi y) \exp(-8\pi^2 \nu t),\\
p(x,y,t) = & -\frac{U_0^2}{4} \left[ \cos(4\pi x) + \cos(4\pi y) \right] \exp(-16\pi^2\nu t),
\end{align}
where $U_0$ is a constant indicating the kinetic energy of the initial flow field, $\nu$ is the shear viscosity, $\bm u = (u,v)$ is the velocity, and $p$ is the pressure.
The computation domain is $0<x<L$ and $0<y<L$ with $L=1$.
We set $U_0=1/\sqrt{3}\times 10^{-2}$ and $\nu=1/\sqrt{3}\times 10^{-4}$.
The corresponding Reynolds number and Mach number are $\text{Re}=U_0 L/\nu = 100$ and $\text{Ma}=U_0/c_s=0.01$, respectively.
The initial distribution function is computed from the Chapman-Enskog expansion at the Navier-Stokes order <cit.>
\begin{equation}
f_\alpha(\bm x, 0) = f^{\text{eq}}_\alpha - \tau [\partial_t f^{\text{eq}}_\alpha
+ \bm \xi_\alpha \cdot \bm \nabla_x f^{\text{eq}}_\alpha],
\label{}
\end{equation}
where the equilibrium distribution functions are evaluated from the initial analytical solution.
We first evaluate the spatial accuracy of the DUGKS and BKG scheme by simulating the flow with varies mesh resolutions ($N\times N$).
As we are analyzing the spatial accuracy, the time step is set to a very small value ($\Delta t=2\tau$) to suppress the errors caused by the time step size.
The $L_2$-error of the velocity filed is measured,
\begin{equation}
E_u(t) = \frac{\sqrt{ \sum_{x,y}|\bm u_n(x,y,t) - \bm u_a(x,y,t)|^2}}{\sqrt{ \sum_{x,y}|\bm u_a(x,y,t)|^2}},
\label{}
\end{equation}
where $\bm u_a$ and $\bm u_n$ are the analytical solution and numerical solution respectively.
The $L_2$-errors at the half-life time $t_c=\ln(2)/(8\nu\pi^2)$ of the two schemes are measured and listed in Table <ref>.
The results are listed in Table <ref>.
It can be seen that both of the methods are of second order accuracy in space.
But the errors computed from DUGKS results are smaller than those of the BKG scheme on the same mesh resolutions.
$L_2$-errors of the velocity filed for the Taylor-Green vortex flow
N 16 32 64 128
1l2[0]*DUGKS $E_u(t_c)$ 4.1416E-03 1.0852E-03 2.6829E-04 6.1103E-05
1l order - 1.93 2.02 2.13
1l2[0]*BKG scheme $E_u(t_c)$ 1.7025E-02 4.3950E-03 1.1015E-03 2.6945E-04
1l order - 1.95 2.00 2.03
Since Both the DUGKS and BKG methods can admit a time step larger than the relaxation time,
we now investigate their performance at large values of $\Delta t/\tau$ .
We fix the mesh size ($N=64$) and the relaxation time but change the time step.
The $L_2$-errors at $t_c$ are shown in Fig. <ref>, from which
we can see that the errors scale almost linearly with the time step size for both methods.
Particularly, the two methods still give reasonably accurate results $\Delta t/\tau$ is as large as 50, as shown in Fig. <ref>.
And again, the errors of the DUGKS are smaller than those of the BKG scheme in all cases.
The two methods both blow up as $\Delta t/\tau=100$ since the CFL number goes beyond 1 at this condition.
Errors of the velocity field at $t_c$ using varies $dt/\tau$ on a $64\times 64$ mesh.
Velocity profile along the line $x=0.5$ at $t_c$ on a $64\times 64$ mesh, the time step is $\Delta t= 50\tau $.
Here, we also discuss the computational efficiencies of the DUGKS and the BKG scheme
when implementing both of the schemes in FV framework, their only difference is the computing of numerical flux.
The DUGKS introduces two sets of additional DFs and macro variables at cell faces are required.
So it can be expected that the DUGKS's computing cost is obviously higher than that of the BKG scheme.
For example on an Intel Xeon [email protected] CPU,
the computation times for 10,000 steps of the BKG and the DUGKS with $64\times 64$ mesh are 10.5s and 19.7s respectively,
meaning the DUGKS is about one time more expansive than the BKG scheme.
§.§ Lid-driven cavity flow
Incompressible two dimensional lid-driven cavity flow is a popular benchmark problem for numerical schemes.
Here, we use it to evaluate the accuracy and stability of the two schemes at different Reynolds numbers.
The flow domain is a square cavity with length $L$.
The top wall moves with a constant velocity $U_w$, while other walls are kept fixed.
The Reynolds number is defined as $\text{Re} = U_w L /\nu$ with $\nu$ being the viscosity of the fluid.
In the computation, we set $L=1$, $U_w = 0.1$, and the viscosity of the fluid is adjusted to achieve different Reynolds numbers.
Uniform Cartesian meshes with grid number $N\times N$ are used in our simulations.
We first simulate the flow at $\text{Re}=1000, 5000$ and $10000$ with different mesh resolutions to compare the accuracy and stability of the DUGKS and BKG methods.
The velocity profiles at steady states along the vertical and horizontal center lines
predicted by the two schemes are presented in Figs. <ref>-<ref>.
The benchmark solutions <cit.> are also included for comparison.
It should be noted that, the grid numbers used in the BKG schemes are doubled from those in the DUGKS at each Reynolds number, in that
the BKG computations are unstable at the coarsest meshes used in the DUGKS.
From these results, we can clearly observe that the DUGKS scheme gives more accurate results than the BKG scheme, especially at large Reynolds numbers.
Furthermore, the results show that the DUGKS is insensitive to mesh resolutions, while the BKG scheme is rather sensitive.
Generally, much finer meshes should be used in the BKG scheme to obtained accurate results.
Specifically, the horizontal velocity profiles in the boundary layer of the top wall departure from the benchmark solutions severely at high Reynolds numbers (see Figs. <ref> and <ref>) with coarser meshes,
which was also observed in <cit.>.
Contrary to the BKG scheme, the DUGKS gives surprisingly good results with the same meshes even at $\text{Re} = 10000$.
[BKG scheme]
Velocity profiles along the cavity center lines at $\text{Re}=1000$ with different mesh resolutions.
[BKG scheme]
Velocity profiles along the cavity center lines at $\text{Re}=5000$ with different mesh resolutions.
[BKG scheme]
Velocity profiles along the cavity center lines at $\text{Re}=10000$ with different mesh resolutions.
As has been analyzed in Sec. <ref>,
the only difference between the DUGKS and the BKG scheme is the treatment of the quadrature for the collision term
in the reconstruction of the cell-interface distribution function, and the difference scales with the time step,
which have been confirmed in the test of Taylor-Green vortex flow.
Now we explore the effect of time step on the solution of this steady flow for the BKG scheme.
We simulate the flow at $\text{Re}=5000$ and $10000$ using various CFL numbers with a fixed grid ($N=128$).
The calculated velocity profiles are shown in Fig. <ref>.
We can see that the errors decrease with decreasing CFL number.
But even with CFL=0.1, the errors are still much larger than those of the DUGKS.
Velocity profiles predicted by the BKG scheme at with different CFL numbers on a $128\times 128$ mesh. (a) $\text{Re}=5000$, (b) $\text{Re}=10000$.
We also use the cavity flow to assess and compare the stability of the two schemes.
Generally, the stability of the numerical schemes for the BGK equation is affected by the treatments of both the advection term and the collision term.
The stability for an explicit discretization of the advection term is controlled by the CFL number,
while the stability due to the collision term treatment depends on the ratio of $\Delta t$ and the collision time $\tau$, i.e., $\Delta t/\tau$.
The maximum values of $\Delta t/\tau$ at varies CFL numbers for a stable computation on the $32\times 32$ and $64\times 64$ meshes are measured
and presented in Fig. <ref> with error ranges.
It can be seen that there exists a clear distinction between the BKG and the DUGKS methods.
For the BKG scheme, the computation is unstable at moderately large $\Delta t/\tau$ even though $\text{CFL}<1$,
while for the DUGKS, the stability is almost not affected by the CFL number as long as $\text{CFL}<1.1$.
This observation confirms to the analysis in Sec. <ref> that the numerical stability is also affected by the treatment of the collision term in the evaluation of numerical flux.
Computing the collision term implicitly both in Eq. (<ref>) and Eq. (<ref>) makes the DUGKS a rather robust scheme.
Maximum values of $\Delta t/\tau$ for stable computations of the cavity flow.
§.§ Laminar boundary layer over a flat plate
In the cavity flow, it is observed that the BKG scheme fails to capture the boundary layer accurately near the top wall of the cavity for large Reynolds numbers.
In this subsection, we use the laminar flow over a flat plate as a stand-alone case to check this phenomenon and therefore,
evaluate the dissipation characteristics of the BKG scheme and the DUGKS.
The flow configuration of this problem is sketched in Fig. <ref>.
A uniform flow with horizontal velocity $U_0$ flows past a flat plate with length $L$.
This steady problem has an analytical self-similar Blasius solution.
The Reynolds number is defined as $\text{Re} = U_0 L /\nu$, where $\nu$ is the kinematic viscosity.
In the simulations, we set $U_0=0.1$, $L=94.76$ and $\text{Re}=10^5$.
The boundary layer is very thin at such a high Reynolds number,
so non-uniform structured meshes stretched in the vertical direction are employed (Fig.<ref>).
The cell size along $Y$ direction increases with a ratio $A_y=1.1$,
and the height of the first layer is $\Delta y_{\text{min}}$.
The grid number in the $Y$ direction is adjusted according to $\Delta y_{\text{min}}$ to make sure the height of the computation domain is right beyond 50.
The cell size in the $X$ direction is refined at the leading edge of the plate, with $\Delta x_{\text{min}} = 0.1$ to account for the singularity of the flow behavior there.
The increasing ratios of the cell size to the downstream and upstream from the leading edge are $A_r=1.05$ and $A_l=1.1$, respectively.
The total cell number in the $X$ direction is 120, with 80 cells distributed on the plate.
Free stream condition is applied to the left and top boundaries.
Outflow boundary condition is applied to the right boundary, and symmetric boundary condition is used at the section before the plate at the bottom boundary.
No-slip boundary condition is imposed at the bottom wall and is realized by the method described in Sec. <ref>.
Mesh for the laminar boundary layer.
Horizontal velocity profiles in the boundray layer calculated by the BKG scheme and the DUGKS with different mesh resolutions, CFL=0.5.
Top: BGK scheme; Bottom: DUGKS; Left: results at x=6.4381; Right: results at x=21.5082.
We simulate the flow with different mesh resolutions by adjusting the parameter $\Delta y_{\text{min}}$ from $0.02$ to $0.1$.
The CFL number is fixed at 0.5.
The velocity profiles at $X_1=6.4381$ and $X_2=21.5082$ predicted by the BKG and the DUGKS methods together with the Blasius solutions are shown in Figs. <ref>-<ref>.
The horizontal velocity is scaled by $U_0$, and the vertical velocity is scaled by $U_0/2\sqrt{\text{Re}_x}$, where $\text{Re}_x$ is the local Reynolds number defined by $U_0x/\nu$.
Vertical velocity profiles in the boundary layer calculated by the BKG scheme and the DUGKS with different mesh resolutions, CFL=0.5.
Top: BGK scheme; Bottom: DUGKS; Left: results at x=6.4381; Right: results at x=21.5082.
Horizontal and vertical velocity profiles at $x=21.5082.$ for boundary layer flow case, calculated by the BKG scheme using $\Delta y_\text{min} = 0.1$ and different CFL numbers.
(a) Horizontal velocity profile. (b) Vertical velocity profile.
From these results, we can observe that,
the boundary layer can be captured accurately by the DUGKS with the three meshes, particularly, with the coarsest mesh ($\Delta y_{\text{min}}=0.1$)
there are only 4 cells in the boundary layer at $x=X_1$.
On the other hand, the BKG scheme can't give satisfactory results even with the finest mesh ($\Delta y_\text{min} = 0.02$),
as shown in Fig. <ref>(a), Fig. <ref>(b), Fig. <ref>(a) and Fig. <ref>(b).
It is also observed that the results of the BKG scheme are quite sensitive to the mesh employed.
These results suggest again that the DUGKS is more robust than the BKG scheme.
Like the cavity flow case, we reduce the time step in the BKG simulation to examine the effects of time step.
The computation is carried out on the mesh of $\Delta y_{\text{min}}=0.1$ and the CFL number varies from $0.5$ to $0.01$.
The velocity profiles are presented in Fig. <ref>.
It can be seen that the use of a small time step can improve the accuracy, but the deviations from the Blasius solution are still obvious even with CFL=0.01.
§ CONCLUSIONS
In this paper, the performance of two kinetic schemes, i.e., the BKG scheme and the DUGKS is compared.
Both of them can remove the time step restriction which is commonly seen in many off-lattice Boltzmann schemes.
A theoretical analysis in the finite-volume framework demonstrates that the two methods differ only in the constructions of numerical flux.
The BKG scheme treats the collision integral with the one-point quadrature when integrate the BGK equation along the characteristic line to evaluate the numerical flux, while DUGKS computes it with the trapezoidal quadrature.
Consequently, the DUGKS is more accurate and stable than the BKG scheme.
The numerical results of three test cases, including unsteady and steady flows, confirm that the
the DUGKS is more accurate and stable than the BKG scheme on the same computing configurations, especially for high Reynolds number flows.
It is also observed that the DUGKS is stable as long as $\text{CFL}<1$, while the BKG scheme's stability degrades quickly as the CFL number goes beyond 0.5.
We attribute this to the implicit treatment of the collision term of the DUGKS when evaluating the numerical flux.
Furthermore, the results show that the DUGKS is more insensitive to mesh resolutions than the BKG method.
Numerical results also demonstrate that the BKG scheme is about one time faster than the DUGKS on a same mesh.
However, it should be noted that the latter can achieve an accurate solution with a much finer mesh, suggesting that it can be more efficient for flow computations than the BGK scheme.
In summary, the theoretical analysis and numerical results demonstrate that the DUGKS can serve as an efficient method for simulating continuum flows, although it is not limited to such flow regime.
§ ACKNOWLEDGMENTS
The authors thank Prof. Kun Xu for many helpful discussions, and thank Dr. Weidong Li for providing the benchmark data of the laminar boundary layer.
This study is financially supported by the National Science Foundation of China (Grant No. 51125024)
and the Fundamental Research Funds for the Central Universities (Grant No. 2014TS119).
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firstpage–lastpage 2015
The Kepler Mission has detected dozens of compact planetary systems
with more than four transiting planets. This sample provides a
collection of close-packed planetary systems with relatively little
spread in the inclination angles of the inferred orbits. A large fraction of the observational sample contains limited multiplicity, begging the question whether there is a true diversity of multi-transiting systems, or if some systems merely possess high mutual inclinations, allowing them to appear as single-transiting systems in a transit-based survey.
This paper
begins an exploration of the effectiveness of dynamical mechanisms in
exciting orbital inclination within exoplanetary systems of this class.
For these tightly packed systems, we determine that the orbital inclination angles are not spread out appreciably
through self-excitation.
In contrast, the two Kepler multi-planet systems with additional non-transiting planets are susceptible to oscillations of their inclination angles, which means their currently observed configurations could be due to planet-planet interactions alone. We also provide constraints and predictions for the expected transit duration variations (TDVs) for each planet. In these multi-planet compact Kepler systems, oscillations
of their inclination angles are remarkably hard to excite; as
a result, they tend to remain continually mutually transiting
(CMT-stable). We study this issue further by augmenting the planet masses and determining the enhancement factor required for oscillations to move the systems out of transit. The oscillations of inclination found here inform the recently suggested dichotomy in the sample of solar systems observed by Kepler.
planets and satellites: dynamical evolution and stability — planetary systems
§ INTRODUCTION
The Kepler mission has discovered a large number of compact extrasolar
systems containing multiple planets that can be observed in transit
<cit.>. Roughly forty of these such systems have
four or more planets. The inventory of these four-plus planet systems
includes mostly super-Earth sized planets, which have radii $R_P =
2-5 R_{\earth}$ and orbital periods in the range 1 – 100 d.
Moreover, the orbital periods of the planets within a given system are
regularly spaced (roughly logarithmically uniform in period or
semimajor axis). Because all of the planets were observable by Kepler
at their times of discovery, these systems have an additional
stringent dynamical constraint: they must have retained a relatively
narrow spread in their orbital inclination angles. On the other hand,
orbital inclination can often be excited in close-packed planetary
systems. The goal of this paper is thus to explore the oscillations of
orbital inclination within solar systems of this class. Excitation of
inclination can be driven by a variety of mechanims, incluing unseen
additional companions, perturbations from stellar encounters in
clusters <cit.>, and self-excitation through
interactions among the observed planets. This paper focuses on this
latter mechanism.
Slight deviations from true coplanarity in these systems (e.g., as
observationally supported in ; and others)
allow for the possibility of oscillations in the inclination angles of
the planetary orbits, e.g., due to secular interactions between the
planets <cit.>. If such oscillations were common, and had sufficient
amplitude, then not all members of a solar system could be seen in
transit at every epoch. As a result, multi-planet systems would
display evidence for “missing” planets, i.e., exceptions to the
(roughly) logarithmically even spacing of orbits that are often observed.
The ubiquity of this class of exoplanetary systems places strong
constraints on both their architectures and dynamical histories (see
also ). We note that the inclination angle oscillations for Jupiter and Saturn in our own solar system are large enough to periodically move the orbits out of a mutually transiting configuration.
Statistical analyses of the Kepler system architectures suggest that
there could exist two distinct populations of planetary systems
<cit.>, namely, a population with single-transiting
planets and an additional population of multi-planet systems. The
existence of these two distinct populations could
be explained by either two true distributions of solar systems (e.g.,
created by two different formation histories) or a single distribution
in which some systems exhibit a high degree of scatter in orbital
inclination angles. Excitation of inclination in nearly coplanar
systems could shift some planets out of a transiting configuration,
thereby leading to the population of single-transit systems. In this
case, the single-transit systems would be a subset of the
multi-transiting group rather than a distinct population.
This paper explores possible oscillations of the inclination angles in
compact extrasolar systems. The measured planetary radii
$R_P = 2 - 5 R_{\earth}$ imply planetary masses $M_P = 4 - 30 M_{\earth}$,
where we use
a conversion law based primarily on the
probabilistic mass-radius relationship derived in <cit.>:
\begin{equation}
\frac{M}{M_{\earth}} \sim \rm{Normal} \left( \mu = 2.7 \left( \frac{R}{R_{\earth}}\right)^{1.3} , \sigma = 1.9 \right)
\end{equation}
where $M$ refers to the mass of a body, $R$ its radius, and this expression represents a $r^{1.3}$ scaling law with a normal distribution of scatter due to potential planetary composition variation.
The Wolfgang relationship describes the a distribution of the potential masses for planets in the range $R_P = 1.5 - 4 R_{\earth}$. Since a small number of planets in our sample lie outside these bounds, we supplement the Wolfgang relation in two ways: for planets with radii $R_P < 1.5 R_{\earth}$, we
supplement with the rocky relation from <cit.>; for planets with radii $R_P > 4 R_{\earth}$, we determine starting density using the Wolfgang relation, then add a scatter and choose a radius anomaly to account for varying core masses and inflation due to thermal effects <cit.>. Of the 208 planets in our sample, only 9 fall above the regime described by the Wolfgang relation.
relatively large masses and close proximity, planet-planet
interactions can be significant. On the other hand, these planetary
systems orbit relatively old stars (with ages of $\sim1-6$ Gyr,
weighted toward the lower end of this range; see ), so
that they are expected to be dynamically stable over $\sim$Gyr time scales. These systems are also generally non-resonant. These
considerations — significant interactions coupled with long-term
stability and non-resonance — suggest that the planetary systems are subject to
secular interactions
<cit.>. In the present context, we are interested in secular
oscillations of the inclination angles of the orbits. If such
oscillations have sufficient amplitudes, the resulting spread of
inclinations angles in the system will sometimes be large enough that
not all of the planets can be seen in transit. When observed in such
a configuration, the system will appear to have gaps in the regular
spacing of planetary orbits that these systems usually exhibit. The
goal of this paper is to understand the amplitude of self-excitation
of inclination angle oscillations and provide limits on transit
duration variations, an observable with amplitude directly related to
inclination evolution over time, for observed Kepler systems with no
unseen companions. This analysis will allow future observations of transit
durations for these systems to inform the presence of massive outer
companions in these systems.
We note that spreads in the inclination angles can be produced by a
variety of astronomical processes. This work will focus on secular
oscillations of the inclination angles by the compact solar system
planets themselves (with semi-major axes $a\lta0.5$ AU). Future work
will focus on the effect of possible additional bodies in the outer
part of the solar system (where $a\approx5-30$ AU), roughly analogous to the giant planets in our outer Solar System.
We stress that oscillations of inclination angles are not rare.
Within our Solar System, for example, the orbital inclinations of
Jupiter and Saturn oscillate with a period of about 51,000 years and
an amplitude of about $1^\circ$ (see Figure 7.1 in ). The
inclination angles of the two orbits coincide every half period
(25,500 years), so that an observer oriented in that plane would see
both planets in transit at that epoch. However, the amplitude of the
oscillation is sufficient to move both planets out of transit for an
appreciable fraction of the secular cycle.
This paper focuses on the case of self-excitation of inclination
angles for Kepler systems with four or more planets, where the secular
dynamics of such systems are considered in Section 3. An analysis of
the observed compact, mutually transiting systems is presented in
Section <ref>, which shows that the systems are consistently mutually
transiting over time. An orbital architecture that is continually
mutually transiting is denoted here as CMT-stable (which should not be
confused with dynamical stability). We consider a generalized class of
systems in Section 3.2, and study compact systems which have been
discovered to host an additional non-transiting planet in Section 3.3
(where these systems are shown to be more active).
We also compare these results with numerical simulations in Section
Section <ref> presents observables for the compact Kepler systems discovered to
date; specifically, the transit durations are predicted to vary and the magnitude of these variations
are determined. In Section <ref>, we study the stability of the observed Kepler systems by considering how the predicted oscillation amplitudes would vary if planet masses are scaled upward: the systems are found to be remarkably dynamically stable. The paper concludes, in Section <ref>,
with a summary of our results and a discussion of their implications,
as well as a statement on our plans for future work.
§ SECULAR THEORY FOR INCLINATION ANGLES
To evaluate the behavior of mutual inclination for these isolated
systems, we apply Laplace–Langrange secular theory <cit.>. This
formalism allows the use of the long-period terms of the disturbing
function to describe orbital motion over many secular periods.
§.§ Review of the Theory
We expand to second order in inclination and eccentricity, and first
order in mass. With this expansion, inclination and eccentricity are
decoupled, so we can write the disturbing function as a function of inclination alone:
ℛ^(sec)_j = n_ja_j^2
[ 1/2 B_jj I_j^2
+∑_k=1^N ( B_jk I_jI_k cos(Ω_j -Ω_k))
] ,
where $j$ is the planet number, $n$ is the mean anomaly, $I$ is the
inclination, $\omega$ is argument of pericenter,
and $\Omega$ is the longitude of the ascending node. The coefficients $B_{ij}$ are defined by
B_{jj} = -n_{j} \Biggl[ \frac{3}{2} J_{2}
\left( \frac{R_{c}}{a_{j}} \right)^{2} - \frac{27}{8} J_{2}^{2}
\left( \frac{R_{c}}{a_{j}} \right)^{4} - \frac{15}{4} J_{4}^{2}
\left( \frac{R_{c}}{a_{j}} \right)^{4}
+ 1/4 ∑m_k/M_c + m_j
α_jk α̅_jk b^(1)_3/2(α_jk) ] ,
B_jk = n_j [ 1/4m_k/M_c + m_j
α_jk α̅_jk b^(1)_3/2(α_jk) ] ,
where $J_{2}$ and $J_{4}$ describe the oblateness of the central star (which we set to be $=0$ in all our analysis), $R_{c}$ is the stellar radius, $m_{k}$ indicates the mass of the $k$th planet, $M_{c}$ denotes the mass of the central star, $\alpha_{jk}$ denotes the semi-major axis ratio $a_{j}/a_{k}$, and $\bar{\alpha}_{jk}$ denotes the semi-major axis ratio $a_{j}/a_{k} < 1$. The quantities $b_{3/2}^{(1)}$ is
the Laplace coefficient, which is defined by
b_3/2^(1) = 1/π ∫_0^2π
cosψ d ψ/(1-2 αcosψ + α^2)^3/2 ,
(as given in ). All of the coefficients $B_{jk}$ can be considered as frequencies that describe the
interaction between each pair of planets, and are elements of the
matrix denoted as B. This application of
secular theory allows us to evaluate the problem analytically, but
neglects higher-order terms. In this formulation, the only terms in
the disturbing function are those that do not depend on the mean
longitudes, as we assume that the short-period terms average out over long timescales. The coefficient matrix B describes
inclination evolution. Solving for the matrix
elements of B allow us to determine the time evolution of
The matrix B defines an eigenvalue problem <cit.>, where
the eigenvalues describe the interaction frequencies between any pair
of planets. The eigenfrequencies of this matrix, denoted here as
$f_{i}$, along with the eigenvectors ${\cal I}_{jk}$, can be used to
describe the time evolution of the system. Given the matrix
B, we can solve for the eigenvalues and eigenvectors using
standard methods. With these quantities specified, we also need the
initial conditions to specify the full solution for the time evolution
of the inclination angles $I_j$ and the angles $\Omega_j$. It is
convenient to transform the dependent variables according to
p_j = I_j sinΩ_j and q_j = I_j sinΩ_j ,
so that the solutions take the form
p_j (t) = ∑_k=1^N I_jk sin(f_k t + γ_k)
q_j (t) = ∑_k=1^N I_jk cos(f_k t + γ_k) ,
where the phases $\gamma_k$, along with the overall amplitudes, are
determined by the initial conditions. The quantities $I_{jk}$ are
eigenvectors, where we use the standard (but awkward) notation such
that the first index $j$ specifies the planet number and hence the
components of the eigenvector and the second index $k$ runs over the
different eigenvectors. It is also useful to define normalized
eigenvectors ${\cal I}_{jk}$ and corresponding scaling factors $T_k$
such that
I_jk = T_k I_jk .
The initial conditions then specify the scaling factors through
the expressions
p_j (t=0) = ∑_k=1^N T_k I_jk sinγ_k
q_j (t=0) = ∑_k=1^N T_k I_jk cosγ_k .
The scaled eigenvectors $I_{jk}$ (which conform to the system's
boundary conditions), the eigenvalues $f_{k}$, and the phases
$\gamma_{k}$ are sufficient to specify the time evolution of the
orbital inclination of each body in the system, i.e.,
I_j(t) = √([ p_j(t) ]^2 +
[ q_j(t) ]^2 ) ,
where the solutions $p_j(t)$ and $q_j(t)$ are given by equations
(<ref>) and (<ref>). Implicit in this solution is the
linear dependance on the interaction coefficients (the matrix elements
given by equations [<ref>]). From this
solution, we note that the inclination evolution has a linear dependance on mass
ratio and a second order dependence on the semi-major axis ratio
between the planet in question and each planet exterior to its orbit.
§ INCLINATION OSCILLATIONS DUE TO SELF-EXCITATION
The compact Kepler systems with four or more planets are tightly
packed systems with minimal mutual inclinations. From this population,
it appears that planets in multi-transiting systems generally have
non-null values of mutual impact parameter, and subsequently
inclination <cit.>. Systems with non-null mutual inclinations
exhibit non-parallel angular momentum vectors, allowing the
possibility of excitation in inclination and other orbital
elements. To test the magnitude of these excitations, we take the
population of all Kepler systems with four or more transiting planets as examples of compact, multi-body, transiting systems.
We obtain our data from the NASA Exoplanet Archive[http://exoplanetarchive.ipac.caltech.edu], updating system parameters when newer values have been found <cit.>.
Plotted here are the inclination evolutions of five roughly coplanar
planets, with initial conditions drawn from the priors of Kepler-256. Although the inclinations of the planets
generally stay within a plane, there is also instantaneous variation,
which manifests as a range in the inclinations of plane of planets. This variation may lead
planets to be knocked out of a transiting configuration. The mutual inclination, shown on the right panel, changes as planets precess, meaning that the width of the plane containing all the planets oscillates over time.
There are observational biases inherent in the Kepler systems, as a photometric transit survey is by definition more likely to find systems with low mutual inclinations and aligned argument of pericenters <cit.>. The Kepler multi-planet systems are likely more aligned and more compact in inclination plane width than an 'average' system, but the sample found by Kepler is
representative of the type of system we would expect to see
from photometric transit surveys such as Kepler <cit.>, K2 <cit.>, and TESS <cit.>.
It is not currently clear, however, whether the Kepler multi-planet systems that we do see in transit are CMT-stable or if we are catching them at a lucky moment in which all planets appear to be in transit. This differentiation is important because the former possibility describes a much less dynamically active system than the latter.
To test the
stability against exciting planets out of the transiting plane, we
used the secular theory described in Section <ref> to
numerically evolve each system in the Kepler multi-planet sample for
several secular periods. This procedure results in a measure of the spread in impact parameter $\Delta
b(t)$ (see below). We also compute the probability that the system is mutually transiting, marginalized over all trials and realizations in our simulations. If $\Delta b(t) <2$ for an entire secular period and the probability of all planets transiting simultaneously for a random time-step in a random realization of the system is high ($P(\rm{transit}) > 0.85$) then the system is said to be CMT-stable in a
transiting configuration. Note that the condition of being CMT-stable against oscillating planets out of transit is much more confining that being dynamically stable against planet ejection. For a given Kepler system, we can use a
Monte Carlo analysis to evaluate $\Delta b(t)$ not just once, but many
times, with starting orbital elements for each realization selected
from observationally motivated priors. For parameters that have been
measured (for transiting systems, the radius of the planet $r_{p}$ and
the semi-major axis $a_{p}$, and sometimes the inclination $I_{p}$ or
eccentricity $e_{p}$), we draw each planet's orbital element from a
normal distribution with mean and standard deviation determined from
observations. For orbital elements not measured, we draw a value from
priors summarized in Table <ref>.
Observationally measured inclinations have been fit from photometric light curves, and for these planets there is a degeneracy between angles over 90and under 90. The literature reports inclination angles as $< 90\degree$, so when we use a literature measurement, we choose a value not only from that planet's measured posterior but also choose its orbit to fall above or below the midplane of the star with equal probability.
For planets without measured inclinations, we choose a plane width from a Rayleigh distribution with width 1.5<cit.>, subject to the constraint that all planets must be transiting. This choice of distribution follows work done by <cit.>. In these recent works, Rayleigh distributions with varying widths are used to describe the size of the plane containing the planets. The value we use here, 1.5, is within the range suggested by the work of <cit.>.
We note that the
argument of the ascending node is not necessarily independent of the
value of inclination angle as assumed here. As planetary systems
evolve to attain nonzero inclination angles, modeled here by a
Rayleigh distribution, the nodes will evolve into some other
distribution, which should be characterized in future work.
Orbital Element Distributions
Parameter Prior
$\omega$ uniform on $(0^\circ, 360^\circ) $
$\Omega$ uniform on $(0^\circ, 360^\circ) $
$e$ uniform on $(0, 0.1) $
$I$ Rayleigh distribution with width $\sigma = 1.5\degree$
When orbital elements have not been measured observationally,
we draw their values randomly from the prior distributions summarized
in this table.
Once we have the initial conditions for each Kepler system, we can evolve orbits as according to the secular theory described in Section <ref>. This must be done individually for each realization of initial conditions for each system.
§.§ Evaluating the Secular Behavior of the Compact, Multi-Planet Kepler Systems
A tightly packed, roughly coplanar system of planets will trade
angular momentum as the system evolves (while keeping the total angular momentum vector of the system constant). The magnitude of this
exchange determines the magnitude of the variations in orbital
elements of each body. Equation (<ref>) describes the
inclination evolution for each body in a system. Once the inclination
solutions for each planet in a system have been found using equation
(<ref>), a comparison between them (see Figure <ref>, which illustrates how the mutual inclination can change over time)
yields a measure of the mutual inclination between all planets in the
system. This mutual inclination describes the width of the plane
containing all the planets.
As the condition for transiting is more
rigorous than approximate coplanarity (even as planets' inclinations
vary in concert, the planets with larger orbital separations are more
likely to cease transiting), we remove the dependence on orbital and stellar properties by working in terms of impact parameter, $b$, which is defined as:
\begin{equation}
b_{j} = \frac{a_{j}}{R_{*}} \cos{(I_{j})}
\label{impact_parameter}
\end{equation}
where $j$ is planet number, $a$ is the semi major axis, $R_{*}$ the
radius of the central star, and $I$ the inclination. When $-1 < b_{j}
< 1$, planet $j$ will transit. Using the analytic expression for inclination evolution (Equation <ref>), we can describe the long-term behavior of not only individual planets but the range of their
respective impact parameters.
The process of extracting the mutual impact parameter
$\Delta b$ is shown in Figure <ref>.
The parameterization of mutual impact parameter, as illustrated
by test case Kepler-11.
First, a plot of inclination for all bodies in a system
(upper left) is generated by solving equation (<ref>) with
the initial parameters of the system as boundary conditions. The semi-major axis dependency
is removed using equation (<ref>), and the result,
impact parameter over time for each planet, is shown (upper
right). The inclinations attained by each planet result in vastly different impact parameters due to the
differences in semi-major axis. Planets closer to the star can attain
more inclination with less effect on their impact parameter. Finally,
the range between impact parameters is calculated (lower right) as
was done for mutual inclination in Figure 1. The result is a measure
of the range of the mutual impact parameter over time, $\Delta b(t)$. As long as this width describes a plane that lies entirely within the limbs of the star, the planets will be CMT-stable.
Using this technique, we explored the evolution of orbits for the entire initial condition parameter space for each Kepler multi-planet system. For a given system, we conducted 4000 Monte Carlo trials for each Kepler system, resulting in
4000 realizations of $\Delta b(t)$, with different initial conditions drawn from the observational priors, supplemented with the values in Table <ref>. This sample can be used to calculate
the mean range of the impact parameter over time for the Kepler system, as well as the width of the plane of planets in impact-parameter space.
We repeated this process of 4000 Monte Carlo trials for each for the 43 systems in our sample of multi-planet Kepler systems, resulting in a measure of the inclination evolution behavior for each system.
Figure <ref> visualizes the results of these trials, where each point represents the mean mutual impact parameter for a different Kepler system. Mean mutual impact parameter is the typical width of the plane containing all planets in the system, and must be smaller than the diameter of the star for all planets to transit. An impact parameter plane width of $\Delta b=2$, marked on the plot, is the upper limit for all planets in the plane to be transiting.
For each multi-planet Kepler system, the parameters of
the system were sampled 4000 times and evolved forward in time. The
resulting inclination angles for the planetary orbits were converted
to a mutual impact parameter (see text). The mean and scatter of these
values are plotted here for each system as a function of the total
mass of the transiting planets, given in earth masses. The dotted
horizontal line indicates the level above which it is not possible to
observe all the planets in transit.
For each point in Figure <ref>, the height of the point as compared to the transit limit ($\delta b=2$) corresponds to the width of the plane containing all the planets. The scatter (represented by error bars) corresponds to the width of the distribution due to the variations between realizations.
For all systems, the projected plane containing the planets is much smaller than the diameter of the star, which means we would expect to see all the planets in transit at for the majority of the secular history of the system.
This parameterization represents the average behavior of each system over time. The plane width demonstrates how much range in impact parameter is normal for each architecture of system. However, we care about the transiting behavior of each system with respect to a single line of sight: that of the observer (Kepler) who originally identified the planets as mutually transiting. For example, it would be possible for a system's impact parameter range to be small enough for it to be possible for all planets to transit, but for the plane to be situated in such a location that only some planets transit. To understand how likely this is to happen, we plot in Figure <ref> the mutual transit probability for each observed system as blue circular points. This probability is defined as the probability that a random time-step from a random realization, chosen from the sample of all 4000 realizations considered in the construction of Figure <ref>, will have all planets transiting along the line of sight to Earth. A probability of 1 would mean that the planets never left a transiting configuration in any time-step in any of our simulations, while a probability of 0 means that the system was never mutually transiting in any time-step in any realization.
Figure <ref> shows that for the observed Kepler systems, all planets are expected to be transiting more than 85% of the time. Indeed, for most systems the probability of mutual transit is even closer to 100%. This demonstrates that not only do we expect the Kepler multi-planet systems to have plane widths small enough to potentially be transiting (Figure <ref>), the majority of the time they should maintain these transiting configurations with respect to our line of sight (Figure <ref>).
From an analysis of the results in Figure <ref> and Figure <ref>, it appears that while
Kepler systems do excite mutual inclinations due to their dynamical
interactions with each other (as their mutual impact parameters do change over time), the magnitude of these
interactions are small enough that although an initially non-null
mutual inclination exists, it remains, through the process of secular evolution, smaller than the threshold necessary for
planets to not be observed in transit. From this, we can state that the observed Kepler systems are generally CMT-stable.
The Kepler systems with four or more planets do not exhibit
sensitivity to self-excitation of inclination due to dynamic
interactions between the inner, roughly coplanar planets. This result
indicates that self-excitation (in the mode considered here) is not a dominant
mechanism in knocking planets out a transiting plane and thereby creating
tightly-packed systems in which only some planets transit.
It it important to note that the analysis of these observed Kepler system is limited by several factors: the measured mutual inclinations will be artificially low compared random systems drawn from the true distribution of planetary architectures, as these are systems with narrow enough ranges in inclination to be discovered in transit in the first place; the impact parameters of observed systems are likely artificially low due to the signal-to-noise bias against higher impact parameters; the deviation between measured planetary arguments of pericenter will also be artificially low <cit.>. These systems are not a representative sample of the true distribution of systems. As a result, the analysis presented here for the observed Kepler systems is not an analysis of the underlying planet population, but only of this particular class of heretofore discovered systems.
For all realizations considered in Sections, <ref>, <ref>, and <ref>, we plot as circles the probability that a randomly chosen time-step from a randomly chosen realization will have all planets transiting along the line of sight to Earth. For the observed Kepler systems, all systems are mutually transiting more than 85% of the time. This result indicates that statistically the observed Kepler systems are seen in transit an overwhelming majority of the time. The generalized systems, plotted as crosses, are mutually transiting a much lower fraction of the time, as are Kepler-48 and -68, the observed currently non transiting systems.
§.§ Inclination Oscillations in Generalized Kepler Systems
The Kepler systems that we see are observationally biased in that they likely have unusually low mutual inclinations and aligned arguments of pericenter <cit.>. As we have shown in Section <ref>, the observed Kepler systems are remarkably CMT-stable in their transiting configurations. We are not simply lucky to see these systems in transit, merely viewing them at an opportune time: instead, we are seeing systems that will likely be consistently transiting over many secular timescales.
The Kepler systems are indeed a special class of system. It would also be interesting to compare their behavior with that of generalized Kepler systems, with a wider range of starting orbital parameters.
To construct these systems, we repeat the following process for each Kepler system in our sample:
* Generate a compact planetary system based on the target Kepler system. To do this, we draw each orbital parameter from an inflated distribution, treating measured 3$\sigma$ errors as the width of our prior from which to draw orbital parameters. We convert radii to masses using the extended Wolfgang relation.
* We evaluate the system for dynamical stability using the Hill-radii criteria outlined in <cit.>. We compute the separation between two orbits ($\Delta$) in terms of their Hill radii:
\begin{equation}
\Delta = (a_{out} - a_{in}) / R_{H}
\end{equation}
when the mutual Hill radius is given by:
\begin{equation}
R_{H} = \left( \frac{M_{in} + M_{out}}{3 M_{*}} \right)^{1/3} (a_{out} - a_{in}) / 2
\end{equation}
and for a system to be considered dynamically stable, $\Delta > 2 \sqrt{3}$ and for each pair of planets, $\Delta_{in} + \Delta_{out} > 18$ <cit.>.
* If the system is dynamically stable according to these Hill arguments, we evolve the system and repeat the process for another set of starting parameters.
Once this process is completed for each Kepler system, we have a sample of analog Kepler systems, which are based on the observed systems but no longer exactly the systems that we observe. This sample allows us to compute the mean mutual impact parameter over time, just as we did for the observed Kepler systems in the previous section.
The result is shown in Figure <ref>, which shows the same statistic plotted in Figure <ref> computed from the generalized Kepler systems. For these generalized systems, the range of the impact parameter over time is higher, suggesting that the Kepler systems we observe are a particularly CMT-stable subset of the dynamically possible compact systems that could exist. Figure <ref> shows as red crosses the mutual transit probability (over all time-steps and all realizations) for these generalized systems, demonstrating that the generalized systems spend significant amounts of time in non-mutually transiting configurations, as their plane widths imply they should.
For the generalized multi-planet Kepler systems, the parameters of
the system were sampled 4000 times and evolved forward in time, just as in Figure <ref>. The
resulting inclination angles for the planetary orbits were converted
to a mutual impact parameter (see text). The mean and scatter of these
values are plotted here for each system as a function of the total
mass of the transiting planets, given in earth masses. The dotted
horizontal line indicates the level above which it is not possible to
observe all the planets in transit.
§.§ Inclination Oscillations in Systems with Non-transiting Planets
Long-term RV followup to systems with transiting planets has not only found masses for Kepler planets, but has also resulted in the characterization of additional, non-transiting companions to some transiting systems <cit.>. Additionally, transit-timing variation analysis <cit.> has both confirmed masses of planets and provided additional candidate planets <cit.>.
The current state of these systems provides insight to their dynamical history: assuming that systems form from roughly coplanar protoplanetary disks, something in the evolution of these systems has resulted in sufficiently large spread in inclinations to prevent all planets from being seen in transit.
As shown in Section <ref>, the
observed multi-transiting Kepler systems are CMT-stable against
self-perturbation (mutual inclinations excited by dynamical
interactions between the transiting planets). Furthermore, the generalized Kepler systems are more likely than not to be seen in mutual transit. For multi-planet systems
with some planets transiting and additional non-transiting companions,
something in the dynamical history of the systems has resulted in misalignment in inclination between the
planets. This effect could be explained in one of many ways: it could be due to a difference in formation
mechanism between the purely multi-transiting systems and the systems
with some planets outside the transiting plane; it could be due to some other perturbation, such as an as-yet undiscovered stellar or massive planetary companion (a possibility beyond the scope of this paper); or finally, it could be due to
the effect of self-excitation between all (known) planets in the system.
Our analysis probes this final possibility, which would apply if all discovered planets (both those that are currently transiting and those that are currently non-transiting) in a system had started out roughly coplanar, in a potentially transiting configuration, and then through secular
interactions some planets had been perturbed out of the transiting
We can test this explanation for the currently observed misalignment of Kepler systems that have been
discovered to have multiple transiting planets and additional,
non-transiting companions using the same method that was used to evaluate the transit stability of the most tightly packed Kepler systems in Section <ref>. Two examples of systems of this architecture are Kepler-48 and
Kepler-68. By starting the planets of these systems in
transiting configuration, we force the starting conditions to be a roughly coplanar disk containing all the planets.
Kepler-48 <cit.> is a four planet
system with three inner transiting planets and one non-transiting
companion at more than 1 AU (a minimum mass 657 $M_{\earth}$ companion with a period of roughly a 980 day period).
Kepler-68 <cit.> is a three
planet system with two transiting planets and one non-transiting
planet, also outside of 1 AU (with a minimum mass of 0.95 $M_{jup}$ companion in roughly a 580 day period).
To evaluate the transit stability of Kepler-48 and Kepler-68, we performed the same Monte Carlo
evolution described in Section <ref>, with all orbital
parameters drawn from observationally constrained priors except
Though the true orbital inclination of the outer planets in the Kepler-48 and Kepler-68 systems is not known, we choose the orbital inclinations for the giant outer planets in each system by drawing a mutual inclination plane width from a Rayleigh distribution with a width of 1.5<cit.>. We constrain this choice of plane width such that the planets all start out mutually transiting, to mimic the starting conditions of the compact Kepler systems. With these starting conditions, we are probing what would happen to the observability of these systems over time, if they did start on feasibly observable architectures.
Through 4000 trials, Kepler-48 and Kepler-68 exhibited significantly more range in their mutual impact parameters than the other compact Kepler systems. Figure <ref> plots the behavior of Kepler-48 and Kepler-68 overlaid on the previous result for the compact Kepler systems. Kepler-68's mean mutual impact parameter is well above the limit for a mutually transiting system, while Kepler-48 spends about 60% of its orbits in a transiting configuration (marginalized over starting parameters).
We treat Kepler-48 and Kepler-68 as isolated systems. In other words, in our experiments, the only perturbation available to excite oscillations in inclination is that of the interactions between known bodies in each system.
Thus, by generating the mean mutual impact parameter over one
secular period for these systems after they start in a transiting
configuration, we can make a statement about the amplitude of
self-excitation in these compact systems. As shown in Figure <ref>, both Kepler-48 and Kepler-68 would be expected to develop significant mutual inclinations that prevent all planets from being seen in transit purely through excite self-excitations
of inclination. Figure <ref> shows as salmon points the mutual transit probability for these two systems, confirming that it is unlikely that the magnitude of the secular interactions would allow these two planets to be seen in transit.
This result indicates that even if these systems were to begin their secular evolution in a roughly coplanar configuration, they would be expected to self-excite sufficient oscillations to produce the current orbital state (where not all planets transit - we do not have sufficient limits on the observed inclinations to make a stronger comparison). Kepler-48 and Kepler-68 are examples of systems that `make sense' dynamically: it is not required to add additional effects (such as a perturbing companion or stellar flyby) to their systems to explain their current non-transiting nature.
It is important to note that the outer planets in these two systems are significantly external to the standard compact systems described in Section <ref>, which generally fell within 0.5 AU of their host star. Kepler-48 and -68 have outer companions at roughly 1.4 and 1.8 AU, respectively. It is possible that part of the reason for the activity of these systems is the lower transit probability of these outer companions, but the presence of Kepler-90 (which has an outer companion semi-major axis of roughly 1 AU) in the CMT-stable sample indicates that external companions do not ensure non-transiting configurations.
Kepler systems in which all discovered planets are transiting
are plotted as black points (they correspond to the same data
presented in Figure <ref>), while Kepler systems where additional
non-transiting planets have been discovered are plotted as red points.
Kepler-48, marked <cit.> is a four planet
system with three inner transiting planets and one non-transiting
companion outside of 1 AU. Kepler-68 <cit.>, marked, is a
three planet system with two transiting planets within 0.1 AU and one
additional non-transiting planet at 1.4 AU.
§.§ Comparison to Numerical Integrations
An illustrative realization of Kelper-341b, with the result from the numerical N-body code plotted in black and the secular theory evolution plotted in red.
The discussion thus far has considered inclination oscillations as
described by second-order Laplace-Lagrange theory. Although the
amplitudes of the oscillations are small, so that the second order
theory is expected to be accurate, in this section we compare the
results to numerical simulations. These latter calculations, by
definition, include interactions to all orders.
For these compact systems, eccentricity and inclination are generally low, but to evaluate the error inherent in our second-order expansion, we evolved each compact system using hybrid symplectic and Bulirsch-Stoer integrator (Chambers 1999). The numerical integrator should provide the effectively `right' answer, and significant deviations between the second-order theory and numerical results would indicate that second order secular theory is insufficient to describe the evolution of the orbital architectures. We compared 400 numerical N-body realizations with 400 secular evolutions (see the visualization of one realization of the comparison in Figure <ref>) to compute the deviations plotted in Figure <ref>, which describe the mean deviation, in degrees, between secular theory and the numerical results. This comparison yielded a standard deviation of the difference in inclination angle obtained using secular theory and numerical results; this value was found to be less than 0.01.
For our use of second order second theory to be adequate for further analysis, we would want this variation between the numerical result and secular result to be much smaller than the threshold for significant inclination (which can cause a planet to become non-transiting). The planet in our sample with the largest semi-major axis and largest number of planets in the system, Kepler-11g, orbits a star with a radius $R_{*} = 0.0053$ AU. This planet would need to attain an inclination of 0.65$\degree$ out of the plane of the other planets to no longer transit. Planets with semi-major axes less than this value would need an even larger range of inclinations to be no longer seen as mutually transiting. Given that the typical deviation between the numerical results and secular theory is less than 0.01$\degree$, the match between secular theory and N-body numerics is good enough to use the second order secular theory for these compact systems.
We additionally note that although there is variation in the period of secular effects between numerical and second-order secular theory <cit.>, this does not affect our result, as we are concerned with the amplitude rather than period of inclination oscillations, and these amplitudes are well-predicted to a reasonable precision. If we were concerned with the exact period of secular effects, second-order Laplace-Lagrange theory would not always be sufficient.
Finally, for completeness we note that the standard deviation of the
residuals between the secular and numerical results is not the only
measure of the difference (e.g., one could use the difference between
the ranges of inclination angles instead). In this case, however,
the differences between the two approaches is small: The differences
would have to be nearly 100 times larger in order to change our main
conclusion, i.e., that the Kepler compact systems remain CMT-stable.
A residuals plot of (top panel) the deviation in inclination over several secular periods for each planet in our sample and (bottom panel) the deviation in eccentricity for the sample sample of realizations. The averaged deviation in inclination between the numerical and secular methods is generally below 0.01$\degree$ for all planets.
§ TRANSIT DURATION VARIATIONS
Oscillations of the orbital inclination angles, as described in
secular theory through equation (<ref>), result in planets
taking different paths across the face of the star as a function of
time. These changing chords, in turn, result in the duration of the
planetary transit varying with inclination and hence with time. For
the case of vanishing eccentricity, we can write $\tau_{T}$, the time
from first to fourth contact (the transit duration) for a single
transit analytically (see ), in the form
τ_T(t) = P/π arcsinΘ
where we have defined the effective angle
Θ≡R_*/a [
(1 + r_p/R_*)^2 - (a/R_*cos^2i)/1-cos^2i
]^1/2 ,
where $P$ is the period of the planet, $a$ is its semi-major axis,
$R_{*}$ the radius of the central star, $r_{p}$ is the radius of the
planet, and $i$ is the inclination of the plane; note that the
inclination angle is a function of the time $t$ at which the duration
is evaluated (so that the duration will also be a function of time). We also assume that orbital elements are effectively constant during a single transit, but that variations occur from transit to transit.
Substituting equation (<ref>) into this expression then yields a
measure of the transit duration, $\tau$, at any point during a
planet's secular evolution. The second order secular theory used in
this work computes motions with the evolution of inclination decoupled
from that of eccentricity, so the null eccentricity approximation for
extracting transit durations from our derived transit parameters is
sufficient. A product of our stability study of the Kepler systems is
time series of $I(t)$ and subsequently $\Delta b(t)$. From these
expressions, we can compute the times series $\tau_{T}(t)$, evaluated
at each transit epoch for each planet in a system.
Thus far, observational study of secular TDVs has been limited by two main
factors: (1) the signature of TDVs caused by even massive planets is
generally small due to small yearly changes in inclination and
eccentricity, and (2) to find TDVs to good precision, the cadence of
photometric measurements must be high enough such that durations can
be extracted from individual transits. Through TTVs can been used to
determine dynamical quantities of multi-planet systems with good success
<cit.>, TDVs in multi-planet systems are generally as much as several order of
magnitude smaller in amplitude <cit.>. However, there has been recent
success measuring the amplitude of planetary TDVs
<cit.>. Since transit duration depends on the chord a planet
takes across its star in our line of sight and oscillating inclination
can directly change this chord, secular interactions exciting
inclinations will also lead to potentially observable transit duration
Transit duration variations are thought to be one of the
few (but currently feasible) promising ways to find moons around extrasolar planets
<cit.>, as the perturbing effect of a moon would alter both
the time of center transit and the duration of said transit for a
transiting planet. Secular TDVs can also be used to constrain the oblateness
of the central body, which has been done observationally for the
KOI-13 system <cit.>. In this context, the stellar
oblateness leads to precession of the orbital elements and thereby
mimics the effects of secular interactions among multiple planets
(see equation <ref>, which depends on the stellar
oblateness $J_{2}$). In order for TDVs to be a useful method to detect exomoons or measure stellar oblateness, the amplitude due to these effects must be large compared to the intrinsic variation which we determine here. We also note that
TDVs are now being compiled from the Kepler data <cit.>,
with more data expected in the near future.
The time series $\tau_{T}(t)$ yields two useful measures: first, it yields the transit duration variation rate, which can be parameterized as $\delta \tau_{T,t}$, the change in duration per unit time (in Table 1, we parameterize this as as a variation per year. For example: a TDV of 1 sec yr$^{-1}$ would mean that over one year, the expected duration would change by one second, regardless of when or how frequently the transits occur). Second, it yields the duration variation per orbit, $\delta \tau_{T,n}$, which can be directly compared to the magnitude of other effects that can also cause TDVs.
Both of these measures provide useful constraints on the properties of the system: the yearly TDVs provide approximate limits for the signal due to secular interactions between planets only. The duration variation per orbit allows for a fit to a series of durations over time, where:
\begin{equation}
\tau_{T}(t) = \tau_{T}(0) + \delta \tau_{T,n}\ n
\label{pal}
\end{equation}
where $n$ is the number of orbits observed. If this is done, then variation accumulates as $(\Delta \tau_{T}) = \delta \tau_{T,n} \ n$ when $(\Delta \tau_{T})$ is the total change in duration over an extended baseline of time. In this case, when the time series contains $N$ independent measurements, the precision in fitting $\delta \tau_{T,n}$, as given in Equation (<ref>), is increased. The uncertainty scales like $\sigma \propto N^{-3/2}$, with one factor of $N^{-1}$ being due to the number of observed transits, a factor of $N^{-1/2}$ being due to the independent nature of these observations <cit.>. In this way, a large number of transit duration measurements can better constrain the TDV per orbit than would be possible looking at yearly drift alone using two widely separated transits <cit.>.
A histogram of the derived annual TDV values in this work, for Kepler systems with four or more coplanar planets. Data is presented in (upper panel) TDV year$^{-1}$ and (lower panel) TDV orbit$^{-1}$.
The bulk of the transit duration variations range from 0.01 to 10 seconds per orbit. The data visualized here is also given in Table 2. This histogram includes only compact mutually transiting systems with four or more planets (Kepler-37, -48, and -68 are not included).
The effect of secular interactions between planets in a multi-planet
system can occlude observations of other parameters traced by transit
durations (such as the presence of exomoons or solar oblateness), but
it can also provide evidence for additional planets in the system, as
non-transiting planets contribute to the duration variations even if
they are not directly observable.
In Table 2, we present expected yearly TDVs for each planet considered in this
These values are also presented in histograms in Figure <ref>.
Though these values are small because the yearly change in
inclination for each planet is very small, they provide limits for the
kind of TDVs expected in the observed Kepler multi-planet systems
without the presence of a perturber. The presence of a perturbing
secondary in any of these systems would lead to transit durations
outside the expected range. For example, circumbinary planets can exhibit TDVs on the order of hours <cit.>. For exomoons, the TDV amplitude is expected to scale with $M_{s} a_{s}^{-1/2}$, when $s$ denotes a satellite <cit.>. This amplitude is typically on the order of tens of seconds, being 13.7 seconds for the Earth-Moon system <cit.>. In comparison, typical values for the secular interactions within a compact system are a bit smaller (being typically between $10^{-2} - 10^{1}$ seconds per orbit).
Significant deviation in
transit durations above these predicted values would suggest the presence of an
additional effect (perturbing planet, extreme solar oblateness,
exomoon, etc.) in the system. The range of transit duration variations summarized in Figure <ref> thus serves as a baseline of the expected TDV distribution for tightly packed, coplanar, multi-planet systems.
§ PLANETARY MASS CONSTRAINTS
The observed current coplanarity of the Kepler multi-planet systems is a stringent constraint on the planets' orbital properties.
For most of the planets in the Kepler system, the ratio $R_{p} /R_{*}$ is well-known. Combined with a value of the stellar radius (determined from either spectroscopy or interferometry), this value yields a measure of the planetary radius.
To perform a dynamical analysis, these measured radii must be converted to mass. Although some Kepler planets have masses measured via long-term radial velocity surveys <cit.>, the population of four-plus planet systems generally do not have measured masses due to the difficultly of measuring masses for small planets in multi-planet systems. Much recent work has been conducted aimed at finding a mass-radius relationship for exoplanets <cit.>. When testing the CMT-stability of the compact Kepler systems, we use a supplemented version of the Wolfgang relation <cit.>. This relation introduces a large amount of scatter in density for planets that could be gaseous or rocky, which is useful for exploring the entire extent of parameter space in which the real planets could be living. However, another question that the apparent relative CMT-stability of the Kepler systems engenders is the effect of systematic mass enhancement <cit.>. To test the effect of such systematic radius errors, we will inflate the masses of the constituent planets in the Kepler compact systems and examine the dynamical and CMT-stability of the systems.
For this experiment, we make a different choice in converting radii to masses: we use conversion law $M_P = M_{\earth} (R_P/R_{\earth})^{2.1}$ inferred from results of the Kepler mission <cit.>. Using this relation removes the scatter due to composition, enabling a qualitative study of the general stability status of the Kepler multi-planet systems, without noise from differing compositions between trials.
Determining the effect of planetary mass enhancement with respect to roughly estimated values would help determine if the parameter space of CMT-stable systems (which we have shown includes all the systems in our sample) changes if the planetary masses are systematically underestimated.
To determine the extent of this parameter space, we evaluate the dynamical stability of the Kepler systems with varying mass enhancement factors, which places constraints on the maximum ratio by which the masses can be enhanced without losing the currently observing transiting configuration of the systems.
To evaluate the effect of having larger planets in each system, we performed 40 numerical simulations of each system using for each mass enhancement factor. The integration time for each system was $10^{6}$ dynamical times. This full treatment accounts for effects ignored in the secular theory such as the coupling of eccentricity and inclination, and instabilities due to orbit crossing or other effects. A mass enhancement factor describes the factor by which we increase all planetary masses within a single system. Although we alter the masses of the planets, we do not alter starting semi-major axes. The systems for each enhancement factor were created using observationally constrained orbital parameters supplemented with orbital parameters drawn from the standard priors (see Table <ref>). When a system remains CMT-stable for the entire time, this means that it is observable in transit and the system as a whole does not go dynamically unstable (e.g., by ejecting a planet).
There are two potential causes of instability in these systems. First, increased inclination oscillations can cause a some planets in a system to lie outside a mutual line of sight, even as a system remains dynamically stable. For the purposes of our analysis, we consider this to be an CMT-unstable system. Second, true dynamical instability (in the form of ejected/star-consumed planets or orbit crossing) also results in an CMT-unstable system. When either of these criteria (large inclination oscillations or true dynamical instability) is met for a certain mass enhancement factor, we categorize that system as unstable.
We parameterize the dynamical fullness of a system in terms of the surface density of a disk consisting of the mass of its constituent planets spread over an annulus with an inner radius equal to the semi-major axis of the most interior planet, and an outer radius equal to the semi-major axis of the most exterior planet:
\begin{equation}
\Sigma = \frac{\sum_{i=1}^{i=n} m_{i}}{\pi (a_{n}^{2} - a_{1}^{2})}
\label{surfden}
\end{equation}
where $n$ is the number of planets in a system, $a$ is the semi-major axis, $m$ is the planetary mass, and $i$ denotes the planet number.
In Figure <ref>, we plot the mass enhancement factor required to make a system CMT-unstable against the the surface density of the planet annuli. This plot is essentially a comparison of the dynamical fullness of the system (surface density) to the stability against excitation (mass enhancement factor required to knock a system out of transit).
The observed result appears to intuitively support that a higher surface density of material leads to a less CMT-stable system (for which a lower mass enhancement factor can excited oscillations out of the plane). The large scatter of the data could also be explained by the existence of two distinct populations (one containing the disks where planet surface density is below 200 $M_{\earth} /$ AU$^{2}$, and another where the density is above 200 $M_{\earth} /$ AU$^{2}$, where the former are significantly less sensitive to mass enhancement).
For many systems with a surface density $\Sigma >$ 200 $M_{\earth} /$ AU$^{2}$, hot or warm Jupiter-like planets would be CMT-stable even in a multiple-planet system. This finding suggests that Jovian-size planets can exist in tightly-packed multi-planet systems with semi-major axis similar to those of the discovered Kepler systems (although this result holds only for Myr timescales, as discussed below).
The mass enhancement factor required to render the systems CMT-unstable may seem higher than expected. On one hand, the integrations are carried out for only $10^{6}$ dynamical times, which generally works out to be a few million years, which is short compared to the system ages. The critical enhancement factor appropriate for the ages of the systems are thus lower, but we assume here that the short-time values provide a good relative measure of stability. On the other hand, these systems are in CMT-stable configurations, even though their surface densities are much larger than that of out solar system (the analogous value for our solar system is 0.49 $M_{\earth} /$ AU$^{2}$). For comparison, we note that the GJ 876 system (one of the most dynamically active systems discovered to date) has a surface density $\Sigma = 2750\ M_{\earth} / $ AU$^{-2}$, which is much larger than the systems considered here.
The mass enhancement factor required to knock a system out of a CMT-stable transiting configuration, plotted by the surface density of the annulus containing all the planets (which is defined in Equation <ref>). The points are shaded based on the ratio of the total planet mass to stellar mass ($M_{disk} / M_{*}$). The shape of the trend can be explained two ways. It could be explained by the existence of two distinct populations (one containing the disks where planet surface density is below 200 $M_{\earth} /$ AU$^{2}$, and another where the density is above 200 $M_{\earth} /$ AU$^{2}$, where the former are significantly less sensitive to mass enhancement), or it could be explained as a monotonic (but high-scatter) decreasing trend with surface density.
§ CONCLUSIONS
This paper has explored the dynamics of compact solar systems
undergoing oscillations in their orbital inclination angles. If such
oscillations occur with sufficient amplitude, then not all of the
planets in a multi-planet system are expected to transit at a given
epoch. By comparing the conditions required for the excitation of
inclination angles with the observed properties of compact
multi-planet systems, we can put constraints on their dynamical
history. In this work, we have provided measures of $\Delta b(t)$, the spread in impact parameters, and characterized the potential dynamical history of compact extrasolar systems. We have also utilized our method to test the dynamical and CMT-stability of a small sample of systems with additional non-transiting planets. From our derived $\Delta b(t)$, we have extracted subsequently the expected TDVs for observed systems in the case
that these systems have no additional non-transiting companions. Finally, we have explored the effect of enhancing the mass of planets in these tightly packed systems, with an aim at determining how robustly the transit stability holds as planetary masses increase.
We have done this analysis by examining the multi-planet Kepler systems with
the greatest number of transiting planets and analyzing their
long-term stability, using a combination of secular (Sections <ref>, <ref>, and <ref>) and numerical techniques (Sections <ref> and <ref>). Using the Kepler systems with the greatest number of transiting planets as our sample, we derived $\Delta b(t)$ for each planet using Monte Carlo techniques to marginalize over potential values of present orbital elements. We have determined that the compact Kepler
systems are CMT-stable against being excited into non-mutually-transiting
Compact solar systems could have configurations that allow for a
significant spread in the orbital inclinations through secular
interactions between the constituent planets (Section
<ref>). However, for the types of architectures
observed in the Kepler sample of multi-planet systems, the expected
range of inclination angles is almost always small. As shown in Figure 3, the
typical spread in the mean mutual impact parameter is typically less
than $\sim0.5$, whereas impact parameters greater than 2 are required
for planets to move out of transit. This result can also be expressed
in terms of inclination angles: self-excitation generally produces
$\Delta i \lta 0.5^\circ$, whereas angles of 1 – 2$^\circ$ are
required to compromise transit in these compact systems. As a result, for most of the systems
discovered by the Kepler mission, the self-excitation of inclination
angle oscillations is generally not large enough to prevent planets
from being observed in transit.
We have also tested the behavior of generalized Kepler systems. For these generalizations, we drew orbital parameters for each system from expanded but observationally inspired posteriors, then tested the dynamical stability. For dynamically stable analogs, we proceeded with the analysis used for the observed Kepler systems. We found that the generalized systems are experience significantly more action in mutual impact parameter excitation, resulting in these systems being on average less CMT-stable than the observed Kepler systems. The observed Kepler systems are remarkably CMT-stable, even compared to their analogs.
Our derived result that self-excitation of inclination
angle oscillations is generally not large enough to prevent planets
from being observed in transit holds for the Kepler systems, but not their analogs; even then, it has an important exception. We have also considered another type of Kepler system that contains 2
or 3 transiting planets and an additional planet not seen in transit
(where the additional body was discovered by radial velocity
follow-up). Kepler 48 and Kepler 68 are examples of this type of system.
These systems are CMT-unstable to significant oscillations in inclination
angle, so that the expected spread in inclination angle is generally
large enough to move planets out of transit. We found this result by
secularly evolving these systems after starting them in a nearly
coplanar configuration. Even starting roughly coplanar, the magnitude of these systems' self-excitation is large enough that not all planets can be seen in transit simultaneously for most of each system's orbital history. This finding indicates that the current Kepler systems with
non-transiting companions could have started roughly coplanar and subsequently had some of their planets
excited out of the plane via dynamical interactions between the planets that we know about. Specifically, it is not necessary to introduce additional bodies into these systems to recreate the currently observed architectures.
We have focused on the secular interactions of compact systems of
planets, and derived observables corresponding to the current known
properties of these systems. These observables, the transit duration
variations for Kepler systems with the observationally determined
properties, are given in Table 2. Implicit in the motivation behind the
calculation of these TDVs is the idea that there could be additional
bodies in the systems we are considering, leading to true TDVs
deviating from those that we have found here. An additional massive
companion or an exomoon, for example, could cause transit duration variations with a larger amplitude than those derived in this work. If future
observations of TDVs in these systems are vastly different than
expected, it could potentially be evidence for either an exomoon or additional, exterior,
non-transiting bodies in these compact systems.
We have also explored the effect of planetary mass enhancement in these systems. The stability of systems is related to how much the constituent planets' masses must be enhanced to result in a system that will no longer mutually transit. Generally, systems with higher effective surface density (calculated by spreading the mass of discovered planets within an annulus with inner and outer radii equal to the inner and outer planet's orbital radii) do not allow mass enhancement factors as high as those with lower surface density. This result suggests that dynamically `full' systems would not be mutually transiting if they hosted Jovian-mass planets. However, some systems with
lower surface densities would be CMT-stable in a transiting configuration
even with Jovian-mass planets (at least over time scales of $\sim10$
Myr), indicating that it might be possible to see multi-transiting
compact systems with Jovian-mass planets if they existed.
The stability boundaries – over longer
time scales – should be explored further in future work.
Spreads in the inclination angles in compact systems can be
produced by a variety of astronomical processes, in addition to those considered in this work. Excitation by the
compact solar system planets themselves (with semi-major axes
$a\lta0.5$ AU) is not generally a significant effect, but we have not (yet)
calculated the effect caused by possible additional bodies in the
outer part of the solar system (where $a\approx5-30$ AU). Since planet
formation is a relatively efficient process, the additional giant
planets, not seen in transit by Kepler, are not only possible but
likely. The orbits of these outer planets can be endowed with high
inclination angles through a variety of dynamical mechanisms. For
example, most solar systems form within clusters, and inclinations can
be excited through dynamical interactions between solar systems and
other cluster members <cit.>. In
addition, a range of inclination angles can be realized through the
formation of planets in warped disks. The observed angular momentum
vectors in star-forming cores do not point in the same direction as a
function of radius <cit.>. This heterogeneity can
lead to differences in angular momentum vector of the disk plane as a
function of radius (for disks produced through collapse of the cores),
which in turn will influence the inclination angles of forming
planets <cit.>. These various mechanisms can lead to inclined, massive, outer
secondaries to the compact systems that we have considered in this
work. The presence of such secondaries would alter the stability of
these systems, and this effect could be evident in the TDVs.
Additionally, it is possible that a system of planets would have only some planets mutually transiting, instead of the condition of all planets in a system transiting that we have considered in this work. For example, although we see four planets in a system discovered by Kepler, it is possible that another short-period companions exists in such a system, resulting in our picture of the system being incomplete. Extensions on our calculation that account for this possibility could potentially be explored by using techniques such as the semi-analytical code CORBITS (Brakensiek & Ragozzine, 2015).
In summary, we have determined that self-excitation is not usually a dominant mechanism in
exciting mutual inclination in tightly packed, multi-planet
systems. Self-excitation does operate in some solar system architectures, where Kepler-48 and Kepler-68 are prime examples. Subsequent analysis of the effect of perturbing secondaries
and stellar fly-bys in a dense cluster environment will complete the
picture of how and when mutual inclinations are excited in
exoplanetary systems.
§ ACKNOWLEDGEMENTS
We would like to thank Konstantin Batygin, Kathryn Volk, Ben Montet, Andrew
Vanderburg, and
Doug Lin
for useful conversations. We would like to additionally thank Konstantin Batygin for his careful review of the manuscript and helpful suggestions.
We would like to thank the referee, Darin Ragozzine, for his thoughtful and helpful suggestions.
J.B. is supported by the
National Science Foundation Graduate Research Fellowship, Grant
No. DGE 1256260.
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Transit Duration Variations for Kepler Compact Systems
Planet Orbital Period, days $\tau_{T,n}$, TDV (s yr$^{-1}$) $\tau_{T,t}$, TDV (s orbit$^{-1}$)
Kepler 11 10.3039 9.322660729 0.263177435
Kepler 11 13.0241 8.478858911 0.302546593
Kepler 11 22.6845 27.49182915 1.708598352
Kepler 11 31.9996 9.333648315 0.818282226
Kepler 11 46.6888 47.07565441 6.021659764
Kepler 11 118.3807 195.5615046 63.42659673
Kepler 20 3.6961219 1.914759018 0.019389542
Kepler 20 6.098493 9.46333942 0.158115368
Kepler 20 10.854092 4.0055073 0.119112725
Kepler 20 19.57706 50.15850045 2.690290337
Kepler 20 77.61184 73.36444964 15.59986281
Kepler 24 4.244384 0.743325382 0.008643722
Kepler 24 8.1453 1.056218584 0.023570458
Kepler 24 12.3335 1.232153241 0.041634964
Kepler 24 18.998355 3.605506001 0.187667625
Kepler 26 3.543919 1.859049714 0.018050196
Kepler 26 12.2829 3.710922038 0.124879135
Kepler 26 17.2513 6.100397395 0.28832818
Kepler 26 46.827915 37.7182322 4.839085401
Kepler 32 0.74296 2.028539272 0.004129106
Kepler 32 2.896 0.980507587 0.007779589
Kepler 32 5.90124 1.7046195 0.027559914
Kepler 32 8.7522 3.609925168 0.08656106
Kepler 32 22.7802 5.932320689 0.370245073
Kepler 33 5.66793 4.848962172 0.075297474
Kepler 33 13.17562 5.861904438 0.211600617
Kepler 33 21.77596 2.870395649 0.171248276
Kepler 33 31.7844 10.62434609 0.925173879
Kepler 33 41.02902 10.88222417 1.223252036
Kepler 49 2.576549 1.222377872 0.008628812
Kepler 49 7.2037945 2.229154696 0.043995541
Kepler 49 10.9129343 3.964566165 0.118534384
Kepler 49 18.596108 16.67313555 0.84946693
Kepler 55 2.211099 0.74535018 0.004515186
Kepler 55 4.617534 2.075484138 0.026256489
Kepler 55 10.198545 7.323822446 0.204636528
Kepler 55 27.9481449 7.955549215 0.609158472
Kepler 55 42.1516418 23.59027592 2.724298248
Kepler 62 5.714932 6.558019889 0.102681199
Kepler 62 12.4417 10.787751 0.367720443
Kepler 62 18.16406 7.847335247 0.390519091
Kepler 62 122.3874 126.4774292 42.40888689
Kepler 62 267.291 200.7871286 147.0372394
Kepler 79 13.4845 12.8033798 0.473005959
Kepler 79 27.4029 28.46032201 2.136699611
Kepler 79 52.0902 20.63076067 2.944275204
Kepler 79 81.0659 89.08814913 19.78633147
Kepler 80 3.072186 2.130009061 0.017928175
Kepler 80 4.645387 1.943170254 0.024730898
Kepler 80 7.053 1.714628021 0.03313225
Kepler 80 9.522 1.971262692 0.051425653
Kepler 82 2.382961 0.965129741 0.006301004
Kepler 82 5.902206 2.222408585 0.035937297
Kepler 82 26.444 17.76864363 1.287326061
Kepler 82 51.538 10.89753388 1.538731784
Kepler 84 4.224537 4.439153917 0.051379096
Kepler 84 8.726 2.002291695 0.047868486
Kepler 84 12.883 3.658674592 0.129136177
Kepler 84 27.434389 10.50305215 0.789437858
Kepler 84 44.552169 29.82330904 3.640255081
Kepler 85 8.306 1.592408413 0.036237108
Kepler 85 12.513 2.059758945 0.070613051
Kepler 85 17.91323 15.62065426 0.766620199
Kepler 85 25.216751 26.0037509 1.796520854
Transit Duration Variations for Kepler Compact Systems (continued)
Planet Orbital Period, days $\tau_{T,n}$, TDV (s yr$^{-1}$) $\tau_{T,t}$, TDV (s orbit$^{-1}$)
Kepler 90 7.008151 44.64259007 0.857156198
Kepler 90 8.719375 51.72347308 1.235606461
Kepler 90 59.73667 122.8193078 20.10086701
Kepler 90 91.93913 163.2018666 41.10859624
Kepler 90 124.9144 196.2255593 67.15451508
Kepler 90 210.60697 160.6239691 92.6808971
Kepler 90 331.60059 200.873026 182.492093
Kepler 102 5.28696 21.67804261 0.314002587
Kepler 102 7.07142 5.624323905 0.108964265
Kepler 102 10.3117 3.349150204 0.094617622
Kepler 102 16.1457 0.736673755 0.032586612
Kepler 102 27.4536 42.68179251 3.210325641
Kepler 106 6.16486 3.817466082 0.064477107
Kepler 106 13.5708 7.56602341 0.281306823
Kepler 106 23.9802 35.33153826 2.321253024
Kepler 106 43.8445 18.90988445 2.271491586
Kepler 107 3.179997 0.746863983 0.006506918
Kepler 107 4.901425 0.979622681 0.013154924
Kepler 107 7.958203 4.806241849 0.104791913
Kepler 107 14.749049 1.19795885 0.048407545
Kepler 122 5.766193 3.958098992 0.062529213
Kepler 122 12.465988 0.796688261 0.027209606
Kepler 122 21.587475 24.6377173 1.457167415
Kepler 122 37.993273 64.29884982 6.692941794
Kepler 150 3.428054 1.867968128 0.017543824
Kepler 150 7.381998 1.135021003 0.022955405
Kepler 150 12.56093 4.078868983 0.140368186
Kepler 150 30.826557 18.32964457 1.548054337
Kepler 169 3.250619 2.003755495 0.017845057
Kepler 169 6.195469 3.473185364 0.058953458
Kepler 169 8.348125 4.542706201 0.103898847
Kepler 169 13.767102 4.044452569 0.152549017
Kepler 169 87.090195 21.57958002 5.148958444
Kepler 172 2.940309 0.46849388 0.003774019
Kepler 172 6.388996 0.965003916 0.016891524
Kepler 172 14.627119 3.850997739 0.154326033
Kepler 172 35.118736 8.863030127 0.852762781
Kepler 186 3.8867907 2.174369234 0.023154296
Kepler 186 7.267302 2.671048208 0.053181682
Kepler 186 13.342996 5.6398171 0.206170019
Kepler 186 22.407704 21.23861165 1.303858968
Kepler 186 129.9441 127.2638103 45.30734602
Kepler 197 5.599308 1.746693303 0.026795271
Kepler 197 10.349695 1.664512376 0.047197796
Kepler 197 15.677563 2.946426313 0.126555573
Kepler 197 25.209715 19.27829892 1.331508004
Kepler 208 4.22864 0.327987695 0.003799841
Kepler 208 7.466623 1.085357765 0.022202623
Kepler 208 11.131786 2.145926971 0.065446575
Kepler 208 16.259458 1.939023031 0.086376612
Kepler 215 9.360672 5.214125094 0.133719767
Kepler 215 14.667108 7.140540403 0.286934458
Kepler 215 30.864423 13.54786835 1.145608602
Kepler 215 68.16101 73.36673268 13.70068658
Kepler 220 4.159807 1.08595549 0.012376343
Kepler 220 9.034199 0.726348102 0.017978009
Kepler 220 28.122397 31.65200923 2.438713341
Kepler 220 45.902733 28.54622634 3.589999468
Kepler 221 2.795906 1.622302263 0.012426862
Kepler 221 5.690586 1.258209005 0.019616292
Kepler 221 10.04156 3.116385606 0.085735269
Kepler 221 18.369917 7.160429778 0.360373975
Transit Duration Variations for Kepler Compact Systems (continued)
Planet Orbital Period, days $\tau_{T,n}$, TDV (s yr$^{-1}$) $\tau_{T,t}$, TDV (s orbit$^{-1}$)
Kepler 223 7.384108 6.794688993 0.1374595
Kepler 223 9.848183 4.741440907 0.12793035
Kepler 223 14.788759 3.494873231 0.141602296
Kepler 223 19.721734 11.9376658 0.645017725
Kepler 224 3.132924 1.856809099 0.015937649
Kepler 224 5.925003 0.971515147 0.015770494
Kepler 224 11.349393 3.646370338 0.113381068
Kepler 224 18.643577 13.12338292 0.67032
Kepler 235 3.340222 0.564712568 0.00516785
Kepler 235 7.824904 7.345399102 0.15747135
Kepler 235 20.060548 6.816955222 0.374662623
Kepler 235 46.183669 39.50423791 4.998494925
Kepler 238 2.090876 0.884749492 0.005068223
Kepler 238 6.155557 2.172310058 0.036635009
Kepler 238 13.233549 3.287882421 0.119206447
Kepler 238 23.654 1.177061595 0.076280041
Kepler 238 50.447 39.55458755 5.466877474
Kepler 251 4.790936 3.555780556 0.04667265
Kepler 251 16.514043 6.199763248 0.2805018
Kepler 251 30.133001 10.65335888 0.879500476
Kepler 251 99.640161 96.79454965 26.4236288
Kepler 256 1.620493 0.323635851 0.001436848
Kepler 256 3.38802 0.427493436 0.003968099
Kepler 256 5.839172 0.884567462 0.014151073
Kepler 256 10.681572 2.074977703 0.060723353
Kepler 265 6.846262 3.529407786 0.066200686
Kepler 265 17.028937 6.886638378 0.32129351
Kepler 265 43.130617 31.42320459 3.713156719
Kepler 265 67.831024 60.40531736 11.22562885
Kepler 282 9.220524 15.78120361 0.398660183
Kepler 282 13.638723 20.28678008 0.758043217
Kepler 282 24.806 25.75148128 1.750112999
Kepler 282 44.347 26.5277244 3.223082175
Kepler 286 1.796302 0.713292306 0.003510379
Kepler 286 3.468095 1.26801456 0.012048205
Kepler 286 5.914323 3.526176454 0.05713684
Kepler 286 29.221289 13.69225381 1.09617892
Kepler 296 5.841648 2.228023946 0.035658443
Kepler 296 10.21457 15.68127047 0.438842287
Kepler 296 19.850242 16.85583517 0.916691527
Kepler 296 34.14204 64.98518257 6.078703295
Kepler 296 63.336 82.07825957 14.24248945
Kepler 299 2.927128 1.224694325 0.009821471
Kepler 299 6.885875 1.810056081 0.034147452
Kepler 299 15.054786 12.26546764 0.505901344
Kepler 299 38.285489 27.25944274 2.859290672
Kepler 306 4.646186 3.606915153 0.045913421
Kepler 306 7.240193 3.698312183 0.073360257
Kepler 306 17.326644 10.59158411 0.502785225
Kepler 306 44.840975 39.39149661 4.839323602
Kepler 338 9.341 3.640821013 0.093175093
Kepler 338 13.726976 3.068493345 0.115400369
Kepler 338 24.310856 7.040320215 0.468921126
Kepler 338 44.431014 13.81305425 1.681446594
Kepler 341 5.195528 1.217989712 0.017337259
Kepler 341 8.01041 1.039299363 0.022808806
Kepler 341 27.666313 22.75719988 1.724952917
Kepler 341 42.473269 12.26030004 1.426671292
Kepler 402 4.028751 1.081566673 0.01193798
Kepler 402 6.124821 0.917643194 0.015398357
Kepler 402 8.921099 3.453650264 0.084411934
Kepler 402 11.242861 2.886419391 0.088908526
Transit Duration Variations for Kepler Compact Systems (continued)
Planet Orbital Period, days $\tau_{T,n}$, TDV (s yr$^{-1}$) $\tau_{T,t}$, TDV (s orbit$^{-1}$)
Kepler 444 3.6001053 2.256350153 0.022255063
Kepler 444 4.5458841 1.40080575 0.017446303
Kepler 444 6.189392 1.916669517 0.032501422
Kepler 444 7.743493 2.784776198 0.059079164
Kepler 444 9.740486 1.368811474 0.036528463
Predicted values of the transit duration variations (TDVs) for
the current sample of Kepler compact systems containing only the planets that have been discovered so far. Duration variations are presented both per orbit as well as per year. Errors are typically on the order of 1% of reported values, but are not reported for brevity.
I_int, i cosΩ_i =
∑_k=1^N T_i [
ℛ(𝐲_1, ϵ)cosγ_ϵ -
ℐ(𝐲_1, ϵ) sinγ_ϵ
I_int, i sinΩ_i = ∑_ϵ= 1^N T_i [
ℛ(𝐲_1, ϵ)cosγ_ϵ +
ℐ(𝐲_1, ϵ) sinγ_ϵ
when $j$ denotes the planet in question, $I_{ji} = T_{i} \bar{I_{ji}}$
is the scaled eigenvector (when $\bar{I_{ji}}$ is the unscaled version
of y), $\gamma_{i}$ is the phase, and $N$ is the total number
of planets.
|
1511.00366
|
Long wavelength limit
for the quantum Euler-Poisson equation
Huimin Liu Department of Mathematics, Chongqing University, Chongqing 401331, P.R.China
Xueke Pu Department of Mathematics, Chongqing University, Chongqing 401331, P.R.China [email protected]
This work is supported in part by NSFC (11471057) and Natural Science Foundation Project of CQ CSTC (cstc2014jcyjA50020).
[2000]35M20; 35Q35
In this paper, we consider the long wavelength limit for the quantum Euler-Poisson equation. Under the Gardner-Morikawa transform, we derive the quantum Korteweg-de Vries (KdV) equation by a singular perturbation method. We show that the KdV dynamics can be seen at time interval of order $O(\epsilon^{-3/2})$. When the nondimensional quantum parameter $H=2$, it reduces to the inviscid Burgers equation.
§ INTRODUCTION
In this paper, we consider a one-dimensional two species quantum plasma system made by one electronic and one ionic fluid, in the electrostatic approximation <cit.>. For simplicity, we only consider the continuity and momentum equations and ignore the energy transport equation, which are sufficient to describe the classical ion-acoustic waves <cit.>. The system is governed by the following equations
∂_tu_e+u_e∂_xu_e =e/m_e∂_xϕ-1/m_en_e∂_xP
+ħ^2/2m_e^2 ∂_x (∂_x^2√(n_e)/√(n_e)),
where $n_{e,i}$ are the electronic and ionic number densities, $u_{e,i}$ the electronic and ionic velocities, $\phi$ the scalar potential, $m_{e,i}$ the electron and ion masses, $-e$ the electron charge, $\hbar=\frac{h}{2\pi}$, where $h$ is Planck's constant and $\epsilon_{0}$ the vacuum permittivity. The electron fluid pressure $P=P(n_{e})$, modeled by the equation of state for a one dimensional zero-temperature Fermi gas, is given by
\begin{equation}\label{equ1'''}
\begin{split}
\end{split}
\end{equation}
where $n_{0}$ is the equilibrium density for both electrons and ions, and $v_{F_{e}}$ is the electrons Fermi velocity, related to the Fermi temperature $T_{F_{e}}$ by $m_{e}v_{F_{e}}^{2}=\kappa_{B}T_{F_{e}}$, where $\kappa_{B}$ is the Boltzmann constant. Throughout this paper, we assume such a cubic law for the electron fluid pressure, which is the most important significant physical case, as pointed out by Jackson <cit.>.
Equations (<ref>) and (<ref>) represent conservation of charge and mass. Equations (<ref>) and (<ref>) account for momentum balance. The third order term in (<ref>), proportional to $\hbar^{2}$, takes into account the influence of quantum diffraction effects. However, the motion of ion can be taken as classical in view of the high ion mass in comparison to the electron mass. Accordingly, (<ref>) contains no quantum terms. Finally, (<ref>) is Poisson's equation, describing the self-consistent electrostatic potential.
Take the following rescaling,
\begin{equation}\label{equ11''}
\begin{split}
&\bar{x}=\frac{\omega_{p_{e}}x}{v_{F_{e}}}, \ \ \bar{t}=\omega_{p_{i}}t,\ \ \bar{n_{e}}=\frac{n_{e}}{n_{0}}, \ \ \bar{n_{i}}=\frac{n_{i}}{n_{0}}, \\
&\bar{u_{e}}=\frac{u_{e}}{c_{s}}, \ \ \bar{u_{i}}=\frac{u_{i}}{c_{s}}, \ \ \bar{\phi}=\frac{e\phi}{\kappa_{B}T_{F_{e}}},
\end{split}
\end{equation}
where $\omega_{p_{e}}$ and $\omega_{p_{i}}$ are the corresponding electron and ion plasma frequencies and $c_s$ is the quantum ion-acoustic velocity, given by
\begin{equation}\label{equ12''}
\begin{split}
\omega_{p_{e}}=\left(\frac{n_{0}e^{2}}{m_{e} \epsilon_{0}}\right)^{1/2}, \ \ \omega_{p_{i}}=\left(\frac{n_{0}e^{2}}{m_{i} \epsilon_{0}}\right)^{1/2}, \ \ c_{s}=\left(\frac{\kappa_{B}T_{F_{e}}}{m_{i}}\right)^{1/2}.
\end{split}
\end{equation}
In addition, consider nondimensional parameter $H={\hbar\omega_{p_{e}}}/{\kappa_{B}T_{F_{e}}}$. Physically, $H$ is the ratio between the electron plasmon energy and the electron Fermi energy. Using the new variables and dropping bars for simplifying natation, we obtain from (<ref>)
\begin{equation}\label{equ14'}
\begin{split}
\frac{m_{e}}{m_{i}}(\partial_{t}u_{e}+u_{e}\partial_{x}u_{e})=\partial_{x}\phi-n_{e}\partial_{x}n_{e}
\end{split}
\end{equation}
Since ${m_{e}}/{m_{i}}\ll1$, we let the left-hand side of (<ref>) to be zero and then integrate about $x$ with the boundary conditions $n_{e}=1$, $\phi=0$ at infinity, to obtain
\begin{equation}\label{equ14''}
\begin{split}
\phi=-\frac{1}{2}+\frac{1}{2}n_{e}^{2}-\frac{H^{2}}{2\sqrt{n_{e}}}\partial_{x}^{2}\sqrt{n_{e}}.
\end{split}
\end{equation}
This last equation is the electrostatic potential in terms of the electron density and its derivatives. Even when the quantum diffraction effects are negligible $(H=0)$, the electron equilibrium is given by a Fermi-Dirac distribution and not by a Maxwell-Boltzmann one.
Applying the rescaling (<ref>) to (<ref>), (<ref>) and (<ref>), we have by dropping the bars
Equations (<ref>)-(<ref>), together with (<ref>), provide a reduced model of four equations with four unknown quantities, $n_{i}$, $u_{i}$, $n_{e}$ and $\phi$. This reduced model is the basic model to be studied in the following, which will lead to the quantum Korteweg-de Vries (KdV) equation (<ref>) under the Gardner-Morikawa transform <cit.>.
Obviously, the reduced system (<ref>)-(<ref>) admits the homogeneous equilibrium solution $(n_e,n_i,u_i,\phi)=(1,1,0,0)$. Global existence of smooth solutions around the equilibrium is an outstanding difficult problem for the Euler-Poisson problem. Without quantum effects, Guo <cit.> firstly obtained global irrotational solutions with small velocity for the 3D electron fluid, based on the Klein-Gordon effect. Then, Jang, Li, Zhang and Wu <cit.> obtained global smooth small solutions for the 2D electron fluid in Euler-Poisson system. Very recently, Guo, Han and Zhang <cit.> finally completely settled this problem and proved that no shocks form for the 1D Euler-Poisson system for electrons. For Euler-Poisson equation for ions, Guo and Pausader <cit.> constructed global smooth irrotational solutions with small amplitude for ion dynamics. For the Euler-Poisson system (<ref>) with quantum effects, there is no existence result, to the best knowledge of the authors.
To access weakly nonlinear solutions for the quantum ion-acoustic system (<ref>)-(<ref>), a singular perturbation method can be applied to the weakly nonlinear classical waves, which finally leads to the quantum KdV equation. For details, see Section 2. To this aspect, one may refer to the recent papers <cit.>. In particular, Guo and Pu established rigorously the KdV limit for the ion Euler-Poisson system in 1D for both the cold and hot plasma case, where the electron density satisfies the classical Maxwell-Boltzmann law. This result was generalized to the higher dimensional case in <cit.>, and the 2D Kadomtsev-Petviashvili-II (KP-II) equation and the 3D Zakharov-Kuznetsov equation are derived for well-prepared initial data under different scalings. Almost at the same time, <cit.> also established the KdV limit in 1D and the Zakharov-Kuznetsov equation in 3D from the Euler-Poisson system. Han-Kwan <cit.> also introduced a long wave scaling for the Vlasov-Poisson equation and derived the KdV equation in 1D and the Zakharov-Kuznetsov equation in 3D using the modulated energy method. For other studies for the Euler-Poisson system or related models, the interested readers may refer to <cit.>, to list only a few. For derivation of the KdV equation from the water waves without surface tension, see <cit.> and the references therein.
In the present paper, we will continue to study the long wavelength limit for the reduced system (<ref>) for ions with quantum effects. Under the Gardner-Morikawa transform, the quantum KdV equation is derived when $H>0$ and $H\neq2$. But when $H=2$, the quantum KdV equation (<ref>) reduces to the inviscid Burger's equation. The formal derivation of the quantum KdV equation can be found in <cit.> and is given in the next section. The main interest in this paper is to make such a formal derivation rigorous. To do so, we need to obtain uniform (in $\epsilon$) estimates for the remainders $\left(n_{eR}^{\epsilon},n_{iR}^{\epsilon}, u_{iR}^{\epsilon}\right)$ and then recover the uniform estimates of $\phi^{\epsilon}_R$ from the relation (<ref>). To apply the Gronwall inequality to complete the proof, we define the triple norm
\begin{equation}\label{tri}
\begin{split}
|\!|\!|(N_i,N_{e},U)|\!|\!|_{\epsilon}^{2}= &\|(N_i,N_{e},U)\|_{H^{2}}^{2} +\epsilon\|(\partial_{x}^{3}N_{e}, \partial_{x}^{3}U)\|_{L^{2}}^{2}\\
&+\epsilon^{2}\|\partial_{x}^{4}N_{e}\|_{L^{2}}^{2} +\epsilon^{3}\|\partial_{x}^{5}N_{e}\|_{L^{2}}^{2}
\end{split}
\end{equation}
which depends on the parameter $\epsilon$ in the Gardner-Morikawa transform. But we regard $H$ as a fixed constant. After careful computations, we finally close the estimates in this triple norm, which gives uniform (in $\epsilon$) estimates for the remainders $(N_i,N_{e},U)$ in $H^2$ and completes the proof. The main result is stated in Theorem <ref>. Furthermore, this implies that
\begin{equation}
\sup_{[0,\epsilon^{-3/2}\tau]}
\left\|\left(
\begin{array}{c}
(n_{i}-1)/\epsilon\\ (n_{e}-1)/\epsilon\\ u_{i}/\epsilon
\end{array}
\right)-KdV\right\|_{H^2}\leq C\epsilon,
\end{equation}
for some $C>0$ independent of $\epsilon>0$, for any fixed $\tau>0$ of order $O(1)$. Here the `KdV' stands for the first approximation of $(n_i,n_e,u_i)$ under the Gardner-Morikawa transform in (<ref>). It shows that the KdV dynamics can be seen at time interval of order $O(\epsilon^{-3/2})$. The result also applies to the case when $H=2$, where the inviscid Burger's equation is derived.
The results in this paper can be generalized to the following general cases. Firstly, for definiteness, we let the electron pressure satisfies the cubic law in (<ref>), but the result in this paper can be generalized to general $\gamma$-law, which will lead to a different relation between $\phi$ and $n_e$ in (<ref>). Secondly, the ion momentum equation (<ref>) does not contain ion pressure, which generally depends on ion density with the form $P_i(n_i)=T_i\ln n_i$. This paper corresponds to the cold ion case $T_i=0$. But the result in this paper can be generalized to general case $T_i>0$, and indeed, the proof will be slightly simpler since in this case, the system is Friedrich symmetrizable. The result in this paper can be also generalized to the general $\gamma$-law of the ion pressure, i.e., when $P_i(n_i)=T_in_{i}^{\gamma}$ for $\gamma\geq1$. For clarity, we will not mention these general cases in the rest of the paper and concentrate on the case $P(n_e)\sim n_e^3$ in (<ref>) and zero ion temperature case $T_i=0$.
This paper is organized as follows. In Section 2, we present the formal derivation of the quantum KdV equation (<ref>) and state the main result in Theorem <ref>. In Section 3, we present uniform estimates for the remainders in (<ref>). The main estimates are stated in Proposition <ref> and <ref>. Finally, we complete the proof in Section 4.
§ FORMAL EXPANSION AND MAIN RESULTS
§.§ Formal KdV expansion
By the classical Gardner-Morikawa transformation <cit.>
\begin{equation}\label{equ39}
x\rightarrow\epsilon^{\frac{1}{2}}(x-t),\ \,t\rightarrow\epsilon^{\frac{3}{2}}t,
\end{equation}
we obtain from (<ref>) the parameterized system
\begin{equation}\label{equ2}
\begin{cases}
\epsilon\partial_{t}n_{i}-\partial_{x}n_{i} +\partial_{x}(n_{i}u_{i})=0,\\
\epsilon\partial_{t}u_{i}-\partial_{x}u_{i} +u_{i}\partial_{x}u_{i}=-\partial_{x}\phi,\\
\epsilon\partial_{x}^{2}\phi=n_{e}-n_{i},
\end{cases}
\end{equation}
where $\epsilon$ is the amplitude of the initial disturbance and is assumed to be small compared with unity and (<ref>) is rescaled into the following relation
$$\phi=-\frac{1}{2}+\frac{1}{2}n_{e}^{2} -\frac{\epsilon H^{2}}{2\sqrt{n_{e}}}\partial_{x}^{2}\sqrt{n_{e}}.$$
We consider the following formal expansion around the equilibrium solution $(n_{i},n_{e},u_{i})=(1,1,0)$,
\begin{equation}\label{expan-formal}
\begin{cases}
n_{i}=1+\epsilon n_{i}^{(1)}+\epsilon^{2}n_{i}^{(2)}+\epsilon^{3}n_{i}^{(3)}+\epsilon^{4}n_{i}^{(4)}+\cdots,\\
n_{e}=1+\epsilon n_{e}^{(1)}+\epsilon^{2}n_{e}^{(2)}+\epsilon^{3}n_{e}^{(3)}+\epsilon^{4}n_{e}^{(4)}+\cdots,\\
u_{i}=\epsilon u_{i}^{(1)}+\epsilon^{2}u_{i}^{(2)}+\epsilon^{3}u_{i}^{(3)}+\epsilon^{4}u_{i}^{(4)}+\cdots.
\end{cases}
\end{equation}
Plugging (<ref>) into (<ref>), we get a power series of $\epsilon$, whose coefficients depend on $(n_{i}^{(k)},n_{e}^{(k)},u_{i}^{(k)})$ for $k=1,2,\cdots$.
At the order $O(1)$, the coefficients are automatically balanced.
At the order $O(\epsilon)$, we obtain
This enables us to assume the relation
\begin{equation}\label{equ3}
{(\mathcal L_1):\ \ \ \ }
\end{equation}
which makes (<ref>) valid and shows that the mode is quasi-neutral in a first approximation. Then only $n_{i}^{(1)}$ needs to be determined.
At the order $O(\epsilon^2)$, we obtain
Differentiating (<ref>) with respect to $x$, and then adding the resultant and (<ref>) to (<ref>) together, we deduce that $n_{i}^{(1)}$ satisfies the quantum Kortweg-de Vries equation
\begin{equation}\label{kdv}
\partial_{t}n_{i}^{(1)}+2n_{i}^{(1)}\partial_{x}n_{i}^{(1)}+\frac{1}{2}(1-\frac{H^{2}}{4})\partial_{x}^{3}n_{i}^{(1)}=0,
\end{equation}
where we have used the relation (<ref>). We note that the system (<ref>), (<ref>) are self-contained, which do not depend on $(n_{i}^{(j)}, n_{e}^{(j)}, u_{i}^{(j)})$ for $j\geq2$. We also note that (<ref>) is different from the classical KdV equation due to the presence of the parameter $H$. When $H\neq2$, it can be transformed into the classical KdV equation, while when $H=2$, it reduces to the inviscid Burger's equation, which is drastically different from the KdV equation. For derivation of KdV from water waves, see <cit.>.
Much of the properties of the KdV equation follow from the interplay between advection and dispersion. One can see that the quantum effects can even invert the sign of dispersion for large $H$. However, this sign is immaterial since we can apply the transform $t\to-t,x\to x, n_i^{(1)}\to-n_i^{(1)}$. This implies that for $H>2$, the localized solutions (bright solitons) with $n_i^{(1)}>0$ of the original equation correspond also to localized solutions, but with inverted polarization ($n_i^{(1)}<0$, dark solitons) and propagating backward in time. But when $H=2$, the dispersive term vanishes, which eventually yields the formation of a shock in the Burger's equation. For details of the solitons, one may refer to <cit.>.
When $H\neq2$, we have the following existence theorem <cit.>.
Let $H\neq2$ and $\tilde s_1\geq2$ be a sufficiently large integer. Then for any given initial data $n_{i0}^{(1)}\in H^{\tilde s_1}(\mathbb R)$, there exists $\tau_*>0$ such that the initial value problem (<ref>) has a unique solution
$n_{i}^{(1)}\in L^{\infty}\big(-\tau_*,\tau_*; H^{\tilde s_1}(\Bbb R)\big)$. Furthermore, by using the conservation laws of the KdV equation, we can extend the solution to any time interval $[-\tau,\tau]$.
There is also an existence theorem for $H=2$, see <cit.>.
Let $H=2$ and $\tilde s_2\geq2$ be a sufficiently large integer. Then for any given initial data $n_{i0}^{(1)}\in H^{\tilde s_2}(\Bbb R)$, there exists $\tilde{\tau}_*>0$ such that the initial value problem (<ref>) with $H=2$ has a unique solution $n_{i}^{(1)}\in L^{\infty}\big(0,\tilde{\tau_*}; H^{\tilde s_2}(\Bbb R)\big)$ with initial data $n_{i0}^{(1)}$.
To find out the equation satisfied by $(n_{i}^{(2)},n_{e}^{(2)},u_{i}^{(2)})$ assuming $(n_{i}^{(1)},n_{e}^{(1)},u_{i}^{(1)})$ is known form (<ref>) and (<ref>), we express $(n_{i}^{(2)},n_{e}^{(2)},u_{i}^{(2)})$ in terms of $(n_{i}^{(1)},n_{e}^{(1)},u_{i}^{(1)})$ from (<ref>),
u_i^(2)=n_i^(2)+g^(1), g^(1)=-∫_0^x𝔤^(1)(t,ξ)dξ,
which makes (<ref>) valid. Thus only $n_{i}^{(2)}$ needs to be determined.
At the order $O(\epsilon^3)$, we obtain
Differentiating (<ref>) with respect to $x$, and then adding the resultant and (<ref>) to (<ref>) together, we deduce that $n_{i}^{(2)}$ satisfies the linearized inhomogeneous quantum KdV equation
\begin{equation}\label{kdv1}
\partial_{t}n_{i}^{(2)}+2\partial_{x}(n_{i}^{(1)}n_{i}^{(2)})+\frac{1}{2}(1-\frac{H^{2}}{4})\partial_{x}^{3}n_{i}^{(2)}=G^{(1)},
\end{equation}
where we have used (<ref>) and $G^{(1)}$ depending on only $n_{i}^{(1)}$. Again, the system (<ref>) and (<ref>) for $(n_{i}^{(2)},n_{e}^{(2)},u_{i}^{(2)})$ are self contained, which do not depend on $(n_{i}^{(j)},n_{e}^{(j)},u_{i}^{(j)})$ for $j\geq3$.
Inductively, at the order $O(\epsilon^{k})$, we obtain a system $(\mathcal S_{k-1})$ for $(n_{i}^{(k-1)},n_{e}^{(k-1)},u_{i}^{(k-1)})$, from which we obtain
\begin{equation}\label{relation}
{(\mathcal L_k):}\ \ \ \
n_{e}^{(k)}=n_{i}^{(k)}+h^{(k-1)},\ \ \ u_{i}^{(k)}=n_{i}^{k}+g^{(k-1)},
\end{equation}
where $h^{(k-1)}$ and $g^{(k-1)}$ depend only on $(n_{e}^{(j)})$ for $1\leq j\leq k-1$. Thus we need only to determine $n_{i}^{(k)}$. At the order $O(\epsilon^{k+1})$, we obtain a system $(\mathcal{S}_{k})$ for $(n_{i}^{(k)},n_{e}^{(k)},u_{i}^{(k)})$, from which we obtain the linearized inhomogeneous KdV equation for $n_{i}^{(k)}$,
\begin{equation}\label{linearized}
\begin{split}
\partial_{t}n_{i}^{(k)}+2\partial_{x}(n_{i}^{(1)}n_{i}^{(k)})+\frac{1}{2}(1-\frac{H^{2}}{4})\partial_{x}^{3}n_{i}^{(k)}=G^{(k-1)}, \ \ \ k\geq3,
\end{split}
\end{equation}
where $G^{(k-1)}$ depends only on $n_{i}^{(1)},n_{i}^{(2)},\cdots,n_{i}^{(k-1)}$, which are “known" from the first $(k-1)^{th}$ steps. Also, it is important to note that the system (<ref>) and (<ref>) for $n_{i}^{(k)},n_{e}^{(k)},u_{i}^{(k)}$ are self contained, which do not depend on $(n_{i}^{(j)},n_{e}^{(j)},u_{i}^{(j)})$ for $j\geq k+1$.
For the solvability of $(n_{i}^{(k)},n_{e}^{(k)},u_{i}^{(k)})$ for $k\geq2$, we state the following
Let $k\geq 2$, $\tilde s_k\leq \tilde s_1-3(k-1)$ be sufficiently large integers and $n_{i0}^{(k)}\in H^{\tilde s_k}(\Bbb R)$. Then when $H\neq 2$, the initial value problem (<ref>) with initial data $n_{i0}^{(k)}$ has a unique solution $n_{i}^{(k)}\in L^{\infty}(-\tau,\tau; H^{\tilde s_k}(\Bbb R))$ for any $\tau>0$. When $H=2$, the initial value problem (<ref>) has a unique solution $n_{i}^{(k)}\in L^{\infty}(0,\tilde\tau_*; H^{\tilde s_k}(\Bbb R))$, where $\tilde\tau_*$ is given in Theorem <ref>.
The proof of Theorem <ref> is standard. Based on this theorem, we will assume that these solutions $(n_{i}^{(k)},n_{e}^{(k)},u_{i}^{(k)})$ for $1\leq k\leq 4$ are as smooth as we want. The optimality of $\tilde s_k$ will not be addressed in this paper.
§.§ Main result
To show that $n_{i}^{(1)}$ converges to a solution of the KdV equation as $\epsilon\rightarrow0$, we must make the above procedure rigorous. Let $(n_{e},n_{i},u_{i})$ be the solution of the scaled system (1.3) of the following expansion
\begin{equation}\label{cut}
\begin{cases}
n_{i}&=1+\epsilon n_{i}^{(1)}+\epsilon^{2}n_{i}^{(2)}+\epsilon^{3}n_{i}^{(3)}+\epsilon^{4}n_{i}^{(4)} +\epsilon^{3}N_i,\\
n_{e}&=1+\epsilon n_{e}^{(1)}+\epsilon^{2}n_{e}^{(2)}+\epsilon^{3}n_{e}^{(3)} +\epsilon^{4}n_{e}^{(4)}+\epsilon^{3}N_e,\\
u_{i}&=\epsilon u_{i}^{(1)}+\epsilon^{2}u_{i}^{(2)}+\epsilon^{3}u_{i}^{(3)} +\epsilon^{4}u_{i}^{(4)}+\epsilon^{3}U,
\end{cases}
\end{equation}
where $(n_{i}^{(1)}$,$n_{e}^{(1)}$,$u_{i}^{(1)})$ satisfies (<ref>) and (<ref>), ($n_{i}^{(k)}$,$n_{e}^{(k)}$,$u_{i}^{(k)}$) satisfies (<ref>) and (<ref>) for $2\leq k\leq4$, and $(N_{i},N_{e},U)$ is the remainder. To simplify the notation slightly, we set
\begin{equation}
\begin{cases}
\tilde{n_i}= n_i^{(1)}+\epsilon n_i^{(2)}+\epsilon^{2}n_i^{(3)}+\epsilon^{3}n_i^{(4)},\\
\tilde{n_e}= n_e^{(1)}+\epsilon n_e^{(2)}+\epsilon^{2}n_e^{(3)}+\epsilon^{3}n_e^{(4)},\\
\tilde{u_i}= u_i^{(1)}+\epsilon u_i^{(2)}+\epsilon^{2}u_i^{(3)}+\epsilon^{3}u_i^{(4)}.
\end{cases}
\end{equation}
After careful computations, we obtain the following remainder system for $(N_{i},N_{e},U)$,
\begin{eqnarray}
\left\{
\begin{array}{llll}
\mathcal{R}_{1}=\partial_{x}(n_{i}^{(1)}u_{i}^{(4)})+\partial_{x}[n_{i}^{(2)}(u_{i}^{(3)}
+\epsilon u_{i}^{(4)})]+\partial_{x}[n_{i}^{(3)}(u_{i}^{(2)}\\ \ \ \ \ \ \ \ \
+\epsilon u_{i}^{(3)}+\epsilon^{2} u_{i}^{(4)})]+\partial_{x}(u_{i}^{(4)}\tilde{u_{i}})-\epsilon\partial_{t}n_{i}^{(4)},\\
\mathcal{R}_{2}^{1}=-\partial_{t}u_{i}^{(4)}+u_{i}^{(1)}\partial_{x}u_{i}^{(4)}+u_{i}^{(2)}(\partial_{x}u_{i}^{(3)}
+\epsilon\partial_{x}u_{i}^{(4)})\\ \ \ \ \ \ \ \ \ \ \
\mathcal{R}_{2}^{2}=n_{e}^{(1)}\partial_{x}n_{e}^{(4)}+n_{e}^{(2)}(\partial_{x}n_{e}^{(3)}+\epsilon\partial_{x}n_{e}^{(4)})\\ \ \ \ \ \ \ \ \ \ \
+n_{e}^{(3)}(\partial_{x}n_{e}^{(2)}+\epsilon\partial_{x}n_{e}^{(3)}+\epsilon^{2} n_{e}^{(4)})+n_{e}^{(4)}\tilde{n_{e}},\\
\mathcal{R}_{2}^{3}=\mathcal{R}_{2}^{3}(n_{e}^{(1)},n_{e}^{(2)},n_{e}^{(3)},n_{e}^{(4)}),\\
\mathcal{R}_{2}^{4}=\mathcal{R}_{2}^{4}(\epsilon N_{e}),\\
\mathcal{R}_{3}^{1}=\mathcal{R}_{3}^{1}(n_{e}^{(1)},n_{e}^{(2)},n_{e}^{(3)},n_{e}^{(4)}),\\
\mathcal{R}_{3}^{2}=\mathcal{R}_{3}^{2}(n_{e}^{(1)},n_{e}^{(2)},n_{e}^{(3)},n_{e}^{(4)}),\\
\mathcal{R}_{3}^{3}=\mathcal{R}_{3}^{3}(N_{e}),
\end{array}
\right.
\end{eqnarray}
where $\mathcal{R}_{2}^{4}$ and $\mathcal{R}_{3}^{3}$ are smooth functions of $N_{e}$, and do not involve any derivatives of $N_{e}$. The mathematical key difficulty is to derive uniform in $\epsilon$ estimates for the remainder $(N_{e},N_{i},U)$.
For convenient usage, we give the following
For $\alpha=0,1,\cdots$ integers, there exists some constant $C=C(\|n_{e}^{(i)}\|_{H^{\tilde s_i}})$ such that
\begin{equation}\label{equ28}
\begin{split}
\|\mathcal R_1,\mathcal R_2^{1,2,3},\mathcal R_3^{1,2}\|_{H^{\alpha}}\leq C ,\ \ \ \alpha=0,1,\cdots,
\end{split}
\end{equation}
\begin{equation}\label{equ29}
\begin{split}
&\|\mathcal R_2^{4}\|_{H^{\alpha}}\leq C \epsilon\|N_{e}\|_{H^{\alpha}}, \ \ \ \alpha=0,1,\cdots,\\
&\|\mathcal R_3^{3}\|_{H^{\alpha}}\leq C \|N_{e}\|_{H^{\alpha}}, \ \ \ \ \alpha=0,1,\cdots,
\end{split}
\end{equation}
\begin{equation}\label{equ34}
\begin{split}
&\|\partial_tR_2^{4}\|_{H^{\alpha}}\leq C \epsilon\|\partial_tN_{e}\|_{H^{\alpha}}, \ \ \ \alpha=0,1,\cdots,\\
&\|\partial_t R_3^{3}\|_{H^{\alpha}}\leq C \|\partial_tN_{e}\|_{H^{\alpha}}, \ \ \ \alpha=0,1,\cdots.
\end{split}
\end{equation}
Recalling the fact that $H^1$ is an algebra, the estimate for Lemma <ref> is straightforward. The details are hence omitted.
Our main result of this paper is the following
Let $\tilde{s_{i}}$ be sufficiently large and $(n_{i}^{(1)},n_{e}^{(1)},u_{i}^{(1)})\in H^{\tilde{s_{1}}}$ be a solution constructed in Theorem <ref> for the quantum KdV equation with initial data $(n_{i0},n_{e0},u_{i0})\in H^{\tilde{s_{1}}}$ satisfying (<ref>). Let $(n_{i}^{(j)},n_{e}^{(j)},u_{i}^{(j)})\in H^{\tilde{s_{j}}}$ (i=2,3,4) be solutions of (<ref>) and (<ref>) constructed in Theorem <ref> with initial data
$(n_{i0}^{j},n_{e0}^{j},u_{i0}^{j})\in H^{\tilde{s_{j}}}$ satisfying (<ref>). Let
$(N_{i0},N_{e0},U_{0})$ satisfy (<ref>) and assume
\begin{equation*}
\begin{split}
n_{i0}&=1+\epsilon n_{i0}^{(1)}+\epsilon^{2}n_{i0}^{(2)}+\epsilon^{3}n_{i0}^{(3)}+\epsilon^{4}n_{i0}^{(4)}+\epsilon^{3}N_{i0},\\
n_{e0}&=1+\epsilon n_{e0}^{(1)}+\epsilon^{2}n_{e0}^{(2)}+\epsilon^{3}n_{e0}^{(3)}+\epsilon^{4}n_{e0}^{(4)}+\epsilon^{3}N_{e0},\\
u_{i0}&=\epsilon u_{i0}^{(1)}+\epsilon^{2}u_{i0}^{(2)}+\epsilon^{3}u_{i0}^{(3)}+\epsilon^{4}u_{i0}^{(4)}+\epsilon^{3}U_{0}.\\
\end{split}
\end{equation*}
Then for any $\tau>0$, there exists $\epsilon_{0}>0$ such that if $0<\epsilon<\epsilon_{0}$, the solution of the EP system (<ref>) with initial data $(n_{i0},n_{e0},u_{i0})$ can be expressed as
\begin{equation*}
\begin{split}
n_{i}&=1+\epsilon n_{i}^{(1)}+\epsilon^{2}n_{i}^{(2)}+\epsilon^{3}n_{i}^{(3)}+\epsilon^{4}n_{i}^{(4)}+\epsilon^{3}N_{i},\\
n_{e}&=1+\epsilon n_{e}^{(1)}+\epsilon^{2}n_{e}^{(2)}+\epsilon^{3}n_{e}^{(3)}+\epsilon^{4}n_{e}^{(4)}+\epsilon^{3}N_{e},\\
u_{i}&=\epsilon u_{i}^{(1)}+\epsilon^{2}u_{i}^{(2)}+\epsilon^{3}u_{i}^{(3)}+\epsilon^{4}u_{i}^{(4)}+\epsilon^{3}U,
\end{split}
\end{equation*}
such that for all $0<\epsilon<\epsilon_{0}$,
\begin{equation}\label{ineq}
\begin{split}
\sup_{[0,\tau]}\Big\{\|(N_{i},N_{e},U)\|_{H^{2}}^{2} &+\epsilon\|(\partial_{x}^{3}N_{e}, \partial_{x}^{3}U)\|_{L^{2}}^{2} +\epsilon^{2}\|\partial_{x}^{4}N_{e}\|_{L^{2}}^{2} +\epsilon^{3}\|\partial_{x}^{5}N_{e}\|_{L^{2}}^{2}\\
+\epsilon^{4}\|\partial_{x}^{6}N_{e}\|_{L^{2}}^{2} \Big\}\leq &C_{\tau}\Big(1+\|(N_{i0},N_{e0},U_{0})\|_{H^{2}}^{2} +\epsilon\|(\partial_{x}^{3}N_{e0},\partial_{x}^{3}U_{0})\|_{L^{2}}^{2}\\
&+\epsilon^{2}\|\partial_{x}^{4}N_{e0}\|_{L^{2}}^{2} +\epsilon^{3}\|\partial_{x}^{5}N_{e0}\|_{L^{2}}^{2} +\epsilon^{4}\|\partial_{x}^{6}N_{e0}\|_{L^{2}}^{2}\Big).
\end{split}
\end{equation}
From (<ref>), we see that the $H^2$-norm of the remainder $(N_i,N_e,U)$ is bounded uniformly in $\epsilon$. Note also the Gardner-Morikawa transform (<ref>), we see that
\begin{equation}
\sup_{[0,\epsilon^{-3/2}\tau]}
\left\|\left(
\begin{array}{c}
(n_{i}-1)/\epsilon\\ (n_{e}-1)/\epsilon\\ u_{i}/\epsilon
\end{array}
\right)-KdV\right\|_{H^2}\leq C\epsilon,
\end{equation}
for some $C>0$ independent of $\epsilon>0$. Here `KdV' is the equation satisfied by the first approximation $(n_i^{(1)},n_e^{(1)},u_i^{(1)})$.
The basic plan is to first estimate some uniform bound for $(N_{e},U)$ and then recover the estimate for $N_{i}$ from the estimate of $N_{e}$ by the equation (<ref>). We want to apply the Gronwall lemma to complete the proof. To state clearly, we first introduce
\begin{equation}\label{|||}
\begin{split}
|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}=&\|(N_{e},U)\|_{H^{2}}^{2} +\epsilon\|(\partial_{x}^{3}N_{e},\partial_{x}^{3}U)\|_{L^{2}}^{2} +\epsilon^{2}\|\partial_{x}^{4}N_{e}\|_{L^{2}}^{2} +\epsilon^{3}\|\partial_{x}^{5}N_{e}\|_{L^{2}}^{2} +\epsilon^{4}\|\partial_{x}^{6}N_{e}\|_{L^{2}}^{2}.\
\end{split}
\end{equation}
As we will see, the zeroth order, the first order to the second order estimates for $(N_{e},U)$ and the third order estimates for $\epsilon(N_{e},U)$ all can be controlled in terms of $|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$.
For convenience, we introduce the following lemma
Let $m\geq1$ be an integer, and then the commutator which is defined by the following
\begin{equation}
\begin{split}\label{w}
\end{split}
\end{equation}
can be bounded by
\begin{equation}
\begin{split}\label{w11}
\|[\nabla^{m},f]g\|_{L^{p}}\leq \|\nabla f\|_{L^{p_{1}}}\|\nabla^{m-1}g\|_{L^{p_{2}}}+\|\nabla^{m} f\|_{L^{p_{3}}}\|g\|_{L^{p_{4}}},
\end{split}
\end{equation}
where $p,p_{2},p_{3}\in(1,\infty)$ and
\begin{equation*}
\begin{split}
\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{_{_{4}}}}.
\end{split}
\end{equation*}
The proof can be found in <cit.>, for example.
§ UNIFORM ENERGY ESTIMATES
In this section, we give the energy estimates uniformly in $\epsilon$ for the remainder $(N_{e},N_{i},U)$, which requires a combination of energy method and analysis of the remainder equation (<ref>). To simplify the proof slightly, we assume that (<ref>) has smooth solutions in $[0,\tau_{\epsilon}]$ for $\tau_{\epsilon}>0$ depending on $\epsilon$.
Let $\tilde{C}$ be a constant independent of $\epsilon$, which will be determined later, much larger than the bound $|\!|\!|(N_{e},U)(0)|\!|\!|_{\epsilon}^{2}$ of the initial data. It is classical that there exists $\tau_{\epsilon}>0$ such that on $[0,\tau_{\epsilon}]$,
\begin{equation}
\begin{split}\label{priori}
\|N_{i}\|_{H^{2}}^{2},\ \ |\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}\leq\tilde{C}.
\end{split}
\end{equation}
As a direct corollary, there exists some $\epsilon_{1}>0$ such that $n_{e} \ and \ n_{i}$ are bounded from above and below, say $\frac{1}{2}<n_{i},n_{e}<\frac{3}{2}$ and $u_{i}$ is bounded by $|u_{i}|<\frac{1}{2}$ when $\epsilon<\epsilon_{1}$. Since $\mathcal{R}_{2}^{4},\mathcal{R}_{3}^{3}$ are smooth functions of $N_{e}$, there exists some constant $C_{1}=C_{1}(\epsilon\tilde{C})$ for any $\alpha,\beta\geq0$ such that
\begin{equation*}
\begin{split}
\left|\partial_{n_{e}^{(j)}}^{\alpha} \partial_{N_{e}}^{\beta}(\mathcal{R}_{2}^{4}, \mathcal{R}_{3}^{3})\right|\leq C_{1}=C_{1}(\epsilon\tilde{C}),
\end{split}
\end{equation*}
where $C_{1}(\cdot)$ can be chosen to be nondecreasing in its argument. We will show that for any given $\tau>0$ there is some $\epsilon_{0}>0$, such that the existence time $\tau_{\epsilon}>\tau$ for any $0<\epsilon<\epsilon_{0}$.
The purpose of this section is to prove Proposition <ref> and <ref>. Since the proof of Proposition <ref> will be almost the same to that of Proposition <ref>, we will omit the proof of Proposition <ref>. In Subsection <ref>, we first show three lemmas that will be frequently used later. In Subsection <ref> and Subsection <ref>, we present and prove the two main propositions, while estimates of some crucial terms are postponed to Subsection <ref> and Subsection <ref>.
§.§ Basic estimates
We first prove the following Lemma <ref>-<ref>, in which we bound $N_{i}$ and $\partial_{t}N_{e}$ in terms of $N_{e}$.
Let $(N_{i},N_{e},U)$ be a solution to (<ref>) and $\alpha\geq0$ be an integer. There exist some constants $0<\epsilon_{1}<1$ and $C_{1}=C_{1}(\epsilon\tilde{C})$ such that for every $0<\epsilon<\epsilon_{1}$,
\begin{equation}\label{eqaL1}
\begin{split}
C_{1}^{-1}\|\partial_{x}^{\alpha}N_{i}\|^{2} \leq &\|\partial_{x}^{\alpha}N_{e}\|^{2}
\leq C_{1}\|\partial_{x}^{\alpha}N_{i}\|^{2}.
\end{split}
\end{equation}
When $\alpha=0$, taking inner product of (1.17c) with $N_{e}$ and integration by parts, we have
\begin{equation}
\begin{split}\label{3.1}
\|N_{e}&\|^{2}+\epsilon\|\partial_{x}N_{e}\|^{2}+\frac{\epsilon^{2}H^{2}}{4}\int\frac{1}{n_{e}}(\partial_{x}^{2}N_{e})^{2}\\
+\epsilon^{2}\int \mathcal{R}_{3}^{1}N_{e}\\
=&:\sum_{i=1}^{23}A_{i} \ .
\end{split}
\end{equation}
Since $\frac{1}{2}<n_{e}<\frac{3}{2}$ and $H$ is a fixed constant, there exists a fixed constant $C$ such that
\begin{equation*}
\begin{split}
\frac{\epsilon^{2}H^{2}}{4}\int\frac{1}{n_{e}}(\partial_{x}^{2}N_{e})^{2}
\geq C\epsilon^{2}\|\partial_{x}^{2}N_{e}\|^{2}.
\end{split}
\end{equation*}
Thus the LHS of (<ref>) is equal or greater than $C(\|N_{e}\|^{2}+\epsilon\|\partial_{x}N_{e}\|^{2} +\epsilon^{2}\|\partial_{x}^{2}N_{e}\|^{2})$. Next, we estimate the RHS of (<ref>). For $A_{1}$, since $\tilde{n_{e}}$ is known and bounded in $L^{\infty}$, there exists some constant $C$ such that
\begin{equation*}
\begin{split}
&\leq C(1+\epsilon^{2}\|N_{e}\|_{L^{\infty}})(\epsilon\|N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2})\\
&\leq C(1+\epsilon^{2}\|N_{e}\|_{H^{1}})(\epsilon\|N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2})\\
&\leq C(1+\epsilon^{2}\tilde{C})(\epsilon\|N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2})\\
&\leq C(\epsilon\|N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2}),
\end{split}
\end{equation*}
where we have used Hölder's inequality, Sobolev embedding $H^1\hookrightarrow L^{\infty}$, the priori assumption (<ref>) and Cauchy inequality.
Note that
\begin{equation}
\begin{split}\label{one order}
\left|\partial_{x}\left(\frac{1}{n_{e}}\right)\right|
\leq C\left(\epsilon|\partial_{x}\tilde{n_{e}}| +\epsilon^{3}|\partial_{x}N_{e}|\right),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{two orders}
\left|\partial_{x}^{2}\left(\frac{1}{n_{e}}\right)\right|
\leq C\big(\epsilon+\epsilon^{3}(|\partial_{x}N_{e}|+|\partial_{x}^{2}N_{e}|)+\epsilon^{6}|\partial_{x}N_{e}|^{2}\big).
\end{split}
\end{equation}
Since $\partial_{x}\tilde{n_{e}},\ \partial_{x}^{2}\tilde{n_{e}}$ are bounded in $L^{\infty}$, similar to $A_{1}$, we have
\begin{equation*}
\begin{split}
A_{2\sim16,18\sim20}\leq C_{1}(\epsilon\|N_{e}\|^{2}+\epsilon^{2}\|\partial_{x}N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2}).
\end{split}
\end{equation*}
Now we estimate $A_{17}$.
By integration by parts, we obtain
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
Similar to (<ref>), we have
\begin{equation}
\begin{split}\label{g1}
\left|\partial_{x}\left(\frac{\partial_{x}\tilde{n_{e}}}{n_{e}^{2}}\right)\right|
\leq C(1+\epsilon^{3}|\partial_{x}N_{e}|).
\end{split}
\end{equation}
Similar to $A_{1}$, by applying Hölder's inequality, Sobolev embedding $H^1\hookrightarrow L^{\infty}$, the priori assumption (<ref>) and Cauchy inequality again, we have
\begin{equation*}
\begin{split}
A_{17}\leq C_{1}(\epsilon\|N_{e}\|^{2}+\epsilon^{2}\|\partial_{x}N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2}).
\end{split}
\end{equation*}
The term $A_{21}$ can be similarly bounded by
\begin{equation*}
\begin{split}
A_{21}\leq C_{1}(\epsilon^{2}\|\partial_{x}N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{2}N_{e}\|^{2}).
\end{split}
\end{equation*}
According to the form of $\mathcal{R}_{3}^{2} \ and \ \mathcal{R}_{3}^{3}$ in (1.18), by applying Cauchy inequality, we have
\[
A_{22}\leq C_{1}\| N_{e}\|^{2}.
\]
By Young inequality, we have
\[
\int N_{e}N_{i}\leq\delta\|N_{e}\|^{2}+C_{\delta}\|N_{i}\|^{2},
\]
for arbitrary $\delta>0$.
Hence, there exists some $\epsilon_1>0$ such that for $0<\epsilon<\epsilon_1$,
\begin{equation}
\begin{split}\label{31}
\|N_{e}\|^{2}+\epsilon\|\partial_{x}N_{e}\|^{2}+\epsilon^{2}\|\partial_{x}^{2}N_{e}\|^{2}\leq C_{1}\|N_{i}\|^{2}.
\end{split}
\end{equation}
Taking inner product of (1.17c) with $\epsilon \partial_{x}^{2}N_{e}$ and $\epsilon^{2}\partial_{x}^{4}N_{e}$, applying Hölder inequality and integration by parts, we have similarly
\begin{equation}
\begin{split}\label{32}
\epsilon\|\partial_{x}N_{e}\|^{2}+\epsilon^{2}\|\partial_{x}^{2}N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{3}N_{e}\|^{2}
\leq C_{1}\|N_{i}\|^{2},
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{33}
\epsilon^{2}\|\partial_{x}^{2}N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{3}N_{e}\|^{2}+\epsilon^{4}\|\partial_{x}^{4}N_{e}\|^{2}
\leq C_{1}\|N_{i}\|^{2}.
\end{split}
\end{equation}
By the estimates (<ref>), (<ref>) and (<ref>), we obtain
\begin{equation}
\begin{split}\label{40}
\|N_{e}\|^{2}+\epsilon\|\partial_{x}N_{e}\|^{2}+\epsilon^{2}\|\partial_{x}^{2}N_{e}\|^{2}+\epsilon^{3}\|\partial_{x}^{3}N_{e}\|^{2}
\leq C_{1}\|N_{i}\|^{2}.
\end{split}
\end{equation}
On the other hand, from the equation (<ref>), there exist some $C$ such that
\begin{equation}
\begin{split}\label{34}
\|N_{i}\|^{2}\leq C_{1}(\|N_{e}\|^{2}+\epsilon\|\partial_{x}N_{e}\|^{2}
\end{split}
\end{equation}
Putting (<ref>)-(<ref>) together, we deduce the inequality for $\alpha=0$.
For higher order inequalities, we differentiate (<ref>) with $\partial_{x}^{\alpha}$ and then take inner product with $\partial_{x}^{\alpha}N_{e}$, $\epsilon\partial_{x}^{\alpha+2}N_{e}$ and $\epsilon^{2}\partial_{x}^{\alpha+4}N_{e}$ separately. The Lemma then follows by the same procedure of the case $\alpha=0$.
Recall $|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}$ in (<ref>). We remark that only $\|N_{i}\|_{H^{2}}$ can be bounded in terms of $|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}$ and no higher order derivatives of $N_{i}$ are allowed in Lemma <ref>. In fact, we only need $0\leq\alpha\leq2$ in Lemma <ref>.
Let $(N_{i},N_{e},U)$ be a solution to (<ref>). There exist some constants $C$ and $C_{1}=C_{1}(\epsilon\tilde{C})$ such that
\begin{equation}\label{eqaL21}
\begin{split}
\|\epsilon\partial_{t}N_{i}\|^{2}\leq & C\big(\|N_{e}\|_{H^{1}}^{2}+\|U\|_{H^{1}}^{2} +\epsilon\|\partial_{x}^{2}N_{e}\|^{2} +\epsilon^{2}\|\partial_{x}^{3}N_{e}\|^{2}\\
&+\epsilon^{3}\|\partial_{x}^{4}N_{e}\|^{2} +\epsilon^{4}\|\partial_{x}^{5}N_{e}\|^{2}\big) +C\epsilon,
\end{split}
\end{equation}
\begin{equation}\label{eqaL22}
\begin{split}
\|\epsilon\partial_{tx}N_{i}\|^{2}\leq & C_{1}(\|N_{e}\|_{H^{2}}^{2} +\|U\|_{H^{2}}^{2} +\epsilon\|\partial_{x}^{3}N_{e}\|^{2} +\epsilon^{2}\|\partial_{x}^{4}N_{e}\|^{2}\\
&+\epsilon^{3}\|\partial_{x}^{5}N_{e}\|^{2} +\epsilon^{4}\|\partial_{x}^{6}N_{e}\|^{2}) +C\epsilon.
\end{split}
\end{equation}
In terms of $|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}$, we can rewrite (<ref>) and (<ref>) as
\[
\|\epsilon\partial_{t}N_{i}\|_{H^{1}}^{2}\leq C_{1}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}+C\epsilon.
\]
From (<ref>), we have
\[
\epsilon\partial_{t}N_{i}=(1-u_{i})\partial_{t}N_{i}-n_{i}\partial_{x}U
\]
Since $\frac{1}{2}<n_{i}<\frac{3}{2}$ and $|u_{i}|<\frac{1}{2}$, taking $L^{2}$-norm yields
\begin{equation*}
\begin{split}
\|\epsilon\partial_tN_{i}\|^2\leq&\|(1-u_{i})\partial_xN_{i}\|^2 +\|n_{i}\partial_xU\|^2 +\epsilon^2\|\partial_x\tilde u_{i}N_{i}\|^2+ \epsilon^2\|\partial_x\tilde n_{i}U\|^2 +\epsilon^4\|\mathcal R_1\|^2\\
\leq & C(\|\partial_xN_{i}\|^2+\|\partial_xU\|^2) +C\epsilon^2(\epsilon^2+\|N_{i}\|^2+\|U\|^2).
\end{split}
\end{equation*}
Applying Lemma <ref> with $\alpha=1$, we deduce (<ref>).
To prove (<ref>), we take $\partial_{x}$ of (<ref>) to obtain
\begin{equation*}
\begin{split}
\|\epsilon&\partial_{tx}N_{i}\|^{2}
\leq C(\|U\|_{H^{2}}^{2}+\|N_{i}\|_{H^{2}}^{2})+C\epsilon^{6}\int|\partial_{x}{N_{i}}|^{2}|\partial_{x}U|^{2}+C\epsilon^{4}.
\end{split}
\end{equation*}
We note that
\begin{equation*}
\begin{split}
\leq C\epsilon^{6}\|U\|_{H^{2}}^{2}\|N_{i}\|_{H^{1}}^{2}
\leq C(\epsilon\tilde{C})\|U\|_{H^{2}}^{2}.
\end{split}
\end{equation*}
Applying Lemma <ref> with $\alpha=2$, we deduce (<ref>).
The Lemma then follows from Lemma <ref>.
Let $(N_{i},N_{e},U)$ be a solution to (<ref>) and $\alpha\geq0$ be an integer. There exist some constants $C_{1}=C_{1}(\epsilon\tilde{C})$ and $\epsilon_{1}>0$ such that for every $0<\epsilon<\epsilon_{1}$,
\begin{equation}\label{eqaL3}
\begin{split}
\epsilon^{4}\|\partial_{t}\partial_{x}^{\alpha+4}N_{e}\|^{2}&+\epsilon^{3}\|\partial_{t}\partial_{x}^{\alpha+3}N_{e}\|^{2}
+\epsilon^{2}\|\partial_{t}\partial_{x}^{\alpha+2}N_{e}\|^{2}\\ \ \ \ \ \ \ \ \
\leq C\|\partial_{t}\partial_{x}^{\alpha}N_{i}\|^{2}+C_{1}.
\end{split}
\end{equation}
The proof is similar to that of Lemma <ref>. When $\alpha=0$, by first taking $\partial_{t}$ of (<ref>) and then taking inner product with $\partial_{t}N_{e}$ and integration by parts, we have
\begin{equation}
\begin{split}\label{m1}
\|\partial_{t}&N_{e}\|^{2}
+\epsilon\int n_{e}(\partial_{tx}N_{e})^{2}+\frac{\epsilon^{2}H^{2}}{4}\int\frac{1}{n_{e}}(\partial_{t}\partial_{x}^{2}N_{e})^{2}\\
=&:\sum_{i=1}^{25}B_{i} \ .
\end{split}
\end{equation}
Estimate of the LHS of (<ref>).
Since $\frac{1}{2}<n_{e}<\frac{3}{2}$ and $H$ is a fixed constant, there exists a fixed constant $C$ such that
\begin{equation*}
\begin{split}
\epsilon\int n_{e}(\partial_{tx}N_{e})^{2} +\frac{\epsilon^{2}H^{2}}{4}\int\frac{1}{n_{e}} (\partial_{t}\partial_{x}^{2}N_{e})^{2}
\geq C(\epsilon\|\partial_{tx}N_{e}\|^{2} +\epsilon^{2}\|\partial_{t} \partial_{x}^{2}N_{e}\|^{2}).
\end{split}
\end{equation*}
Thus the LHS of (<ref>) is equal or greater than $C(\|\partial_{t}N_{e}\|^{2} +\epsilon\|\partial_{tx}N_{e}\|^{2} +\epsilon^{2} \|\partial_{t}\partial_{x}^{2}N_{e}\|^{2})$. Next, we estimate the righthand side terms. For $B_{1}$, by applying Hölder's inequality, Cauchy inequality and Sobolev embedding $H^1\hookrightarrow L^{\infty}$, we have
\begin{equation*}
\begin{split}
B_{1}&=\epsilon^{2}\int(\partial_{x}\tilde{n_{e}} +\epsilon^{2}\partial_{x}N_{e})\partial_{tx}N_{e} \partial_{t}N_{e}\\
&\leq C\epsilon(1+\epsilon^{2} \|\partial_{x}N_{e}\|_{L^{\infty}}) (\epsilon\|\partial_{t}N_{e}\|^{2} +\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})\\
&\leq C(\epsilon\tilde{C}) (\epsilon\|\partial_{t}N_{e}\|^{2} +\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})\\
&\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2} +\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}),
\end{split}
\end{equation*}
where we have used (<ref>). Similarly,
\begin{equation*}
\begin{split}
B_{2}\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2} +\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})+C_{1}.
\end{split}
\end{equation*}
By (<ref>), Sobolev embedding theorem and Cauchy inequality, we have
\begin{equation*}
\begin{split}
B_{3}+B_{4}\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}
\end{split}
\end{equation*}
where we have used (<ref>) and (<ref>).
Estimate of $B_{5}$. Similar to (<ref>), we note that
\begin{equation}
\begin{split}\label{one}
\leq C(\epsilon|\partial_{t}\tilde{n_{e}}|+\epsilon^{3}|\partial_{t}N_{e}|).
\end{split}
\end{equation}
Therefore, we have
\begin{equation*}
\begin{split}
B_{5}&\leq\epsilon^{2} \|\partial_{x}^{4}N_{e}\|
&\leq \epsilon^{2}\tilde{C}(\epsilon C+\epsilon^{3}\|\partial_{t}{N_{e}}\|_{H^{1}})\|\partial_{t}N_{e}\|\\
&\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})+C_{1}.
\end{split}
\end{equation*}
Estimate of $B_{6}$. By direct computation, we have
\begin{equation*}
\begin{split}
\partial_{t}(\partial_{x}\tilde{n_{e}}\partial_{x}N_{e})
\end{split}
\end{equation*}
which yields that
\begin{equation*}
\begin{split}
\|\partial_{t}(\partial_{x}\tilde{n_{e}}\partial_{x}N_{e})\|
\leq C(\|\partial_{x}N_{e}\|+\|\partial_{tx}N_{e}\|),
\end{split}
\end{equation*}
where $C$ is a fixed constant. By applying Hölder inequality and Young inequality, we have
\begin{equation*}
\begin{split}
B_{6}\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})+C_{1}.
\end{split}
\end{equation*}
$B_{8}$ is similar to $B_{6}$.
Estimate of $B_{7}$. We note that
\begin{equation*}
\begin{split}
\|\partial_{t}[(\partial_{x}N_{e})^{2}]\|
\leq C(\|\partial_{x}N_{e}\|_{L^{\infty}}\|\partial_{tx}N_{e}\|).
\end{split}
\end{equation*}
Thus, we have
\begin{equation*}
\begin{split}
B_{7}\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}),
\end{split}
\end{equation*}
thanks to Hölder inequality, Cauchy inequality and Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and (<ref>).
Estimate of $B_{9}$. Since $\mathcal{R}_{3}^{1}$ is known, thus by Cauchy inequality, we have
\begin{equation*}
\begin{split}
B_{9}\leq C_{1}\epsilon\|\partial_{t}N_{e}\|^{2}+C_{1}.
\end{split}
\end{equation*}
Estimate of $B_{20}$. By direct computation, we have
\begin{equation*}
\begin{split}
\partial_{t}\left[\frac{1}{n_{e}^{4}} (\partial_{x}N_{e})^{4}\right]
\end{split}
\end{equation*}
Similar to (<ref>), we have
\begin{equation}
\begin{split}\label{repeat}
\left|\partial_{t}\left(\frac{1}{n_{e}^{4}}\right)\right|
\leq C(\epsilon+\epsilon^{3}|\partial_{t}N_{e}|).
\end{split}
\end{equation}
Thus by applying Hölder inequality, Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and (<ref>) again, we have
\begin{equation*}
\begin{split}
B_{20}&\leq C\epsilon^{11}\|\partial_{t}[\frac{1}{n_{e}^{4}}(\partial_{x}N_{e})^{4}]\|\|\partial_{t}N_{e}\|\\
&\leq C\epsilon^{11}\|\partial_{x}N_{e}\|_{L^{\infty}}^{4}(\epsilon+\epsilon^{3}\|\partial_{t}N_{e}\|)\|\partial_{t}N_{e}\|
&\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})+C_{1}.
\end{split}
\end{equation*}
The estimates of $B_{10\sim13},B_{15} \ and \ B_{17}$ are similar to that for $B_{20}$.
Estimate of $B_{21}$. By direct computation, we have
\begin{equation*}
\begin{split}
\partial_{t}\left[\frac{1}{n_{e}^{3}} (\partial_{x}N_{e})^{2}\partial_{x}^{2}N_{e}\right]
\end{split}
\end{equation*}
Thus similarly, we have
\begin{equation*}
\begin{split}
B_{21}\leq &C\epsilon^{8}\|\partial_{t}\big[\frac{1}{n_{e}^{3}}(\partial_{x}N_{e})^{2}\partial_{x}^{2}N_{e}\big]\|\|\partial_{t}N_{e}\|\\
\leq &C\epsilon^{8}[\|\partial_{x}N_{e}\|_{L^{\infty}}^{2}\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}
\leq &C_{1}\|N_{e}\|_{H^{2}}^{2}(1+\epsilon\|\partial_{x}^{3}N_{e}\|^{2})\epsilon\|\partial_{t}N_{e}\|^{2}\\
\leq &C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}+\epsilon^{3}\|\partial_{t}\partial_{x}^{2}N_{e}\|^{2})+C_{1}.
\end{split}
\end{equation*}
The estimates of $B_{14},B_{16},B_{18}\ and \ B_{19}$ are similar to that for $B_{21}$.
Estimate of $B_{23}$. By direct computation, we have
\begin{equation*}
\begin{split}
\partial_{t}\left[\frac{1}{n_{e}^{2}} \partial_{x}N_{e}\partial_{x}^{3}N_{e}\right]
\end{split}
\end{equation*}
Thus $B_{23}$ is divided three terms
\begin{equation*}
\begin{split}
B_{23}=\frac{3\epsilon^{5}H^{2}}{4}\sum_{i=1}^{3}\int G_{i}\partial_{t}N_{e}=:B_{231}+B_{232}+B_{233}.
\end{split}
\end{equation*}
The first two terms $B_{231}$ and $B_{232}$ can be easily estimated by $C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2} +\epsilon^{2}\|\partial_{tx}N_{e}\|^{2})$. For the last term $B_{233}$, we integrate by parts and use Hölder inequality, Cauchy inequality and (<ref>) again to obtain
\begin{equation*}
\begin{split}
\leq &C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}+\epsilon^{3}\|\partial_{t}\partial_{x}^{2}N_{e}\|^{2}),
\end{split}
\end{equation*}
where we also have used (<ref>).
Thus we have
\begin{equation*}
\begin{split}
B_{23}\leq C_{1}(\epsilon\|\partial_{t}N_{e}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}
\end{split}
\end{equation*}
$B_{22}$ is similar to $B_{23}$.
Estimate of $B_{24}$. Since $\mathcal{R}_{3}^{2}$ is known, by
using (<ref>) in Lemma <ref>, we have
\[
B_{24}\leq C_{1}(1+\epsilon\|\partial_{t}N_{e}\|^{2}).
\]
Estimate of $B_{25}$. Applying Young inequality, we have
\[
\]
where for arbitrary small $\gamma>0$. Hence, we have shown that there exists some $\epsilon_1>0$ such that for $0<\epsilon<\epsilon_1$, we have
\begin{equation}\label{lemma31}
\begin{split}
\|\partial_{t}N_{e}\|^{2} +\epsilon\|\partial_{tx}N_{e}\|^{2} +\epsilon^{2}\|\partial_{t} \partial_{x}^{2}N_{e}\|^{2}\leq C\|\partial_{t}N_{i}\|^{2}+C_{1}.
\end{split}
\end{equation}
Similarly, taking $\partial_{tx}$ of (<ref>) and then taking inner product with $\epsilon\partial_{tx}N_{e}$, we have
\begin{equation}\label{lemma32}
\begin{split}
\epsilon\|\partial_{tx}N_{e}\|^{2}+\epsilon^{2}\|\partial_{t}\partial_{x}^{2}N_{e}\|^{2}
\leq C_{\alpha_{2}}\|\partial_{t}N_{i}\|^{2} +\epsilon\|\partial_{t}N_{e}\|^{2}+C_{1}.
\end{split}
\end{equation}
Taking $\partial_{t}\partial_{x}^{2}$ of (<ref>) and then taking inner product with $\epsilon^{2}\partial_{t}\partial_{x}^{2}N_{e}$, we have
\begin{equation}\label{lemma33}
\begin{split}
\epsilon^{2}\|\partial_{t}\partial_{x}^{2}N_{e}\|^{2}+\epsilon^{3}\|\partial_{t}\partial_{x}^{3}N_{e}\|^{2}
\leq &C_{\alpha_{3}}\|\partial_{t}N_{i}\|^{2}+\epsilon^{2}\|\partial_{tx}N_{e}\|^{2}\\
\end{split}
\end{equation}
Putting (<ref>), (<ref>) and (<ref>) together, let $C=\max\{C_{\alpha_{1}},C_{\alpha_{2}},C_{\alpha_{3}}\}$, we obtain
\begin{equation*}
\begin{split}
\|\partial_{t}N_{e}\|^{2}&+\epsilon\|\partial_{tx}N_{e}\|^{2}+\epsilon^{2}\|\partial_{t}\partial_{x}^{2}N_{e}\|^{2}
\leq C\|\partial_{t}N_{i}\|^{2}+C_{1}.
\end{split}
\end{equation*}
Thus we have proven (<ref>) for $\alpha=0$. The case of $\alpha\geq1$ can be proved similarly.
§.§ Zeroth, first and second order estimates
The zeroth, first and second order estimates can be summarized in the following
Let $(N_{i},N_{e},U)$ be a solution to (<ref>) and $\gamma=0,1,2$, then
\begin{equation}\label{equ10}
\begin{split}
\frac{1}{2}\frac{d}{dt}&\|\partial_{x}^{\gamma}U\|^{2} +\frac{1}{2}\frac{d}{dt} \left[\int\frac{n_{e}}{n_{i}}(\partial_{x}^{\gamma} N_{e})^{2} +\epsilon\int\frac{n_{e}^{2}}{n_{i}} (\partial_{x}^{\gamma+1}N_{e})^{2} +\frac{\epsilon^{2}H^{2}}{4} \int\frac{1}{n_{i}}(\partial_{x}^{\gamma+2}N_{e})^{2}\right]\\
&+\frac{1}{2}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\left[\int \frac{(\partial_{x}^{\gamma+1}N_{e})^{2}}{n_{e}n_{i}} +\epsilon\int\frac{1}{n_{i}} (\partial_{x}^{\gamma+2}N_{e})^{2} +\frac{\epsilon^{2}H^{2}}{4}\int\frac{1}{n_{e}^{2} n_{i}}(\partial_{x}^{\gamma+3}N_{e})^{2}\right]\\
\leq & C_{1}(1+\epsilon|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation}
This proposition can be proved after long tedious calculations, which can be done by the same procedure that used in the proof of Proposition <ref>. Hence we omit the details here for simplicity.
§.§ Third order estimates
Let $(N_{i},N_{e},U)$ be a solution to (<ref>)Ł¬ then
\begin{equation}\label{equ111}
\begin{split}
\frac{\epsilon}{2}\frac{d}{dt}&\|\partial_{x}^{3}U\|^{2}+\frac{\epsilon}{2}\frac{d}{dt}\left[\int\frac{n_{e}}{n_{i}}(\partial_{x}^{3}N_{e})^{2}
+\epsilon\int\frac{n_{e}^{2}}{n_{i}} (\partial_{x}^{4}N_{e})^{2} +\frac{\epsilon^{2}H^{2}}{4} \int\frac{1}{n_{i}}(\partial_{x}^{5}N_{e})^{2} \right]\\
&+\frac{\epsilon}{2}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\left[\int\frac{1}{n_{e}n_{i}}(\partial_{x}^{4}N_{e})^{2}
\leq & C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation}
We take $\partial_{x}^{3}$ of (<ref>) and then take inner product of $\epsilon\partial_{x}^{3}U$. We obtain
\begin{equation}\label{equ11}
\begin{split}
\frac{\epsilon}{2}\frac{d}{dt}&\|\partial_{x}^{3}U\|^{2}-\int\partial_{x}^{3}\big((1-u_{i})\partial_{x}U\big)\partial_{x}^{3}U
+\epsilon\int\partial_{x}^{3}\big(\partial_{x}\tilde u_{i}U\big)\partial_{x}^{3}U\\
+\frac{\epsilon H^{2}}{4}\int\partial_{x}^{3}\big(\frac{\partial_{x}^{3}N_{e}}{n_{e}}\big)\partial_{x}^{3}U
-\epsilon\int\partial_{x}^{3}\big(\partial_{x}\tilde n_{e}N_{e}\big)\partial_{x}^{3}U\\
=&:\sum_{i=1}^{11}F_{i} \ .
\end{split}
\end{equation}
Estimate of the LHS of (<ref>). First, we estimate the second term on the LHS of (<ref>). Using commutator notation (<ref>) to rewrite it as
\begin{equation*}
\begin{split}
-\int\partial_{x}^{3}\big((1-u_{i}) \partial_{x}U\big)\partial_{x}^{3}U
&=-\int\big([\partial_{x}^{3},1-u_{i}]\partial_{x}U +(1-u_{i})\partial_{x}^{4}U\big)\partial_{x}^{3}U =:M_{1}+M_{2}
\end{split}
\end{equation*}
We first estimate $M_1$. By commutator estimate of Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},1-u_{i}]\partial_{x}U\|
\leq\|\partial_{x}(1-u_{i})\|_{L^{\infty}}\|\partial_{x}^{3}U\|
\end{split}
\end{equation*}
Thus by Hölder inequality, Cauchy inequality and Sobolev embedding theorem $H^1\hookrightarrow L^{\infty}$, we have
\begin{equation}
\begin{split}\label{G2}
| M_{1}|
\leq &\|[\partial_{x}^{3},1-u_{i}]\partial_{x}U\|\|\partial_{x}^{3}U\|\\
\leq &C(1+\epsilon^{2}\|\partial_{x}U\|_{L^{\infty}}^{2})(\epsilon\|\partial_{x}^{3}U\|^{2})
\leq &C_{1}\big(1+\epsilon^{2}(\|U\|_{H^{2}}^{2}+\epsilon\|\partial_{x}^{3}U\|^{2})\big)(\|U\|_{H^{2}}^{2}+\epsilon\|\partial_{x}^{3}U\|^{2})\\
\leq &C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2},
\end{split}
\end{equation}
where $|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$ is given in (<ref>). Next, we estimate $M_{2}$. By integration by parts, we have
\begin{equation}
\begin{split}\label{G3}
&\leq C(1+\epsilon^{2}\|\partial_{x}U\|_{L^{\infty}})(\epsilon\|\partial_{x}^{3}U\|^{2})\\
&\leq C(1+\epsilon^{2}\|U\|_{H^{2}})(\epsilon\|\partial_{x}^{3}U\|^{2}),
\end{split}
\end{equation}
where we have used Sobolev embedding theorem $H^1\hookrightarrow L^{\infty}$. In light of (<ref>) and (<ref>), we find the second term on the LHS of (<ref>) can be bounded by $C(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}$. The third term on the LHS of (<ref>) is bilinear in the unknowns and can be bounded by
\begin{equation*}
\begin{split}
\epsilon\int\partial_{x}^{3}(\partial_{x}\tilde u_{i}U)\partial_{x}^{3}U
\leq C(\|U\|_{H^{2}}^{2}+\epsilon\|\partial_{x}^{3}U\|^{2})
\leq C|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
Next, we estimate of the RHS of (<ref>). We first estimate the terms $F_{i}$ for $3\leq i\leq11$.
Estimate of $F_{3}$. Since $F_{3}$ is bilinear in the unknowns, it can be bounded by
\begin{equation*}
\begin{split}
F_{3}\le C(\|N_{e}\|_{H^{2}}^{2}+\epsilon\|\partial_{x}^{3}N_{e}\|^{2}+\epsilon\|\partial_{x}^{3}U\|^{2}),
\end{split}
\end{equation*}
where we have used Cauchy inequality.
Estimate of $F_{8}$. Using commutator notation (<ref>), we write
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
By commutator estimates (<ref>) in Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},\frac{1}{n_{e}^{3}}](\partial_{x}N_{e})^{3}\|
\leq\|\partial_{x}(\frac{1}{n_{e}^{3}})\|_{L^{\infty}}\|\partial_{x}^{2}\left((\partial_{x}^{3}N_{e})^{3}\right)\|
\end{split}
\end{equation*}
By direct computation and Sobolev embedding theorem, we note that
\begin{equation}
\begin{split}\label{use 1}
\|\partial_{x}(\frac{1}{n_{e}^{3}})\|_{L^{\infty}}
\leq C(\epsilon+\epsilon^{3}\|\partial_{x}N_{e}\|_{L^{\infty}})
\leq C(\epsilon+\epsilon^{3}\|N_{e}\|_{H^{2}}),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{use 2}
\leq &C(\epsilon+\epsilon^{3}(|\partial_{x}N_{e}|+|\partial_{x}^{2}N_{e}|+|\partial_{x}^{3}N_{e}|)\\
&+\epsilon^{6}(|\partial_{x}N_{e}|^{2} +|\partial_{x}N_{e}||\partial_{x}^{2}N_{e}|) +\epsilon^{9}|\partial_{x}N_{e}|^{3}),
\end{split}
\end{equation}
which yields that
\begin{equation}
\begin{split}\label{use 21}
\|\partial_{x}^{3}(\frac{1}{n_{e}^{3}})\|
\leq &C\big(\epsilon+\epsilon^{3}(\|\partial_{x}N_{e}\|+\|\partial_{x}^{2}N_{e}\|+\|\partial_{x}^{3}N_{e}\|)\\
\end{split}
\end{equation}
By direct computation, we have
\begin{equation}
\begin{split}\label{use 9}
\|\partial_{x}^{2}\left[(\partial_{x}N_{e})^{3}\right]\|
\leq C(\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}^{2}\|\partial_{x}N_{e}\|+\|\partial_{x}N_{e}\|_{L^{\infty}}^{2}\|\partial_{x}^{3}N_{e}\|).
\end{split}
\end{equation}
Therefore, by (<ref>), (<ref>) and (<ref>), and using Hölder inequality and Sobolev embedding $H^1\hookrightarrow L^{\infty}$, we can obtain
\begin{equation}
\begin{split}\label{s1}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation}
On the other hand, by direct computation, we have
\begin{equation}
\begin{split}\label{use 10}
\|\partial_{x}^{3}\left[(\partial_{x}N_{e})^{3}\right]\|
\leq &C(\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}^{2} \|\partial_{x}^{2}N_{e}\| \\
&+\|\partial_{x}N_{e}\|_{L^{\infty}} \|\partial_{x}^{2}N_{e}\|_{L^{\infty}} \|\partial_{x}^{3}N_{e}\| +\|\partial_{x}N_{e}\|_{L^{\infty}}^{2} \|\partial_{x}^{4}N_{e}\|).
\end{split}
\end{equation}
By applying Hölder inequality and Sobolev embedding theorem again, we have
\begin{equation}
\begin{split}\label{s2}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
Adding the estimates (<ref>) and (<ref>), we have
\begin{equation*}
\begin{split}
F_{8}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{5})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
The estimates of $F_{4}\sim F_{6}$ are similar to $F_{8}$ and can be bounded by
\begin{equation*}
\begin{split}
F_{4\sim6}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
Estimate of $F_{9}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
\Big\{[\partial_{x}^{3},\frac{1}{n_{e}^{2}}]\partial_{x}N_{e}\partial_{x}^{2}N_{e}
\end{split}
\end{equation*}
By commutator estimates (<ref>) of Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},\frac{1}{n_{e}^{2}}]\partial_{x}N_{e}\partial_{x}^{2}N_{e}\|
\leq\|\partial_{x}(\frac{1}{n_{e}^{2}})\|_{L^{\infty}}\|\partial_{x}^{2}\left(\partial_{x}N_{e}\partial_{x}^{2}N_{e}\right)\|
\end{split}
\end{equation*}
By direct computation, we note that $\|\partial_{x}(\frac{1}{n_{e}^{2}})\|_{L^{\infty}}$, $|\partial_{x}^{3}(\frac{1}{n_{e}^{2}})|$ and $\|\partial_{x}^{3}(\frac{1}{n_{e}^{2}})\|$ have similar estimates to (<ref>), (<ref>) and (<ref>). Hence we have
\begin{equation}
\begin{split}\label{use 9'}
\|\partial_{x}^{2}(\partial_{x}N_{e}\partial_{x}^{2}N_{e})\|
\leq C(\|\partial_{x}N_{e}\|_{L^{\infty}}\|\partial_{x}^{4}N_{e}\|+\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}\|\partial_{x}^{3}N_{e}\|).
\end{split}
\end{equation}
Therefore, by Hölder inequality and Sobolev embedding theorem, we obtain
\begin{equation}
\begin{split}\label{s3}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2},
\end{split}
\end{equation}
where we have used (<ref>), (<ref>) and (<ref>). On the other hand,
\begin{equation*}
\begin{split}\label{use 10'}
\|\partial_{x}^{3}(\partial_{x}N_{e} \partial_{x}^{2}N_{e})\| \leq C(\|\partial_{x}N_{e}\|_{L^{\infty}} \|\partial_{x}^{5}N_{e}\| +\|\partial_{x}^{2}N_{e}\|_{L^{\infty}} \|\partial_{x}^{4}N_{e}\| +\|\partial_{x}^{3}N_{e}\|_{L^{\infty}} \|\partial_{x}^{3}N_{e}\|).
\end{split}
\end{equation*}
Therefore, by applying Hölder inequality again, we have
\begin{equation}
\begin{split}\label{s4}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
Adding the estimates (<ref>) and (<ref>), we have
\begin{equation*}
\begin{split}
F_{9}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
$F_{7}$ is similar to $F_{9}$.
From equation (<ref>) and (<ref>) in Lemma <ref>, we can obtain
\begin{equation*}
\begin{split}
&F_{10}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2},\\
&F_{11}\leq C_{1}(1+\epsilon|\!|\!|(N_{e},U)|\!|\!|).
\end{split}
\end{equation*}
Estimate of $F_{1}+F_{2}$. By direct computation, we have
\begin{equation*}
\begin{split}
F_{1}+F_{2}=&\int\left\{\partial_{x}^{2}(n_{e}\partial_{x}N_{e})-\frac{\epsilon H^{2}}{4}\partial_{x}^{2}\left(\frac{\partial_{x}^{3}N_{e}}{n_{e}}\right)\right\}\partial_{x}^{4}U\\
=&\int\left(n_{e}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{\partial_{x}^{5}N_{e}}{n_{e}}\right)\partial_{x}^{4}U
&-\frac{\epsilon H^{2}}{4}\int\left(\sum_{\beta=1}^{2}C_{2}^{\beta} \partial_{x}^{\beta}\left(\frac{1}{n_{e}}\right)\partial_{x}^{5-\beta}N_{e}\right)\partial_{x}^{4}U\\
\end{split}
\end{equation*}
Estimates of $K_{2}$ and $K_{3}$. By integration by parts, we have
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
K_{3}=\frac{\epsilon H^{2}}{4}\int\big(\sum_{\beta=1}^{2}C_{2}^{\beta}\partial_{x}^{\beta+1}(\frac{1}{n_{e}})\partial_{x}^{5-\beta}N_{e}
\end{split}
\end{equation*}
For $K_{2}$, we have
\begin{equation*}
\begin{split}
K_{2}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
Combining (<ref>), (<ref>) and (<ref>), we obtain
\begin{equation*}
\begin{split}
K_{3}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{3})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2},
\end{split}
\end{equation*}
where we have used Hölder inequality and Sobolev embedding theorem.
Estimates of $K_{1}$. By $\eqref{rem-1}$, we have
\begin{equation*}
\begin{split}
\partial_{x}^{4}U =&\frac{1}{n_{i}}\Big\{\partial_{x}^{3}((1-u_{i})\partial_{x}N_{i})-\epsilon\partial_{t}\partial_{x}^{3}N_{i}\\
=&:\sum_{i=1}^{6}E_{i} \ .
\end{split}
\end{equation*}
Accordingly, $K_{1}$ is decomposed into
\begin{equation}\label{biaohao}
\begin{split}
K_{1}=\sum_{i=1}^{6}\int(n_{e}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}}\partial_{x}^{5}N_{e})E_{i}=:\sum_{i=1}^{6}K_{1i}.
\end{split}
\end{equation}
We first estimate the terms $K_{1i}$ for $3\leq i\leq6$ and leave $K_{11}$ and $K_{12}$ in the next two subsections. By Lemma <ref> and Lemma <ref> in the next two subsections, we have
\begin{equation*}
\begin{split}
K_{11}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}),\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
K_{12}\leq &-\frac\epsilon2\frac{d}{dt}\left[\int\frac{n_{e}}{n_{i}}(\partial_{x}^{3}N_{e})^{2}
&-\frac\epsilon2\frac{\epsilon H^{2}}{4}\frac{d}{dt}\left[\int\frac{1}{n_{e}n_{i}}(\partial_{x}^{4}N_{e})^{2}
\end{split}
\end{equation*}
Estimate of $K_{13}$. It can be decomposed that
\begin{equation*}
\begin{split}
K_{13}=\int\left(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e}\right)
\sum_{\beta=1}^{3}C_{3}^{\beta}\partial_{x}^{\beta}n_{i}\partial_{x}^{4-\beta}U.
\end{split}
\end{equation*}
When $\beta=1,2$, $K_{13}$ can be easily bounded by $C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$ by Hölder inequality, Cauchy inequality and Lemma <ref>.
When $\beta=3$, by integration by parts, we have
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\partial_{x}(\frac{1}{n_{e}n_{i}})\partial_{x}^{5}N_{e}\big)
\partial_{x}^{2}n_{i}\partial_{x}U\\
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{6}N_{e}\big)
\partial_{x}^{2}n_{i}\partial_{x}U\\
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e}\big)
\partial_{x}^{2}n_{i}\partial_{x}^{2}U\\
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use 11}
\left|\partial_{x}\left(\frac{1}{n_{e}n_{i}}\right)\right|, \ \ \left|\partial_{x}\left(\frac{n_{e}}{n_{i}}\right)\right|\leq C\left(\epsilon+\epsilon^{3}(|\partial_{x}N_{e}| +|\partial_{x}N_{i}|)\right).
\end{split}
\end{equation}
Therefore, by Hölder inequality, Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and Lemma <ref>, we have
\begin{equation*}
\begin{split}
K_{131}\leq C_{1}&\left(1+\epsilon^{2}(\|\partial_{x}N_{e}\|_{L^{\infty}}^{2}+\|\partial_{x}N_{i}\|_{L^{\infty}}^{2}
+\|\partial_{x}U\|_{L^{\infty}}^{2})\right) \left(\epsilon\|\partial_{x}^{3}N_{e}\|^{2} +\|\partial_{x}^{2}N_{i}\|^{2} +\epsilon^{3}\|\partial_{x}^{5}N_{e}\|^{2}\right)\\
\leq C_{1}&\big(1+\epsilon^{2}(\|N_{e}\|_{H^{2}}^{2}+\|N_{i}\|_{H^{2}}^{2}
\leq C_{1}&(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
On the other hand, by Hölder inequality and Lemma <ref>, $K_{132}\ and\ K_{133}$ can be bounded by $C_{1}(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$. Thus $K_{13}$ is bounded by $C_{1}(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}$.
Estimate of $K_{14}$. By Hölder inequality and Lemma <ref>, $K_{14}$ can be bounded easily by $C_{1}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$.
Estimate of $K_{15}$. By applying integration by parts and (<ref>), we have
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\partial_{x}(\frac{1}{n_{e}n_{i}})\partial_{x}^{5}N_{e}\big)
\partial_{x}^{2}(\partial_{x}\tilde{u_{i}}N_{i})\\
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{6}N_{e}\big)
\partial_{x}^{2}(\partial_{x}\tilde{u_{i}}N_{i})\\
\leq &C_{1}(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2},
\end{split}
\end{equation*}
where we have used Hölder inequality and the Lemma <ref>.
Estimate of $K_{16}$. Since $\mathcal{R}_{1}$ is known, thus we have
\begin{equation*}
\begin{split}
K_{16}\leq C_{1}(\epsilon\|\partial_{x}^{3}N_{e}\|+\epsilon^{3}\|\partial_{x}^{5}N_{e}\|).
\end{split}
\end{equation*}
Summarizing all the estimates, we complete the proof of Proposition <ref>.
§.§ Estimate of $K_{11}$
Next we estimate $K_{11}$ in (<ref>).
Let $(N_{i},N_{e},U)$ be a solution to (<ref>)Ł¬ then
\begin{equation*}
\begin{split}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
Recall that in (<ref>),
\begin{equation*}
\begin{split}
K_{11}&=\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})\partial_{x}^{3}\big((1-u_{i})\partial_{x}N_{i})\big)\\
&=\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})
\sum_{\gamma=0}^{3}C_{3}^{\gamma}\partial_{x}^{3-\gamma}(1-u_{i})\partial_{x}^{\gamma+1}N_{i}.
\end{split}
\end{equation*}
When $\gamma=0,1$, by Hölder inequality, Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and Lemma <ref>,
\begin{equation*}
\begin{split}
K_{11}|_{\gamma=0,1}&\leq C_{1}(1+\epsilon^{2}(\epsilon\|\partial_{x}^{3}U\|^{2}))
&\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
By integration by parts for $\gamma=2$ and (<ref>), we have
\begin{equation*}
\begin{split}
K_{11}|_{\gamma=2}=&3\int(\partial_{x}(\frac{n_{e}}{n_{i}})\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\partial_{x}(\frac{1}{n_{e}n_{i}})\partial_{x}^{5}N_{e})\partial_{x}u_{i}\partial_{x}^{2}N_{i}\\
&+3\int(\frac{n_{e}}{n_{i}}\partial_{x}^{4}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{6}N_{e})\partial_{x}u_{i}\partial_{x}^{2}N_{i}\\
&+3\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})\partial_{x}^{2}u_{i}\partial_{x}^{2}N_{i}\\
\leq &C(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2},
\end{split}
\end{equation*}
where we have used Hölder inequality and Lemma <ref>.
In the following we estimate $K_{11}$ for $\gamma=3$, by (<ref>), we have
\begin{equation*}
\begin{split}
\partial_{x}^{4}N_{i}=&\partial_{x}^{4}N_{e}-\epsilon\partial_{x}^{4}(n_{e}\partial_{x}^{2}N_{e})
=:&\sum_{i=1}^{22}F_{i}\ .
\end{split}
\end{equation*}
Accordingly, $K_{11}|_{\gamma=3}$ is decomposed into
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e}\big)(1-u_{i})F_{i}
=:\sum_{i=1}^{22}J_{i}\ .
\end{split}
\end{equation*}
Estimate of $J_{1}$. By integration by parts, we have
\begin{equation*}
\begin{split}
+\frac{\epsilon H^{2}}{8}\int(\partial_{x}^{4}N_{e})^{2}\partial_{x}(\frac{1-u_{i}}{n_{e}n_{i}}).
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use 12}
\left|\partial_{x}\frac{1-u_{i}}{n_{e}n_{i}}\right|,\ \ \left|\partial_{x}\frac{n_{e}(1-u_{i})}{n_{i}}\right|\leq C\left(\epsilon+\epsilon^{3}(|\partial_{x}N_{e}| +|\partial_{x}N_{i}|+|\partial_{x}U|)\right).
\end{split}
\end{equation}
Hence by Hölder inequality, Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and Lemma <ref>, we have
\begin{equation*}
\begin{split}
J_{1}\leq &C(1+\epsilon^{2}(\|N_{e}\|_{H^{2}}^{2} +\|N_{i}\|_{H^{2}}^{2}+\|U\|_{H^{2}}^{2})) (\epsilon\|\partial_{x}^{3}N_{e}\|^{2} +\epsilon^{2}\|\partial_{x}^{4}N_{e}\|^{2})\\
\leq &C((1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
Estimate of $J_{2}$. By integration by parts, we have
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\big(\partial_{x}\frac{1-u_{i}}{n_{e}n_{i}}\big)
\partial_{x}^{5}N_{e}\Big]\sum_{\alpha=0}^{3}C_{\alpha}^{3}\partial_{x}^{\alpha}n_{e}\partial_{x}^{5-\alpha}N_{e}\\
-\frac{\epsilon H^{2}}{4}\frac{1-u_{i}}{n_{e}n_{i}}
\partial_{x}^{6}N_{e}\Big]\sum_{\beta=0}^{3}C_{\beta}^{3}\partial_{x}^{\beta}n_{e}\partial_{x}^{5-\beta}N_{e}\\
\end{split}
\end{equation*}
Using the equation (<ref>), and by Hölder inequality and Sobolev embedding theorem,
\begin{equation*}
\begin{split}
\leq C((1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
When $\beta=0,1,2$, $J_{22}$ can be easily estimated by $C(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$. When $\beta=3$, by integration by parts, we have
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\big(\partial_{x}\frac{1-u_{i}}{n_{i}}\big)
\end{split}
\end{equation*}
Similar to (<ref>), we have
\begin{equation}
\begin{split}\label{use131}
\left|\partial_{x}\frac{n_{e}^{2}(1-u_{i})}{n_{i}}\right|\leq & C\left(\epsilon+\epsilon^{3}(|\partial_{x}N_{e}|+|\partial_{x}N_{i}|+|\partial_{x}U|)\right),\\
\left|\partial_{x}\frac{1-u_{i}}{n_{i}}\right|\leq & C\left(\epsilon+\epsilon^{3}(|\partial_{x}N_{i}|+|\partial_{x}U|)\right).
\end{split}
\end{equation}
Therefore, $J_{22}$ can be estimated by $C(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}$.
As a result, $J_{2}$ can be estimated by $C(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}$.
$J_{3}\sim J_{5}$ can be also estimated by
$C(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}$.
Estimate of $J_{17}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
J_{17}=&-\frac{3\epsilon^{11}H^{2}}{4}\int\big(\frac{n_{e}(1-u_{i})}{n_{i}}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1-u_{i}}{n_{e}n_{i}}\partial_{x}^{5}N_{e}\big)\\
\end{split}
\end{equation*}
By commutator estimates (<ref>), we have
\begin{equation*}
\begin{split}
\left\|\left[\partial_{x}^{4}, \frac{1}{n_{e}^{4}}\right](\partial_{x}N_{e})^{4} \right\|
\leq \left\|\partial_{x}\left(\frac{1}{n_{e}^{4}}\right)\right\|_{L^{\infty}}\|\partial_{x}^{3}(\partial_{x}N_{e})^{4}\|
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use211}
\left\|\partial_{x}^{4}\left(\frac{1}{n_{e}^{4}}\right)\right\|
\leq &C\Big(1+\epsilon^{3}(\|\partial_{x}N_{e}\| +\|\partial_{x}^{2}N_{e}\|+\|\partial_{x}^{3}N_{e}\| +\|\partial_{x}^{4}N_{e}\|)\\
&+\epsilon^{6}(\|\partial_{x}N_{e}\|_{L^{\infty}}\|\partial_{x}N_{e}\|_{H^2} +\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}\|\partial_{x}^{2}N_{e}\|)\\
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{use 17}
\|\partial_{x}^{3}(\partial_{x}N_{e})^{4}\|
\leq &C(\|\partial_{x}N_{e}\|_{L^{\infty}}^{2}\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}\|\partial_{x}^{3}N_{e}\|\\
\end{split}
\end{equation}
Therefore, using (<ref>), (<ref>) and (<ref>), by Hölder inequality and Sobolev embedding theorem, we obtain
\begin{equation}
\begin{split}\label{s5}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{8})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
On the other hand, by direct computation, we have
\begin{equation}
\begin{split}\label{use 18}
\|\partial_{x}^{4}(\partial_{x}N_{e})^{4}\|
\leq &C\big(\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}^{3}\|\partial_{x}^{2}N_{e}\|
\end{split}
\end{equation}
Therefore, by applying Hölder inequality again, we have
\begin{equation}
\begin{split}\label{s6}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
Adding the estimates (<ref>) and (<ref>), we have
\begin{equation*}
\begin{split}
J_{17}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{8})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
$J_{7}\sim J_{10},\ J_{12},\ J_{14}$ are similar to $J_{17}$.
Estimate of $J_{19}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
J_{19}=&\frac{1}{2}\epsilon^{5}H^{2}\int\left(\frac{n_{e}(1-u_{i})}{n_{i}}\partial_{x}^{3}N_{e}-\frac{\epsilon H^{2}}{4}\frac{1-u_{i}}{n_{e}n_{i}}\partial_{x}^{5}N_{e}\right)\\
+\frac{1}{n_{e}^{2}}\partial_{x}^{4}\big((\partial_{x}^{2}N_{e})^{2}\big)\right\} =:J_{191}+J_{192}.
\end{split}
\end{equation*}
By commutator estimates (<ref>), we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{4},\frac{1}{n_{e}^{2}}](\partial_{x}^{2}N_{e})^{2}\|
\leq \|\partial_{x}\frac{1}{n_{e}^{2}}\|_{L^{\infty}}\|\partial_{x}^{3}(\partial_{x}^{2}N_{e})^{2}\|
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use222}
\|\partial_{x}^{3}(\partial_{x}^{2}N_{e})^{2}\|
\leq C(\|\partial_{x}^{3}N_{e}\|_{L^{\infty}}\|\partial_{x}^{4}N_{e}\|
\end{split}
\end{equation}
Therefore, using (<ref>), (<ref>) and (<ref>), by Hölder inequality and Sobolev embedding theorem, we can obtain
\begin{equation}
\begin{split}\label{s7}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
On the other hand, by direct computation, we have
\begin{equation}
\begin{split}\label{use 22}
\|\partial_{x}^{4}(\partial_{x}^{2}N_{e})^{2}\|
\leq C(\|\partial_{x}^{4}N_{e}\|_{L^{\infty}}\|\partial_{x}^{4}N_{e}\|
\end{split}
\end{equation}
Therefore, by applying Hölder inequality again, we have
\begin{equation}
\begin{split}\label{s8}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
Adding the estimates (<ref>) and (<ref>), we have
\begin{equation*}
\begin{split}
J_{19}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
$J_{11},\ J_{13},\ J_{15},\ J_{18}$ are similar to $J_{19}$.
Estimate of $J_{21}$.
\begin{equation*}
\begin{split}
J_{21}=\frac{\epsilon^{2} H^{2}}{4}\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})(1-u_{i})\partial_{x}^{4}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})
=:J_{211}+J_{212}\ .
\end{split}
\end{equation*}
By integration of parts twice and commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
J_{211}=&\frac{\epsilon^{2} H^{2}}{4}\int\left(\frac{n_{e}(1-u_{i})}{n_{i}}\partial_{x}^{5}N_{e}
\end{split}
\end{equation*}
By commutator estimates (<ref>) of Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{2},\frac{1}{n_{e}}]\partial_{x}^{4}N_{e}\|
\leq \|\partial_{x}\frac{1}{n_{e}}\|_{L^{\infty}}\|\partial_{x}^{5}N_{e}\|
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use223}
\|\partial_{x}^{2}(\frac{n_{e}(1-u_{i})}{n_{i}})\|
\leq &C\big(\epsilon+\epsilon^{3}(\|\partial_{x}N_{e}\|+\|\partial_{x}N_{i}\|+\|\partial_{x}U\|)\\
\end{split}
\end{equation}
Therefore, by applying Hölder inequality, Sobolev embedding theorem and Lemma <ref>,
\begin{equation*}
\begin{split}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
By integration by parts and commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
&=\frac{\epsilon^{3} H^{4}}{16}\int(\frac{1-u_{i}}{n_{e}n_{i}}
\partial_{x}^{6}N_{e}+\partial_{x}(\frac{1-u_{i}}{n_{e}n_{i}})\partial_{x}^{5}N_{e})
\end{split}
\end{equation*}
Similar to $J_{211}$, using (<ref>) and (<ref>), Sobolev embedding, Cauchy inequality and Lemma <ref>, we have
\begin{equation}
\begin{split}\label{s9}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
By integration by parts, we have
\begin{equation*}
\begin{split}
J_{2122}=&-\frac{\epsilon^{3} H^{4}}{16}\int\left(\frac{1}{2}\partial_{x}(\frac{1-u_{i}}{n_{e}^{2}n_{i}})
\end{split}
\end{equation*}
Note that
\begin{equation}
\begin{split}\label{use224}
\|\partial_{x}\left(\frac{1}{n_{e}}\partial_{x}(\frac{1-u_{i}}{n_{e}n_{i}})\right)\|_{L^{\infty}}
\leq &C(\epsilon+\epsilon^{3}(\|\partial_{x}N_{e}\|_{L^{\infty}}+\|\partial_{x}N_{i}\|_{L^{\infty}}+\|\partial_{x}U\|_{L^{\infty}})\\
\end{split}
\end{equation}
Thus, by (<ref>), (<ref>), Hölder inequality, Sobolev embedding theorem and Lemma <ref>,
\begin{equation}
\begin{split}\label{s10}
\leq C(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation}
Adding the estimates (<ref>) and (<ref>), we have
\begin{equation*}
\begin{split}
J_{21}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
$J_{16}$ and $J_{20}$ are similar to $J_{21}$ and can be bounded by $C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$ˇŁ
Estimate of $J_{22}$. By using (<ref>) and (<ref>) in Lemma <ref>, similarly we have
\begin{equation*}
\begin{split}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
The proof of Lemma (<ref>) is then complete.
§.§ Estimate of $K_{12}$.
Next, we estimate $K_{12}$ in (<ref>).
Let $(N_{i},N_{e},U)$ be a solution to (<ref>), then there holds
\begin{equation*}
\begin{split}
K_{12}\leq &-\frac\epsilon2\frac{d}{dt}\left[\int\frac{n_{e}}{n_{i}}(\partial_{x}^{3}N_{e})^{2}
&-\frac\epsilon2\frac{\epsilon H^{2}}{4}\frac{d}{dt}\left[\int\frac{1}{n_{e}n_{i}}(\partial_{x}^{4}N_{e})^{2}
\end{split}
\end{equation*}
Recall that in (<ref>)
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e}\right)\partial_{t}\partial_{x}^{3}N_{i}.
\end{split}
\end{equation*}
By the $\eqref{rem-3}$, we have
\begin{equation*}
\begin{split}
\partial_{t}&\partial_{x}^{3}N_{i}=\partial_{t}\partial_{x}^{3}N_{e}-\epsilon\partial_{t}\partial_{x}^{3}\big(n_{e}\partial_{x}^{2}N_{i}\big)
=:&\sum_{i=1}^{22}D_{i}\ .
\end{split}
\end{equation*}
Accordingly, $K_{12}$ is decomposed into
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})D_{i}
=:\sum_{i=1}^{22}I_{i}\ .
\end{split}
\end{equation*}
Estimate of $I_{1}$.
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})\partial_{t}\partial_{x}^{3}N_{e}=:I_{11}+I_{12}.
\end{split}
\end{equation*}
By integration by parts, we have
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use 23}
\|\partial_{t}\frac{n_{e}}{n_{i}}\|_{L^{\infty}}
\leq C\big(\epsilon+\epsilon^{3}(\|\partial_{t}N_{e}\|_{L^{\infty}}+\|\partial_{t}N_{i}\|_{L^{\infty}})\big).
\end{split}
\end{equation}
Thus by Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
\frac{\epsilon}{2}\int(\partial_{t}\frac{n_{e}}{n_{i}})(\partial_{x}^{3}N_{e})^{2}
&\leq C\|\partial_{t}\frac{n_{e}}{n_{i}}\|_{L^{\infty}}(\epsilon\|\partial_{x}^{3}N_{e}\|^{2})\\
&\leq C_{1}(1+\epsilon^{2}(\|\epsilon\partial_{tx}N_{e}\|^{2}+\|\epsilon\partial_{tx}N_{i}\|^{2}))(\epsilon\|\partial_{x}^{3}N_{e}\|^{2})\\
&\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
Applying integration by parts again twice, we have
\begin{equation*}
\begin{split}
&-\frac{\epsilon^{2} H^{2}}{4}\int\partial_{x}(\frac{1}{n_{e}n_{i}})\partial_{x}^{4}N_{e}\partial_{t}\partial_{x}^{3}N_{e}\\
=&:I_{121}+I_{122}+I_{123}\ .
\end{split}
\end{equation*}
Note that the estimate of $\|\partial_{t}(\frac{1}{n_{e}n_{i}})\|_{L^{\infty}}$ is similar to that for (<ref>), thus similarly $I_{122}$ can be estimated by $C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}$. By (<ref>), Sobolev embedding theorem and Cauchy inequality, we have
\begin{equation*}
\begin{split}
I_{123}&\leq C\big(1+\epsilon^{2}(\|\partial_{x}N_{e}\|_{L^{\infty}}^{2}
&\leq C\big(1+\epsilon^{2}(\|N_{e}\|_{H^{2}}^{2}+\|N_{i}\|_{H^{2}}^{2})\big)
&\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2},
\end{split}
\end{equation*}
where we have used Lemma <ref>-<ref>.
Therefore, we obtain
\begin{equation}
\begin{split}\label{w1}
\end{split}
\end{equation}
Estimate of $I_{2}$. Recall that
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})\partial_{t}\partial_{x}^{3}(n_{e}\partial_{x}^{2}N_{e})
=:I_{21}+I_{22}\ .
\end{split}
\end{equation*}
Estimate of $I_{21}$. By integration by parts, we have
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
Estimate of $I_{211}$. By direct computation, we have
\begin{equation*}
\begin{split}
\big(2\partial_{x}n_{e}\partial_{x}^{3}N_{e}+\partial_{x}^{2}n_{e}\partial_{x}^{2}N_{e}\big)\\
\end{split}
\end{equation*}
Note that the estimate of $\|\partial_{t}({n_{e}^{2}}/{n_{i}})\|_{L^{\infty}}$ is similar to (<ref>), thus by integration by parts,
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
By Hölder inequality, Cauchy inequality, Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
By (<ref>) and direct computation, we have
\begin{equation*}
\begin{split}
I_{212}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2})(1+|\!|\!|(N_{e}, U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
Estimate of $I_{22}$. By direct computation, we have
\begin{equation*}
\begin{split}
\big(n_{e}\partial_{x}^{5}N_{e}+\sum_{\beta=1}^{3}C_{3}^{\beta}\partial_{x}^{\beta}n_{e}\partial_{x}^{5-\beta}N_{e}\big)\\
\big(\sum_{\beta=1}^{3}C_{3}^{\beta}\partial_{x}^{\beta}n_{e}\partial_{x}^{5-\beta}N_{e}\big)\\
\end{split}
\end{equation*}
By integration by parts in $t$, we have
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
Note that
\begin{equation}
\begin{split}\label{use 5}
\|\partial_{t}\frac{1}{n_{i}}\|_{L^{\infty}}\leq C(\epsilon+\epsilon^{3}\|\partial_{t}{N_{i}}\|_{L^{\infty}}).
\end{split}
\end{equation}
Therefore, by Sobolev embedding theorem and Lemma <ref>, we have
\begin{equation*}
\begin{split}
\frac{H^{2}}{4}\frac{\epsilon^{2}}{2}\int(\partial_{x}^{5}N_{e})^{2}\partial_{t}\frac{1}{n_{i}}
&\leq C_{1}(1+\epsilon^{2}\|\epsilon\partial_{t}N_{i}\|_{H^{1}}^{2})(\epsilon^{3}\|\partial_{x}^{5}N_{e}\|^{2})\\
&\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
Similarly, by Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{222}\leq C_{1}(1+\epsilon^{2}\|\epsilon\partial_{t}N_{e}\|_{H^{1}}^{2})(\epsilon^{3}\|\partial_{x}^{5}N_{e}\|^{2})
\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
By Cauchy inequality, Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{223}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
Therefore, we have
\begin{equation}
\begin{split}\label{w2}
\end{split}
\end{equation}
Estimate of $I_{3}$.
\begin{equation*}
\begin{split}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})\partial_{t}\partial_{x}^{3}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})
\end{split}
\end{equation*}
Estimate of $I_{31}$.
By integration by parts twice,
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
By integration by parts in $t$, Sobolev embedding, (<ref>) and Lemma <ref>, we have
\begin{equation*}
\begin{split}
\leq &-\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{d}{dt}\int\frac{1}{n_{i}}(\partial_{x}^{5}N_{e})^{2}
\end{split}
\end{equation*}
Note that
\begin{equation}
\begin{split}\label{use 4}
\|\partial_{t}\frac{1}{n_{e}}\|_{L^{\infty}}
\leq C(\epsilon+\epsilon^{3}\|\partial_{t}N_{e}\|_{L^{\infty}}),
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{use 27}
\|\partial_{tx}\frac{1}{n_{e}}\|_{L^{\infty}}
\leq C\big(\epsilon+\epsilon^{3}(\|\partial_{t}N_{e}\|_{L^{\infty}}+\|\partial_{x}N_{e}\|_{L^{\infty}})
\end{split}
\end{equation}
By Sobolev embedding $H^1\hookrightarrow L^{\infty}$, Cauchy inequality and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
Therefore, we obtain
\begin{equation*}
\begin{split}
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation*}
\begin{split}
\big(\frac{1}{n_{e}}\partial_{t}\partial_{x}^{5}N_{e}+\partial_{t}\frac{1}{n_{e}}\partial_{x}^{5}N_{e}
\end{split}
\end{equation*}
By Sobolev embedding, Cauchy inequality and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{312}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}),
\end{split}
\end{equation*}
where we have used (<ref>), (<ref>) and (<ref>).
$I_{313}$ is similar to $I_{312}$, thus we have
\begin{equation}
\begin{split}\label{a2}
\end{split}
\end{equation}
Estimate of $I_{32}$. By integration by parts, we have
\begin{equation*}
\begin{split}
I_{32}&=\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\frac{1}{n_{e}n_{i}}
\partial_{x}^{5}N_{e}\partial_{t}\partial_{x}^{3}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})\\
&=-\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\frac{1}{n_{e}n_{i}}\partial_{x}^{6}N_{e}
\partial_{t}\partial_{x}^{2}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})
-\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int(\partial_{x}\frac{1}{n_{e}n_{i}})\partial_{x}^{5}N_{e}
\partial_{t}\partial_{x}^{2}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})\\
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation*}
\begin{split}
I_{321}=&-\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\frac{1}{n_{e}n_{i}}\partial_{x}^{6}N_{e}\partial_{t}\partial_{x}^{6}N_{e}\\
&-\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\frac{1}{n_{e}n_{i}}\partial_{x}^{6}N_{e}
\big((\partial_{t}\frac{1}{n_{e}})\partial_{x}^{6}N_{e}+2(\partial_{tx}\frac{1}{n_{e}})\partial_{x}^{5}N_{e}\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +2(\partial_{x}\frac{1}{n_{e}})\partial_{t}\partial_{x}^{5}N_{e}+(\partial_{x}^{2}\frac{1}{n_{e}})\partial_{t}\partial_{x}^{4}N_{e}
\end{split}
\end{equation*}
By integration by parts in $t$, we have
\begin{equation*}
\begin{split}
I_{3211}=-\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\int\frac{1}{n_{e}^{2}n_{i}}(\partial_{x}^{6}N_{e})^{2}
+\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\partial_{t}(\frac{1}{n_{e}^{2}n_{i}})(\partial_{x}^{6}N_{e})^{2}.
\end{split}
\end{equation*}
Similar to (<ref>), $\|\partial_{t}({1}/{n_{e}^{2}n_{i}})\|_{L^{\infty}}$ has same bound with $\|\partial_{t}({n_{e}}/{n_{i}})\|_{L^{\infty}}$. Therefore, by Sobolev embedding $H^1\hookrightarrow L^{\infty}$ and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\partial_{t}(\frac{1}{n_{e}^{2}n_{i}})(\partial_{x}^{6}N_{e})^{2}
&\leq C(1+\epsilon^{2}(\|\epsilon\partial_{t}N_{e}\|_{L^{\infty}}^{2}
&\leq C(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}.
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use 235}
\|\partial_{t}\partial_{x}^{2}\frac{1}{n_{e}}\|_{L^{\infty}}
\leq &C\big(\epsilon+\epsilon^{3}(\|\partial_{x}N_{e}\|_{L^{\infty}}+\|\partial_{t}N_{e}\|_{L^{\infty}}
\end{split}
\end{equation}
By (<ref>), (<ref>), (<ref>) and (<ref>), we have
\begin{equation*}
\begin{split}
I_{3212}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}),
\end{split}
\end{equation*}
where we have used Sobolev embedding theorem, Lemma <ref> and <ref>. Thus, we have
\begin{equation}
\begin{split}\label{two}
I_{321}\leq-\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\int\frac{1}{n_{e}^{2}n_{i}}(\partial_{x}^{6}N_{e})^{2}
\end{split}
\end{equation}
By integration by parts, we have
\begin{equation*}
\begin{split}
I_{322}=&-\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\partial_{x}^{5}N_{e}(\partial_{x}\frac{1}{n_{e}n_{i}})
\partial_{t}\partial_{x}^{2}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})\\
=&\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\partial_{x}^{6}N_{e}(\partial_{x}\frac{1}{n_{e}n_{i}})
\partial_{t}\partial_{x}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})\\
&+\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int\partial_{x}^{5}N_{e}(\partial_{x}^{2}\frac{1}{n_{e}n_{i}})
\partial_{t}\partial_{x}(\frac{\partial_{x}^{4}N_{e}}{n_{e}})\\
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation*}
\begin{split}
I_{3221}=\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\int&\partial_{x}^{6}N_{e}(\partial_{x}\frac{1}{n_{e}n_{i}})
\Big(\frac{1}{n_{e}}\partial_{t}\partial_{x}^{5}N_{e}+(\partial_{t}\frac{1}{n_{e}})\partial_{x}^{5}N_{e}\\
\end{split}
\end{equation*}
By (<ref>), (<ref>), (<ref>) and (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{3221}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{use233}
\|\partial_{x}^{2}(\frac{1}{n_{e}n_{i}})\|_{L^{\infty}}
\leq &C\big(\epsilon+\epsilon^{3}(\|\partial_{x}N_{e}\|_{L^{\infty}}+\|\partial_{x}N_{i}\|_{L^{\infty}})
\end{split}
\end{equation}
Thus by (<ref>), (<ref>), (<ref>) and (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{3222}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
Therefore, we have
\begin{equation}
\begin{split}\label{a1}
I_{322}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation}
Adding (<ref>) and (<ref>), we have
\begin{equation}
\begin{split}\label{three}
I_{32}\leq-\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\int\frac{1}{n_{e}^{2}n_{i}}(\partial_{x}^{6}N_{e})^{2}
\end{split}
\end{equation}
Combining to (<ref>) and (<ref>), we have
\begin{equation}
\begin{split}\label{w3}
-\frac{1}{2}\frac{\epsilon^{3}H^{2}}{4}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\int\frac{1}{n_{e}^{2}n_{i}}(\partial_{x}^{6}N_{e})^{2}\\
\end{split}
\end{equation}
Thus combining (<ref>), (<ref>) and (<ref>), we obtain
\begin{equation*}
\begin{split}
\sum_{i=1}^{3}I_{i}
&-\frac{1}{2}\frac{\epsilon H^{2}}{4}\frac{d}{dt}\left\{\int\frac{1}{n_{e}n_{i}}(\partial_{x}^{3}N_{e})^{2}+\epsilon\int\frac{1}{n_{i}}(\partial_{x}^{4}N_{e})^{2}
+\frac{\epsilon^{2} H^{2}}{4}\int\frac{1}{n_{e}^{2}n_{i}}(\partial_{x}^{5}N_{e})^{2}\right\}\\
\end{split}
\end{equation*}
By Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{4,5,6,7}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation*}
Estimate of $I_{18}$. By direct computation, we have
\begin{equation*}
\begin{split}
I_{18}&=\frac{3 H^{2}}{4}\epsilon^{12}\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})
\partial_{x}^{3}\big\{(\partial_{t}\frac{1}{n_{e}^{4}})(\partial_{x}N_{e})^{4}
&=:I_{181}+I_{182}\ .
\end{split}
\end{equation*}
Estimate of $I_{181}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
\partial_{x}^{3}(\partial_{t}(\frac{1}{n_{e}^{4}})(\partial_{x}N_{e})^{4}
\end{split}
\end{equation*}
By commutator estimate (<ref>), we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},\partial_{t}(\frac{1}{n_{e}^{4}})](\partial_{x}N_{e})^{4}\|
\leq\|\partial_{tx}(\frac{1}{n_{e}^{4}})\|_{L^{\infty}}\|\partial_{x}^{2}(\partial_{x}N_{e})^{4}\|
\end{split}
\end{equation*}
Note that the estimate of $\|\partial_{tx}(\frac{1}{n_{e}^{4}})\|_{L^{\infty}}$ is similar to that for (<ref>). We note that
\begin{equation}
\begin{split}\label{h1}
\|\partial_{x}^{2}(\partial_{x}N_{e})^{4}\|\leq C(\|\partial_{x}N_{e}\|^{2}_{L^{\infty}}\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}\|\partial_{x}^{2}N_{e}\|
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{h2}
\|\partial_{t}&\partial_{x}^{3}(\frac{1}{n_{e}^{4}})\| \leq C\big(\epsilon+\epsilon^{3}(\|\partial_{x}N_{e}\|+\|\partial_{t}N_{e}\|+\|\partial_{x}^{2}N_{e}\|
\end{split}
\end{equation}
Thus by (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{181}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})
\end{split}
\end{equation*}
Estimate of $I_{182}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
\partial_{x}^{3}(\frac{1}{n_{e}^{4}}(\partial_{x}N_{e})^{3}\partial_{tx}N_{e})
\end{split}
\end{equation*}
By commutator estimate (<ref>) of Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},\frac{1}{n_{e}^{4}}](\partial_{x}N_{e})^{3}\partial_{tx}N_{e})\|
\leq&\|\partial_{x}(\frac{1}{n_{e}^{4}})\|_{L^{\infty}}\|\partial_{x}^{2}((\partial_{x}N_{e})^{3}\partial_{tx}N_{e})\|\\
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{m3}
\|\partial_{x}^{2}&((\partial_{x}N_{e})^{3}\partial_{tx}N_{e})\|
\leq C(\|\partial_{x}N_{e}\|_{L^{\infty}}\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}^{2}\|\partial_{tx}N_{e}\|\\
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{m4}
\|\partial_{x}^{3}&((\partial_{x}N_{e})^{3}\partial_{tx}N_{e})\|
\leq C(\|\partial_{x}N_{e}\|_{L^{\infty}}\|\partial_{x}^{2}N_{e}\|_{L^{\infty}}\|\partial_{x}^{3}N_{e}\|_{L^{\infty}}\|\partial_{tx}N_{e}\|\\
\end{split}
\end{equation}
Thus by (<ref>), (<ref>), (<ref>) and (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{182}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})
\end{split}
\end{equation*}
Therefore, we have
\begin{equation*}
\begin{split}
I_{18}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})
\end{split}
\end{equation*}
The estimates of $I_{8}\sim I_{11},\ I_{13}\ and\ I_{15}$ are similar to that for $I_{18}$.
Estimate of $I_{21}$. By direct computation, we have
\begin{equation*}
\begin{split}
I_{21}&=\frac{3 H^{2}}{4}\epsilon^{6}\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})
\partial_{x}^{3}\big\{(\partial_{t}\frac{1}{n_{e}^{2}})\partial_{x}N_{e}\partial_{x}^{3}N_{e}
\end{split}
\end{equation*}
Estimate of $I_{211}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
\partial_{x}^{3}((\partial_{t}\frac{1}{n_{e}^{2}})\partial_{x}N_{e}\partial_{x}^{3}N_{e}
\end{split}
\end{equation*}
By commutator estimate (<ref>) in Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},\partial_{t}(\frac{1}{n_{e}^{2}})]\partial_{x}N_{e}\partial_{x}^{3}N_{e}\|
\leq&\|\partial_{tx}(\frac{1}{n_{e}^{2}})\|_{L^{\infty}}\|\partial_{x}^{2}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\|\\
\end{split}
\end{equation*}
Note that the estimate of $\|\partial_{tx}(\frac{1}{n_{e}^{2}})\|_{L^{\infty}}$ is similar to that for (<ref>). By direct computation, we note that
\begin{equation}
\begin{split}\label{m7}
\|\partial_{x}^{2}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\|
\leq C(\|\partial_{x}^{3}N_{e}\|_{L^{\infty}}\|\partial_{x}^{3}N_{e}\|
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{m8}
\|\partial_{x}^{3}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\|
\leq C(\|\partial_{x}^{3}N_{e}\|_{L^{\infty}}\|\partial_{x}^{4}N_{e}\|
\end{split}
\end{equation}
Thus by (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{211}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})
\end{split}
\end{equation*}
Estimate of $I_{212}$. Using commutator notation (<ref>), we have
\begin{equation*}
\begin{split}
I_{212}=&\frac{3 H^{2}}{4}\epsilon^{6}\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})
\partial_{x}^{3}\big\{
\frac{1}{n_{e}^{2}}\partial_{t}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\big\}\\
=&\frac{3 H^{2}}{4}\epsilon^{6}\int(\frac{n_{e}}{n_{i}}\partial_{x}^{3}N_{e}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}n_{i}}\partial_{x}^{5}N_{e})
\big\{[\partial_{x}^{3},\frac{1}{n_{e}^{2}}]\partial_{t}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{n_{e}^{2}}\partial_{t}\partial_{x}^{3}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\big\}\\
\end{split}
\end{equation*}
By commutator estimate (<ref>) in Lemma <ref>, we have
\begin{equation*}
\begin{split}
\|[\partial_{x}^{3},\frac{1}{n_{e}^{2}}]\partial_{t}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\|
\leq&\|\partial_{x}(\frac{1}{n_{e}^{2}})\|_{L^{\infty}}\|\partial_{x}^{2}(\partial_{t}(\partial_{x}N_{e}\partial_{x}^{3}N_{e}))\|\\
\end{split}
\end{equation*}
By direct computation, we have
\begin{equation}
\begin{split}\label{m88}
\|\partial_{t}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\|_{L^{\infty}}
\leq C(\|\partial_{tx}N_{e}\|_{L^{\infty}}\|\partial_{x}^{3}N_{e}\|_{L^{\infty}}
\end{split}
\end{equation}
\begin{equation}
\begin{split}\label{m9}
\|\partial_{t}\partial_{x}^{2}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\|
\leq &C(\|\partial_{x}N_{e}\|_{L^{\infty}}\|\partial_{t}\partial_{x}^{5}N_{e}\|
\end{split}
\end{equation}
Thus by (<ref>), (<ref>), (<ref>) and (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>,
\begin{equation*}
\begin{split}
I_{2121}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})
\end{split}
\end{equation*}
By integration by parts, we have
\begin{equation*}
\begin{split}
I_{2122}=&-\frac{3 H^{2}}{4}\epsilon^{6}\int\big\{\partial_{x}(\frac{1}{n_{e}n_{i}})\partial_{x}^{3}N_{e}
-\frac{\epsilon H^{2}}{4}\partial_{x}(\frac{1}{n_{e}^{3}n_{i}})\partial_{x}^{5}N_{e}\big\}
\partial_{t}\partial_{x}^{2}(\partial_{x}N_{e}\partial_{x}^{3}N_{e})\\
&-\frac{3 H^{2}}{4}\epsilon^{6}\int\big\{\frac{1}{n_{e}n_{i}}\partial_{x}^{4}N_{e}
-\frac{\epsilon H^{2}}{4}\frac{1}{n_{e}^{3}n_{i}}\partial_{x}^{6}N_{e}\big\}
\partial_{t}\partial_{x}^{2}(\partial_{x}N_{e}\partial_{x}^{3}N_{e}).
\end{split}
\end{equation*}
By direct computation, we note that $\partial_{x}\frac{1}{n_{e}^{3}n_{i}}$ has same estimate with (<ref>), thus by (<ref>), Sobolev embedding theorem and Lemma <ref>-<ref>, we have
\begin{equation*}
\begin{split}
I_{2122}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2})
\end{split}
\end{equation*}
Therefore, we have
\begin{equation*}
\begin{split}
I_{212}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})
\end{split}
\end{equation*}
and hence
\begin{equation*}
\begin{split}
I_{21}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})
\end{split}
\end{equation*}
The estimates of $I_{12},\ I_{14},\ I_{16},\ I_{17},\ I_{19}\ and\ I_{20}$ are similar to that of $I_{18}$. According to the Lemma <ref>, we have
\begin{equation*}
\begin{split}
I_{22}\leq C_{1}(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{4})
\end{split}
\end{equation*}
The proof of Lemma <ref> is then complete.
§ PROOF OF THEOREM <REF>
Adding Propositions <ref> with $\gamma=0,1,2$ and Proposition <ref> together, we obtain
\begin{equation}\label{Gron}
\begin{split}
\frac{1}{2}&\frac{d}{dt}(\|U\|_{H^{2}}^{2}+\epsilon\|\partial_{x}^{3}U\|_{L^{2}}^{2})
+\frac{1}{2}\frac{d}{dt}\Big\{\int\frac{n_{e}}{n_{i}}(\sum_{i=0}^{2}| \partial_{x}^{i}N_{e}|^{2}+\epsilon|\partial_{x}^{3}N_{e}|^{2})\Big\}\\
&+\frac{1}{2}\frac{d}{dt}\Big\{\int\epsilon(\frac{n_{e}^{2}}{n_{i}}+\frac{H^{2}}{8n_{e}n_{i}})(\sum_{i=0}^{3}| \partial_{x}^{i}N_{e}|^{2}+\epsilon|\partial_{x}^{4}N_{e}|^{2})\Big\}\\
&+\frac{3H^{2}}{16}\frac{d}{dt}\Big\{\int\frac{\epsilon^{2}}{n_{i}}(\sum_{i=0}^{4}| \partial_{x}^{i}N_{e}|^{2}+\epsilon|\partial_{x}^{5}N_{e}|^{2})\Big\}\\
&+\frac{H^{2}}{8}\frac{d}{dt}\Big\{\int\frac{\epsilon^{3}}{n_{e}^{2}n_{i}}(\sum_{i=0}^{5}| \partial_{x}^{i}N_{e}|^{2}+\epsilon|\partial_{x}^{6}N_{e}|^{2})\Big\}\\
\leq &C(1+\epsilon^{2}|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{6})(1+|\!|\!|(N_{e},U)|\!|\!|_{\epsilon}^{2}).
\end{split}
\end{equation}
Integrating the inequality (<ref>) over $(0,t)$ yields
\begin{equation*}
\begin{split}
&\leq C|\!|\!|(N_{e},U)(0)|\!|\!|_{\epsilon}^{2}
&\leq C_{1}|\!|\!|(N_{e},U)(0)|\!|\!|_{\epsilon}^{2}+\int_{0}^{t}C_{1}
\end{split}
\end{equation*}
where $C$ is an absolute constant.
Recall that $C_1$ depends on $|\!|\!|(N_{e},U)|\!|\!|^2_{\epsilon}$ through $\epsilon|\!|\!|(N_{e},U)|\!|\!|^2_{\epsilon}$ and is nondecreasing. Let $C_1'=C_1(1)$ and $C_2>C\sup_{\epsilon<1}|\!|\!|(u^{\epsilon}_R,\phi^{\epsilon}_R) (0)|\!|\!|^2_{\epsilon}$. For any arbitrarily given $\tau>0$, we choose $\tilde C$ sufficiently large such that $\tilde C>e^{4C_1'\tau}(1+C_2)(1+C_1')$. Then there exists $\epsilon_0>0$ such that ${\epsilon}\tilde C\leq 1$ for all $\epsilon<\epsilon_0$, we have
\begin{equation}\label{e999}
\begin{split}
\sup_{0\leq t\leq\tau}|\!|\!|(N_{e},U)(t)|\!|\!|^2_{\epsilon}\leq e^{4C_1'\tau}(C_2+1)<\tilde C.
\end{split}
\end{equation}
In particular, we have the uniform bound for $(N_{e},U)$,
\begin{equation}
\begin{split}\label{final}
\sup_{0\leq t\leq\tau}\Big(&\|(N_{e},U)\|_{H^{2}}^{2} +\epsilon\|(\partial_{x}^{3}N_{e}, \partial_{x}^{3}U)\|_{L^{2}}^{2}\\
&+\epsilon^{2}\|\partial_{x}^{4}N_{e}\|_{L^{2}}^{2} +\epsilon^{3}\|\partial_{x}^{5}N_{e}\|_{L^{2}}^{2} +\epsilon^{4}\|\partial_{x}^{6}N_{e}\|_{L^{2}}^{2}\Big) \leq \tilde C.
\end{split}
\end{equation}
On the other hand, by Lemma <ref> and (<ref>), we have
\begin{equation*}
\begin{split}
\sup_{0\leq t\leq\tau}\|N_{i}\|_{H^2}^2\leq \tilde C.
\end{split}
\end{equation*}
It is now standard to obtain uniform estimates independent of $\epsilon$ by the continuity method.
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Dynamic surface control (DSC) method uses high gain filters to avoid the "explosion of complexity" issue inherent in backstepping based controller designs. As a result, the closed loop system and filter dynamics possess time scale separation between them. This paper attempts to design a novel disturbance observer based dynamic surface controller using contraction framework. In doing so the steady state error bounds are obtained in terms of design parameters which are exploited to tune the closed loop system performance. The results not only show that DSC technique recover the performance of a backstepping controller for a small range of filter parameter but also derive the maximum bound for it. Furthermore the stability bounds are also derived in the presence of disturbances and convergence of trajectories to a small penultimate bound is proved. The convergence results are shown to hold for less conservative choice of filter parameter and observer gain. The effectiveness of the proposed controller is verified through simulation example.
Contraction Theory, Singular Perturbation, Dynamic Surface Control, Disturbance Observer
§ INTRODUCTION
Designing backstepping controller for nonlinear system in parametric strict feedback forms involves calculating multiple numbers of virtual control variables ($\alpha_i$) and its derivatives <cit.>. The computations of these derivatives become cumbersome and unreliable with increase in order of the system. This problem is popularly termed as explosion of complexity. To overcome this problem a new methodology called dynamic surface control was introduced in <cit.>. The core concept of dynamic surface control (DSC) is to make the virtual control variables to pass through high gain filters thereby avoiding the differentiations involved <cit.>. Moreover the differentiability assumption on the dynamics is somewhat relaxed and the technique is also able to deal with plant uncertainties.
Many variants of DSC are proposed in literature depending on the uncertainties involved in system dynamics and practical implementation of the technique. An observer based DSC is proposed in <cit.> using a Lyapunov based LMI approach, a neural network based DSC is proposed in <cit.> and a DSC frameowrk robust to unknown deadzone is proposed in <cit.>. Some of the interesting application of DSC can be found in <cit.> and references there in. However all the existing literatures utilize a high gain filter to overcome the analytical calculation of derivatives which is central concept of DSC. Recently a singular perturbation based analysis is provided in <cit.> for DSC which elaborately discuss the time scale separation in it. But the semi-global results obtained in <cit.> are valid for a small range of perturbation (filter) parameter and it points out that the steady state behavior of the closed loop system does not depend on selection of large control gains.
Relation between closed loop system response and selection of controller parameters is the most critical issue in DSC based controller designs. In this paper we propose contraction theory based approach to answer this question and quantify the stability bounds in terms of design parameters. The overall control scheme proposed here comprises of a standard DSC law along with a high gain disturbance observer. The resulting closed loop system is shown to be in three time scale. A Composite Lyapunov function based stability analysis can be carried out to find a conservative bound on the filter parameter and observer gain. However for this purpose many interconnection conditions <cit.> need to be satisfied which are not easy to search in presence of disturbances. Instead of this approach, we adopt contraction theory which does not require any interconnection conditions and relax the conservative bound on filter parameter. We exploit the partial contraction tools and robustness property of contractive system to derive exact relationship between design parameters and closed loop system performance which is helpful in tuning the controller. Moreover the stability bounds obtained here does not limit the filter parameter to be within a small range. The advantage of using contraction framework is more evident in presence of disturbances where arbitrarily reducing the magnitude of filter parameter and increasing the observer gain may lead to peaking phenomenon. The results shows that by proper selection of controller gains, the steady state error bounds can be reduced even for less conservative value of observer gain and filter parameter.
Throughout this paper, we adopt the following notations and symbols. $B_x, B_z$ denote compact subsets, $R^m$ denotes a m-dimensional real vector space. For real vectors $v$, $||v||$ denotes the Euclidean norm and for real matrix $||E||$ denotes induced matrix norm. The matrix measure induced by infinity norm is denoted as $\rho_\infty$. A metric $\Theta$ denotes a symmetric positive definite matrix and $I_n$ is an $n \times n$ identity matrix.
§.§ Prerequisites From Contraction Theory
Contraction is a stability tool to study differential behavior of dynamical systems <cit.>. A system of the form $\dot{x}=f(x,t)$ is said to be contracting if all trajectories starting inside some region in state space will converge to each other <cit.>. A region in the state space is called a contraction region if the following inequality is satisfied for the system dynamics.
\begin{equation}\label{cont1}
F=(\dot{\Theta}+\Theta\frac{\partial{f}}{\partial{x}})\Theta^{-1} \leq -\lambda I
\end{equation}
\begin{equation}\label{econt}
\Rightarrow (\dot{M}+M\frac{\partial{f}}{\partial{x}}+\frac{\partial{f}}{\partial{x}}^{T}M) \leq -2\lambda M
\end{equation}
where $\Theta$ is a nonsingular matrix, $\lambda$ is referred as contraction rate and $M=\Theta^T\Theta$ is called contraction metric. The quantity $F$ is called generalized Jacobian which should be negative definite for (<ref>) to be satisfied. The quantity $-\lambda$ refers to the presence of a maximum negative eigen value or existence of a uniformly negative matrix measure <cit.>. Some important results from previous literatures which are used in this paper are outlined in the form of following lemmas. Detailed proofs of these results can be found in <cit.>Lemma-1: Suppose an autonomous system $\dot{\it \bf x}=\it \bf f(\it \bf x)$ is globally contracting with a nonsingular metric $\rm \bf \Theta(\it \bf x)$ then all the trajectories of this system will converge to an unique equilibrium point.
The robustness property of contraction in a perturbed nonlinear system is summarized in the form of a lemma. Lemma-2: Define a perturbed system of the following form.
\begin{equation} \label{pertsys}
\dot{\it \bf x_p}= \it \bf f(\it \bf x_p, t)+\it \bf d(\it \bf x_p,w, t)
\end{equation}
where $w$ is external parameter. Suppose the system $\dot{\it \bf x}=\it \bf f(\it \bf x, t)$ is contracting using a nonsingular metric $\rm \bf \Theta$ with a rate $\it \lambda$. Then the following two cases arise.
a) Suppose the Jacobian of the perturbation term $\||\frac{\partial{\it \bf d(\it \bf x_p, w, t)}}{\partial{\it \bf x_p}}\|| \leq \it \lambda$ for $\forall t > 0$, then the perturbed system is still contracting and the trajectories of the perturbed system will exponentially converge to the trajectories of nominal(unperturbed) system. i.e
\begin{equation}\label{lem21}
\lim_{t \to \infty} ||\it \bf x_p(t)-x(t)|| \rightarrow 0
\end{equation}
b) If the perturbation term is bounded in any norm, then the difference between trajectories of the perturbed system and the nominal system will converge to a steady state bound given as:
\begin{equation}\label{lem22}
\lim_{t \to \infty} ||\it \bf x_p(t)-x(t)|| \leq \frac{\it \Upsilon \it d}{\it \lambda}
\end{equation}
where $\it \Upsilon$ is the condition number of the metric $\rm \bf \Theta$ and $||\it \bf d(\it \bf x_p, w, t)|| \leq \it d$.
Apart from these properties, contraction framework provides a very useful tool called partial contraction <cit.>, which find application in filter design and synchronization problems <cit.>.
Definition-1: Consider the dynamical system in (<ref>). Define a auxiliary/copy system
\begin{equation}\label{defpar}\dot{y}=f(y, t)+d(x_p, w, t).\end{equation}
Replacing the variable $y$ in (<ref>) by $x_p$ we retrieve (<ref>). So the variable $x_p(t)$ of (<ref>) is a particular solution of (<ref>). Hence (<ref>) will be referred as a virtual system for (<ref>).
A System $\dot{\it \bf x}=\it \bf f(\it \bf x, \it \bf y, t)$ is said to be partially contracting in $\it \bf x$ if an auxiliary system defined by $\dot{\it \bf z}=f(\it \bf z, \it \bf y,t)$ is contracting for any value of $\it \bf y, \forall t>0$. If the auxiliary system verifies a smooth specific property, then the trajectories of original system will also verify that property exponentially.
§ SYSTEM DESCRIPTION
We plan to discuss dynamic surface control for the systems in the following form.
\begin{equation}\label{rec1}
\begin{split}
& \dot{x}_1 = f_1(x_1)+b_1(x_1)x_2+d_1\\
& \dot{x}_2 = f_2(x_1,x_2)+b_2(x_1,x_2)x_3+d_2\\
& \dots\\
& \dot{x}_n = f_n(x_1,x_2,x_3....x_n)+b_n(x_1,x_2,\hdots,x_n)u+d_n\\
\end{split}
\end{equation}
where $f(x)=\begin{bmatrix}f_1 & f_2 &\hdots &f_n\end{bmatrix}^T\in R^n$ is a smooth vector field, $b_i(.) \geq 0 \ \forall x \in R^n$ and $u \in R$ is the control input. The unknown disturbances $d_i$ are assumed to be slowly varying.
The problem is to design a control law such that the output $x_1(t)$ tracks the desired signal $x_d(t)$.
Assumption 1: The desired signal $x_d(t)$ and all its derivatives are continuous and bounded.
Assumption 2: The disturbance dynamics is assumed to be in the form of:
\begin{equation}\label{ddy1}
\dot{d}_i=\Xi(d_i)
\end{equation}
$\Xi(.)$ is unknown but bounded, i.e. $|\Xi(d_i)| \leq c_1$ in the region of interest where $c_1$ is a positive constant. The assumption is not restrictive in nature and satisfied for wide range of disturbances like friction, saturation etc <cit.>.
§.§ Backstepping Design
Integrator backstepping design provides an elegant method for stabilization of strict feedback and parametric strict feedback nonlinear systems <cit.>. The controller design is recursive in nature and can deal with uncertainties in system dynamics. We briefly summarize the methodology of backstepping controller design in contraction framework in the following lemma for (<ref>) in the absence of disturbances. The detailed proof can be found in <cit.>.
Lemma-4: In the absence of any disturbances in (<ref>), suppose a control law is selected recursively as
\begin{equation}\label{bcs1}
\begin{split}
+ \dot{\alpha}_{(n)}
&z_i=x_{i}-\alpha_{i}, (i=1,2......m)\\
& \alpha_{1}=x_d(t) \ \text{and} \ \text{for}\ i\in [2 \hdots n]\\
& \qquad \qquad \qquad -b_{i-2}z_{i-2}
& \text{ $\chi_i(z_i)$ are smooth functions for which $\frac{\partial{\chi_i(z_i)}}{\partial{z_i}} > 0$.}
\end{split}
\end{equation}
Then the closed loop system is in the following form.
\begin{equation}\label{sysbrs}
\begin{split}
& \dot{z}_1=-\chi_1(z_1)+b_1z_2\\
& \dot{z}_i=-b_{i-1}z_{i-1}-\chi_i(z_i)+b_{i}z_{i+1},\ \text{for}\ i \in[2, n-1]\\
& \dot{z}_n=-b_{n-1}z_{n-1}-\chi_n(z_n)
\end{split}
\end{equation}
With the proper selection of $\chi_i(.)$, (<ref>) is contracting in an identity metric as its Jacobian is uniformly negative definite.$\diamond$
From (<ref>), it is observed that, computation of the virtual control inputs analytically become cumbersome with the increase in the order of system. However the dynamic surface control which will be discussed in subsequent sections does not encounter this problem.
§ DYNAMIC SURFACE CONTROL
In this approach the derivative of virtual control inputs $\alpha_i(.)$ are replaced by their filtered counterparts to realize the control law (<ref>). The control law is given by
\begin{equation}\label{fbcont2}
\begin{split}
+ \dot{\alpha}_{(n)f}-\hat{d}_n
&z_i=x_{i}-\alpha_{if}, (i=1,2......m)\\
& \alpha_{1f}=\alpha_1=x_d(t) \ \text{ and for}\ i\in [2 \hdots n]\\
& \qquad \qquad \qquad -b_{i-2}z_{i-2}
\end{split}
\end{equation}
The scalar $\hat{d}_i$ is the estimate for $d_i$ which is obtained using a suitable observer. The signals $\alpha_{if}$ are obtained through a first order filter and expressed as:
\begin{equation}\label{fil1}
\mu\dot{\alpha}_{if}=-\alpha_{if}+\alpha_i \quad i\in[2,\hdots,n]
\end{equation}
where the initial condition $(\alpha_{if}(0)=\alpha_i(0))$ and $\mu\in[0,1]$ is a filter parameter. The closed loop system can be expressed as following form.
\begin{equation}\label{sysb}
\begin{split}
& \dot{z}_1=-\chi_1(z_1)+b_1z_2+b_1\tilde{\alpha}_2+\tilde{d}_1\\
& \dot{z}_n=-b_{n-1}z_{n-1}-\chi_n(z_n)+\tilde{d}_n
\end{split}
\end{equation}
where $\tilde{\alpha}_i=\alpha_{if}-\alpha_i$ and $\tilde{d}_i=d_i-\hat{d}_i$ for $i\in[2, \hdots, n-1]$.
§.§ High Gain Disturbance Observer
The observer structure for the estimation of disturbances in (<ref>) is given by
\begin{equation}\label{hgdo1}
\begin{split}
&\dot{\xi}_i=-k(\xi_i+kx_i-f_i(x_1,x_2,\hdots,x_i)-b_ix_{i+1}) \\
&\dot{\xi}_n=-k(\xi_n+kx_n-f_n(x_1,x_2,\hdots,x_n)-b_nu) \\
\end{split}
\end{equation}
where $i \in [1,2,\hdots,n-1]$, $k >> 0$ is a positive constant, $\hat{d}_i$ is the estimate of $d_i$ and $\xi_i$ is an intermediate variable. The dynamics of disturbance estimate is given by
\begin{equation}\label{hgdo2}
\dot{\hat{d}}_i=h(\hat{d}_i,d_i)=-k(\hat{d}_i-d_i)
\end{equation}
For a given $d_i$, denote ${d}_{ids}$ as the root of $h(\hat{d}_{ids},d_i)=0$ and define a virtual system
\begin{equation}\label{hgvs1}
\dot{{d}}_{ids}=h({d}_{ids},d_i)+\Xi(d_i)
\end{equation}
The dynamics (<ref>) can be regarded as a perturbed copy of $\hat{d}_i$. Moreover it is partially contracting in ${d}_{ids}$ in a metric $\Theta_d=I$ with a rate $\lambda_d=k$. As the perturbation term $\Xi(d_i)$ is bounded, we can use lemma 2 to obtain the following inequality.
\[\lim_{t \to \infty}||\hat{d}_i-d_{ids}|| \leq \frac{c_1}{k}\]
Hence the disturbance observer proposed in (<ref>) will converge to a small bound depending on the gain constant $k$.
Remark 1: In case of slowly varying disturbances for which $\dot{d}_i \approx 0$, the scalar $c_1=0$. Therefore in this special case $\hat{d}_i \rightarrow d_{ids}$.
§ PROBLEM DESCRIPTION
The closed loop dynamics of the overall system comprising of equations (<ref>), (<ref>) and (<ref>) can be expressed as:
\begin{equation}\label{snp1}
\dot{{z}}=f({z},\alpha_f,\alpha,\hat{d},d)
\end{equation}
\begin{equation}\label{fil2}
\mu{\dot{\alpha}_f}=g(\alpha_f,\alpha)=-\alpha_f+\alpha
\end{equation}
\begin{equation}\label{hgdo3}
\epsilon {\dot{\hat{d}}}=h(\hat{d}, d)=-\hat{d}+d
\end{equation}
where $z=[z_1, z_2, \hdots, z_n]^T$, $\alpha_f=[\alpha_{2f}, \alpha_{3f}, ..., \alpha_{nf}]^T$, $\alpha=[\alpha_{2}, \alpha_{3}, ..., \alpha_{n}]^T$, $\hat{d}=[\hat{d}_1, \hat{d}_2, \hdots, \hat{d}_n]$, $d=[d_1, d_2, \hdots, d_n]$ and $\epsilon=\frac{1}{k} \in [0, 1]$.
The dynamics (<ref>), is in singularly perturbed form and hence the convergence of the system trajectories depends on the selection of filter parameter $\mu$ and high gain parameter $\epsilon$. The focus of this paper is to show that, the performance of a DSC law is comparable to a backstepping controller (Performance Recovery) under proper selection of filter parameter $\mu$, control gains (tuning functions) $\chi_i(.)$ and observer gain $1/\epsilon$. Using contraction theory we will derive the steady state bounds of the closed loop system as a function of controller parameters and show that these bounds can be changed according to design goal.
Assumption 3: There exist two compact sets, $B_z \subset R^n$ and $B_{\alpha} \subset R^{n-1}$ respectively for $z$ and $\alpha_f$ such that $f$ and its partail derivatives are continuous and bounded.
§.§ Performance Recovery Without Disturbance Observer
Theorem 1: If all the assumptions (1, 2 and 3) are satisfied, then the following statement is true.
In the absence of disturbances there exists a $\mu^* \in [0, 1]$ such that the control law (<ref>) recovers the performance of a backstepping controller for $\mu \in [0, \mu^*]$ inside $(B_z \times B_{\alpha})$ and it can be selected following (<ref>).
Proof: In the absence of disturbance terms, the control law (<ref>) will not depend on $\hat{d}_i$. In this case the closed loop system can be written as
\begin{equation}\label{snp11}
\dot{{z}}=f({z},\alpha_f,\alpha)
\end{equation}
\begin{equation}\label{fil21}
\mu{\dot{\alpha}_f}=g (\alpha_f,\alpha)=-\alpha_f+\alpha
\end{equation}
where $f(.)=[f_1(.), f_2(.), \hdots, f_n(.)]^T$ and
\begin{equation*}\label{sysbad}
\begin{split}
& f_1(.)=-\chi_1(z_1)+b_1z_2+b_1\tilde{\alpha}_2\\
& f_n(.)=-b_{n-1}z_{n-1}-\chi_n(z_n)
\end{split}
\end{equation*}
where $\tilde{\alpha}_i=\alpha_{if}-\alpha_i$ and $i \in [2, \hdots n-1]$. For any given $\alpha$, the equation $g(\alpha_{f},\alpha)=0$ has an unique root given by $\alpha_{f}=\alpha$, which we denote as $\alpha_{des}$. Using $\alpha_{des}$ we define a virtual system,
\begin{equation}\label{snp2}
\mu\dot{\alpha}_{des}=g (\alpha_{des},\alpha)+\mu Q(z,\alpha,\alpha_{des})
\end{equation}
where $Q(z,\alpha_{des},\alpha)=\frac{\partial{\alpha_{des}}}{\partial{z}}\dot{z}$ and $g(.)=-\alpha_{des}+\alpha$. The auxiliary system (<ref>) can be regarded as perturbed copy of (<ref>). From the structure of $g(.)$, the dynamics of (<ref>) is partially contracting in $\alpha_{des}$ when the perturbation term $Q(.)$ is absent.
From assumption 3 the term $f(.)$ and its partial derivatives are bounded inside $(B_z \times B_{\alpha})$, so it is reasonable to assume $Q(.)$ and its partial derivatives are bounded inside $(B_z \times B_{\alpha})$. Suppose there exists a positive constant $c_2$ such that
\begin{equation}\label{infn1}
||\frac{\partial{Q(z,\alpha,\alpha_{des})}}{\partial{\alpha_{des}}}||\leq c_2 \ \text{inside} (B_z \times B_{\alpha}).
\end{equation}
The differential dynamics of (<ref>) can be written as
\begin{equation}\label{fil3}
\begin{split}
& \mu \ \delta \dot{\alpha}_{des}=\{\frac{\partial{g(\alpha_{des},\alpha)}}{\partial{\alpha_{des}}}+\mu\frac{\partial}{\partial{\alpha_{des}}}(Q(z,\alpha,\alpha_{des}))\}\delta {\alpha}_{des}\\
& \Rightarrow \mu \ \delta \dot{\alpha}_{des}=\{-I+\mu\frac{\partial}{\partial{\alpha_{des}}}(Q(z,\alpha,\alpha_{des}))\}\delta {\alpha}_{des}
\end{split}
\end{equation}
Select a small positive constant $\mu^*$ such that
\begin{equation}\label{fils1}
\mu^*c_2 \leq 1.
\end{equation}
Using lemmas 2 and 3, the virtual system (<ref>) is partially contracting in $\alpha_{des}$ with a contraction rate $\frac{1-||\mu c_2||}{\mu}$ for all $\mu \in [0, \mu^*]$. Therefore, the trajectories of system (<ref>) and (<ref>) will converge to each other.
\begin{equation}\label{bod1}
\lim_{t \to \infty} ||\alpha_f(t)-\alpha_{des}(t)|| \rightarrow 0
\end{equation}
From (<ref>), $\alpha_{des} = \alpha$ and therefore
\begin{equation}\label{bd1}
\lim_{t \to \infty} ||\alpha_f(t)-\alpha(t)|| = \lim_{t \to \infty} || \tilde \alpha|| \rightarrow 0 \end{equation}
Replacing (<ref>) in (<ref>), the reduced system obtained can be written in compact form as
\begin{equation}\label{snp3}
\dot{\bf{z}}_r=f(\bf{z}_r,\alpha_{des},\alpha)
\end{equation}
where $z_r$ denotes the states of the reduced slow system. The reduced system is in same form that will result from a integrator backstepping design (<ref>) and therefore is contracting in $z_r$.
Lets rewrite (<ref>) as
\begin{equation}\label{snp4}
\dot{\bf{z}}=f(\bf{z},\alpha_{des},\alpha)+f(\bf{z},\alpha_f,\alpha)-f(\bf{z},\alpha_{des},\alpha)
\end{equation}
This equation can also be regarded as a perturbed version of the reduced slow system (<ref>). Suppose (<ref>) is contracting in a metric $\Theta_z$ with a rate $\beta$ and the function $f$ is Lipschitz in $\alpha_f$ with a constant $L_1$. The perturbation term in (<ref>) verify the following bound.
\begin{equation}\label{snp5}
||f(\bf{z},\alpha_f,\alpha)-f(\bf{z},\alpha_{des},\alpha)|| \leq L_1||\alpha_f (t)-\alpha_{des} (t)||
\end{equation}
Using lemma 2 and (<ref>), we obtain the following result.
\begin{equation}
\lim_{t \to \infty} ||z(t)-z_r(t)|| \rightarrow 0
\end{equation}
Therefore the DSC design will recover the performance of a backstepping controller exponentially if the filter parameter $\mu$ is chosen such that $\mu \leq \mu^*$ according to theorem 1. $\diamond$
Remark 2: The authors of <cit.> has shown the dynamics of DSC possesses two time scale property and proved the closed loop system stability inside a small range of filter parameter $\mu$. However Theorem 1 not only proves the closed loop system stability but also answers, how to select the filter parameter for full performance recovery. The bound obtained in (<ref>), provides the maximum value of filter parameter for which a DSC law will recover the performance of a backstepping controller in absence of any disturbances. A quadratic Lyapunov function <cit.> approach can also be utilized to obtain the stability bounds, but it requires many interconnection conditions to be satisfied.
§.§ DSC With Disturbance Observer
Presence of disturbances in (<ref>) complicates the selection of filter parameter $\mu$ which can not be made arbitrary small due to sampling requirements and noise. However it is possible to guarantee convergence of trajectories to a ultimate bound for a wider range of filter parameter and observer gain. The main result of this paper is summarized in the form of a theorem.
Theorem 2: Let the assumptions (1, 2 and 3) are satisfied for the system (<ref>) and there exists a positive constant $c_3$ such that $|\frac{\partial{\alpha_{des}}}{\partial{z}}\dot{z}| \leq c_3$ inside $B_z \times B_{\alpha}$. Then the trajectories of the closed loop system (<ref>) will follow the bounds given in (<ref>) and (<ref>).
Express the fast subsystem (<ref>), (<ref>) as:
\begin{equation}\label{fss1}
\mu \dot{v}=Q(v, \alpha, d, \kappa)
\end{equation}
where $\epsilon=\mu \kappa$, $v=[\alpha_f ^ T, \hat{d} ^T]^T$ and $Q(.)=[g^T(.), \frac{1}{\kappa}h^T(.) ]^T.$
The Jacobian of (<ref>) is given by
\[J_v=\frac{\partial{Q}}{\partial{v}}=\begin{bmatrix}J_\alpha\\J_d\end{bmatrix}=\begin{bmatrix}\frac{\partial{g(.)}}{\partial{v}}\\\frac{1}{\kappa}\frac{\partial{h(.)}}{\partial{v}}\end{bmatrix}\]
We will show that $J_v$ is negative definite and therefore (<ref>) is partially contracting in $v$. From (<ref>) and (<ref>), the matrix measure $\rho_{\infty}$ corresponding to infinity norm for $J_\alpha$ and $J_d$ are given by
\[\rho_\infty^\alpha=max_i(J_\alpha^{ii}+\sum_{i\neq j}||J_\alpha^{ij}||)=-1.\]
\[\rho_\infty^d=max_i(J_d^{ii}+\sum_{i\neq j}||J_d^{ij}||)=-1/\kappa.\]
It is important to note that $\alpha$ is treated as an external variable in (<ref>) while evaluating $J_\alpha$.
Therefore the matrix measure $\rho_{\infty}^v$ is given by
\[\rho_{\infty}^v=max(-1, \frac{-1}{\kappa}).\]
As $\kappa$ is positive constant, $\rho_\infty^v$ is negative. Hence (<ref>) is partially contracting in $v$.
Note: Depending on the magnitude of $\kappa$ which may be greater than or less than unity, the dynamics of disturbance observer is slower or faster than the filter dynamics. However we will show that it is not necessary to select a very small value of $\epsilon$ (or a very large value of observer gain) for better performance.
Following lemma I, there exists an unique root of the equation $Q(v, \alpha, d, \kappa)=0$, which is denoted as:
\[v_{ds}=[\alpha^T, d^T]^T\]
Define a virtual system given by:
\begin{equation}\label{virfs1}
\mu \dot{v}_{ds}=Q(v_{ds},\kappa)+\mu \dot{v}_{ds}
\end{equation}
In the absence of the perturbation $\dot{v}_{ds}$, (<ref>) is partially contracting in $v_{ds}$ in identity metric with a rate $1/\mu$. Following similar line of arguments as in theorem 1, the system (<ref>) can be regarded as perturbed virtual system of (<ref>). Suppose there exists a positive constant $c_3$ such that
\begin{equation}\label{infa1}
||\frac{\partial{\alpha_{}}}{\partial{z}}\dot{z}|| \leq c_3 \ \text{inside $B_z \times B_{\alpha}$.}
\end{equation}
From assumption 2 and (<ref>), the perturbation term $\dot{v}_{ds}$ is bounded inside the region $\rm B_z \times B_\alpha$ i.e,
\[||\dot{v}_{ds}|| \leq max(c_1,c_3) \]
Exploiting the boundedness of the perturbation term and using lemma 2, the trajectories of (<ref>) satisfies the following bound.
\begin{equation}\label{bound1}
||v(t)-v_{ds}(t)|| \leq ||v(0)-v_{ds}(0)||e^{(-1/\mu)t}+ max(c_1, c_3)
\end{equation}
replacing $v_{ds}$ in (<ref>), the reduced system can be written as:
\begin{equation}\label{redss1}
\dot{{z}}_r=f({z}_r,v_{ds},\kappa,\alpha,d)
\end{equation}
The system (<ref>) is in same form as (<ref>) and hence is contracting in $z_r$ with a metric $\Theta_z$.
The dynamics of (<ref>) can be expressed as following virtual system.
\begin{equation}\label{virss1}
\dot{{z}}=f({z},v_{ds},\kappa,\alpha,d)+f({z}_r,v,\kappa,\alpha,d)-f({z},v_{ds},\kappa,\alpha,d)
\end{equation}
Exploiting the Lipschitz property of $f(.)$, the perturbation term in (<ref>) satisfy the following inequality.
\begin{equation}\label{virss1}
||f({z}_r,v,\kappa,\alpha,d)-f({z},v_{ds},\kappa,\alpha,d)|| \leq L_v|v(t)-v_{ds}(t)|
\end{equation}
where $L_v$ is the Lipschitz constant. From the bound (<ref>) and lemma 2, the trajectories of (<ref>) converge to the following steady state bound.
\begin{equation}\label{bound2}
\lim_{t\rightarrow \infty}|z(t)-z_{ds}(t)| \leq \mu\frac{C_z L_v (max(c_1,c_3))}{\lambda_z ||max(-1, -1/\kappa)||}
\end{equation}
where $C_z$ is the condition number of the metric $\Theta_z$ and $\lambda_z$ is contraction rate of the reduced slow system (<ref>).
Parameter Selection: Theorem 2 discusses the effect of disturbances in the performance of DSC technique. It provides precise steady state error bounds for closed loop system trajectories depending on selection of observer and controller gains. The effect of disturbances can be reduced for sufficiently high value of observer gain but it may lead to peaking phenomenon inherent in high gain observers. We can use a moderate value of observer gain and still reduce the performance deficit by properly choosing the tuning functions $\chi(.)$. One can select the tuning functions as $\chi_i(.)=k_c z_i$ with the proper choice of $k_c$ using (<ref>). Larger control gains $k_c$ leads to large contraction rate $\lambda_z$ thereby small steady state error. The bound can also be decreased by selecting a low value of filter parameter $\mu$, but it can not be selected arbitrarily small.
Remark 3: It is shown in theorem 2 that, the disturbance observer based DSC can not fully recover the performance of a backstepping controller in presence of rate bounded disturbances. However when the disturbance is slowly varying and can be approximated by $\dot{d} \approx 0$, full performance recovery is possible within a small range of perturbation parameter. This can be proved using the similar approach as theorem 1.
§ SIMULATION EXAMPLES
Consider an example of D.C motor driven manipulator <cit.>.
\begin{equation}\label{dcrm1}
\begin{split}
& \dot{x}_1=x_2\\
& \dot{x}_2=\frac{-N}{M} sin x_1 -\frac{B}{M}x_2+\frac{1}{M}x_3\\
&\dot{x}_3=-\frac{K_b}{L}x_2-\frac{R}{L}x_3+\frac{1}{L}u \\
\end{split}
\end{equation}
Where the three states $x_1, x_2, x_3$ correspond to position, velocity and armature current respectively and $u$ is the input voltage. The parameters of the system (<ref>) is given in appendix. The controller design objective is position tracking of a desired signal $x_d= (\pi/2) sin (8 \pi t/ 5)$.
The constant parameters of this system are given by,
$K_b=0.90\ N m/A, R=5 \Omega, L=0.025 H, M=0.0640, B=0.0044, N=2.2816.$
Initially the system (<ref>) is simulated without the disturbance terms. The tuning functions are chosen as
$\chi_i(.)=-5z_i$. The expression of $\alpha_i$ are calculated analytically for comparing the performance of DSC with integrator backstepping control law. Using (<ref>) the value of $\mu^*$ obtained is given by $\mu^*=1/83$. The simulation is done using a value of $\mu=0.01$ and initial conditions $[2\pi, 0, 0]$. The simulation results are given in figure 1.
Closed loop System Response without disturbances
The closed loop system trajectories for backstepping controller and DSC are denoted as $[e_1, e_2, e_3]$ and $[z_1, z_2, z_3]$ respectively. From the figures 1, it is observed that DSC law recovers the performance of a backstepping based control law for the given choice of design parameters. The magnitude of filter parameter $\mu$ can be increased up to a threshold value of $\mu^*$, but beyond that full performance recovery is difficult to achieve.
The simulation is also performed with disturbances in (<ref>). The disturbances are the combination of both columb friction and periodic disturbances and given by
\[d_1=0.2sgn(x(2))+10sin(2t+1)+10t, d_2=10cos(2t+1)\]
The observer gain in (<ref>) is selected as $k=50$ and the control gains (tuning functions) are kept same as that of previous case. It is observed from figure 2 that, the steady state response for these choice of parameters are not satisfactory.
The simulation is then repeated by selecting another set of tuning functions (control gains)
$\chi_i(.)=-40z_i$. Figure 3 confirms the improvement of response inequality, which justify the selection of $\chi_i(.)$ to reduce the steady state error bound. The procedure does not need the observer gain to be very large and hence peaking phenomenon can be avoided.
Closed loop System Response with disturbances (low control gains)
Closed loop System Response with disturbances (high control gains)
§ CONCLUSION
In this paper we proposed a new contraction theory based approach to design DSC law and tune its parameters. Steady state error bounds for DSC law are derived in terms of design parameters. The methodology presented here relaxes the conservative bound on filter parameter which can not be made arbitrarily small due to sampling requirements and noise. Moreover it is proved that proper selection of control gains can reduce the dependency on higher observer gain for better performance of the DSC law. As a result the peaking phenomenon inherent in high gain observers can be avoided. The proposed contraction based method has important practical application in robotics, electro-hydraulic systems etc which is the subject of future works.
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1511.00284
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Eigenvalue Based Testing for Structural Breaks]Empirical eigenvalue based testing for structural breaks in linear panel data models
Department of Mathematics, University of Utah, Salt Lake City, UT, USA
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada
Testing for stability in linear panel data models has become an important topic in both the statistics and econometrics research communities. The available methodologies address testing for changes in the mean/linear trend, or testing for breaks in the covariance structure by checking for the constancy of common factor loadings. In such cases when an external shock induces a change to the stochastic structure of panel data, it is unclear whether the change would be reflected in the mean, the covariance structure, or both. In this paper, we develop a test for structural stability of linear panel data models that is based on monitoring for changes in the largest eigenvalue of the sample covariance matrix. The asymptotic distribution of the proposed test statistic is established under the null hypothesis that the mean and covariance structure of the panel data's cross sectional units remain stable during the observation period. We show that the test is consistent assuming common breaks in the mean or factor loadings. These results are investigated by means of a Monte Carlo simulation study, and their usefulness is demonstrated with an application to U.S. treasury yield curve data, in which some interesting features of the 2007-2008 subprime crisis are illuminated.
§ INTRODUCTION
We consider in this paper the problem of testing for the presence of a structural break in linear panel data models. Structural breaks in panel data may result from any of a number of sources. For example, if the data under consideration consists of U.S. macroeconomic indicators, then the onset of a recession, or the introduction of a new technology, may be evidenced by changes in the correlations between indicators or linear model parameters fitted from the data.
Change point analysis has been extensively developed to study such features in data; we refer to Aue and Horváth (2012) for a recent survey of the field in the context of time series. Adapting change point methodology to the panel data setting presents a difficulty since the dimension, or number of cross sectional units ($N$), may be larger in relation to the sample size ($T$) than is typical in classical change point analysis. This encourages asymptotic frameworks in which both $N$ and $T$ tend jointly to infinity.
Most of the literature in this direction address either testing for changes in the mean, or testing for changes in the correlation structure as measured by changes in common factor loadings. With regards to testing for and estimating changes in the mean, we refer to Bai (2010), who derives a least squares change point estimator. Kim (2011, 2014), and Baltagi et al. (2015) extend this methodology to account for changes in linear trends in the presence of cross sectional dependence modeled by common factors. Horváth and Hušková (2012) develop a test for a structural change in the mean based on the CUSUM estimator. Li et al. (2014) and Qian and Su (2014) consider multiple structural breaks in panel data, and Kao et al. (2014) considers break testing under cointegration.
Estimating and testing for changes in the covariance of scalar and vector valued time series of a fixed dimension are considered in Galeano and Peña (2007), Aue et al. (2009), and Wied et al (2012). With regards to testing for changes in the factor structure of panel data, Breitung and Eickmeier (2011) develop methodology that relies on testing for constancy of the least squares estimates obtained by regression on the principal component factors. Their test depends on estimating the number of common factors according to the information criterion developed in Bai and Ng (2002). In both the testing procedure, and the method used to determine the number of common factors, it is presumed that the mean remains constant.
In such instances when external shocks induce a change to the stochastic structure of panel data, it is unclear whether or not the change would affect the mean, the covariance structure, or both. Methods for detecting changes in the mean appear to be somewhat robust to small changes in the covariance structure of the panels, however the methods proposed in Breitung and Eickmeier (2011) to test for changes in the common factor loadings are sensitive to both changes in the mean, and large changes in the covariance, evidenced by non-monotonic power. This was recently addressed in Yamamoto and Tanaka (2015), in which a correction is proposed, but it raises the question of whether alternative methods to estimating principal components, and the number of common factors, might be effective in terms of detecting instability in panel data.
The alternative that we explore here relies on analyzing the largest eigenvalues of the covariance matrix. Using the largest eigenvalues of a covariance matrix as a simplified summary of the covariance structure of multivariate time series has served an important role in finance and econometrics for quite some time. This idea is utilized in Markowitz portfolio optimization (cf. Markowitz (1952, 1956)), and to model co–movements of markets and stocks as a barometer for risk (cf. Keogh et al. (2004) and Zovko and Farmer (2007)), among other applications.
In this paper, we propose methodology for testing structural stability in linear panel data models that is based on a process derived from the largest eigenvalue of the covariance matrix based on an increasing proportion of the total sample. The asymptotic distribution of the eigenvalue process is established assuming structural stability. Furthermore, we show that functionals of the eigenvalue process diverge when there is a common break in the mean or covariance as measured by the common factor loadings.
The rest of the paper is organized as follows. In Section <ref>, we present the linear panel data models and assumptions considered in the paper, as well as the main asymptotic results for the largest eigenvalue under the null hypothesis of stability of the model parameters. Section <ref> contains the details of applying the results of Section <ref> to the change point problem, including asymptotic consistency results under the mean break and factor loading break alternatives. In Section <ref>, we discuss the practical implementation of the test, and present the results of a Monte Carlo simulation study. Section <ref> contains an application of the methodology developed in the paper to US treasury yield curve data. Analogous results for smaller eigenvalues are considered in Section <ref>. All proofs of the technical results are collected in Section <ref>.
§ MODELS, ASSUMPTIONS, AND ASYMPTOTICS UNDER $H_0$
We consider the model
X_i,t=(μ_i+δ_iI{t≥t^*})+(γ_i+ψ_iI{t≥t^*})η_t+e_i,t, 1≤i ≤N, 1≤t ≤T,
where $X_{i,t}$ denotes the $i^{\mbox{th}}$ cross section of the panel at time $t$, $\mu_i$ denotes the initial mean of the $i^{\mbox{th}}$ cross section that changes to $\mu_i+\delta_i$ at the unknown time $t^*$, $\eta_t$ denotes a real valued common factor with initial loadings $\gamma_i$ that may change to $\gamma_i+\psi_i$, and $e_{i,t}$ denote the idiosyncratic errors. It is presumed that both the common factor and idiosyncratic errors may be serially correlated. As we develop asymptotics, we assume that the number of cross sections $N$ depends on the observation period $T$, and $N$ is allowed to tend to infinity with $T$. We make the assumption that $\eta_t \in {\mathbb R}$ for the sake of simplicity; these results could be extended to the more general case of a vector valued common factor and factor loading.
We are interested in testing the null hypothesis that the model parameters remain stable during the observation period $1\leq t \leq T$, i.e.
H_0:\; t^*>T.
When $H_0$ holds, the model of (<ref>) reduces to
X_i,t=μ_i+γ_iη_t+e_i,t, 1≤i ≤N, 1≤t ≤T.
Let $\cdot^\T$ denote the matrix transpose, and define the vectors $\X_t=(X_{1,t}, X_{2,t},\ldots ,X_{N,t})^\T\in {\mathbb R}^N$. We define
_N,T(u)=1/Tu ∑_t=1^Tu(_t-_T)(_t-_T)^, 1/T≤u≤1,
to be the sample covariance matrix based on the proportion $u$ of the sample, where
\baX_T=\frac{1}{T}\sum_{t=1}^T\X_t.
In order to test $H_0$, we utilize the processes derived from the $K$ largest eigenvalues $\hat{\lambda}_{1}(u)\geq \hat{\lambda}_{2}(u)\geq\ldots \geq \hat{\lambda}_{K}(u)$ of $\hat{C}_{N,T}(u)$. We focus our attention at first on the process derived from the largest eigenvalue, and make the primary objective of this section is to establish the weak convergence of $\hat{\lambda}_{1}(u)$ under $H_0$. Analogous results for processes derived from the smaller eigenvalues are provided in Section <ref>. We note that an alternative to using $\hat{\lambda}_i(u)$ is to use $\tilde{\lambda}_i(u)=(\lf Tu\rf/T)\hat{\lambda}_i(u)$, which are equivalent with the largest eigenvalues of
_N,T(u)=1/ T∑_t=1^Tu(_t-_T)(_t-_T)^, 0≤u≤1.
Assuming that $H_0$ holds, $\C=\mbox{cov}(\X_t)$ does not depend on $t$, and in this case we define the eigenvalues and eigenvectors of $\C$ by
λ_i_i=_i, 1≤i ≤N,
where $\|\fe_i\|=1,\;\;1\le i \le N$, and $\| \cdot \|$ denotes the Euclidean norm in ${\mathbb R}^N$. Since $N$ is allowed to depend on $T$, both the eigenvalues $\lambda_i$ and eigenvectors $\fe_i$ may evolve as $T\to \infty$. Throughout this paper, we make use of the following assumptions:
The eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_K$ satisfy that $\min_{1\leq i \leq K}(\lambda_{i}-\lambda_{i+1})\geq c_0$ for some constant $c_0>0$.
The common factor loadings satisfy that $|\gamma_i|\leq c_1\;\;\mbox{for all}\;\;1\leq i \leq N\;\;\mbox{with some}\;\; c_1>0.$
The main goal of our paper is to establish the weak convergence of the $K$–dimensional process
\A_{N,T}(u)=(A_{N,T,1}(u), A_{N,T,2}(u), \ldots ,A_{N,T,K}(u))^\T,
where for $1\leq i \leq K$,
A_{N,T,i}(u)=T^{1/2}u(\hat{\lambda}_i(u)-\lambda_i),\;\;1/T\leq u \leq 1\;\;\mbox{and}\;\;A_{N,T,i}(u)=0,\;\;0\leq u<1/T.
Assuming that the eigenvalues of $\C$ are distinct is necessary to derive a normal approximation for their estimates, and is a common assumption in the literature. We assume that the common factors and idiosyncratic errors satisfy a fairly general weak dependence condition.
We say that a stationary time series $\{\varepsilon_t,\; -\infty < t < \infty\}$ is an $L^p-m-$approximable Bernoulli shift with rate function $\chi$ if $E\varepsilon_t=0, \;\;E\varepsilon_t^{p}<\infty,$ and $\varepsilon_t=g(\vare_t, \vare_{t-1},\ldots)$ for some measurable function $g:{\mathbb R}^\infty \to {\mathbb R}$ where $\{\vare_s, -\infty<s<\infty\}$ are independent and identically distributed random variables, and $(E(\eta_t-\eta_t^{(m)})^{p})^{1/p}=\chi(m)$ with $\eta_t^{(m)}=g(\vare_t,\vare_{t-1}, \ldots, \vare_{t-m}, \vare^*_{t-m-1,t,m},\vare^*_{t-m-2,t,m},\ldots)$ and the $\vare^*_{i,j,\ell}$ are independent and identically distributed copies of $\vare_0$.
The space of stationary processes that may be represented as Bernoulli shifts is enormous; we refer to Wu (2005) for a discussion. Examples include stationary ARMA, ARCH, and GARCH processes. The rate function describes the rate at which such processes can be approximated with sequences exhibiting a finite range of dependence. In many examples of interest, the rate function may be taken to decay exponentially in the lag parameter.
(a) $\{\eta_t,\;\; -\infty < t < \infty\}$ is $L^{12}-m-approximable$ with rate function $\chi_\eta(m) = c_2 m^{-\alpha_\eta}$ for constants $c_2>0$ and $\alpha_\eta>1$, and $E\eta_t^2=1$.
(b) The sequences $\{e_{i,t},\;\; -\infty < t < \infty\},\; 1 \le i \le N$, are each $L^{12}-m-approximable$ with rate functions $\chi_{e,i}(m) \le c_3 m^{-\alpha_e}$ for constants $c_3>0$ and $\alpha_e>1$. There exist constants $c_4$ and $c_5$ such that $0 < c_4 \le Ee^2_{i,_t}=\sigma_i^2\le c_5 <\infty$.
(c) The sequences $\{\eta_t,\;\; -\infty < t < \infty\},$ and $\{e_{i,t},\;\; -\infty < t < \infty\},\; 1 \le i \le N$ are independent.
The least restrictive moment condition that could be assumed in order to obtain a normal approximation for the empirical eigenvalues is four moments. Our assumption of twelve moments comes from the fact that we apply a third order Taylor series expansion for the difference between the empirical eigenvalue process $\hat{\lambda}_i(u)$ and $\lambda_i$, (cf. Hall and Hosseini–Nasab (2009)) and twelve moments are needed to get an upper bound for the highest order term that is uniform with respect to $u$. The condition in Assumption <ref> that $E \eta_t^2 =1 $ is nonrestrictive; it makes the model (<ref>) identifiable. In order to state the main result, we define
\xi_{i,t} = \fe_i^\T (\bX_t - E \bX_0 ) (\bX_t - E \bX_0 )^\T \fe_i.
Assumptions Assumption <ref>(a) and Assumption <ref>(b) mean that the sequences $\eta_t, e_{i,t}, 1\leq i \leq N$ are Bernoulli shifts and therefore they are stationary sequences. We refer to Aue and Horváth (2013) for a general discussion of Bernoulli shifts and several examples satisfying Assumption <ref>(a) and Assumption <ref>(b).
Let $\bgamma=(\gamma_1,\gamma_2, \ldots, \gamma_N)^\T$. Throughout this paper $\bgamma=\bgamma(T)$. We assume
|γ_i|≤c_4 1≤i ≤N c_4.
Since $N$ as well as $\bgamma$ might depend on $T$, the covariance matrix $\C$ and therefore ${\fe}_1$, ${\fe}_2, \ldots, \fe_N$, the orthonormal eigenvectors associated with the eigenvalues $\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_N$ are functions of $T$. The eigenvalues and eigenvectors are defined by
λ_i_i=_i, 1≤i ≤N.
V_1=∑_ℓ=-∞^∞(η_0^2, η^2_ℓ),
_2={∑_s=-∞^∞lim_T→∞∑_k=1^N_i(k)_j(k)(η_0, η_s)(e_k,0, e_k,s), 1≤i,j≤K},
\begin{align}\label{v-def-3}
\V_3
&=\left\{\sum_{s=-\infty}^\infty\lim_{T\to \infty}\left[\sum_{k=1}^N\fe_i^2(k)\fe^2_j(k)\cov (e^2_{k,0}, e^2_{k,s}) \right.\right.\\
&\hspace{1.5cm}+2\left(\sum_{k=1}^N\fe_i(k)\fe_j(k)\cov (e_{k,0}, e_{k,s})\right)^2 \notag\\
&\hspace{1.5cm}\left. \left. -2\sum_{k=1}^N\fe_i^2(k)\fe_j^2(k)(\cov (e_{k,0}, e_{k,s}))^2
\right], 1\leq i,j\leq K
\right\}.\notag
\end{align}
We use the notation $\V_2=\{V_2(i,j), 1\leq i,j\leq K\}$ and $\V_3=\{V_3(i,j), 1\leq i,j\leq K\}$.
If, for example, we assume that $r(s)=\cov (e_{k,0}, e_{k,s})$ for all $1\leq k\leq N$, then $\V_2$ is a diagonal matrix with
V_2(i,i)=\sum_{s=-\infty}^\infty \cov (\eta_0, \eta_s) r(s).
The expression for $\V_3$ also simplifies since by the orthonormality of the $\fe_i$'s we have
\sum_{k=1}^N\fe_i(k)\fe_j(k)\cov (e_{k,0}, e_{k,s})=r(s)I\{i=j\}.
If we further assume that each of the $\{e_{k,s}, -\infty<s<\infty\}$ sequences are Gaussian, then $\mbox{cov} (e^2_{k,0}, e^2_{k,s})=2r^2(s)$, and $\V_3$ also reduces to a diagonal matrix with $V_3(i,i)=2\sum_{s=-\infty}^\infty r^2(s)$.
We allow that $N$ might increase with $T$ but we require that
N(logT)^1/3/T^1/2→0, T→∞.
a_i=lim_T→∞_i^, 1≤i≤K
and define $\G=\{G(i,j), 1\leq i,j\leq K\}$ with $G(i,j)=a_i^2a_j^2V_1+4a_ia_jV_2(i,j)+V_3(i,j)$.
Lemma <ref> demonstrates that the limit in (<ref>) is finite assuming that it exists.
If assumptions (<ref>), Assumption <ref>(a)–Assumption <ref>, Assumption <ref>, (<ref>) hold, and
=O(1), T→∞,
then we have that
$\A_{N,T}(u)\;\;\mbox{converges weakly in}\;\;{\mathcal D}^K[0,1]\;\;\mbox{to}\;\;\W_{\G}(u),$
where $\W_{\G}(u)$ is a $K$–dimensional Wiener process, i.e. $\W_{\G}(u)$ is Gaussian with $E\W_{\G}(u)={\bf 0}$ and $E\W_{\G}(u)\W_{\G}^\T(u')=\min(u,u') \G$.
If $H_0$ and Assumptions <ref> and <ref> hold, and
=O(1), T→∞,
then we have that
$\A_{N,T}(u)\;\;\mbox{converges weakly in}\;\;{\mathcal D}^K[0,1]\;\;\mbox{to}\;\;\W_{\G}(u),$
where $\W_{\G}(u)$ is a $K$–dimensional Wiener process, i.e. $\W_{\G}(u)$ is Gaussian with $E\W_{\G}(u)={\bf 0}$ and $E\W_{\G}(u)\W_{\G}^\T(u')=\min(u,u') \G$.
If $H_0$ and Assumptions <ref>, <ref>, and <ref> hold, and
N(logT)^1/3/T^1/2→0, T→∞,
$$\frac{T^{1/2}}{\sigma_1}u(\hat{\lambda}_1(u)-\lambda_1)\stackrel{{\mathcal D[0,1]}}{\longrightarrow} W(u),$$
where $W(u)$ is a Wiener process, $\stackrel{{\mathcal D[0,1]}}{\longrightarrow}$ denotes weak convergence in the Skorokhod topology, and
\sigma_1^2 =\sigma_1^2(T)= \sum_{t = -\infty}^\infty \mbox{{\rm cov}}(\xi_{1,0},\xi_{1,t}).
Theorem <ref> shows that the distribution of the largest eigenvalue process may be approximated by a Brownian motion. We note that the norming sequence $\sigma_1^2$, which is essentially the long run variance of the quadratic forms $\xi_{1,t}$, may change with $N$. In fact, we show in Section <ref> that if $\bgamma=(\gamma_1,\gamma_2, \ldots, \gamma_N)^\T$, then under $H_0$ $\sigma_1^2 \to \infty$, as $T\to \infty$, if $\|\bgamma\| \to \infty$. The necessity of including the logarithm term in the rate condition (<ref>) comes from the fact that we establish weak convergence on the entire unit interval. This condition can be improved by considering convergence on an interval that is bounded away from zero.
If the conditions of Theorem <ref> are satisfied and (<ref>) is replaced with
N/T^1/2→0, T→∞,
then for all $c \in (0,1]$,
$$\frac{T^{1/2}}{\sigma_1}u(\hat{\lambda}_1(u)-\lambda_1)\stackrel{{\mathcal D[c,1]}}{\longrightarrow} W(u).$$
where $\sigma_1^2$ is defined as in Theorem <ref>.
Conditions (<ref>) and (<ref>) require that the sample size $T$ is asymptotically larger than the squared dimension $N^2$. The case when $N$ is proportional to $T$ has received considerable attention in the probability and statistics literature. Assuming that $\hat{C}_{N,T}(1)$ is based on independent and identically distributed entries, the distribution of $\hat{\lambda}_1(1)$ converges to a Tracy-Widom distribution (cf. Johnstone (2008)). For a survey of the theory of eigenvalues of large random matrices, we refer to Aue and Paul (2014).
If $\|\bgamma\|\to 0$, as $T\to \infty$, then $a_i=0$ according to Lemma <ref>. In this case the weak limit of $\A_{N,T}(u)$ is the $K$–dimensional Wiener process $\W_{\V_3}(u)$, since $\G=\V_3$.
To state the next result we introduce the covariance matrix $\bH=\{H(i,j), 1\leq i ,j\leq K\}$: $H(1,1)=V_1, H(1,i)=H(i,1)=a_i^2V_1$ and $H(i,j)=a_i^2a_j^2V_1+4a_ia_j V_2(i,j)+ V_3(i,j), 2\leq i,j\leq K$.
If assumptions (<ref>), Assumption <ref>(a)–Assumption <ref>, Assumption <ref>, (<ref>) hold, and
→∞, T→∞,
then we have that $\left\{\|\bgamma\|^{-2}A_{N,T;1}(u), A_{N,T;i}(u), 2\leq i\leq K \right\}$ converges weakly in
${\mathcal D}^K[0,1]$ to $\W_{\bH}(u)$,
where $\W_{\bH}(u)$ is a $K$–dimensional Wiener process, i.e. $\W_{\bH}(u)$ is Gaussian with $E\W_{\bH}(u)={\bf 0}$ and $E\W_{\bH}(u)\W_{\bH}^\T(u')=\min(u,u') \bH$.
We show in Lemma <ref> that in case of (<ref>), $\lambda_1$, the largest eigenvalue of $\C$ satisfies
\left|\frac{\lambda_1}{\|\bgamma\|^2}-1\right|=O(1)
Thus Theorem <ref>(ii) yields that $\hat{\lambda}_{N,T;1}(u)/\|\bgamma\|^2\to 1$ in probability for all $u>0$.
Theorems <ref> and <ref> provide the limits of the weighted differences $T^{1/2}u(\hat{\lambda}_i(u)$ $-\lambda_i)=T^{1/2}(\tilde{\lambda}_i-u\lambda_i), 1\leq i \leq K$. It is easy to get the limits of $T^{1/2}(\hat{\lambda}_i(u)-\lambda_i),1\leq i \leq K$.
Namely, if the conditions of Theorem <ref> are satisfied but (<ref>) is replaced with
N/T^1/2→0, T→∞,
then $T^{1/2}(\hat{\lambda}_i(u)-\lambda_i), 1\leq i \leq K$ converges weakly in ${\mathcal D}^K[c,1]$ to $\W_{\G}(u)/u$ for any $0<c\leq 1$
where $\W_{\G}(u)$ is defined in Theorem <ref>.
Similarly, under the conditions of Theorem <ref> but replacing (<ref>) with (<ref>) we have that
$\|\bgamma\|^{-2}T^{1/2} (\hat{\lambda}_1(u)-\lambda_1), T^{1/2}(\hat{\lambda}_i(u)-\lambda_i), 2\leq i \leq K$ converges weakly in ${\mathcal D}^K[c,1]$ to $\W_{\bH}(u)/u$ for any $0<c\leq 1$, where $\W_{\bH}(u)$ is defined in Theorem <ref>.
Horváth and Hušková (2012) proved that (<ref>) is a necessary and sufficient condition to detect change in the means of the panels so if (<ref>) is not satisfied one cannot detect the change in the means using CUSUM or quasi likelihood based tests. The analysis of the eigenvalues of covariance matrices has received a great amount of attention when $N$ much larger than or proportional to $T$. For a review of the “large $N$, small $T$" problem we refer to Aue and Paul (2014).
§ CHANGEPOINT DETECTION
Hence for all $1\leq i \leq K$ we get that
G^-1/2(i,i)B_N,T;i(u) 𝒟[0,1]⟶ B(u),
where $B(u)$ denotes a Brownian bridge. Also, under the conditions of Remark <ref>, we have for $1\leq i \leq K$ that
G^-1/2(i,i)T^1/2(λ̂_i(u)-λ̂_i(1)) 𝒟[c,1]⟶ B(u)/u, 0<c≤1.
If the conditions of Theorem <ref> and Remark <ref> are satisfied, we obtain that $(\|\bgamma\|^{-2}B_{N,T;1}(u),$ $B_{N,T;i}(u), 2\leq i \leq K)^\T$ converges weakly in ${\mathcal D}^K[0,1]$ to $\W^0_{\bH}(u),$
where $\W^0_{\bH}(u)$ is a $K$–dimensional Brownian bridge, i.e. $\W^0_{\bH}(u)$ is Gaussian with $E\W^0_{\bH}(u)={\bf 0}$ and $E\W^0_{\bH}(u)(\W^0_{\bH}(u'))^\T=
V_1^-1/2^-2B_N,T;1(u) 𝒟[0,1]⟶ B(u),
V_1^-1/2^-2T^1/2(λ̂_1(u)-λ̂_1(1)) 𝒟[c,1]⟶ B(u)/u, 0<c≤1,
H^-1/2(i,i)B_N,T;i(u) 𝒟[0,1]⟶ B(u),
H^-1/2(i,i)T^1/2(λ̂_i(u)-λ̂_i(1)) 𝒟[c,1]⟶ B(u)/u, 0<c≤1.
§.§ Estimating the norming sequence
Consistent estimation of $\sigma_1^2$ is required in order to apply Theorems <ref> and <ref> to test $H_0$. As $\sigma_1^2$ is defined as the long run covariance of the quadratic forms $\xi_{i,t}$, we propose a natural nonparametric estimator. We define $\hat{\fe}_i$ by
\hat{\lambda}_i(1)\hat{\fe}_i=\hat{\C}_{N,T}(1)\hat{\fe}_i,\;\;1\leq i \leq N.
Let $\hat{\xi}_{i,t}=(\hat{\fe}_i^\T(\X_t-\bar{\X}^*_{T,t}))^2,$ where
1/t̂^*∑_t=1^t̂^* _t , 1 ≤t ≤t̂^*
1/T- t̂^*∑_t=t̂^*+1^T _t , t̂^*+1 ≤t ≤T,
and $\hat{t}^*$ is the least squares change point estimator for a change in the mean defined in Section 3 of Bai (2010). Estimating the mean under the alternative of a mean change is done to ensure monotonic power in that case. Let $J$ be a kernel/weight function that is continuous and symmetric about the origin in ${\mathbb R}$ with bounded support, and satisfying $J(0)=1$. Examples of such functions include the Bartlett and Parzen kernels; further examples and discussion may be found in Taniguchi and Kakizawa (2000). We define the estimator $\hat{v}^2_{1,T}$ for $\sigma_1^2$ by
J(u)=0, |u|>c_* c_*>0,
J ℝ.
\begin{align}\label{var-est-def}
\hat{v}^2_{1,T}=\sum_{s=-N+1}^{N-1}J\left(\frac{s}{h}\right)\hat{r}_{1,s},
\end{align}
where $h$ denotes a smoothing bandwidth parameter, and
1/T-s∑_t=1^T-s (ξ̂_1,t-ξ̅_1,T)(ξ̂_1,t+s-ξ̅_1,T), s≥0
1/T-|s|∑_t=-s^T (ξ̂_1,t-ξ̅_1,T)(ξ̂_1,t+s-ξ̅_1,T), s< 0,
\bar{\xi}_{1,T}=\frac{1}{T}\sum_{t=1}^T\xi_{1,t}.
If $H_0$ and the conditions of Theorem <ref> are satisfied, and
h=h(T)→∞ hN^3/T^1/2→0, T→∞,
v̂^2_1,T/σ_1^2 P→ 1, T→∞.
The results in Theorems <ref>, <ref>, and <ref> can be used to test for the stability of the largest eigenvalue, which, as we show below, suggests stability of the model parameters.
\hat{B}_{T,1}(u)=\frac{T^{1/2}}{\hat{v}_{1,T}}u(\hat{\lambda}_1(u)-\hat{\lambda}_1(1)),\;\;0\leq u \leq 1.
Under the conditions of Theorems <ref> and <ref>, $\hat{B}_{T,1}(u) \stackrel{{\mathcal D[0,1]}}{\longrightarrow} W^0(u)$, where $W^0$ is a standard Brownian bridge.
The continuous mapping theorem and Corollary <ref> imply that
\begin{align}\label{imp-form}
\sup_{0\le t \le 1} |\hat{B}_{T,1}(u)| \stackrel{{\mathcal D}}{\to} \sup_{0 \le t \le 1} |W^0(t)|.
\end{align}
The limiting distribution on the right hand side of (<ref>) is commonly referred to as the Kolmogorov distribution. An approximate test of size $\alpha$ of $H_0$ is to reject if $\sup_{0\le t \le 1} |\hat{B}_{T,1}(u)| $ is larger than the $\alpha$ critical value of the Kolmogorov distribution. One could also consider alternate functionals of $\hat{B}_{T,1}$ to test $H_0$. The distributions of many functionals of $W^0$ are well–known (cf. Shorack and Wellner (1986), pp. 142–149).
§.§ Consistency under alternatives
We now turn our attention to studying the consistency of tests for $H_0$ based on $\sup_{0 \le t \le 1} |\hat{B}_{T,1}(u)|$ under the mean break and factor loading break alternatives. Following the literature, we assume that the change does not occur too close to the end points of the sample:
t^*=Tθ 0<θ<1.
First we consider the case of a break in the mean, i.e. the model
X_i,t=(μ_i+δ_iI{t≥t^*})+γ_iη_t+e_i,t, 1≤i ≤N, 1≤t ≤T,
holds. Let $\balpha=\balpha_T=(\delta_1, \delta_2, \ldots, \delta_N)^\T$ and assume
Under (<ref>), Assumptions <ref>, <ref>, and <ref>, and assuming that (<ref>), (<ref>), and (<ref>) are satisfied, then we have that
sup_0≤u≤1|B̂_T,1(u)| P→ ∞
We note that assumptions (<ref>) and (<ref>) also appeared in Horváth and Hušková (2012) where the optimality of these
conditions are discussed. It is clear if $N$ is large, relatively small changes can be detected by $\hat{\lambda}_1(u)$. As a consequence of the proof of Theorem <ref>, it follows that
\max_{2\leq i \leq K}\sup_{0\leq u\leq 1}T^{1/2}|\hat{\lambda}_i(u)-\hat{\lambda}_i(1)|=O_P(1),
i.e. a change in the mean is asymptotically entirely captured by the largest eigenvalue of the partial covariance matrices.
The condition (<ref>) suggests how a local change in the mean alternative may be considered. For example, if $\delta_1=\delta_2=\ldots =\delta_N=\delta(N,T)$ and $\gamma_1=\gamma_2=\ldots =\gamma_N=\gamma$, $\gamma$ is fixed , we need that $(T/N)^{1/2}|\delta(N,T)|\to \infty$ for (<ref>) to hold, which describes at what rate $\delta(N,T)$ may tend to zero while maintaining consistency.
Next we consider the model when the loadings can change during the observation period:
X_i,t=(μ_i+δ_iI{t≥t^*})+∑_ℓ=1^Lγ_i,ℓη_tI{t^*_ℓ-1<t≤t^*_ℓ}+e_j,t, 1≤i ≤N, 1≤t ≤T,
where $0=t_0<t_1<t_2<\ldots <t_L=T$. Let $\bgamma_\ell=(\gamma_{1,\ell}, \gamma_{2,\ell}, \ldots , \gamma_{N,\ell})^\T, 1\leq \ell\leq L$. If
then the results of Theorems <ref>, <ref>(i) and <ref> remain true, i.e. changes in the means can be deducted even if the loadings change assuming that (<ref>) holds.
We showed that our method will detect changes in the mean even if the loadings are changing too.
Next we consider the model
X_i,t=μ_i+(γ_i+ψ_iI{t≥t^*})η_t+e_i,t, 1≤i ≤N, 1≤t ≤T,
i.e. the means of the panels remain the same but the loadings change at time $t^*$. Let $\bdelta=(\psi_1,\psi_2, \ldots,\psi_N)^\T$.
Under (<ref>), Assumptions <ref>, <ref>, and <ref>, and assuming that (<ref>), (<ref>) and
lim_T→∞(1-θ)[^2+2|^|] +(^)^2/^2/^2+max_1≤i ≤Nσ_i^2>1
hold, then $\sup_{0\leq u\leq 1}|\hat{B}_{T,1}(u)|\;\;\stackrel{P}{\to}\;\;\infty$, as $T \to \infty$.
Roughly speaking, it is possible that the covariance might change on a subspace that is orthogonal to the first eigenvector (or more generally the first $K$ eigenvectors), and then if this change is not sufficiently large, the first eigenvalue cannot have power to detect it. Condition (<ref>) is sufficient to imply that this does not occur.
§ FINITE SAMPLE PERFORMANCE
In order to demonstrate how the result in (<ref>) is manifested in finite samples, we present here the results of a Monte Carlo simulation study involving several different data generating processes (DGP's) that follow (<ref>). All simulations were carried out in the R programming language (cf. R Development Core Team (2010)). In order to compute the long run variance estimate $\hat{{\it v}}^2_{1,T}$ defined in (<ref>), we used the “sandwich" package (cf. Zeileis (2006)), in particular the “kernHAC" function. The Parzen kernel with corresponding bandwidth defined in Andrews (1991) were employed.
§.§ Empirical Size
We begin by presenting the results on the empirical size of the test for stability based on the largest eigenvalue by considering two examples of synthetic data generated according to model (<ref>). We use the notation $Y_{i} \sim Y$ to denote that the sequence of random variables $Y_i$ are independent and identically distributed with distribution $Y$. Let $N_{i,t}(0,1)$ $i\ge0$ and $t \in \mathbb{Z}$ denote iid standard normal random variables, and let $AR_i(1,p)$ $i \ge 0$ denote independent autoregressive one processes with parameter $p$ based on standard normal errors. We generated observations $X_{i,t}$ according to (<ref>) and the DGP's
(IID): $\eta_t= N_{0,t}(0,1),$ $e_{i,t} = s_i N_{i,t}(0,1)$, $s_i \sim Unif(.8,1.2)$, $\gamma_i \sim N(0,1)$,
(AR-1): $\eta_t=AR_0(1,.5),$ $e_{i,t} = s_i AR_i(1,.5)$, $s_i \sim Unif(.8,1.2)$, $\gamma_i \sim N(0,1)$.
The purpose of choosing random parameters $s_i$, which define the standard deviations of the idiosyncratic errors, and $\gamma_i$ is two fold. Firstly, this forces Assumption <ref> to hold. Secondly, this choice highlights that the methodology is relatively robust to variations in the parameter values.
Five simulated paths of the process $\hat{B}_{T,1}(u)$ are shown in the left hand panel of <ref> when $T=100$ and $N=20$, under IID. The most notable feature is that each process always starts with a spike near the origin, i.e. $\hat{\lambda}_i(u)$ is much larger than $\hat{\lambda}_i(1)$ when $u$ is small. The reason for this is that, when $u$ is small, $\hat{\lambda}_i(u)$ is computed from a matrix that is low rank, and hence will tend to be closer to the norm of the observation vectors, which is on the order of $N$, than the eigenvalue that it being estimated. This problem is ameliorated when $N$ decreases or $T$ increases, but significantly affects the results for many practical values of $N$ and $T$.
The left panel illustrates five simulated paths of $\hat{B}_{T,1}(u)$ when $N=20$ and $T=100$ under (IID), and the right panel illustrates five simulated paths of $ \tilde{B}_{T,1}(u)$ under the same conditions with $\epsilon=.05$.
2cDGP 7cIID 7cAR-1
3-9 11-17
2r 3c$\epsilon=.05$ 3c$\epsilon=.1$ 3c$\epsilon=.05$ 3c$\epsilon=.1$
3-57-9 11-13 15-17
N T 10% 5% 1% 10% 5% 1% 10% 5% 1% 10% 5% 1%
10 50 18.1 11.2 3.8 8.8 4.9 1.8 26.7 18.4 10.0 24.7 17.9 8.4
100 8.3 3.5 .7 9.2 3.6 .7 17.1 10.3 3.4 9.2 3.6 .7
200 8.7 4.1 .7 8.6 4.3 1.0 11.7 5.7 2.0 10.4 5.1 1.6
20 50 18.6 12.3 5.5 9.5 4.8 .7 23.7 16.5 8.0 25.8 17.8 8.7
100 8.5 3.6 .6 9.1 4.5 .3 14.9 9.4 3.4 14.9 9.0 3.7
200 8.4 4.2 .6 8.8 3.3 .5 11.8 6.5 2.0 12.4 6.8 1.5
50 50 23.3 13.7 5.3 10.2 3.9 .7 24.8 17.3 8.8 24.4 18.6 .9
100 8.8 3.5 .6 9.0 4.2 1.0 17.8 11.6 4.0 15.3 8.5 3.5
200 10.0 5.0 1.3 8.9 3.8 .5 13.0 7.1 2.1 12.2 6.4 1.7
Empirical sizes with nominal levels of 10%, 5%, and 1% in both the independent (IID) and
dependent (AR-1) cases based on the process $\tilde{B}_{T,1}$.
In order to correct for this, we define
B̃_T,1(u) = {
0 u ∈[0,ϵ]
B̂_T,1(u)-1-u/1-ϵB̂_T,1(ϵ) u ∈(ϵ,1]
for a trimming parameter $\epsilon>0$. Five corresponding paths of $\tilde{B}_{T,1}(u)$ are illustrated in the right panel of Figure <ref>, with $\epsilon=.05$.
Table <ref> contains the percentages of the test statistic $\sup_{0\le u \le 1} |\tilde{B}_{T,1}(u)|$ that are larger than the 10%, 5%, and 1% critical values of the Kolmogorov distribution. The results can be summarized as follows:
* When $T$ is small ($T=50$), then the size of the test may be inflated by two sources. One of them is the spiked effect, and this is particularly pronounced when $\epsilon$ is small and $N$ is large. If the temporal dependence in the data is low, then increasing $\epsilon$ can allow the test to achieve good size even for small $T$ and relatively large $N$. However, strong temporal dependence can cause size inflation for small $T$ that cannot be accounted for by increasing $\epsilon$.
* Another source of size inflation that is present for larger values of $T$ may be attributed to estimating the variance under the alternative of a break in the mean. This may be improved by considering alternative variance estimation approaches, such as those developed in Kejriwal (2009).
* The difference in the results between the IID and AR-1 DGP's were small for larger values of $T$ $(T=100,200)$, indicating the variance estimation is performing well.
* For $T=200$, the empirical sizes are close to nominal in all cases.
§.§ Empirical Power
In order to study the power of our test under both the mean break and loading break alternatives, we considered two processes that satisfy (<ref>) with $t^*=T\theta$ with $\theta \in (0,1)$. Throughout the simulations below, we set $t^*=T/2$, i.e. the break was in the middle of the sample. We also studied the situation in which breaks occured towards the endpoints of the sample. The results in those cases tended to be worse, but not more so than expected. We define the DGP's
MB($\delta$): $X_{i,t}=\delta_iI\{t\geq T/2\})+\gamma_i\eta_t+e_{i,t},\;\;1\leq i \leq N, 1\leq t \leq T$, where $\delta_i \sim \mbox{Unif}(-\delta,\delta)$
LB($\Delta$): $X_{i,t}=(\gamma_i+\psi_iI\{t\geq T/2\})\eta_t+e_{i,t},\;\;1\leq i \leq N, 1\leq t \leq T$, where $\psi_i \sim N(0,\Delta^2)$
In each case we take the other terms in (<ref>), i.e. the idiosyncratic errors, common factor, and factor loadings, to satisfy AR-1. We let the parameters $\delta$ and $\Delta$ vary between $0$ and $4$ at increments of $.5$, and let $N=10,20,50$, and $T=50,100,200$. The results are displayed in terms of power curves in Figures <ref> and <ref> in case of a mean break alternative (MB($\delta$)) and in Figures <ref> and <ref> in case of breaks in the factor loadings (LB($\Delta$)) when the size of the significance level of the test was fixed at 5%. We summarize the results as follows:
Power curves generated from data following MB$(\delta)$ for fixed $N$ and varying $T$. The horizontal axis measures $\delta$, and the vertical axis measures the empirical power when the significance level is fixed at 5%.
Power curves generated from data following MB$(\delta)$ for fixed $T$ and varying $N$. The horizontal axis measures $\delta$, and the vertical axis measures the empirical power when the significance level is fixed at 5% .
Power curves generated from data following LB$(\Delta)$ for fixed $N$ and varying $T$. The horizontal axis measures $\Delta$, and the vertical axis measures the empirical power when the significance level is fixed at 5%.
Power curves generated from data following LB$(\Delta)$ for fixed $T$ and varying $N$. The horizontal axis measures $\Delta$, and the vertical axis measures the empirical power when the significance level is fixed at 5%.
Mean Break:
* In the case of a mean break, for each value of $T$ and $N$ that we considered there is a substantial gain in power for $\delta$ exceeding 1.5. We note that data generated according to AR-1 have cross-sectional standard deviations of on average 1.6, and, when $\delta=2$, the average squared size of the change in the mean of each cross section is 1.33. Thus testing based on the largest eigenvalue seemed very sensitive to detect changes in the mean.
* Due to the estimation of the variance under a mean break, the test exhibited monotonic power.
* Increasing $T$ with fixed $N$ improved the empirical power, as expected, and the same was observed when $T$ was fixed and $N$ increased. The latter occurrence is likely attributable to the fact that as $N$ increases, changes in the mean occur in more cross sections, and the size is inflated in these cases due to the spiked effect.
Loading Break
* In the case of a break in the factor loadings, even smaller changes relative to the size of the standard deviation $(\Delta=1)$ of the idiosyncratic errors resulted in dramatic increases in power.
* We noticed that for smaller values of $T$ $(T=50)$ the power seemed to level off for larger breaks in the common factors, and never reached more than $90\%$.
* For larger $T$ $(T=100,200)$, the power approached 1 at a much faster rate for breaks in the factor loadings, and this occurrence seemed to be independent of the value of $N$.
* Increasing $N$ resulted in reduced power in this case, although the effects of changing $N$ were not particularly pronounced.
The results are summarized as follows:
* Increasing $N$ and $T$ has the effect of increasing the power in all cases.
* Under AR-1 the standard deviation of each cross–sectional unit is approximately 1.6. In comparison, the average squared size of the change in the mean under MB($2$) is 1.33, and in this case the power is over 90% for all values of $N$ and $T$ that we considered.
* (INCLUDE RESULTS FOR LOADING BREAK)
Power Results.
§ APPLICATION TO U.S. YIELD CURVE DATA
Following Yamamoto and Tanaka (2015), we consider an application of our methodology to test for structural breaks in U.S. Treasury yield curve data considered in Gürkaynak et al. (2007), which is available at http://www.federalreserve.gov/
econresdata, and which the authors graciously maintain. The data consists of yields for fixed interest securities with maturities between one and thirty years with one year increments ($N=30$). We studied a portion of this data set spanning from January 1st, 1990 to August 28th, 2015, that we further reduced from daily to monthly observations by considering only the data from the last day of each month. Figure <ref> illustrates the yield curves corresponding to 1, 5, 10, and 30 year maturities.
Yield curves at a 1-month resolution between January, 1990 and August, 2015 correpsonding to 1 year, 5 year, 10 year, and 30 year maturities.
In order to remove the effects of stochastic trends, and to allow for a comparison of our results to Yamamoto and Tanaka (2015), we first differenced each series. We applied the hypothesis test for stability of the largest eigenvalue based on $\sup_{0 \le t \le 1} |\tilde{B}_{T,1}(t)|$ with trimming parameter $\epsilon=.05$ to sequential blocks of the first differenced data of length 10 years, corresponding to 120 monthly observations in each sample ($T=120$). The first block contained data spanning from January, 1990 to December, 1999, and the last block contained data spanning from September, 2005 to August, 2015, which constituted a total of 172 tests. The P-value from each test is plotted against the end date of the corresponding 10 year block in Figure <ref>.
P-values corresponding to 10 year blocks of the first differenced yield curve data. The vertical axis measures the magnitude of the P-value, and the horizontal axis indicates the concluding month of the 10 year block. P-values below the horizontal line are below .05.
The most notable result of this analysis is the persistent instability of the largest eigenvalue evident in the samples that end in late 2008 to early 2009. This seems to correspond with the subprime crisis, which sparked what has been termed the “Great Recession". The stability of the largest eigenvalue seems to return near the end of 2013. This may be indicative of the economic recovery, and provides a way of dating the end of the recession. The findings of structural breaks in the correlation structure of the yield curves during the 2007-2009 recession are consistent with those of Yamamoto and Tanaka (2015).
Also notable is the lack of persistent instability in relation to the 2001 economic recession. This illuminates a difference between the two recessions: The 2001 recession may be better modeled as a first order structural break, which is not as evident in the first differenced yield curve series, whilst the 2009 recession, which generated numerous policy changes and endured for a longer period, is manifested as a change in the largest eigenvalue.
currency exchange rate data of 22 currencies with respect to the United States Dollar(USD). The data spans from 01/03/2000 to 12/31/2013 thirteen years of daily (weekdays) data. In order that the exchange rates have similar scales throughout the sample, we divided each currencies exchange rate by its initial rate on 1/03/2000. A plot of the resulting scaled exchange rates for seven countries are displayed in Figure <ref>. We applied the stability test based on the largest eigenvalue to 13 segments corresponding to each year represented within the data set. The p-values of each test the individual segements are given in the second column of Table <ref>.
In order to remove potential effects due to changes in the mean in the exchange rate, we computed the first differenced series for each cross-sectional unit of the observations. Seven first differenced series corresponding to the series in Figure <ref> are displayed in Figure <ref>. These series exhibit typical GARCH type behavior. We applied the test based on the largest eigenvalue to 18 segments of length 200. The p-values of the test and corresponding dates are displayed in Table <ref>.
Year Scaled Exchange Rate p-values First Differenced SER p-values
Empirical sizes with
nominal levels of 10% and 5% and 1% in both the independent and
dependent cases.
§ RESULTS FOR SMALLER EIGENVALUES
In this section, we provide analogous results to Theorems <ref> and <ref> for the smaller eigenvalues. Namely, we aim to establish the weak convergence of the $K$–dimensional process
\A_{N,T}(u)=(A_{N,T,1}(u), A_{N,T,2}(u), \ldots ,A_{N,T,K}(u))^\T,
where for $1\leq i \leq K$,
A_{N,T,i}(u)=T^{1/2}u(\hat{\lambda}_i(u)-\lambda_i),\;\;1/T\leq u \leq 1\;\;\mbox{and}\;\;A_{N,T,i}(u)=0,\;\;0\leq u<1/T.
V_1=∑_ℓ=-∞^∞(η_0^2, η^2_ℓ),
_2={∑_s=-∞^∞lim_T→∞∑_k=1^N_i(k)_j(k)(η_0, η_s)(e_k,0, e_k,s), 1≤i,j≤K},
\begin{align}\label{v-def-3}
\V_3
&=\left\{\sum_{s=-\infty}^\infty\lim_{T\to \infty}\left[\sum_{k=1}^N\fe_i^2(k)\fe^2_j(k)\cov (e^2_{k,0}, e^2_{k,s}) \right.\right.\\
&\hspace{1.5cm}+2\left(\sum_{k=1}^N\fe_i(k)\fe_j(k)\cov (e_{k,0}, e_{k,s})\right)^2 \notag\\
&\hspace{1.5cm}\left. \left. -2\sum_{k=1}^N\fe_i^2(k)\fe_j^2(k)(\cov (e_{k,0}, e_{k,s}))^2
\right], 1\leq i,j\leq K
\right\}.\notag
\end{align}
We use the notation $\V_2=\{V_2(i,j), 1\leq i,j\leq K\}$ and $\V_3=\{V_3(i,j), 1\leq i,j\leq K\}$.
If, for example, we assume that $r(s)=\cov (e_{k,0}, e_{k,s})$ for all $1\leq k\leq N$, then $\V_2$ is a diagonal matrix with
V_2(i,i)=\sum_{s=-\infty}^\infty \cov (\eta_0, \eta_s) r(s).
The expression for $\V_3$ also simplifies since by the orthonormality of the $\fe_i$'s we have
\sum_{k=1}^N\fe_i(k)\fe_j(k)\cov (e_{k,0}, e_{k,s})=r(s)I\{i=j\}.
If we further assume that each of the $\{e_{k,s}, -\infty<s<\infty\}$ sequences are Gaussian, then $\mbox{cov} (e^2_{k,0}, e^2_{k,s})=2r^2(s)$, and $\V_3$ also reduces to a diagonal matrix with $V_3(i,i)=2\sum_{s=-\infty}^\infty r^2(s)$.
a_i=lim_T→∞_i^, 1≤i≤K
and define $\G=\{G(i,j), 1\leq i,j\leq K\}$ with $G(i,j)=a_i^2a_j^2V_1+4a_ia_jV_2(i,j)+V_3(i,j)$.
Lemma <ref> demonstrates that the limit in (<ref>) is finite.
If $H_0$ and the conditions of Theorem <ref> hold, and
=O(1), T→∞,
then we have that
$\A_{N,T}(u)\;\;\mbox{converges weakly in}\;\;{\mathcal D}^K[0,1]\;\;\mbox{to}\;\;\W_{\G}(u),$
where $\W_{\G}(u)$ is a $K$–dimensional Wiener process, i.e. $\W_{\G}(u)$ is Gaussian with $E\W_{\G}(u)={\bf 0}$ and $E\W_{\G}(u)\W_{\G}^\T(u')=\min(u,u') \G$.
If $\|\bgamma\|\to 0$, as $T\to \infty$, then $a_i=0$ according to Lemma <ref>. In this case the weak limit of $\A_{N,T}(u)$ is the $K$–dimensional Wiener process $\W_{\V_3}(u)$, since $\G=\V_3$.
To state the next result we introduce the covariance matrix $\bH=\{H(i,j), 1\leq i ,j\leq K\}$: $H(1,1)=V_1, H(1,i)=H(i,1)=a_i^2V_1$ and $H(i,j)=a_i^2a_j^2V_1+4a_ia_j V_2(i,j)+ V_3(i,j), 2\leq i,j\leq K$.
If $H_0$ and the conditions of Theorem <ref> hold, and
→∞, T→∞,
then we have that $\left\{\|\bgamma\|^{-2}A_{N,T;1}(u), A_{N,T;i}(u), 2\leq i\leq K \right\}$ converges weakly in
${\mathcal D}^K[0,1]$ to $\W_{\bH}(u)$,
where $\W_{\bH}(u)$ is a $K$–dimensional Wiener process, i.e. $\W_{\bH}(u)$ is Gaussian with $E\W_{\bH}(u)={\bf 0}$ and $E\W_{\bH}(u)\W_{\bH}^\T(u')=\min(u,u') \bH$.
We note that $\sigma_1^2$ defined in Theorem <ref> coincide with $G(1,1)$ and $H(1,1)\|\gamma\|^2$ in the cases when $\|\bgamma\|=O(1)$ and $\|\bgamma\|\to \infty$ as $T \to \infty$, respectively, and so Theorems <ref> and <ref> imply Theorem <ref>.
We show in Lemma <ref> that in case of (<ref>), $\lambda_1$, the largest eigenvalue of $\C$ satisfies
\left|\frac{\lambda_1}{\|\bgamma\|^2}-1\right|=O(1)
Thus Theorem <ref> yields that $\hat{\lambda}_(u)/\|\bgamma\|^2\to 1$ in probability for all $u>0$.
Theorems <ref>, <ref> and <ref> provide the limits of the weighted differences $T^{1/2}u(\hat{\lambda}_i(u)$ $-\lambda_i)=T^{1/2}(\tilde{\lambda}_i-u\lambda_i), 1\leq i \leq K$. If the conditions of Theorem <ref> are satisfied but (<ref>) is replaced (<ref>) as in Theorem <ref>, then $T^{1/2}(\hat{\lambda}_i(u)-\lambda_i), 1\leq i \leq K$ converges weakly in ${\mathcal D}^K[c,1]$ to $\W_{\G}(u)/u$ for any $0<c\leq 1$ where $\W_{\G}(u)$ is defined in Theorem <ref>.
§ TECHNICAL RESULTS
§.§ Proof of Theorems <ref>, <ref>, <ref> and <ref>
Throughout these proofs we use the terms of the form $c_{i,j}$ to denote unimportant numerical constants. We can assume without loss of generality that $E\X_t={\bf 0}$, and so we define
\C_{N,T}(u)=\frac{1}{T}\sum_{t=1}^{\lf Tu\rf}\X_t\X_t^\T.
If (<ref>) and Assumptions <ref>, <ref>, and <ref> hold, then we have, as $T\to\infty$,
\sup_{0\leq u \leq 1}\left\| \tC_{N,T}(u)-\C_{N,T}(u) \right\|=O_P\left( \frac{N}{T} \right).
It is easy to see that
\tC_{N,T}(u)=\C_{N,T}(u)-\baX_T\left(\frac{1}{T}\sum_{t=1}^{\lf Tu\rf}\X_t\right)^\T-\left(\frac{1}{T}\sum_{t=1}^{\lf Tu\rf}\X_t\right)\baX_T^\T+
\baX_T\baX_T^\T,
and therefore
\begin{align*}
\sup_{0\leq u \leq 1}\left\| \tC_{N,T}(u)-\C_{N,T}(u) \right\|&\leq 2\sup_{0\leq u \leq 1}\left\| \baX_T\left(\frac{1}{T}\sum_{t=1}^{\lf Tu\rf}\X_t\right)^\T \right\|+\left\|\baX_T\baX_T^\T\right\|\\
&\leq 3\sup_{0\leq u \leq 1}\left\| \baX_T\left(\frac{1}{T}\sum_{t=1}^{\lf Tu\rf}\X_t\right)^\T \right\|.
\end{align*}
Using assumption (<ref>) we obtain that
\begin{align*}
&T^2\left\|\sum_{t=1}^{\lf Tu\rf}\X_t\baX_T^\T\right\|^2\\
&=\sum_{\ell=1}^N\sum_{p=1}^N \left(
\gamma_\ell\sum_{t=1}^{T}\eta_t+\sum_{t=1}^{T}e_{\ell,t}\right)^2\left(\gamma_p\sum_{t=1}^{\lf Tu\rf}\eta_t+\sum_{t=1}^{\lf Tu\rf}e_{p,t}\right)^2\\
&=\left(\sum_{\ell=1}^N\gamma_\ell^2\right)^2\left(\sum_{t=1}^{\lf Tu\rf}\eta_t\right)^2
\left(\sum_{t=1}^{ T}\eta_t\right)^2+2\sum_{\ell=1}^N\gamma_\ell^2\left(\sum_{t=1}^T\eta_t\right)^2\left(\sum_{v=1}^{\lf Tu\rf}\eta_v\right)
\left(\sum_{s=1}^{\lf Tu\rf}\sum_{p=1}^N\gamma_pe_{p,s}\right)\\
&\hspace{.5cm}+\sum_{\ell=1}^N\gamma_\ell^2\left(\sum_{t=1}^T\eta_t\right)^2\sum_{p=1}^N\left( \sum_{t=1}^{\lf Tu\rf}e_{p,t} \right)^2
+\sum_{p=1}^N\gamma_p^2\left(\sum_{t=1}^{\lf Tu\rf}\eta_t\right)^2\sum_{\ell=1}^N\left(\sum_{s=1}^Te_{\ell,s}\right)^2
\\
&\hspace{.5cm}+2\sum_{\ell=1}^N\left(\sum_{s=1}^Te_{\ell,s}\right)^2\left(\sum_{s=1}^{\lf Tu\rf}\eta_s\right)\left(\sum_{s=1}^{\lf Tu\rf}\sum_{p=1}^N \gamma_p e_{p,s}\right)+\sum_{\ell=1}^N\left(\sum_{s=1}^Te_{\ell,s}\right)^2\sum_{p=1}^N\left( \sum_{s=1}^{\lf Tu\rf} e_{p,s} \right)^2
\\
&\hspace{.5cm}+2\sum_{t=1}^T\eta_t\sum_{\ell=1}^N\gamma_\ell\left(\sum_{s=1}^Te_{\ell,s}\right)\sum_{p=1}^N\gamma_p^2\left(\sum_{v=1}^{\lf Tu\rf}\eta_v\right)^2+2\sum_{t=1}^T\eta_t\sum_{\ell=1}^N\gamma_\ell\left(\sum_{s=1}^Te_{\ell,s}\right)\sum_{p=1}^N\left( \sum_{s=1}^{\lf Tu\rf} e_{p,s} \right)^2\\
&\hspace{.5cm}+4\sum_{t=1}^T\eta_t\sum_{\ell=1}^N\left(\sum_{s=1}^T\gamma_\ell e_{\ell,s}\right)\sum_{z=1}^{\lf Tu\rf}\eta_z\sum_{p=1}^N\left(\sum_{s=1}^{\lf Tu\rf}\gamma_p e_{p,s}\right)
\\
&=R_{T,1}(u)+R_{T,2}(u)+\ldots +R_{T,9}(u).
\end{align*}
First we prove that
sup_0≤u ≤1| ∑_t=1^Tuη_t |=O_P(T^1/2).
It follows from Proposition 4 of Berkes et al. (2011) that under conditions Assumption <ref>(a) and Assumption <ref>(a) we have for any $2<\kappa\leq 12 $ that
E(∑_t=Tv^Tuη_t)^κ≤c_1,1(Tu-Tv)^κ/2 0≤v≤u ≤1,
and therefore the maximal inequality of Móricz et al. (1982) implies (<ref>).
Next we show that
sup_0≤u ≤1|∑_s=1^Tu∑_p=1^Nγ_pe_p,s|=O_P(1)T^1/2.
Following the arguments leading to (<ref>) one can verify that for any $2<\kappa\leq 12$
E|∑_s=Tv^Tue_p,s|^κ≤c_1,2(Tu-Tv)^κ/2 0≤v≤u ≤1,
with some constant $c_{1,2}$ for all $1\leq p \leq N$.
Hence for any $0\leq v<u\leq 1$ we have via Rosenthal's inequality (cf. Petrov (1995), p. 59) and (<ref>) that
\begin{align*}
E\left|\sum_{s=\lf Tv\rf}^{\lf Tu\rf}\sum_{p=1}^N\gamma_pe_{p,s}\right|^\kappa
&=E\left|\sum_{p=1}^N\sum_{s=\lf Tv\rf}^{\lf Tu\rf}\gamma_pe_{p,s}\right|^\kappa\\
&\leq c_{1,3}\left\{\sum_{p=1}^N |\gamma_p|^\kappa E\left|\sum_{s=\lf Tv\rf}^{\lf Tu\rf}e_{p,s}\right|^\kappa+\left(\sum_{p=1}^N\gamma_p^2 E\left(\sum_{s=\lf Tv\rf}^{\lf Tu\rf}e_{p,s}\right)^2\right)^{\kappa/2}
\right\}\\
&\leq c_{1,4} (\lf Tu\rf -\lf Tv\rf)^{\kappa/2}\left\{\sum_{p=1}^N |\gamma_p|^\kappa +\left(\sum_{p=1}^N \gamma_p^2 \right)^{\kappa/2}
\right\}.
\end{align*}
Using again the maximal inequality of Móricz et al. (1982) we conclude
\begin{align*}
E\sup_{0\leq u \leq 1}\left|\sum_{s=1}^{\lf Tu\rf}\sum_{p=1}^N\gamma_pe_{s,p}\right|^\kappa &\leq c_{1,5}T^{\kappa/2}\left\{\sum_{p=1}^N |\gamma_p|^\kappa +\left(\sum_{p=1}^N \gamma_p^2 \right)^{\kappa/2}
\right\}\\
&\leq c_{1,6} T^{\kappa/2}\left\|\bgamma\right\|^{\kappa},
\end{align*}
by Assumption <ref>. This completes the proof of (<ref>).
Similarly to (<ref>) we show that
sup_0≤s ≤1∑_ℓ=1^N(∑_s=1^Tue_ℓ,s)^2=O_P(NT).
First we note
E\sup_{0\leq u \leq 1}\sum_{\ell=1}^N\left(\sum_{s=1}^{\lf Tu\rf}e_{\ell,s}\right)^2\leq \sum_{\ell=1}^NE\sup_{0\leq u \leq 1}\left(\sum_{s=1}^{\lf Tu\rf}e_{\ell,s}\right)^2
and by Jensen's inequality we have
E\sup_{0\leq u \leq 1}\left(\sum_{s=1}^{\lf Tu\rf}e_{\ell,s}\right)^2\leq \left(E\sup_{0\leq u \leq 1}\left|\sum_{s=1}^{\lf Tu\rf}e_{\ell,s}\right|^\kappa\right)^{2/\kappa}.
Using again Proposition 4 of Berkes et al. (2011) we get for all $0\leq v \leq u \leq 1$ that
E\left|\sum_{s=\lf Tv\rf}^{\lf Tu\rf}e_{\ell,s}\right|^\kappa\leq c_{1,7}(\lf Tu\rf-\lf Tv\rf)^{\kappa/2}
and therefore the maximal inequality of Móricz et al. (1982) yields
\left(E\sup_{0\leq u \leq 1}\left|\sum_{s=1}^{\lf Tu\rf}e_{\ell,s}\right|^\kappa\right)^{2/\kappa}\leq c_{1,8}T^{1/2}.
This completes the proof of (<ref>).
The upper bounds in (<ref>)–(<ref>) imply
\sup_{0\leq u \leq 1}|R_{T,i}(u)|=O_P((\|\bgamma\|^4+\|\bgamma\|^3)T^2),\quad\mbox{if}\;\;i=1, 2, 7,
\sup_{0\leq u \leq 1}|R_{T,i}(u)|=O_P((\|\bgamma\|^2+\|\bgamma\|)NT^2),\quad\mbox{if}\;\;i=3, 4, 5, 8, 9.
\sup_{0\leq u \leq 1}|R_{T,6}(u)|=O_P(N^2T^2).
Assumption <ref> implies that
$\|\bgamma\|\leq c_{1,9} N,$
the proof of Lemma <ref> is complete.
Let $\bar{\lambda}_1(u)\geq \bar{\lambda}_2(u)\geq \ldots \geq \bar{\lambda}_K(u)$ denote the $K$ largest eigenvalues of $\C_{N,T}(u)$.
If (<ref>) and Assumptions <ref>, <ref>, and <ref> hold, then we have, as $T\to\infty$,
\max_{1\leq i \leq K}\sup_{0\leq u \leq 1}| \tilde{\lambda}_i(u)-\bar{\lambda}_i(u)|=O_P\left(\frac{N}{T}\right).
It is well–known (cf. Dunford and Schwartz (1988)) that
\max_{1\leq i \leq K}\sup_{0\leq u \leq 1}| \tilde{\lambda}_i(u)-\bar{\lambda}_i(u)|\leq c_{2,1}\sup_{0\leq u\leq 1}\| \tilde{\C}_T(u)-\C_T(u)\|
with some absolute constant $c_{2,1}$ and therefore the result follows from Lemma <ref>.
\begin{align*}
Z_{N,T;i}(u)%=\sum_{\ell\neq i}^N\frac{u}{\lambda_i-\lambda_\ell}\left(\fe_i^\T(\hat{\C}_{N,T}(u)-\C)\fe_\ell\right)^2
=\sum_{\ell\neq i}^N\frac{1}{u(\lambda_i-\lambda_\ell)}\left(\fe_i^\T(\tilde{\C}_{N,T}(u)-u\C)\fe_\ell\right)^2,\;\;1\leq i \leq K.
\end{align*}
If (<ref>), Assumptions <ref>, <ref>, and <ref> hold, then we have, as $T\to\infty$,
\begin{align*}
\sup_{0\leq u \leq 1}\left|\tilde{\lambda}_i(u)-\frac{\lf Tu\rf}{T}\lambda_i-\fe_i^\T(\tilde{\C}_{N,T}(u)-u\C)\fe_i-
%\sum_{\ell\neq i}^N\frac{1}{u(\lambda_i-\lambda_\ell)}\left(\fe_i^\T(\tilde{\C}_{N,T}(u)-u\C)\fe_\ell\right)^2
\end{align*}
According to formula (5.17) of Hall and Hosseini–Nasab (2009) we have for all $1/T\leq u\leq 1$ that
\begin{align*}
\sum_{\ell\neq i}^N\frac{1}{\lambda_i-\lambda_\ell}\left(\fe_i^\T(\hat{\C}_{N,T}(u)-\C)\fe_\ell\right)^2
\right|\\
&\hspace{.5cm}\leq c_{3,1} \hat{\Delta}^{3}(u),
\end{align*}
\hat{\Delta}(u)=\max_{1\leq \ell \leq N}\left( \sum_{j=1}^N(\hat{C}_{N,T;j,\ell}(u) -C_{j,\ell})^2 \right)^{1/2},
and $\hat{C}_{N,T;j,\ell}(u)$ and $ C_{j,\ell}$ denote the $(k,\ell)^{\mbox{th}}$ element of $\hat{\C}_{N,T}(u)$ and $\C$, respectively. Hence
\begin{align*}
\sup_{0\leq u \leq 1}\left|\tilde{\lambda}_i(u)-\frac{\lf Tu\rf}{T}\lambda_i-\fe_i^\T(\tilde{\C}_{N,T}(u)-u\C)\fe_i-Z_{N,T;i}(u)
%\sum_{\ell\neq i}^N\frac{1}{u(\lambda_i-\lambda_\ell)}\left(\fe_i^\T(\tilde{\C}_{N,T}(u)-u\C)\fe_\ell\right)^2
\right|
\leq c_{3,1}\sup_{0\leq u \leq 1}\Delta^{3}(u),
\end{align*}
\Delta(u)=\max_{1\leq \ell \leq N} R_{N,T;\ell}(u)
R_{N,T;\ell}(u)=\left( \sum_{j=1}^N(\tilde{C}_{N,T;j,\ell}(u) -\frac{\lf Tu\rf}{T}C_{j,\ell})^2 \right)^{1/2},
where $\tilde{C}_{N,T;j,\ell}(u)$ denotes the $(j,\ell)^{\mbox{th}}$ element of the matrix $\tilde{\C}_{N,T}(u)$.
By inequality (2.30) in Petrov (1995, p. 58) we conclude
R_{N,T;\ell}^6(u)\leq N^2\sum_{j=1}^N\left(\tilde{C}_{N,T;j,\ell}(u) -\frac{\lf Tu\rf}{T}C_{j,\ell}\right)^6
and hence
E\sup_{0\leq u \leq 1}\left( R_{N,T;\ell}(u) \right)^6\leq N^2\sum_{j=1}^NE\sup_{0\leq u \leq 1}\left(\tilde{C}_{N,T;j,\ell}(u)- \frac{\lf Tu\rf}{T}C_{j,\ell}\right)^6.
Using the definitions of $\tilde{C}_{N,T;j,\ell}(u)$ and $ C_{j,\ell}$ we write
\begin{align*}
\biggl(\tilde{C}_{N,T;j,\ell} &(u)-\frac{\lf Tu \rf}{T}C_{j,\ell} \biggl)^6\\
&=T^{-6}\left(\sum_{s=1}^{\lf Tu \rf}\left[\gamma_\ell\gamma_j(\eta_s^2-1)+\gamma_\ell\eta_se_{j,s}+\gamma_j\eta_se_{\ell,s}+e_{\ell,s}e_{j,s}-Ee_{\ell,s}e_{j,s}
\right]\right)^6\\
&\leq 4^6T^{-6}\left[\gamma_\ell^6\gamma_j^6\left(\sum_{s=1}^{\lf Tu\rf }(\eta_s^2-1)\right)^6+\gamma_\ell^6\left(\sum_{s=1}^{\lf Tu\rf}\eta_se_{j,s}\right)^6
+\gamma_j^6\left(\sum_{s=1}^{\lf Tu\rf}\eta_se_{\ell,s}\right)^6\right.\\
&\hspace{2cm}\left. +\left( \sum_{s=1}^{\lf Tu\rf}(e_{\ell,s}e_{j,s}- Ee_{\ell,s}e_{j,s}) \right)^6\right].
\end{align*}
Utilizing Assumption <ref>(a), we obtain along the lines of (<ref>) that $E(\sum_{s=1}^t(\eta_s^2-1))^6\leq c_{3,2} t^3$, so by the stationarity of $\eta_t^2,-\infty<t<\infty$ and the maximal inequality of Móricz et al. (1982) we obtain that
E\sup_{0\leq u \leq 1}\left(\sum_{s=1}^{\lf Tu\rf }(\eta_s^2-1)\right)^6\leq c_{3,3}T^3
Similarly, for all $1\leq j, \ell\leq N$
E\sup_{0\leq u \leq 1}\left(\sum_{s=1}^{\lf Tu\rf}\eta_se_{\ell,s}\right)^6\leq c_{3,4}T^3
\;\;\;\mbox{and}\;\;\;
E\sup_{0\leq u \leq 1}\left( \sum_{s=1}^{\lf Tu\rf}(e_{\ell,s}e_{j,s}- Ee_{\ell,s}e_{j,s}) \right)^6\leq c_{3,5}T^3.
Hence for all $1\leq \ell\leq N$ we have by Assumption <ref> that
E(sup_0≤u ≤1R_N,T;ℓ(u))^6≤c_3,6T^-3N^3.
Using (<ref>) we conclude for all $x>0$
\begin{align*}
P\left\{ \sup_{0\leq u \leq 1}\max_{1\leq \ell \leq N}R_{N,T;\ell}(u)>xN^{2/3}T^{-1/2}
\right\}
&\leq \sum_{\ell=1}^NP\left\{\sup_{0\leq u \leq 1}R_{N,T;\ell}(u)>xN^{2/3}T^{-1/2}
\right\} \\
&\leq \sum_{\ell=1}^N \frac{T^3}{x^6N^6}E\left(\sup_{0\leq u \leq 1}R_{N,T;\ell}(u)\right)^6\\
&\leq\sum_{\ell=1}^N \frac{T^3}{x^6N^6}C_5T^{-3}N^3,
\end{align*}
which shows that
sup_0≤u ≤1Δ^3(u)=O_P(N^2T^-3/2) sup_0≤u ≤1uΔ̂^3(u)=O_P(N^2T^-3/2).
Since $\fe_1$ is defined via (<ref>) up to a sign, we can assume without loss of generality that $\bgamma^\T\fe_1\geq 0$.
If (<ref>), Assumptions <ref>, <ref>, and <ref> hold
and $\|\bgamma\|\to \infty$ hold, then we have
max_2≤i ≤N|^_i|≤c_4,1 c_4,1,
max_2≤i ≤Nλ_i≤c_4,2 c_4,2.
By (<ref>) we have
\C=\bgamma\bgamma^T+\bLambda,
where $\bLambda$ is the $N\times N$ diagonal matrix with $\sigma_1^2, \sigma_2^2, \ldots, \sigma_N^2$ in the diagonal. We can write
\fe_1=\bar{\alpha}_1\frac{\bgamma}{\|\bgamma\|}+\bar{\beta}_1\br_1,\;\;\mbox{with some}\;\;\bar{\alpha}_1\geq 0,\;\;\mbox{where}\;\; \bar{\alpha}_1^2+\bar{\beta}_1^2=1, \bgamma^\T\br_1=0\;\;\mbox{and}\;\;\|\br_1\|=1.
It follows from the definition of $\lambda_1$ and $\fe_1$ that
\begin{align*}
\lambda_1=\fe_1^\T\C\fe_1\geq \|\bgamma\|^2
\end{align*}
\begin{align}\label{lin-125}
\fe_1^\T\C\fe_1=\bar{\alpha}_1^2\|\bgamma\|^2+\fe_1^\T\bLambda\C\fe_1,\;\;\fe_1^\T\bLambda\C\fe_1\leq \sum_{\ell=1}^N\fe_i^2(\ell)\sigma_\ell^2\leq c_5
\end{align}
where $c_5$ is defined in Assumption <ref>(b). Thus we conclude
\|\bgamma\|^2\leq \bar{\alpha}_1^2\|\bgamma\|^2+c_5.
By assumption $\bgamma^\T\fe_1\geq 0$ and therefore $0\leq \bar{\alpha}_1\leq 1$. Hence $(1-\bar{\alpha}_1)^2\leq 1-\bar{\alpha}_1^2\leq c_5/\|\bgamma\|^2$ and $\bar{\beta}_1^2\leq c_5/\|\bgamma\|^2$. Thus we get
completing the proof of (<ref>). Since $\bar{\alpha}_2^2\geq 1-c_5/\|\bgamma|^2$, (<ref>) follows from (<ref>).
For all $i\geq 2$ we have
|\bgamma^\T\fe_i|=\|\bgamma\|\left|\left(\frac{\bgamma}{\|\bgamma\|}-\fe_1\right)^\T\fe_i\right|\leq \|\bgamma\|\left\|\frac{\bgamma}{\|\bgamma\|}-\fe_1\right\|\leq 2c_5
by (<ref>) which gives (<ref>). Since $\lambda_i=\fe_i^\T\C\fe_i=(\fe_i^\T\bgamma)^2 +\fe_i^\T\bLambda\fe_i$ and $\fe_i^\T\bLambda\fe_i= \sum_{\ell=1}^N\fe_i^2(\ell)\sigma^2_\ell\leq c_5$ by Assumption <ref>(b), the last claim of this lemma follows from (<ref>).
If (<ref>), and Assumptions <ref>, <ref>, and <ref> hold, then we have
max_1≤i ≤Ksup_0≤u≤1|Z_N,T;i(u)|=O_P(N(logT)^1/3/T).
It follows from (<ref>) that $\fe_i\C\fe_\ell=0$, if $i\neq \ell$. Hence we get
\begin{align*}
\fe_i^\T\left(\tilde{\C}_{N,T}(u)-\frac{\lf Tu\rf}{T}\C\right)\fe_\ell&=\frac{1}{T}\sum_{s=1}^{\lf Tu\rf}\fe_i^\T\X_s\X_s^\T\fe_\ell.
\end{align*}
First we assume that $\|\bgamma\|=O(1)$. It follows from the definition of $Z_{N,T;i}$ that
\begin{align*}
|Z_{N,T;i}(u)|&=\left|\sum_{\ell\neq i}^N\frac{1}{u(\lambda_i-\lambda_\ell)}\left(\fe_i^\T(\tC_{N,T}(u)-u\C)\fe_\ell \right)^2\right|\\
&\leq \frac{1}{c_5}\frac{1}{T}\sum_{\ell\neq i}^N\left(\frac{1}{(Tu)^{1/2}}\sum_{s=1}^{\lf Tu\rf}\fe_i^\T \X_s\X_s^\T\fe_\ell\right)^2,
\end{align*}
where $c_0$ is defined in Assumption <ref>. Let $\rho>1$ and write with $c=\lf 1/\log \rho\rf +1$
\begin{align*}
\max_{1\leq v \leq T}v^{-1/2}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right|&\leq \max_{1\leq k\leq c\log T}\max_{\rho^{k-1}<v\leq \rho^k}v^{-1/2}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right|\\
&\leq \max_{1\leq k\leq c\log T}\rho^{-(k-1)/2}\max_{1\leq v\leq \rho^k}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right|.
\end{align*}
Thus we get for any $x>0$ via Markov's inequality that
\begin{align}\label{maxx}
P&\left\{\max_{1\leq v \leq T}v^{-1/2}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right|>x\right\}\\
\sum_{k=1}^{c\log T}P\left\{\max_{1\leq v\leq \rho^k}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right|>x\rho^{(k-1)/2}
\right\}\notag\\
&\hspace{1cm}\leq \sum_{k=1}^{c\log T}x^{-6}\rho^{-3(k-1)}E\left( \max_{1\leq v\leq \rho^k}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right| \right)^6.\notag
\end{align}
Using (<ref>) we obtain with $\fe_i=(\fe_i(1), \fe_i(2), \ldots, \fe_i(N))^\T$ that
\begin{align*}
\fe_i^\T\X_s\X_s^\T\fe_\ell
\end{align*}
since for $i\neq \ell$ we have $E\fe_i^\T\X_s\X_s^\T\fe_\ell=\fe_i^\T\C\fe_\ell=0$. Clearly, on account of $\|\fe_i\|=1$, the Cauchy–Schwarz inequality implies
\begin{align*}
\left|\sum_{k=1}^N\gamma_{k}\fe_i(k)\right|\leq \|\bgamma\|.
\end{align*}
Following the proofs of (<ref>), we get that from Assumption <ref>(a) that
\begin{align}\label{fo-1}
\leq c_{5,1}v^{3}\|\bgamma\|^{12}.
\end{align}
\tau_{s}=\eta_s\sum_{n=1}^Ne_{n,s}\fe_\ell(n)\;\;\;\mbox{and}\;\;\;\tau_{s}^{(m)}=\eta_s^{(m)}\sum_{n=1}^Ne_{n,s}^{(m)}\fe_\ell(n),
where $\eta_s^{(m)}$ and $e_{n,s}^{(m)}$ are defined in Assumption <ref>(a) and Assumption <ref>(b), respectively. By independence we have
\begin{align*}
&\leq 2^{6} E\left|\eta_0-\eta_0^{(m)}\right|^{6} E\left|\sum_{n=1}^Ne_{n,0}\fe_\ell(n)\right|^{6}
+2^{6} E\left|\eta_0^{(m)}\right|^{6} E\left|\sum_{n=1}^N(e_{n,0}-e_{n,0}^{(m)})\fe_\ell(n)\right|^{6}.
\end{align*}
By the independence of the variables $e_{n,0}, 1\leq n \leq N$ and the Rosenthal inequality (cf. Petrov (1995)) we conclude
\begin{align*}
E\left|\sum_{n=1}^Ne_{n,0}\fe_\ell(n)\right|^6 &\leq c_{5,2}\left\{\sum_{n=1}^NE|e_{n,0}|^6 |\fe_\ell(n)|^6+
\left(\sum_{n=1}^NE e_{n,0}^2\fe_\ell^2(n) \right)^{3}
\right\}\\
&\leq c_{5,3}\sup_{1\leq n<\infty}Ee^6_{n,0}\\
&\leq c_{5,4},
\end{align*}
where $c_{5,4}$ is a constant, on account of Assumption <ref>(b) and $\|\fe_\ell\|=1$. Due to the independence of $e_{n,0}-e_{n,0}^{(m)}$ and $ e_{r,0}-e_{r,0}^{(m)}$, if $n\neq r$, we can apply again the Rosenthal inequality to get
\begin{align*}
&\leq c_{5,5}\left\{\sum_{n=1}^NE|e_{n,0}-e_{n,0}^{(m)}|^6 |\fe_\ell(n)|^6
\right\}\\
&\leq c_{5,6}m^{-6\alpha},
\end{align*}
resulting in
\begin{align}\label{hor-1}
E\left|\tau_{0}-\tau_0^{(m)}\right|^6\leq c_{5,7}m^{-6\alpha} .
\end{align}
Hence the moment inequality in Berkes et al. (2011) yields
Similarly to (<ref>) we have
E|∑_s=1^v η_s∑_k=1^Ne_k,s_i(k) |^6≤c_5,9v^3.
\bar{\tau}_s=\sum_{n=1}^N\sum_{k=1}^N(e_{k,s}e_{n,s}-Ee_{k,s}e_{n,s})\fe_{i}(k)\fe_\ell(n)=\sum_{n=1}^Ne_{n,s}\fe_\ell(n)\sum_{k=1}^Ne_{k,s}\fe_{i}(k)
\begin{align*}
\bar{\tau}_s^{(m)}
\end{align*}
where $e_{n,s}^{(m)}$ defined in Assumption <ref>(b). Clearly,
\left|\sum_{n=1}^NEe_{n,s}^2\fe_{i}(n)\fe_\ell(n)\right|\leq \sup_{1\leq n<\infty}Ee_{n,0}^2,
\begin{align*}
\bar{\tau}_s-\bar{\tau}_s^{(m)}=\left(\sum_{n=1}^N(e_{n,s}-e_{k,s}^{(m)})\fe_\ell(n)\right)\sum_{k=1}^Ne_{k,s}\fe_{i}(k)+\left(\sum_{k=1}^N(e_{k,s}-e_{k,s}^{(m)} )\fe_{i}(k)\right)\sum_{n=1}^Ne_{n,s}^{(m)}\fe_\ell(n).
\end{align*}
Thus we get by the Cauchy–Schwarz inequality that
\begin{align*}
E|\bar{\tau}_0-\bar{\tau}_0^{(m)}|^{6}\leq 2^{6} &\left\{\left(E\left| \sum_{n=1}^N(e_{n,0}-e_{k,0}^{(m)})\fe_\ell(n)\right|^{12}
E\left| \sum_{k=1}^Ne_{k,s}\fe_{i}(k) \right|^{12}\right)^{1/2} \right.\\
& \hspace{1cm} +E\left. \left(\left|\sum_{k=1}^N(e_{k,0}-e_{k,0}^{(m)} )\fe_{i}(k)\right|^{12}E\left|\sum_{n=1}^Ne_{n,0}^{(m)}\fe_\ell(n)\right|^{12}\right)^{1/2}
\right\}.
\end{align*}
Using again Rosenthal's and Jensen's inequalities, we obtain that
\begin{align*}
E\left| \sum_{n=1}^N(e_{n,0}-e_{k,0}^{(m)})\fe_\ell(n)\right|^{12}&\leq c_{5,10}\left\{\sum_{n=1}^NE|e_{n,0}-e_{k,0}^{(m)}|^{12}|\fe_\ell(n)|^{12}\right. \\
& \left. \hspace{1cm}+
\left( \sum_{n=1}^NE(e_{n,0}-e_{k,0}^{(m)})^2\fe_\ell^2(n)\right)^{6}
\right\}\\
&\leq c_{5, 11}m^{-12\alpha},
\end{align*}
and similarly
\begin{align*}
E\left| \sum_{k=1}^Ne_{k,0}\fe_{i}(k) \right|^{12}&\leq c_{5, 12}\left\{\sum_{k=1}^NE|e_{k,0}|^{12}|\fe_{i}(k) |^{12}
+\left( \sum_{k=1}^NEe_{k,0}^2\fe_{i}^2(k) \right)^{6}
\right\}\\
&\leq 2 c_{5,12}\sup_{1\leq k <\infty}E|e_{k,0}|^{12}.
\end{align*}
Thus we have
E|τ̅_0-τ̅_0^(m)|^6≤c_5, 13m^-6α,
and therefore Proposition 4 of Berkes et al. (2011) implies
E|∑_s=1^vτ̅_s|^6≤c_5, 14v^3.
Putting together (<ref>)–(<ref>) we conclude
E|∑_s=1^v_i^_s_s^_ℓ|^6≤c_5, 15v^3(1+^6+^12).
Since $\fe_i^\T\X_s\X_s^\T\fe_\ell, -\infty<s<\infty$ is a stationary sequence, (<ref>) and the maximal inequality of Móricz et al. (1982) imply
\begin{align}\label{ineq**}
E\max_{1\leq v\leq z}\left|\sum_{s=1}^v\fe_i^\T\X_s\X_s^\T\fe_\ell\right|^6\leq c_{5, 16}z^3(1+\|\bgamma\|^6+\|\bgamma\|^{12}).
\end{align}
Now we use (<ref>) with $x=u(\log T)^{1/6}$ resulting in
\begin{align*}
P\left\{\max_{1\leq v \leq T}v^{-1/2}\left|\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right|>u(\log T)^{1/6}\right\}\leq c_{5, 17}u^{-6},
\end{align*}
E\left(\max_{1\leq v \leq T}v^{-1/2}\sum_{s=1}^{v}\fe_i^\T\X_s\X_s^\T\fe_\ell\right)^2\leq c_{5, 18}(\log T)^{1/3}.
This completes the proof of (<ref>).
Next we assume that $\|\bgamma\|\to \infty$. It is easy to see that for for $2\leq i\leq K$
\begin{align*}
|Z_{N,T;i}(u)|\leq \frac{1}{T}&\left|\sum_{\ell\neq i, \ell\neq 1}^N\frac{1}{\lambda_i-\lambda_\ell}\left(\frac{1}{(Tu)^{1/2}}\sum_{s=1}^{\lf Tu\rf} \fe_i^\T\X_s\X_s^\T\fe_\ell^\T\right)^2\right|\\
&+\frac{1}{T}\frac{1}{\lambda_1-\lambda_2}\left(\frac{1}{(Tu)^{1/2}}\sum_{s=2}^{\lf Tu\rf} \fe_i^\T\X_s\X_s^\T\fe_1\right)^2.
\end{align*}
If $2\leq i \leq K$, then the proof of (<ref>) shows that
\sum_{\ell\neq i, \ell\neq 1}^N\left(\frac{1}{(Tu)^{1/2}}\sum_{s=1}^{\lf Tu\rf} \fe_i^\T\X_s\X_s^\T\fe_\ell\right)^2=O_P(N(\log T)^{1/3}),
and therefore by Assumption <ref> for any $2\leq i \leq K$ we have
\begin{align*}
\left|\sum_{\ell\neq i, \ell\neq 1}\frac{1}{\lambda_i-\lambda_\ell}\left(\frac{1}{(Tu)^{1/2}}\sum_{s=1}^{\lf Tu\rf} \fe_i^\T\X_s\X_s^\T\fe_\ell^\T\right)^2\right|=O_P(N(\log T)^{1/3}).
\end{align*}
By (<ref>) we have along the lines of the proof of (<ref>)
\begin{align}\label{horem-1}
E\sup_{1\leq v \leq T}\frac{1}{v}&\left(\sum_{s=1}^v\fe_i^\T\X_s\X_s^\T\fe_\ell -\bgamma^\T\fe_i\bgamma^\T\fe_\ell\sum_{s=1}^v(\eta_s^2-1)-
\bgamma^\T\fe_i\sum_{s=1}^v\sum_{n=1}^Ne_{n,s}\fe_\ell(n) \right.\\
&\hspace{2cm}\left.-\bgamma^\T\fe_\ell\sum_{s=1}^v\sum_{k=1}^Ne_{k,s}\fe_i(k)\right)^2\leq c_{5, 19}(\log T)^{1/3},\notag
\end{align}
where in the last step we used (<ref>).
Also, (<ref>) and (<ref>) imply via the maximal inequality in Móricz et al. (1982) that
Esup_1≤v ≤T(1/v∑_s=1^v(η_s^2-1))^2≤c_5,20(logT)^1/3,
Esup_1≤v ≤T1/v( ∑_s=1^v∑_k=1^Ne_k,s_i(k))^2≤c_5, 21(logT)^1/3.
Using now (<ref>) and (<ref>) we conclude that
\begin{align*}
\frac{1}{\lambda_1-\lambda_2}\left(\frac{1}{(Tu)^{1/2}}\sum_{s=1}^{\lf Tu\rf} \fe_i^\T\X_s\X_s^\T\fe_1\right)^2=\frac{(\fe_1^\T\bgamma)^2}{\lambda_1-\lambda_2}O_P((\log T)^{1/3}).
\end{align*}
Since by Lemma <ref> we have that $(\fe_1^\T\bgamma)^2/(\lambda_1-\lambda_2)=O(1)$, the proof of (<ref>) is complete when $2\leq i \leq K$. It is easy to see that by (<ref>) and Lemma <ref>
\begin{align*}
\sup_{0\leq u \leq 1}|Z_{N,T;1}(u)|&\leq \frac{1}{T}\frac{1}{\lambda_1-\lambda_2}\sup_{0\leq u \leq 1}\sum_{\ell=2}^N\left(\frac{1}{(Tu)^{1/2}}\sum_{s=1}^{\lf Tu\rf}\fe_1^\T\X_s\X_s^\T\fe_\ell\right)^2\\
&=\frac{1}{T}\frac{N}{\lambda_1-\lambda_2}\left(O_P((\log T)^{1/3} +(\fe_1^\T\gamma)^2 E\max_{1\leq v \leq T}\left(v^{-1/2}\sum_{s=1}^v(\eta_s^2-1)\right)^2\right.\\
+E\max_{2\leq i\leq N}\left(v^{-1/2}\sum_{s=1}^v\sum_{k=1}^Ne_{k,s}\fe_i(k)\right)^2 \right)\\
&=\frac{(\fe_1^\T\gamma)^2}{\lambda_1-\lambda_2}\frac{N(\log T)^{1/3}}{T}
\end{align*}
an account of (<ref>) and (<ref>). According to Lemma <ref> we have that $(\fe_1^\T\gamma)^2/(\lambda_1-\lambda_2)=O(1)$, completing the proof of Lemma <ref>.
Using the definition of $\tilde{\C}_{N,T}(u)$ and (<ref>) we get for any $1\leq i \leq K$,
\begin{align*}
T\fe_i^\T(\tilde{\C}_{N,T}(u)-(\lf Tu\rf/T)\C)\fe_i=(\fe_i^\T\bgamma)^2&\sum_{t=1}^{\lf Tu\rf}(\eta_t^2-1)+2\fe_i^T\bgamma\sum_{t=1}^{\lf Tu\rf}\eta_t\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}\\
&+\sum_{t=1}^{\lf Tu\rf}\left(\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}\right)^2-\lf Tu\rf\sum_{\ell=1}^N\fe_i^2(\ell)\sigma^2_{\ell}.
\end{align*}
D_{N,T}(u)=\frac{1}{T^{1/2}}\sum_{t=1}^{\lf Tu\rf}(\eta_t^2-1),\;\;\;F_{N,T;i}(u)=\frac{1}{T^{1/2}}\sum_{t=1}^{\lf Tu\rf}\eta_t\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t},\;1\leq i\leq K,
G_{N,T;i}(u)=\frac{1}{T^{1/2}}\left\{\sum_{t=1}^{\lf Tu\rf}\left(\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}\right)^2-\lf Tu\rf\sum_{\ell=1}^N\fe_i^2(\ell)\sigma^2_{\ell}\right\},\;1\leq i\leq K.
If (<ref>) and Assumptions <ref>, <ref>, and <ref> hold, then $\{D_{N,T}(u), F_{N,T;i}(u), G_{N,T;i}(u), 0\leq u \leq 1, 1\leq i \leq K\}$ converges in ${\mathcal D}^{2K+1}[0,1]$ to the Gaussian process $\bGamma(u)=(\Gamma_1(u), \Gamma_2(u),\ldots, \\
\Gamma_{2K+1}(u))^\T,0\leq u \leq 1$, $E\bGamma(u)={\bf 0}$, and
V_1 0^ 0^
0 _2
0 _3
First we define the $m$–dependent processes
D_{N,T}^{(m)}(u)=\frac{1}{T^{1/2}}\sum_{t=1}^{\lf Tu\rf}((\eta_t^{(m)})^2-1),\;\;\;F_{N,T;i}^{(m)}(u)=\frac{1}{T^{1/2}}\sum_{t=1}^{\lf Tu\rf}\eta_t\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}^{(m)},\;1\leq i\leq K,
G_{N,T;i}^{(m)}(u)=\frac{1}{T^{1/2}}\left\{\sum_{t=1}^{\lf Tu\rf}\left(\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}^{(m)}\right)^2-\lf Tu\rf\sum_{\ell=1}^N\fe_i^2(\ell)\sigma^2_{\ell}\right\},\;1\leq i\leq K,
where $\eta_t^{(m)}$ and $e_{\ell,t}^{(m)}$ are defined in Assumption <ref>(a) and Assumption <ref>(b), respectively. We show that for any $x>0$
\begin{align}\label{c-1}
\lim_{m\to\infty}\limsup_{T\to \infty}P\left\{|D_{N,T}(u)-D_{N,T}^{(m)}(u)|>x\right\}=0,
\end{align}
\begin{align}\label{c-2}
\lim_{m\to\infty}\limsup_{T\to \infty}P\left\{| F_{N,T;i}(u)- F_{N,T;i}^{(m)}(u) |>x\right\}=0,
\end{align}
\begin{align}\label{c-3}
\lim_{m\to\infty}\limsup_{T\to \infty}P\left\{| G_{N,T;i}(u)- G_{N,T;i}^{(m)}(u) |>x\right\}=0,
\end{align}
for all $0<u\leq 1$ and $1\leq i \leq K$. It follows from Assumption <ref>(a) and the Cauchy–Schwarz inequality that
\begin{align}\label{c-4}
E\left|\eta_0^2-(\eta^{(m)}_0)^2\right|^6&= E\left\{|\eta_0+\eta^{(m)}_0||\eta_0-\eta^{(m)}_0|\right\}^6\\
&\leq 2^4 (E\eta_0^{12})^{1/2}(E|\eta_0-\eta^{(m)}|^{12})^{1/2}\notag\\
&\leq c_{6,1}m^{-6\alpha}.\notag
\end{align}
By stationarity, we get that
\begin{align*}
\mbox{var}&\left(T^{-1/2}\sum_{s=1}^{\lf Tu\rf}(\eta_s^2-(\eta^{(m)}_s)^2)^2\right)\\
&\leq \frac{1}{T}\sum_{s=1}^TE(\eta_s^2-(\eta^{(m)}_s)^2)
+2\sum_{s=1}^T E(\eta_0^2-(\eta^{(m)}_0)^2)(\eta_s^2-(\eta^{(m)}_s)^2)\\
&\leq E(\eta_0^2-(\eta^{(m)}_0)^2)^2+2\sum_{s=1}^T |E(\eta_0^2-(\eta^{(m)}_0)^2)(\eta_s^2-(\eta^{(m)}_s)^2)|.
\end{align*}
Since $\eta_0^2-(\eta^{(m)}_0)^2$ is independent of $\eta^{(m)}_s$, if $s>m$, we obtain that
\begin{align*}
\sum_{s=m+1}^T &|E(\eta_0^2-(\eta^{(m)}_0)^2)(\eta_s^2-(\eta^{(m)}_s)^2)|\\
&\leq \sum_{s=m+1}^T |E(\eta_0^2-1)\eta_s^2|+\sum_{s=m+1}^T |E((\eta^{(m)}_0)^2)-1)\eta_s^2|.
%+\sum_{s=m+1}^T |E(\eta_0^2-1)(\eta^{(m)}_0)^2|+\sum_{s=m+1}^T |E((\eta^{(m)}_s)^2)-1)(\eta^{(m)}_0)^2|.
\end{align*}
The independence of $\eta_0$ and $\eta_s^{(s)}$, (<ref>), and Hölder's inequality yield
\begin{align*}
\sum_{s=m+1}^T |E(\eta_0^2-1)\eta_s^2|&=\sum_{s=m+1}^T |E(\eta_0^2-1)(\eta_s^2-(\eta_s^{(s)})^2|\\
&\leq \sum_{s=m+1}^\infty
&\leq c_{6,2}m^{-(\alpha-1)}
\end{align*}
with $c_{6,2}=(c_{6,1}/(\alpha-1)) (E|\eta_0^2-1|^{6/5})^{5/6} $. The same argument gives that
\sum_{s=m+1}^T |E((\eta^{(m)}_0)^2)-1)\eta_s^2|\leq c_{6,2}m^{-(\alpha-1)}.
On the other hand, applying again (<ref>) and the Cauchy–Schwarz inequality we conclude
\begin{align*}
\sum_{s=1}^m |E(\eta_0^2-(\eta^{(m)}_0)^2)(\eta_s^2-(\eta^{(m)}_s)^2)|\leq \sum_{s=1}^m E(\eta_0^2-(\eta^{(m)}_0)^2)^2 %^{1/2}(E(\eta_s^2-(\eta^{(m)}_s)^2)^2)^{1/2}
\leq c_{6,1}m^{-(\alpha-1)}.
\end{align*}
Chebyshev's inequality now implies (<ref>). The proofs of (<ref>) and (<ref>) go along the lines of (<ref>), we only need to replace (<ref>) with (<ref>) and (<ref>), respectively.
Next we show that for each $m$, $\{D_{N,T}^{(m)}(u), F_{N,T;i}^{(m)}(u), G_{N,T;i}^{(m)}(u), 0\leq u \leq 1, 1\leq i \leq K\}$ converges in ${\mathcal D}^{2K+1}[0,1]$ to the Gaussian process $\bGamma^{(m)}(u)=(\Gamma_1^{(m)}(u), \Gamma_2^{(m)}(u),\\
\ldots, \Gamma_{2K+1}^{(m)}(u))^\T,0\leq u \leq 1$, with $E\bGamma^{(m)}(u)={\bf 0}$, and
V_1^(m) 0^ 0^
0 _2^(m)
0 _3^(m)
V_1^(m)=∑_ℓ=-m^m((η_0^(m))^2, (η^(m)_ℓ)^2),
_2^(m)={∑_s=-m^mlim_N→∞∑_k=1^N_i(k)_j(k)(η_0^(m), η_s^(m))(e_k,0^(m), e_k,s^(m)), 1≤i,j≤K},
\begin{align}\label{v-def-3-m}
\V_3^{(m)}
&=\left\{\sum_{s=-m}^m\lim_{N\to \infty}\left(\sum_{k=1}^N\fe_i^2(k)\fe^2_j(k)\cov ((e_{k,0}^{(m)})^2, (e_{k,s}^{(m)})^2) \right.\right.\\
&\hspace{1.5cm}+2\left(\sum_{k=1}^N\fe_i(k)\fe_j(k)\cov (e_{k,0}^{(m)}, e_{k,s}^{(m)})\right)^2 \notag\\
&\hspace{1.5cm}\left. \left. -2\sum_{k=1}^N\fe_i^2(k)\fe_j^2(k)(\cov (e_{k,0}^{(m)}, e_{k,s}^{(m)}))^2
\right), 1\leq i,j\leq K
\right\}.\notag
\end{align}
Let $0\leq u_1<u_2<\ldots <u_M\leq 1$ and $\mu_{i,k,\ell}, 1\leq i \leq M, 1\leq k,\ell\leq K$. We can write
\begin{align*}
&\sum_{k=1}^M \mu_{k,1,1}(D_{N,T}^{(m)}(u_k) -D_{N,T}^{(m)}(u_{k-1}))+\sum_{k=1}^M \sum_{i=1}^K \mu_{k,2,i}(F_{N,T,i}^{(m)}(u_k)-F_{N,T,i}^{(m)}(u_{k-1}))\\
&\hspace{2cm}+\sum_{k=1}^M \sum_{i=1}^K \mu_{k,3,i}(G_{N,T,i}^{(m)}(u_k)-G_{N,T,i}^{(m)}(u_{k-1}))\\
&={\mathcal S}_1+\ldots +{\mathcal S}_M,
\end{align*}
{\mathcal S}_k=\sum_{s=\lf Tu_{i-1}\rf +1}^{\lf Tu_i\rf}\xi_{N,T;s}(k),\;\;1\leq i\leq M.
The variables $\xi_{N,T;s}(k),\lf Tu_{k-1}\rf +1\leq s \leq \lf Tu_k\rf, 1\leq k \leq M$ are $m$–dependent and therefore $T^{-1/2}{\mathcal S}_1$, $T^{-1/2}{\mathcal S}_2,\ldots$, $T^{-1/2}{\mathcal S}_M$ are asymptotically independent. Hence we need only show the asymptotic normality of $T^{-1/2}{\mathcal S}_k$ for all $1\leq k \leq M$. For every fixed $k$ the variables $\xi_{N,T;s}(k),\lf Tu_{k-1}\rf +1\leq s \leq \lf Tu_k\rf$ form an $m$–dependent stationary sequence with zero mean,
\begin{align*}
\lim_{T\to\infty}&\mbox{var}\left(T^{-1/2}{\mathcal S}_k\right)\\
&=\mbox{var}\left( \mu_{k,1,1} \Gamma_{1}^{(m)}(u_k) -(\Gamma_1^{(m)}(u_{k-1})) +
\sum_{i=1}^K \mu_{k,2,i}(\Gamma_{i+1}^{(m)}(u_k)-\Gamma_{i+1}^{(m)}(u_{k-1}))\right. \\
&\hspace{3cm}\left. +\sum_{i=1}^K \mu_{k,3,i}(\Gamma_{i+K+1}^{(m)}(u_k)-\Gamma_{i+K+1}^{(m)}(u_{k-1}))
\right)
\end{align*}
and $E|\xi_{N,T;s}(k)|^3\leq C_{1}$, where $C_{1,1}$ does not depend on $N$ nor on $T$. Due to the $m$–dependence, these properties imply the asymptotic normality of $T^{-1/2}{\mathcal S}_k$. Applying the Cramér–Wold device (cf. Billingsley (1968)), we get that the finite dimensional distributions of $\{D_{N,T}^{(m)}(u),F_{N,T;i}^{(m)}(u), G_{N,T;i}^{(m)}(u), 0\leq u \leq 1, 1\leq i \leq K\}$ converge to that of $\bGamma^{(m)}(u)$.
Since $\|\V^{(m)}-\V\|\to 0$ as $T\to \infty$, and $\bGamma(u)$ and $\bGamma^{(m)}(u)$ are Gaussian processes we conclude that that $\bGamma^{(m)}(u)$ converges in ${\mathcal D}^{2K+1}[0,1]$ to $\bGamma(u)$. On account of (<ref>)–(<ref>) we obtain that the finite dimensional distributions of $\{D_{N,T}(u),F_{N,T;i}(u), G_{N,T;i}(u), 0\leq u \leq 1, 1\leq i \leq K\}$ converge to that of $\bGamma(u)$. It is shown in the proof of Lemma <ref>
E\left|\sum_{t=1}^v(\eta_t^2-1)\right|^3\leq c_{6,3}v^{3/2},\;\; E \left|\sum_{t=1}^{v}\eta_t\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}\right|^3\leq c_{6,4}v^{3/2}\;\;
E\left|\sum_{t=1}^{v}\left(\sum_{\ell=1}^N\fe_i(\ell)e_{\ell,t}\right)^2-v\sum_{\ell=1}^N\fe_i^2(\ell)\sigma^2_{\ell}\right|^3\leq c_{6,5}v^{3/2}.
Due to the stationarity of $\eta, e_{i,t}, 1\leq i \leq N$, the tightness follows from Theorem 8.4 of Billingsley (1968).
Proof of Theorem <ref>. By Lemmas <ref>, <ref> and <ref> we have that
\sup_{0\leq u \leq 1}\left|T^{1/2}\left(\tilde{\lambda_i}(u)-\frac{\lf Tu\rf}{T}\lambda_i\right)-T^{1/2}\fe_i^\T\left(\tilde{\C}_{N,T}(u)-\frac{\lf Tu\rf}{T}\C\right)\fe_i\right|=o_P(1).
\begin{align*}
\sup_{0\leq u \leq 1}&\left|T^{1/2}\fe_i^\T\left(\tilde{\C}_{N,T}(u)-\frac{\lf Tu\rf}{T}\C\right)\fe_i-G_{N,T;i}(u)\right|\\
(\fe_i^\T\bgamma)^2\sup_{0\leq u \leq 1}|D_{N,T}(u)|+2|\fe_i^\T\bgamma|\sup_{0\leq u \leq 1}|F_{N,T;i}(u)|\\
\end{align*}
since by Lemma <ref>
\sup_{0\leq u \leq 1}|D_{N,T}(u)|=O_P(1)\;\;\;\mbox{and}\;\;\;\sup_{0\leq u \leq 1}|F_{N,T;i}(u)|=O_P(1).
By the Cauchy–Schwarz inequality we have that $|\fe_i^\T\bgamma|\leq \|\bgamma\|$ and therefore
\sup_{0\leq u \leq 1}\left|T^{1/2}\fe_i^\T\left(\tilde{\C}_{N,T}(u)-\frac{\lf Tu\rf}{T}\C\right)\fe_i-G_{N,T;i}(u)\right|=o_P(1).
The weak convergence of $G_{N,T;i}(u), 0\leq u \leq 1, 1\leq i \leq K$ is proven in Lemma <ref>, which completes the proof of Theorem <ref>.
Proof of Theorem <ref>.
Lemmas <ref> and <ref> yield
\begin{align*}
\sup_{0\leq u \leq 1}\left|T^{1/2}\|\bgamma\|^{-2}\Bigl(\tilde{\lambda}_1(u)-
\frac{\lf Tu\rf}{T}\lambda_1\Bigl)-T^{1/2}\|\bgamma\|^{-2}\fe_1^\T\Bigl(\tilde{\C}_{N,T}(u)-\frac{\lf Tu\rf}{T}\C\Bigl)\fe_1\right|
\end{align*}
Thus Lemma <ref> yields
\begin{align*}
\sup_{0\leq u \leq 1}\left|T^{1/2}\|\bgamma\|^{-2}\left(\tilde{\lambda}_1(u)-\frac{\lf Tu\rf}{T}\lambda_1\right)-\frac{(\fe_1^\T\bgamma)^2}{\|\bgamma\|^2}D_{N,T}(u)
\right|=o_P(1).
\end{align*}
According to Lemma <ref> $\sup_{0\leq u \leq 1}|D_{N,T}(u)|=O_P(1)$ and since $(\fe_1^\T\bgamma)^2/\|\bgamma\|^2\to 1$ by Lemma <ref>, we conclude
\begin{align}\label{fin-1}
\sup_{0\leq u \leq 1}\left|T^{1/2}\|\bgamma\|^{-2}\left(\tilde{\lambda}_1(u)-\frac{\lf Tu\rf}{T}\lambda_1\right)-D_{N,T}(u)
\right|=o_P(1).
\end{align}
Lemmas <ref> and <ref> imply
\begin{align}\label{fin-2}
\sup_{0\leq u \leq 1}&\left|T^{1/2}(\tilde{\lambda}_i(u)-u\lambda_i)-((\fe_i^\T\bgamma)^2D_{N,T}(u)+2\fe_i^\T\bgamma F_{N,T;i}(u)+G_{N,T;i}(u))
\right|\\
\end{align}
Combining (<ref>) and (<ref>) with Lemma <ref>, we obtain that $\{T^{1/2}|\|\bgamma\|^{-2}(\tilde{\lambda}_1(u)-u\lambda_1), T^{1/2}(\tilde{\lambda}_i(u)-u\lambda_i), 2\leq i \leq K\}$ converges weakly in ${\mathcal D}^K[0,1]$ to $\bGamma^0(u)=(\Gamma^0_1(u), \Gamma^0_2(u),\\
\ldots ,\Gamma^0_K(u))^\T$, where $\Gamma_1^0(u)=\Gamma_1(u)$ and
$\Gamma_i^0(u)=a_i^2\Gamma_1(u)+2a_i\Gamma_{i+1}(u)+\Gamma_{i+K+1}(u), 2\leq i \leq K$. The computation of the covariance function of $\bGamma^0(u)$ finishes the proof of Theorem <ref>.
Proof of Theorem <ref>. Theorem <ref> is implied by Theorems <ref> and <ref> by Remark <ref>.
Proof of Theorem <ref> and Remark <ref>. Theorem <ref> follows from Remark <ref>. Remark <ref> follows from Theorems <ref> and <ref> when the condition (<ref>) is replaced with (<ref>). This requires replacing Lemma <ref> with the result that for all $c>0$
\max_{1\leq i \leq K}\sup_{c\leq u\leq 1}|Z_{N,T;i}(u)|=O_P\left(\frac{N}{T}\right),
which follows from (<ref>) and Markov's inequality.
§.§ Proof of Theorems <ref> and <ref>
We prove a more general result concerning consistent estimates for norming sequences for each eigenvalue process. Let
\hat{\xi}_{i,t}=(\hat{\fe}_i^\T(\X_t-\bar{\X}_T))^2,\;\;1\leq t\leq T\;\;\;1\leq i \leq K,
and define
\begin{align*}%\label{var-est-def-all}
\hat{v}^2_{i,T}=\sum_{s=-N+1}^{N-1}J\left(\frac{s}{h}\right)\hat{r}_{i,s},
\end{align*}
1/T-s∑_t=1^T-s (ξ̂_i,t-ξ̅_i,T)(ξ̂_i,t+s-ξ̅_i,T), s≥0
1/T-|s|∑_t=-s^T (ξ̂_i,t-ξ̅_i,T)(ξ̂_i,t+s-ξ̅_i,T), s< 0,
\bar{\xi}_{i,T}=\frac{1}{T}\sum_{t=1}^T\xi_{i,t}.
We show that if $\|\bgamma\|=O(1)$ as $T\to \infty$, then
v̂^2_i,T/G(i,i) P→ 1, T→∞.
Moreover, if $\|\bgamma\|\to \infty$ as $T \to \infty$, then
v̂^2_1,T/V_1^4 P→ 1, T→∞,
and for $2\leq i \leq K$,
v̂^2_i,T/H(i,i) P→ 1, T→∞.
We can assume without loss of generality that $E\bX_t={\bf 0}$. Elementary algebra gives that
\begin{align*}
(\bX_t-&\bar{\bX}_T)( \bX_t-\bar{\bX}_T)^\T-\frac{1}{T}\sum_{u=1}^T(\bX_u-\bar{\bX}_T)( \bX_u-\bar{\bX}_T)^\T\\
-\bX_t\bar{\bX}_T^\T-\bar{\bX}_T \bX_t^\T.
\end{align*}
It is easy to see that
\begin{align*}
&E\left|\hfe_i^\T\left[\frac{1}{T}\sum_{t=1}^T(\bX_t\bX_t^\T-E\bX_0\bX_0^\T)\right]\hfe_i\left[ \hfe_i^\T\frac{1}{T}\sum_{u=1}^T\left(\bX_u\bX_s^\T-E\bX_0\bX_0^\T\right)\hfe_i\right]\right|\\
&\hspace{.5cm}\leq E\left\|\frac{1}{T}\sum_{t=1}^T(\bX_t\bX_t^\T-E\bX_0\bX_0^\T)\right\|^2\\
\frac{1}{T^2}\sum_{\ell=1}^N\sum_{\ell'=1}^NE\left(\sum_{u=1}^T(X_{\ell,u}X_{\ell',u}-EX_{\ell,u}X_{\ell',u})\right)^2\\
\end{align*}
and therefore by Markov's inequality we have
\begin{align}\label{ps-00}
&\lim_{T\to\infty}\left|\left[\frac{1}{T}\sum_{t=1}^T\hfe_i^\T(\bX_t\bX_t^\T-E\bX_0\bX_0^\T)\hfe_i\right]\left[ \hfe_i^\T\frac{1}{T}\sum_{u=1}^T\left(\bX_u\bX_s^\T-E\bX_0\bX_0^\T\right)\hfe_i\right]\right|\\
\end{align}
Using the same arguments as above, for every $c_{7,1}$ one can find $c_{7,2}$ such that
\begin{align}\label{ps-0}
\lim_{T\to\infty}P\left\{\frac{1}{T}\sum_{t=1}^T\hfe_i^\T(\bX_t\bX_t^\T-E\bX_t\bX_t^\T)\hfe_i\hfe_i^\T\X_{t+s}\bar{\bX}_T\hfe_i^\T\geq c_{7,2}N^2/T\right\}\leq c_{7,1}
\end{align}
for every $c_{7,3}$ there is $c_{7,4}$ such that
\begin{align}\label{ps01}
\lim_{T\to\infty}P\left\{\left|\hfe_i^\T\frac{1}{T}\sum_{u=1}^T(\bX_u\bX_u^\T-E\bX_u\bX_u^\T)\hfe_i\frac{1}{T}\sum_{t=1}^T\hfe_i^\T\bX_{t+s}\bar{\X}_T\hfe_i
\right|\geq c_{7,4}N^2/T\right\}\leq c_{7,3}.
\end{align}
We note
\begin{align*}
\left|\frac{1}{T}\sum_{t=1}^{T-s}\hfe_i^\T\bX_t\bar{\bX}_T\hfe_i\hfe_i^\T\bX_{t+s}\bar{\bX}_T\hfe_i\right|\leq
\|\bar{\bX}_T\|^2\frac{1}{T}\sum_{t=1}^{T}\|\bX_t\|\|\bX_{t+s}\|.
\end{align*}
By (<ref>) and assumption $\mu_i=0$ we get that from Assumption <ref>(a)–Assumption <ref>(b) and Assumption <ref>
\begin{align*}
&=\frac{1}{T^2}\sum_{u,v=1}^T\sum_{\ell=1}^N(\gamma_\ell^2E\eta_u\eta_v+Ee_{\ell, u}e_{\ell,v})\\
&=O\left(\frac{N}{T} \right)
\end{align*}
using the arguments in the proof of Lemma <ref>. Due to stationarity we have
E\|\bX_t\|\|\bX_{t+s}\|\leq (E\|\bX_t\|^2E\|\bX_{t+s}\|^2)^{1/2}=E\|\bX_0\|^2
\begin{align*}
E\|\bX_0\|^2=\sum_{\ell=1}^N (\gamma_\ell^2 +Ee_{\ell,0}^2)=O(N)
\end{align*}
Hence for every $c_{7,5}$ there is $c_{7,6}$ such that
\begin{align}\label{ps-1}
\lim_{T\to\infty}P\left\{\left|\frac{1}{T}\sum_{t=1}^{T-s}\hfe_i^\T\bX_t\bar{\bX}_T\hfe_i\hfe_i^\T\bX_{t+s}\bar{\bX}_T\hfe_i\right|\geq c_{7,6}N^2/T\right\}\leq c_{7,5}.
\end{align}
Putting together (<ref>)–(<ref>) we conclude
\begin{align*}
\hat{v}_{i,T}^2=\tilde{v}_{i,T}^2+O_P\left( \frac{hN^2}{T} \right),
\end{align*}
\tilde{v}^2_{i,T}=\sum_{s=-N+1}^{N-1}J\left(\frac{s}{h}\right)\tilde{r}_{i,s},
1/T-s∑_t=1^T-s ξ̃_i,tξ̃_i,t+s, s≥0
1/T-|s|∑_t=-s^T ξ̃_i,tξ̃_i,t+s, s< 0
with $\tilde{\xi}_{i,t}=\hfe_i^\T(\bX_t\bX_t^\T-E(\bX_0\bX_0^\T))\hfe_i$.
It follows from Dunford and Schwartz (1988) and Assumption <ref> that with some constant $c_{7,6}$
\begin{align}\label{dusch}
\max_{1\leq i \leq K}\|\hfe_i-\hat{c}_i\fe_i\|\leq c_{7,6}\|\hat{\C}_{N,T}(1)-\C\|,
\end{align}
where $\hat{c}_i, 1\leq i \leq K$ are random signs. We write
\|\hat{\C}_{N,T}(1)-\C\|\leq \left\| \frac{1}{T}\sum_{t=1}^T(\X_t\X_t^\T-\C)\right\|+\left\| \bar{\X}_T\bar{\X}^\T \right\|,
and since we can assume without loss of generality that $E\X_t={\bf 0}$ we get from the proof of Lemma <ref>
\left\| \bar{\X}_T\bar{\X}_T^\T \right\|=O_P\left(\frac{N}{T}\right).
\begin{align*}
E\left\| \sum_{t=1}^T(\X_t\X_t^\T-\C)\right\|^2&=E\sum_{\ell,\ell'=1}^N\left(\sum_{t=1}^T(X_{\ell,t}X_{\ell',t}-EX_{\ell,t}X_{\ell',t})\right)^2\\
\end{align*}
γ_ℓγ_ℓ', ℓ≠ℓ'
γ_ℓ^2+Ee^2_ℓ,0, ℓ=ℓ'.
+Ee_ℓ,te_ℓ,t' Ee_ℓ',te_ℓ',t', ℓ≠ℓ'
γ_ℓ^4Eη_t^2η_t'^2+2γ_ℓ^2Eη_0^2Ee^2_ℓ,0+4γ_ℓ^2Eη_tη_t'Ee_ℓ, te_ℓ,t'+Ee^2_ℓ,te^2_ℓ,t',
Thus we have
\begin{align*}
\sum_{\ell=1}^N&\sum_{t,t'=1}^T(EX^2_{\ell,t}X^2_{\ell,t'}-(EX^2_{\ell,0})^2)\\
+ 4\sum_{\ell=1}^N\gamma_\ell^2\sum_{t,t'=1}^TE\eta_t\eta_{t'}
\end{align*}
\begin{align*}
&\sum_{\ell,\ell'=1, \ell\neq \ell'}^N\sum_{t,t'=1}^T(EX_{\ell,t}X_{\ell,t'}X_{\ell',t}X_{\ell',t'}-EX_{\ell,t}X_{\ell,t'}EX_{\ell',t}X_{\ell',t'})\\
&\hspace{.5cm}=\sum_{\ell,\ell'=1, \ell\neq \ell'}^N\gamma_\ell^2\gamma^2_{\ell'}\sum_{t,t'=1}^T(E\eta_t^2\eta^2_{t'}-1)
+2\sum_{\ell,\ell'=1, \ell\neq \ell'}^N\gamma_\ell^2\sum_{t,t'=1}^TE\eta_t\eta_{t'}Ee_{\ell',t}e_{\ell',t'}\\
+\sum_{\ell,\ell'=1, \ell\neq \ell'}^N\sum_{t,t'=1}^TEe_{\ell,t}e_{\ell,t'} Ee_{\ell',t}e_{\ell',t'}\\
\end{align*}
We conclude from (<ref>) that
\begin{align}\label{dusch-f}
\max_{1\leq i \leq K}\|\hfe_i-\hat{c}_i\fe_i\|=O_P\left({N}{T^{-1/2}}\right).
\end{align}
Next we define
1/T-s∑_t=1^T-s ξ_i,tξ_i,t+s, s≥0
1/T-|s|∑_t=-s^T ξ_i,tξ_i,t+s, s< 0,
where ${\xi}_{i,t}=\fe_i^\T(\bX_t\bX_t^\T-E(\bX_0\bX_0^\T))\fe_i$.
We write
\begin{align*}
\tilde{v}^2_{i,T} - v^2_{i,T} = \sum_{j=1-N}^{N-1} J\left(\frac{j}{h}\right)(\tilde{r}_{j,s}-r_{j,s}).
\end{align*}
For $j \ge 0$,
\begin{align*}
\tilde{r}_{i,s}-r_{i,j}&= \frac{1}{T-j}\sum_{t=1}^{T-j} \tilde{\xi}_{i,t}\tilde{\xi}_{i,t+j} - \xi_{i,t}\xi_{i,t+j} \\
&= \frac{1}{T-j}\sum_{t=1}^{T-j} (\tilde{\xi}_{i,t} - \xi_{i,t})\tilde{\xi}_{i,t+j} + \frac{1}{T-j}\sum_{t=1}^{T-j} (\tilde{\xi}_{i,t+j} - \xi_{i,t+j})\xi_{i,t}.
\end{align*}
According to the definitions of $\tilde{\xi}_{i,t}$ and $\xi_{i,t}$,
\begin{align*}
\tilde{\xi}_{i,t} - \xi_{i,t} = (\hfe_i^\T - \fe_i^\T)U_t\hfe_i + \fe_i^\T U_t(\hfe_i - \fe_i),
\end{align*}
where $U_t=\bX_t \bX_t^\T - E\bX_0 \bX_0^\T$, from which it follows that,
\begin{align}
(\tilde{\xi}_{i,t} - \xi_{i,t})\tilde{\xi}_{i,t+j} = (\hfe_i^\T - \fe_i^\T)U_t\hfe_i \hfe_i^\T U_{t+j} \hfe_i + \fe_i^\T U_t(\hfe_i - \fe_i) \hfe_i^\T U_{t+j} \hfe_i.
\end{align}
According to the Cauchy-Schwarz and triangle inequalities,
\begin{align*}
\left|\sum_{j=0}^{N-1} J\left(\frac{j}{h}\right)\frac{1}{T-j}\sum_{t=1}^{T-j} (\hfe_i^\T - \fe_i^\T)U_t\hfe_i \hfe_i^\T U_{t+j} \hfe_i \right| \le \|\hfe_i - \fe_i\| \left|\sum_{j=0}^{N-1} J\left(\frac{j}{h}\right)\frac{1}{T-j}\sum_{t=1}^{T-j} \|U_t\| \|U_{t+j} \| \right|.
\end{align*}
According to <ref> $\|\hfe_i - \fe_i\|=O_P(NT^{-1/2})$. Furthermore, since $E\|U_0\|^2=O(N^2)$, and $J$ has bounded support, the Cauchy-Schwarz and triangle inequalities imply that
\begin{align}\label{lcalc-1}
E\left|\sum_{j=0}^{N-1} J\left(\frac{j}{h}\right)\frac{1}{T-j}\sum_{t=1}^{T-j} \|U_t\| \|U_{t+j} \| \right| \le c_1 h E\|U_0\|^2= O(hN^2),
\end{align}
For some constant $c_1$. Hence, according to (<ref>) and Markov's inequality, we obtain that
\begin{align}
\left|\sum_{j=0}^{N-1} J\left(\frac{j}{h}\right)\frac{1}{T-j}\sum_{t=1}^{T-j} (\hfe_i^\T - \fe_i^\T)U_t\hfe_i \hfe_i^\T U_{t+j} \hfe_i \right| = O_P(hN^3T^{-1/2}).
\end{align}
Similar arguments applied to the remaining terms in $\tilde{v}^2_{i,T} - v^2_{i,T}$ show that
\begin{align}
|\tilde{v}^2_{i,T} - v^2_{i,T}|=O_P( hN^3T^{-1/2}).
\end{align}
It follows from (<ref>) and Assumption <ref>(b) that
\begin{align*}
\lim_{T\to\infty}\frac{1}{G(i,i)}\sum_{s=-\infty}^\infty E{\xi}_{i,t}{\xi}_{i,t+s}=1,
\end{align*}
\begin{align*}
\lim_{T\to\infty}\frac{1}{G(i,i)}\sum_{s=-\infty}^\infty K\left( \frac{s}{h} \right)E {\xi}_{i,t}{\xi}_{i,t+s}=1.
\end{align*}
Ev^2_i,T=∑_s=-∞^∞K( s/h )E ξ_i,tξ_i,t+s,
if we show that
we get immediately that
\frac{{v}^2_{i,T}}{G(i,i)}\;\;\;\stackrel{P}{\to}\;\;\;1,\quad{as }\;\;\;T\to \infty.
To this end, we have that
\begin{align*}
\mbox{var}({v}^2_{i,T})
=\sum_{s,s'=-\infty}^\infty J\left(\frac{s}{h}\right)J\left(\frac{s'}{h}\right)({r}_{i,s}-E\xi_{i,0}\xi_{i,s})({r}_{i,s'}-E\xi_{i,0}\xi_{i,s'})
\end{align*}
\begin{align*}
&\left|\sum_{s,s'=0}^\infty J\left(\frac{s}{h}\right)J\left(\frac{s'}{h}\right)({r}_{i,s}-E\xi_{i,0}\xi_{i,s})({r}_{i,s'}-E\xi_{i,0}\xi_{i,s'})\right|\\
&\hspace{.5cm}\leq \sum_{s,s'=0}^\infty \Biggl|J\left(\frac{s}{h}\right)J\left(\frac{s'}{h}\right)\Biggl|\frac{1}{T-s}\frac{1}{T-s'}\sum_{t=1}^{T-s}\sum_{t'=1}^{T-s'}\biggl|
&\hspace{.5cm}\leq c_{7,7}\frac{1}{T^2} \sum_{s,s'=0}^h \sum_{t=1}^{T-s}\sum_{t'=1}^{T-s'}\biggl|
\end{align*}
with some constant $c_{7,7}$, since we can assume without loss of generality that $J(u)=0$ if $|u|\geq 1$.
Now we assume that the conditions of Theorem <ref> are satisfied. First we prove (<ref>). It follows from (<ref>) and (<ref>) that
\lim_{T\to \infty}\frac{1}{\|\bgamma\|^4}Er^2_{1,T}=V_1.
Following the proof of one can verify that
\mbox{var}\left( \frac{1}{\|\bgamma\|^4}r^2_{1,T} \right)=0,
completing the proof of (<ref>). The proof of (<ref>) goes along the lines of that of (<ref>) and therefore the details are omitted.
Proof of Theorem <ref>. We can assume without loss of generality that $\mu_i=0, 1\leq i \leq N$. Using (<ref>) we have
s(T-t^*/T)^2^+∑_u=1^s_u,T, 0≤s≤t^*
t^*≤s ≤T,
\begin{align*}
\U_{u,T}=&(\bgamma\eta_u+\be_u)(\bgamma\eta_u+\be_u)^\T-\frac{T-t^*}{T}(\bgamma\eta_u+\be_u)\balpha^\T-(\bgamma\eta_u+\be_u)\bZ_T^\T
&+\frac{T-t^*}{T}\balpha\bZ_T^\T-\bZ_T(\bgamma\eta_u+\be_u)^\T+\frac{T-t^*}{T}\bZ_T\balpha^\T+\bZ_T\bZ_T^\T,\;\;\mbox{if}\;\;1\leq u \leq t^*,
\end{align*}
\bZ_T=\bgamma\frac{1}{T}\sum_{v=1}^T\eta_v+\frac{1}{T}\sum_{v=1}^T\be_v, \quad\be_v=(e_{1,v}, e_{2,v},\ldots ,e_{N,v})^\T
\begin{align*}
\U_{u,T}=&(\bgamma\eta_u+\be_u)(\bgamma\eta_u+\be_u)^\T+\frac{t^*}{T}(\bgamma\eta_u+\be_u)\balpha^\T-(\bgamma\eta_u+\be_u)\bZ_T^\T
&-\frac{t^*}{T}\balpha\bZ_T^\T-\bZ_T(\bgamma\eta_u+\be_u)^\T-\frac{t^*}{T}\bZ_T\balpha^\T+\bZ_T\bZ_T^\T,\;\;\mbox{if}\;\;t^*\leq u \leq T.
\end{align*}
It follows along the same lines as the proof of (<ref>) that
\sup_{0\leq s\leq T}\left|\fe ^\T\sum_{u=1}^s\U_{u,T}\fe\right|=O_P(T^{1/2})
for $\fe\in R^N$ with $\|\fe\|=1$. Hence
\frac{\hat{\lambda}_{1,T}(t^*/T)}{\|\balpha\|}\;\;\stackrel{P}{\to}\;\;\theta(1-\theta)^2
\frac{\hat{\lambda}_{1,T}(1)}{\|\balpha\|}\;\;\stackrel{P}{\to}\;\;\theta(1-\theta),
which completes the proof of Theorem <ref>.
The proof of Theorem <ref> is based on the following lemma.
We assume that model (<ref>) holds, Assumptions <ref>, <ref>, <ref>, (<ref>), (<ref>) are satisfied.
If for some $0<\epsilon<1$ there exists an $N_0$ such that for all $N\geq N_0$
|sup{^[ θ^+(1-θ)(+)(+)^+]: ∈R^N, =1}/sup{
^[^+]: ∈R^N, =1} -1
Since the means of the panels do not change during the observation period in (<ref>), we can assume without loss of generality that $\mu_i=0, 1\leq i \leq N$. It follows from Theorems <ref> and <ref> that for any $u^*\in (0,\theta ]$ that
\left|{\hat{\lambda}_1 (u^*)}-{\lambda_1}\right|=O_P\left(NT^{-1/2}\right).
One can show that Lemmas <ref> and <ref> hold with minor modifications under model (<ref>), and thus
\left|{\hat{\lambda}_1 (1)}-{\bar{\lambda}_1(1)}\right|=O_P\left(NT^{-1}\right),
where $\bar{\lambda}_1(1)$ is the largest eigenvalue of $\sum_{t=1}^T\X_t\X_t^\T/T$. Simple arithmetic shows that
\frac{1}{T}\sum_{t=1}^T\X_t\X_t^\T=\C_T^{(1)}+\bG_{1,T}+\bG_{2,T},
\C_T^{(1)}=\bgamma\bgamma^\T\frac{1}{T}\sum_{t=1}^T\eta_t^2+\frac{1}{T}\sum_{t=1}^T\be_t\be_t^\T+(\bdelta\bdelta^\T +\bgamma\bdelta^\T +\bdelta\bgamma^\T)\frac{1}{T}\sum_{t=t^*}^T\eta_t^2,
\bG_{1,T}=\frac{1}{T}\sum_{t=1}^T(\bgamma\be_t^\T+\be_t\bgamma^\T)\eta_t
\bG_{2,T}=\frac{1}{T}\sum_{t=t^*}^T(\be_t\bdelta^\T+\bdelta\be_t^\T)\eta_t.
It follows along the lines of the proof of (<ref>) that
\|\bG_{i,T}\|=O_P(NT^{-1/2}),\quad i=1,2,
and thus if $\lambda_T^{(1)}$ denotes the largest eigenvalue of $\C_T^{(1)}$, then we also have that
\left|{\bar{\lambda}_1(1)}-{\lambda_T^{(1)}} \right|=O_P(NT^{-1/2}).
Let $\phi_T$ be the largest eigenvalue of $(\bdelta\bdelta^\T+\bdelta\bgamma^\T+\bgamma\bdelta^\T)(1-\theta)+\bgamma\bgamma^\T+\bLambda$. Then one can show using the arguments establishing Theorems <ref> and <ref> that
\left| {\lambda_T^{(1)}}-{\phi_T} \right|=O_P(NT^{-1/2}).
Assumption (<ref>) implies that there is an $\epsilon >0$ for all $T$ sufficiently large
\left|\frac{\lambda_1}{\phi_T}-1 \right|>\epsilon,
and therefore there is a constant $c_{8,1}$ such that
\left|{\lambda_1}-{\phi_T} \right|>c_{8,1}.
Observing that $\hat{v}_{1,T}=O_P(h^{1/2})$ and $\sup_{0\leq u \leq 1}|\hat{B}_{T,1}(u)|\geq |\hat{B}_{T,1}(u^*)|$, the proof of (<ref>) is complete.
Proof of Theorem <ref>. It is clear that assumption (<ref>) implies (<ref>), and therefore Theorem <ref> follows from Lemma <ref>.
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\begin{align*}
&\varepsilon_t=g(\vare_t, \vare_{t-1},\ldots)\;\;\mbox{with some measurable function}\;g,\;\mbox{where}\;\vare_s, -\infty<s<\infty\;\;\\
&\mbox{are independent and identically distributed random variables and}\notag\\
&(E(\eta_t-\eta_t^{(m)})^{p})^{1/p}=\chi(m)\;\;\mbox{with}\;\; \notag\\
& \eta_t^{(m)}=g(\vare_t,\vare_{t-1}, \vare_{t-m}, \vare_{t-m-1,t,m},\vare_{t-m-2,t,m},\ldots),\;\mbox{and}\; \vare_{i,j,\ell}\;\mbox{are independent} \notag\\
&\mbox{and identically distributed copies of}\;\vare_0,\notag
\end{align*}
\begin{align}\label{eta-mod-1}
E\eta_t=0, E\eta_t^2=1,\;\mbox{and}\;\;E\eta_t^{12}<\infty
\end{align}
\begin{align}\label{eta-mod-2}
&\eta_t=g(\vare_t, \vare_{t-1},\ldots)\;\;\mbox{with some measurable function}\;g,\;\mbox{where}\;\vare_s, -\infty<s<\infty\;\;\\
&\mbox{are independent and identically distributed random variables and}\notag\\
&(E(\eta_t-\eta_t^{(m)})^{12})^{1/12}=O(m^{-{\alpha}})\;\;\mbox{with some}\;\; {\alpha}>1,\;\; \notag\\
& \eta_t^{(m)}=g(\vare_t,\vare_{t-1}, \vare_{t-m}, \vare_{t-m-1,t,m},\vare_{t-m-2,t,m},\ldots),\;\mbox{where}\; \vare_{i,j,\ell}\;\mbox{are independent} \notag\\
&\mbox{and identically distributed copies of}\;\vare_0,\notag
\end{align}
The assumption $E\eta_t^2=1$ is not a restriction, it is needed the identify the loading.
Similar conditions are assumed on the error terms $e_{i,t}$:
\begin{align}\label{e-1}
&Ee_{i,t}=0, c_1\leq Ee^2_{i,_t}=\sigma_i^2\leq c_2\;\mbox{with some $0<c_1\leq c_2$ and}\;\;Ee_{i,t}^{12}\leq c_3\;\;\\
&\mbox{for all}\;\;1\leq i \leq N\notag
\end{align}
\begin{align}\label{e-2}
&e_{i,t}=g_i(\vare_t(i), \vare_{t-1}(i),\ldots)\;\;\mbox{with some measurable function}\;g_i,\;\mbox{where}\;\vare_s, \vare_s(i), \\
&-\infty<s<\infty,1\leq i \leq N\;\mbox{are independent}\;\mbox{and}\; \vare_s(i), -\infty<s<\infty \;\mbox{are }\notag\\
&\mbox{identically distributed random variables,}\;(E(e_{i,t}-e_{i,t}^{(m)})^{12})^{1/12}\leq cm^{-{\alpha}}\;\mbox{with}\;\notag\\
&\mbox{some}\; {\alpha}>1, \mbox{and}\; e_{i,t}^{(m)}=g_i(\vare_t(i),\vare_{t-1}, \vare_{t-m}(i), \vare_{t-m-1,t,m}(i),\vare_{t-m-2,t,m}(i),\ldots),\;\notag\\
&\mbox{where}\; \vare_{i,j,\ell}(i)\;\mbox{are independent}\;
\mbox{and} \; \vare_{i,j,\ell}(i) \;\mbox{are identically distributed copies of}\; \notag\\
&\vare_0(i)\;\mbox{for each} \;1\leq i \leq N. \notag
\end{align}
|
1511.00550
|
The object of the present is a proof of the existence of functorial resolution of tame quotient singularities for quasi-projective varieties over algebraically closed fields.
FEDERICO BUONERBAFUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES
The role of quotient singularities, in the general problem of the resolution of singularities in positive characteristic, has been highlighted in the fundamental work of de Jong <cit.>, where the general problem has been reduced to the more specific one of understanding singularities created by inseparable morphisms and group actions.
[<cit.>, 7.4]
Let $X$ be a projective variety over an algebraically closed field. There exists a radicial morphism $Y\to X$ and a modification $Z\to Y$ such that $Z$ has at worst quotient singularities.
A huge class of group actions enjoy the property of being linearizable, i.e. the action is formally equivalent to that of a group of linear endomorphisms of vector space. In this situation the singularities created by the action are easier to handle, since the existence of formal coordinates along which the action is linear provides a rich amount of information. By way of examples, all tame, <ref>, abelian group actions create singularities that live, étale-locally, in the world of toric varieties, see <ref>, and their resolution is in fact a key step for those positive characteristic oriented constructions that lead to a resolution of singularities in characteristic zero, such as those appearing in <cit.>,<cit.>. As well known, étale-locally the resolution problem for tame abelian quotient singularities is completely understood, i.e. there are explicit algorithms, such as <cit.>, theorems 11 and 11*, and <cit.>, III.iii.4.bis, that provide us with a resolution. The issue is therefore the local-to-global step (and this is by no means a new story, <cit.> 3.14.9.), where the toric machinery is not sufficiently strong without additional global assumptions, for example the existence of a toroidal embedding. Consequently it seems convenient to look for a resolution procedure that is functorial under étale localization, following the philosophy of <cit.>, <cit.>.
Our main result is:
Let $k$ be an algebraically closed field, and $X/k$ a quasi-projective variety with tame quotient singularities, with an étale cover $\bigsqcup_j X_j\to X$ such that $\max_{x\in X_j}|G_x|$ is finite for every $j$. Then there exists a resolution functor $X\to (M(X),r_X)$ where $M(X)$ is a smooth, quasi projective variety, and $r_X:M(X)\to X$ is a proper, birational, relatively projective morphism, which is an isomorphism over the smooth locus of $X$. The resolution functor commutes with étale base change, that is to say for every étale morphism $f:Y\to X$ there is a unique isomorphism $\phi_f:f^*M(X)\to M(Y)$.
Now we can discuss the ideas employed. Following the general pattern, <cit.>, that the construction of a resolution functor is algorithmic in its own nature, we define an invariant $X\to i(X)\in \mathbf N$, such that $i(X)=0$ if and only if $X$ is smooth. Then we proceed by induction on $i$, by performing birational operations - smooth and weighted blow-ups along smooth centers - that eventually decrease $i$. The proof consists of two steps: the first step <ref> constructs a birational modification whose resulting space has only tame cyclic quotient singularities, and the invariant does not increase. Here the full power of the philosophy “functoriality in the étale topology" appears, and most of the logical difficulties, apart from some routine technical issues, are hidden in the inductive step of the process. This inductive step resolves the singularities of a Deligne-Mumford stack with smaller invariant, and the variety we are looking for is its GC quotient, whose existence as an algebraic space is granted by <cit.>. At this point the inductive hypotheses show up again and force the GC quotient to be a quasi-projective variety, if we started with such.
The second step <ref> is an algorithm that reduces the invariant if we start with tame cyclic quotient singularities and an extra structure of “global character for the local geometric stabilizers", more precisely an equivariant divisor, on the Vistoli covering stack, along which the stabilizers act with a faithful character.
The logic how the two steps fit together and provide a resolution is as follows: starting with a variety with tame quotient singularities, the first step produces a variety with tame cyclic quotient singularities and a marked irreducible divisor, <ref>. Properties of this pair are such that we can employ it as input for our second step, whose output is a variety with tame cyclic quotient singularities whose invariant is strictly smaller than the one we started with.
Functoriality in the étale topology implies, as explained for example in <cit.>, that the resolution procedure applies to algebraic spaces, Deligne Mumford-stacks and analytic spaces without any compactness assumption, <ref>, <ref>, <ref>.
It is worth remarking that, in the course of the proof, we will encounter singularities created by diagonal actions of group schemes of roots of unity, which are not necessarily tame. This must happen if one wants to tackle the problem using weighted modifications/toric geometry in positive characteristic. It turns out that, in this specific circumstance, Mumford's resolution process for toroidal singularities can be carried over by weakening a bit the assumption on the existence of a toroidal embedding, requiring instead the existence of a divisor along which diagonal cyclic stabilizers act with faithful character, see <ref> for the precise condition.
Unfortunately, the methods developed here seem to completely lose their efficiency if one drops the tameness assumption, in fact the crucial fact we use profusely is that there exist regular parameters along which tame abelian actions are diagonalized, while of course this never happens with non-trivial $p$-group actions in characteristic $p$. We suspect that a resolution functor that deals with all possible quotient singularities simultaneously must proceed via a completely different strategy.
Acknowledgments: This problem has been suggested by Fedor Bogomolov. His encouragement, and the enlightening conversations we had, have been of great help. This work could never be completed without the support of Michael McQuillan, whose restless and careful explanations permeate and shape the core ideas building our proof. Grateful thanks are extended to Gabriele Di Cerbo, for reading a first draft of this paper and for his editing, that drastically improved our presentation.
§ GENERALITIES
§.§ Tame group actions
Let $k$ be an algebraically closed field and $G$ be a finite group acting by $k$-automorphism on a noetherian, regular, local $k$-algebra $\mathcal O$. The action is said tame if $(|G|,\cha(k))=1$. If $G$ acts on a smooth algebraic variety by $k$-automorphisms, we say that the action is tame if it is generically free, and the stabilizer at any geometric point acts tamely on the corresponding Zariski local ring. From the point of view of singularities we could request more, i.e. the action to be free in codimension one, by way of the celebrated Chevalley-Shephard-Todd theorem:
If $G$ is a finite group acting tamely on a smooth algebraic variety, then for any geometric point $x$ with stabilizer $G_x$, the ring of invariants $\mathcal O_x^{G_x}\subset \mathcal O_x$ is a regular local ring if and only if $G_x$ is generated by pseudoreflections, i.e every element $g\in G_x$ fixes pointwise a smooth divisor depending on $g$.
Tame actions enjoy some remarkable properties, with respect to the local algebra they are the simplest possible. We can recollect those properties that we need in the following simple propositions:
Let $G$ be a cyclic group, generated by $g\in G$, acting tamely on the regular local $k$-algebra $\mathcal O$ with maximal ideal $m$. Then there are elements $\zeta_1,...,\zeta_n\in k^*$ and a regular system of parameters $x_1,...,x_n$ generating $m$ such that $x_i^g=\zeta_ix_i$.
As $g$ acts on the $k$-vector space $m/m^2$, by assumption we can certainly find parameters and eigenvalues satisfying the identities $x_i^g=\zeta_ix_i$ mod $m^2$. We can replace each $x_i$ by $y_i=|G|^{-1}\sum_{k=0}^{k=|G|-1}x_i^{g^k}\zeta_i^{-k}$, and observe that $y_i=x_i$ mod $m^2$, and $y_i$ satisfies the required relation.
If $G$ acts tamely on the smooth algebraic variety $X$ then the subvariety $X_G$ of points fixed by $G$ is smooth.
Let $\mathcal O$ be the local ring at a geometric point $x$ fixed by $G$, then (essentially) as in the previous proof we can average any isomorphism $\hat{\mathcal O}\to k[[x_1,...,x_n]]$ in order to make it equivariant with respect to the induced action of $G$ on the power series ring. Here the action is by linear automorphism, whence the ideal defining $X_G$ is easily seen to be generated by the linear forms $(x_i^g-x_i)_{g\in G}$, and consequently the subvariety $X_G$ is smooth at $x$.
Let $\mathcal O$ be a regular local strictly henselian $k$-algebra with maximal ideal $m$, $G$ a finite group acting tamely on it. If $I$ is a principal ideal which is fixed by $G$ then there exists a character $\chi :G\to k^*$ and a generator $f$ of $I$, such that $f^g=\chi (g) f$ for any $g\in G$.
Let $f$ be a generator of $I$. By assumption, for every $g$ there is $u_g\in \mathcal O^*$ such that $f^g=u_gf$. It follows that $f^{gh}=u_g^hu_hf$, whence the set map $G\to \mathcal O^*$ given by $g\to u_g$ is a 1-cocycle representing a class in $H^1(G,\mathcal O^*)$. Consider the exact sequence $0\to (1+m)\to \mathcal O^*\to k^*\to 0$ of multiplicative groups. By assumption on $\mathcal O$ being strictly henselian, since $|G|$ is coprime to $\cha(k)$ we see that $1+m$ is $|G|$-divisible, whence $H^1(G,1+m)=H^2(G,1+m)=0$. Consequently we have a natural isomorphism $H^1(G,\mathcal O^*)\to H^1(G,k^*)$. Moreover since the $G$-action on the residue field $k$ is trivial, we get $H^1(G,k^*)=\Hom (G,k^*)$, whence, upon replacing $f$ by $uf$, for some invertible $u$, the map $g\to u_g$ is a character of the group $G$, and of course $uf$ is the required generator.
Let $q:\mathcal O\to \mathcal O'$ be an injective morphism of rings, equivariant by the action of a cyclic group $C$. Assume $C$ acts with faithful character on $f\in \mathcal O$, then $C$ acts with faithful character on $q(f)$.
Let $s$ be the minimal integer such that $\chi^sq(f)=q(f)$, then $q(\chi^sf-f)=0$, and since $q$ is injective we deduce $s=|C|$, that is the action is faithful.
§.§ Stacks with quotient singularities
Let $k$ be an algebraically closed field. An algebraic variety $X/k$ has analytic quotient singularities if it is normal and for any geometric point $x$, there exist a finite group $G_x$ acting on a smooth complete local $k$-algebra $k[[x_1,...,x_n]]$, an inclusion $i_x:\hat{\mathcal O}_{x}\hookrightarrow k[[x_1,...,x_n]]$ such that $i_x(\hat{\mathcal O}_{x})=k[[x_1,...,x_n]]^{G_x}$. By Artin approximation this is equivalent to the fact that $X$ admits an étale cover $\bigsqcup V_i\to X$, where $V_i=U_i/G_i$ for smooth varieties $U_i$ and finite groups $G_i$, whose action on $U_i$ is generically free. $X$ has tame quotient singularities if each $G_i$ acts on $U_i$ tamely. Similarly $X$ has algebraic quotient singularities if it admits such an open cover in the Zariski topology.
For a closed point $x\in X$, the group $G_x$ is called the geometric stabilizer at $x$.
A Deligne-Mumford stack $\mathcal X$ has tame quotient singularities if for some étale atlas (whence for all) $U\to \mathcal X$ the algebraic variety $U$ has tame quotient singularities.
Before getting into the properties of DM stack with tame quotient singularities, we recall an extremely useful local description of DM stacks:
[<cit.> 2.8]
Let $\mathcal X$ be a DM stack over any field, then it admits an étale cover by classifying stacks, i.e. stacks of the form $[Y/G]$ with $G$ a finite group.
By <cit.> 1.1, $\mathcal X$ admits a GC quotient $\mathcal X\to X$. Let $x$ be a geometric point in $X$, with étale local ring $\mathcal O_x$, and $U\to \mathcal X$ an étale atlas. Consider the following diagram with fibered squares:
@VVV @VVV \\
\mathcal X@<<<\mathcal X_x\\
@VVV @VVV\\
X@<<<\Spec(\mathcal O_x)
\end{CD}$$
where of course $S$ is a strictly henselian local ring and $Y=\Spec (S)\to \mathcal X_x$ is an étale atlas.
As the residue field of $S$ is separably closed, $R=Y\times_{\mathcal X_x}Y$ must be a finite disjoint union of copies of $Y$, whence the set $G$ of connected components of $R$ inherits canonically a group structure if we declare that $s$ is the natural projection. Thus $(s,t):R=Y\times G\rightrightarrows Y$is a groupoid with an étale morphism $[Y/G]\to \mathcal X$.
In the situation where $\mathcal X$ is a DM stack with tame quotient singularities there exists a canonical, smooth cover of $\mathcal X$, which we will refer to as Vistoli covering stack, constructed as follows:
[<cit.> 2.8]
Let $\mathcal X$ be a DM stack with tame quotient singularities. There exists a smooth DM stack $\mathcal X^V$ with a morphism $\mathcal X^V\to \mathcal X$ which is étale in codimension one, satisfying the following universal property: if $\mathcal Y$ is a smooth DM stack and $\mathcal Y\to \mathcal X$ is étale in codimension one, then there exists a unique factorization $\mathcal Y\to \mathcal X^V\to \mathcal X$.
Let $W\to \mathcal X$ be an étale atlas with tame quotient singularities, say $W=\bigsqcup U_i/G_i$ and let $(s,t): R_{\mathcal X}=W\times_{\mathcal X}W\rightrightarrows W$. Denote by $H_i$ the normal subgroup of $G_i$ generated by pseudo-reflections, let $W^V=\bigsqcup U_i/H_i$ and $R^V$ be the normalization of $W^V\times_{\mathcal X} W^V$
U_i/H_i@<<< s^* (U_i/H_i) @<<< U_i/H_i\times_{\mathcal X}U_j/H_j @<\text{normalization}<< R^V\\
@VVV @VVV @VVV \\
U_i/G_i @<\text{s}<< R_{\mathcal X} @<<< t^*(U_j/H_j)\\
@VVV @V\text{t}VV @VVV \\
\mathcal X@<<< U_j/G_j@<<< U_j/H_j
\end{CD}$$
$R^V$ is a normal algebraic variety since the diagonal of $\mathcal X$ is representable. The two projections $(s^V,t^V):R^V\rightrightarrows W^V$ are finite and étale in codimension one, since the action of $G_i/H_i$ on $U_i/H_i$ is free in codimension one for every $i$. $W^V$ is smooth by <ref>, whence the Zariski-Nagata purity theorem (<cit.>, X.3.4) implies that the projections $s^V,t^V$ are everywhere étale and the groupoid $\mathcal X^V=[W^V/R^V]$ is a smooth DM stack satisfying the required properties.
Similarly in the situation where $\mathcal X$ is a normal DM stack and $\mathcal D$ is a $\mathbf Q$-Cartier divisor, such that for any geometric point $x\in \mathcal X$, the minimal integer number $n(x)$ such that $n(x)\mathcal D_{|x}$ is Cartier satisfies $(n(x),\cha(k))=1$, there is a canonical cyclic cover of $\mathcal X$ in which $\mathcal D$ becomes everywhere Cartier. We will denote this stack by $\mathcal {X(D)}$, the Cartification of $\mathcal D$. The construction is a copy-and-paste extension of that of Gorenstein covering stack:
There exists a cyclic cover $p_D:\mathcal{X(D)}\to \mathcal X$ such that $p^*\mathcal D$ is Cartier, satisfying the following universal property: if $\mathcal Y$ is a DM stack and $q:\mathcal Y\to \mathcal X$ is such that $q^*\mathcal D$ is Cartier, then there exists a unique factorization $\mathcal Y\to \mathcal {X(D)}\to \mathcal X$.
Let $x$ be a geometric point in $\mathcal X$, $V$ an étale neighborhood and $V^o$ its smooth locus. Up to shrinking, we can assume that $n\mathcal D$ is generated, over $V$, by a local section $f$ and that $\mathcal D$ is generated, over $V^o$, by a local section $t$. Inside the geometric line bundle $\Spec\ (\Sym\ \mathcal O_{V^o}(D)^{\vee})\to V^o$ consider the subvariety $W^o$ defined by the ideal $T^n-f=0$, where $T$ is a generator of $\Sym\ \mathcal O_{V^o}(D)^{\vee}$ as an $\mathcal O_{V^o}$-algebra. This is a cover of $V^o$ corresponding to the extraction of an $n$-th root of the unity $u\in \mathcal O_{V^o}^*$ satisfying $t^n-uf=0$, which is étale since $(n,\cha(k))=1$. This cover extends canonically to a ramified cover $V(\mathcal D)\to V$ by taking the integral closure of $\mathcal O_V$ in the function field of $W^o$. This gives a local description, with respect to an atlas of $\mathcal X$, of an atlas for $\mathcal {X(D)}$, in particular the étale local ring of $\mathcal{X(D)}$ at a geometric point $x$ is $\mathcal O_{x}[T]/(T^{n(x)}-f)$. To deduce the groupoid relation, we proceed as in the Vistoli situation, thus we need to check that the natural morphism $(V_i(\mathcal D)\times_{\mathcal X} V_j(\mathcal D))^{\norm}\to V_i(\mathcal D)$ is étale. After étale localization around a geometric point, this amounts to proving that the natural morphisms
$$\mathcal O[T]/(T^n-f)\to \mathcal O[T,S]/(T^n-f,S^m-g)^{\norm}$$
$$\mathcal O[T]/(S^m-g)\to \mathcal O[T,S]/(T^n-f,S^m-g)^{\norm}$$
are étale, where $\mathcal O$ is a strictly henselian local ring with a height one prime ideal $I$ such that $(f)=I^n$ and $(g)=I^m$. Let $d=\gcd(n,m)$, $n=n'd$, $m=m'd$ and $w=nm/d$. There is a unit $u\in \mathcal O^*$ such that $f^{m'}=ug^{n'}$, thus we have a relation $T^w=uS^w$, and since $\mathcal O$ is strictly henselian and $(w,\cha(k))=1$ we see that $u^{1/w}\in \mathcal O$. Consequently we deduce a relation $\prod_{\zeta^{w}=1}T-\zeta u^{1/w}S=0$ in $\mathcal O[T,S]/(T^n-f,S^m-g)$. Taking its normalization we get a product of rings
$$\prod_{\zeta^{w}=1}\mathcal O[T,S]/(T^n-f,S^m-g, T- \zeta u^{1/w}S)\simeq \prod_{\zeta^{w}=1} \mathcal O[T]/(T^n-f)$$
which implies that the natural morphisms defining the Cartification groupoid
$$\mathcal O[T]/(T^n-f)\to \prod_{\zeta^{w}=1} \mathcal O[T]/(T^n-f)$$
$$\mathcal O[S]/(S^m-g)\to \prod_{\zeta^{w}=1} \mathcal O[T]/(T^n-f)$$
given respectively by $T\to \prod_{\zeta^{w}=1} T$ and $S\to \prod_{\zeta^{w}=1} \zeta u^{1/w}T$ are indeed étale, thus proving the existence of $\mathcal {X(D)}$ as a DM stack. The universal property follows by the local nature of the construction.
The construction of the Cartification morphism depends heavily on the assumption that the Cartier index of the divisor is coprime to the characteristic of the base field. Without this assumption the local Cartification factors through a non trivial inseparable quotient, and there is no hope to create a DM stack out of it.
In the sequel we will need a reformulation of the universal property of the Cartification.
Cartification is stable under pullbacks, that is if $f:\mathcal Y\to \mathcal X$ is any morphism and $\mathcal D$ is $\mathbf Q$-Cartier on $\mathcal X$ then there exists a canonical isomorphism $i_f:\mathcal Y(f^*\mathcal D)\to f^*\mathcal X(\mathcal D)$.
This is an easy application of the universal property: $\mathcal D$ becomes Cartier on $\mathcal Y(f^*\mathcal D)$ and this affords a morphism $i_f:\mathcal Y(f^*\mathcal D)\to f^*\mathcal X(\mathcal D)$. Since $\mathcal D$ is Cartier on $\mathcal X(\mathcal D)$, so it is after pullback to $f^*\mathcal X(\mathcal D)$, whence by the universal property we deduce an inverse to $i_f$.
§.§ Weighted blow-up and characters
Let $\mathcal X$ be a smooth DM stack, and $(a_1,...,a_r)$ an $r$-tuple of natural numbers. A blow-up with weights $(a_1,...,a_r)$ is the projectivization $\Proj_{(\oplus_{k\geq 0}I_k/I_{k+1})}\mathcal X\to \mathcal X$, where $I_k$ is a sheaf of ideals such that, étale locally, there exist functions $x_1,...,x_r$ forming part of a regular system of parameters and $I_k$ is generated by monomials $x_1^{\alpha_1}...x_r^{\alpha_r}$ with $a_1\alpha_1+...+a_r\alpha_r\geq k$. Then $\oplus_{k\geq 0}I_k/I_{k+1}$ is a sheaf of finitely generated graded algebras.
To give a local description let's assume we are blowing up the origin in the affine space over $k$ with weights $(a_1,...,a_n)$. It is covered by affine open sets $V_i=\Spec(R_i)$ where $R_i=k[ x_1^{\alpha_1}...x_n^{\alpha_n}/x_i^{c_i}$ s.t. $a_ic_i=a_1\alpha_1+...+a_n\alpha_n]$, and the morphism $R_i\to k[y_1,...,y_n]$ given by $x_i\to y_i^{a_i}$, $x_j\to y_jy_i^{a_j}$ induces an isomorphism $\Spec(k[y_1,...,y_n]^{\mu_{a_i}})=\mathbf A^n_k/\mu_{a_i}\to \Spec(R_i)$ where the roots of unity act by way of a generator $g\in \mu_{a_i}$ as follows: $y_i^g=\zeta y_i$, $y_j^g=\zeta^{-a_j}y_j$ with $\zeta$ an actual root of unity in $k^*$. Moreover the exceptional divisor has equation $y_i=0$ in the $i$-th chart. Plainly this local description doesn't make any sense whenever $\cha(k)=p> 0$ since $p$ might divide some $a_i$, so we need more care in defining cyclic group actions:
Let $\mathcal O$ be a regular local $k$-algebra of dimension $n$, and $l>0$ any natural number. A diagonal action of the cyclic group $C_l$ of order $l$ on $\mathcal O$ with characters $(a_1,..., a_n)$ is a morphism $\mathcal O\to \mathcal O[T]/(T^l-1)$ such that there exists a regular system of parameters $x_1,...,x_n$ with $x_i\to T^{a_i}x_i$. The equalizer of this action will be denoted by $\mathcal O^C$.
An action of the cyclic group of order $l$ on the smooth DM stack $\mathcal X$ is an action of the group scheme $\Spec(\mathbf Z[T]/(T^l-1))$, i.e. a morphism $\mathcal X\times_{\mathbf Z}\Spec(\mathbf Z[T]/(T^l-1))\to \mathcal X$. It is clear what is means for the action to be diagonal with characters $(a_1,...,a_n)$ around a fixed geometric point.
The definition gives a description of the affine cover of the weighted blow-up in full generality, namely the open sets $V_i$ will be $\Spec(R_i)$ where $R_i$ is the algebra obtained as equalizer of the $\mu_{a_i}$-action on $k[y_1,...,y_n]$ with characters $(-a_1,...,-a_{i-1},1,-a_{i+1},...,a_n)$. Such equalizer is generated, as a $k$-algebra, by monomials, and indeed there is a clear interpretation in terms of toric geometry, <cit.>.
We will be interested in weighted blow-ups induced by diagonal actions of cyclic groups.
Notation as in <ref>, the weighted algebra induced by the characters $(a_1,...,a_n)$ is the sheaf of algebras $A(a_1,...,a_n)=\oplus_{k\geq 0} I_k/I_{k+1}$ on $\Spec(\mathcal O)$, where $I_k=(x_1^{\alpha_1}...x_n^{\alpha_n}|a_1\alpha_1+...+a_n\alpha_n\geq k)$. $C$ acts naturally on $A(a_1,...,a_n)$ and its fixed sub-algebra $B(a_1,...,a_n)=A(a_1,...,a_n)^C$ is canonically a sheaf of algebras on $\Spec(\mathcal O^C)$.
At which point one might question the dependence of such filtration by ideals $I_k$ on the choices made, which unfortunately turns out to be non-trivial:
Let $C$ be a cyclic group of order $l$ acting tamely on $k[x,y]$ by way of $x\to \chi x,\ y\to \chi^{-1}y$ for some faithful character $\chi: C\to k^*$. The ring of invariants is $R=k[x^l,y^l,xy]$ whence the action of $\mu_2$ on the plane by way of $x\to y,\ y\to x$ naturally descends to an action on $R$. Observe however that if $l\neq 2$ the weighted algebra induced by the characters $(1,-1)$ of the $C$-action on the plane is not invariant under $\mu_2$, indeed the weighted filtration by ideals is
$$I_k=(x^ay^b\ |\ a+(l-1)b\geq k)\subset k[x,y]$$
and this filtration is clearly not invariant under $\mu_2$.
From a global perspective, say on a smooth projective variety $X$ endowed with a tame action of $C$, the example is telling us that local choice of characters (around each fixed component) is by no means sufficient to deduce that the corresponding weighted algebras will patch as we travel around an open cover of $X$. However it turns out that a "global" choice of faithful character is enough to insist that local algebras glue, <ref>.
Let $\mathcal O$ be a regular strictly henselian local ring with a diagonal action of a cyclic group $C$. Let $f\in \mathcal O$ be a non-unit, then we say that the action is faithful across the divisor $D(f)=(f=0)$ if, following <ref>, there exists a choice of characters for the action of $C$, such that $f\to Tf$. The weighted algebra corresponding to the choice of characters making the action faithful across $D(f)$ will be denoted by $A(D(f))$. The sub-algebra of $C$-invariants will be denoted by $B(D(f))$.
The following proposition is, broadly speaking, the solution to all our problems, in that it gives a sufficient condition for local weighted algebras to glue to a sheaf of algebras over an algebraic variety:
Let $X$ be a variety with diagonal cyclic quotient singularities, $D$ a reduced, irreducible $\mathbf Q$-Cartier divisor. Assume that for every $x\in D$, $C_x$ is faithful across $D$. Then the local weighted algebras $B(D_x)$ glue to a sheaf of algebras $B$ on $X$.
First some notation: fix $x\in X$ lying on the support of $D$ and denote by $I_D$ the ideal defining $D$ around $x$. Set $l=|C_x|$. Let $\mathcal O_x^V$ be a strictly henselian, regular local ring such that $\mathcal O_x=(\mathcal O_x^V)^{C_x}$ and $x_1,...,x_n$ a regular system of parameters in $\mathcal O_x^V$ such that $I_D\cdot \mathcal O_x^V=(x_n=0)$, and $C_x$ acts by characters $x_i\to T^{a_i}x_i$ with $a_n=1$. Let $\phi$ be an automorphism of $\mathcal O_x$ fixing the ideal $I_D$. Observe that $I_{D}^l=(x_n^l)$ whence $\phi(x_n^l)=ux_n^l$ for some unit $u$. Consider the elements $y_i=x_n^{l-a_i}x_i\in I_{D}$. We have
On the other hand $\phi(y_i)\in I_{D}$ whence $\phi(y_i)=x_n^{b_i}p_i$ for some $b_i>0$ and $p_i\in \mathcal O_x^V$ not a zero divisor in $\mathcal O_x^V/(x_n)$, so then $lb_i=l(l-a_i)$, i.e. $\phi(x_n^{l-a_i}x_i)=x_n^{l-a_i}p_i$. Thus $x_n^{l-a_i}p_i$ is fixed by $C_x$, and the action must be $p_i\to T^{a_i}p_i$ for every $i$. Moreover, the identity
inside the fraction field $f.f.(\mathcal O_x^V)$ implies more generally that
$$\phi(x_1^{\alpha_1}...x_n^{\alpha_n})=u^{l^{-1}\sum_{i=1}^n a_i\alpha_i -\sum_{i=1}^{n-1}\alpha_i}p_1^{\alpha_1}...p_{n-1}^{\alpha_1}x_n^{\alpha_n}\\\ (l|\sum_{i=1}^n a_i\alpha_i)$$
which clearly implies that $\phi$ preserves the weighted algebra induced by these characters. Whence local weighted algebras $B(D_x)$ patch to a sheaf of algebras on $X$.
§.§ Toric varieties
Here we recall basic notation of the theory of toric varieties, following <cit.>, Chap. 1. The scope of this section is to give a combinatorial interpretation of diagonal cyclic quotient singularities, and re-interpret the observations in <ref> from this perspective.
Let $V$ be an $n$-dimensional real vector space. The choice of a basis gives a lattice $\mathbf Z^n\subset \mathbf R^n\simeq V$. Let $\sigma\subset V$ be a cone which is generated by finitely many elements in $\mathbf Z^n$. The affine toric variety $X_{\sigma}$ is defined as $\Spec(k[\sigma^*\cap \mathbf Z^n])$, where $\sigma^*$ is the dual cone to $\sigma$, and $k[\sigma^*\cap \mathbf Z^n]$ is the algebra generated by monomials $x_1^{a_1}\dots x_n^{a_n}$ for $(a_1,...,a_n)\in \sigma^*\cap \mathbf Z^n$. It is easy to see that toric open subsets of $X_{\sigma}$ correspond similarly to the faces of $\sigma$, whence whenever $\tau\subset \mathbf R^n$ is a fan (i.e. a polyhedron which is obtained as union of finitely many, finitely generated cones intersecting along common faces) we have a toric variety $X_{\tau}$ obtained by glueing the affine toric varieties corresponding to the cones defining $\tau$.
A fundamental construction admitting an interpretation in terms of the toric cone is that of blow-up: given a toric cone $\sigma =<v_1,...,v_m>$ and a vector $v\in \sigma$ which is not in $\sum_{i=1}^m \mathbf N v_i$, we can consider the fan $\sigma_v$ obtained as the union of the cones $\sigma_i=< v_1,...,v_{i-1},v,v_{i+1},...,v_n >$. Then there exists a sheaf of ideals $I_v$ in $X_{\sigma}$ such that the normalization of $Bl_{I_v}X_{\sigma}$ is isomorphic to $X_{\sigma_v}$, and the toric morphism induced by the inclusion $\sigma_v\to \sigma$ corresponds to the natural projection $(Bl_{I_v}X_{\sigma})^{\norm}\to X_{\sigma}$ under this isomorphism. Now we turn our attention to the class of cyclic quotient singularities by observing that a formal germ of cyclic quotient singularity is always toric, by a trivial extension of <cit.>, pp. 16-18:
Let $e_1,...,e_n$ be a basis of a given real $n$-dimensional vector space $V$ and let $a_1,...,a_{n-1},l$ be natural numbers, $l\neq 0$. Then the cone $\sigma$ generated by $e_1,...,e_{n-1},le_n-(a_1e_1+...+a_{n-1}e_{n-1})$ corresponds to the quotient of $\mathbf A^n_k$ by the action of the cyclic group of order $l$ with characters $a_1,...,a_{n-1},1$.
The basis $e_1,...,e_n$ determines a free group $\mathbf Z^n\subset V$. Let $N\subset \mathbf Z^n$ be the subgroup of index $l$ generated by $e_1,...,e_{n-1},le_n-(a_1e_1+...+a_{n-1}e_{n-1})$, and let $M$ be its dual lattice in $V^*$. Observe that $\sigma^*$ is generated by
$$e_i^*+a_il^{-1}e_n^*\ \quad (i\leq n-1), \quad l^{-1}e_n$$
which is basis of $M$, whence $\Spec(k[\sigma^*\cap M])=\mathbf A^n_k$ with coordinates $x_1,...,x_n$ corresponding to this choice of generators for $\sigma^*$. Also the choice of a generator $\zeta\in \mu_l$ gives an isomorphism $\mathbf Z^n/N\simeq \mu_l$ obtained by sending $e_n\to \zeta$. Consider now the $\mu_l$ action on $\Spec(k[\sigma^*\cap M])=\mathbf A^n_k$ with characters $(a_1,..,a_{n-1},1)$. On monomials the action is given by
$$\mathbf x^m\to T^{lm(e_n)}\mathbf x^m\ \quad (m\in \sigma^*\cap M)$$
whence the fixed subalgebra is generated by monomials $\mathbf x^m$ such that $m(e_n)\in \mathbf Z$, which is evidently $k[\sigma^*\cap \mathbf Z^n]$.
The next observation expands the toroidal view on diagonal cyclic quotient singularities <ref>, by giving a toroidal interpretation of weighted blow-ups induced by characters of diagonal cyclic actions:
Let $C$ be a cyclic group of order $l$ acting on $\mathbf A^n_k$ diagonally with characters $a_1,...,a_{n-1},1$, and let $X_{\sigma}$ be the quotient variety. Let $A=A(a_1,...,a_{n-1},1)$ and $B=B(a_1,...,a_{n-1},1)$, cf <ref>. Then $\Proj_B(X_{\sigma})$ has only diagonal cyclic quotient singularities of orders $a_1,...,a_{n-1}$. Moreover such modification coincides with the toric blow-up obtained by way of the decomposition of $\sigma$, as defined in <ref>, by adding the vector $e_n$.
By the local description given at the beginning of <ref>, $\Proj_A(\mathbf A^n_k)$ has diagonal cyclic quotient singularities of orders $a_1,...,a_{n-1}$. The action of $C$ lifts to $\Proj_A(\mathbf A^n_k)$, and it acts by pseudoreflections across the exceptional divisor of $\Proj_A(\mathbf A^n_k)\to \mathbf A^n_k$. It follows that $\Proj_A(\mathbf A^n_k)/C=\Proj_{B}(X_{\sigma})$ has diagonal cyclic quotient singularities of orders $a_1,...,a_{n-1}$. Inspection of the local structure of $\Proj_{B}(X_{\sigma})$ around the exceptional divisors reveals immediately that the said toric blow-up gives the same output.
§ RESOLUTION
The aim of this section is to prove the main theorem.
Let $k$ be an algebraically closed field, and $X/k$ a quasi-projective variety with tame quotient singularities, with an étale cover $\bigsqcup_j X_j\to X$ such that $\max_{x\in X_j}|G_x|$ is finite for every $j$. Then there exists a resolution functor $X\to (M(X),r_X)$ where $M(X)$ is a smooth, quasi projective variety, and $r_X:M(X)\to X$ is a proper, birational, relatively projective morphism which is an isomorphism over the smooth locus of $X$. The resolution functor commutes with étale base change, that is to say for every étale morphism $f:Y\to X$ there is a unique isomorphism $\phi_f:f^*M(X)\to M(Y)$.
Before getting into the proof, let's observe a few immediate corollaries in the spirit of <cit.>, 3.42, 3.43, 3.44:
Let $X$ be an analytic space with tame quotient singularities. Then there exists a relatively projective, proper and birational morphism $X'\to X$ with $X'$ smooth.
We can find an étale cover of $X$, consisting of at most countably many analytic spaces satisfying the condition that each of them has bounded geometric stabilizers.
Let $\mathcal X$ be a DM stack with tame quotient singularities. Then there exists a representable, proper, birational, relatively projective morphism $\mathcal X'\to \mathcal X$ with $\mathcal X'$ smooth.
We can represent $\mathcal X$ as an étale groupoid $(s,t):R\rightrightarrows X$. Functoriality in the étale topology gives unique isomorphisms $\phi_s:M(R)\to s^*M(X)$ and $\phi_t: M(R)\to t^*M(X)$, and the induced morphisms $(s\phi_s,t\phi_t):M(R)\rightrightarrows M(X)$ inherit canonically the structure of a groupoid, whose projection onto $\mathcal X$ is clearly representable.
The resolution functor commutes with smooth base change.
This is <cit.> 3.9.2.
The rest of the manuscript will be devoted to the proof of <ref>.
§.§ Step 1: Reduction to tame cyclic quotient singularities.
We can assume that $X=X_j$ for some index $j$. The invariant on which the inductive argument builds upon is $i(X)=\max_{x\in X}|G_x|$. By <ref> the subvariety $Z$ consisting of points with geometric stabilizer of maximal cardinality is smooth, so let $p_Z:Y=Bl_ZX\to X$ be its blow-up with exceptional divisor $E$. If $i(Y)<i(X)$ we are done, otherwise necessarily $i(Y)=i(X)$ and $E$ is $\mathbf Q$-Cartier but not Cartier, whence there is a non trivial Cartification cover $q_E:Y(E)\to Y$.
Consider the natural projection $q_V:Y^V\to Y(E)$ and let $y\in Y^V$ be a geometric point with $|G_y|=i(Y)$. The projection induces a morphism of strictly henselian local rings at $y$, $\mathcal O_{Y(E),q_V(y)}\to \mathcal O_{Y^V,y}$. The action of $G_y$ on $\mathcal O_{Y^V,y}$ leaves $E_y$ invariant, so by <ref> it is given by a non-trivial character $\chi_{E,y}:G_y\to k^*$. We see immediately that $\mathcal O_{Y,q_Eq_V(y)}$ is the subring of $\mathcal O_{Y(E),q_V(y)}$ of elements fixed by the cyclic action of $Im(\chi_{E,y})$. Thus $G_{q_V(y)}=ker(\chi_{E,y})$ and the claim follows.
Now we can apply our inductive hypothesis, in the formulation <ref>, and obtain a representable, proper, birational, relatively projective morphism $r_Y:Y_1\to Y(E)$ with $Y_1$ smooth DM stack, along with a GC quotient morphism $m:Y_1\to X_1$, <cit.> 1.1. Since $Y$ is quasi-projective and $r_Y$ is relatively projective, $X_1$ is quasi-projective. The total pullback of $r_Y^*q_E^*E$ naturally descends to a $\mathbf Q$-Cartier divisor $E_1$ on $X_1$, i.e. $m^*E_1=r_Y^*q_E^*E$. The pair $(X_1,E_1)$ enjoys particularly good properties, that we can summarize as follows:
In our setting the GC quotient morphism $m:Y_1\to X_1$ is locally a cyclic cover, say given by the tame cyclic action of $C_y$ in a neighborhood of a geometric point $y\in Y_1$. The assertion we need to prove is that $C_y$ acts faithfully across $r_Y^*q_E^*E$. Observe that $C_y$ is faithful on $q_E^*E\subset Y(E)$ in a neighborhood of $r_Y(y)$, and since $Y_1\to Y(E)$ is birational, whence injective on local rings, we conclude by <ref> that $C_y$ is faithful on $r_Y^*q_E^*E$ as well.
It is pretty straightforward to control the behaviour of our construction under étale base change:
Let $f:X'\to X$ be an étale morphism, then the square
X'_1 @>>>X_1\\
@VVV @VVV\\
X' @>>>X
\end{CD}$$
is fibered.
Consider the diagram
X'@<<< Y'=Bl_{f^*Z}X' @<<< Y(f^*E)'@<\text{Resolution}<< Y_1'@>\text{GC quotient}>> X_1'\\
@V\text{\'etale}VV @VVV @VVV @VVV @VVV \\
X@<<< Y=Bl_Z(X) @<<< Y(E)@<\text{Resolution}<< Y_1@>\text{GC quotient}>> X_1\\
\end{CD}$$
The two leftmost squares are fibered, by <ref>. The third square from the left is also fibered, by our inductive hypothesis, thus $Y_1'= Y_1\times_X X'$, and we conclude since the GC quotient functor commutes with flat base change - as it follows by definition, <cit.> 1.8.
We conclude the first step of the proof by summarizing what we got so far.
: Given a quasi-projective variety $X$ satisfying the assumptions of <ref>, there exists a functor $X\to (X_1,E_1=E_1(X),p_X)$ where $X_1$ is a quasi projective variety with tame cyclic quotient singularities with $i(X_1)\leq i(X)$, $E_1$ is a $\mathbf Q$-Cartier divisor such that $X_1^V\to X_1(E_1)$ is an isomorphism. $p_X:X_1\to X$ is a proper, relatively projective, birational morphism. The functor commutes with étale base change, that is to say for every étale morphism $f:Y\to X$ there is a unique isomorphism $\psi_f:f^*X_1\to Y_1$ such that $\psi_f^*p_Y^*E_1(Y)=p_X^*E_1(X)$. Here is a diagram summarizing our construction:
q_V^*q_E^*E\subset Y_V\\
q_E^*E\subset Y(E) @<\text{resolution}<< m^*E_1\subset Y_1\\
@V\text{Cartification}VV @V\text{GC quotient=Cartification}VV\\
E\subset Bl_ZX=Y@<<< E_1\subset X_1\\
Z\subset X
\end{CD}$$
§.§ Step 2: Reduction of the invariant for tame cyclic quotient singularities.
The second step of the proof works out an algorithmic procedure to reduce the invariant $i(X)$, if we are given a pair $(X,E)$ where $X$ is a quasi-projective variety with tame cyclic quotient singularities, $E$ is a reduced divisor whose connected components are irreducible, and the canonical projection $X^V\to X(E)$ is an isomorphism - meaning that the local cyclic groups defining the Vistoli cover $X^V\to X$ act with a faithful character on the pullback of $E$ in $X^V$. The key observation we need to construct a resolution is <ref>, that for such a pair $(X,E)$ there exists a choice of characters for the cyclic action whose corresponding local weighted algebras form a globally well defined sheaf of algebras. Blowing it up, <ref>, we obtain a quasi-projective variety with cyclic quotient singularities whose stabilizers are smaller, but these need not be tame anymore. However the cyclic quotients need not be tame for weighted blow-ups to make sense, whence further applications of <ref>, in the form <ref>, can be used to remove the non-tame stabilizers that might appear after the first blow-up. More precisely let $Z$ be the smooth subvariety of points with maximal geometric stabilizers. By assumption, the stabilizers are faithful across $E$, whence by <ref> there is a sheaf of algebras $B_Z$ induced by the faithful actions of the generic stabilizers at $Z$ along $E$. Let $p_w:X^w:=\Proj_{B_Z}X\to X$ and consider the pair $(X^w,E^w:=p_w^*E)$. We have two cases:
(i) $X^w$ has tame cyclic quotient singularities. In this case $i(X^w)<i(X)$ and the main theorem follows.
(ii) $X^w$ has non-tame, diagonal cyclic quotient singularities. In this case we can eliminate the non-tame cyclic stabilizers by way of the following.
Let $(X,D)$ be a pair where $X$ is a quasi-projective variety with diagonal cyclic quotient singularities and $D=\sum_{i=1}^t D_i$ is a reduced divisor. Assume that, if $x\in X$ is such that $C_x$ is not tame, then $x\in D$ and $C_x$ is faithful across at least one irreducible component of $D$. Then there exists a resolution functor $(X,D)\to (M(X,D),u_X)$ where $M(X,D)$ is a quasi projective variety with tame cyclic quotient singularities such that $i(M(X,D))\leq \max_{x\in X}|C_x|$, $u_X:M(X,D)\to X$ is a proper, birational, relatively projective morphism which is an isomorphism over the smooth locus of $X$. The functor commutes with étale base change, that is to say for every étale morphism $f:(Y,D_Y)\to (X,D)$ there is a unique isomorphism $\phi_f:f^*M(X,D)\to M(Y,D_Y)$.
Let $j(X)=\max\{|C_x|$, $x\in X$ s.t. $C_x$ is not tame$\}$. Clearly the subvariety $Z$ of points with maximal non-tame geometric stabilizer is a smooth subvariety of $D$. We will proceed by induction on $j(X)$. Let $Y$ be a connected component of $Z$, and denote by $h(Y)$ the maximal index such that $Y\subset D_{h(Y)}$ and the stabilizer $C_Y$ is faithful across $D_{h(Y)}$. By <ref> for every such $Y$ there is a sheaf of algebras $B_Y$ supported on $Y$ and induced by the faithful action of the geometric stabilizers across $D_{h(Y)}$. Let $B_Z=\sum_Y B_Y$, and denote by $p_1:X_1=\Proj_{B_Z}X\to X$ with exceptional divisor $D_{t+1}$ and set $D_1=p_1^*D=\sum D_i^1+D_{t+1}$ where of course $D_i^1$ is the strict transform of $D_i$ under $p_1$. By <ref> $j(X_1)<j(X)$, and we claim that the pair $(X_1,D_1)$ still satisfies the assumptions of <ref>. Indeed let $Y_1$ be a connected component of $Z_1$, if $Y_1$ is not contained in $D_{t+1}$ then $Y_1$ is the proper transform of some component with non-tame stabilizer $Y\subset X$, and faithfulness across $D_{h(Y)}^1$ is granted by <ref>. Next if $Y_1\subset D_{t+1}$, then the local description of a weighted blow-up, as given at the beginning of <ref>, forces the geometric stabilizers along $Y_1$ to be faithful across $D_{t+1}$. By induction on $j(X)$ we conclude the existence of $(M(X,D),u_X)$. Functoriality in the étale topology is a trivial consequence of our iterative construction.
We are ready to conclude the proof of the main theorem: starting with a quasi-projective variety with tame quotient singularities $X$, <ref> gives a pair $(X_1,E_1)$ as in <ref>. Running <ref> with input $(X_1,E_1)$ we get a quasi-projective variety with tame cyclic quotient singularities $M(X_1^w,E_1^w)$ - where of course $M(X_1^w,E_1^w)=X^w_1$ in case i) above - such that $i(M(X_1^w,E_1^w))<i(X)$, along with a proper, relatively projective, birational morphism $M(X_1^w,E_1^w)\to X$.
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1511.00102
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Astrophysics, Cosmology and Gravity Centre (ACGC), Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa.
Astrophysics, Cosmology and Gravity Centre (ACGC), Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa.
South African Astronomical Observatory, Observatory 7925, Cape Town, South Africa
Astrophysics, Cosmology and Gravity Centre (ACGC), Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa.
Departamento de Física & Instituto de Astrofísica e Ciências do Espaço,
Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, P-1749-016
Lisbon, Portugal
04.50.Kd, 98.80.-k, 98.80.Cq, 12.60.-i
Modified gravity has attracted much attention over the last few years and remains a potential candidate for dark energy. In particular, the so-called viable $f(R)$ gravity theories, which are able to both recover General Relativity (GR) and produce late-time cosmic acceleration, have been widely studied in recent literature. Nevertheless, extended theories of gravity suffer from several shortcomings which compromise their ability
to provide realistic alternatives to the standard cosmological $\Lambda$CDM Concordance model.
We address the existence of cosmological singularities and the conditions that guarantee late-time acceleration,
assuming reasonable energy conditions for standard matter in the so-called Hu-Sawicki $f(R)$ model, currently among the most widely studied modifications to General Relativity.
Then using the Supernovae Ia Union 2.1 catalogue, we further constrain the free parameters of this model. The combined analysis of both theoretical and observational constraints sheds some light on the viable parameter space of these models and the form of the underlying effective theory of gravity.
§ INTRODUCTION
Over the last few years, a great deal of effort related to the problem of the origin of late-time cosmic acceleration has been devoted to the so-called $f(R)$ theories of gravity. This is due to the fact that by a choosing the Lagrangian of the gravitational interaction to be an appropriate function of the Ricci scalar, the late-time acceleration of the universe expansion can be reproduced without the need of introducing a dark energy field (for a review see Ref. <cit.>). Contrary to the first $f(R)$ models from the 80's, for example Starobinsky's $R^2$ inflation model, much of the current work on $f(R)$ gravity is aimed at obtaining a description of the late-time history of the universe, when the curvature is very small. Roughly speaking, these models provide an infrared correction to General Relativity (GR), which may be inspired by string theories
<cit.>. Moreover, the analysis of inflation within the framework of modified gravity and even the unification of late-time acceleration and inflation still draws a great deal of attention, particularly after the last release of Planck data and the success of Starobinsky inflation <cit.>. It is therefore possible that these types of modifications to general relativity could lead to a complete picture of the evolution of the universe
<cit.>. Unfortunately, in general, such modifications of GR are plagued by a number of problems, such as violations of local gravity tests, the absence of a matter dominated era and antigravity regimes among others. In order to deal with these shortcomings, some $f(R)$ models referred to as viable have been proposed in the last few years (see Refs. <cit.>). Those models are able to introduce corrections at cosmological scales, while GR is recovered on local scales and the usual predictions of GR remain the same. To do so, the authors of these works extended the so-called Chameleon mechanism <cit.>, initially applied to scalar-tensor theories, to $f(R)$ gravity. The Chameleon mechanism basically introduces a scale hierarchy over the additional terms of the gravitational action so that on local scales, for example the Earth or the Solar System, GR is effectively recovered. On the other hand, these terms become important on cosmological scales, whereby the appropriate choice of theory parameters, the late-time acceleration can be reproduced. In addition, these viable modified theories of gravity are also able to evade the Ostrogradski and Dolgov-Kawasaki instabilities <cit.>. All these features make these theories a promising candidate for dark energy.
However, a common issue of every viable $f(R)$ gravity is the presence of a number of theoretical shortcomings such as back-reaction averaging effects, absence of smooth junction conditions in astrophysical context <cit.>, faster growth of structures in disagreement with large-scale structure catalogues <cit.>, the appearance of unexpected singularities and existence of anti-gravity regimes, among others. The latter two issues form part of the study presented in this manuscript.
In more detail, a common feature of every viable $f(R)$ gravity is the presence of sudden cosmological singularities both in the past and the future (for a classification of singularities, see Ref. <cit.>). The existence of such singularities within General Relativity is connected to violations of the energy conditions by the matter content, particularly dark energy. Nevertheless, within modified gravity, the energy conditions may be violated naturally through the extra geometrical terms that appear in the field equations <cit.>. The occurrence of such singularities have been explored extensively in the literature, since observations do not discard an equation of state parameter for dark energy $w_{}<-1$ <cit.>. One of these cosmological singularities is the so-called sudden singularity, where the first derivative of the Hubble parameter diverges <cit.>. Viable $f(R)$ gravity, in general contains this type of cosmological singularity. Furthermore, the singularity represents an asymptotically stable point and therefore its avoidance depends entirely upon the initial conditions and the election of the free parameters (see Refs. <cit.>).
On the other hand, it is well known that in the context of GR without a cosmological constant, a non-positive contribution for the space-time geometry to the Raychaudhuri equation, or in other words the attractiveness of gravitational interaction, is obtained once standard fluids are assumed and regardless of the solution of the Einstein's equations <cit.>. However, this result can be reversed in the context of extended theories of gravity, where depending on the theory and parameter choice, the subsequent convergence (or divergence) of geodesics for fundamental observers can be obtained without invoking the presence of exotic fluids.
Moreover, an upper bound to the contribution of space-time geometry can be provided both in terms of the gravitational model and the metric under consideration. Using this upper bound and assuming usual energy conditions, restrictions on $f(R)$ models can be derived in order to constrain their cosmological viability <cit.>.
Consequently, the careful analysis of the geometrical terms in the Raychaudhuri equation for extended theories of gravity plays a critical role in the demonstration of the singularities theorems <cit.>, as well as in the context of the so-called Holographic Principle <cit.>.
An analysis of this geometrical contribution <cit.> showed that it can be interpreted as the mean curvature in the direction of the congruence <cit.>. It can also be easily verified that for a Robertson-Walker model with a negative deceleration parameter, this contribution is positive <cit.> and the attractive character of gravity vanishes.
Actually. as shown in <cit.>, the mean curvature for a given geodesics turns out to be positive for almost all timelike directions in a Robertson-Walker model with the present value of the deceleration parameter.
The present manuscript is devoted to the analysis of a class of viable $f(R)$ models – the so-called Hu-Sawicki gravity model <cit.> – which has received a lot of attention lately <cit.>. Here we investigate the possible constraints on the free parameters of this model by using both theoretical limits and observational data, particularly the Union2.1 catalogue of Supernovae Ia <cit.>. Then, the free parameters are constrained by obtaining the region of parameter-space which is free of cosmic singularities and also has a positive contribution to the Raychaudhuri equation at late times. Initial conditions for the background evolution are fixed at large ($z\sim 10$) redshifts to be the same as in the $\Lambda$CDM model in order to guarantee that the high redshift cosmology is compatible with BBN and CMB constraints. After obtaining a region free of singularities, which provides a smooth evolution from the past until today, the free parameters are then fitted by using Supernovae Ia data.
The paper is organised as follows: In Sec. <ref> we provide an overview of $f(R)$ theories of gravity in the metric formalism in general and the Hu-Sawicki model in particular, providing a brief review of the dynamical system approach which enables us to easily solve the background cosmological equations. Then in Sec. <ref> we discuss the emergence of sudden singularities in these kind of models using the equivalent picture of scalar-tensor theories. We also present the theoretical analysis leading to upper bounds on the positive geometrical contributions to the Raychaudhuri equation for $f(R)$ models. The statistical analysis using supernovae data is performed in Sec. <ref> enabling us, together with other gravitational and cosmological tests to constrain the viable parameters space. We end the paper in Sec. <ref> presenting the main results of this investigation. A brief appendix <ref> at the end of the paper gives details on the process to find the apparent magnitude statistical minimum.
Unless otherwise specified, natural units ($\hbar=c=k_{B}=8\pi G=1$) will be used throughout this paper.
§ COSMOLOGICAL EVOLUTION IN HU-SAWICKI $F(R)$ MODEL
$f(R)$ gravity usually refers to a set of theories whose Lagrangian is given by a general function of the Ricci scalar,
S=∫d^4x √(-g)[f(R)+2ℒ_m] ,
$\mathcal{L}_m$ is the Lagrangian of the matter content. It is straightforward to obtain the field equations by varying the action with respect to the metric field $g_{\mu\nu}$, leading to
R_μν f_R- 1/2 g_μν f(R) + (g_μν - ∇_μ ∇_ν)f_R = T^(m)_μν .
where $f_R\equiv \frac{{\rm d}f}{{\rm d}R}$. Higher derivatives of $f$ with respect to $R$ will be denoted as $f_{2R}$, $f_{3R}$, etc.
We are primarily interested in studying spatially flat Robertson-Walker cosmologies, whose metric, expressed in the usual co-moving coordinates, is given by
ds^2=-dt^2+a(t)^2∑_i=1^3(dx^i)^2 .
Then, the corresponding field equations obtained from (<ref>) and corresponding to a dust-dominated Universe become
H^2 = 1/3f_R( ρ_m +Rf_R-f/2-3HṘf_2R),
-3H^2-2Ḣ = 1/f_R[
+ .1/2(f-Rf_R)] ,
where the Hubble parameter is $H(t)=\dot{a}/a$, the dot denotes a derivative with respect to cosmic time, and $\rho_m$
denotes the standard matter energy density.
We can also use the continuity equation,
ρ_m=0 to reduce the number of independent equations.
It has been shown recently <cit.> that it is convenient (and numerically more stable) to express the cosmological equations as a set of autonomous first order equations in order to study the expansion history of a general class of $f(R)$ theories. Taking advantage of this fact, we rewrite equations (<ref>) - (<ref>) in terms of the following dynamical system variables
\begin{align}\label{ncvariables}
{x}&= \frac{\dot{R}f_{2R}}{H f_{R}},~~~~{v}=\frac{R}{6H^{2}}, \nonumber\\
\\
{y}&=\frac{f}{6H^2 f_{R}}, ~~~~{\Omega} =\frac{\rho_{m}}{3H^2 f_{R}}. \nonumber
\end{align}
Written in terms of (<ref>), the Friedmann and Raychaudhuri equations become
1=Ω + v -x -y
dh/dz = h/z+1 ( 2- v ) ,
where $h=H/H_{0}$ and we obtain the following set of first order differential equations directly from the dynamical system variables.
\begin{align}
\frac{{\rm d} {x}}{{\rm d}z}=&\frac{1}{(z+1)}\left[ - {\Omega}+{ {x}}^{2}+ \left( 1+ {v} \right) {x} -2 {v}+4 {y}\right] , \label{Neqx}\\
& \nonumber \\
\frac{{\rm d} {y}}{{\rm d}z}=&-\frac{1}{(z+1)}\left({ {v} {x}{\Gamma}- {x} {y}+4\, {y}-2\, {y} {v}}\right)\;,\\
& \nonumber \\
\frac{{\rm d} {v}}{{\rm d}z}=&-\frac{ {v}}{(z+1)} { \left( {x}{\Gamma}+4-2\, {v} \right) } ,\label{Neqv}\\
& \nonumber \\
\frac{{\rm d} {\Omega}}{{\rm d}z}=&\frac{1}{(z+1)}\left[{ {\Omega}\, \left( -1+ {x}+2\, {v} \right) }\right]\;.
\label{dodz}
\end{align}
These equations describe the cosmological evolution of a general $f(R)$ theory of gravity, where $\Gamma \equiv \frac{f_{R}}{Rf_{2R}}$ specifies the theory <cit.>.
In this paper we focus on the analysis of the so-called viable $f(R)$ theories of gravity, which, in addition to producing the late time accelerated era of expansion, also recovers results consistent with General Relativity on local scales. To illustrate our analysis, let us consider the model of this kind proposed in Ref. <cit.>,
HS(R) ,
where $\{b,c,d,n\}$ are constants to be determined by both theoretical and observational constraints, while $H_0$ is the $\Lambda$CDM Hubble parameter evaluated today. For this model, the $\Gamma$ term in the dynamical systems equations takes the following form
Γ= -( d r^n + 1 ) [ r( d r^n +1)^2 - bnc r^n ] /bnc[ r^n(n-1) - d r^2n(n+1) ] ,
where $r=R/cH_0^2$ is the dimensionless Ricci scalar.
The success of this model lies in its ability to produce an effective cosmological constant at late times, thus mimicking the expansion history of the $\Lambda$CDM model, as well as avoiding violations of local gravity tests. To do so, the extra scalar degree of freedom - known as the scalaron - behaves like a Chameleon field, whose mass is given by:
m_f_^HS=√(3f_2R^HS) .
Summarising, since the mass of the scalaron (<ref>) depends on the scale via the Ricci scalar, roughly speaking, the so-called thin-shell condition (a smooth transition from high to low curvature regimes) is satisfied provided the mass (<ref>) is large enough in the high curvature regime, such that deviations from General Relativity are avoided. For further details about the chameleon mechanism c.f. Ref.<cit.> and for its extension to $f(R)$ gravity, see Ref. <cit.>.
In spite of the great success of models of this kind, they are plagued with several shortcomings such as the presence of antigravity regimes or the occurrence of cosmic curvature singularities. Both of these issues are analysed in detail below, but before doing so, let us first illustrate their cosmological behaviour, given by (<ref>) and how they are able to mimic the cosmological constant behaviour at late times.
In order to illustrate this qualitatively, Fig. <ref> depicts the shape of $f^{HS}(R)$ for a set of the free parameters of the model. The free parameter $n$ controls the slope of the transition to a constant plateau. The amplitude of the correction is directly determined by the free parameter $c$, so that when $R\ll cH_0^2$, corrections to GR are negligible, and in the high curvature limit, $R\gg cH_0^2$, $f^{HS}(R)$ behaves effectively like a cosmological constant, namely[In expression (<ref>) the limit must be understood as $R\gg cH_0^2$, i.e., eras with high Ricci curvature, such as matter/radiation dominated eras.]
lim_cH_0^2/R →0 f^HS(R) ≈- b/dm^2 .
As was originally presented in <cit.>, we limit the choices of the free parameters by requiring that this theory must mimic the $\Lambda$CDM model. In order for this to occur, we require that
c=6(1-Ω_m)d/b .
In this way, the amplitude of the plateau is controlled by the free parameters $\{b,\, d\}$ and the matter density today $\Omega_m\equiv\rho_{m,0}/3H_0^2$.
In the top central panel of Fig. <ref>, the Hubble parameter evolution is compared with the $\Lambda$CDM model, while the central bottom panel depicts the deceleration parameter $q=-\ddot{a}/aH^2$. This clearly shows that for this choice of parameters, the expansion history is indistinguishable from the $\Lambda$CDM model, a feature which makes this class of theories such a popular parameterisation of dark energy. Having said this, in what follows, we will show that not all values of the parameters lead to viable expansion histories due to the presence of curvature singularities at physically relevant redshifts nor they guarantee cosmological expansion at late times.
The Hu-Sawicki model for a sample of the free parameters $\{n=1,b=200\}$. We show the redshift evolution of the Hubble parameter and the deceleration parameter in the inner upper panel and inner lower panel, respectively, for the Hu-Sawicki (blue) model and the $\Lambda$CDM model (red). Both the Hubble parameter and the deceleration parameter corresponding to this model are clearly indistinguishable from $\Lambda$CDM.
§ SINGULARITIES AND THE NON-ATTRACTIVE CHARACTER OF GRAVITY IN VIABLE $F(R)$ THEORIES
§.§ Singularities
One of the main shortcomings of viable $f(R)$ theories of gravity is the occurrence of cosmological singularities, in particularly the appearance of a sudden singularity, where $\dot{H}\rightarrow\infty$ in a finite time $t_s$ (see <cit.>). This is a feature which can be easily analysed within the scalar-tensor framework of $f(R)$ gravity, where the action (<ref>) takes the form
S=∫d^4x √(-g)[ϕR-V(ϕ)+2ℒ_m] ,
by means of the relations
ϕ=f_R ; V(ϕ)=Rf_R-f(R) .
For the model (<ref>), the scalar field $\phi$ and its potential in terms of the Ricci scalar become
\begin{eqnarray}
\phi&=&1-b n\frac{(R/cH_0^2)^{n-1}}{\left[1+d(R/cH_0^2)^n\right]^2}\ ,\\
\label{1.10}
\end{eqnarray}
In general it is not possible to get the explicit expression of the scalar potential in terms of the scalar field $V=V(\phi)$ since the first expression in (<ref>) is not analytically invertible for a general $n$. Nevertheless, this is possible for the case $n=1$, such that the scalar potential yields
V(ϕ)=cH_0^2b+(1-ϕ)±2√(b(1-ϕ))/d .
Note that in this case the potential is not univocally defined, as depicted in Fig. <ref>. It is straightforward to check that the sudden singularity, where $R\rightarrow\infty$, occurs for
ϕ→1 , V→bc/dH_0^2 ,
since $\dot{H}\propto V'(\phi)$ and the first derivative of the potential $V'(\phi\rightarrow 1)\rightarrow \infty$.
Hence, in order to construct a consistent and smooth cosmological evolution for the $f(R)$ model (<ref>), the occurrence of such singularity has to be avoided.
Note that the two branches of the scalar potential, Fig. <ref>, contain different asymptotically stable points. While the upper branch ends at the singular point $\phi=1$, and any cosmological evolution located initially on that branch, the other branch ends in an asymptotically stable de Sitter evolution (see Ref. <cit.>). Therefore, depending upon the initial conditions and the model parameters values, the singularity may be avoided, as shown in the following Section.
Evolution of the scalar potential (<ref>) for $n=1$. The singular behaviour lies at $\phi=1$. The lower branch is singularity free whereas the upper branch leads inevitably towards the singularity. The potential corresponds to the free parameteres used in Fig. <ref>, $\{n=1,b=200\}$.
§.§ Attractive character
In this section we focus on finding inequalities which provide an upper bound for the positive contribution to the space-time geometry of the Raychaudhuri equation for timelike geodesics[The analysis for null geodesics is much simpler as shown in <cit.>.], rendering the gravitational interaction attractive. Let us express the Raychaudhuri equation for timelike geodesics as <cit.>
\begin{eqnarray}
\frac{\text{d}\theta}{\text{d}\tau}=-\frac{1}{3}\,\theta^2-\sigma_{\mu\nu}\sigma^{\mu\nu}+\omega_{\mu\nu}\omega^{\mu\nu}-R_{\mu\nu}\xi^\mu\xi^\nu\,,
\label{ray}
\end{eqnarray}
where $\theta$, $\sigma_{\mu\nu}$ and $\omega_{\mu\nu}$ are respectively the expansion, shear and rotation of the congruence of timelike geodesics generated by the tangent vector field $\xi^\mu$ and $\tau$ is an affine parameter. One of the standard interpretations of the Raychaudhuri equation is that, once the Strong Energy Condition (SEC) is assumed[Note that both dust matter and radiation satisfy the SEC. For a discussion about cases where this condition does not hold see <cit.>. In particular, a stress-energy tensor corresponding to a cosmological constant $\Lambda$ fluid does not fulfill the SEC.]
\begin{eqnarray}
T_{\mu\nu}\xi^\mu\xi^\nu\geq-\frac{1}{2}T,%\ \ \ \ \ \ \ \ \ \ \ \text{SEC}\,.
\label{SEC}
\end{eqnarray}
Provided that GR is considered as the underlying theory, the SEC immediately implies that $R_{\mu\nu}\xi^\mu\xi^\nu\geq 0$, which may be interpreted as a manifestation of the attractive character of gravity.
It therefore follows that the mean curvature <cit.> in every timelike direction defined by
\begin{eqnarray}
{\cal M}_{\xi}\,\equiv\,-R_{\mu\nu}\xi^\mu\xi^\nu
\label{Mean_curvature}
\end{eqnarray}
is negative or zero in GR provided that the SEC holds.
The utility of the Raychaudhuri equation in the singularity theorems is based on the following result: if one chooses a congruence of timelike geodesics whose tangent vector field is locally hypersurface-orthogonal, then one gets $\omega_{\mu\nu}=0$ for all the congruences. Since the term $\sigma_{\mu\nu}\sigma^{\mu\nu}$ is non-negative and whenever $R_{\mu\nu}\xi^\mu\xi^\nu \geq 0$ is assumed, then
\begin{eqnarray}
\frac{\text{d}\theta}{\text{d}\tau}+\frac{1}{3}\theta^{2}\leq 0\;
%which implies
\rightarrow\; \theta^{-1}(\tau) \geq \theta^{-1}_0 +\frac{1}{3}\tau\,.
\end{eqnarray}
This inequality
tells us that a congruence initially converging ($\theta_0\leq 0$) will converge to zero in a finite time.
Reverse reasoning backwards in time can be easily formulated.
Let us stress at this stage that the requirement for the previous reasoning to be true for any general theory of gravity does not need any energy condition to hold, but rather that $R_{\mu\nu}\xi^\mu\xi^\nu \geq 0$ for every non-spacelike vector.
In what follows, we focus on timelike geodesics, referring the reader to <cit.>, where details on null geodesics were presented. We also consider the aforementioned constraint in late-time cosmological scenarios, i.e., assuming a de Sitter phase and subdominant contributions from both radiation and dust. Thus, the Ricci scalar $R=R_0$ will be approximately constant for situations where we require cosmological expansion
of timelike geodesics in order to match observations.
Following the general results in <cit.>, one can prove that
\begin{eqnarray}
R_{\mu\nu}\xi^\mu\xi^\nu \geq \frac{f(R_0)-R_{0}f_R(R_{0})}{2(1+f_R(R_0))}\;,
\label{Ib}
\end{eqnarray}
where we have just considered Eqn. (<ref>) with constant scalar curvature and all standard matter sources - if any - to satisfy the SEC. Therefore, the r.h.s. of (<ref>) must be negative in order to allow $R_{\mu\nu}\xi^\mu\xi^\nu < 0$ or equivalently ${\cal M}_{\xi} > 0$. It follows that ${\cal M}_{\xi}$ must be bounded from above. Hence the necessary condition for timelike geodesics to diverge at late times becomes:
\begin{eqnarray}
\frac{f(R_0)-R_{0}f_R(R_{0})}{2(1+f_R(R_0))} < 0\,,
\end{eqnarray}
and provided that $1+f_R(R_0)>0$, we obtain
\begin{eqnarray}
f(R_0)-R_{0}f_R(R_{0}) < 0\,.
\label{lacondicion}
\end{eqnarray}
If we now consider equation (<ref>) in vacuum ($T=0$) for constant scalar curvature solutions, the value of $R_0$ satisfies
\begin{eqnarray}
\label{ec_curvatura_constante}
\end{eqnarray}
Although in general this algebraic equation cannot be solved analytically, some $f(R)$ models exist (depending upon the parameters of the model) for which a closed solution can be found. Thus rearranging terms in the equation (<ref>) one can prove that the equation above implies[Here $1-f_R(R_0)>0$ has been assumed in agreement with usual viability conditions on $f(R)$ theories.]
\begin{eqnarray}
R_0 > 0\,.
\label{Rmayor0}
\end{eqnarray}
Hence, a positive contribution to the Raychaudhuri equation from the space-time geometry ${\cal M}_{\xi}$
for every timelike direction is obtained provided that $R_{0}>0$. This condition will
constrain the parameters of different Hu-Sawicki models. As mentioned above constant curvatures $R_0$ usually cannot be determined analytically[For $n=1$, Eqn. (<ref>) can be exactly solved <cit.>.] from (<ref>) although numerical solutions do generally exist, as we shall illustrate for $n=2,3$ in the upcoming section.
In conclusion, the combination of the analyses described in Sections
<ref> and <ref> provide two complementary independent ways of constraining viable $f(R)$ models. We have applied those results to the Hu-Sawicki class of models for different exponents $n=2,3$ and summarise the results in Fig. <ref>.
This information can be then used in MCMC analyses in order to avoid regions in the parameter-space which we know possess singular points in their cosmological evolution or do not provide late-time accelerated expansion.
[justification=justified,singlelinecheck=false]Regions in the $b-d$ plane, for $n=2$ (upper panel) and $n=3$ (lower panel) containing
singular/regular sets of parameters for different values of $\Omega_{m}$, and regions with different signs of $R_0$. In both panels, the non-meshed zone represents $R_0>0$, and the grey regions represent entirely singular regions (regardless of the value of $\Omega_{m}$); for $n=2$ this corresponds to $d<0$, and for $n=3$ this corresponds to $b<0$. Other singular regions in the $b-d$ plane do depend upon the value of $\Omega_m$ and have been represented in different colours (see legends in the panels). Note that this analysis focuses on the past cosmological evolution $z\geq 0$, thus this does not ensure a whole regular condition for $n>1$.
For the case $n=2$, the closer $\Omega_{m}$ gets to its best-fit value $\Omega_m=0.27$ (see Section <ref>), the narrower the aforementioned upper singular parabolic region becomes. When $\Omega_{m}=0.27$, the phase space is completely regular for all values of $d>0$ and all values of $b \neq 0$. For the case $n=3$, there appears not to be any improvement as we get closer to the $\Omega_m$ best-fit value but the singular region located at $b>0$ grows with $\Omega_m$.
The best fit values found in Table <ref> lie in the blank regions for both $n=2,3$.
§ FITTING THE HU-SAWICKI MODEL WITH SNE I${\RM A}$
We implement a Markov Chain Monte Carlo (MCMC) routine to estimate the parameters of the Hu-Sawicki model, by fitting to the Union 2.1 supernovae data (see Ref. <cit.>), consisting of 557 sources. Using a Metropolis-Hastings algorithm, we sample from a three-dimensional parameter space $\{b,d,\Omega_{m}\}$, while holding $n$ fixed at three integer values of 1,2,3. However, the occurrence of singularities in a model's resulting expansion history makes any numerical analysis, such as a parameter optimisation routine, more complicated, as the statistics and posterior distributions may be compromised whenever encountering singular evolutions. We therefore attempt to manage this difficulty as follows.
§.§ Numerical detection of singularities
Given that singularities are expected within the parameter space, it is useful to determine the regions of parameter-space containing regular solutions, so that appropriate priors on the free parameters may be considered.
Since the integration of the cosmological equations is needed for the optimisation routine, the detection and avoidance of any singularities is a mandatory step at this stage.
The case $n=1$ is particularly simple as the appearance of singularities solely depends upon two free parameters $\{b,\Omega_m\}$, and the scalar potential is obtained exactly in (<ref>). Consequently the region of parameter-space leading to regular solutions can easily be found <cit.>. As pointed out above, we need to stay initially on the lower branch of the scalar potential Fig. <ref> in order to avoid the singularity, which is located at $\phi=1$ where $V(\phi=1)=\frac{bc}{d}H^2_0$. This leads to the condition:
V_ϕ=1<bc/dH^2_0 ,
Then, by imposing the initial conditions to match the model with $\Lambda$CDM at a particular redshift and using the expression (<ref>), we get the following constraint on $b$,
b<3(Ω_m-1)H_0^2/R_0,ΛCDM=Ω_m-1/[z_0 (z_0^2+3z_0+3)-3]Ω_m +4 ,
where $z_0$ is the initial redshift.
For $n>1$, we need to resort to numerical techniques in order to determine the singularity free regions in the parameter space. We proceed by testing a reasonably large grid of the sampling region, within a chosen redshift range. The dynamical system equations (<ref>) - (<ref>) are integrated from $z=10$ to the present era[As in <cit.>, we see that choosing $z0\geq 10$ provides good initial conditions, since for earlier times, the $\Lambda$CDM model is indistinguishable from the Hu-Sawicki model.
The solution at every point in the grid is examined to determine any singular behaviour. We present two-dimensional representations of this grid, showing the $b-d$ plane, for fixed values of $\Omega_{m}$ in Fig. <ref> for $n=2$ and $n=3$. According to this, for higher values of $n$ the singular regions in the phase space are more complicated. While the filled regions in the plot represent regions which contain the singularities in this range of parameter space, there may exist regular parameter sets for different values of $\Omega_m$ within those regions as well. Similarly, the white space gives singularity free regions that also depend on the value of $\Omega_m$. This analysis ensures an smooth cosmological evolution for $z\geq 0$ but is unable to ensure a future cosmological evolution in absence of singularities. However, note that many other dark energy models allowed by the observations contain future cosmological singularities <cit.>.
§.§ SNIa fit to the Hu-Sawicki model
In this section
we present the results of a Markov Chain Monte Carlo method performed to fit for the free parameters of the Hu-Sawicki model subject to the theoretical constraints presented above. The results obtained in the previous section aid in the avoidance of highly dense singular regions, as well as the interpretation of the MCMC chains. The regions for which the cosmological evolution does not guarantee expansion
excluded a priori in the calculations here, although once the maximum likelihood is obtained, we are able to determine whether the corresponding points in the parameter-phase space lie in the allowed regions, i.e., singularity-free and late-time expansion ones.
The observable to be compared with the catalog of Union2 is the apparent magnitude, which is defined as follows
\begin{eqnarray}
m^{th}(z;\Omega_m^0,z_0,x_{i})&=&{\bar M} (M,H_0) +\nonumber\\
&& 5\,{\rm log}_{10} \left[D_L (z;\Omega_m^0,z_0,x_{i})\right]
\label{SN2}
\end{eqnarray}
where $x_{i}$ are the free parameters of the model and ${\bar M}$ is the magnitude zero point offset, which is given by
\begin{eqnarray}
{\bar M} = M + 5\, {\rm log}_{10}\left[\frac{c\,H_0^{-1}}{\rm Mpc}\right] + 25\ .
\label{SN3}
\end{eqnarray}
Here $M$ is the absolute magnitude and $H_0$ is the Hubble parameter evaluated today, while $D_L (z;\Omega_m^0, z_0 ,x_i)$ is the corresponding free luminosity distance:
\begin{equation}
D_L^{} (z;\Omega_m^0,z_0,x_{i})= (1+z) \int_0^z {\rm d}z'\frac{H_0}{H(z';\Omega_m^0,z_0,x_{i})}\ .
\label{SN1}
\end{equation}
Then, for a particular set of the free parameters $\{\Omega_m^0,x_{i}\}$, the Hubble parameter
$H(z;\Omega_m^0,z_0,x_{i})$ is obtained by solving equations (<ref>) - (<ref>). Thus, the theoretical value of the apparent magnitude (<ref>) can be determined, and compared with the observational data from <cit.>, which provides the observed apparent magnitudes $m^{obs}(z)$ of the SN Ia with the corresponding redshifts $z$ and errors $\sigma_{m(z)}$. Then, the best fit is determined by studying the probability distribution
\begin{equation}
P({\bar M}, \Omega_m^0, w_0,z_0)= {\cal N} {\rm e}^{- \chi^2/2}\ ,
\label{SN4}
\end{equation}
where $\chi^2\equiv\chi^2({\bar M}, \Omega_m^0, z_0, x_i)$ and
\begin{eqnarray}
\chi^2 =\sum_{i=1}^{557} \frac{(m^{obs}(z_i) - m^{th}(z_i;{\bar M}, \Omega_m^0, z_0, x_i))^2} {\sigma_{m^{obs}(z_i)}^2}\,.
\label{SN5}
\end{eqnarray}
Here ${\cal N}$ is a normalisation factor. Those free parameters $\{{\bar \Omega}_m^0, {\bar z}_0, {\bar x}_i\}$ minimising the $\chi^2$ expression (<ref>) will correspond to what we call the best
fit. On the other hand, the parameter $\bar{M}$ can be minimised and dropped out of the $\chi^2$ expression. Details on such a process are provided in Appendix <ref>.
$n$ $b$ $b_{\rm best\, fit}$ $d$ $d_{\rm best\, fit}$
1 $347\pm{300}$ 745 - -
2 $825\pm 200 $ 1052 $303\pm{200}$ 49
3 $947\pm 300$ 1388 $3515\pm 500 $ 3675
$n$ $\Omega_{m}$ $\Omega_{m\,{\rm best\,fit}}$ $\chi_{min}^{2}$ $\chi_{red}^{2}$
1 $0.270 \pm 0.020$ 0.270 542.683 0.979
2 $0.272\pm 0.020 $ 0.270 542.683 0.981
3 $0.264\pm 0.018$ 0.270 542.689 0.981
$\Lambda$CDM $0.27 \pm 0.02$ 0.27 542.685 0.978
MCMC analysis results for the fitting of Hu-Sawicki model to Union 2 SNIa data. The free parameters $b$,$d$ and $\Omega_{m}$ are estimated, for each case where $n$ is fixed, $n=1,2,3$. We include the results for $\Lambda$CDM for comparison. Each free parameter is represented by two columns, the left showing the mean and $1\sigma$ of the resulting posterior, and the right showing its best fit value.
The best fit values lie in the white regions in Fig. <ref> for both $n=2,3$ exponents, therefore providing the appropriate cosmological expansion behaviour at late time.
The MCMC analysis for the Hu-Sawicki model was performed by fixing integer values of $n=1,2,3$, sampling for the posterior distributions of the remaining free parameters $b$, $d$ and $\Omega_{m}$.
For each of $n=1,2,3$, twenty chains were generated, comprising $2.5\times10^5$ sampled points in the respective parameter space. The obvious prior
$\Omega_{m} \in (0,1]$ was imposed. For the sake of simplicity, each parameter was sampled following a normal distribution centered at zero with standard deviations[Having initially no information about the scales of $b$ and $d$, the sampling distributions were chosen so as to scan the available phase space efficiently.
The relatively large values of $\sigma_{b}$ and $\sigma_{d}$ were settled upon in order
to optimise the computing time.
More conservative values for these quantities were tested and the results
did not significantly differ from those presented here.]
$\sigma_{b}=5$, $\sigma_{d}=5$ and $\sigma_{\Omega_{m}}=0.03$ respectively.
Results are depicted in Fig. <ref>.
Each chain was initialised at unique points in the phase space, and
for each Markov Chain, convergence of the matter density fraction of the universe today, i.e., $\Omega_{m}$ occurred fairly quickly. In fact, $\Omega_{m}$ is very well described by a Gaussian posterior distribution of all three values of $n$, with an error comparable to that of a similar analysis done for $\Lambda $CDM. Table <ref> summarises the results for each value of $n$. We include the best fit values for each parameter corresponding to each value of the exponent $n$, as well as the mean and 1-$\sigma$ standard deviation of their sample distribution. We find in all cases that the best fit values do in fact lie in the $R>0$ regions.
For the case $n=1$, the parameter space is 2-dimensional as $d$ factors out of the system entirely. In this simple scenario, the convergence of the $b$ parameter is remarkably bad (left panel in Fig. <ref>). We find, consistently for each Markov chain generated, that a range of
$b$ values minimising $\chi^{2}$ exists. The $\chi^{2}$ surface is extremely flat, and we find that the variation in the values of the $\chi^{2}$ is small ($\sigma_{\chi^{2}} = 1.470$).
When $n=2$, the parameter space is 3-dimensional. Once again, $\Omega_{m}$ converges quickly to $\Omega_{m}=0.27 \pm 0.020$, however, $b$ and $d$ show no acceptable convergence in general (mid panel Fig. <ref>). In both cases the standard deviations of the posterior distributions are very large. As can be seen from Table <ref>, the best fit values of $b$ and $d$ are not similar to their mean values. The variation in the $\chi^{2}$ values, $\sigma_{\chi^2}=1.530$, is small in this case as well, implying that a wide range of values for $b$ and $d$ perform similarly when fitting the supernovae data. It is therefore possible for the best fit value, which minimises the $\chi^{2}$, to be quite different to the mean of the posterior.
Finally, for $n=3$, where $\sigma_{\chi^{2}}=1.364$, it can be seen that the results are very similar to those of $n=2$. Whereas $\Omega_{m}$ successfully converges, $b$ and $d$ remain unconstrained (right panel Fig. <ref>). The standard deviations of these two free parameters are large, so that the values which minimise $\chi^{2}$ is not reflected in the statistics of the posteriors.
At this stage we must emphasise that although all the generated MCMC chains
gave identical results for $\Omega_{m}$, they provided inconsistent results for $b$ and $d$. The distributions of $b$ and $d$ were highly sensitive to the initial points of the various chains, which reiterates the fact that a wide range of values form part of an acceptable optimum region for the values of $b$ and $d$, some of which are not necessarily connected within the phase space. We have depicted the chain-dependence of the results for the $\{b, d\}$ parameters in Fig. <ref> showing the results of four different chains for the cases $n=2$ and $n=3$. As can be seen, $b$ and $d$ show no tendency to converge to a preferred state. We are led to conclude that supernovae data does not impose strong enough constraints on the free parameters of the Hu-Sawicki model.
§ CONCLUSIONS
In this paper, we have investigated, through a combination of theoretical and statistical tests, several issues which must be considered when trying to constrain the parameter space of viable $f(R)$ theories of gravity, and by extension any
extended theory of gravity. We focused our study on the Hu-Sawicki model, which is considered to provide a reasonable parameterisation of the required features of effective extensions of the Hilbert-Einstein gravitational Lagrangian which are consistent with the $\Lambda$CDM expansion history and astrophysical tests of gravity.
We first considered two theoretical constraints, which have been widely overlooked in previous literature, namely the appearance of singularities and upper bounds ensuring the cosmological expansion at late times. As discussed previously, these kinds of viable $f(R)$ models analysed in this manuscript contain a cosmological singularity, where the first derivative of the Hubble parameter diverges <cit.>. By analysing the phase space of this model, particularly in the scalar-tensor framework, we found that the singularity is actually an asymptotically stable point, which can be avoided by an appropriate choice of the free parameters together with convenient initial conditions <cit.>.
We then investigated the requirements needed to obtain a positive contribution in the space-time geometry term appearing in the Raychaudhuri equation for time-like geodesics. This upper bound for $f(R)$ models guarantees the non-attractive character of gravity at late-times on cosmological scales, i.e., the cosmological expansion by purely gravitational means. We paid special attention to the asymptotic case of (de Sitter) constant scalar curvature with the sole assumption being that the usual energy conditions for standard fluids hold. The Hu-Sawicki model proved to have free parameters capable of satisfying both constraints. For example, for the exponent $n=2$, when these two analysis were combined, we were able to exclude models with $d<0$ and large regions of parameter space with $b<0$. The necessary conditions for the free parameters which give rise to both singular-free and accelerated de Sitter regimes were presented in Fig. <ref>, where the singular regions are presented for several values of $\Omega_{m}$. For the case $n=3$, we found that we needed to exclude regions where $b<0$ and large regions where $d>0$. In this case, the larger the value of $\Omega_m$, the larger the singular region turns out to be.
Our aim in this paper was therefore to constrain the parameters space region that leads to both a smooth and regular Hubble evolution and late-time expansion, and then use these theoretical constraints to determine priors for the free parameters when fitting with Supernovae Ia data. Using this reduced parameter space, we then looked at what further constraints would be obtained when the expansion history of these models was compared to Supernovae Ia data, using an extensive Markov Chain Monte Carlo analysis. In order to do so, a full resolution of the cosmological background equations was performed using the dynamical system approach <cit.>.
Thus, for exponents $n=2, 3$ the best-fit values that were found for the free parameters lie in both the singularity-free and accelerated regions.
We also found that while the density parameter of matter $\Omega_{m}$ is well described by a Gaussian posterior distribution of the studied values of exponent $n$, the remaining free parameters $b$ and $d$ cannot be properly constrained by the sole use of supernovae data, with large intervals in the parameter space providing almost the same statistical significance. Consequently, for the studied exponents ($n=1,2,3$), we were not able to improve on what is obtained by the $\Lambda$CDM model. In other words, the supernovae analysis by itself remains a weak tool to constrain $f(R)$ models able to provide an explanation for the accelerated universe beyond the Concordance $\Lambda$CDM model. In fact, this weakness was illustrated by the fact that the best-fit statistical distributions for free parameters did depend upon the starting point of the various Monte Carlo chains as illustrated in Fig. <ref>, which gives the results of four different chains for the cases $n=2$ and $n=3$.
We are therefore led to the conclusion that while the theoretical analysis conducted (the avoidance of singularities in the cosmological expansion history and the non-attractive character at late times) can indeed be used as a powerful tool to constrain the parameter space of viable $f(R)$ models, when combined with observational constraints coming from supernovae catalogues, does not lead to a significant reduction in the parameter space consistent with a $\Lambda$CDM expansion history. Further observational data, for example large-scale structure surveys, the density contrast at different redshifts and the integrated Sachs-Wolfe effect will be needed in order to improve the exclusion maps we provided in the investigation. Work in this direction is currently in progress.
Constraints for the posterior distributions and contours for each free parameter, in each case of $n$, $n=1$ (left panel), $n=2$ (central panel) and $n=3$ (right panel).
For $n=1$, parameter $d$ vanishes from the system. We find $\Omega_{m}=0.27 $, and $b$ is unconstrained for a wide range of values. For the case $n=2$, again $\Omega_{m}=0.27$ and $b$ is unconstrained for a wide range of values.
For $n=3$, $\Omega_{m}=0.264$, whereas $b$ and $d$ remain unconstrained.
Above, we plot the results of four MCMC chains corresponding to the cases $n=2$ and $n=3$ to illustrate the lack of hard constraints on the best fit values for the parameters $b$ and $d$ in both cases.
§ ACKNOWLEDGMENTS
We would like to thank David Bacon for a comprehensive reading of the manuscript and for his useful comments.
A.d.l.C.D. acknowledges financial support from the University of Cape Town (UCT) Launching Grants programme and MINECO (Spain) projects
FIS2014-52837-P, FPA2014-53375-C2-1-P and Consolider-Ingenio MULTIDARK CSD2009-00064.
P. K. S. D. thanks the National Research Foundation (NRF) for financial support.
S.K. is grateful to the NRF and the Faculty of Science, University of Cape Town for financial support.
D.S.-G. acknowledges support from a postdoctoral fellowship Ref. SFRH/BPD/95939/2013 by Fundação para a Ciência e a Tecnologia (FCT, Portugal) and the support through the research grant UID/FIS/04434/2013 (FCT, Portugal). D.S.-G. also acknowledges the NRF for financial support.
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In this Appendix we give details on the process to find the apparent magnitude statistical minimum.
Indeed, by expanding (<ref>) in terms of $\bar{M}$ as
\begin{equation}
\chi^2 (\Omega_m^0, z_0, x_i)= A - 2 {\bar M} B + {\bar M}^2 C\ ,
\label{SN6}
\end{equation}
\begin{eqnarray}
A(\Omega_m^0, z_0, x_i)&=&\sum_{i=1}^{557} \frac{(m^{obs}(z_i) - m^{th}(z_i ; \Omega_m^0, z_0, x_i))|_{\bar{M}=0}^2}{\sigma_{m^{obs} (z_i)}^2} \,,
\label{SN7.1} \nonumber \\
B(\Omega_m^0, z_0, x_i)&=&\sum_{i=1}^{557} \frac{(m^{obs}(z_i) - m^{th}(z_i ; \Omega_m^0, z_0, x_i)|_{\bar{M}=0})}{\sigma_{m^{obs}(z_i)}^2} \,,
\label{SN7.2} \nonumber \\
C&=&\sum_{i=1}^{557}\frac{1}{\sigma_{m^{obs}(z_i)}^2 } \,.
\label{SN7.3}
\end{eqnarray}
The minimum of equation (<ref>) is located at ${\bar M}={B}/{C}$, such that the $\chi^2$ turns out
\begin{eqnarray}
{\tilde\chi}^2(\Omega_m^0, z_0, x_i)=A(\Omega_m^0, z_0, x_i)- \frac{B(\Omega_m^0, z_0, x_i)^2}{C}
\label{SN8}
\end{eqnarray}
Hence, minimising ${\tilde\chi}^2(\Omega_m^0, z_0, x_i)$ independently of ${\bar M}$, is enough to find the best fit since $\chi_{min}^2={\tilde\chi}_{min}^2$.
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PRESENTED AT
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1511.00011
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reionization – intergalactic medium – galaxies: formation, high redshift, luminosity function – Local Group – radiative transfer – methods: numerical
§ INTRODUCTION
The epoch of reionization (hereafter, "EoR") resulted from
the escape of ionizing radiation from the first generation of
star-forming galaxies into the otherwise cold, neutral gas of the
primordial intergalactic medium ("IGM"). This radiation created
intergalactic H II regions of ever-increasing size, surrounding the
galaxies that created them, leading eventually to the complete overlap
of neighboring H II regions and the end of the EoR within the first
billion years after the big bang. The growth and geometry of the
H II regions reflected that of the underlying galaxies as the radiation
sources and their spatial clustering, and that of the
density fluctuations in the intergalactic gas as the absorbing medium.
The story of cosmic reionization is inseparable, therefore, from
that of the emergence of galaxies and large-scale structure in the
universe. Recently, the observational frontier has begun to push
the look-back-time horizon accessible to direct observation back into
this period. High-redshift galaxies and quasars have been observed,
for example, which constrain the evolution of the IGM opacity to
H Lyman alpha resonance scattering, the mean ionizing flux density,
the UV luminosity function of galaxies, and the cosmic star formation
history, while observations of the cosmic microwave background ("CMB")
constrain the mean elecron scattering optical depth integrated
thru the entire IGM over time, back to the epoch of recombination, and
give some information about the evolution of the mean ionized fraction.
Observational probes of the large-scale structure of reionization and
its evolving "patchiness" are just beginning. The cosmic 21cm background
from the evolving patchwork of intergalactic H I regions, not yet reionized,
is a prime example, which is an important science-driver for the development
of a new generation of low-frequency radio telescope arrays, such as LOFAR, MWA, PAPER, and SKA.
The EoR is also thought to be important for its impact on galaxy formation. It has been suggested that the rising
intergalactic UV radiation field is responsible for suppressing the star formation of low-mass galaxies,
by removing gas from the smallest galaxies — minihalos — by photoevaporation <cit.> and through the suppression of gas infall onto low-mass galaxies above the minihalo mass range, by photoheating the intergalactic gas and raising its pressure <cit.>, affecting their star formation efficiency.
This process could provide a credible solution to the “missing satellites problem” <cit.>, by inhibiting star formation in low mass galaxies at early times <cit.>. In this framework, a number of semi-analytical models (hereafter SAMs) have been shown to reproduce well the satellite population of the Milky Way (hereafter MW), such as <cit.>.
They suggest that the ultra-faint dwarf galaxies (hereafter UFDs) discovered by the SDSS <cit.> are effectively reionisation fossils, living in dark matter sub-haloes of about $10^{6-9} \Msun$. Deep HST observations have recently made it possible to start testing this idea. While this is currently a matter of debate, it seems that at least the low-mass satellites of the MW and M31 may have star formation histories compatible with a very early suppression <cit.>, e.g. by reionization.
In this context, we are led to the exciting possibility that low-mass satellites in the Local Group can be used as probes of the local reionization process.
While the reionization of the Universe is often considered to be complete at $z=6$ <cit.>, further quasar Lyman $\alpha$ observations have shown that extended opaque hydrogen troughs may still exist below this redshift, highlighting the patchiness of the process and suggesting that the reionization may actually still be ongoing at $z=6$ and end as low as $z\sim5$ <cit.>.
Given this observational evidence, one can only expect that the reionization of the Local Group must have been very different from a uniform or instantaneous event.
Indeed, <cit.> described two distinct reionization scenarios for the Local Group, internal (and slow, driven by sources within the Local Group) or external and fast, driven by a massive, rare object such as Virgo. Many more scenarios can be conceived, depending on the physics considered (e.g. quasars, X-ray binaries), and it would be desirable, as a long term goal, to establish which properties of the local satellites population could allow us to discriminate between these scenarios.
Already, <cit.> showed that the structure of the UV background during reionization has a strong impact on the properties of the satellite population of galaxies. In particular, they showed that an internally-driven reionization led to significant changes in the radial distribution of the satellites of the MW. This prediction was then further supported by using more refined models <cit.>, built upon high resolution zoom simulations of the formation of the Local Group performed by the CLUES project[Constrained Local UniversE Simulations, <http://www.clues-project.org/index.html>]. However, these studies suffered from several limitations.
First of all, these studies were performed in the so-called “post-processing” framework, and therefore only crudely account for stellar feedback (radiative and SNe). Furthermore, since the gas density field is frozen in the post-processing paradigm, the effect of radiation on the distribution of gas in the IGM (smoothing of gas overdensities due to photo-heating) can not be accounted for either, while it has been shown to have a significant impact on photon consumption <cit.>. Second, the gas distribution is only known in the zoom region, which is only a few Mpc wide, and mainly contains the progenitors of the MW, M31 and M33, and the IGM in-between. Beyond this region, the box is populated with low resolution pure dark matter particles. This dual description makes it very challenging to describe external (i.e. pure dark matter, low resolution) and internal (i.e. zoom region, including gas particles) reionization sources in a homogeneous, consistent way.
Our ultimate challenge for modelling reionization, then, is to be able
to simulate the coupled multi-scale problem of global reionization and individual
galaxy formation, with gas and gravitational dynamics and radiative transfer,
simultaneously. And to predict the relic impact of this global reionization on the
universe today, to make comparisons possible with the observationally-accessible
nearby universe, including the Local Group and its satellites, it is further necessary
to start from initial conditions preselected to produce the observed galaxies and
large-scale structure of the local universe. For this simulation to characterize the
evolving "patchiness" of reionization in a statistically meaningful way requires a
a comoving simulation volume as large as $\sim (100$ Mpc$)^3$ [e.g. <cit.>].
Accounting for the millions of galaxies in this volume over the full mass range of
galactic haloes which are thought to contribute most significantly to reionization,
the so-called "atomic-cooling haloes" (henceforth, "ACHs"), those with virial temperatures
above $\sim 10^4$ K (corresponding to halo masses above $\sim 10^8$ ) while modelling
the impact of reionization on these individual galaxies and the IGM, requires a physical resolution of a few kpc's over the entire volume.
To meet this challenge and
satisfy all these requirements, we
have developed a new, hybrid CPU-GPU, fully-coupled, cosmological radiation-hydrodynamics-
gravity code, RAMSES-CUDATON, capable of simulating the EOR in a comoving volume 91 Mpc
on a side, with $4096^3$ dark matter particles and a cubic-lattice of $4096^3$ cells for the
gas and radiation field. Our simulation self-consistently models BOTH global reionization
AND the formation and reionization of the Local Group, by starting from a "constrained
realization" of initial conditions for the local universe which reproduces all the familiar
features of the local universe in a volume centered on the Local Group, such as the Milky
Way, M31, and the Fornax and Virgo clusters — i.e. CLUES initial conditions. We call this
simulation "CoDa", for Cosmic Dawn.
The main goal of this first paper is to introduce the simulation and compare the results with current observational constraints on the EOR, such as the evolution of the cosmic neutral fraction, the cosmic ionizing flux density, the cosmic star formation rate, the Thomson optical depth measured by cosmic microwave background experiments and high redshift UV luminosity functions (Sec. <ref>).
It is laid out as follows: first we describe the code and the simulation setup (Sec. <ref>). We then proceed to our results and compare to available observational constraints of the EoR (Sec. <ref>), and finish with a short summary.
§ METHODOLOGY
The Cosmic Dawn simulation uses the fully coupled radiation hydrodynamics code RAMSES-CUDATON. This section describes the principles of the code and its deployment. For quick reference, the parameters of the simulation are summarized in Tab. <ref>.
Temperature distribution in a 45 x 45 x 0.03 slice of the simulation at redshift $=6.15$. Orange regions show photo-heated, ionized material, while the cold, still neutral medium appears in blue. The slice is situated in the supergalactic YZ plane with its origin shifted to the center on the Local Group progenitor at $=7$. The circles of decreasing radius indicate the positions and approximate sizes of the progenitors of, respectively, Virgo, Fornax, the Milky Way and M31.
2cCosmology (WMAP5+BAO+SN)
Dark energy density $\Omega_{\Lambda}$ 0.721
Matter density $\Omega_{\rm{m}}$ 0.279
Baryonic matter density $\Omega_{\rm{b}}$ 0.046
Hubble constant $h={\rm H}_0/(100 \, {\rm km/s})$ 0.70
Power spectrum
$\,$ Normalization $\sigma_8$ 0.817
$\,$ Index $n$ 0.96
Number of nodes (GPUs) 8192 (8192)
Grid size $4096^{3}$
Box size $\Lbox$ 91 Mpc
Grid cells per node 128x256x256
DM particle number $N_{DM}$ $4096^3$
DM particle mass $M_{DM}$ 3.49 x $10^5$
Initial redshift $z_{start}$ 150
End redshift $z_{end}$ 4.23
2cStar formation
Density threshold $\delta_{\star}$ $50 \, \langle \rho \rangle$
Temperature threshold $T_{\star}$ $2 \times 10^4$K
Efficiency $\epsilon_{\star}$ $10^{-2}$
Stellar particle mass (post-SN) $M_{\star}$ 3194
Massive star lifetime $t_{\star}$ 10 Myr
2cSupernova feedback
Mass fraction $\eta_{SN}$ 10%
Energy $E_{SN}$ $10^{51}$ erg
Stellar ionizing emissivity 1.824$\times 10^{47}$ ph/s/
Stellar particle escape fraction $\fesc$ 0.5
Effective photon energy 29.61 eV
Effective H cross-section $\sigma_E$ 1.097 x $10^{-22}$m$^2$
Speed of light $c$ 299 792 458 m/s
Cosmic Dawn simulation parameters summary
§.§ RAMSES-CUDATON
§.§.§ RAMSES
RAMSES is a code for simulating the formation of large scale structures, galaxy formation and self-gravitating hydrodynamics in general <cit.>. Collisionless N-body dynamics are solved via a particle-mesh integrator. The adaptive mesh refinement was turned off in this run (see Sec. <ref> for more explanations on this).
Gas dynamics is modeled using a second-order unsplit
Godunov scheme <cit.> based on the HLLC Riemann solver <cit.>.
We assume a perfect gas Equation of State (hereafter EoS) with $\gamma = 5/3$.
We consider star
formation using a phenomenological approach. In each
cell with gas density larger than a gas overdensity $\delta_{\star}=50$, we spawn new star particles
at a rate given by
\begin{equation}
\dot{\rho_{\star}} = \epsilon_{\star} \frac{\rho_{gas}}{t_{ff}} \,\, {\rm with} \, \, t_{ff}=\sqrt{\frac{3 \pi}{32 G \rho}}
\end{equation}
where $t_{ff}$ is the free-fall time of the gaseous component and
$\epsilon {\star}$= 0.01 is the star formation efficiency.
We also require the cell temperature to be lower than $T_{\star}=2 \times 10^4$ K in order to form stars: above this temperature, the gas is fully ionized, suffers inefficient hydrogen cooling and can not form stars. This temperature criterion is widely used in hydrodynamical simulations <cit.>. A common variation in simulations of the EoR, producing a similar effect, is to use a threshold in ionized fraction, as in <cit.>.
The star particle mass at birth depends on the cell gas density, but is always a multiple of a fixed elementary mass $M_{\star}^{birth}$, chosen to be a small fraction, $\sim 5$% of the baryonic mass resolution. In this framework, with the box size and resolution of CoDa (see Sec. <ref>), we have $M_{\star}^{birth}=3549$ . This mass is small enough to sample adequately the star formation history of even low mass galaxies, and still large enough to mitigate stochastic variations in the number of massive stars per star particle.
For each star particle, we assume that $\eta_{SN}$=10% of its mass is in the form of massive stars which will go supernova after a lifetime $t_{\star}=10$ Myr, leaving no remnant. We consider a supernova energy $E_{SN}=10^{51}$ erg per 10 of progenitor, and this feedback was implemented in the RAMSES code using the kinetic feedback of <cit.>. After the massive stars have exploded, the remaining low mass stellar population is represented by a long-lived stellar particle of mass
\begin{equation}
M_{\star}=(1-\eta_{SN})M_{\star}^{birth}=3194 M_{\odot} \, .
\end{equation}
No chemical enrichment was implemented: indeed, extrapolating the results of <cit.> to $z=6-10$ yields very low metallicities for the gas ($Z/\Zsun = 1/100 - 1/1000$), in a similar range to the simulations of <cit.>. At these metallicities, the cooling rates are still rather close to those for metal-free gas in most of the temperature range <cit.> and therefore we do not expect chemical enrichment to make a dramatic impact on our results.
In cosmological simulations, it is customary to rely on subgrid models,
providing an effective EoS that captures the basic turbulent and
thermal properties of the gas in turbulent, multiphase, centrifugally supported disks. Models with various degrees of
complexity have been proposed in the literature, for instance using a polytropic EoS <cit.>.
We refrain from using this approach, because these subgrid models account for star formation and radiation from massive stars in star forming complexes, which are already explicitly accounted for in our feedback models. Therefore we set the RAMSES parameters so that the polytropic EoS does never kick in.
To summarize, we used for this simulation rather standard galaxy formation recipes, which have proven quite successful in reproducing various properties of galaxy evolution <cit.>. The main novelty of this study is the inclusion of the coupling between the hydrodynamics and the radiation produced by the stars, treated by our radiative transfer module ATON, described in Sec. <ref>.
At the galactic scale, since galaxies can be smaller than one CoDa cell, their ISM is not resolved. However, the goal of our star formation model is to produce sources with suitable number density and ionizing photon output to reionize the box in a reasonable time, and we must calibrate it to do so.
The overall star formation efficiency and feedback
effect that result from our modelling are subject to adjustment of the
adopted efficiency parameters, $\delta_\star$ and $\epsilon_\star$. While the previous
experience of simulating galaxy formation with such prescriptions is a useful
guide, there is some additional tuning required to simulate the EoR. For a
self-consistent simulation of the EoR, these local parameters for the
internal efficiency of individual galaxies should be chosen so as to achieve
whatever global efficiency over time is required to make the simulated EoR
satisfy the known observational constraints on the EoR. We discuss some of
these global EoR constraints in more detail in Sec. <ref>. Unfortunately, it
is not possible to perform multiple runs of the size and resolution of the CoDa
simulation, each with different values for these efficiency parameters, since the
computational cost is prohibitive. Each such simulation would have to run all
the way to the end of reionization, and one cannot substitute for this by doing
coarse-grained, lower-resolution versions of this large volume to reduce the
computational cost enough per simulation, since lower resolution simulations do not
yield the same outcome as the CoDa simulation with its higher resolution.
In practice, we solve this problem, instead, by performing a large suite of smaller-box
simulations (e.g. as small as 4 on a side) with the same particle mass and
grid resolution as CoDa but much smaller volumes, since each of these can be run on
the same computer as CoDa but require many fewer CPU-GPU hours. In this way, we were able
to adjust the efficiency parameters for CoDa in advance, to best guess the outcome of the
much larger-volume simulation.
There are limitations that are inevitable in this procedure, however.
For example, small boxes tend to reionize much more quickly
(i.e. over a more narrow redshift range) than do large-boxes and to have fewer
rare haloes, thereby delaying the first appearance of sources. Periodic boundary
conditions for a small box also tend to make its reionization history differ from that
in a large box, since, e.g., radiation from sources inside the box that escapes from
it must re-enter that box, but there is no realistic accounting of more distant
sources. In addition, cosmic variance makes it difficult to guarantee that the outcome
of a random realization of Gaussian-random-noise initial conditions in such a small
volume predicts the outcome of a different realization in a much larger volume.
current observational constraints on the global EoR are still highly uncertain and
continue to change with time as more data becomes available, and these do not,
themselves, exclude the possibility that the reionization of the local universe
differed in some way from that of the universe-at-large.
§.§.§ ATON
ATON is a UV continuum radiative transfer code, which relies on a moment-based description of
the radiative transfer equations and uses the M1 closure relation <cit.>. It tracks the out-of-equilibrium ionisations
and cooling processes involving atomic hydrogen <cit.>. Radiative quantities (energy
density, flux and pressure) are described on a fixed grid and evolved
according to an explicit scheme under the constraint of a Courant-Friedrich-Lewy condition (hereafter CFL), i.e. timesteps must be no larger than some fraction of a cell-crossing time for a signal travelling at the characteristic speed for radiative transport. In the case of interest here, involving ionization fronts in the low-density IGM, this speed can approach the speed of light.
Each stellar particle is considered to radiate for one massive star lifetime $t_{\star}=10$ Myr, after which the massive stars die (triggering a supernova explosion) and the particle becomes UV-dark. We used a mono-frequency treatment of the
radiation with an effective frequency of 29.61 eV for a $10^5$ K black body
spectrum as in <cit.>. This corresponds to a $\sim 100 $pop III star, although the mass scale of such primordial stars is currently very much debated (cf the introduction of <cit.> for a summary).
Finally, we assume each star particle to have an intrinsic emissivity of 4800 ionizing photons/Myr per stellar baryon (i.e. $1.824\times 10^{47}$ photons/s/), as in <cit.>, although the latter used a 50 000 K spectrum. Here the exact temperature of the stellar sources is actually of little importance: this temperature affects the effective ionizing photon frequency, which sets the effective cross-section of the photo-ionization reaction $\sigma_E$, and therefore the penetration of UV photons. In such EoR simulations, all UV, H-ionizing photons are consumed within a few cells of the ionization front (hereafter I-front), and variations of the UV photons' energy mostly affects the thickness of the I-front, but not its average position. Therefore, the emissivity of the stellar sources turns out to be more important than the effective frequency. Using a $50 000 $K black body spectrum would yield results rather similar to the $10^5$K we used here, provided the emissivity in ionizing photon number is kept constant.
Because of the relatively high spatial resolution of the CoDa simulation for an EoR simulation, we do not make any correction in terms of a clumping factor, to account for small-scale inhomogeneity in the gas density that was unresolved by the grid spacing, as was done for the largest boxes of <cit.>.
A price which must be paid for such a high spatial resolution, however, is that the CFL
timestep limiter of the light-crossing time per cell is exceedingly small and would
normally make the number of timesteps required to span the evolution of the system
over a given physical time prohibitively large if all of the gas dynamical quantities
had to be advanced with this same small timestep. The sound-crossing and gravity
time-scales are orders of magnitude larger than this radiation transport time,
however, so there is a great mismatch between the number of timesteps required to account
accurately for mass motions and that required to evolve the radiation field and ionization
state of the gas. We have solved this computational problem without sacrificing accuracy
by separately advancing the radiative transfer and ionization balance rate equations
on GPUs while simultaneously advancing those for the gas and gravitational dynamics
on the CPUs. By programming those GPUs so that each performs hundreds of these
smaller radiative transfer timesteps in the same wall-clock time as it takes the CPU
to perform one dynamical step, we are able to overcome this computational barrier.
ATON has been ported on multi-GPU architecture, where each GPU handles a Cartesian sub-domain and communications are dealt with using the MPI protocol <cit.>. By achieving an x80 acceleration factor compared to CPUs, the CFL
condition is satisfied at high resolution within short wall-clock computing
times. As a consequence, no reduced speed of light approximation is required.
We then have to determine an escape fraction for our stellar particles, i.e. the fraction of UV radiation able to escape the stellar birth cloud represented by the particle, into the interstellar medium (hereafter ISM) of the cell it belongs to, which is different from the escape fraction from the entire galaxy. Here we discuss two potential sources of absorption in the ISM: Hydrogen, and dust. We will therefore consider the two corresponding escape fractions, $\fesc^{H}$ and $\fesc^{dust}$ (we use the ${}_{,\star}$ subscript to remind the reader at all times that this escape fraction applies to a stellar particle and its cell and not to a galaxy and the IGM). Their product gives the stellar particle escape fraction $\fesc=\fesc^{H} \times \fesc^{dust}$. While there is some guidance (although somehow controversial) in the literature for the choice of the escape fraction, we can consider two limiting cases:
* the large scale, low spatial resolution regime, where the radiative transfer grid cells size is larger than the galaxies, e.g. dx$=0.44$ as in <cit.>. This regime prescribes the use of a “galactic escape fraction”, which accounts for the ionizing photons lost to the unresolved interstellar medium of each galaxy. In this case, $\fesc$ can span a wide range of values, from almost 0 to about $40\%$, as shown by high resolution simulations <cit.>.
* the very high resolution regime, which is closer to resolving the interstellar medium and star forming molecular clouds, such as <cit.>, where the cell can be as small as dx$\sim$ 1 pc in the most refined regions. In this case the totality of the stellar photons produced reaches the interstellar medium. Therefore the stellar particle escape fraction $\fesc=1$ although the galactic escape fraction can be much smaller.
In CoDa, as galaxies form, their progenitor dark matter haloes span volumes
from several to hundreds of grid cells across. Even after they are fully collapsed,
these haloes are still typically a few to a few tens of cells across. As a result,
the effects of internal radiative transfer of UV photons through the interstellar
hydrogen are accounted for, although in a rather coarse way. Hence, we set $\fesc^{H}=1$, so as not to over-count the attenuation caused by the interstellar hydrogen gas inside the galaxy.
Dust may also play an important role: according to <cit.>, dust optical depths as high as $0.8$ in the UV could be common within galaxies by the end of the EoR, which amounts to a throughput of about $\fesc^{dust}=0.5$ of the UV flux. With these assumptions, we obtain a stellar particle escape fraction of $\fesc=0.5$.
Finally, we neglect any possible Active Galactic Nuclei (hereafter AGN) phase of our galaxies. Such sources could already be in place in rare massive proto-clusters during reionization <cit.>. They are very rare and thought to be minor contributors to the cosmic budget of hydrogen-ionizing photons <cit.>, although they could be important for explaining the line of sight variations of the properties of the Ly $\alpha$ forest just after reionization <cit.>.
§.§.§ Radiation-hydrodynamics coupling
We developed an interface that enables data exchange between RAMSES and ATON on the fly, leading effectively to a coupling between dynamics (handled by RAMSES) and radiation (handled by ATON).
At the end of a dynamical time-step, RAMSES sends to ATON the gas density, temperature and ionized fraction. Radiative transfer is then performed by ATON using these inputs, via sub-cycling of 100s to 1000s of radiative sub-steps. Once this sub-cycling is completed, ATON sends the temperature and ionized fraction back to RAMSES.
However, CUDATON (the GPU port of ATON) can only handle regular Eulerian grids. For this reason, AMR is not used for this simulation, and RAMSES is effectively used as a hydro-PM code. Furthermore, CUDATON assumes a Cartesian domain decomposition for the MPI-parallelism when similar domains are treated by each GPU. Meanwhile, RAMSES relies on a space-filling curve decomposition. As a consequence, data must also be reorganized at each transfer between the 2 applications.
This coupling method is similar to RAMSES-RT <cit.> but with one photon group only and no AMR. RAMSES-RT has been extensively tested and passes the test suite of <cit.>. RAMSES-CUDATON was similarly tested, as described in <cit.>[<http://www.physikstudium.uzh.ch/fileadmin/physikstudium/Masterarbeiten/Stranex_2010.pdf>].
As expected due to the similarity of the codes, RAMSES-CUDATON performs similarly well with these tests:
various instances of Strömgren spheres and the photo-evaporation of a gas cloud, the hallmark of radiation-hydrodynamics coupling in galaxy formation.
One of the great advantages of RAMSES-CUDATON is its ability to run with the true speed of light, which stems from the combination of a moment-based method with GPU acceleration. This is quite unique among the handful of existing radiation-hydrodynamics codes. Some of them, mostly those involving ray-based photon propagation schemes, use infinite speed of light. Others, in particular those using moment-based methods such as ATON for the RT (e.g. RAMSES-RT), or <cit.>, use the reduced speed of light approximation (as low as 1/100 of the real speed of light) in order to reduce the computational cost of the simulation. This is because in such a framework, the time-step of the code is set by the fastest physical process (gravitational, hydrodynamical and radiative). Since the speed of light is typically 100 to 1000 times faster than the speed of sound or bulk matter motions in cosmological simulations, the time-step must be about 100's of times shorter and therefore the simulation will be 100's of times slower than its pure hydrodynamical counterpart. While this is probably a good enough approximation in the dense interstellar medium of galaxies, it is not valid in the low density intergalactic medium, i.e. most of the volume of a cosmological simulation as we perform in the present paper<cit.>. Not only would the speed of ionization fronts in the IGM be misrepresented by the reduced speed of light approximation, but long-range feedback effects, like the impact of one massive object or a distant cluster of objects on the properties of low-mass galaxies may be impossible to capture in such a "slow light" framework.
In RAMSES-CUDATON this problem is alleviated thanks to the GPU optimization. The x80 boost almost cancels out the added cost of the RT, and allows us to work efficiently with the real speed of light.
§.§ Simulation setup
§.§.§ Initial conditions
The initial conditions (hereafter ICs) were produced by the CLUES project (Constrained simulations of the local universe), assuming a WMAP5 cosmology <cit.>, i.e. $\Omega_{\rm{m}}=0.279$, $\Omega_{\rm{b}}=0.046$, $\Omega_{\Lambda}=0.721$ and $h=0.7$. A power spectrum with a normalization of $\sigma_8=0.817$ and $n=0.96$ slope was used.
The comoving box is 91 (i.e.64$h^{-1}$) Mpc on a side, with 4096$^3$ dark matter particles on 4096$^3$ cells and with the same cubic-lattice of 4096$^3$ cells for the gas and radiation properties. The mass of each dark matter particle is then $3.49$ x $10^5$ . The CoDa initial conditions have the average universal density for the chosen cosmology.
These ICs are tailored so as to reproduce the structure of the local universe at z=0, using constrained realizations of a Gaussian random field, in the spirit of <cit.>.
In essence, the method takes as its input, galaxy observations of the local universe today: galaxy catalogs with distances and peculiar velocities, galaxy group catalogs, such as the MARKIII <cit.>, SBF <cit.> and the Karachentsev catalog <cit.>. These are used to infer the z $=$ 0 dark matter distribution. Using the Hoffman-Ribak algorithm <cit.>, a set of ICs (particle positions and velocities) at high redshift are produced, which will evolve into this present matter distribution over a Hubble time. This technique reproduces the large scale structure around the LG, although with some scatter, whereas small scales below 1 are essentially random. In order to make sure that a realistic LG is obtained, a second step was performed.
A large number of similarly constrained realizations of the initial conditions, of order 200, was produced and a selection made for the one which best fit the LG and its relation to other known features of the local universe at z = 0, with masses and locations of objects like the LG and its largest galaxies, the MW and M31, and a cluster such as Virgo. The best realization is then enriched with higher order random k-modes to an arbitrary resolution, limited in practice by computer memory. The resulting ICs therefore contain:
* constrained low k-modes, driving the emergence of the large scale structures and galaxy clusters,
* unconstrained but “chosen” intermediate k-modes, picked from a sample of many realizations to produce a LG,
* fully random high k-modes, added by the resolution enhancement procedure.
This procedure is detailed in <cit.>, and it produced the original CLUES ICs up to a resolution of 2048$^3$ in the full box. Here we have used Ginnungagap[<https://github.com/ginnungagapgroup/ginnungagap>] to increase further the resolution of the ICs up to that required by CoDa, i.e. 4096$^3$.
A CoDa temperature map at z=6.15 in the supergalactic YZ plane is shown in Fig. <ref>. The white circles of decreasing radius denote the approximate locations and sizes of the progenitors of Virgo and Fornax galaxy clusters, M31 and the MW. Massive objects such as galaxy cluster progenitors appear as large bubbles of photo-heated gas in the temperature maps. Two galaxy group progenitors also appear on the map between the Local Group and Virgo, reminiscent of the CenA and M81 groups, as described for instance in <cit.>. The exact occurence, position and growth of these particular groups are not strongly constrained in our ICs generation procedure, but such groups are statistically expected to emerge in the filament that harbors the LG and that points to the Virgo cluster <cit.>. The impact of such groups on the formation and evolution of the LG could be quite significant and therefore their existence in the simulation is one of the virtues of using such tailored initial conditions.
Because of its location with respect to Fornax and Virgo, the LG shown in Fig. <ref> can be considered as “the” LG of the box. While the fairly large size of the box is a requirement in order to reproduce well enough the local Universe without artifacts due to the periodic boundary conditions, and properly model reionization sources external to the Local Group, it has a third benefit: it will allow us find many galaxy pairs similar to the Local Group in mass ratios and separation through the box. This should therefore allow us to study “what if” scenarios, where analogs of the MW-M31 system would have evolved in a different environment.
The baryonic ICs were generated assuming a uniform temperature and the gas and dark matter are assumed to have identical velocity fields. The startup ionized fraction was set to the freeze-out value at $z=150$. They are computed following standard recipes such as in RECFAST <cit.>.
§.§.§ Code deployment
The CoDa simulation was performed on the Titan supercomputer at Oak Ridge National Laboratory.
The computational domain spans a comoving cube 91 Mpc on a side, sampled on a fixed grid of 4096$^3$ cells. The code was deployed on 8192 cores, with each core coupled to 1 GPU, therefore requiring 8192 Titan nodes (1 GPU per node). Each node hosted one MPI process which managed a 128x256x256 volume.
§.§ Run and data management
The run was performed from August to December 2013, and took about 10 days wall clock, using a total of about 2 million node hours, i.e. 60 million hours according to the Titan charging policy (1 node hour = 30 core hours). A total of 138 snapshots were written, every 10 Myr, from z=150 down to z=4.23, each one of size $\sim$16 TBytes, summing up to more than 2 PBytes of data in total.
Since it is not possible to save such a large amount of data in long-term storage, we decided to keep the following reduced data products:
* Full-box, low resolution (2048$^3$ grid) versions of the gas field (mass and volume-weighted temperature, volume-weighted pressure, density, velocities, ionized fraction), ionizing flux density, and of the dark matter density field, plus a list of all the star particles in the box, with their mass, coordinates and ages.
* Cutouts: for a set of 495 cubic subvolumes of 4 h$^{-1}$ Mpc on a side, centered on regions of interest, the original, full-resolution data of the simulation, each with 256$^3$ cells, was saved. The coordinates of these cutout regions were pre-computed from a pure N-body simulation (no gas dynamics) by GADGET, with 2048$^3$ particles, using the same CLUEs initial conditions, but coarsened for this lower resolution (i.e. instead of CoDa's $4096^3$ particles and cells), which we ran down to z $=$ 0, to identify interesting regions in the present-day local universe. We will refer to this dark matter only simulation as DM2048. Each cutout contains all the gas quantities (same as full-box), the ionizing flux density, and the dark matter particles, at maximum resolution. The regions of interest we picked include the LG as determined by the CLUES constrained initial conditions. They also include 62 LG analogs, i.e. pairs of galaxies with masses, separations and relative velocities compatible with the present-day MW-M31 system. They will be used in the future as examples of LGs forming in different environments. Each of these systems require 2 cutouts, one for the MW and one for M31. Finally, we added 399 regions following halo progenitors of various masses, from large galaxies to dwarfs, embedded in different environments, from cluster and galaxy groups to voids.
* Full box halo catalogs (see Sec. <ref>) containing the number of particles, position and velocity of each halo, as well as the list of dark matter particles it contains (ids and positions).
This strategy allowed us to bring down the data volume to a more manageable $\sim 200$ TBytes, while still allowing us to achieve our scientific goals.
§.§ Processing: friends-of-friends halo catalogs
We used the massively parallel friends-of-friends (hereafter FoF) halo finder of <cit.> with a standard linking length of b=0.2 to detect dark matter haloes in the CoDa simulation. They are reliably detected down to $\sim 10^8$ . More details on the resulting CoDa mass functions can be found in Appendix <ref>.
§.§ Online data publication
We plan to make a subset of the data and higher level products publicly available through the cosmosim database hosted by Leibnitz Institut für Astrophysik Potsdam[<https://www.cosmosim.org/cms/simulations/cosmic-dawn/>] and the VizieR database at CDS Strasbourg[<http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=VI/146>].
§ RESULTS
In this section we show that the simulation effectively captures the main physical processes we intended to describe, and show that the global behaviour of the simulation is correct. We emphasize the radiative-hydrodynamic processes, since it is the main novelty of this simulation, and finish by discussing the role of various halo mass ranges in the overall reionization process. Our intention is to introduce these results and highlight a few of them, while leaving further analysis and interpretation for future papers. Note: In what follows, in order to make comparisons possible between simulation results at a given redshift and some observable properties of the universe at a different redshift, we will, henceforth, refer to simulation redshift as , while z will refer to the observed, or "real world" redshift.
§.§ Maps: Zoom-In on Galaxies and the Cosmic Web during the EoR
§.§.§ Example: The M31 Progenitor Cut-out
Maps of the physical properties properties of the gas in a slice through the comoving 4 h$^(-1)$ Mpc$^3$ subvolume that contains the progenitor of M31. From left to right: $=11$ to $=4.28$ (end of the simulation).Top: co-moving gas density, 2nd row: Temperature, 3rd row: Neutral gas fraction, bottom: Ionizing flux density. Gas density and ionizing flux density are a projection based upon averaging over a slice 40 cells (or 625 h$^{-1}$ kpc) thick, while the temperature and neutral fraction shown are taken from a slice just 1 cell thick (i.e. 15.625 h$^{-1}$ kpc).
Illustration of the sheathed temperature structure of gas filaments. Left: a xz slab (1 cell thick) of the cutout of Fig. <ref>, chosen to highlight the filamentary structure. Right: gas overdensity, temperature, Mach number, x axis velocity, neutral Hydrogen fraction and ionizing flux density $J_{21}$ along the black segment of the left panel. The vertical dashed lines mark the position of the two temperature peaks, for reference.
Mass-weighted maps (xy projection) of the neutral fraction (left) mass-weighted fractional excess of the ionizing flux density (right) for the M31 cutout at $=4.28$. The fields are averaged over a 40 cells (625 ) thick slice.
The physical properties of the gas for the M31 progenitor cutout at 3 different epochs are shown in Fig. <ref> . The gas density and ionizing flux density are averaged over a 625 thick slice, while the temperature and neutral fraction shown are taken from a 1 cell thick slice (i.e. 15.625 ).
The gas is distributed along sheets, filaments and knots, which become more pronounced with time. The small scale structure visible in the high redshift maps progressively disappears, due to a combination of hierarchical merging and smoothing due to heating by UV radiation. Meanwhile, the voids get less dense, as their gas is pulled towards higher density regions. Let us recall that the filamentary structures seen in these maps are more often 2d planes extending perpendicular to the map than actual 1D filaments. Statistically, <cit.> showed that the volume in gas sheets is 7-8 times larger than the total volume of filaments, although the latter end up hosting a similar fraction of the total dark matter mass. Nevertheless we will call these filaments because that is what they look like in a 2D map, but it should be kept in mind that these are often sheets rather than actual filamentary structures of gas.
In the high redshift cutout ($\sim 11$), the first star forms in the highest density clump, located at $(x,y)=(25.5, 27)$, and immediately starts to photo-heat its surroundings, as can be seen in the leftmost temperature map. Since the UV radiation is responsible both for ionizing the hydrogen gas and heating it to a few $10^4$ K, it is not surprising that temperature, neutral fraction and ionizing flux density maps look very similar in structure. The ionization front propagates inside-out, faster in low density regions and slower in high density regions because the I-front speed is proportional to the number density ratio of ionizing photons to H atoms at the location of the front. This gives rise to the typical butterfly-shaped ionized regions seen in the middle temperature and neutral fraction maps. Radiation leaves the source's host halo, first into low-density regions, and later on manages to penetrate denser nearby gas sheets and filaments.
The $=6.15$ temperature map also demonstrates the very different scales of the feedback processes implemented: the yellow-orange regions are photo-heated, while the much hotter, reddest region near the highest density peak is also subject to supernova feedback. Therefore stellar formation in this simulation impacts the gas in two very different ways: UV radiation produces long range photo-heating, while supernovae form very local, but very hot bubbles. These supernovae-heated regions are significantly more highly ionized than regions which are only photo-heated.
The bottom middle panel shows that the photon propagation is restricted to photo-heated, ionized regions, as expected since the mean-free-path of ionizing UV radiation is very small in the neutral gas ahead of the I-fronts, so the boundaries of the H II regions are sharply defined. Additionally, this panel shows the location of young star particles, appearing as peaks in the ionizing flux density distribution. Their distribution follows closely the gas density peaks, in which they form.
The last redshift (post-reionization) panel displays several interesting features, as follows.
§.§.§ Sheathed gas filaments
First of all, the temperature map shows that gas filaments after reionization (or after reionization is completed locally) are rather thick, as also seen for instance in <cit.>, although using a uniform background, and in <cit.>.
Moreover, the filaments display a sheathed structure in temperature, with a hot, tube-like envelope surrounding a cooler core. This is also seen, for instance, in <cit.>, although the aspect of the sheath looks thicker in the latter, perhaps because of the larger cell size in low density regions (i.e. the outskirts of the filament). More recently, <cit.> obtained sheathed filamentary structures very similar to CoDa's filaments in physical properties and thickness (about $\sim 200$ in both cases).
We investigated the origin of this structure. It could simply be due to the difference in cooling rates between the denser, more neutral core and the more diffuse filament envelope. There could also be a more dynamical process causing this temperature gradient: the filament could expand radially due to the outward pressure force caused by its photo-heating. During this expansion, its outer layers would get shocked, forming a high temperature layer similar to the contact discontinuity seen in the adiabatic shock tube experiment <cit.>. To gain better insight into the process, we detail the properties of a gas filament in Fig. <ref>. The left panel shows an xz slab (one cell thick) of the cutout of Fig. <ref>, chosen to highlight the filamentary gas structure. The right panel shows the gas properties of individual cells along the short black segment crossing the filament of the left panel. The density is highest at the filament's center, and decreases outwards. The temperature shows a double peaked profile, surrounding the cooler, high density center. The temperature peaks, combined with the decreasing density, result in 2 dips in the Hydrogen neutral fraction. Moreover, the gas velocity field reveals that all of the gas is in-flowing towards the filament, rather than out-flowing, and the temperature peaks are located at the position showing the strongest velocity gradients, i.e. they demark a strong deceleration region in the gas, and the transition from a supersonic to subsonic flow, as shows the Mach number plot. Therefore, this hot layer is not a contact discontinuity (which would be flowing outwards) but rather an accretion shock formed by gas accreting onto the filament or plane. This is similar to the accretion shocks found for 1D pancake collapse by <cit.>, in which dark-matter-dominated gravity drives a cold, supersonic baryonic infall toward the central density peak, and this causes strong shocks that decelerate the baryons and convert their kinetic energy of infall to thermal energy (i.e. high T). It is also similar to the halo accretion shocks described by <cit.>. However, instead of spherical accretion as considered by <cit.>, we here have either planar, perpendicular accretion onto a sheet or radial, cylindrical accretion onto a filament.
§.§.§ Self-shielding
The filament's core is also significantly more neutral than its hot envelope. This is just the result of the higher density and therefore higher cooling and recombination rates: indeed, the bottom right panel of Fig. <ref> shows that the ionizing flux density is completely flat across the filament: the UV background does not “see” the filament. It is not able to self-shield.
Here we investigate further the occurrence of self-shielding in CoDa. The low redshift ionizing flux density map of Fig. <ref> shows that except at the location of the 3 most massive density peaks still forming stars, the pockets of radiation seem to have given way to an almost perfectly uniform UV background. To get a sharper view of this, we computed the ratio of the gas-mass-weighted ionizing flux density to the average $J_{21}$:
\begin{equation}
\langle \rho J_{21} \rangle_z / \langle \rho \rangle_z \langle J_{21}\rangle_{xyz} \, ,
\end{equation}
where the subscript $_z$ denotes an average in the $z$ direction while $\langle J_{21} \rangle _{xyz}$ denotes the average value over the whole cutout volume. The resulting map is shown in the right panel of Fig. <ref>. Thanks to the mass-weighting, both sources and sinks appear. While the largest gas over-densities seem to host sources, the smaller ones tend to be dark and act as sinks. Interestingly, most haloes hosting sources seem to be also partly sinks, and the mass-weighted neutral fraction map shows that all photon sources are located next to rather neutral cells. This illustrates how galaxies are all at least marginally resolved in CoDa, i.e. even when they are just a few cells across, the code can still capture, albeit coarsely, their strongly heterogeneous internal structure.
A filament of gas joins the 2 main sources of the map. The lower half of it consists of a chain of small gas overdensities, and seems able to self-shield, as shows the photon-density map. On the other hand, the upper half is less dense, and does not appear self-shielded: its ionizing flux density is identical to the background. This link between density and self-shielding is made more obvious by the phase diagram ($J_{21}$ versus gas density) shown in Fig. <ref>: the majority of the cells see a UV background at $J_{21}\sim -1$. The high density part of the diagram has 2 branches. In the upper branch, the cells are under the influence of a nearby source or contain one.
The lower branch arises from self-shielding, and starts at an overdensity of $\sim 100$, similar to what was found by <cit.>. This value is at the very high end of the density range of filaments, and most filaments have lower densities, including that of fig. <ref>. Therefore, while haloes are at least partially self-shielded, filaments are not.
Filaments can indeed resist the I-front propagation at the onset of their own local reionization, but not for long. The $=6.15$ neutral fraction map indeed shows a few neutral clumps and filaments still partially neutral, surrounded by ionized gas and pockets of radiation. However this does not last and in the last snapshot they end up with neutral fractions between $x_{\rm HI}=10^{-2}-10^{-4}$.
Distribution of the ionizing flux density versus gas overdensity for the M31 progenitor cutout, last snapshot.
§.§ Global properties
Global properties of the simulation, and comparison to observations. The dot-dashed black line shows the simulation results, while the solid black line shows a temporal rescaling of the CoDa simulation, i.e. for instance $x_{\rm HI}($$=z/1.3)$, so as to shift the end of the EoR to z=6 instead of z=4.6. This transformation brings all the fields in agreement with the available high-z observations.
In this section we show that the global evolution of the simulation is physically consistent and in line with our theoretical understanding of the EoR, and to some extent with the observational constraints available.
The most basic quantities to consider when gauging the success of a global EoR simulation are the evolution of the cosmic means of the neutral fraction, ionizing flux density, and cosmic star formation history. These are shown in Fig. <ref>, along with several observational constraints from <cit.>, <cit.>, <cit.>, and the Planck CMB thomson optical depth $\tau$ <cit.>. The observed cosmic star formation rate (hereafter SFR) constraints are taken from <cit.>: the grey area shows the envelope including the dust-corrected and dust-uncorrected SFRs. As a first remark, we note that the cosmic SFR in CoDa increases at all times, unlike the simulation of <cit.>, which shows a decline at late times. In the latter, the authors suggested this could be due to the small box size they used. This may indeed be the case, since CoDa is more than 95 times larger in volume.
The neutral fraction plot shows a characteristic very steep decrease, down to $x_{\rm HI}\sim 10^{-4.2}$, where the slope becomes more gentle. This transition marks the end of the EoR, and correlates with the end of the surge in ionizing flux density seen in the middle panel. Following this definition, reionization is complete in the simulation at redshift $=4.6$, which is about 300 Myr later than indicated by observational constraints. This discrepancy is consistent with the lower level of star formation in the simulation as compared to the observations of <cit.>, shown in the right panel.
These observational constraints are global ones, of course, and do not directly constrain the history of the local universe simulated by CoDa, to the extent that the local universe might be statistically differentiated from the global average.
In CoDa, the EoR lasts about 30% longer than global EoR observational constraints suggest. This could be a problem when using CoDa for producing mock observations of the Lyman $\alpha$ forest or the Gunn-Peterson trough for instance, for which the large size of the simulation is a desirable property. In an attempt to correct for this, we also show the effect of shrinking the redshift axis of the simulation: the solid black line in Fig. <ref> shows the global properties of the CoDa simulation at the rescaled redshift $=z/1.3$. This simple rescaling mimics the effect of a modest increase of the star formation efficiency parameter. It brings the neutral fraction evolution in reasonable agreement with the observational constraints, as well as the ionizing flux density, the cosmic star formation history and the CMB Thomson scattering optical depth $\tau$. Therefore the simulation should remain viable for producing mock observations, although other quantities, such as the collapsed mass fraction in halos, could be offset because of this temporal shift. We plan to explore this in future papers. In the rest of the paper, we carefully, explicitly state when such rescaled redshifts are used.
The post-reionization neutral hydrogen fraction in CoDa is about 0.5 dex lower than observed. Such an offset is not uncommon and the literature shows that simulations of the EoR can exhibit a variety of such small departures from the observed evolution of the ionized fraction measured from quasar lines of sight (see for instance <cit.>), in particular in the fully coupled radiation-hydrodynamics regime, due to the cost and difficulty of tuning such simulations. In the case of CoDa, this offset may be reduced by adopting a lower black-body temperature for our stellar sources, i.e. less massive stars, leading to a smaller effective ionizing photon energy, and therefore a smaller injection of energy per photo-ionization event, resulting in a lower average temperature of the large scale IGM, which would make it slightly more neutral. Indeed, Fig. 9 of <cit.> shows that changing the heat injection per photo-ionization event has a strong impact on the post-reionization neutral fraction. However, decreasing the black body temperature of our sources to e.g. 50 000 K would also increase the effective H cross-section $\sigma_E$, leading to more ionizations and therefore may lead to an even larger ionized fraction at equilibrium. More simulations will be required to test this in future work. Finally, <cit.> showed that modelling the gas clumping at sub-grid resolution could help in reproducing the observed trend. Just how much resolution is required to capture this additional gas clumping fully, beyond that of the CoDa simulation, was demonstrated, for example, by <cit.>, who calculated the transfer of ionizing UV radiation through a static snapshot of the density field from a sub-Mpc-volume cosmological simulation. More recently, <cit.> used a fully-coupled radiation-hydro
simulation of such a small volume to show that hydrodynamical back-reaction must be included, as well, since this causes the clumping to be time-dependent; eventually, even the denser, self-shielded gas ionizes and photoevaporates.
Large simulations such as CoDa, which can be run only once because of their cost, are very difficult to calibrate. As described in Section 2.1, for instance, tuning the subgrid star formation efficiency parameters by performing a large suite of small-box simulations (e.g. 4 boxes) to find parameters that satisfy observational constraints does not guarantee that a large 91 Mpc box will follow the same evolution, because of cosmic variance, and because the large box will contain voids which can be orders of magnitude larger in volume than the small 4 calibration run. Moreover, it is unclear how periodicity (photons exit through one face and come back through the opposite face) affects the calibration of such small boxes. Taking a step back, it now appears that the strategy for calibrating such a large run should be more hierarchical. In the future, we can use the CoDa simulation, itself, to help re-calibrate, as well.
§.§ Impact of radiative feedback on galaxy formation
Instantaneous star formation rate per halo as a function of instantaneous halo mass, for various redshifts. The instantaneous SFR is computed as the stellar mass formed within an $R_{200}$ radius sphere centered on the dark matter halo center of mass, during the last 10 Myr, divided by a duration of 10 Myr. Notice the sharp suppression at low mass.
In this subsection we investigate the impact of the radiation-hydrodynamics coupling on galaxy formation. Since early works of <cit.>, several authors have addressed the impact of radiative feedback on galaxy formation, at low and high mass and in a number of contexts, with a fixed uniform UV background from <cit.>, and more recently with fully coupled RHD simulations <cit.>, as a reduction and possibly a suppression of star formation at low masses.
We computed the instantaneous SFR of CoDa haloes as the stellar mass formed within a sphere of radius,
$R_{200}$ (i.e. that within which the average density of the dark matter is 200 times the cosmic mean density) centered on the dark matter halo center of mass, during the last 10 Myr, divided by a duration of 10 Myr. Fig. <ref> shows the instantaneous SFR that results, as a function of the instantaneous mass of the dark matter halo, for several redshifts.
There is a general trend for haloes at all masses to form fewer stars as time goes by, which is linked to a wide-spread decrease in accretion rate, as seen in <cit.>. Here we find a trend: SFR $\propto M^{\alpha}$ for $M>10^{10}$ with a slope $\alpha \sim 5/3$. This slope is compatible with the values found in the literature. i.e $1<\alpha<2.5$, in numerical and semi-analytical studies, such as <cit.>.
However, the most striking feature of Fig. <ref> is the very sharp decrease in SFR for the low mass haloes, around $\sim 2 \times 10^{9}$ . Before z$\sim6$, low mass haloes sit on the same global trend as the high mass haloes. However, during the EoR, they transition from this “normal” state to a strongly suppressed state: at z=4.2, the $10^9$ haloes form almost 1000 times less stars than they did at z$=6$. This suppression reflects the great reduction of the gas fraction inside the galaxies that is below 20,000 K (as required for star formation by our star formation criterion described in Sec. <ref>), once the galaxy and its environment are exposed to photoionization during reionization. In contrast, the gas core of high mass haloes is dense enough to remain cool and/or cool down fast enough to keep forming stars, even if in bursts.
To check this, we ran two 8 RAMSES-CUDATON simulations: one with the exact same parameters and resolution ($512^3$ grid) as the CoDa simulation and a second without radiative transfer (therefore vanilla RAMSES, with SN feedback only), and no UV background. The star formation histories of haloes of various masses[the halo masses are those measured at the last snapshot, contrary to those of Fig. <ref>, which are instantaneous] are shown in Fig. <ref> for these 2 simulations. For the RHD simulation (left panel), there is again a very strong suppression of star formation at the end of the EoR for the low mass haloes. We note that more massive haloes are also affected, although in a less spectacular, if still significant way: even the $10^{10-11}$ mass bin sees its star formation decrease by a factor of $\sim2$ when compared to the pure hydro run (right panel).
This experiment confirms that the addition of radiative feedback via photoheating associated with hydrogen photoionization is the cause of the sharp suppression of star formation seen in low-mass haloes.
While this trend is clear and extends well up into the intermediate halo mass range, far above the minimum mass resolved in our halo mass function, the quantitative details may be affected by the limits of our spatial resolution. For instance, higher resolution simulations could form denser clumps inside a given halo, which would better resist ionization and photo-heating. Comparing with literature, we find for instance that <cit.>, using the renaissance simulations, offering up to 10 times higher mass resolution and better spatial resolution, but in a volume thousands of times smaller than CoDa, reports SFR suppression at halo masses typically 10 times smaller, i.e. $\sim 2 \times 10^8$ . Meanwhile, <cit.>, using the CROC simulations, find a very moderate impact of radiative feedback on the properties of low mass galaxies during reionization. However, they check for this effect by examining variations in the faint end of the UV luminosity function (hereafter LF). While this is valuable from an observational perspective, the impact of radiative feedback on star formation or the lack thereof is more easily assessed by looking at the SFR - halo mass relation as we do here, than at the UV LF. This difference in methodology makes a direct comparison with <cit.> difficult. Moreover, their treatment of star formation relies on modelling molecular hydrogen formation. This raises the density threshold for star formation, and therefore pushes star formation to higher mass haloes. The smallest galaxies, which are the ones most-strongly suppressed by radiative feedback in CoDa, would probably not have formed any stars at all within the CROC formalism. Therefore the question of the impact of radiative feedback on low mass haloes is tied, not only to spatial and mass resolution, but also to the choices made for the star formation recipe.
While performing a higher resolution CoDa would be prohibitively expensive with the current code and facilities, we performed a simple resolution study spanning 2 times higher and up to 4 times coarser spatial resolution with respect to CoDa, using $512^3$ boxes of 4 to 32 on a side. Details can be found in Sec. <ref>. Preliminary results indicate that degrading resolution beyond CoDa's produces a radiative suppression of star formation at increasingly higher masses. However, the test boxes at CoDa resolution and 2 times higher resolution yield the same suppression mass scale of about $\sim 2 \times 10^9$ . We note that this is compatible with the results of <cit.>, obtained with a mass resolution comparable to CoDa but better spatial resolution due to the use of an SPH-based Lagrangean hydro code instead of CoDa's unigrid scheme. Their study reports a SFR suppression mass similar to ours, of $\sim 10^9 $ .
Total star formation histories of 4 halo mass bins for two simulations in a test box 8 $\hmo$ Mpc on a side. Left: with full radiation hydrodynamics. Right: SN feedback only, no radiation. The mass bins correspond to the haloes' final mass (i.e. halo mass measured in the final simulation timestep, at $\sim 3$).
§.§ Contribution to the cosmic star formation density
We now investigate the contribution of haloes of various masses to the total star formation density of the box. While Fig. <ref> already made clear that individual halo SFR increases with mass, one needs to multiply their SFR by the halo number density to obtain their contribution to the star formation density. This is shown in Fig. <ref>: each line shows the star formation density due to each mass bin (each mass bin is a decade wide). As expected, the lowest mass haloes contribute almost equally to the other mass bins at high z ($>$11) but their contribution decreases sharply during the late stages of the EoR, as they become increasingly suppressed. This decrease, though more gentle, is also seen for the $10^{9-10}$ bin. However, for the 3 most massive bins, we find a constant rise of the star formation density. This rise is faster for increasing halo mass. This $10^{12-13}$ mass bin can only be traced at $<7$, as before this time the simulation does not contain any halo that massive. The $10^{10-11}$ haloes dominate the cosmic star formation density for most of the simulation time, before being overtaken by the $10^{11}$ mass bin. The hierarchy between mass bins and the overall evolution is very similar to that shown in Fig. 2 of <cit.>, although the details of the methodology and physics implemented differ vastly. The total SFR is however lower than the observational estimate, as was already pointed out in Sec. <ref>. This reflects the underestimated value adopted for the subgrid parameters which control the efficiency of star formation over-all, but not the relative contributions for different halo masses.
We will refrain, at this stage, from drawing conclusions as to which mass bin contributes most to large scale reionization. Indeed, while all stellar particles have the same intrinsic specific emissivity, we do not know how much of the ionizing photons produced made it into the IGM. We will come back to this question by measuring the circum-galactic escape fraction in a future study.
Cosmic SFR density as a function of redshift for various bins of instantaneous halo mass: each line shows the contribution of all galaxies within a given mass bin to the total cosmic SFR density.
§.§ UV luminosity function
UV luminosity functions and comparison with observations. The full circles and crosses with error bars are the observations from Bouwens et al. 2015 and Finkelstein et al. 2015, respectively, at z=$[6,7,8,10]$, while the shaded area and the thick line show the envelope and the median of the LFs of 5 equal, independent, rectangular sub-volumes taken in the CoDa simulation, at zcoda $=[6,7,8,10]/1.3$. For clarity, the LFs have been shifted downwards by 0, 2, 4 and 6 dex.
In order to gain more insight into the balance of bright vs faint galaxies in our simulation, we computed the luminosity function (hereafter LF) of CoDa haloes. We computed the $M_{AB1600}$ magnitudes using the lowest metallicity stellar population models of <cit.>.
We ensured consistency with RAMSES-CUDATON's source model by rescaling Conroy's single stellar population models in flux so as to obtain the same ionizing photon output over 10 Myr. The results are shown in Fig. <ref>, along with observational constraints. The latter are taken from <cit.> and <cit.>, which have been shown to be in broad agreement with a number of other studies including <cit.> and <cit.>.
The LFs have been shifted vertically for clarity. The shaded area shows the envelope of the LFs obtained for 5 rectangular independent sub-volumes of the CoDa simulation. Each of these sub-volumes spans $\sim 150,000$ Mpc$^3$ ($\Delta x=\Lbox/5$, $\, \Delta y=\Delta z=\Lbox$, i.e. 1/5 of the full box volume), which is similar to the volume probed by CANDELS-DEEP at $z=6$. The resulting envelope therefore illustrates the expected effect of cosmic variance at $M>-20$. The thick solid line shows the median of these 5 LFs. The observed $z=6-8$ LFs are in rather good agreement with our simulation for $M>-21$.
However, the simulation seems to predict an overabundance of bright galaxies, at all redshifts. Due to the small simulated volume compared to the observations (the survey volume at the bright end is 4 times larger than CoDa), it is not clear if this is a robust prediction of the simulation or just a statistical accident. There could however be several reasons for such an overabundance:
* Missing physics:
* dust: the only impact of dust extinction in CoDa is through a constant stellar escape fraction $\fesc^{dust}=0.5$. However, dust extinction becomes increasingly important at the bright end <cit.>. Moreover, observed high redshift galaxies, even during the EoR, could be fairly dusty <cit.>.
* AGN feedback: the radiative and/or mechanical feedback from AGNs is believed to be responsible for the drop-off of the bright end of galaxy LF at low redshift. Could it be that early super-massive black holes in massive high redshift galaxies regulate the bright end of the galaxy LF as well? CoDa does not include AGN feedback, and it could help explain the overabundance in our LFs.
* overlinking: the FoF algorithm we used for halo detection is notorious for producing too massive haloes at high redshift, when compared to other halo-finding algorithms, such as the spherical overdensity. This is known as the “overlinking problem” <cit.>. However, we checked that using a shorter FoF linking length b=0.15 instead of the usual b=0.2, which could be more adequate at high redshift, did not improve the LFs.
§.§.§ Halo Mass and UV luminosity:
Galaxy mass - magnitude distribution at z=8 (i.e. =6.15). The color indicates the galaxy number density in N/Mpc$^3$/Mag/log(M$_{\odot}$). The red line indicates the average Magnitude for each mass bin. Left: original stellar particle masses (i.e. quantized). Right: “smoothed” stellar particle masses.
We also investigated the relation between UV galaxy luminosity and halo mass. There is a general trend of increasing UV continuum luminosity with halo mass. This is shown in the left panel of Fig. <ref>, which corresponds to the galaxy population of the $z=8$ UV LF of Fig. <ref>. Moreover, for a given halo mass, the luminosity can fluctuate significantly, as shows the vertical spread of the distribution. The vertical dispersion increases with decreasing halo mass, and is largest for the mass range sensitive to radiative feedback. The horizontal overdensity at $(M_{AB1600} \sim -10,\, M<10^9$ $)$ is due to the quantization of stellar mass: the haloes located in this region have exactly one young UV-bright stellar particle, which has a fixed minimum mass. Below this limit, haloes are populated by star particles older than 10 Myr, therefore fainter in the UV.
Indeed, in RAMSES-CUDATON, the mass of a stellar particle is always a multiple of the elementary star particle mass $M_{\star}=3194$ . This quantization of the stellar mass leads to a quantization in the magnitudes and therefore to possible artifacts in the faint-end LF, whereas the bright end LF is not affected, due to the larger stellar masses involved.
We can mitigate the effect of this quantization by modifying the stellar particles' masses by adding to them a random number taken from a uniform distribution between $[-M_{\star}/2,M_{\star}/2]$, where $M_\star$ is the stellar particles elementary mass. A stellar particle of mass $M_\star$ is therefore assigned a mass between $0.5 M_{\star}$ and $1.5 M_\star$, while a stellar particle of mass $10 M_\star$ is assigned a mass between $9.5 M_\star$ and $10.5 M_\star$. Therefore this “smoothing” of the stellar particles' mass is stronger, relatively, for haloes with lower stellar mass, but on average the total added stellar mass is 0.
This simple procedure reduces quantization artifacts in the faint end of the LF, as shows the right panel of Fig. <ref>, and will help its interpretation.
§ DISCUSSION
§.§ Observing the End of Reionization: Depressing the Faint-End of the Luminosity Function
Same as Fig. 10, extended to include the faint end, using the “smoothed” stellar particle masses as explained in Sec. <ref>
It has long been thought that reionization must have exerted a negative
feedback on the ability of low-mass galactic haloes to form stars,
by suppressing the infall of intergalactic baryons from the photoheated IGM
following its local reionization (e.g. <cit.>. As we mentioned in Section 1, this effect may help explain why
there are so many fewer dwarf galaxy satellites observed in the Local Group
than there are low-mass haloes predicted by N-body simulations of the CDM model.
However, it has also been suggested that the suppression of star formation in
low-mass galaxies by reionization feedback might be observable in the evolution
of the global star formation rate and galactic luminosity function at high
redshift. <cit.>, for example, used a semi-analytical approach
to show that if the local IGM Jeans mass filter scale jumped up when a patch
became an H II region, a sharp drop in the global star formation rate would
appear as reionization overtook a significant fraction of the volume of the
universe. Large-scale N-body + radiative transfer simulations of patchy
reionization like those by <cit.>, <cit.>, and
<cit.>, for example, which also make this assumption about
reionization suppression of low-mass galaxies inside H II regions find that
reionization is, in fact, self-regulated, such that the rise of the suppressible
galaxies limits their ability to finish reionization. As a result, the dominant
contribution, they find, comes from the larger-mass halos which are too big to be
suppressed and whose number density is growing exponentially fast as reionization
approaches completion. In that case, the global star formation rate does NOT
decline sharply as reonization ends. However, the signature of this low-mass
suppression may still be observable in ways that reflect the earlier onset and
longer ramping up of reionization when low-mass galaxies dominate the early
phase, before they saturate and the more massive galaxies dominate (e.g. <cit.>). Since the low-mass suppressible dwarf galaxies whose
contribution eventually drops off are also the ones at the faint end of the
luminosity function, these expectations are consistent with another, possibly
observable effect suggested by <cit.>, of a drop in the galaxy
LF at the faint end, marking the end of reionization. Similar conclusions have
also been drawn more recently by, for example, <cit.>.
We can use our CoDa simulation results to address both of these questions:
Does the global star formation history show a strong drop as reionization approaches
the end? And is there a depression of the faint end of the LF at this time?
According to our results for the global SFR density plotted
in Fig. <ref>, there is no sharp downturn in the SFR as reionization
runs to completion, which is not surprising, given the
relative contribution to this SFR from haloes of different
mass plotted in Fig. 9. The galaxies above 10$^{10}$ ,
which Fig. 7 indicates are basically too massive to be
strongly suppressed by external radiative feedback alone
during reionization, are found to dominate that SFR.
Of course, we cannot say what the SFR history would have
been if we had neglected radiative feedback. Our
comparison in Fig. 8 of the SFRs in the two test box simulations (SN-feedback only, no radiation, for one, and SN-feedback and radiative transfer in the other) as a function of the final masses of the
haloes within which the stars reside at the end of the simulations
at z = 3, shows that there can, in fact, be a depression of the
SFR even in the higher mass bins when radiation is included, but
much less dramatic than for the lower-mass haloes. And whether
this is internal feedback or external reionization feedback is
not evident from that plot alone. Nevertheless, it is a fact
in the CoDa simulations that the global SFR continues to rise
even up to the end of reionization, as seen in Fig. <ref>, and contrary to the expectations
of Barkana and Loeb (2000).
However, since radiative star formation suppression is very efficient in CoDa, it results in a change of shape of the faint end of the galaxy UV LF, as shown in Fig. <ref>, which uses the smoothed stellar particle masses, as explained in Sec. <ref>. The $M_{AB1600} = -10 $ to $-12$ galaxies are hosted by haloes affected by radiative suppression by the UV background. As their SFR drops post-reionization, so does the abundance of galaxies in this magnitude range. Therefore, even though we find no sudden drop in the cosmic SFR density, we do see a sharp change in the shape of the LF at the very faint end, due to reionization, as proposed by e.g. <cit.>. With respect to a similar exploration by <cit.>, we find, like them, that the total cosmic SFR evolution is smooth throughout reionization because it is dominated by the more massive, radiation-immune galaxies, but unlike them, we find that reionization affects the shape of the faint end LF.
The bright end of the LF increases over time continuously, reflecting the convolution of the rising halo mass function with the halo-mass dependence of the star formation rate for haloes too massive to be suppressed by reionization. At fainter magnitudes but still brighter than -10, the feedback from reionization on lower-mass haloes is reflected in the depression of the LF toward the end of reionization, in the magnitude range -10 to -12.
For magnitudes even fainter than -10 in Fig. <ref>, the LF reflects the passively evolving stars inside the low-mass suppressible haloes which have stopped forming new stars after they are overtaken by reionization.
The section fainter than -10 shifts the peak to fainter magnitudes over time, as the stars formed inside low-mass haloes before they were fully suppressed get older.
§.§ The dark, not missing, satellites
Fraction of bright halos as a function of mass and redshift. The original CoDa redshifts corresponding to the labels are $=[5.5,6,7,8,10]$/1.3, i.e. as in Fig. <ref>. The S14 curves show the results of Sawala et al. 2014.
The suppression of SFR we measured in low mass haloes is an important process in reducing the number of bright haloes, and therefore towards a remedy to the missing satellites problem. Indeed, Fig. <ref> shows that the fraction of bright haloes (i.e. haloes hosting at least one stellar particle) is a steep function of the dark matter halo mass. Below $10^8$ , more than 99% of the haloes are dark. The transition between dark and luminous haloes takes place between $10^8$ and $10^9$ . The transition mass (i.e. the mass for which half of the haloes are dark) does not evolve much prior to reionization but shifts quickly to higher masses near the end of the EoR and just afterwards ($z=6$ and $z=5.5$). This evolution is similar to that seen in <cit.>, although the latter used an instantaneous and uniform reionization model, implemented as a uniform heating of the gas (i.e. no coupled radiation-hydrodynamics was performed as opposed to CoDa). Thanks to this simplification in the treatment of the EoR, they were able to carry out their simulation down to z=0 and show that such a mass-dependent reduction in the fraction of luminous haloes is able to quantitatively match the observed abundance of satellites <cit.>. Therefore, although CoDa did not run down to z=0, the trends we measured in the suppression of star formation in low mass haloes suggest that CoDa's z=0 low mass halo population will be predominantly dark, alleviating the missing satellites problem.
§ CONCLUSIONS
CoDa (Cosmic Dawn) is a very large, fully coupled radiation-hydrodynamics simulation of galaxy formation in the local universe during the Epoch of Reionization. It was performed on Titan at Oak Ridge National Laboratory using RAMSES-CUDATON deployed on 8192 nodes, using 1 core and 1 GPU per node. This is the first time a GPU-accelerated, fully coupled RHD code has been used on such a scale.
The simulation accurately describes the properties of the gas and its interplay with ionizing radiation, in particular the growth of typical butterfly-shaped ionized regions around the first stars and first galaxies, accompanied by photo-heating and the subsequent progressive smoothing of small scale gas structures. Gas filaments tend not to be self-shielded once the reionization radiation sweeps across them; the flux density of the ionizing radiation internal to these filaments thereafter is identical to that of the background. However, they are indeed slightly more neutral than surrounding voids. Furthermore, they develop a sheathed temperature structure, showing up as a hot tubular envelope surrounding a cooler core, similar in nature to the more spherical accretion shocks seen around forming galaxies. On the other hand, haloes hosting gas denser than $100 \, \langle \rho \rangle$, when they do not host an ionizing source, show up as photon sinks in ionizing flux density maps, and are therefore self-shielded.
The low star formation efficiency assumed in our simulation leads to late reionisation. This can be corrected by a simple contraction of the redshift axis, designed to make reionization complete at $z=6$. This remapping of the redshifts mimics the effect of a modest increase of the star formation efficiency parameter, and brings the simulation in agreement with several other observational constraints of the EoR, including the Thomson scattering optical depth measured by the Planck mission, the evolution of the ionizing flux density and the cosmic star formation rate.
The star formation rate of individual galaxies of a given mass is on average higher at high redshift and decreases as the Universe expands.
Galaxies below $\sim 2.10^{9}$ are strongly affected by the spreading, rising UV background: their star formation rate drops by a factor as high as 1000. This suppression reflects the great reduction of the gas fraction inside the galaxies that is below 20,000 K, once the galaxy and its environment are exposed to photoionization during reionization.
This produces a large number of sterile, low-mass haloes by the end of reionization, and a corresponding depression of the faint end of the UV luminosity function in the magnitude range M$_{\rm AB1600} = [-12, -10]$. The latter is similar to an effect suggested by <cit.>, although the accompanying sharp drop in the overall star formation rate they also suggested is not supported by the CoDa simulation results. This is explained by the fact that the total star formation rate is dominated at that time by more massive galaxies, too massive to be subject to this strong reionization feedback suppression.
The halo mass scale below which the CoDa simulation finds galactic halo star formation to be reduced by radiative feedback during the EOR was checked for numerical resolution effects by a set of smaller-box simulations of different mass resolution, described in Appendix B. The results showed that, while lower resolution than that of the CoDa simulation would increase the value of the threshold halo mass for suppression, a finer resolution by a factor of two in spatial resolution and eight in mass resolution (replacing one cell by 8 cells) finds the same threshold mass, suggesting that the value of the suppression mass is not an artifact of limited numerical resolution.
Although CoDa did not run down to z=0, the trends we measured in the suppression of star formation suggest that CoDa's z=0 low mass halo population would be predominantly dark, alleviating the missing satellites problem.
In contrast, the gas core of high mass haloes is dense enough to remain cool and/or cool down fast enough to keep forming stars, even if in bursts.
Overall, star formation in the whole box is dominated by $10^{10}$ haloes for most of the EoR, except at the very end where $10^{11}$ haloes become frequent enough to take the lead.
The CoDa luminosity functions are in broad agreement with high redshift observations, except for the most luminous objects, where the number counts are subject to strong cosmic variance and may be affected by additional processes, such as evolving dust content and AGN feedback.
§ ACKNOWLEDGEMENTS
This study was performed in the context of several French ANR (Agence Nationale de la Recherche) projects. PO acknowledges support from the French ANR funded project ORAGE (ANR-14-CE33-0016). NG and DA acknowledge funding from the French ANR for project ANR-12-JS05-0001 (EMMA). The CoDa simulation was performed at Oak Ridge National Laboratory / Oak Ridge Leadership Computing Facility on the Titan supercomputer (INCITE 2013 award AST031). Processing was performed on the Eos, Rhea and Lens clusters. Auxiliary simulations used the PRACE-3IP project (FP7 RI-312763) resource curie-hybrid based in France at Très Grand Centre de Calcul. ITI was supported by the Science and Technology Facilities Council [grant number ST/L000652/1]. SG and YH acknowledge support by DFG grant GO 563/21-1. YH has been partially supported by the Israel Science Foundation (1013/12).
AK is supported by the Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional (MINECO/FEDER, UE) in Spain through grants AYA2012-31101 and AYA2015-63810-P as well as the Consolider-Ingenio 2010 Programme of the Spanish Ministerio de Ciencia e Innovación (MICINN) under grant MultiDark CSD2009-00064. He also acknowledges support from the Australian Research Council (ARC) grant DP140100198. GY also acknowledges support from MINECO-FEDER under research grants AYA2012-31101 and AYA2015-63810-P. PRS was supported in part by U.S. NSF grant AST-1009799, NASA grant NNX11AE09G, NASA/JPL grant RSA Nos. 1492788 and 1515294, and supercomputer resources from NSF XSEDE grant TG-AST090005 and the Texas Advanced Computing Center (TACC) at the University of Texas at Austin. PO thanks Y. Dubois, F. Roy and Y. Rasera for their precious help dealing with SN feedback in RAMSES and various hacks in pFoF. NG thanks J. Dorval for useful discussions regarding k-d trees which helped with the analysis of this simulation.
§ CODA HALO MASS FUNCTIONS
Mass functions of our simulations and comparison to literature, at =10.28 and =4.41. Top: the black and red solid lines show the mass functions obtained for CoDa and DM2048 respectively. Over-plotted are the theoretical mass function of <cit.> and the <cit.> FoF universal fit (dotted and dashed lines, respectively). The gray area shows the Poisson error bars expected for the Watson FoF mass function. The vertical lines show the mass corresponding to 100 and 1000 particles haloes in CoDa.
Bottom: The ratio of the the CoDa and the DM2048 mass functions is shown (black solid line).
The mass functions (hereafter MF) obtained with FoF <cit.> at =10.28 and =4.41 are shown in Fig. <ref>, along with a Sheth-Tormen MF <cit.> and a FoF universal fit from <cit.>. At =4.41, the CoDa MF is fairly well represented by both fits for haloes larger than 1000 particles, but sits slightly above the Poissonian error bars of the Watson FoF MF (gray area) at the high mass end. To check the origin of this excess, we also plot the MF of the DM2048 simulation, a dark matter only companion simulation run with the N-body code Gadget 2 using the same CLUES initial conditions as CoDa degraded to $2048^3$ resolution. This comparison is useful because DM2048 has about $\sim 20$ times better force resolution than CoDa: tree codes such as Gadget 2 <cit.> perform well with a force resolution set to $1/20$ to $1/40$ of their average interparticle distance, while CoDa's force resolution is equal to $\sim 1.5$ times the cell size (equal to the average interparticle distance), due to the unigrid scheme, as shown for instance in Fig. 1 of <cit.>. The DM2048 can therefore be used as reference: it can inform us on possible artefacts due to CoDa's unigrid gravity solving. The DM2048 MF also displays some excess at the high mass end, comparable to CoDa's. This is confirmed by the bottom panel, showing the ratio of CoDa MF to DM2048 MF, which is close to 1 at the high mass end.
In contrast, the MFs differ significantly at low masses, with CoDa showing an increasing deficit of low mass haloes compared to DM2048.
We attribute this deficit to CoDa's limited force resolution, which hinders the proper resolution and detection of the smallest haloes. This deficit is present in the $=10.28$ MFs as well (left panels), with a similar amplitude. Above 1000 particles, on the other hand, CoDa displays an excess of haloes compared to DM2048. This is also likely caused by the limited force resolution of CoDa: the unresolved low mass haloes provide a large pool of “free”, untagged particles which the FoF may spuriously link to massive haloes and therefore increase their particle numbers. All in all, in order to mitigate these effects in our analysis, we will refrain from analysing haloes less massive than $10^8$ . Above this mass, our MF is uncertain by no more than a factor of two on average, but much better in general, in particular above 1000 particles and at lower redshifts.
§ NUMERICAL RESOLUTION AND SUPPRESSION MASS
Box size Grid size Suppression mass
() ($10^9$ )
4 $512^3$ 1.7
8 $512^3$ 1.7
16 $512^3$ 8
32 $512^3$ 40
Parameters of the simulations of the resolution study. The gray row corresponds to the CoDa resolution. The suppression mass is defined as the intersection between the high mass SFR fit and the low mass SFR fit of each simulation.
As described above, CoDa finds that star formation is suppressed in
low-mass haloes by the feedback associated with reionization, for haloes in
the range roughly below $\sim 2 \times 10^9$ . To investigate the dependence
of this suppression mass on the size and mass resolution of the simulation,
we performed a series of smaller-box simulations,
with input parameters identical to those of CoDa, but a range of resolutions,
both higher and lower than that of CoDa. The simulation parameters are summarized
in Table <ref>, along with the resulting suppression mass obtained in each case.
As seen in Table <ref>, a fixed number of particles and cells are adopted for
a hierarchy of different box sizes, so their (space, mass)-resolutions
range from (2,8) to (1/4,1/64) times those of CoDa.
Since we kept the input parameters for star formation efficiency the same
in all simulations, the higher resolution cases produced a higher star formation
rate (by resolving higher density gas and thereby triggering the subgrid star formation
criterion more often) and, hence, ended reionization earlier,
as shown in Fig. <ref>. In order to compare the halo mass scales of suppression at
different resolutions, therefore, it is necessary to make an adjustment for this
displacement of the reionization time-histories relative to each other
for the different cases. Fortunately, this is a well-defined operation.
The global reionization histories in Fig. <ref> all have in common a very sharp
drop in the neutral fraction at the end of reionization, followed by a much
flatter, slow decline thereafter in the post-reionization era, controlled then
by the average UV background and IGM density. A well-defined epoch of comparison
is that which corresponds to a fixed interval of time just after this sharp end
of reionization. Otherwise, if we compared them, instead, at the same cosmic
time, haloes in different cases would have spent a very different amount of time
experiencing the feedback effects of reionization, exposed to the UV background
radiation. For instance, the end of reionization in the 32 simulation happens
at $\sim$ 4.6, so at = 4.5, haloes have seen the post-reionization
UV background for about 35 Myr. By contrast, for the highest resolution simulation,
in the 4 box, reionization ends at $\sim$ 5.8, so at = 4.5, haloes would
have seen the post-reionization UV background for about 320 Myr, i.e. almost ten times
longer. In order to make a meaningful comparison, then, we pick the time of comparison
to be at the same interval of time just after each case's reionization ends. The
redshifts chosen to compare halo properties for different cases are shown by the
dot on each simulation's neutral fraction evolution (Fig. <ref>, left panel) and
listed on the plot in the right panel, of the average SFRs as a function of halo
mass for each case at its corresponding redshift.
The higher star formation rates for cases with higher resolution also mean that
the vertical scale of the SFRs in Fig. <ref> should be adjusted to make a direct
comparison of the mass-dependence of the SFRs for different cases. This, too, is
straightforward, since all of the cases have in common a universal shape for this
mass dependence, which shares the slope of the SFR at high mass, above the
mass scale where suppression occurs, and a turn-over at low-mass where suppression
is occurring. This is obvious in Fig. <ref>, where we have renormalized the curves
in Fig. <ref> (right panel) by adjusting their vertical heights so as to make their
high-mass SFRs lie on top of the power-law fit to this high-mass end of the CoDa SFR, given by $\log_{10}({\rm SFR})=5/3 \log_{10}({\rm M}) -18.6$, and shown as a straight line on the log-log plot in both figures. It is clear from
Fig. <ref>, now, as we compare the cases from lowest resolution to highest resolution,
that the turn-over, reflecting the suppressed mass range (where Fig. <ref> labels the
value of the mass at which the low-mass turn-over segment joins the high-mass segment
in each case), moves to successively lower mass, until the CoDa resolution is reached in
the 8 box. When the resolution is increased yet again so as to exceed that
of the CoDa simulation, by a factor of two in length and a factor of eight in mass,
in the 4 box, the SFR dependence on mass is identical with that
for the CoDa resolution case, as the curves for both cases completely overlap at
all halo masses. This demonstrates that the suppression mass found by the CoDa simulation
is well-enough resolved and not an artifact of numerical resolution.
Neutral fraction and average SFR per halo for the resolution study.
Average SFR per halo renormalized to match the high-mass branch of the 8 simulation (CoDa resolution, solid black line, given by $\log_{10}({\rm SFR})=5/3 \log_{10}({\rm M}) -18.6$). The dashed lines show a power law fit to the low mass branch of each simulation, with a slope ${\rm M}^4$. The numbers next to the dashed lines give the suppression mass as the mass where the low mass fit intersects the high mass fit.
|
1511.00476
|
Non-free ID-modules]Non-free iterative differential modules
Andreas Maurischat, Lehrstuhl A für Mathematik, RWTH Aachen University, Germany
[2010]12H20, 13B05
In <cit.> we established a Picard-Vessiot theory over differentially simple rings which may not be fields. Differential modules over such rings were proven to be locally free but don't have to be free as modules. In this article, we give a family of examples of non-free differential modules, and compute Picard-Vessiot rings as well as Galois groups for them.
§ INTRODUCTION
Differential Galois theory – and also difference Galois theory – is a generalisation of classical Galois theory to transcendental extensions. Instead of polynomial equations, one considers differential resp. difference equations, and even iterative differential equations in positive characteristic.
One branch of differential Galois theory is Picard-Vessiot theory, the study of linear differential equations. In this case the Galois group turns out to be a linear algebraic group or more general an affine group scheme of finite type over the field of constants.
Originally, one considered extension of fields as in the finite Galois theory.
However, this had to be extended for two main reasons: In difference Galois theory, zero-divisors may occur in minimal solution rings. The Galois group scheme does not act algebraically on the solution field but on the Picard-Vessiot ring, an important subring whose field of fractions is the solution field.
Therefore, it is natural also to replace the base field by a base ring with “nice” properties. This has been done in several settings (see e.g. <cit.>, <cit.>, <cit.>) where the base ring is a simple ring, i.e. has no nontrivial ideals stable under the extra structure. In <cit.>, we even gave an abstract setting covering all existing Picard-Vessiot theories, the base ring being again a simple object.
Having a base ring instead of a field, the first point is that not all modules have to be free.
In <cit.>, it has been shown that all iterative differential modules (ID-modules) over a simple iterative differential ring are locally free which is enough to obtain all the machinery of Picard-Vessiot-rings and Galois groups.[Local freeness of the modules have also been shown for other settings e.g. in <cit.> or <cit.>.]
But this raises the question whether there really exist ID-modules which are not free.
In this article, we answer this question by providing a family of ID-modules which are not free as modules. We also compute Picard-Vessiot rings and Galois groups for them. As we don't assume that the field of constants is algebraically closed, Picard-Vessiot rings for a fixed module are not unique, and our example will also show this effect.
As in characteristic zero simple iterative differential rings are the same as simple differential rings this also provides examples for differential modules in characteristic zero which are not free.
The article is organized as follows.
In Section 2, the basic notation, and some basic examples of ID-rings are given. We proceed in Section 3 with recalling some properties of ID-simple rings and ID-modules over ID-simple rings. In our definition, ID-modules are finitely generated as modules.
Section 4 is dedicated to Picard-Vessiot rings for ID-modules and their Galois groups.
Finally in Section 5, we give an example of a non-free ID-module over some ID-simple ring, and compute some of its Picard-Vessiot rings as well as the corresponding Galois groups.
§ BASIC NOTATION
All rings are assumed to be commutative with unit and different from $\{0\}$.
We will use the following notation (see also <cit.>). An iterative
derivation on a ring $R$ is a family of additive maps $(\th{n})_{n\in \NN}$ on $R$ satisfying
* $\th{0}=\id_R$,
* $\th{n}(rs)=\sum_{i+j=n}\th{i}(r)\th{j}(s)$ for all $r,s\in R$, $n\in \NN$, as well as
* $\th{i}\circ \th{j}=\binom{i+j}{i}\th{i+j}$ for all $i,j\in \NN$.
Most times we will consider the map
$$\theta:R\to R[[T]], r\mapsto \sum_{n=0}^\infty \th{n}(r)T^n,$$
where $R[[T]]$ is the power series ring over $R$ in one variable $T$.
The conditions that the $\th{n}$ are additive, and condition <ref> are then equivalent to $\theta$ being a homomorphism of rings, and condition <ref> is equivalent to the commutativity of the diagram
R [r]^θ_U [d]_θ R[[U]]
R[[T]] [r]^θ_U[[T]] R[[U,T]],
where $\theta_U$ is the map $\theta$ with $T$ replaced by $U$ and $\theta_U[[T]]$ is the $T$-linear extension of $\theta_U$.
The pair $(R, (\th{n})_{n\in \NN})$ or the pair $(R,\theta)$ is then called an ID-ring and
$$C_R:=\{ r \in R\mid \theta(r)=r\}=\{r \in R\mid \th{n}(r)=0\, \forall\, n>0\}$$
is called the ring of
constants of $(R,\theta)$. An ideal $I\ideal R$ is called an
ID-ideal if $\theta(I)\subseteq I[[T]]$ and $R$ is
ID-simple if $R$ has no
ID-ideals apart from $\{0\}$ and $R$. An ID-ring which is a field is called an ID-field.
Iterative derivations are extended to localisations by
$\theta(\frac{r}{s}):=\theta(r)\theta(s)^{-1}$ and to tensor products
$$\theta^{(k)}(r\otimes s)=\sum_{i+j=k} \theta^{(i)}(r)\otimes
\theta^{(j)}(s)$$
for all $k\geq 0$.
A homomorphism of ID-rings $f:S\to R$ is a ring homomorphism $f:S\to R$ s.t. $\theta_R^{(n)}\circ f=f\circ \theta_S^{(n)}$ for all $n\geq 0$.
An ID-module $(M,\theta_M)$ over an ID-ring $R$ is a finitely generated $R$-module $M$ together with an iterative derivation $\theta_M$ on $M$, i.e. an additive map $\theta_M:M\to M[[T]]$ such that $\theta_M(rm)=\theta(r)\theta_M(m)$, $\theta_M^{(0)}=\id_M$ and
$\theta_M^{(i)}\circ \theta_M^{(j)}=\binom{i+j}{i}\theta_M^{(i+j)}$ for all $i,j\geq 0$.
A subset $N\subseteq M$ of an ID-module $(M,\theta_M)$ is called ID-stable, if $\theta_M^{(n)}(N)\subseteq N$ for all $n\geq 0$. An ID-submodule of $(M,\theta_M)$ is an ID-stable $R$-submodule $N$ of $M$ which is finitely generated as $R$-module.[If $R$ is an ID-simple ring, then ID-stable submodules are always ID-submodules.]
For an ID-module $(M,\theta_M)$ and an ID-stable $R$-submodule $N\subseteq M$, the factor module $M/N$ is again an ID-module with the induced iterative derivation.
The free $R$-module $R^n$ is an example of an ID-module over $R$ with iterative derivation given componentwise. An ID-module $(M,\theta_M)$ over $R$ is called trivial if $M\isom R^n$ as ID-modules, i.e. if $M$ has a basis of constant elements.
For ID-modules $(M,\theta_M)$, $(N,\theta_N)$, the direct sum $M\oplus N$ is an ID-module with iterative derivation given componentwise, and the tensor product $M\otimes_R N$ is an ID-module with iterative derivation $\theta_\otimes$ given by $\theta_\otimes^{(k)}(m\otimes n):=\sum_{i+j=k} \theta_M^{(i)}(m)\otimes \theta_N^{(j)}(n)$ for all $k\geq 0$.
For ID-modules $(M,\theta_M)$, $(N,\theta_N)$, a morphism $f:(M,\theta_M)\to(N,\theta_N)$ of ID-modules is a homomorphism $f:M\to N$ of the underlying modules such that $\theta_N^{(k)}\circ f=f\circ \theta_M^{(k)}$ for all $k\geq 0$. For a morphism $f:(M,\theta_M)\to(N,\theta_N)$, the kernel $\Ker(f)$ and the image $\Ima(f)$ are ID-stable $R$-submodules of $M$ resp. $N$. The image $\Ima(f)$ is indeed an ID-submodule, since it is isomorphic to $M/\Ker(f)$. Also the cokernel $\Coker(f)$ is an ID-module.
* For any field $C$ and $R:=C[t]$, the homomorphism of $C$-algebras
$\theta_t:R \to R[[T]]$
given by $\theta_t(t):=t+T$ is an iterative derivation on $R$ with field of constants $C$. This
iterative derivation will be called the iterative derivation with respect to $t$. $R$ is indeed an ID-simple ring, since for any polynomial $0 \ne f\in R$ of degree $n$, $\theta^{(n)}(f)$ equals the leading coefficient of $f$, and hence is invertible in $R=C[t]$.
* For any field $C$, $C[[t]]$ also is an ID-ring with the iterative derivation with respect to $t$, given by $\theta_t(f(t)):=f(t+T)$ for $f\in C[[t]]$. The constants of $(C[[t]],\theta_t)$ are $C$, and $(C[[t]],\theta_t)$ also is ID-simple, since for
$f=\sum_{i=n}^\infty a_i t^i\in C[[t]]$ with $a_n\ne 0$, one has
$$\theta_t^{(n)}(f)= \sum_{i=n}^\infty a_i \binom{i}{n} t^{i-n}\in C[[t]]^\times.$$
Hence, every non-zero ID-ideal contains a unit.
This ID-ring will play an important role, since every ID-simple ring can be ID-embedded into $C[[t]]$ for an appropriate field $C$.
* For any ring $R$, there is the trivial iterative derivation on $R$ given by
$\theta_0:R\to R[[T]],r\mapsto r\cdot T^0$. Obviously, the ring of constants of $(R,\theta_0)$ is
$R$ itself.
* Given a differential ring $(R,\partial)$ containing the rationals (i.e. a $\QQ$-algebra $R$ with a derivation $\partial$), then $\th{n}:=\frac{1}{n!}\partial^n$ defines an iterative derivation on $R$. On the other hand, for an iterative derivation $\theta$, the map $\partial:=\theta^{(1)}$ always is a derivation, and $\th{n}$ equals $\frac{1}{n!}\partial^n$ by the iteration rule. Hence, differential rings containing $\QQ$ are special cases of ID-rings.
Since for a differentially simple ring in characteristic zero, its ring of constants always is a field (same proof as for ID-simple rings), we see that differentially simple rings in characteristic zero are a special case of ID-simple rings in arbitrary characteristic.
§ PROPERTIES OF ID-SIMPLE RINGS AND ID-MODULES OVER ID-SIMPLE RINGS
We summarize some properties of ID-simple rings. Proofs can be found in <cit.> or <cit.>.
Throughout the section, let $(S,\theta)$ denote an ID-simple ring with constants $C$.
Then $S$ is an integral domain, and its ring of constants $C$ is a field. Furthermore, the field of fractions of $S$ has the same constants as $S$ (cf. <cit.>).
If $\m\ideal S$ is a maximal ideal, and $k:=S/\m$ is the residue field. Then $(S,\theta)$ can be embedded into $(k[[t]],\theta_t)$ as ID-rings via
$$S\to k[[t]], s\mapsto \sum_{n=0}^\infty \overline{\th{n}(s)}t^n,$$
where $\overline{\th{n}(s)}$ denotes the image of $\th{n}(s)\in S$ in the residue field $k$
(cf. <cit.>). This will be important later on, in the case that $k=C$.
Now consider an ID-module $(M,\theta_M)$ over the ID-simple ring $(S,\theta)$.
Such an ID-module is projective as $S$-module, i.e. locally free (cf. <cit.>). In particular, if $S$ is a local ring, the ID-module is free as a module.
For the special ID-simple ring $(C[[t]],\theta_t)$, we even have:
(cf. <cit.>)
Let $C$ be a field. Then every ID-module over $(C[[t]],\theta_t)$ is trivial.
§ PICARD-VESSIOT RINGS AND GALOIS GROUPS
Throughout the section, let $(S,\theta)$ denote an ID-simple ring, and $(M,\theta_M)$ an ID-module over $S$.
A solution ring for $M$ is an ID-ring $0\ne (R,\theta_R)$ together with a homomorphism of ID-rings $f:S\to R$ s.t. the natural homomorphism
\begin{equation}\label{eq:natural-homom}
R\otimes_{C_R} C_{R\otimes_S M}\longrightarrow R\otimes_S M \tag{$\dagger$}
\end{equation}
is an isomorphism.[Take care that the definition of a solution ring given here is different to that in <cit.>, but consistent with the one in <cit.>. However, the definitions of a PV-ring are all equivalent.]
A Picard-Vessiot ring (PV-ring) for $M$ is a minimal solution ring $0\ne (R,\theta_R)$ which is ID-simple and has the same field of constants as $S$.
Here, minimal means that if $0\ne (\tilde{R},\theta_{\tilde{R}})$ with $\tilde{f}:S\to \tilde{R}$ is another solution ring, then any monic ID-homomorphism $g:\tilde{R}\to R$ (if it exists) is an isomorphism.
* Since the kernel of an ID-homomorphism is an ID-ideal, and $S$ is ID-simple, the homomorphism $f$ is always injective. Therefore, we can view any solution ring $R$ as an extension of $S$, and we will omit the homomorphism $f$.
* If $R$ is an ID-simple ring, then the homomorphism (<ref>) is always injective (cf. <cit.>), and therefore $R\otimes_{C_R} C_{R\otimes_S M}$ can be seen as a free ID-submodule of $R\otimes_S M$. Hence, an ID-simple ring is a solution ring if and only if $R\otimes_S M$ has a basis of ID-constant elements.
* Assume that $M$ is a free $S$-module with basis $\vect{b}=(b_1,\dots, b_r)$, and $R$ is an ID-simple solution ring for $M$, then there is a matrix $X\in \GL_r(R)$ s.t. $\vect{b}X$ is a basis of constant elements in $R\otimes_S M$. Such a matrix will be called a fundamental solution matrix for $M$ (with respect to $\vect{b}$).
* Every PV-ring is faithfully flat over $S$ (cf. <cit.>). This will have a nice effect on the Galois groups. Namely, the Galois group of $R/S$ (definition given later) is the same as the one of $\Quot(S)\otimes_S R$ over the field of fractions $\Quot(S)$.
* If $\m\ideal S$ is a maximal ideal, $k:=S/\m$, and $(S,\theta)\to (k[[t]],\theta_t)$ the embedding given above. Then $k[[t]]$ is a solution ring for any ID-module $M$ over $S$.
The next proposition shows the importance of this remark in case that $k=C$. In this case the ring $\tR$ of the proposition can be chosen to be the ring $C[[t]]$.
Let $\tR$ be an ID-simple solution ring for $M$ with the same constants as $S$ (i.e. $C_{\tR}=C_S$). Then there is a unique Picard-Vessiot ring $R$ inside $\tR$.
* If $M$ is a free $S$-module, then $R$ is the $S$-subalgebra of $\tR$ generated by the coefficients of a fundamental solution matrix and the inverse of its determinant.
* For general $M$ (i.e. $M$ “only” locally free), $R$ is obtained as follows:
Let $x_1,\dots, x_l\in S$ such that $\gener{x_1,\dots, x_l}_S=S$ and $M[\frac{1}{x_i}]$ is free over $S[\frac{1}{x_i}]$ for all $i=1,\dots, l$. Let $\vect{e}=(e_1,\dots, e_r)$ be a $\tR$-basis of $\tR\otimes_S M$ consisting of ID-constant elements, and for all $i$, let $\vect{b_i}$ be a basis of $M[\frac{1}{x_i}]$ over $S[\frac{1}{x_i}]$ consisting of elements in $M$.
Define the matrices $Y_i\in \Mat_{r\times r}(\tR)$ via $\vect{b_i}=\vect{e}Y_i$ ($i=1,\dots,l$), and choose $n_i\in\NN$ such that $x_i^{n_i}M\subseteq \gener{\vect{b_i}}_S$. Then
$$R:=S[Y_j, \det(x_j^{n_j}Y_j^{-1})\mid j=1,\dots l].$$
In particular, every Picard-Vessiot ring for $M$ is a finitely generated $S$-algebra.
For the proofs see <cit.>.
* The first part of the proposition shows that for an ID-module which is free as $S$-module, the definition given here coincides with the usual one given for example in <cit.> (if the constants are algebraically closed) resp. in <cit.>. The second part gives a receipt to compute PV-rings in general.
* If $S$ has a maximal ideal $\m\ideal S$ such that $S/\m=C$, then $(C[[t]],\theta_t)$ is an ID-simple solution ring for $M$ with same constants as $S$. Hence, the existence of a Picard-Vessiot ring is guaranteed in this case.
* Different maximal ideals of $S$ with residue field $C$ lead to different ID-embeddings $S\to C[[t]]$. If the field of constants is not algebraically closed, this can lead to different non-isomorphic PV-rings, as in the example given in Section <ref>.
* If the field of constants is algebraically closed, then as in the case of differential fields, for every ID-module there exists a PV-ring, and furthermore, all PV-rings for a given ID-module are isomorphic (cf. <cit.>).
* Picard-Vessiot rings behave well under extension of the base field by constants. By this we mean the following: Let $D/C$ be a field extension, $D$ equipped with the trivial iterative derivation, and $S_D:=S\otimes_C D$. If $R/S$ is a Picard-Vessiot ring for some ID-module $M$, then $R_D:=R\otimes_C D$ is a
Picard-Vessiot ring for the $S_D$-ID-module $M \otimes_C D$.
In view of the previous remark (with $D$ being the algebraic closure of $C$) this also implies that all Picard-Vessiot rings for some ID-module become isomorphic over the algebraic closure of the constants $C$.
§.§ The differential Galois group scheme
In the classical case over a differential field with algebraically closed field of constants the differential Galois group is defined as the group of differential automorphisms of a Picard-Vessiot ring over the base differential field. This group turns out to be a Zariski-closed subgroup of some $\GL_n(C)$ where $C$ denotes the field of constants.
If the constants are not algebraically closed, and in the iterative differential setting in positive characteristic, this group of differential automorphisms might be “too small”, and one has to consider a group valued functor instead.
Let $S$ be an ID-simple ring with field of constants $C$, and let $R$ be a PV-ring for some ID-module $M$ over $S$. Then we define the group functor
$\underline{\Aut}^{\theta}(R/S):\cat{Algebras}/C\to \cat{Grps}$ associating to each $C$-algebra $D$ the group $\Aut^\theta(R\otimes_C D/S\otimes_C D)$ of ID-automorphisms of $R\otimes_C D$ that fix the elements of $S\otimes_C D$. Here, $D$ is equipped with the trivial iterative derivation.
We call $\G=\underline{\Aut}^{\theta}(R/S)$ the ID-Galois group (scheme) of $R/S$ and also denote it by $\uGal(R/S)$.
The term “Galois group scheme” is justified by the following
(cf. <cit.>)
The group functor $\uGal(R/S)$ is represented by the algebra $C_{R\otimes_S R}$ which is a finitely generated $C$-algebra. Hence, $\uGal(R/S)=\Spec(C_{R\otimes_S R})$ is an affine group scheme of finite type over $C$.
* If $R$ is a PV-ring for a free module $M$ of rank $n$, hence generated by the entries of a fundamental solution matrix $X\in \GL_n(R)$ and the entries of $X^{-1}$, then the ring of constants $C_{R\otimes_S R}$ is generated by the entries of $Z:=(X^{-1}\otimes 1)(1 \otimes X)\in \GL_n(C_{R\otimes_S R})$ [By $1\otimes X$ we mean the matrix whose $(i,j)$-th entry is $1\otimes X_{ij}$, and similar for $X^{-1}\otimes 1$.] and of $Z^{-1}=(1\otimes X^{-1})(X\otimes 1)$.
This representation of $C_{R\otimes_S R}$ provides a closed embedding $\uGal(R/S)
%=\Spec(C_{R\otimes_S R})
\hookrightarrow \GL_n$, and the action of $\uGal(R/S)$ on $R$ is given by multiplying the fundamental solution matrix $X$ by a matrix of $\uGal(R/S)\subseteq \GL_n$ from the right.
* As a Picard-Vessiot ring $R$ is faithfully flat over $S$ (cf. <cit.>), it is linearly disjoint over $S$ to the field of fractions $F=\Quot(S)$. In particular, $R_F=F\otimes_S R$ is the localisation of $R$ by all non-zero elements in $S$, and hence is a Picard-Vessiot ring over $F$. Therefore,
$\uGal(R_F/F)=\uGal(R/S)$. Hence, the ID-Galois group can be computed over any localisation of $S$. This simplifies its computation, since one can reduce to the case of a free module as given in <ref>.
§ EXAMPLE
We now give an example of an ID-module which is not free as a module. Therefore, we first need an ID-simple ring for which non-free projective modules exist. The most standard examples for non-free projective modules are non-principle ideals of Dedekind domains. This will be our example after having attached iterative derivations to the Dedekind domain as well as the module.
§.§ An ID-simple ring having non-free projective modules
Let $C$ be any field and let
which is the localisation of an integral extension of $C[t]$ of degree $3$. $S$ is integrally closed, and hence $S$ is a Dedekind domain.
Since we inverted $3s^2-1$, $S$ is étale over $C[t]$, and hence the iterative derivation $\theta_t$ by $t$ on $C[t]$ can be uniquely extended to an iterative derivation $\theta$ on $S$ (cf. <cit.>). The $\th{n}(s)$ can be computed successively using the equation
obtained from $s^3-s=t^2$ by applying $\theta$.
In particular,
The ID-ring $(S,\theta)$ is ID-simple.
If $0\ne I\lneq S$ is an ideal, then $I\cap C[t]$ is an ideal of $C[t]$. Since, $S$ is the localisation of an integral extension, the ideal $I\cap C[t]$ also is nontrivial. Furthermore, if $I$ is an ID-ideal, then obviously $I\cap C[t]$ is also ID-stable, hence an ID-ideal of $C[t]$. But $(C[t],\theta_t)$ is ID-simple by example <ref>. Hence, $S$ also contains no nontrivial ID-ideals.
§.§ A non-free ID-module over $S$ in characteristic zero
We first restrict to the case of ${\rm char}(C)=0$. In this case, an iterative derivation $\theta_M$ on $M$ is uniquely determined by the derivation $\partial_M:=\theta_M^{(1)}$.
We consider the $S$-module $M$ generated by two elements $f_1$ and $f_2$ subject to the relations $tf_1-sf_2=0$ and $(s^2-1)f_1-tf_2=0$. As $S$-module $M$ is isomorphic to the ideal $I=\gener{s,t}_S\subseteq S$ by mapping $f_1$ to $s$ and $f_2$ to $t$.
Since $I$ is a non-principal ideal of $S$, $I$ and hence $M$ is a non-free projective $S$-module of rank $1$.
For any $b\in S$,
$$\partial_M(f_1):=bf_1+\frac{3s^2+1}{3s^2-1}f_2\quad \text{ and } \quad
\partial_M(f_2):=sf_1+bf_2$$
defines a derivation on $M$.
Furthermore, every derivation on $M$ can be written in this form.
Using the definition, one obtains
\begin{eqnarray*}
\partial_M(tf_1-sf_2) &=& \partial(t)f_1+t\partial_M(f_1)- \partial(s)f_2-s\partial_M(f_2) \\
&=& f_1+tbf_1+t\frac{3s^2+1}{3s^2-1}f_2 - \frac{2t}{3s^2-1}f_2-s^2f_1-sbf_2 \\
&=& b(tf_1-sf_2)+(1-s^2)f_1+\left( \frac{3s^2+1}{3s^2-1}-\frac{2}{3s^2-1}\right)tf_2\\
&=& b(tf_1-sf_2)-\left((s^2-1)f_1-tf_2\right)=0,
\end{eqnarray*}
and similarly $\partial_M\left((s^2-1)f_1-tf_2\right)=0$. Hence, the derivation is a well-defined derivation on $M$.
On the other hand, given a derivation $\partial_M$ on $M$, we obtain a derivation on the $\Quot(S)$-vector space $\tilde{M}:=\Quot(S)\otimes_S M$ by scalar extension. The element $f_2$ is a basis of that vector space, and $f_1=\frac{s}{t}f_2\in \tilde{M}$.
Hence, $\partial_M(f_2)=af_2$ for some $a\in \Quot(S)$ which can also be written as
$\partial_M(f_2)=sf_1+bf_2$ for $b=a-\frac{s^2}{t}$.
\begin{eqnarray*}
\partial_M(f_1) &=& \partial_M\left( \frac{s}{t}f_2\right)= \partial\left( \frac{s}{t}\right)f_2+
\frac{s}{t}\partial_M(f_2)
&=&\left(\frac{2}{3s^2-1}-\frac{s}{t^2}\right)f_2+ \frac{s^3}{t^2}f_2+bf_1
= bf_1+ \frac{3s^2+1}{3s^2-1}f_2.
\end{eqnarray*}
Therefore, $\partial_M$ is of the form above for some $b\in \Quot(S)$. But, $M$ is stable under this derivation if and only if $bf_2\in M$ as well as $bf_1\in M$. So $M$ is stable under the derivation if and only if $bM\subseteq M$, i.e. $b\in S$.
§.§ Picard-Vessiot rings and Galois groups for this ID-module
The ID-ring $S$ has a $C$-rational point, e.g. the ideal $\m=(s-1,t)$, and we obtain an ID-embedding $S\to (S/\m)[[t]]\isom C[[t]]$.[Using the variable $t$ in the power series ring is justified by the fact, that $t\in S$ indeed maps to $t\in C[[t]]$ via the given embedding.] So by Prop. <ref> there exists a Picard-Vessiot ring for $M$ inside $C[[t]]$, and we follow the explicit description of the Picard-Vessiot ring given there.
First at all, we choose $x_1:=s$ and $x_2:=s^2-1$. Then $M[\frac{1}{x_1}]$ is free over $S[\frac{1}{x_1}]$ with basis $b_1:=f_1$, and $M[\frac{1}{x_2}]$ is free over $S[\frac{1}{x_2}]$ with basis $b_2:=f_2$. Further, $x_1M=sM\subseteq \gener{b_1}_S$ and $x_2M=(s^2-1)M\subseteq \gener{b_2}_S$, hence we can choose $n_1=n_2=1$.
Let $0\ne e\in C[[t]]\otimes_S M$ be a constant element, and $y\in C[[t]]$ such that $f_1=ye$.
As $s\not\in \m$, $s$ is invertible in $S_\m\isom C[[t]]$, and $f_2=\frac{t}{s}f_1
\in C[[t]]\otimes_S M$. In particular, $f_1$ is a basis of $C[[t]]\otimes_S M$.
Actually, this also implies that $y$ is invertible in $C[[t]]$, as it is the base change matrix between the bases $f_1$ and $e$ of $C[[t]]\otimes_S M$. As
$$\partial(y)e=\partial_M(ye)=\partial(f_1)=bf_1+\frac{3s^2+1}{3s^2-1}f_2=\left( b+\frac{3s^2+1}{3s^2-1}\frac{t}{s}\right)ye,$$
$y$ is a solution of the differential equation
\begin{equation}\label{eq:diff-eq-for-y}
\partial(y)=\left( b+\frac{3s^2+1}{3s^2-1}\frac{t}{s}\right)y.
\end{equation}
Furthermore we get (with notation as in Prop. <ref>)
$Y_1=y, Y_2=\frac{yt}{s}$, $\det(x_1^{n_1}Y_1^{-1})=\frac{s}{y}$ as well as
$\det(x_2^{n_2}Y_2^{-1})=\frac{(s^2-1)s}{yt}=\frac{t}{y}$. Hence,
$$R=S[y,\frac{yt}{s}, \frac{s}{y}, \frac{t}{y}].$$
Be aware that the inverse of $y$ is not in $R$.
As $M$ is a module of rank $1$, the Galois group is a subgroup of $\GL_{1}=\GG_{m}$. Hence, the Galois group is $\GG_{m}$ or one of the groups $\mu_n$ of $n$-th roots of unity. The Galois group is $\GG_{m}$ if $y$ is transcendental over $S$, and it is $\mu_n$ if $n$ is the least positive integer such that $y^n\in S$.
Whether $y$ is transcendental over $S$ or not, depends on the choice of $b$.
* If we take, $b=\frac{-3st}{3s^2-1}$, then $\partial(y)=\frac{t}{(3s^2-1)s}y$, and hence
Therefore, $\frac{y^2}{s}$ is a constant, i.e. $y$ is a square root in $C[[t]]$ of $cs$ for some $0\ne c\in C$.
Actually, any $c\ne 0$ such that $cs$ is a square in $C[[t]]$ will do, as different choices just correspond to different choices of the constant basis $e$. As in $C[[t]]$, $s\equiv 1\mod t$, there exists a square root $\sqrt{s}\in C[[t]]$ of $s$ with $\sqrt{s}\equiv 1\mod t$. Hence, we can choose $c=1$, and $y=\sqrt{s}$, and obtain
an extension of degree $2$ and Galois group $\mu_2$.
If we would have taken the maximal ideal to be $\m=(s+1,t)$, and the corresponding embedding $S\hookrightarrow (S/\m)[[t]]\isom C[[t]]$, then in the last step $s\equiv -1\mod t$ inside $C[[t]]$, and we have a square root $\sqrt{-s}$ of $-s$ in $C[[t]]$ with $\sqrt{-s} \equiv 1\mod t$. This leads to the Picard-Vessiot ring
which is not isomorphic as an $S$-algebra to $R$ above, if $-1$ is not a square in $C^\times$.
The Galois group, however, is again $\mu_2$.
* If we take, $b=0$, then inside $\Quot(S)$ we have
$$\partial\left(\frac{y}{s}\right)= \frac{\partial(y)}{s}-y\frac{\partial(s)}{s^2}
= \frac{(3s^2+1)t}{(3s^2-1)s}\frac{y}{s}-\frac{2t}{(3s^2-1)s}\frac{y}{s}=\frac{t}{s}\frac{y}{s}$$
If $\frac{y}{s}$ was not transcendental over $\Quot(S)$, then some $n$-th power $w=\left(\frac{y}{s}\right)^n$ would be in $\Quot(S)$. For $w$ we get the differential equation
$$\partial(w)= \frac{nt}{s} w.$$
Writing $w=w_0(s)+w_1(s)t$ with $w_0,w_1\in C(s)$, we calculate
\begin{eqnarray*}
\partial(w)&=& \partial(w_0(s))+\partial(w_1(s))t+w_1(s)
&=& \left( w_1(s)+ w'_1(s)\frac{2(s^3-s)}{3s^2-1} \right)+ \frac{2w'_0(s)}{3s^2-1}t,
\end{eqnarray*}
as well as
$$ \frac{nt}{s} w= \frac{nt}{s}w_0(s)+ \frac{nt^2}{s}w_1(s)
= n(s^2-1)w_1(s) + \frac{nw_0(s)}{s}t.$$
Here $w'_0(s)$ and $w'_1(s)$ denote the usual derivatives of rational functions.
By comparing coefficients of $t$, we obtain
\begin{eqnarray*}
\frac{nw_0(s)}{s} &=& \frac{2w'_0(s)}{3s^2-1} \qquad \text{and} \\
(ns^2-n-1)w_1(s) &=& w'_1(s)\frac{2(s^3-s)}{3s^2-1}.
\end{eqnarray*}
If $w_0,w_1\ne 0$, this implies
$$\deg_s(w_0(s))=\deg_s\left( \frac{nw_0(s)}{s}\right)+1
= \deg_s\left( \frac{2w'_0(s)}{3s^2-1}\right) +1
=\deg_s(w'_0(s)) -1,$$
$$\deg_s(w_1(s))= \deg_s\left( (ns^2-n-1)w_1(s)\right) -2
= \deg_s\left( w'_1(s)\frac{2(s^3-s)}{3s^2-1}\right) -2
=\deg_s(w'_1(s)) -1.$$
But $\deg_s(f'(s))\leq \deg_s(f(s))-1$ for all $0\ne f(s)\in C(s)$, and hence $w_0(s)=w_1(s)=0$, i.e. $w=0$ which is impossible.
Hence, there is no such $w$, and $\frac{y}{s}$ and also $y$ are transcendental over $S$.
§.§ A non-free ID-module over $S$ in positive characteristic
Finding an example in positive characteristic is harder, since one is not done by giving just $\theta_M^{(1)}$, but by giving all $\theta_M^{(p^j)}$ which moreover have to commute and have to be nilpotent of order $p$.
We will follow a different approach here.
We start with the example in characteristic zero given in the previous paragraph.
The iterative derivation on $C[t]$ is already defined on $\ZZ[t]$ and extends to the ring
since the latter is étale over the former.
Therefore, the ID-ring $S$ from above (with constants $C$) is obtained as $S=C\otimes_\ZZ S_\ZZ$. And this holds in any characteristic.
For constructing an ID-module $M$ over $S$, one can start with a projective module $M'$ over $S_\ZZ$, and define a derivation on $M:=S\otimes_{S_\ZZ} M'$. If the corresponding iterative derivation stabilizes $M'$, one can reduce modulo $p$, to obtain an iterative derivation on $M'/pM'$.
This is then an ID-module over $\mathbb{F}_p\otimes_\ZZ S_\ZZ$.
Therefore take the ID-module over $S_\QQ$ from above with $b=\frac{-3st}{3s^2-1}$. Then we know that $e=\frac{1}{y}f_1$ is a constant basis of $R\otimes M$, where $y=\sqrt{s}$.
Hence, $\theta_M(f_1)=\theta_M(ye)=\theta(y)e=\frac{\theta(y)}{y}f_1$.
Replacing $y$ by $\sqrt{s}$ and using the chain rule (cf. <cit.>)
one obtains:
=\sqrt{s}\cdot \sum_{k=0}^\infty \binom{1/2}{k} \left(\frac{\theta(s)}{s}-1\right)^k.$$
Therefore, all appearing rational numbers only have powers of $2$ in the denominator, and we can reduce modulo any prime $p$ different from $2$,
obtaining a non-free ID-module in characteristic $p$.
Katsutoshi Amano and Akira Masuoka.
Picard-Vessiot extensions of Artinian simple module algebras.
J. Algebra, 285(2):743–767, 2005.
Yves André.
Différentielles non commutatives et théorie de Galois
différentielle ou aux différences.
Ann. Sci. École Norm. Sup. (4), 34(5):685–739, 2001.
Tobias Dyckerhoff.
The inverse problem of differential Galois theory over the field
Preprint, 2008.
Hideyuki Matsumura.
Commutative ring theory, volume 8 of Cambridge Studies in
Advanced Mathematics.
Cambridge University Press, Cambridge, second edition, 1989.
Translated from the Japanese by M. Reid.
B. Heinrich Matzat and Marius van der Put.
Iterative differential equations and the Abhyankar conjecture.
J. Reine Angew. Math., 557:1–52, 2003.
Andreas Maurischat.
Infinitesimal group schemes as iterative differential Galois
J. Pure Appl. Algebra, 214(11):2092–2100, 2010.
Andreas Maurischat.
Picard-Vessiot theory of differentially simple rings.
J. Algebra, 409:162–181, 2014.
Andreas Maurischat.
A categorical approach to Picard-Vessiot theory.
Preprint available from arXiv at http://arxiv.org/abs/1507.04166,
Juli 2015.
Andreas Röscheisen.
Iterative Connections and Abhyankar's Conjecture.
PhD thesis, Heidelberg University, Heidelberg, Germany, 2007.
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1511.00330
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Multiple stellar populations in the Milky Way globular clusters manifest themselves with a large variety.
Although chemical abundance variations in light elements, including He, are ubiquitous, the amount of these variations is different in different globulars.
Stellar populations with distinct Fe, C+N+O and slow-neutron capture elements have been now detected in some globular clusters, whose number will likely increase. All these chemical features correspond to specific photometric patterns.
I review the chemical+photometric features of the multiple stellar populations in globular clusters and discuss how the interpretation of data is being more and more challenging. Very excitingly, the origin and evolution of globular clusters is being a complex puzzle to compose.
§ INTRODUCTION
The modern picture of globular clusters (GCs) is different from the idea of these objects being a simple stellar population.
Thanks to high precision photometry we know that the color-magnitude diagram (CMD) of all investigated Milky Way GCs exhibit multiple photometric sequences along all evolutionary stages, from the main sequence (MS) up to the red giant branch (RGB) and the horizontal branch (HB) (e.g. <cit.>; <cit.>).
Multiple photometric sequences are the effect of the presence of different stellar populations, with their distinct chemical composition and stellar internal structure (e.g. <cit.>).
The list of the most investigated chemical elements that have been observed to vary among stars in a given GC includes: He, Li, C, N, O, C+N+O, Na, Mg, Al, Si, K, Fe, and many neutron-capture ($n$-capture) elements.
Not all GCs exhibit the same chemical pattern, and most of them do not show evidence for significant abundance variations in many of these elements.
In the same way, each photometric split along different evolutionary stages of the CMD tells us a different story about the nature of the multiple stellar populations hosted in GCs.
I review the chemical abundance variations observed in the Milky Way GCs and the way we identify them photometrically. I will focus on the more massive GCs, that are those exhibiting more complex and puzzling chemical/photometric patterns.
§ LIGHT ELEMENTS VARIATIONS
The most common observed star-to-star abundance patterns in Galactic GCs (GGCs) are the C-N and O-Na anticorrelations. Chemical variations in CNONa have been found to be ubiquitous among the GGCs investigated so far (e.g. <cit.>).
On the other hand, the size of these variations appears to change with the mass, as more massive GCs are generally those having the more extended anticorrelations.
Chemical variations in light elements are responsible for many of the multiple sequences observed along the CMDs.
As displayed in Fig. <ref>, carbon, nitrogen and oxygen form strong CN, NH, CH, and OH molecular bands in the ultraviolet region of the spectrum and cause the split RGBs and MSs observed in CMDs constructed by using the UV bands.
A clear example of this behaviour is M 4, represented in Fig. <ref>, which clearly exhibits two RGBs populated by C-O-rich/N-Na-poor and C-O-poor/N-Na-rich stars. Indeed, stars with different content in these elements define different sequences along the CMD when the $(U-B)$ color is used (e.g. <cit.>).
The huge effort, through the Legacy GO-13297 (<cit.>), to analyse the GCs multiple stellar populations in UV bands with HST is based on this finding.
Helium variations have been observed within GCs.
Multi-wavelength photometry of GCs has been often used to infer the relative helium abundances in GCs. Multiple MSs and RGBs, detected in most clusters (<cit.>), are interpreted as due to the presence of stellar populations with different He content (e.g. <cit.>; <cit.>).
A few GCs have highly He-enhanced second populations. Among the GCs with highest He enhancements one of the most studied is NGC2 2808. Photometry suggests the presence of at least five distinct stellar populations that span a wide range of He abundance, up to $Y=0.40$ (<cit.>).
The degree of internal variations in He correlates with the mass of the cluster as shown in <cit.>.
On the spectroscopic side, the determination of He contents is difficult because IR He lines are subject to chromospheric effects and lines in the optical can only be observed at high temperatures. To date, evidence for the presence of significant He variations using spectroscopic analysis have been provided for NGC 2808 and $\omega$ Centauri (<cit.>; <cit.>). For NGC 2808, high He abundances have been inferred for blue HB stars (<cit.>).
Some GCs also exhibit Mg-Al anticorrelations (e.g. <cit.>; <cit.>; <cit.>; <cit.>). In many GCs only variations in Al are observed, while Mg does not change significantly.
Right panels: The $U$-$(U-B)$ CMD from ground-based photometry for M 4, with the location of N-Na-poor/C-O-rich and N-Na-rich/C-O-poor stars (<cit.>) (upper panel). The lower panel represents the HST $m_{F275W}$-$c_{F275W,F336W,F438W}$ diagram around the MS of M 4 (<cit.>). Both the RGB (upper panel) and MS (lower panel) splits in these bands are due to light elements variations. Left panels: Synthetic spectra for RGB stars computed with the mean C/N/O abundances of the first (blue) and second (red) population in M 4, with the location of the relevant molecular bands (upper panel). The middle panel displays the difference between the two spectra. The bandpasses of the HST filters is shown in the lower panel. The HST Survey of multiple populations observes GCs in filters $F275W$, $F336W$ and $F438W$.
§ ANOMALOUS GCS
It is a long time that high resolution spectroscopy revealed large internal variations in Fe and in the elements produced via slow $n$-capture reactions ($s$-elements) in the most massive GGC $\omega$ Centauri (e.g. <cit.>; <cit.>).
More recently we realised that, although this phenomenon is quite rare in Milky Way GCs, it is not confined to $\omega$ Centauri.
Indeed, a few GCs (“anomalous") have been found to show a genuine internal variation in the bulk metallicity (e.g. <cit.>, <cit.>; <cit.>; <cit.>, <cit.>; <cit.>; <cit.>).
Some of the anomalous GCs, specifically NGC 1851, M 22, M 2, NGC 5286 and M 19 exhibit internal variations in $s$-process elements (<cit.>; <cit.>; <cit.>; <cit.>; <cit.>); and for NGC 1851 and M 22 significant variations in the total C+N+O have been also detected (<cit.>; <cit.>).
The $s$-elements and C+N+O abundances are positively correlated to Fe variations; the common light elements variations, typical of normal GGCs, are independently present within each population with different Fe/$s$-elements/C+N+O (<cit.>).
Both the $s$-elements and the total C+N+O chemical contents are instead uniform in typical GGCs.
Photometrically, the most striking feature of anomalous GCs can be identified in the SGB region, which is split (<cit.>; <cit.>). This feature is also peculiar of anomalous GCs, since typical GGCs exhibit a single SGB in visual bands.
On theoretical background split SGBs are the indication of stellar populations with different C+N+O or different age (<cit.>).
Spectroscopic+photometric campaigns have confirmed that the SGB splits are produced by stars with different interior structures due to different overall C+N+O and metallicities (<cit.>; <cit.>; <cit.>). The RGB of anomalous GCs is also peculiar, as they show well-separated multiple branches in bands where typical GGCs exhibit single sequences, e.g. in the index $c_{BVI}$ and in the $U$-$(U-V)$, where the faint and the bright SGB are clearly connected with the red and the blue RGB, respectively (<cit.>).
A summary of some spectroscopic and photometric features of anomalous GCs is presented in Fig. <ref>, where the double SGB and RGB of M 22 due to the C+N+O variations are clearly visible. Chemical abundances for stars distributed on the split SGB has been provided for M 22 and NGC 1851, demonstrating that in both cases the $s$-rich stars distribute on the anomalous faint SGB (see CMDs on the right panels of Fig. <ref>). Some chemical variations in $s$-elements for M 22, M 2 and NGC 5286 have been represented in the lower panels of Fig. <ref>.
The similarity of these objects with $\omega$ Centauri is remarkable, not just because of the metallicity variations but also for the chemical pattern is $s$-elements. <cit.> presented a comparison between M 22 and $\omega$ Centauri and both the GCs have a very similar rise of the $s$-elements content as a function of Fe in the common Fe-range ([Fe/H]$\lesssim-$1.5, see right panels in Fig. <ref>). Additionally both these clusters present similar patterns in the Na-O anticorrelation present among stars with different Fe and C+N+O variations (<cit.>).
Upper panels: CMDs obtained from HST photometry for the anomalous GCs M 22 and NGC 1851. On the left panel the $m_{F336W}$-$(m_{F336W}-m_{F606W})$ for M 22 clearly shows the SGB split. The anomalous faint SGB is connected with a redder RGB sequence (red). The right panels display the $m_{F606W}$-$(m_{F606W}-m_{F814W})$ CMD for NGC 1851 and M 22 around the SGB region. In both clusters $s$-poor and $s$-rich stars, as inferred from spectroscopy (<cit.>; <cit.>), have been represented in blue and red, respectively.
Lower panels: [La/Eu] as a function of [Y/Eu] for the anomalous GCs M 22, M 2 and NGC 5286 (<cit.>; <cit.>; <cit.>). $s$-poor and $s$-rich stars have been represented with blue triangles and red dots, respectively.
§.§ On the Fe variations of M 22
Recent claim has been made that the anomalous GC M 22 does not host stellar populations with different Fe (<cit.>).
The argument is that, if photometric gravities are used instead of those from the ionisation equilibrium, the difference in Fe ii between the two stellar populations with different $s$/C+N+O content disappears.
The Fe i difference instead cannot be removed whatever technique is used to derive the atmospheric parameters. The reason supplied by <cit.> for this behaviour was the presence of non-local thermodynamic effects (NLTE) affecting Fe i.
Such an effect, if real, would strongly affect what we know about anomalous GCs, including $\omega$ Centauri. Indeed, the well-known rise in $s$-process elements with Fe in $\omega$ Centauri lies on the same Fe-range spanned by M 22 stars. Given the similar Fe and chemical patterns of these two GCs, if the common Fe-regime is considered, there is no reason why the same effects should not apply to $\omega$ Centauri (see left panels of Fig <ref>). This would imply that there is no rise of $s$-elements abundances with Fe also in $\omega$ Centauri.
As it will be presented in <cit.>, however, uncounted NLTE effects on Fe i are not responsible for the appearance of the Fe spread in M 22 (see <cit.> for discussion of NLTE effects on Fe).
This result was already suggested by the variations in CaT lines found in M 22 (<cit.>), that should not be affected by the same NLTE effects.
On the other hand, the disappearance of the Fe ii difference in the Mucciarelli et al. analysis is due to the fact that they neglect the C+N+O variations in M 22 discussed in Sect. <ref> and get significant systematic offset in gravities, that affect only the $s$-poor population.
As shown in Fig. <ref>, this offset makes the two populations of M 22 lying on the same isochrone, which is unrealistic given their different C+N+O content, as suggested both from spectroscopy and photometry.
Such systematic in surface gravity affects the abundances obtained from ionised Fe, but does not affect Fe i significantly.
On the other hand, the main advantage of stellar parameters independent on photometry is that they do not introduce systematic errors that affect one stellar population only. This is suggested by the temperature-gravity distribution obtained from <cit.>, which agrees with that expected from two stellar populations with different C+N+O.
Thus, the Fe variations in M 22 and its similarity with $\omega$ Centauri are confirmed. There is no support for NLTE affecting this result, but systematics on the atmospheric parameters due to the neglecting of C+N+O variations may introduce spurious results.
Left panels: [Ba/Fe] as a function of [Fe/H] for $\omega$ Centauri (<cit.>) and M 22 (<cit.>; shifted by 0.15 dex in [Fe/H]). For M 22 the $s$-poor and $s$-rich stars have been plotted as blue triangles and red dots, respectively. A straight line (green dashed line) tracing the rise of $s$-elements with Fe for stars with [Fe/H]$\lesssim -$1.5 in $\omega$ Centauri is super-imposed to the M 22 data. If there are no Fe variations in M 22, they will also disappear in $\omega$ Centauri for stars with [Fe/H]$\lesssim -$1.5 dex, removing the well-known $s$-pattern in this GC.
Right panels: Theoretical isochrones from BaSTI (<cit.>) for a CNO-poor and CNO-rich population. Superimposed to the isochrones are the atmospheric parameters for M 22 RGBs from <cit.> (middle-panel) and <cit.> (right-panel). The atmspheric parameters from <cit.> force the stars on a single sequence.
§ THE CHEMICAL ENRICHMENT IN GCS
Explaining the whole observational scenario of multiple stellar populations in GCs is difficult.
Simply based on the chemical abundances, we may think that normal and anomalous GCs have experienced a different chemical evolution.
Although we know from recent photometric results that GCs host more than two stellar populations (<cit.>), a “two-populations” scenario can still approximatively explain the light elements chemical patterns observed in typical GCs, as sketched in Fig. <ref> (left panel).
We may assume a self-pollution scenario, e.g. the first generation (1G) is enhanced in O, depleted in Na (just as typical of Galactic halo stars), and, later on, a second generation (2G) forms.
If we suppose a similar scenario, 2G stars form from hot-H burning processed material, which has been realised from some kind of polluter. The proposed candidate polluters are fast-rotating massive stars (<cit.>), asymptotic giant branch stars (<cit.>), or supermassive stars (<cit.>).
If we instead do not invoke multiple star-formation episodes, we have to find other mechanisms that can account for the chemical variations. In this context, massive interacting binaries have been proposed to be able of reproduce some of the chemical variations (<cit.>).
None of these scenarios at the moment is accepted, as all of them have serious shortcomings in explaining the observations (see <cit.>).
If explaining normal GCs is complex, anomalous GCs present even more challenges. The chemical pattern displayed by these objects is complex and we do not have, at the moment, a solution to interpret the observations in terms of a reasonable chemical enrichment history. As an example, the right panel of Fig. <ref> shows the Na and O abundances in M 22. In this GC internal variations in Na and O exist in stars with different Fe/$s$-elements/C+N+O content. This means that, for this object, as well as in the other anomalous GCs, we do not know how the intra-cluster chemical enrichment proceeded, e.g. if the first channel to be active was the enrichment in the hot H-burning products (producing light elements patterns) or the enrichment in Fe. Surely, by assuming the self-pollution scenario, polluters of different mass have contributed to the intra-cluster pollution, including low-mass AGBs that produce $s$-elements and increase the C+N+O (e.g. <cit.>; <cit.>).
The enrichment in Fe suggests that anomalous GCs have been able of retaining Supernovae ejecta. Metallicity variation, which was thought to be an exclusive feature of $\omega$ Centauri, proposed to be the remnant of a dwarf galaxy (e.g. <cit.>), is a more widespread phenomenon in GCs. Thus, it is tempting to speculate that the anomalous GCs may be nuclei of disrupted dwarf galaxies, as suggested for $\omega$ Centauri.
[Na/Fe] versus [O/Fe] for M 4 (<cit.>) and M 22 (Marino et al. 2009, 2011a). For M 4 a two-generation model is sufficient with the second generation (2G), being Na-rich and O-poor as it formed from the first generation (1G) polluters.
For the anomalous GC M 22, the observational scenario is more complex: the Na and O variations are present in both the Fe/$s$-elements/C+N+O-poor (blue) and Fe/$s$-elements/C+N+O rich (red) populations.
We do not know yet which is the 2G that formed directly after the 1G, e.g. which enrichment occurred first (in Fe or in hot H-burning products). Similar challenges affect the scenario of massive interacting binaries.
§ CONCLUSIVE REMARKS
Thanks to the large observational material, in the last decade the formation and evolution of GCs has became more difficult to understand.
We do not have any model which is able to satisfy all the observational constraints. I conclude by listing some relevant open issues that we should address in the future:
* in general, the origin of the multiple stellar populations in GCs. Will one or more of the proposed scenarios (self-pollution, fast-rotating massive stars, supermassive stars, mass exchange in massive interacting binaries) able to explain the whole observational scenario?
* could the heterogeneity of the multiple population zoo be reconciled with a unique scenario?
* is there some GCs' property, such as mass, determining the length of star formation, and hence the level of chemical enrichment?
* which is the origin of anomalous GCs? Why they exhibit different chemical enrichment with respect to the majority of GGCs? Does their peculiar chemical pattern simply imply a longer star-formation history? Or, could have they originated as nuclei of disrupted galaxies?
I acknowledge the organisers of the Symposium for inviting me to review results on the multiple stellar populations in globular clusters.
I am grateful to all the collaborators who contribute to the presented results.
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1511.00054
|
Gaussian processes have been successful in both supervised and
unsupervised machine learning tasks, but their computational
complexity has constrained practical applications. We introduce a
new approximation for large-scale Gaussian processes, the Gaussian
Process Random Field (GPRF), in which local GPs are coupled via
pairwise potentials. The GPRF likelihood is a simple, tractable, and
parallelizeable approximation to the full GP marginal likelihood,
enabling latent variable modeling and hyperparameter selection on
large datasets. We demonstrate its effectiveness on synthetic
spatial data as well as a real-world application to seismic event
§ INTRODUCTION
Many machine learning tasks can be framed as learning a function given
noisy information about its inputs and outputs. In
regression and classification, we are given inputs and asked to
predict the outputs; by contrast, in latent variable modeling we are
given a set of outputs and asked to reconstruct the inputs that could
have produced them. Gaussian processes (GPs) are a flexible class of
probability distributions on functions that allow us to approach
function-learning problems from an appealingly principled and clean
Bayesian perspective. Unfortunately, the time complexity of exact GP
inference is $O(n^3)$, where $n$ is the number of data points. This makes exact GP calculations infeasible for
real-world data sets with $n > 10000$.
Many approximations have been
proposed to escape this limitation. One particularly simple
approximation is to partition the input space into smaller blocks,
replacing a single large GP with a multitude of local ones. This gains
tractability at the price of a potentially severe independence assumption.
In this paper we relax the strong independence assumptions of
independent local GPs, proposing instead a Markov random field (MRF) of
local GPs, which we call a Gaussian Process Random Field (GPRF). A
GPRF couples local models via pairwise potentials that incorporate
covariance information. This yields a surrogate for the full GP
marginal likelihood that is simple to implement and
can be tractably evaluated and optimized on
large datasets, while still enforcing a smooth covariance
structure. The task of approximating the marginal likelihood is
motivated by unsupervised applications such as the GP latent variable
model <cit.>, but examining the predictions made by our model also
yields a novel interpretation of the Bayesian Committee Machine <cit.>.
We begin by reviewing GPs and MRFs, and
some existing approximation methods for large-scale GPs. In
Section <ref> we present the GPRF objective and examine its
properties as an approximation to the full GP marginal likelihood. We
then evaluate it on synthetic data as well as an application to seismic event location.
§ BACKGROUND
§.§ Gaussian processes
Gaussian processes <cit.> are
distributions on real-valued functions. GPs are
parameterized by a mean function $\mu_\theta(\v{x})$, typically assumed
without loss of generality to be $\mu(\v{x})=0$, and a covariance function (sometimes called a kernel)
$k_\theta(\v{x}, \v{x}')$, with hyperparameters $\theta$. A common choice is the squared
exponential covariance, $k_{SE}(\v{x}, \v{x}') =
\sigma^2_f\exp\left(-\frac{1}{2}\|\v{x}-\v{x}'\|^2 / \ell^2\right)$, with
hyperparameters $\sigma^2_f$ and $\ell$ specifying respectively a
prior variance and correlation lengthscale.
We say that a random function $f(x)$ is Gaussian process distributed if, for
any $n$ input points $X$, the vector of function values $\v{f} = f(X)$ is
multivariate Gaussian, $\v{f} \sim \N(\v{0}, k_\theta(X, X)).$ In many applications we
have access only to noisy observations $\v{y} = \v{f} + \v{\eps}$ for some
noise process $\v{\eps}$. If the noise is iid Gaussian, i.e., $\v{\eps}\sim
\N(\v{0}, \sigma_n^2 \I)$, then the observations are themselves Gaussian, $\v{y} \sim \N(\v{0}, K_y)$, where $K_y = k_\theta(X, X) + \sigma^2_n\I.$
The most common application of GPs is to Bayesian regression
<cit.>, in which we attempt to predict the
function values $\v{f}^*$ at test points $X^*$ via the conditional
distribution given the training
data, $p(\v{f}^* | \v{y}; X, X^*, \theta)$. Sometimes, however, we do not observe the training inputs $X$,
or we observe them only partially or noisily. This setting is known as the Gaussian Process
Latent Variable Model (GP-LVM) <cit.>; it uses
GPs as a model for unsupervised
learning and nonlinear dimensionality reduction. The GP-LVM
setting typically involves multi-dimensional observations, $Y = (\v{y}^{(1)}, \ldots,
\v{y}^{(D)})$, with each output dimension $\v{y}^{(d)}$ modeled as an
independent Gaussian process. The input locations and/or hyperparameters are typically sought via maximization of the marginal likelihood
\begin{align}
\mathcal{L}(X, \theta) = \log p(Y ; X, \theta) &= \sum_{i=1}^D -\frac{1}{2}\log |K_y| - \frac{1}{2} \v{y}_i^T K_y^{-1} \v{y}_i + C\nonumber\\
&= -\frac{D}{2}\log |K_y| - \frac{1}{2}\tr(K_y^{-1} YY^T) + C,\label{eqn:mlik}
\end{align}
though some recent work <cit.> attempts to recover an
approximate posterior on $X$ by maximizing a variational bound. Given a
differentiable covariance function, this
maximization is typically performed by gradient-based methods,
although local maxima can be a significant concern as
the marginal likelihood is generally non-convex.
§.§ Scalability and approximate inference
Full GP.
Local GPs.
Bayesian committee machine.
Predictive distributions on a toy regression problem.
The main computational difficulty in GP methods is
the need to invert or factor the kernel matrix $K_y$, which requires time cubic
in $n$. In GP-LVM inference this must be done at every optimization step to evaluate
(<ref>) and its derivatives.
This complexity has inspired
a number of approximations. The most commonly studied are inducing-point methods, in which the unknown function is
represented by its values at a set of $m$ inducing
points, where $m \ll n$. These points can be chosen by maximizing the marginal
likelihood in a surrogate model <cit.> or by
minimizing the KL divergence between the approximate and exact GP posteriors
<cit.>. Inference in such models can
typically be done in $O(nm^2)$ time, but this comes at the price
of reduced representational capacity: while smooth functions with long lengthscales may
be compactly represented by a small number of inducing points, for
quickly-varying functions with significant local
structure it may be difficult to find any faithful representation more
compact than the complete set of training observations.
A separate class of approximations, so-called “local” GP methods
<cit.>, involves
partitioning the inputs into blocks of $m$ points each, then modeling
each block with an independent Gaussian process. If the partition is
spatially local, this corresponds to a covariance function that imposes independence
between function values in different regions of the input
space. Computationally, each block requires only $O(m^3)$ time; the
total time is linear in the number of blocks. Local approximations
preserve short-lengthscale structure within each block, but their harsh independence assumptions can lead to
predictive discontinuities and inaccurate uncertainties
(Figure <ref>). These assumptions are problematic for
GP-LVM inference because the marginal likelihood becomes
discontinuous at block boundaries. Nonetheless, local GPs sometimes
work very well in practice, achieving results comparable to more
sophisticated methods in a fraction of the time <cit.>.
The Bayesian Committee Machine (BCM) <cit.> attempts
to improve on independent local GPs by averaging the predictions of
multiple GP experts. The model is formally equivalent to an inducing-point model in which the test points are the inducing points,
i.e., it assumes that the training blocks are conditionally
independent given the test data. The BCM can yield high-quality
predictions that avoid the pitfalls of local GPs
(Figure <ref>), while
maintaining scalability to very large datasets
<cit.>. However, as a purely predictive
approximation, it is unhelpful in the GP-LVM setting, where we are
interested in the likelihood of our training set irrespective of any
particular test data. The desire for a BCM-style approximation to the
marginal likelihood was part of the motivation for this current work;
in Section <ref> we show that the GPRF proposed in this
paper can be viewed as such a model.
Mixture-of-experts models <cit.>
extend the local GP concept in a different direction: instead of
deterministically assigning points to GP models based on their spatial
locations, they treat the assignments as unobserved random variables
and do inference over them. This allows the model to adapt to different
functional characteristics in different regions of the space, at the
price of a more difficult inference task. We are not aware of
mixture-of-experts models being applied in the GP-LVM setting, though
this should in principle be possible.
Simple building blocks are often combined to create more complex
The PIC approximation <cit.> blends a global
inducing-point model with local block-diagonal covariances,
thus capturing a mix of global and local
structure, though with the same boundary discontinuities as in
“vanilla” local GPs. A related approach is the use of covariance
functions with compact support <cit.> to capture local
variation in concert with global inducing points. <cit.> surveys and compares
several approximate GP regression methods on synthetic and real-world
Finally, we note here the similar title of <cit.>,
which is in fact orthogonal to the present work: they use a random
field as a prior on input locations, whereas this paper defines a
random field decomposition of the GP model itself, which may be
combined with any prior on $X$.
§.§ Markov Random Fields
We recall some basic theory regarding Markov random fields
(MRFs), also known as undirected graphical models <cit.>. A pairwise
MRF consists of an undirected graph $(V, E)$, along with node potentials $\psi_i$ and edge
potentials $\psi_{ij}$, which define an energy function on a random vector $\v{y}$,
\begin{equation}
E(\v{y}) = \sum_{i\in V} \psi_{i}(\v{y}_i) + \sum_{(i,j)\in E}
\psi_{ij}(\v{y}_i, \v{y}_j),\label{eqn:mrf}
\end{equation}
where $\v{y}$ is partitioned into components
$\v{y}_i$ identified with nodes in the graph. This energy in turn defines a
probability density, the “Gibbs
distribution”, given by $p(\v{y}) = \frac{1}{Z}\exp(-E(\v{y}))$ where
$Z = \int \exp(-E(\v{z})) d\v{z}$ is a normalizing constant.
Gaussian random fields are the special case of pairwise MRFs in which
the Gibbs distribution is multivariate Gaussian. Given a partition of
$\v{y}$ into sub-vectors $\v{y}_1, \v{y}_2, \ldots, \v{y}_M$, a
zero-mean Gaussian distribution with covariance $K$ and precision
matrix $J = K^{-1}$ can
be expressed by potentials
\begin{equation}
\psi_i(\v{y}_i) = -\frac{1}{2}\v{y}_i^T
J_{ii} \v{y}_i, \qquad\psi_{ij}(\v{y}_i, \v{y}_j) = -\v{y}_i^T J_{ij}
\v{y}_j \label{eqn:gaussian-mrf}
\end{equation}
where $J_{ij}$ is the submatrix of $J$ corresponding
to the sub-vectors $\v{y}_i$, $\v{y}_j$. The
normalizing constant $Z =
(2\pi)^{n/2}|K|^{1/2}$ involves the determinant of the covariance
matrix. Since edges whose potentials are zero can be dropped without
effect, the nonzero entries of the precision matrix can be seen as
specifying the edges present in the graph.
§ GAUSSIAN PROCESS RANDOM FIELDS
We consider a vector of $n$ real-valued[The extension to
multiple independent outputs is straightforward.] observations
$\v{y} \sim \N(\v{0}, K_y)$ modeled by a GP, where $K_y$ is implicitly
a function of input locations $X$ and hyperparameters $\theta$. Unless
otherwise specified, all probabilities $p(\v{y}_i), p(\v{y}_i,
\v{y}_j)$, etc., refer to marginals of this full GP. We would like to
perform gradient-based optimization on the marginal likelihood
(<ref>) with respect to $X$ and/or $\theta$, but suppose that
the cost of doing so directly is prohibitive.
In order to proceed, we assume a partition $\v{y} = (\v{y}_1, \v{y}_2,
\ldots, \v{y}_M)$ of the observations into $M$ blocks of size at most
$m$, with an implied corresponding partition $X = (X_1, X_2, \ldots,
X_M)$ of the (perhaps unobserved) inputs. The source of this partition
is not a focus of the current work; we might imagine that the blocks
correspond to spatially local clusters of input points, assuming that
we have noisy observations of the $X$ values or at least a reasonable
guess at an initialization. We let $K_{ij} = \cov_\theta(\v{y}_i,
\v{y}_j)$ denote the appropriate submatrix of $K_{y}$, and $J_{ij}$
denote the corresponding submatrix of the precision matrix
$J_y=K_y^{-1}$; note that $J_{ij} \ne (K_{ij})^{-1}$ in general.
§.§ The GPRF Objective
Given the precision matrix $J_y$, we could use
(<ref>) to represent the full GP distribution in
factored form as an MRF. This is not directly useful, since computing
$J_y$ requires cubic time. Instead we propose approximating the
marginal likelihood via a random field in which local GPs are
connected by pairwise potentials. Given an edge set which we will
initially take to be the complete graph, $E =\{(i,j) | 1\le i < j \le
M\}$, our approximate objective is
\begin{align}
q_{GPRF}(\v{y}; X, \theta)&= \prod_{i=1}^M p(\v{y}_i)
\prod_{(i,j)\in E} \frac{p(\v{y}_i, \v{y}_j)}{p(\v{y}_i) p(\v{y}_j)},\label{eqn:gprf-naive}\\
&= \prod_{i=1}^M p(\v{y}_i)^{1-|E_i|} \prod_{(i,j)\in E} p(\v{y}_i, \v{y}_j) \nonumber
\end{align}
where $E_i$ denotes the neighbors of $i$ in the graph, and $p(\v{y}_i)$ and $p(\v{y}_{i},
\v{y}_j)$ are marginal probabilities under the full GP;
equivalently they are the likelihoods of local GPs
defined on the points $X_i$ and $X_i \cup X_j$ respectively. Note that
these local likelihoods depend implicitly on $X$ and $\theta$. Taking
the log, we obtain the energy function of an unnormalized MRF
\begin{equation}
\log q_{GPRF}(\v{y}; X, \theta) = \sum_{i=1}^M (1-|E_i|)\log p(\v{y}_i)
+ \sum_{(i,j)\in E} \log p(\v{y}_i, \v{y}_j) \label{eqn:gprf-log}
\end{equation}
with potentials
\begin{equation}
\psi_i^{GPRF}(\v{y}_i) = (1-|E_i|)\log p(\v{y}_i), \qquad \psi_{ij}^{GPRF}(\v{y}_i, \v{y}_j) =
\log p(\v{y}_i, \v{y}_j).\end{equation}
We refer to the approximate objective (<ref>) as
$q_{GPRF}$ rather than $p_{GPRF}$ to emphasize that it is not in
general a normalized probability density. It can be interpreted as a
“Bethe-type” approximation <cit.>, in which a joint
density is approximated via overlapping pairwise marginals. In the
special case that the full precision matrix $J_y$ induces a tree
structure on the blocks of our partition, $q_{GPRF}$ recovers the
exact marginal likelihood. (This is shown in the supplementary material.) In general this will not be the case, but in the spirit of loopy
belief propagation <cit.>, we consider the
tree-structured case as an approximation for the general setting.
Before further analyzing the nature of the approximation, we first
observe that as a sum of local Gaussian log-densities, the objective (<ref>)
is straightforward to implement and fast to evaluate. Each of the $O(M^2)$
pairwise densities requires $O((2m)^3) = O(m^3)$ time, for an overall complexity of
$O(M^2m^3) = O(n^2m)$ when $M=n/m$. The quadratic dependence on $n$ cannot
be avoided by any algorithm that computes similarities between all
pairs of training points; however, in practice we will consider “local”
modifications in which $E$ is something smaller than all
pairs of blocks. For example, if each block is connected only to a
fixed number of spatial neighbors, the complexity reduces
to $O(nm^2)$, i.e., linear in $n$. In the special case where $E$ is
the empty set, we recover the exact likelihood of independent local GPs.
It is also straightforward to obtain the gradient of
(<ref>) with respect to hyperparameters $\theta$ and inputs $X$, by summing
the gradients of the local densities. The likelihood and gradient for each term in the sum
can be evaluated independently using only local subsets of the
training data, enabling a simple parallel implementation.
Having seen that $q_{GPRF}$ can be optimized efficiently, it remains
for us to argue its validity as a proxy for the full GP marginal
likelihood. Due to space constraints we
defer proofs to the supplementary material, though our results are not
difficult. We first show that, like the full marginal likelihood (<ref>),
$q_{GPRF}$ has the form of a Gaussian distribution, but with a
different precision matrix.
The objective $q_{GPRF}$ has the form of an unnormalized Gaussian density
with precision matrix $\tilde{J}$, with blocks $\tilde{J}_{ij}$ given by
\begin{equation}
\tilde{J}_{ii} = K_{ii}^{-1} + \sum_{j\in E_i} \left(Q^{(ij)}_{11}
- K_{ii}^{-1}\right), \qquad \tilde{J}_{ij} =
\left\{\begin{array}{ll}Q^{(ij)}_{12} & \text{ if } (i,j) \in E\\0
& \text{ otherwise.}\end{array}\right),\label{eqn:approx-precision}
\end{equation}
where $Q^{(ij)}$ is the local precision matrix $Q^{(ij)}$
defined as the inverse of the marginal covariance,
\[Q^{(ij)} = \left(\begin{array}{cc} Q^{(ij)}_{11} & Q^{(ij)}_{12}\\
Q^{(ij)}_{21} & Q^{(ij)}_{22}\end{array}\right) = \left(\begin{array}{cc} K_{ii} & K_{ij}\\
K_{ji} & K_{jj}\end{array}\right)^{-1}.\]
Although the Gaussian density represented by $q_{GPRF}$ is not in general normalized, we show that it is approximately normalized in a certain sense.
The objective $q_{GPRF}$ is approximately normalized in the sense that
the optimal value of the Bethe free energy <cit.>,
\begin{equation}
F_B(b) = \sum_{i\in V} \left(\int_{\v{y}_i} b_i(\v{y}_i) \frac{(1-|E_i|)\ln
b_i(\v{y}_i)}{\ln \psi_i(\v{y}_i)}\right) + \sum_{(i,j)\in E} \left(\int_{\v{y}_i, \v{y}_j} b_{ij}(\v{y}_i,
\v{y}_j) \ln \frac{b_{ij}(\v{y}_i,
\v{y}_j)}{\psi_{ij}(\v{y}_i, \v{y}_j))}\right)
\label{eqn:bethe-energy} \approx \log Z,
\end{equation}
the approximation to the normalizing constant found by loopy belief
propagation, is precisely zero. Furthermore, this
optimum is obtained when the pseudomarginals $b_i, b_{ij}$ are
taken to be the true GP marginals $p_i, p_{ij}$.
This implies that loopy belief propagation run on a GPRF
would recover the marginals of the true GP.
§.§ Predictive equivalence to the BCM
We have introduced $q_{GPRF}$ as a surrogate model
for the training set $(X, \v{y})$; however, it is natural to extend
the GPRF to make predictions at a set of test points $X^*$, by including the
function values $\v{f}^* = f(X^*)$ as an $M+1$st block, with an edge to each of the training blocks. The resulting
predictive distribution,
\begin{align}
p_{GPRF}(\v{f}^* | \v{y}) \propto q_{GPRF}(\v{f}^*, \v{y})
&= p(\v{f}^*) \prod_{i=1}^M \frac{p(\v{y}_i,
\v{f}^*)}{p(\v{y}_i) p(\v{f}^*)} \left(\prod_{i=1}^M p(\v{y}_i) \prod_{(i,j)\in E} \frac{p(\v{y}_i, \v{y}_j)}{p(\v{y}_i)
p(\v{y}_j)}\right) \nonumber \\
%&\propto p(\v{f}^*) \prod_{i=1}^M \frac{p(\v{y}_i,
% \v{f}^*)}{p(\v{y}_i) p(\v{f}^*)} \nonumber\\
&\propto p(\v{f}^*)^{1-M} \prod_{i=1}^M p(\v{f}^* | \v{y}_i),
\end{align}
corresponds exactly to the prediction of
the Bayesian Committee Machine (BCM) <cit.>. This motivates the
GPRF as a natural extension of the BCM as a model for the training
set, providing an alternative to the standard transductive
interpretation of the BCM.[The GPRF is still transductive, in
the sense that adding a test block $\v{f^*}$ will change the
marginal distribution on the training observations $\v{y}$, as
can be seen explicitly in the precision matrix (<ref>). The contribution of the GPRF is that it provides a reasonable model for
the training-set likelihood even in the absence of test
points. ] A similar derivation shows that the conditional distribution of any
block $\v{y}_i$ given all other blocks $\v{y}_{j\ne i}$ also takes the
form of a BCM prediction, suggesting the possibility of
pseudolikelihood training <cit.>, i.e.,
directly optimizing the quality of BCM predictions on held-out blocks
(not explored in this paper).
§ EXPERIMENTS
§.§ Uniform Input Distribution
We first consider a 2D synthetic dataset intended to simulate spatial location tasks such as WiFi-SLAM
<cit.> or seismic event location (below), in which we
observe high-dimensional measurements but have only noisy information
regarding the locations at which those measurements were
taken. We sample $n$ points uniformly from the square of side length
$\sqrt{n}$ to generate the true inputs $X$, then sample 50-dimensional output $Y$ from independent GPs
with SE kernel $k(r) = \exp(-(r/\ell)^2)$ for $\ell=6.0$ and noise standard deviation
$\sigma_n = 0.1$. The observed points
$X^\text{obs} \sim N(X, \sigma^2_\text{obs}I)$ arise by corrupting $X$ with
isotropic Gaussian noise of standard deviation
$\sigma_\text{obs}=2$. The parameters $\ell$, $\sigma_n$, and
$\sigma_\text{obs}$ were chosen to generate problems with
interesting short-lengthscale structure for which
GP-LVM optimization could nontrivially improve the initial noisy locations. Figure <ref> shows a typical sample from this model.
Noisy observed locations: mean error 2.48.
Full GP: 0.21.
GPRF-100: 0.36. (showing grid cells)
FITC-500: 4.86. (with inducing points, note contraction)
Inferred locations on synthetic data
($n=10000$), colored by the first output dimension $\v{y}_1$.
For local GPs and GPRFs, we take the spatial partition to be a
grid with $n/m$ cells, where $m$ is the desired number of points per
cell. The GPRF edge set $E$ connects each cell to its
eight neighbors (Figure
<ref>), yielding linear time
complexity $O(nm^2)$. During optimization, a practical choice
is necessary: do we use a fixed partition of the points, or re-assign
points to cells as they cross spatial boundaries? The latter
corresponds to a coherent (block-diagonal) spatial covariance function, but introduces
discontinuities to the marginal likelihood. In our experiments the GPRF was not sensitive to
this choice, but local GPs performed more reliably with fixed
spatial boundaries (in spite of the discontinuities), so we used this
approach for all experiments.
Mean location error over time for $n=10000$,
including comparison to full GP.
Mean error at convergence as a
function of $n$, with learned
Mean location error over time for $n=80000$.
Results on synthetic data.
For comparison, we also evaluate the Sparse GP-LVM, implemented in GPy
<cit.>, which uses the FITC approximation to the
marginal likelihood <cit.>. (We also considered
the Bayesian GP-LVM <cit.>, but found it to be more
resource-intensive with no meaningful difference in results on this
problem.) Here the approximation parameter $m$ is the number of inducing points.
We ran L-BFGS optimization to recover maximum a posteriori (MAP)
locations, or local optima thereof. Figure <ref> shows
mean location error (Euclidean distance) for $n=10000$ points; at this
size it is tractable to compare directly to the full GP-LVM. The GPRF
with a large block size ($m$=1111, corresponding to a 3x3 grid) nearly matches the solution
quality of the full GP while requiring less time, while the local
methods are quite fast to converge but become stuck at inferior
optima. The FITC optimization exhibits an interesting pathology: it
initially moves towards a good solution but then diverges towards what
turns out to correspond to
a contraction of the space (Figure <ref>); we
conjecture this is because there are not enough inducing points to
faithfully represent the full GP distribution over the entire space. A
partial fix is to allow FITC to jointly optimize over locations and
the correlation lengthscale $\ell$; this yielded a biased lengthscale
estimate $\hat{\ell} \approx 7.6$ but more accurate
locations (FITC-500-$\ell$ in Figure <ref>).
To evaluate scaling behavior, we next considered problems of
increasing size up to $n=80000.$[The astute reader will wonder how we generated
synthetic data on problems that are clearly too large for an exact
GP. For these synthetic problems as well as the seismic example below, the
covariance matrix is relatively sparse, with only ˜2% of entries
corresponding to points within six kernel lengthscales of each other. By considering only these
entries, we were able to draw samples using a sparse Cholesky
factorization, although this required approximately 30GB of RAM. Unfortunately, this approach does not straightforwardly
extend to GP-LVM inference under the exact GP, as the standard
expression for the marginal likelihood
\[\frac{\partial}{\partial \v{x}_{i}} \log p(\v{y})
= \frac{1}{2} \tr\left( \left((K_y^{-1} \v{y}) (K_y^{-1} \v{y})^T -
K_y^{-1}\right) \frac{\partial K_y}{\partial \v{x}_{i}} \right)\]
involves the full
precision matrix $K_y^{-1}$ which is not sparse in general. Bypassing this expression via
automatic differentiation through the sparse Cholesky
decomposition could perhaps allow exact GP-LVM inference to scale to
somewhat larger problems.] Out of generosity to FITC we allowed each method to learn
its own preferred lengthscale. Figure <ref>
reports the solution quality at convergence, showing that even with an
adaptive lengthscale, FITC requires increasingly many inducing points
to compete in large spatial domains. This is intractable for larger
problems due to $O(m^3)$ scaling; indeed, attempts to run at $n>55000$
with 2000 inducing points exceeded 32GB of available memory. Recently,
more sophisticated inducing-point methods have claimed scalability to very large
datasets <cit.>, but they do
so with $m\le 1000$; we expect that
they would hit the same fundamental scaling constraints for problems
that inherently require many inducing points.
On our largest synthetic problem, $n=80000$, inducing-point
approximations are intractable, as is the full GP-LVM. Local GPs converge more quickly
than GPRFs of equal block size, but the GPRFs find higher-quality
solutions (Figure <ref>). After a short initial period, the
best performance always belongs to a GPRF, and at the conclusion of 24 hours the best GPRF
solution achieves mean error 42% lower than the best local
solution (0.18 vs 0.31).
§.§ Seismic event location
Event map for seismic dataset.
Mean location error over time.
Seismic event location task.
We next consider an application to seismic event location, which formed the
motivation for this work. Seismic waves can be
viewed as high-dimensional vectors generated from an
underlying three-dimension manifold, namely the Earth's
crust. Nearby events tend to generate similar waveforms; we
can model this spatial correlation as a Gaussian process. Prior information regarding the event
locations is available from traditional travel-time-based location systems
<cit.>, which produce an independent Gaussian uncertainty ellipse for each event.
A full probability model of seismic waveforms, accounting for background
noise and performing joint alignment of arrival times, is beyond the scope of this
paper. To focus specifically on the ability to approximate GP-LVM inference, we
used real event locations but generated synthetic waveforms by sampling from a 50-output GP using a Matérn kernel
<cit.> with $\nu=3/2$ and a lengthscale of 40km. We also
generated observed location estimates $X^\text{obs}$, by corrupting the
true locations with Gaussian noise of standard deviation
20km in each dimension. Given the
observed waveforms and noisy locations, we are interested in recovering the
latitude, longitude, and depth of each event.
Our dataset consists of 107556 events detected at the Mankachi array
station in Kazakstan between 2004 and 2012. Figure <ref>
shows the event locations, colored to reflect a principle axis tree
partition <cit.> into blocks of $400$ points (tree
construction time was negligible). The GPRF edge set contains all pairs of
blocks for which any two points had initial locations within one
kernel lengthscale (40km) of each other. We also evaluated
longer-distance connections, but found that this relatively local
edge set had the best performance/time tradeoffs: eliminating edges not
only speeds up each optimization step, but in some cases actually
yielded faster per-step convergence (perhaps because denser edge sets
tended to create large cliques for which the pairwise GPRF objective is
a poor approximation).
Figure <ref> shows the quality of recovered locations as a
function of computation time; we jointly optimized over event locations as
well as two lengthscale parameters (surface distance
and depth) and the noise variance $\sigma^2_n$. Local GPs
perform quite well on this task, but the best
GPRF achieves 7% lower mean error than the best local
GP model (12.8km vs 13.7km, respectively) given equal time. An even
better result can be obtained by using the results of a local GP
optimization to initialize a GPRF. Using the same
partition ($m=800$) for both local GPs and the GPRF, this “hybrid” method
gives the lowest final error (12.2km), and is dominant across a wide
range of wall clock times, suggesting it as a promising
practical approach for large GP-LVM optimizations.
§ CONCLUSIONS AND FUTURE WORK
The Gaussian process random field is a tractable and effective
surrogate for the GP marginal likelihood. It has the flavor of
approximate inference methods such as loopy belief propagation, but
can be analyzed precisely in terms of a deterministic approximation to
the inverse covariance, and provides a new training-time
interpretation of the Bayesian Committee Machine. It is easy to
implement and can be straightforwardly parallelized.
One direction for future work involves finding partitions for
which a GPRF performs well, e.g., partitions that induce a block near-tree
structure. A perhaps related question is identifying when the GPRF
objective defines a normalizable probability
distribution (beyond the case of an exact tree structure) and under
what circumstances it is a good approximation to the exact GP likelihood.
This evaluation in this paper focuses on spatial data; however, both local
GPs and the BCM have been successfully applied to high-dimensional
regression problems <cit.>, so
exploring the effectiveness of the GPRF for dimensionality reduction
tasks would also be interesting. Another useful avenue is to integrate the
GPRF framework with other approximations: since the GPRF and inducing-point methods have complementary strengths – the GPRF is useful for
modeling a function over a large space, while inducing points are useful
when the density of available data in some region of the space
exceeds what is necessary to represent the function – an integrated
method might enable new applications for which neither approach alone
would be sufficient.
§.§.§ Acknowledgements
We thank the anonymous reviewers for their helpful
suggestions. This work was supported by DTRA grant #HDTRA-11110026,
and by computing resources donated by Microsoft Research under an Azure for Research grant.
This file contains additional derivations for our NIPS 2015 paper, “Gaussian Process Random Fields”. Notation used here follows the notation of the paper. Code to construct the datasets and reproduce the experimental results is available online at <https://github.com/davmre/gprf/>.
§ BLOCK TREE STRUCTURE
It is straightforward to see that the GPRF objective is exact when the MRF induced by the true precision matrix $J$, with respect to our chosen partition of $\v{y}$, is a tree. For any choice of root node $\v{y}_\text{root}$, the tree structure implies that we can write the true GP distribution as a product of parent-conditional distributions,
\[p(\v{y}) = p(\v{y}_\text{root}) \prod_{i \ne \text{root}} p(\v{y}_i | \v{y}_{\pi(i)})\]
where $\pi(i)$ is the (unique) parent of node $i$ with respect to our chosen root. Then expanding the conditional distribution
\begin{align*}
p(\v{y}) &= p(\v{y}_\text{root}) \prod_{i \ne \text{root}} \frac{p(\v{y}_i , \v{y}_{\pi(i)})}{p(\v{y}_{\pi(i)})}\\
&= p(\v{y}_\text{root}) \prod_{i \ne \text{root}} p(\v{y}_i) \frac{p(\v{y}_i , \v{y}_{\pi(i)})}{p(\v{y}_i) p(\v{y}_{\pi(i)}) }\\
&= \left(\prod_{i} p(\v{y}_i) \right) \left(\prod_{i \ne \text{root}} p(\v{y}_i) \frac{p(\v{y}_i , \v{y}_{\pi(i)})}{p(\v{y}_i) p(\v{y}_{\pi(i)}) }\right)
\end{align*}
yields exactly the GPRF objective for the edge set $E={(i, \pi(i)})$, i.e., the edges that define the tree.
Note that the structure of the MRF induced by the true GP will depend on the partition we choose: a given precision matrix may induce a tree structure for some choices of partition but not for others (e.g., even a fully dense matrix can be viewed as a tree for trivial partitions that split the dataset into only one or two blocks).
In many cases it is easier to reason about the structure of the covariance matrix than that of the precision matrix. Assuming a stationary kernel, nonzero (or non-negligible) entries of the covariance matrix correspond to data points that are nearby to each other, meaning that the sparsity pattern of the covariance matrix reflects the geometry of the data itself. If the data can be viewed as lying on a treelike manifold – for example, seismic fault lines, or even trivial special cases such as time series data which lies on the real line – then for reasonable choices of partition, a graph connecting nearby blocks of data points will have a tree structure. Of course, there is no formal guarantee that this structure will fully carry over to the precision matrix, though intuitively we'd expect that points very distant from each other are also unlikely to interact strongly in the precision matrix.
§ APPROXIMATION TO THE TRUE GAUSSIAN
In this section we prove Theorem 1 from the main text, showing that $q_{GPRF}$ is an unnormalized Gaussian density with a particular precision matrix.
For any pair of blocks $(i,j)$, define the local precision matrix $Q^{(ij)}$ to be the inverse of the marginal covariance,
\[Q^{(ij)} = \left(\begin{array}{cc} Q^{(ij)}_{11} & Q^{(ij)}_{12}\\
Q^{(ij)}_{21} & Q^{(ij)}_{22}\end{array}\right) = \left(\begin{array}{cc} K_{ii} & K_{ij}\\
K_{ji} & K_{jj}\end{array}\right)^{-1},\]
The notation $Q^{(ij)}$ is used to distinguish these local precision
matrices from the blocks $J_{ij}$ of the global precision matrix. Writing $q_{GPRF}$ in terms of unnormalized Gaussian densities,
\begin{align*}
\log q_{GPRF}(\v{y}) &= -\frac{1}{2} \sum_{i=1}^M (1-|E_i|) \v{y}_i^T
K_{ii}^{-1} \v{y}_i -\frac{1}{2} \sum_{(i,j)\in E} \left(\begin{array}{c}
\v{y}_i \\ \v{y}_j\end{array}\right)^T Q^{(ij)}\left(\begin{array}{c}
\v{y}_i \\
\v{y}_j\end{array}\right) + C\\
&= -\frac{1}{2}\sum_{i=1}^M \v{y}_i^T \left(K_{ii}^{-1} - |E_i|
K_{ii}^{-1}\right)\v{y}_i -\frac{1}{2} \left(\sum_{(i,j)\in E} \v{y}_i^T
Q^{(ij)}_{11} \v{y}_i + 2\v{y}_i^T Q^{(ij)}_{12}\v{y}_j + \v{y}_j^T Q^{(ij)}_{22}\v{y}_j\right) + C\\
&= -\frac{1}{2}\sum_{i=1}^M \v{y}_i^T \left(K_{ii}^{-1} + \sum_{j\in E_i}
\left(Q^{(ij)}_{11} - K_{ii}^{-1}\right) \right)\v{y}_i - \sum_{(i,j)\in E}
\v{y}_i^T Q^{(ij)}_{12} \v{y}_j + C
\end{align*}
we obtain the standard form of a Gaussian
MRF (expression (3) from the main text) showing that $q_{GPRF}$ does in fact induce a Gaussian density on
$\v{y}$. Note that in passing from the second to the third line we used the fact that $Q^{ij}_{11} = Q^{ji}_{22}$, by definition. This Gaussian representation allows us to read off the implicit precision matrix $\tilde{J}$ in block wise form
\begin{equation}
\tilde{J}_{ii} = K_{ii}^{-1} + \sum_{j\in E_i} \left(Q^{(ij)}_{11}
- K_{ii}^{-1}\right), \qquad \tilde{J}_{ij} =
\left\{\begin{array}{ll}Q^{(ij)}_{12} & \text{ if } (i,j) \in E\\0
& \text{ otherwise.}\end{array}\right.\label{eqn:approx-precision}
\end{equation}
We see that the off-diagonal blocks of the precision matrix are simply
the corresponding blocks of the pairwise local precisions. Each
diagonal block, by contrast, combines the inverse of the local
covariance matrix with corrections from the pairwise
precisions. Note that the approximate precision $\tilde{J}$ may not be positive definite. In this case $q_{GPRF}$ is not a normalizable density,
although it is still “approximately normalized” in the sense of the next section.
§ APPROXIMATE NORMALIZATION
In this section we prove Theorem 2 from the main text:
The objective $q_{GPRF}$ is approximately normalized in the sense that
the optimal value of the Bethe free energy <cit.>,
\begin{equation}
F_B(b) = \sum_{i\in V} \left(\int_{\v{y}_i} b_i(\v{y}_i) \frac{(1-|E_i|)\ln
b(\v{y}_i)}{\ln \psi_i(\v{y}_i)}\right) + \sum_{(i,j)\in E} \left(\int_{\v{y}_i, \v{y}_j} b_{ij}(\v{y}_i,
\v{y}_j) \ln \frac{b_{ij}(\v{y}_i,
\v{y}_j)}{\psi_{ij}(\v{y}_i, \v{y}_j))}\right)
\label{eqn:bethe-energy} \approx \log Z,
\end{equation}
the approximation to the normalizing constant found by loopy belief
propagation, is precisely zero. Furthermore, this
optimum is obtained when the pseudomarginals $b_i, b_{ij}$ are
taken to be the true GP marginals $p_i, p_{ij}$.
These claims are established rather directly by substituting the GPRF potentials $\psi_i^{GPRF}, \psi_{ij}^{GPRF}$ for the log pseudomarginals $\log \psi_{i}, \log \psi_{ij}$ in (<ref>), yielding
\begin{align}
F_B(b)&= \sum_{i\in V} \left(\int_{\v{y}_i} b_i(\v{y}_i) \frac{(1-|E_i|)\ln
b_i(\v{y}_i)}{(1-|E_i|) \ln p(\v{y}_i) }\right) + \sum_{(i,j)\in E} \left(\int_{\v{y}_i, \v{y}_j} b_{ij}(\v{y}_i,
\v{y}_j) \ln \frac{b_{ij}(\v{y}_i,
\v{y}_j)}{p(\v{y}_i, \v{y}_j))}\right) \nonumber\\
&= \sum_{i\in V} KL[b_i \| p_i] + \sum_{(i,j)\in E} KL[b_{ij}\| p_{ij}],
\end{align}
where $KL[b\|p] = \int b(\v{x}) \ln \frac{b(\v{x})}{p(\v{x})}d\v{x}$
is the Kullback-Liebler divergence between distributions $b$ and
$p$. This is minimized when the distributions are equal, at which
point the divergence is zero. Thus, taking $b_i=p_i$and
$b_{ij}=p_{ij}$ yields the optimal value $F_B=0$.
We might have hoped that, as local GPs match the marginal
distributions of the full GP on individual blocks, perhaps a
higher-order approximation could match the exact marginals on pairs of
blocks. This is not possible, since any Gaussian distribution whose
pairwise marginals match the full GP must in fact be the full GP
(Gaussians are entirely characterized by their covariances). Instead we
can view $q_{GPRF}$ as approximately matching the pairwise
marginals of the full GP, in the sense that the pseudomarginals found
by running loopy belief propagation on $q_{GPRF}$ are in fact the true
marginals of the full GP. This is a consequence of the fact that loopy
BP converges to stationary points of the Bethe energy
|
1511.00213
|
This paper studies theoretically and empirically
a method of turning machine-learning algorithms
into probabilistic predictors that
automatically enjoys a property of validity (perfect calibration)
and is computationally efficient.
The price to pay for perfect calibration is that these probabilistic predictors
produce imprecise (in practice, almost precise for large data sets) probabilities.
When these imprecise probabilities are merged into precise probabilities,
the resulting predictors, while losing the theoretical property of perfect calibration,
are consistently more accurate than the existing methods in empirical studies.
The conference version of this paper
is to appear in Advances in Neural Information Processing Systems 28, 2015.
§ INTRODUCTION
Prediction algorithms studied in this paper belong to the class of Venn–Abers predictors,
introduced in <cit.>.
They are based on the method of isotonic regression <cit.>
and prompted by the observation that when applied in machine learning
the method of isotonic regression
often produces miscalibrated probability predictions
(see, e.g., <cit.>);
it has also been reported (<cit.>, Section 1)
that isotonic regression is more prone to overfitting than Platt's scaling
<cit.> when data is scarce.
The advantage of Venn–Abers predictors is that they are a special case of Venn predictors
(<cit.>, Chapter 6),
and so (<cit.>, Theorem 6.6) are always well-calibrated
(cf. Proposition <ref> below).
They can be considered to be a regularized version of the procedure
used by <cit.>, which helps them resist overfitting.
The main desiderata for Venn (and related conformal, <cit.>, Chapter 2) predictors
are validity, predictive efficiency, and computational efficiency.
This paper introduces two computationally efficient versions of Venn–Abers predictors,
which we refer to as inductive Venn–Abers predictors (IVAPs) and cross-Venn–Abers predictors (CVAPs).
The ways in which they achieve the three desiderata are:
* Validity (in the form of perfect calibration) is satisfied by IVAPs automatically,
and the experimental results reported in this paper suggest that it is inherited by CVAPs.
* Predictive efficiency is determined by the predictive efficiency
of the underlying learning algorithms
(so that the full arsenal of methods of modern machine learning
can be brought to bear on the prediction problem at hand).
* Computational efficiency is, again, determined by the computational efficiency
of the underlying algorithm;
the computational overhead of extracting probabilistic predictions consists of sorting
(which takes time $O(n\log n)$, where $n$ is the number of observations)
and other computations taking time $O(n)$.
An advantage of Venn prediction over conformal prediction,
which also enjoys validity guarantees,
is that Venn predictors output probabilities rather than p-values,
and probabilities, in the spirit of Bayesian decision theory,
can be easily combined with utilities to produce optimal decisions.
When the probability $\{p_0,p_1\}$ is imprecise, we can get an uncertain optimal decision;
in this case, it makes sense to use the minimax principle for choosing
among the available decisions.
The reader whose main interest is in probability
(rather than perfectly calibrated multiprobability) prediction
might ask why calibration is a useful technique:
some machine-learning methods (such as logistic regression, naive Bayes, and neural networks)
are able to output probabilities directly.
The main reason is that the best (as measured by standard loss functions) results
are obtained not by those methods but by non-probabilistic predictors
combined with a calibration method
(see, e.g., Table 2 in <cit.>).
In Sections <ref> and <ref> we discuss IVAPs and CVAPs, respectively.
Section <ref> is devoted to minimax ways of merging imprecise probabilities
into precise probabilities
and thus making IVAPs and CVAPs precise probabilistic predictors.
this paper we concentrate on binary classification problems,
in which the objects to be classified are labelled as 0 or 1.
Most of machine learning algorithms are scoring algorithms,
in that they output a real-valued score for each test object,
which is then compared with a threshold to arrive at a categorical prediction, 0 or 1.
As precise probabilistic predictors, IVAPs and CVAPs
are ways of converting the scores for test objects into numbers in the range $[0,1]$
that can serve as probabilities,
or calibrating the scores.
In Section <ref> we briefly discuss two existing calibration methods,
Platt's <cit.> and the method <cit.> based on isotonic regression,
and compare them with IVAPs and CVAPs theoretically.
Section <ref> is devoted to experimental comparisons
and shows that CVAPs consistently outperform the two existing methods (more extensive experimental studies can be found in <cit.>).
§ INDUCTIVE VENN–ABERS PREDICTORS (IVAPS)
In this paper we consider data sequences (usually loosely referred to as sets)
consisting of observations $z=(x,y)$,
each observation consisting of an object $x$ and a label $y\in\{0,1\}$;
we only consider binary labels.
We are given a training set whose size will be denoted $l$.
This section introduces inductive Venn–Abers predictors.
Our main concern is how to implement them efficiently,
but as functions, an IVAP is defined in terms of a scoring algorithm
(see the last paragraph of the previous section) as follows:
* Divide the training set of size $l$ into two subsets,
the proper training set of size $m$ and the calibration set of size $k$,
so that $l=m+k$.
* Train the scoring algorithm on the proper training set.
* Find the scores $s_1,\ldots,s_k$ of the calibration objects $x_1,\ldots,x_k$.
* When a new test object $x$ arrives, compute its score $s$.
Fit isotonic regression to $(s_1,y_1),\ldots,(s_k,y_k),(s,0)$ obtaining a function $f_0$.
Fit isotonic regression to $(s_1,y_1),\ldots,(s_k,y_k),(s,1)$ obtaining a function $f_1$.
The multiprobability prediction for the label $y$ of $x$
is the pair $(p_0,p_1):=(f_0(s),f_1(s))$
(intuitively, the prediction is that the probability that $y=1$ is either $f_0(s)$ or $f_1(s)$).
Notice that the multiprobability prediction $(p_0,p_1)$ output by an IVAP
always satisfies $p_0<p_1$,
and so $p_0$ and $p_1$ can be interpreted as the lower and upper probabilities,
in practice, they are close to each other for large training sets.
First we state formally the property of validity of IVAPs
(adapting the approach of <cit.> to IVAPs).
A random variable $P$ taking values in $[0,1]$ is perfectly calibrated
(as a predictor) for a random variable $Y$ taking values in $\{0,1\}$
if $\Expect(Y\mid P) = P$ a.s.
A selector is a random variable taking values in $\{0,1\}$.
As a general rule,
in this paper
random variables are denoted by capital letters
(e.g., $X$ are random objects and $Y$ are random labels).
Let $(P_0,P_1)$ be an IVAP's prediction for $X$
based on a training sequence $(X_1,Y_1),\ldots,(X_l,Y_l)$.
There is a selector $S$ such that $P_S$ is perfectly calibrated for $Y$
provided the random observations $(X_1,Y_1),\ldots,(X_l,Y_l),(X,Y)$ are i.i.d.
Our next proposition concerns the computational efficiency of IVAPs; both propositions will be proved later in the section Proposition <ref> will be proved later in this section
while Proposition <ref> is proved in <cit.>.
Given the scores $s_1,\ldots,s_k$ of the calibration objects,
the prediction rule for computing the IVAP's predictions
can be computed in time $O(k\log k)$ and space $O(k)$.
Its application to each test object takes time $O(\log k)$.
Given the sorted scores of the calibration objects,
the prediction rule can be computed in time and space $O(k)$.
Proofs of both statements rely on the geometric representation
of isotonic regression as the slope of the GCM (greatest convex minorant)
of the CSD (cumulative sum diagram):
see <cit.>, pages 9–13 (especially Theorem 1.1).
To make our exposition more self-contained, we define both GCM and CSD below.
First we explain how to fit isotonic regression to $(s_1,y_1),\ldots,(s_k,y_k)$
(without necessarily assuming that $s_i$ are the calibration scores and $y_i$ are the calibration labels,
which will be needed to cover the use of isotonic regression in IVAPs).
We start from sorting all scores $s_1,\ldots,s_{k}$ in the increasing order
and removing the duplicates.
(This is the most computationally expensive step in our calibration procedure,
$O(k\log k)$ in the worst case.)
Let $k'\le k$ be the number of distinct elements among $s_1,\ldots,s_k$,
i.e., the cardinality of the set $\{s_1,\ldots,s_k\}$.
Define $s'_j$, $j=1,\ldots,k'$, to be the $j$th smallest element of $\{s_1,\ldots,s_k\}$,
so that $s'_1<s'_2<\cdots<s'_{k'}$.
Define $w_j:=\left|\left\{i=1,\ldots,k:s_i=s'_j\right\}\right|$
to be the number of times $s'_j$ occurs among $s_1,\ldots,s_k$.
Finally, define
\frac{1}{w_j}
\sum_{i=1,\ldots,k:s_i=s'_j}
to be the average label corresponding to $s_i=s'_j$.
The CSD of $(s_1,y_1),\ldots,(s_k,y_k)$ is the set of points
\begin{equation}\label{eq:CSD}
\left(
\sum_{j=1}^i
\sum_{j=1}^i
y'_j w_j
\right),
\quad
\end{equation}
in particular, $P_0=(0,0)$.
The GCM is the greatest convex minorant of the CSD.
The value at $s'_i$, $i=1,\ldots,k'$, of the isotonic regression
fitted to $(s_1,y_1),\ldots,(s_k,y_k)$ is defined to be the slope of the GCM
between $\sum_{j=1}^{i-1}w_j$ and $\sum_{j=1}^i w_j$;
the values at other $s$ are somewhat arbitrary
(namely, the value at $s\in(s'_{i},s'_{i+1})$ can be set to anything
between the left and right slopes of the GCM at $\sum_{j=1}^i w_j$)
but are never needed in this paper
(unlike in the standard use of isotonic regression in machine learning, <cit.>):
e.g., $f_1(s)$ is the value of the isotonic regression fitted to a sequence that already contains $(s,1)$.
Set $S:=Y$.
The statement of the proposition even holds conditionally
on knowing the values of $(X_1,Y_1),\ldots,(X_m,Y_m)$
and the multiset $\lbag(X_{m+1},Y_{m+1}),\ldots,(X_l,Y_l)$,
this knowledge allows us to compute the scores $\lbag s_1,\ldots,s_k,s\rbag$
of the calibration objects $X_{m+1},\ldots,X_l$ and the test object $X$.
The only remaining randomness
is over the equiprobable permutations of $(X_{m+1},Y_{m+1}),\ldots,(X_l,Y_l),(X,Y)$;
in particular, $(s,Y)$ is drawn randomly
from the multiset $\lbag(s_{1},Y_{m+1}),\ldots,(s_k,Y_l),(s,Y)\rbag$.
It remains to notice that, according to the GCM construction,
the average label of the calibration and test observations
corresponding to a given value of $P_S$ is equal to $P_S$.
The last observation is stated, in a more general case,
as Theorem 1.3.5 in <cit.>.
A more standard approach would be to show that IVAPs are (inductive) Venn predictors
and then use the general validity result for Venn predictors
(as we did in <cit.>),
but in this paper we are using a shortcut avoiding defining inductive Venn predictors.
The idea behind computing the pair $(f_0(s),f_1(s))$ efficiently
is to pre-compute two vectors $F^0$ and $F^1$ storing $f_0(s)$ and $f_1(s)$,
for all possible values of $s$.
Let $k'$ and $s'_i$ be as defined above
in the case where $s_1,\ldots,s_k$ are the calibration scores
and $y_1,\ldots,y_k$ are the corresponding labels.
The vectors $F^0$ and $F^1$ are of length $k'$,
and for all $i=1,\ldots,k'$ and both $\epsilon\in\{0,1\}$,
$F^{\epsilon}_{i}$ is the value of $f_{\epsilon}(s)$ when $s=s'_i$.
Therefore, for all $i=1,\ldots,k'$:
* $F^{1}_{i}$ is also the value of $f_{1}(s)$ when $s$ is just to the left of $s'_i$;
* $F^{0}_{i}$ is also the value of $f_{0}(s)$ when $s$ is just to the right of $s'_i$.
Since $f_0$ and $f_1$ can change their values only at the points $s'_i$,
the vectors $F^0$ and $F^1$ uniquely determine the functions $f_0$ and $f_1$,
For details of computing $F^0$ and $F^1$, see <cit.>.
There are several algorithms
for performing isotonic regression on a partially, rather than linearly, ordered set:
see, e.g., <cit.>, Section 2.3
(although one of the algorithms described in that section, the Minimax Order Algorithm,
was later shown to be defective <cit.>).
Therefore, IVAPs (and CVAPs below) can be defined in the situation
where scores take values only in a partially ordered set;
moreover, Proposition <ref> will continue to hold.
(For the reader familiar with the notion of Venn predictors
we could also add that Venn–Abers predictors will continue to be Venn predictors,
which follows from the isotonic regression being the average of the original function
over certain equivalence classes.)
The importance of partially ordered scores stems from the fact that they enable us
to benefit from a possible “synergy” between two or more prediction algorithms
Suppose, e.g., that one prediction algorithm outputs (scalar) scores $s^1_1,\ldots,s^1_k$ for the calibration objects $x_1,\ldots,x_k$
and another outputs $s^2_1,\ldots,s^2_k$ for the same calibration objects;
we would like to use both sets of scores.
We could merge the two sets of scores into composite vector scores,
$s_i:=(s^1_i,s^2_i)$, $i=1,\ldots,k$,
and then classify a new object $x$ as described earlier using its composite score $s:=(s^1,s^2)$,
where $s^1$ and $s^2$ are the scalar scores computed by the two algorithms
and the partial order between composite scores is defined as usual,
\Longleftrightarrow
(s^1\le t^1)\;\&\;(s^2\le t^2).
Preliminary results reported in <cit.> in a related context
suggest that the resulting predictor can outperform predictors based on the individual scalar scores.
However, we will not pursue this idea further in this paper.
§.§ Computational details of IVAPs
Let $k'$, $s'_i$, and $w_i$ be as defined above
in the case where $s_1,\ldots,s_k$ and $y_1,\ldots,y_k$ are the calibration scores and labels.
The corners of a GCM are the points on the GCM where the slope of the GCM changes.
It is clear that the corners belong to the CSD,
and we also add the extreme points ($P_0$ and $P_{k'}$ in the case of (<ref>))
of the CSD to the list of corners.
We will only explain in detail how to compute $F^1$;
the computation of $F^0$ is analogous and will be explained only briefly.
First we explain how to compute $F^1_1$.
Extend the CSD as defined above
(in the case where $s_1,\ldots,s_k$ and $y_1,\ldots,y_k$ are the calibration scores and labels)
by adding the point $P_{-1}:=(-1,-1)$.
The corresponding GCM will be referred to as the initial GCM;
it has at most $k'+2$ corners.
Algorithm <ref>, which operates with a stack $S$ (initially empty),
computes the corners;
it is a trivial modification of Graham's scan
(<cit.>; <cit.>, Section 33.3).
The corners are returned on the stack $S$,
and they are ordered from left to right
($P_{-1}$ being at the bottom of $S$ and $P_{k'}$ at the top).
The operator “and” in line <ref> is, as usual, short circuiting.
The expression “the angle formed by points $a$, $b$, and $c$
makes a nonleft (resp. nonright) turn”
may be taken to mean that $(b-a)\times(c-b)\le0$ (resp. ${}\ge0$),
where $\times$ stands for cross product of planar vectors;
this avoids computing angles and divisions
(see, e.g., <cit.>, Section 33.1).
Initializing the corners for computing $F^1$
$S.\text{size}>1$ and the angle formed by points
$\Top(S)$, and $P_i$
makes a nonleft turn
Algorithm <ref> allows us to compute $F^1_1$
as the slope of the line between the two bottom corners in $S$,
but this will be done by the next algorithm.
Computing $F^1$
set $F^1_{i}$ to the slope of
$P_{i-1} = P_{i-2} + P_{i} - P_{i-1}$
$P_{i-1}$ is at or above $\overrightarrow{\Top(S'),\NextToTop(S')}$
$S'.\text{size}>1$ and the angle formed by points
$P_{i-1}$, $\Top(S')$, and $\NextToTop(S')$
makes a nonleft turn
The rest of the procedure for computing the vector $F^1$
is shown as Algorithm <ref>.
The main data structure in Algorithm <ref> is a stack $S'$,
which is initialized (in lines <ref>–<ref>)
by putting in it all corners of the initial GCM
in reverse order as compared with $S$
(so that $P_{-1}=(-1,-1)$ is initially at the top of $S'$).
At each point in the execution of Algorithm <ref>
we will have a length-1 active interval
and the active corner,
which will nearly always be at the top of the stack $S'$.
The initial CSD can be visualized by connecting each pair of adjacent points:
$P_{-1}$ and $P_0$, $P_0$ and $P_1$, etc.
It stretches over the interval $[-1,k']$ of the horizontal axis;
the subinterval $[-1,0]$ corresponds to the test score $s$
(assumed to be to the left of all $s'_i$)
and each subinterval $\left[\sum_{j=1}^{i-1}w_j,\sum_{j=1}^{i}w_j\right]$
corresponds to the calibration score $s'_i$,
The active corner is initially at $P_{-1}=(-1,-1)$;
the corners to the left of the active corner are irrelevant and ignored
(not remembered in $S'$).
The active interval is always between the first coordinate of $\Top(S')$
and the first coordinate of $\NextToTop(S')$.
At each iteration $i=1,\ldots,k'$ of the main loop <ref>–<ref>
we are computing $F^1_{i}$, i.e., $f_1(s)$ for the situation
where $s$ is between $s'_{i-1}$ and $s'_{i}$ (meaning to the left of $s'_1$ if $i=1$),
and after that we swap the active interval (corresponding to $s$)
and the interval corresponding to $s'_i$;
of course, after swapping pieces of CSD are adjusted vertically
in order to make the CSD as a whole continuous.
At the beginning of each iteration $i$ of the loop <ref>–<ref>
we have the CSD
\begin{equation}\label{eq:start-CSD}
\end{equation}
corresponding to
\begin{align*}
\text{the points } & s'_1,\ldots, s'_{i-1},s,s'_i,s'_{i+1},\ldots,s'_{k'}\\
\text{with the weights } & w_1,\ldots,w_{i-1},1,w_i,w_{i+1},\ldots,w_{k'} % tweaking: , \text{ respectively};
\end{align*}
the active interval is the projection of $\overrightarrow{P_{i-2},P_{i-1}}$
(onto the horizontal axis, here and later).
At the end of that iteration
we have the CSD which looks identical to (<ref>)
but in fact contains a different point $P_{i-1}$ (cf. line <ref> of the algorithm)
and corresponds to
\begin{align*}
\text{the points } & s'_1,\ldots,s'_{i-1},s'_i,s,s'_{i+1},\ldots,s'_{k'}\\
\text{with the weights } & w_1,\ldots,w_{i-1},w_i,1,w_{i+1},\ldots,w_{k'} % tweaking: , \text{ respectively};
\end{align*}
the active interval becomes the projection of $\overrightarrow{P_{i-1},P_{i}}$.
To achieve this,
in line <ref> we redefine $P_{i-1}$ to be the reflection of the old $P_{i-1}$
across the mid-point $(P_{i-2}+P_{i})/2$.
The stack $S'$ always consists of corners of the GCM of the current CSD,
and it contains all the corners to the right of the active interval
(plus one more corner, which is the active corner).
At each iteration $i$ of the loop <ref>–<ref>:
* We report the slope of the GCM over the active interval as $F^1_{i}$
(line <ref>).
* We then swap the fragments of the CSD corresponding to the active interval and to $s'_i$
leaving the rest of the CSD intact.
This way the active interval moves to the right
(from the projection of $\overrightarrow{P_{i-2},P_{i-1}}$
to the projection of $\overrightarrow{P_{i-1},P_{i}}$).
* If the point $P_{i-1}$ above the left end-point of the active interval
is above (or at) the GCM,
move to the next iteration of the loop.
(The active corner does not change.)
The rest of this description assumes that $P_{i-1}$ is strictly below.
* Make $P_{i-1}$
the active corner.
Redefine the GCM to the right of the active corner
by connecting the active corner
to the right-most corner $C$ such that the slope of the line connecting the active corner and that corner is minimal;
all the corners between the active corner and that right-most corner $C$
are then forgotten.
The worst-case computation time of Algorithms <ref> and <ref>
is $O(k')$.
In the case of Algorithm <ref>,
see <cit.>, Section 33.3.
In the case of Algorithm <ref>,
it suffices to notice that the total number of iterations for the while loop
does not exceed the total number of elements pushed onto $S'$
(since at each iteration we pop an element off $S'$);
and the total number of elements pushed onto $S'$
is at most $k'$ (in the first for loop)
plus $k'$ (in the second for loop).
For convenience of the reader wishing to program IVAPs and CVAPs,
we also give the counterparts of Algorithms <ref> and <ref>
for computing $F^0$:
see Algorithms <ref> and <ref> below.
In those algorithms,
we do not need the point $P_{-1}$ anymore;
however, we need a new point $P_{k'+1}:=P_{k'}+(1,1)$.
The stacks $S$ and $S'$ that they use are initially empty.
Initializing the corners for computing $F^0$
$S.\text{size}>1$ and the angle formed by points $\NextToTop(S)$,
$\Top(S)$, and $P_i$ makes a nonright turn
Computing $F^0$
set $F^0_{i}$ to the slope of $\overrightarrow{\Top(S'),\NextToTop(S')}$
$P_{i} = P_{i-1} + P_{i+1} - P_{i}$
$P_{i}$ is at or above $\overrightarrow{\Top(S'),\NextToTop(S')}$
$S'.\text{size}>1$ and the angle formed by points $P_{i}$,
$\Top(S')$, and $\NextToTop(S')$ makes a nonright turn
Alternatively, we could use the algorithm for computing $F^1$
in order to compute $F^0$,
since, for all $i\in\{1,\ldots,k'\}$,
\begin{multline*}
\left(
\right)\\
\bigl(
w_1,\ldots,w_{k'}, % \\: tweaking
\bigr),
\end{multline*}
where the dependence on various parameters is made explicit.
After computing $F^0$ and $F^1$ we can arrange the calibration scores $s'_1,\ldots,s'_{k'}$
into a binary search tree: see Algorithm <ref>,
where $F^0_0$ is defined to be $0$ and $F^1_{k'+1}$ is defined to be $1$;
we will refer to $s'_i$ as the keys of the corresponding nodes
(only internal nodes will have keys).
Algorithm <ref> is in fact more general than what we need:
it computes the binary search tree for the scores $s'_a,s'_{a+1},\ldots,s'_b$
for $a\le b$;
therefore, we need to run $\BST(1,k')$.
The size of the binary search tree is $2k'+1$;
$k'$ of its nodes are internal nodes corresponding to different values of $s'_i$,
and the other $k'+1$ of its nodes are leaves corresponding to the $k'+1$ intervals
formed by the points $s'_1,\ldots,s'_{k'}$.
Let us compute the function $F(k')$ giving the size of the BST.
Start from
\begin{align*}
\end{align*}
The inductive step is
\begin{align*}
\end{align*}
It is easy to check that the solution is $F(k)=2k+1$.
$\BST(a,b)$ (to create the binary search tree, run $\BST(1,k')$)
construct the binary tree whose root has key $s'_a$ and payload $\{F^0_a,F^1_a\}$,
left child is a leaf with payload $\{F^0_{a-1},F^1_a\}$,
and right child is a leaf with payload $\{F^0_{a},F^1_{a+1}\}$
its root
construct the binary tree whose root has key $s'_a$ and payload $\{F^0_a,F^1_a\}$,
left child is a leaf with payload $\{F^0_{a-1},F^1_a\}$,
and right child is $\BST(b,b)$
its root
construct the binary tree whose root has key $s'_c$ and payload $\{F^0_c,F^1_c\}$,
left child is $\BST(a,c-1)$, and right child is $\BST(c+1,b)$
its root
Once we have the binary search tree it is easy to compute the prediction for a test object $x$
in time logarithmic in $k'$:
see Algorithm <ref>,
which passes $x$ through the tree and uses $N$ to denote the current node.
Formally, we give the test object $x$, the proper training set $T'$, and the calibration set $T''$
as the inputs of Algorithm <ref>;
however, the algorithm uses for prediction
the binary search tree built from $T'$ and $T''$,
and the bulk of work is done in Algorithms <ref>–<ref>.
$\IVAP(T',T'',x)$inductive Venn–Abers predictor
set $N$ to the root of the binary search tree and compute the score $s$ of $x$
$N$ is not a leaf
set $N$ to $N$'s left child
set $N$ to $N$'s right child
if $s=\key(N)$
The worst-case computational complexity of the overall procedure
involves the following components:
* Training the algorithm on the proper training set,
computing the scores of the calibration objects,
and computing the scores of the test objects;
at this stage the computation time is determined by the underlying algorithm.
* Sorting the scores of the calibration objects
takes time $O(k\log k)$.
* Running our procedure for pre-computing $f_0$ and $f_1$
takes time $O(k)$
(by Lemma <ref>).
* Processing each test object takes an additional time of $O(\log k)$
(using binary search).
In principle, using binary search does not require an explicit construction of a binary search tree
(cf. <cit.>, Exercise 2.3-5),
but once we have a binary search tree we can easily transform it into a red-black tree,
which allows us to add new observations to (and remove old observations from)
the calibration set in time $O(\log k)$
(<cit.>, Chapter 13).
A very useful version of PAVA:
in the form of “Up-and-Down-Blocks”,
as explained in <cit.>, Section 2.3, especially Figure 2.2;
see <cit.>, first page, for a much more intuitive description
of this version of PAVA.
In terms of CSD and GCM, this version of PAVA is just Graham's scan
(see, e.g., <cit.>, Section 33.3).
§ CROSS VENN–ABERS PREDICTORS (CVAPS)
A CVAP is just a combination of $K$ IVAPs,
where $K$ is the parameter of the algorithm.
It is described as Algorithm <ref>,
where $\IVAP(A,B,x)$ stands for the output of IVAP applied to $A$ as proper training set,
$B$ as calibration set, and $x$ as test object,
and $\GM$ stands for geometric mean
(so that $\GM(p_1)$ is the geometric mean of $p_1^1,\ldots,p_1^K$
and $\GM(1-p_0)$ is the geometric mean of $1-p_0^1,\ldots,1-p_0^K$).
The folds should be of approximately equal size,
and usually the training set is split into folds at random
(although we choose contiguous folds in Section <ref>
to facilitate reproducibility).
One way to obtain a random assignment of the training observations to folds
(see line <ref>) is to start from a regular array
in which the first $l_1$ observations are assigned to fold 1,
the following $l_2$ observations are assigned to fold 2,
up to the last $l_K$ observations which are assigned to fold $K$,
where $\left|l_k-l/K\right|<1$ for all $k$,
and then to apply a random permutation.
Remember that the procedure Randomize-in-Place
(<cit.>, Section 5.3)
can do the last step in time $O(l)$.
See the next section for a justification of the expression $\GM(p_1)/(\GM(1-p_0)+\GM(p_1))$
used for merging the IVAPs' outputs.
$\CVAP(T,x)$cross-Venn–Abers predictor for training set $T$
split the training set $T$
into $K$ folds $T_1,\ldots,T_K$
$(p_0^k,p_1^k):=\IVAP(T\setminus T_k,T_k,x)$
We have no theoretical guarantees of validity for CVAPs,
but in Section <ref> we might draw calibration pictures
showing empirical calibration (perhaps inherited from its component IVAPs).
To complete the definition of CVAPs,
suppose the underlying algorithm requires parameter tuning.
Following <cit.>,
we can perform parameter tuning on the same hold-out folds
that are used for calibration.
But we need to explore empirically whether the calibration of CVAPs suffers
when we do so (which it might well do).
§.§ Alternative method to try for CVAPs
We could also use isotonic regression for partially ordered sets
for merging upper and lower probabilities (separately the former and the latter).
Each fold can be represented as a linearly ordered (by their scores) chain of objects in that fold
extended adding the test object (if it is not in the chain already).
Since the test object is in each chain,
we get a partial order like the one shown in Figure <ref>
(the order being represented by the arrows;
we will refer to its direction informally as bottom up).
There are two options: the weight of the new observation with a postulated label can be $1$,
or it can be equal to the number $K$ of folds (as in the usual CVAPs).
We consider only the case of postulated label $1$.
A typical partial order arising in the case of 5 folds
Suppose isotonic regression has been performed on each chain
without taking into account the test object with the postulated label;
this can be done before the prediction stage starts.
In this way each chain has been divided into blocks
(level sets of the isotonic regression),
which we will call level 1 blocks.
Replace each block $B$ containing or straddling the test object by the blocks
that are the level sets of the isotonic regression on the part of the block $B$ above the test object
or the level sets of the isotonic regression on the part of the block $B$ below the test object;
we will call them level 2 blocks.
Finally, the test object itself (at the centre of Figure <ref>)
is treated as another block, called the central block.
Let $g$ be the function assigning to each block the average label in this block
(by an average we always mean the weighted average);
we are interested in the closest isotonic function $g^*$.
We can pool each block:
this follows directly from Theorems 2.5 [a later remark: Theorem 2.5 is not applicable since elements of a block are not always poolable]
and 2.6(i) of <cit.>
and the PAVA (<cit.>, Section 1.2).
Therefore, in Figure <ref>
we can assume that each point represents a block,
with the central block consisting of the test object only
(the weight of the central block is the number of times the test object occurs in the training set
plus 1 or $K$).
Notice that the resulting structure then satisfies the following properties:
* The function $g$ is strictly increasing over each lower tentacle,
i.e., linearly ordered chain below the central block.
(This is obvious for the level 1 blocks in the lower tentacle
and for the level 2 blocks in the lower tentacle;
to see that the value of $g$ on the level 2 blocks is greater
than its value on the level 1 blocks
it suffices to apply Theorem 2.4 in <cit.> to the original chain
corresponding to that fold.)
* Similarly,
the function $g$ is strictly increasing over each linearly ordered chain above the central block.
The main part of the rest of the algorithm is shown as Algorithm <ref>,
which arranges the centre of the partial order in Figure <ref>,
where the centre is defined as the central block $C$ plus its neighbours
(referred to simply as neighbours in Algorithm <ref>).
Algorithm <ref> makes the function $g$ isotonic over the centre.
If this changes the value of $g$ at some neighbour $N$ of the central block,
the values of $g$ over the tentacle of $N$
(i.e., the linear chain emanating from $N$ away from the central block)
are replaced by the corresponding isotonic regression.
After making $g$ isotonic over each tentacle,
we check whether it remains isotonic over the centre.
If yes, we are done, and if not, we repeat Algorithm <ref>,
and so on, until $g$ is isotonic both over the centre and over all tentacles.
The correctness of the overall algorithm follows from the results in <cit.>
mentioned earlier and the Minimum Lower Sets algorithm
(see <cit.>, Theorem 2.7).
Isotonising the centre
$g$ is not isotonic over the centre
set $N$ to an upper neighbour with the smallest value of $g$
pool $C$ and $N$ into a new $C$
set $N$ to a lower neighbour with the largest value of $g$
pool $C$ and $N$ into a new $C$
Assuming the number $K$ of folds to be constant,
the worst-case computation time of the overall algorithm is $O(l)$ (per test object),
where $l$ is the size of the training set,
but it is clear that in practice we can expect the algorithm to run much faster.
§ MAKING PROBABILITY PREDICTIONS OUT OF MULTIPROBABILITY ONES
In CVAP (Algorithm <ref>)
we merge the $K$ multiprobability predictions output by $K$ IVAPs.
In this section we design a minimax way
for merging them,
essentially following <cit.>.
For the log-loss function the result is especially simple,
The deficiency of guaranteed calibration:
$\log(\GM(1-p_0)+\GM(p_1))\in[0,1]$ (for binary log).
We need to check how small it is.
Notice that the probability interval $(1-\GM(1-p_0),\GM(p_1))$ (formally, a pair of numbers)
is narrower than the corresponding interval for the arithmetic means;
this follows from the fact that a geometric mean never exceeds the corresponding arithmetic mean
and that we always have $p_0<p_1$.
Let us check that $\GM(p_1)/(\GM(1-p_0)+\GM(p_1))$
is indeed the minimax expression under log loss.
Suppose the pairs of lower and upper probabilities to be merged
are $(p^1_0,p^1_1),\ldots,(p^K_0,p^K_1)$
and the merged probability is $p$.
The extra cumulative loss suffered by $p$
over the correct members $p^1_1,\ldots,p^K_1$ of the pairs
when the true label is $1$ is
\begin{equation}\label{eq:1}
\log\frac{p^1_1}{p}+\cdots+\log\frac{p^K_1}{p},
\end{equation}
and the extra cumulative loss of $p$
over the correct members of the pairs
when the true label is $0$ is
\begin{equation}\label{eq:0}
\log\frac{1-p^1_0}{1-p}+\cdots+\log\frac{1-p^K_0}{1-p}.
\end{equation}
Equalizing the two expressions we obtain
\frac{p^1_1 \cdots p^K_1}{p^K}
\frac{(1-p^1_0)\cdots(1-p^K_0)}{(1-p)^K},
which gives the required minimax expression for the merged probability
(since (<ref>) is decreasing and (<ref>) is increasing in $p$).
For the computations in the case of the Brier loss function,
see <cit.>.
In the case of the Brier loss function,
we solve the linear equation
\begin{equation*} % tweaking
% \begin{multline*}
(1-p)^2 - (1-p^1_{1})^2 + \cdots + (1-p)^2 - (1-p^K_{1})^2
% \\
p^2 - (p^1_{0})^2 + \cdots + p^2 - (p^K_{0})^2
\end{equation*}
in $p$;
the result is
\frac1K
\sum_{k=1}^K
\left(
p^k_1 + \frac12 (p^k_0)^2 - \frac12 (p^k_1)^2
\right).
This expression is more natural than it looks:
see <cit.>,
the discussion after (11);
notice that it reduces to arithmetic mean when $p_0=p_1$.
The argument above (“conditioned” on the proper training set)
is also applicable to IVAP, in which case we need to set $K:=1$;
the probability predictor obtained from an IVAP
by replacing $(p_0,p_1)$ with $p:=p_1/(1-p_0+p_1)$
will be referred to as the log-minimax IVAP.
(And CVAP is log-minimax by definition.)
§ COMPARISON WITH OTHER CALIBRATION METHODS
The two alternative calibration methods that we consider in this paper
are Platt's <cit.> and isotonic regression <cit.>.
§.§ Platt's method
Platt's <cit.> method uses sigmoids
\begin{equation*} % \label{eq:sigmoid}
\frac{1}{1+\exp(As+B)},
\end{equation*}
where $A<0$ and $B$ are parameters,to calibrate the scores.
Platt discusses two approaches:
* run the scoring algorithm and fit the parameters $A$ and $B$ on the full training set,
* or run the scoring algorithm on a subset (called the proper training set in this paper)
and fit $A$ and $B$ on the rest (the calibration set).
Platt recommends the second approach,
especially that he is interested in SVM,
and for SVM
the scores for the training set tend to cluster around $\pm 1$.
(In fact, this is also true for the calibration scores,
as discussed below.)
Platt's recommended method of fitting $A$ and $B$ is
\begin{equation}\label{eq:min}
\left(
t_i \log p_i
(1-t_i) \log (1-p_i)
\right)
\to
\min,
\end{equation}
where, in the simplest case, $t_i:=y_i$ are the labels of the calibration observations
(so that (<ref>) minimizes the log loss on the calibration set).
To obtain even better results,
Platt recommends regularization:
\begin{equation}\label{eq:t+}
\frac{k_++1}{k_++2}
\end{equation}
for the calibration observations labelled 1 (if there are $k_+$ of them) and
\begin{equation}\label{eq:t-}
\frac{1}{k_-+2}
\end{equation}
for the calibration observations labelled 0 (if there are $k_-$ of them).
We can see from (<ref>) and (<ref>) that the predictions of Platt's predictor
are always in the range
Platt uses a regularization procedure ensuring that the predictions of his method are always in the range
\begin{equation}\label{eq:range}
\left(
\frac{1}{k_-+2},
\frac{k_++1}{k_++2}
\right). % tweak , or .
\end{equation}
where $k_-$ is the number of calibration observations labelled 0
and $k_+$ is the number of calibration observations labelled 1.
It is interesting that the predictions output by the log-minimax IVAP
are in the same range (except that the end-points are now allowed):
see <cit.>.
Let us check that the predictions output by the log-minimax IVAP
are in the same range as those for Platt's method (except that the end-points are now allowed):
In the case of IVAP,
$p_1\ge1/(k_-+1)$ and $p_0\le1-1/(k_++1)$,
where $k_-$ and $k_+$ are the numbers of positive and negative observations in the calibration set,
In the case of log-minimax IVAP,
(i.e., $p$ is in the closure of (<ref>)).
In the case of CVAP,
where $k$ is the size of the largest fold.
The statement about IVAP is obvious,
and we will only check that it implies the two other statements.
For concreteness, we will consider the lower bounds.
The lower bound $1/(k_-+2)$ for log-minimax IVAP can be deduced from $p_1\ge1/(k_-+1)$
using the isotonicity of $t/(c+t)$ in $t>0$ for $c>0$:
\begin{equation*}
\frac{p_1}{(1-p_0)+p_1}
\ge
\frac{1/(k_-+1)}{(1-p_0)+1/(k_-+1)} % \\
\ge
\frac{1/(k_-+1)}{1+1/(k_-+1)}
\frac{1}{k_-+2}.
% \end{multline*}
\end{equation*}
In the same way the lower bound $1/(k+2)$ for CVAP follows from $\GM(p_1)\ge1/(k+1)$:
\begin{equation*} % tweaking
% \begin{multline*}
\frac{\GM(p_1)}{\GM(1-p_0)+\GM(p_1)}
\ge
\frac{1/(k+1)}{\GM(1-p_0)+1/(k+1)} % \\
\ge
\frac{1/(k+1)}{1+1/(k+1)}
\frac{1}{k+2}.
% \qedhere % gives a spurious warning
\tag*{$\qed$}
\end{equation*}
It is clear that the end-points of the interval (<ref>)
can be approached arbitrarily closely in the case of Platt's predictor
and attained in the case of IVAPs.
The main disadvantage of Platt's method is that the optimal calibration curve $g$
is quite often far from being a sigmoid;
and if the training set is very big,
we will suffer, since in this case we can learn the best shape of the calibrator $g$.
This is particularly serious in asymptotics as the amount of data tends to infinity.
Zhang <cit.> (Section 3.3) observes that in the case of SVM
and universal <cit.> kernels the scores tend to cluster around $\pm1$
at “non-trivial” objects, i.e., objects that are labelled 1
with non-trivial (not close to 0 or 1) probability.
This means that any sigmoid will be a poor calibrator unless the prediction problem is very easy.
Formally, we have the following statement
(a trivial corollary of known results),
which uses the notation $\eta(x)$ for the conditional probability
that the label of an object $x\in\mathbf{X}$ is 1
and assumes that the labels take values in $\{-1,1\}$, $y_i\in\{-1,1\}$
(rather than $y_i\in\{0,1\}$, as in the rest of this paper).
Suppose that the probability of each of the events
$\eta(X)=0$, $\eta(X)=1/2$, and $\eta(X)=1$ is 0.
Let $f_m$ be the SVM for a training set of size $m$,
i.e., the solution to the optimization problem
\begin{equation}\label{eq:SVM}
C_m \left\|f\right\|^2_H
\sum_{i=1}^m
\phi(f(x_i)y_i)
\to
\min,
\end{equation}
where $\phi(v):=(1-v)^+$ and $H$ is a universal RKHS (<cit.>, Definition 4.52).
$H$ is separable automatically: see Lemma 4.33 in <cit.>.
$H$ consists of bounded functions automatically (since the elements of $H$ are continuous).
The assumptions in <cit.> (Theorem 8.1) are weaker:
e.g., it is enough to assume that $H$ is dense in $L_1(Q_{\mathbf{X}})$,
$Q$ being the data-generating probability measure on $\mathbf{X}\times\{0,1\}$
and $Q_{\mathbf{X}})$ being its marginal on $\mathbf{X}$.
As $m\to\infty$,
\begin{equation*} % \label{eq:solution}
\to
\begin{cases}
-1 & \text{if $\eta(X)\in[0,1/2]$}\\
1 & \text{if $\eta(X)\in(1/2,1]$}
\end{cases}
\end{equation*}
in probability
provided $C_m\to\infty$ and $C_m=o(m)$.
This follows immediately from Theorem 4.4 in <cit.>
for a natural class of universal kernels related to neural networks.
In general, see the proof of Theorem 8.1 in <cit.>.
The intuition behind the SVM decision values clustering around $\pm1$ is very simple.
SVM solves the optimization problem (<ref>);
asymptotically as $m\to\infty$ and under natural assumptions
(such as $C_m\to\infty$ and $C_m=o(m)$),
this solves
\Expect
\phi(f(X)Y)
\to
\min.
We can optimize separately for different values of $\eta(x)$.
Given $\eta(x)=\eta^*$, we have the optimization problem
\eta^*\phi(f) + (1-\eta^*)\phi(-f) \to \min,
whose solutions are
\begin{equation*} % \label{eq:solution}
\in
\begin{cases}
(-\infty,-1] & \text{if $\eta(x)=0$}\\
\{-1\} & \text{if $\eta(x)\in(0,1/2)$}\\
[-1,1] & \text{if $\eta(x)=1/2$}\\
\{1\} & \text{if $\eta(x)\in(1/2,1)$}\\
[1,\infty) & \text{if $\eta(x)=1$}.
\end{cases}
\end{equation*}
As a function of $f$,
\eta^*\phi(f) + (1-\eta^*)\phi(-f)
is a continuous function which is equal to $\eta^*(1-v)$ for $v\in(-\infty,-1]$,
equal to $(1-\eta^*)(1+v)$ for $v\in[1,\infty)$, and linear for $v\in[-1,1]$
(and these conditions determine the function).
Assuming that the probability of each of the events
$\eta(X)=0$, $\eta(X)=1/2$, and $\eta(X)=1$ is 0,
it is easy to check that asymptotically the best achievable excess log loss of a sigmoid
over the Bayes algorithm is
\begin{equation} % tweaking
% \begin{multline}
\label{eq:excess}
\Expect
\Bigl(
\KL
\left(
\eta
\:\middle||\:
\Expect(\eta\mid\eta>1/2)
\right)
\III_{\eta>1/2} % \\
\KL
\left(
\eta
\:\middle||\:
\Expect(\eta\mid\eta<1/2)
\right)
\III_{\eta<1/2}
\Bigr),
\end{equation}
where $\KL$ is Kullback–Leibler divergence defined in terms of base 2 logarithm $\log_2$,
and the conditional expectation $\Expect(\eta\mid E)$
is defined to be $\Expect(\eta\III_E)/\Prob(E)$.
Indeed, the optimal prediction for $\eta>1/2$ is $\Expect(\eta\mid\eta>1/2)$
and the optimal prediction for $\eta<1/2$ is $\Expect(\eta\mid\eta<1/2)$.
Let us check, e.g., the first statement.
We are to minimize over $p\in(0,1)$ the integral of
-\eta\log p + (1-\eta)\log(1-p)
over the set $\eta>1/2$.
Set $C:=\int_{\eta>1/2}\eta\dd P$ and $D:=P(\eta>1/2)$.
So we are to maximize
C\log p + (D-C)\log(1-p).
Differentiating and solving the equation
\frac{C}{p} - \frac{D-C}{1-p} = 0
we obtain
p = C/D = \Expect(\eta\mid\eta>1/2).
On the other hand, there are no apparent obstacles
to it approaching 0 in the case of isotonic regression,
considered in the next subsection.
For illustration, suppose $\eta:=\eta(X)$ is distributed uniformly in $[0,1]$.
It is easy to see that
\begin{align*}
\Expect(\eta\mid\eta>1/2) &= 3/4\\
\Expect(\eta\mid\eta<1/2) &= 1/4;
\end{align*}
therefore, the excess loss (<ref>) is
\begin{multline*}
\Expect
\Bigl(
\KL
\left(
\eta
\:\middle||\:
\right)
\III_{\eta>1/2}
\KL
\left(
\eta
\:\middle||\:
\right)
\III_{\eta<1/2}
\Bigr) % \\ % tweaking
\Expect
\Bigl(
\eta\log_2\eta
\Bigr)\\ % tweaking
\Expect
\left(
\eta 1_{\eta>1/2} \log_2\frac43
\eta 1_{\eta<1/2} \log_2 4
\right) % \\ % tweaking
% =
% - 0.5 + 0.5623351 % for natural logs
% =
% -0.7213475 + 0.8112781
% =
% 0.0899306
\approx
-0.7213 + 0.8113
\end{multline*}
We can see that the Bayes log loss is $72.13\%$,
whereas the best loss achievable by a sigmoid is $81.13\%$,
9 percentage points worse.
§.§ Isotonic regression
There are two standard uses of isotonic regression:
we can train the scoring algorithm using what we call a proper training set,
and then use the scores of the observations in a disjoint calibration (also called validation) set
for calibrating the scores of test objects (as in <cit.>);
alternatively, we can train the scoring algorithm on the full training set
and also use the full training set for calibration
(it appears that this was done in <cit.>).
Alternatively, we could use a cross-validation scheme similar to CVAPs.
In both cases, however, we can expect to get an infinite log loss when the test set becomes large enough (see <cit.>).
Indeed, suppose that we have fixed proper training and calibration sets
(not necessarily disjoint, so that both cases mentioned above are covered)
such that the score $s(X)$ of a random object $X$
is below the smallest score of the calibration objects with a positive probability;
suppose also that the distribution of the label of a random observation
is concentrated at 0 with probability zero.
Under these realistic assumptions
the probability that the average log loss on the test set is $\infty$
can be made arbitrarily close to one
by making the size of the test set large enough:
indeed, with a high probability there will be an observation $(x,y)$ in the test set
such that the score $s(x)$ is below the smallest score of the calibration objects but $y=1$;
the log loss on such an observation will be infinite.
The presence of regularization is an advantage of Platt's method:
e.g., it never suffers an infinite loss when using the log loss function.
There is no standard method of regularization for isotonic regression,
and we do not apply one[One of the reviewers
of the conference version of this paper
proposed complementing the calibration set used in isotonic regression
by two dummy observations:
one with score $+\infty$ and labelled by $0$
and the other with score $-\infty$ and labelled by $1$.].
§ EMPIRICAL STUDIES
The main loss function (cf., e.g., <cit.>)
that we use in our empirical studies is the log loss
\begin{equation}\label{eq:log-loss}
\lambda_{\log}(p,y)
\begin{cases}
-\log p & \text{if $y=1$}\\
-\log(1-p) & \text{if $y=0$},
\end{cases}
\end{equation}
where $\log$ is binary logarithm,
$p\in[0,1]$ is a probability prediction, and $y\in\{0,1\}$ is the true label.
Another popular loss function is the Brier loss
\begin{equation}\label{eq:Brier-loss}
\lambda_{\Br}(p,y)
\end{equation}
We choose the coefficient 4 in front of $(y-p)^2$ in (<ref>)
and the base 2 of the logarithm in (<ref>) in order for the minimax no-information predictor
that always predicts $p:=1/2$ to suffer loss 1.
An advantage of the Brier loss function
is that it still makes it possible to compare the quality of prediction
in cases when prediction algorithms (such as isotonic regression) give a categorical but wrong prediction
(and so are simply regarded as infinitely bad when using log loss).
In the multi-class case
we assume that the label space $\mathbf{Y}$ is finite
and consider probability predictions that are probability measures on $\mathbf{Y}$;
for example, a prediction $p\in[0,1]$ output by a CVAP
is re-interpreted as the probability measure $P$ on $\{0,1\}$
such that $P(\{1\})=p$.
The main loss function that we use is the log loss
\begin{equation}\label{eq:multi-log-loss}
\lambda_{\log}(P,y)
-\log_{\left|\mathbf{Y}\right|} P(\{y\}),
\end{equation}
where we take the size $\left|\mathbf{Y}\right|$ of the label space
as the base of the logarithm.
Another popular loss function is the Brier loss
\lambda_{\Br}(P,y)
\frac{\left|\mathbf{Y}\right|}{\left|\mathbf{Y}\right|-1}
\sum_{y'\in\mathbf{Y}}
\left(
1_{y'=y} - P(\{y'\})
\right)^2,
where the coefficient in front of the sum is chosen in such a way that the minimax no-information predictor
that always predicts $1/\left|\mathbf{Y}\right|$ suffers loss 1
(this is also the reason for our choice of the base of the logarithm in (<ref>)).
The loss of a probability predictor on a test set will be measured
by the arithmetic average of the losses it suffers on the test set,
namely, by the mean log loss (MLL) and the mean Brier loss (MBL)
\begin{equation}\label{eq:losses} % notCONF manual
\MLL
% &:=
\frac1n
\sum_{i=1}^n
\lambda_{\log}(p_i,y_i), % \\
\quad
\MBL
% &:=
\frac1n
\sum_{i=1}^n
\lambda_{\Br}(p_i,y_i), % \notag
% \end{align}
\end{equation}
where $y_i$ are the test labels and $p_i$ are the probability predictions for them.
We will not be checking directly whether various calibration methods
produce well-calibrated predictions,
since it is well known that lack of calibration increases the loss
as measured by loss functions such as log loss and Brier loss
(see, e.g., <cit.> for the most standard decomposition
of the latter into the sum of the calibration error and refinement error).
In the case of the Brier loss,
we might also have taken the square root:
the behaviour of the log loss entropy
(i.e., Shannon entropy) is intermediate between the Brier entropy (i.e., Gini index)
and the square root of the Brier entropy:
see Figure <ref>
(analogous to a standard figure from <cit.>).
Advantages of using the RMBL as our main measure
are that we used it in <cit.>,
and that many prediction algorithm suffer disproportionately large log losses
(as compared to their expected values).
On the other hand, Brier loss is a proper loss function
whereas its square root is not.
Gini index (solid blue), the square root of Gini index (dotted blue),
and Shannon entropy (solid red)
In this section we compare log-minimax IVAPs
(i.e., IVAPs whose outputs are replaced by probability predictions,
as explained in Section <ref>)
and CVAPs with Platt's method <cit.>
and the standard method <cit.> based on isotonic regression;
the latter two will be referred to as “Platt” and “Isotonic”
in our tables and figures.
(Even though for both IVAPs and CVAPs we use the log-minimax procedure
for merging multiprobability predictions,
the Brier-minimax procedure leads to virtually identical empirical results.)
We use the same underlying algorithms as in <cit.>,
namely J48 decision trees (abbreviated to “J48”),
J48 decision trees with bagging (“J48 bagging”),
logistic regression (sometimes abbreviated to “logistic”),
naive Bayes, neural networks, and support vector machines (SVM),
as implemented in Weka <cit.> (University of Waikato, New Zealand).
The underlying algorithms (except for SVM) produce scores in the interval $[0,1]$,
which can be used directly as probability predictions
(referred to as “Underlying” in our tables and figures)
or can be calibrated using the methods of <cit.>
or the methods proposed in this paper (“IVAP” or “CVAP” in the tables and figures).
We start our empirical studies with
the data set
available from the UCI repository <cit.>
(this is the main data set used in <cit.>
and one of the data sets used in <cit.>);
as we will see later,
the picture that we observe is typical for other data sets as well.
We use the original split of the data set into a training set of $N_{\rm train}=32,561$ observations
and a test set of $N_{\rm test}=16,281$ observations.
The results of applying the four calibration methods
(plus the vacuous one, corresponding to just using the underlying algorithm)
to the six underlying algorithms
for this data set are shown in
Figure <ref>Figures <ref> and <ref>.
The six top plots report results for the log loss
(namely, $\MLL$, as defined in (<ref>))
and the six bottom plots for the Brier loss
(namely, $\MBL$).
Figures <ref> reports results for the log loss
(namely, $\MLL$, as defined in (<ref>))
and Figure <ref> for the Brier loss
(namely, $\MBL$).
The underlying algorithms are given in the titles of the plots
and the calibration methods are represented by different line styles,
as explained in the legends.
The marks on the horizontal axis are the ratios of the size of the proper training set to the size of the calibration set
(except for the label , which will be explained later);
in the case of CVAPs, the number $K$ of folds can be expressed as the sum of the two numbers forming the ratio
(therefore, column 4:1 corresponds to the standard choice of 5 folds in the method of cross-validation).
Missing curves or points on curves mean that the corresponding values either are too big
and would squeeze unacceptably the interesting parts of the plot if shown
or are infinite (such as many results for isotonic regression and neural networks under log loss).
In the case of CVAPs, the training set is split into $K$ equal
(or as close to being equal as possible) contiguous folds:
the first $\lceil N_{\rm train}/K\rceil$ training observations are included in the first fold,
the next $\lceil N_{\rm train}/K\rceil$ (or $\lfloor N_{\rm train}/K\rfloor$)
in the second fold, etc.(first $\lceil\cdot\rceil$ and then $\lfloor\cdot\rfloor$ is used
unless $N_{\rm train}$ is divisible by $K$).
In the case of the other calibration methods,
we used the first $\lceil\frac{K-1}{K}N_{\rm train}\rceil$ training observation
as the proper training set (used for training the scoring algorithm)
and the rest of the training observations are used as the calibration set.
The log and Brier losses of the four calibration methods
applied to the six prediction algorithms on the data set.
The log losses of the four calibration methods
applied to the six prediction algorithms on the data set.
The analogue of Figure <ref> for Brier loss.
It seems that our experimental results illustrate a new phenomenon:
the “all” mode often produces best results for isotonic regression
(unlike what <cit.> seem to say [a later remark: I could not find the precise place in <cit.>]).
In the case of log loss, isotonic regression often suffers infinite losses,
which is indicated by the absence of the round marker for isotonic regression;
e.g., only one of the log losses for SVM is finite.
We are not trying to use ad hoc solutions, such as clipping predictions
to the interval $[\epsilon,1-\epsilon]$ for a small $\epsilon>0$,
since we are also using the bounded Brier loss function.
The CVAP lines tend to be at the bottom in all plots;
experiments with other data sets also confirm this.
The column in the plots of
Figure <ref>Figures <ref> and <ref>refers to using the full training set as both the proper training set and calibration set.
(In our official definition of IVAP we require that the last two sets be disjoint,
but in this section we continue to refer to IVAPs modified in this way simply as IVAPs;
in <cit.>, such prediction algorithms were referred to as SVAPs,
simplified Venn–Abers predictors.)
Using the full training set as both the proper training set and calibration set
might appear naive
(and is never used in the extensive empirical study <cit.>),
but it often leads to good empirical results on larger data sets.
However, it can also lead to very poor results,
as in the case of “J48 bagging” (for IVAP, Platt, and Isotonic),
the underlying algorithm that achieves the best performance in
Figure <ref>Figures <ref> and <ref>.
A natural question is whether CVAPs perform better than the alternative calibration methods in
Figure <ref>Figures <ref> and <ref>(and our other experiments) because of applying cross-over (in moving from IVAP to CVAP)
or because of the extra regularization used in IVAPs.
The first reason is undoubtedly important for both loss functions
and the second for the log loss function.
The second reason plays a smaller role for Brier loss for relatively large data sets
(in the lower half of Figure <ref>Figure <ref>the curves for and are very close to each other),
but IVAPs are consistently better for smaller data sets even when using Brier loss.
In Tables <ref> and <ref>
we apply the four calibration methods and six underlying algorithms
to a much smaller training set,
namely to the first $5,000$ observations of the data set as the new training set,
following <cit.>;
the first $4,000$ training observations are used as the proper training set,
the following $1,000$ training observations as the calibration set,
and all other observations (the remaining training and all test observations)
are used as the new test set.
The results are shown in Tables <ref> for log loss
and <ref> for Brier loss.
They are consistently better for IVAP than for IR (isotonic regression).
Results for nine very small data sets are given in Tables 1 and 2 of <cit.>,
where the results for IVAP
(with the full training set used as both proper training and calibration sets,
labelled “SVA” in the tables in <cit.>)
are consistently (in 52 cases out of the 54 using Brier loss) better,
usually significantly better, than for isotonic regression
(referred to as DIR in the tables in <cit.>).
The log loss for the four calibration methods and six underlying algorithms
for a small subset of the data set
algorithm Platt IR IVAP CVAP
J48 0.5226 $\infty$ 0.5117 0.5102
J48 bagging 0.4949 $\infty$ 0.4733 0.4602
logistic 0.5111 $\infty$ 0.4981 0.4948
naive Bayes 0.5534 $\infty$ 0.4839 0.4747
neural networks 0.5175 $\infty$ 0.5023 0.4805
SVM 0.5221 $\infty$ 0.5015 0.4997
The analogue of Table <ref> for the Brier loss
algorithm Platt IR IVAP CVAP
J48 0.4463 0.4378 0.4370 0.4368
J48 bagging 0.4225 0.4153 0.4123 0.3990
logistic 0.4470 0.4417 0.4377 0.4342
naive Bayes 0.4670 0.4329 0.4311 0.4227
neural networks 0.4525 0.4574 0.4440 0.4234
SVM 0.4550 0.4450 0.4408 0.4375
The following information might help the reader in reproducing our results
(in addition to our code being posted on arXiv together with this paper).
For each of the standard prediction algorithms within Weka that we use,
we optimise the parameters by minimising the Brier loss on the calibration set,
apart from the column labelled .
(We cannot use the log loss since it is often infinite in the case of isotonic regression.)
We then use the trained algorithm to generate the scores for the calibration and test sets,
which allows us to compute probability predictions
using Platt's method, isotonic regression, IVAP, and CVAP.
All the scores apart from SVM are already in the $[0,1]$ range and can be used as probability predictions.
In the case of SVM,
we use the Weka sequential minimal optimization algorithm with the option “build logistic models”,
which calibrates the SVM scores into probabilities.
We then apply the other calibration methods on top of those probability predictions,
which is equivalent to applying them to the original SVM scores.
Most of the parameters are set to their default values,
and the only parameters that are optimised are (pruning confidence) for J48 and J48 bagging,
(ridge) for logistic regression,
(learning rate) and (momentum) for neural networks (),
and (complexity constant) for SVM (, with the linear kernel);
naive Bayes does not involve any parameters.
Notice that none of these parameters are “hyperparameters”,
in that they do not control the flexibility of the fitted prediction rule directly;
this allows us to optimize the parameters on the training set for the column.
In the case of CVAPs, we optimise the parameters by minimising the cumulative Brier loss over all folds
(so that the same parameters are used for all folds).
To apply Platt's method to calibrate the scores generated by the underlying algorithms
we use logistic regression, namely the function within MATLAB's Statistics toolbox.
For isotonic regression calibration we use the implementation of the PAVA in the R package
(namely, the function ).
Missing values are handled using the Weka filter ,
which replaces all missing values for nominal and numeric attributes with the modes and means from the training set.
For further experimental results,
see <cit.>.
§.§ Additional experimental results
Figures <ref> and <ref> show Figure <ref> shows our results for the data set
(available from the UCI repository <cit.> and also known as ).
In converting this multiclass classification problem to binary we follow <cit.>:
treat the largest class as 1 and the rest as 0,
and only consider a random and randomly permuted subset consisting of $30,000$ observations;
the first $5000$ of those observations are used as the training set and the remaining $25,000$ as the test set.
The CVAP results are still at the bottom of the plots and very stable;
and the values at the column are still particularly unstable.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
Similar results for the , , ,
and data sets
are shown in
Figures <ref>, <ref>, <ref>, and <ref>, respectively. Figures <ref>–<ref>. The data sets are split into training and test sets in proportion 2:1,
without randomization.
In particular, for the data set we ignore the original split into the training ($5822$ observations)
and test ($4000$ observations) sets.
Since the values for the column are so unstable,
the reader might prefer to disregard them in the case of IVAP, Platt, and Isotonic.
Figures <ref>, <ref>, and <ref> Figures <ref>–<ref> the CVAP results tend to be at the bottom of the plots.
The data set is much more difficult,
and all results in
Figure <ref> Figures <ref>–<ref> are poor and somewhat mixed;
however, they still demonstrate that CVAPs and IVAPs produce stable results and avoid the occasional bad failures
characteristic of the alternative calibration methods.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of
Figure <ref>
for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set.
The analogue of Figure <ref> for the data set .
The analogue of Figure <ref> for the data set .
The analogue of Figure <ref> for the data set .
And finally, Figures <ref> and <ref>
show the results for log loss and Brier loss, respectively, for the data set
and for a wide range of the ratios of the size of the proper training set to the calibration set.
The left-most column of each plot is $1:9$, which means,
in the case of Platt's method, isotonic regression, and IVAPs,
that 10% of the training set was allocated to the proper training set and the rest to the calibration set.
In the case of CVAPs, $1:9$ means that the training set was split into 10 folds,
each of them in turn was used as the proper training set, and the rest were used as the calibration set;
the results were merged using the minimax procedure as described in Section <ref>.
In the case of the underlying algorithm, $1:9$ means that only 10% of the training set
was in fact used for training (the same 10% as for the first three calibration methods).
The other columns are $1:8$, $1:7$,…, $1:2$,
$1:1$ (which corresponds to $1:1$ in
Figure <ref>Figures <ref> and <ref>),…,
$4:1$ (which corresponds to $4:1$ in
Figure <ref>Figures <ref> and <ref>,
i.e., to the standard procedure of 5-fold cross-validation),
$5:1$,…, $9:1$ (the latter corresponds to the other standard cross-validation procedure,
that of 10-fold cross-validation);
the results in those columns are analogous to those in the column $1:9$.
In order not to duplicate the information we gave earlier for the data set,
we give the results for a randomly permuted data set.
There is not much difference between 5 and 10 folds for most underlying algorithms
(logistic regression behaves unusually in that its performance deteriorates
as the size of the proper training set increases,
perhaps because less data are available for calibration).
The log loss on the data set of the six prediction algorithms
and four calibration methods
The analogue of Figure <ref> for the Brier loss function
§.§ Our plans
Experiments that we are doing for the paper:
* Run the precise IVAP
(i.e., the IVAP whose outputs are replaced by probability predictions,
as explained in Section <ref>)
on several data sets (see below, or binary)
and compare the results with Platt's calibration and isotonic regression.
This looks the cleanest comparison of the 3 calibration methods.
The data sets can be split into 3 equal parts
(proper training, calibration, and test sets).
For preliminary results,
see Tables <ref> (for log loss)
and <ref> (for Brier loss);
the data set is binary.
In those tables, the parameters were chosen based on the proper training set
using the functions
(for choosing $C$ in the case of SVM)
and for the number of decision trees in the case of boosting).
The next step: use the same hold-out fold for parameter tuning
as for calibration,
as mentioned earlier (the end of Section <ref>).
* Run CVAPs on the same data sets,
but now split into 2 equal parts unless there is a standard split into training and test sets,
for future reference and without any comparisons with alternative calibration methods.
To apply CVAPs to multi-class data sets,
merge their predictions using one of the standard methods of pairwise coupling,
as reviewed in <cit.>;
we will be using the PKPD method
(<cit.>, Section 2.3; <cit.>).
The experimental results in <cit.>
consist of Tables 2, 3 (artificial data) and 5, 6, 7, 8 (real data).
The main one is Table 8 (log loss on real data),
in which PKPD looks the best algorithm overall.
This is also true about Table 5 (Brier loss on real data)
and Table 3 (MSE on artificial data).
Tables 2, 6, and 7 are about the number of errors
(and so do not evaluate probability predictions).
A big advantage of PKPD is that it is given by a simple formula.
* Compare CVAP with the modified CVAP,
in which we merge the multiprobability predictions made by each component CVAP
and then merge those merged predictions
(i.e., use a 2-step procedure of merging instead of the 1-step procedure used in CVAPs).
This would be especially convincing if IVAP never perform much worse than the Platt and IR methods.
At the second step, there are two natural ways of merging probabilities:
* the minimax method for log loss proposed in Section <ref>
(but now $p_0=p_1$):
where $p$ is the vector of probabilities output by the component IVAPs;
* arithmetic mean,
which is in fact the minimax method for Brier loss proposed in Section <ref>:
\frac1K
\sum_{k=1}^K
however, it is easy to check that this method leads to the same result as the original CVAP
(with the Brier loss-minimax merging).
* Computation time for IVAPs for really large data sets
(using ).
The data sets covered by our experiments:
* The data set.
For the comparison experiments
(Tables <ref> and <ref>),
we use the original split into the training and test sets;
the sizes of the two sets are (after the removal of observations with missing values) $30,162$ and $15,060$,
* The data set
According to Vapnik,
learning is slow for this data set:
after 10K observations, the percentage of correct predictions is $80\%$;
after 500K it becomes $95\%$.
It gives, however, awful results; not exchangeable.
Some of these data sets contain missing values for attributes.
The missing values within Weka are handled by the “Replace Missing Values” method
\text{http://weka.sourceforge.net/doc.dev/weka/filters/unsupervised/attribute/ReplaceMissingValues.html}
The underlying algorithms in R:
* Boosted trees (from the R package ) with the number of trees 100.
The method seems to overfit in the case of Tables <ref> and <ref>:
there are only two different probabilities for the whole test set.
* SVM ( from the package ) with linear kernels and $C = 0.01$.
Other data sets to try:
* The USPS data set, as in <cit.>, Tables 1 and 2,
using the 1-NN conformity score based on tangent distance.
Do this both for the original split into the training and test sets
(especially awkward)
and for a random split.
[Less important: Can the 1-NN VAP be also run on the USPS data set?]
* The NIST data set, which is much bigger.
Is there a standard split into training / test subsets?
* Part of the UCI repository:
CoIL Challenge 2000 (Predicting Caravan Policy Ownership).
* I suggested to Ivan and Valya on 1 March 2015:
Reuters and Web (used by Platt),
SVM should work OK for them since this is what he used
(but the other 5 methods on Ivan's list would also be interesting),
it might, however, require a lot of pre-processing;
KDD-98 in addition to TIC (used by Zadrozny and Elkan);
Zadrozny and Elkan use Pendigits and 20 newsgroups,
but they applied naive Bayes to them, and it might not be easy for us to implement it.
For GBM, Spambase is very good.
If time left or later:
* Run CVAPs on tiny data sets, as in <cit.>, Tables 1 and 2.
* Repeat the experiments in <cit.>.
Do similar experiments for CVAPs.
All data sets in that paper are binary (two classes only).
* Implement CVAPs as part of an R package (using C code) and submit it to CRAN.
* Produce calibration pictures for boosted decision trees
(or whatever the best algorithm is):
* the original algorithm
* Platt calibration <cit.>
* Zadrozny–Elkan calibration <cit.>
* precise IVAP calibration
* not really comparable: CVAP calibration
Valya is working on the USPS data set.
To apply CVAPs to multi-class data sets,
merge their predictions using one of the standard methods of pairwise coupling,
as reviewed in <cit.>;
we will be using the PKPD method
(<cit.>, Section 2.3; <cit.>).
Pairwise coupling is used in the function
in the R package:
it is applied on top of Platt's algorithm.
§ CONCLUSION
This paper introduces two new computationally efficient algorithms for probabilistic prediction,
IVAP, which can be regarded as a regularised form of the calibration method
based on isotonic regression,
and CVAP, which is built on top of IVAP using the idea of cross-validation.
Whereas IVAPs are automatically perfectly calibrated,
the advantage of CVAPs is in their good empirical performance.
This paper does not study empirically upper and lower probabilities produced by IVAPs and CVAPs,
whereas the distance between them provides information about the reliability
of the merged probability prediction.
Finding interesting ways of using this extra information is one of the directions of further research.
§.§ Acknowledgments
We are grateful to the conference reviewers for numerous helpful comments and observations,
to Vladimir Vapnik for sharing his ideas about exploiting synergy between different learning algorithms,
and to participants in the conference Machine Learning: Prospects and Applications
(October 2015, Berlin) for their questions and comments.
The first author has been partially supported by EPSRC (grant EP/K033344/1)
and AFOSR (grant “Semantic Completions”).
The second and third authors are grateful to their home institutions for funding their trips to Montréal to attend NIPS 2015.
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§ IDEAS ABOUT AN R PACKAGE
Our planned programs: in R (perhaps with inner loops in C) and MATLAB.
This appendix is about R programs.
The required programs:
* The underlying algorithm should be wrapped into two R functions:
* one, called (short for “underlying”),
is analogous to R's fundamental function
(and the function from the package ,
which has an identical interface);
its result is an object of a new class ;
* the other, ,
is the method of the generic function that orients itself
to objects of class .
* Venn–Abers predictors for this paper (IVAP and CVAP)
are implemented as the following functions:
* creates a prediction rule for IVAP (an object of class )
from a proper training set and a calibration set;
it only covers the binary case;
* uses
to create a prediction rule for CVAP (an object of class )
from a training set;
covers both the binary case and the multiclass case
(where pairwise coupling is used on top of binary );
* methods and of
orient themselves to objects of classes and ,
We need to split the underlying algorithm into two parts,
creating the underlying prediction rule and exploiting it,
because is used in two places:
for calibration and for prediction.
The signatures of all these functions/methods are:
The is, as for , something like .
The should be of a suitable class (, , or ).
The three methods for the generic function take the usual arguments.
In ,
are the indices of the proper training set in ,
and are the indices of the calibration set in .
In , is the number of folds,
and the option determines whether the predictor is binary
or pairwise coupling is used.
§.§ The outputs
* The function outputs an object of S3 class ,
which is a list whose components must contain ,
the levels of the label (this is the case for ).
* The function outputs an object of S3 class ,
which is a list with the following components:
* an object of class for computing scores
based on the proper training set ;
* the binary search tree;
at this time the following more primitive structure is used instead:
* the list of keys, i.e., the vector $s'$
(the scores of the objects in the calibration set
with duplicates removed);
* the vector $F^0$;
* the vector $F^1$.
* The function outputs an object of S3 class ,
which is vector of length $K$ of objects.
* The method outputs a vector of scores
for the test set.
* The method outputs a list
whose components are a vector of lower probabilities and a vector of upper probabilities
for the test set.
* The method outputs
a vector of probabilities for the test set.
§.§ Testing R code
There is no need to use C for CVAPs at the stage of creating a CVAP object (prediction rule);
it remains to write a C function for
The computational efficiency of the R code for IVAP is being tested on an artificial data set:
$(x_i,y_i)$ are generated in the IID fashion,
$y_i\in\{0,1\}$, $y_i=1$ with probability $1/2$,
and $x_i=y_i+\xi_i$, where $\xi\sim N(0,1)$.
Some toy examples are given in Tables <ref> and <ref>.
The calibration scores are assumed to be $1,2,3$ and $1,2,3,4$, respectively.
labels $F^0$ $F^1$
$(0,0,0)$ $(0,0,0)$ $(1/4,1/3,1/2)$
$(0,0,1)$ $(0,0,1/2)$ $(1/3,1/2,1)$
$(0,1,0)$ $(0,1/3,1/3)$ $(1/2,2/3,2/3)$
$(0,1,1)$ $(0,1/2,2/3)$ $(1/2,1,1)$
$(1,0,0)$ $(1/4,1/4,1/4)$ $(1/2,1/2,1/2)$
$(1,0,1)$ $(1/3,1/3,1/2)$ $(2/3,2/3,1)$
$(1,1,0)$ $(1/2,1/2,1/2)$ $(3/4,3/4,3/4)$
$(1,1,1)$ $(1/2,2/3,3/4)$ $(1,1,1)$
The vectors $F^0$ and $F^1$ for different sets of labels
for three calibration objects with scores $1,2,3$.
labels $F^0$ $F^1$
$(0,0,0,0)$ $(0,0,0,0)$ $(1/5,1/4,1/3,1/2)$
$(0,0,0,1)$ $(0,0,0,1/2)$ $(1/4,1/3,1/2,1)$
$(0,0,1,0)$ $(0,0,1/3,1/3)$ $(1/3,1/2,2/3,2/3)$
$(0,0,1,1)$ $(0,0,1/2,2/3)$ $(1/3,1/2,1,1)$
$(0,1,0,0)$ $(0,1/4,1/4,1/4)$ $(2/5,1/2,1/2,1/2)$
$(0,1,0,1)$ $(0,1/3,1/3,1/2)$ $(1/2,2/3,2/3,1)$
$(0,1,1,0)$ $(0,1/2,1/2,1/2)$ $(1/2,3/4,3/4,3/4)$
$(0,1,1,1)$ $(0,1/2,2/3,3/4)$ $(1/2,1,1,1)$
$(1,0,0,0)$ $(1/5,1/5,1/5,1/5)$ $(2/5,2/5,2/5,1/2)$
$(1,0,0,1)$ $(1/4,1/4,1/4,1/2)$ $(1/2,1/2,1/2,1)$
$(1,0,1,0)$ $(1/3,1/3,2/5,2/5)$ $(3/5,3/5,2/3,2/3)$
$(1,0,1,1)$ $(1/3,1/3,1/2,2/3)$ $(2/3,2/3,1,1)$
$(1,1,0,0)$ $(2/5,2/5,2/5,2/5)$ $(3/5,3/5,3/5,3/5)$
$(1,1,0,1)$ $(1/2,1/2,1/2,3/5)$ $(3/4,3/4,3/4,1)$
$(1,1,1,0)$ $(1/2,3/5,3/5,3/5)$ $(4/5,4/5,4/5,4/5)$
$(1,1,1,1)$ $(1/2,2/3,3/4,4/5)$ $(1,1,1,1)$
The vectors $F^0$ and $F^1$ for different sets of labels
for four calibration objects with scores $1,2,3,4$.
|
1511.00027
|
We show that the Fokker Planck equation can be derived from a
Hypergeometric differential equation. The same applies to a non
linear generalization of such equation.3mm
Keywords: Non linear Fokker-Plank equations,
separation of variables, hypergeometric function.
§ INTRODUCTION
In this paper we uncover the fact that the celebrated
Fokker-Planck (FP) equation <cit.>
-∂/∂x [K(x) F] + Q/2
∂^2 F/∂x^2, can be derived from an
hypergeometric differential equation. In this equation, $F$ is
the distribution function, $K(x)$ the drift coefficient and $Q$
the diffusion coefficient (a positive quantity) <cit.>. The
second term on the r.h.s, describes the effects of the fluctuating
forces (diffusion term). Without it, (<ref>) would describe
deterministic motion (over-damped motion of a particle under the
force $K(x)$). For the time being, we restrict ourselves to the
case $K={\rm constant}$. A similar hypergeometric derivation
applies to a non linear generalization of equation
(<ref>), in the spirit if the one advanced 20 years ago by
Plastino and Plastino <cit.>, that has arisen interest till
today <cit.>.
This papers continues a line of research initiated by
uncovering hypergeometric connotations of quantum equations
§ HYPERGEOMETRIC DERIVATION OF THE FOKKER-PLANK EQUATION
The ordinary hypergeometric function $F_1^2(a,b;c;z)$ is a special
function represented by the hypergeometric series, that includes
many other special functions as specific or limiting cases. It is
a solution of a second-order linear ordinary differential equation
(ODE). Many second-order linear ODEs can be transformed into this
equation. Generalized hypergeometric functions include the
confluent hypergeometric function as a special case, which in turn
have many particular special functions as special instances, such
as elementary functions, Bessel functions, and the classical
orthogonal polynomials.
In particular, the confluent hypergeometric function reads
∑_n=0^∞ a_n/b_n z^n/n!; a, b ∈ℛ, with a_n, b_n the
Pochhamer symbols
a_0=1, a_n=a (a+1) (a+2) ... (a+n-1); same for b.
The confluent hypergeometric equation satisfies the
differential equation <cit.>:
\begin{equation}
\label{eq2.1}
\end{equation}
that, for $a=b$ adopts the appearance
\begin{equation}
\label{eq2.2}
\end{equation}
\begin{equation}
\label{eq2.3} \phi(a,a,z)=e^z.
\end{equation}
Let us consider now the function $\phi$ and the variable $\lambda$
\begin{equation}
\label{eq2.4} \phi\left[a,a,-\left(\lambda t+\frac {x}
{\lambda}\right)\right]= e^{-\left(\lambda t+\frac {x}
\end{equation}
where we express $\lambda$ in terms of an equation involving two
quantities $K$ and $Q$ of Eq. (<ref>)
\begin{equation}
\label{eq2.5} \lambda^3+K\lambda+\frac {Q} {2}=0,
\end{equation}
and we define $z$ as
\begin{equation}
\label{eq2.6} z=-\lambda t-\frac {x} {\lambda}.
\end{equation}
Given that $\phi$ is such that
\begin{equation}
\label{eq2.7} \phi^{''}=\lambda^2\frac {\partial^2\phi} {\partial
x^2}\;\;\; ;\;\;\; \phi^{'}=-\frac {1} {\lambda}\frac
{\partial\phi} {\partial t} \equiv \phi,
\end{equation}
Eq. (<ref>) can be recast as
\begin{equation}
\label{eq2.8} z\lambda^2\frac {\partial^2\phi} {\partial
x^2}+a\phi^{'} +\frac {z} {\lambda}\frac {\partial\phi} {\partial
\end{equation}
Since $\phi^{'}=\phi$, (<ref>) gets simplified to
\begin{equation}
\label{eq2.9} \lambda^3\frac {\partial^2\phi} {\partial x^2}
+\frac {\partial\phi} {\partial t}=0.
\end{equation}
According to (<ref>), Eq. (<ref>) becomes
\begin{equation}
\label{eq2.10} -(K\lambda+\frac {Q} {2})\frac {\partial^2\phi}
{\partial x^2} +\frac {\partial\phi} {\partial t}=0.
\end{equation}
In addition, since $\phi$ verifies
\begin{equation}
\label{eq2.11} \lambda\frac {\partial^2\phi} {\partial x^2}=
-\frac {\partial\phi} {\partial x},
\end{equation}
we are led to the following expression for (<ref>)
\begin{equation}
\label{eq2.12} K\frac {\partial\phi} {\partial x} -\frac {Q}
{2}\frac {\partial^2\phi} {\partial x^2} +\frac {\partial\phi}
{\partial t}=0,
\end{equation}
which is tantamount to
\begin{equation}
\label{eq2.13} \frac {\partial\phi} {\partial t}+ \frac {\partial
(K\phi)} {\partial x} -\frac {Q} {2}\frac {\partial^2\phi}
{\partial x^2}=0,
\end{equation}
i.e., Fokker-Plank's equation para $K$ independent of $x$. Of course, when $K$ does depend upon $x$
one just postulates (<ref>). Note that, by
definition, (<ref>) is a solution of (<ref>).
§ SEPARATION OF VARIABLES IN THE ORNSTEIN-UHLENBECK PROCESS $K=X$
The Ornstein–Uhlenbeck process is a stochastic process that,
loosely, describes the velocity of a massive Brownian particle
under the influence of friction. It is stationary, Gaussian, and
Markovian, being the only nontrivial evolution that satisfies
these three conditions, up to allowing for linear transformations
of the space and time variables. We believe that this well known
process of linear drift <cit.> is worth revisiting for
didactic purposes. We start with
\begin{equation}
\label{eq3.1} \frac {\partial F} {\partial t}+ \frac {\partial
(KF)} {\partial x} -\frac {Q} {2}\frac {\partial^2F} {\partial
\end{equation}
\begin{equation}
\label{eq3.2} F(x,t)=G(t)H(x),
\end{equation}
which leads to
\begin{equation}
\label{eq3.3} \frac {1} {G}\frac {\partial G} {\partial t}= \frac
{1} {H}\left[ \frac {Q} {2}\frac {\partial^2H} {\partial x^2}
-\frac {\partial (KH)} {\partial x}\right]=-\lambda,
\end{equation}
with $\lambda>0$. From here we are immediately led to
\begin{equation}
\label{eq3.4} \frac {\partial G} {\partial t}+\lambda G=0,
\end{equation}
\begin{equation}
\label{eq3.5} \frac {Q} {2}\frac {d^2H} {d x^2} -\frac {d (KH)} {d
x}+\lambda H=0.
\end{equation}
For the linear instance $K=-x$ we first obtain for $G$
\begin{equation}
\label{eq3.6} G(t)=e^{-\lambda t}.
\end{equation}
Applying the Fourier transform to (<ref>) we find
\begin{equation}
\label{eq3.7} \frac {Q} {2}\alpha^2\hat{H}+ \alpha\frac {d\hat{H}}
\end{equation}
where $\hat{H}$ is the Fourier transform of $H$ of variable
$\alpha$. One solves (<ref>) and get
\begin{equation}
\label{eq3.8} \hat{H}(\alpha)=|\alpha|^{\lambda}e^{-\frac
{Q\alpha^2} {4}},
\end{equation}
and from (<ref>) we encounter for $H$
\begin{equation}
\label{eq3.9} H(x)=\frac {1} {2\pi}\int\limits_{-\infty}^{\infty}
|\alpha|^{\lambda}e^{-\frac {Q\alpha^2} {4}} e^{-i\alpha
\end{equation}
Thus we have for $F$ the general expression
\begin{equation}
\label{eq3.10} F(x,t)=\frac {1} {2\pi}
\int\limits_0^{\infty}\int\limits_{-\infty}^{\infty} \lambda
a(\lambda)e^{-\lambda t} |\alpha|^{\lambda}e^{-\frac {Q\alpha^2}
{4}} e^{-i\alpha x}\;d\alpha\;d\lambda,
\end{equation}
where $a(\lambda)$ must verify
\begin{equation}
\label{eq3.11} \int\limits_0^{\infty} a(\lambda)\;d\lambda=1.
\end{equation}
Eq. (<ref>) may have been obtained before, but we were
unable to find such derivation in the vast FP-literature available
to us.
§ NON-LINEAR FOKKER-PLANK EQUATION <CIT.>
Anomalous diffusion is exhibited in a variety of physical
systems and is therefore the subject of much interest. It can be
observed, for example, in general systems such as plasma flow,
porous media, and surface growth, as well as in more specific
situations such as cytltrimethylammonium bromide miscelles
dissolved in salted water and NMR relaxometry of liquids in porous
glasses <cit.>. The main characteristic of anomalous
diffusion is the fact that the mean squared displacement is not
proportional to time $t$ but rather to some power of it. If the
scaling is faster than $t$, then the pertinent system is
superdiffusive while, if it is slower than $t$, it is
subdiffusive. A nonlinear Fokker-Planck diffusion equation has
been proposed for those systems with correlated anomalous
diffusion, beginning with <cit.> and followed afterward by,
for instance, <cit.>. For an excellent overview,
see <cit.>.
For the ordinary hypergeometric function $F_1^2(a,b;c;z)$ we
have <cit.>, using now three Pochhamer symbols,
F_1^2(a,b;c;z) ≡F(a,b;c;z)= ∑_n=0^∞
a_(n)b_(n)/c_(n) z^n/n!; (|z| < 1 ),
where the series terminates if either $a$ or $b$
is a non-zero integer. A particularly important special case is
F(-m,b,b,-z) = (1+z)^m. Eq. (<ref>)
verifies <cit.>
\begin{equation}
\label{eq4.1} z(1-z)F^{''}(\alpha,\beta;\gamma;z)+
\alpha\beta F(\alpha,\beta;\gamma;z)=0.
\end{equation}
If $\beta=\gamma$, then $F$ satisfies <cit.>
\begin{equation}
\label{eq4.2} F(-\alpha,\gamma;\gamma;-z)=(1+z)^{\alpha}.
\end{equation}
Focus attention now upon the function
\begin{equation}
\label{eq4.3} f(x,t)=\left[1+(q-1)\left(\lambda t+\frac {x}
{\lambda} \right)\right]^{\frac {1} {1-q}},
\end{equation}
where $\lambda$ obeys (for $K$ and $Q$ both constants)
\begin{equation}
\label{eq4.4} \lambda^3+K\lambda+\frac {Q} {2}=0.
\end{equation}
Recourse to (<ref>) allows one to write
\begin{equation}
\label{eq4.5} F\left[\frac {1} {q-1},\gamma;\gamma;
(1-q)\left(\lambda t+\frac {x} {\lambda} \right)\right]=
\left[1+(q-1)\left(\lambda t+\frac {x} {\lambda}\right)
\right]^{\frac {1} {1-q}},
\end{equation}
and then
\begin{equation}
\label{eq4.6} z=(1-q)\left(\lambda t+\frac {x} {\lambda}\right).
\end{equation}
For $\beta=\gamma$, $F$ [Cf. (<ref>)] adopts the appearance
\begin{equation}
\label{eq4.7} z(1-z)F^{''}(\alpha,\gamma;\gamma;z)+
\alpha\beta F(\alpha,\gamma;\gamma;z)=0.
\end{equation}
Since $F$ verifies
\begin{equation}
\label{eq4.8} F^{''}=\frac {\lambda^2} {(1-q)^2}\frac {\partial^2
F} {\partial x^2}\;\;\;;\;\;\;F^{'}=\frac {1} {\lambda} \frac
{\partial F} {\partial t},
\end{equation}
then (<ref>) becomes
\begin{equation}
\label{eq4.9} z(1-z)\frac {\lambda^2} {(1-q)^2}\frac {\partial^2
F} {\partial x^2}+ \frac {qz} {\lambda(1-q)^2} \frac {\partial F}
{\partial t}+\gamma(1-z)F^{'}- \frac {\gamma} {q-1}F=0,
\end{equation}
and, adequately simplifying,
\begin{equation}
\label{eq4.10} (1-z)\lambda^3\frac {\partial^2 F} {\partial x^2}+
q\frac {\partial F} {\partial t}+ \frac {\gamma} {z}(1-q)^2
\left[(1-z)F^{'}-\frac {1} {q-1}F\right]=0.
\end{equation}
Again, since $F$ fulfills
\begin{equation}
\label{eq4.11}
(1-z)F^{'}-\frac {1} {q-1}F=0
\end{equation}
Eq. (<ref>) becomes
\begin{equation}
\label{eq4.12} (1-z)\lambda^3\frac {\partial^2 F} {\partial x^2}+
q\frac {\partial F} {\partial t}=0,
\end{equation}
or, equivalently,
\begin{equation}
\label{eq4.13} \lambda^3F^{(1-q)}\frac {\partial^2 F} {\partial
x^2}+ q\frac {\partial F} {\partial t}=0,
\end{equation}
since $F^{(1-q)}(z)=1-z$. Thus, we are in a position to cast
(<ref>) as
\begin{equation}
\label{eq4.14} \lambda^3\frac {\partial^2 F} {\partial x^2}+ \frac
{\partial F^q} {\partial t}=0.
\end{equation}
Utilizing (<ref>) we can recast things as
\begin{equation}
\label{eq4.15} -\left(\lambda K+\frac {Q} {2}\right)\frac
{\partial^2 F} {\partial x^2}+ \frac {\partial F^q} {\partial
\end{equation}
Remembering that $F$ obeys
\begin{equation}
\label{eq4.16} \lambda K\frac {\partial^2 F} {\partial
x^2}=-K\frac {\partial F^q} {\partial x}= -\frac {\partial (KF^q)}
{\partial x},
\end{equation}
we obtain from (<ref>)
\begin{equation}
\label{eq4.17} \frac {\partial F^q} {\partial t}+ \frac
{\partial(KF^q)} {\partial x} -\frac {Q} {2}\frac {\partial^2 F}
{\partial x^2}=0,
\end{equation}
a nonlinear Fokker-Planck equation. We postulate its validity for
$K=K(x)$ as well. If we set
* $g = F^{q}$
* $2-q^*= 1/q$,
we immediately ascertain that Eq. (<ref>), expressed in
terms of $g$ and $q^*$, coincides with the nonlinear FP postulated
by Plastino and Plastino in <cit.>
∂g/∂t + ∂(Kg)/∂x - Q/2∂^2 g^2-q^*/∂x^2=0.
For the stationary case ($F$ independent of
$t$) we have for Eq. (<ref>)
\begin{equation}
\label{eq4.19} \frac {\partial(K(x) F^q)} {\partial x} -\frac {Q}
{2}\frac {\partial^2 F} {\partial x^2}=0,
\end{equation}
whose solution is
\begin{equation}
\label{eq4.20} F(x)=\left[1+\frac {2(q-1)} {Q}V(x)\right]^{\frac
{1} {1-q}},
\end{equation}
where $\frac {dV(x)} {dx}=-K(x)$.
§ CONCLUSIONS
We have shown that the Fokker-Planck equation and its
nonlinear generalization by Plastino and Plastino <cit.> are
contained within the structure of hypergeometric linear
differential equations, for constant drift $K$. The FP-extensions
to general drifts $K(x)$ have to be postulated like in the
ordinary cases.
We have displayed a general solution for the Orstein-Uhlenbeck
equation of constant drift that possibly might be new, although we
cannot ascertain it.
We also give an exact solution of the nonlinear FP equation
when $F$ does not depend upon the time.
risken H. Risken, The Fokker-Planck equation (Springer, Berlin, 1989).
ppFP A. R. Plastino, A. Plastino, Physica A 222
(1995) 347.
tp1 A. Plastino and M. C. Rocca:“Hypergeometric
Connotations of Quantum Equations”. ArXiV: 1505.06365;
A.Plastino and M. C. Rocca: Physics
Letters A, 379 (2015) 2690.
ii1 M. N. Najafi,
Phys. Rev. E 92, 022113 (2015)
ii2 Z. G. Arenas, D. G. Barci, C. Tsallis
Phys. Rev. E 90, 032118 (2014).
ii3 G. A. Casas, F. D. Nobre, E. M. F. Curado
Phys. Rev. E 86, 061136 (2012).
ii4 M. S. Ribeiro, C. Tsallis, F. D. Nobre, Phys. Rev. E 88,
052107 (2013); M. S. Ribeiro, F. D. Nobre, C. Tsallis, ibid. 89, 052135 (2014).
book T. D. Frank, The nonlinear Fokker-Planck
equation (Springer, Berlin, 2005).
otros C. Tsallis, D. J. Bukman, Phys. Rev. E 54
(1996) R2197 1996; A. Compte, D. Jou, J. Phys. A 29 (1996)
4321; A. Compte, D. Jou, Y. Katayama, J. Phys. A 30 (19997)
lisa L. Borland, Phys. Rev. E 57 (1998) 6634.
nobre V. Schwammle, E.M.F. Curado, F.D. Nobre, Eur. Phys. J. B 58 (2007)
tp2 I. S. Gradshteyn and I. M. Ryzhik:
“Table of Integtals, Series and Products”,
9.151, p.1045. Academic Press (1965).
tp3 M. Abramowitz and I. Stegun:
“Handbook of Mathematical Functions”,
15.1.8, p.556. National Bureau of Standards
Applied Mathematics Series 55 (1964).
|
1511.00515
|
Department of Control and Power Engineering,
Gdańsk University of Technology, Poland
A four-dimensional photon polarization space, such that gives a different interpretation of the ladder operators for the time-like degree comparing to the Gupta-Bleuler formulation is presented. This interpretation, coming from the construction of a covariant Hamiltonian, gives positive defined norms for all the four polarization degrees of freedom. This paper is written in the background of reducible representation algebras introduced by Czachor, although the interpretation of the ladder operators as presented here could be done independently. States that reproduce standard electromagnetic fields (i.e. photons with two transverse polarizations) from the four-dimensional covariant formalism (i.e. with two additional longitudinal and time-like polarizations) are shown explicitly. Lastly, Lorentz and gauge transformations are discussed in detail.
§ INTRODUCTION
The Gupta-Bleuler formulation of four-dimensional quantization is well known from literature <cit.> - <cit.>. Mostly, it is assumed that the time-like photon states have negative norms. Unfortunately, states with such norms form a serious difficulty with the probability interpretation of quantum mechanics. There seems to be no experimental verification of such particles, therefore they are often called in literature unphysical or even “ghosts" <cit.>. In this paper the problem of negative norms of the time-like photons is profoundly investigated, introducing a different interpretation of the ladder operators.
For further analysis let us take a convention $c=\hbar=1$. Building a relativistic model for photons, we have to consider momentum and polarization of the photon. In this mathematical model these two quantities will be described in a tensor product structure, i.e.
\begin{eqnarray}
\textrm{photon momentum space}~
\otimes
\textrm{photon polarization space},
\nonumber
\end{eqnarray}
although it should be stressed that in relativistic context spin and momentum are not independent degrees of freedom.
First, in Section <ref> a construction coming from a covariant Hamiltonian for a four-dimensional polarization oscillator is shown.
Then we ask what is the consequence of creating particles on the energy of the whole system, i.e. does this raise the energy level or lower it? This is discussed in sections: <ref> for the space-like polarizations and further in <ref> and <ref> two different interpretations of the time-like polarization are considered.
Next, some preliminary aspects on the Czachor's reducible representations of canonical commutation relations (CCR) are presented. Such model has strong arguments in its favor, mainly because it deals with most of the infrared and ultraviolet divergences. In Section <ref> a basic concept of the model is introduced. Further, in Section <ref> the ladder and the number operators are introduced for $N=1$ oscillator space.
In Section <ref> the covariant Hamiltonian in the reducible representation algebra is presented.
How does the extension to $N$-oscillator space look like and what is the definition of the number operator in an arbitrary $N$ oscillator space are discussed in Section <ref>.
Next, the construction of vacuum is shown in <ref>. Then we ask: how does such a theory, with four polarization degrees of freedom, correspond to Maxwell's electromagnetism theory. It turns out that vectors, denoted here by $\Psi_{EM}$, reproduce standard Maxwell electrodynamics. This is shown in Section <ref>. Finally, the electromagnetic field tensor for such representation is introduced in Section <ref>.
In the last section a homogeneous Lorentz transformation for the four-dimensional oscillator is introduced. When taking into account the four-photon polarization degrees of freedom, the Lorentz transformation is accompanied by another transformation and this manifests itself also on the spin-frame level. In Section <ref> a corresponding transformation of the ladder operators is derived. Further, in Section <ref>, the generators for the irreducible representation are introduced. Next, in Section <ref>, we introduce Lorentz transformation in reducible representation algebra, whereas, in <ref> we show vacuum transformation. In Section <ref> we discuss gauge transformations of the four-vector potential and
in Section <ref> we show that the “ghost" operator and the covariant number operator are invariant in any gauge and any reference frame. Finally, in Section <ref> the four-translations are introduced. Summary and final conclusions are drawn in Section <ref>.
This paper closes with mathematical appendices, introducing the Minkowski and null tetrad in Penrose and Rindler notation <cit.> and proofs involving Lorentz transformation on electromagnetic potential electromagnetic field operator.
§ FOUR-DIMENSIONAL PHOTON POLARIZATION SPACE
§.§ Construction of four-dimensional polarization space
To construct a four-dimensional photon polarization space, let us first introduce a covariant Hamiltonian of the form
\begin{equation}
\frac{p_{{\bf a}}p^{{\bf a}}}{2}
-\frac{q_{{\bf a}}q^{{\bf a}}}{2},~~~ {\bf a}=0,1,2,3.
\end{equation}
Here $p_{{\bf a}}$ and $q_{{\bf a}}$ are some canonical variables such that
\begin{equation}\label{canvar}
p_{{\bf a}}
i\partial_{{\bf a}}
i\frac{\partial}{\partial q^{{\bf a}}}
ig_{{\bf {ab}}}\frac{\partial}{\partial q_{{\bf {b}}}},
\end{equation}
with commutation relations
\begin{eqnarray}
[q_{{\bf a}},p_{{\bf {b}}}]
\end{eqnarray}
The metric here is denoted as diag$(+,-,-,-)$. These canonical variables should not be mistaken with the position and momentum of the photon field. This construction is made strictly for the four degrees of photon polarization. Now let us define non-hermitian operators
\begin{eqnarray}\label{acov}
a_{{\bf a}}
\frac{
q_{{\bf a}}+ip_{{\bf a}}}{\sqrt{2}}
\frac{
q_{{\bf a}}-\partial_{{\bf a}}}{\sqrt{2}},
\\
a_{{\bf a}}^{\dagger}\label{acovdag}
\frac{
q_{{\bf a}}-ip_{{\bf a}}}{\sqrt{2}}
\frac{
q_{{\bf a}}+\partial_{{\bf a}}}{\sqrt{2}},
\end{eqnarray}
which satisfy the following commutation relations
\begin{equation}\label{comaadag}
[a_{{\bf a}},a_{{\bf {b}}}^{\dagger}]
\end{equation}
\begin{equation}
[a_{{\bf a}},a_{{\bf {b}}}]\label{comaa}
[a_{{\bf a}}^{\dag},a_{{\bf {b}}}^{\dag}]
\end{equation}
As a consequence of such covariant formalism, there are four polarization degrees of freedom. They can be written in a four-dimensional tensor product space
\begin{eqnarray}\label{4a}
a_{1}=\textrm{a}_1\otimes 1\otimes 1\otimes 1,~~~~
a_{2}=1\otimes \textrm{a}_2\otimes 1\otimes 1,
\\
a_{3}=1\otimes 1\otimes \textrm{a}_3\otimes 1,~~~~
a_{0}=1\otimes 1\otimes 1\otimes \textrm{a}_0,
\end{eqnarray}
and such space is spanned by kets of the form
\begin{eqnarray}\label{ket4irr}
|n_1\rangle \otimes |n_2\rangle \otimes |n_3\rangle \otimes |n_0\rangle.
\end{eqnarray}
Let us also take a closer look at
\begin{eqnarray}
a_{{\bf a}}^{\dagger}
a^{{\bf a}}
\frac{1}{2}q_{{\bf a}}q^{{\bf a}}
p_{{\bf a}}p^{{\bf a}}.
\end{eqnarray}
This implies that the Hamiltonian operator (<ref>) can be written in terms of operators $a_{{\bf a}}^{\dagger}$ and $a^{{\bf a}}$ in the form
\begin{eqnarray}
-a_{{\bf a}}^{\dagger}
a^{{\bf a}}
~~~~~ {\bf a}=0,1,2,3.
\end{eqnarray}
Furthermore, the following commutation relations hold
\begin{equation}
[H,a_{{\bf a}}]
[ -a_{{\bf {b}}}^{\dagger}
a^{{\bf {b}}}
,a_{{\bf a}}]
-a_{{\bf a}},
\end{equation}
\begin{equation}
[H,a_{{\bf a}}^{\dagger}]
[- a_{{\bf {b}}}^{\dagger}
a^{{\bf {b}}}
,a_{{\bf a}}^{\dagger}]
a_{{\bf a}}^{\dagger}.
\end{equation}
Let us also assume that the eigenvalue of the covariant Hamiltonian operator (<ref>) acting on four-dimensional space is denoted by $E$.
\begin{eqnarray}
\end{eqnarray}
At this point this is really a quantity corresponding to the number of particles, but in the upcoming Section <ref> an extension of this model to reducible representation algebra is made and $E(1)$ will correspond to the total energy operator.
For now it can be shown that indeed $a_{{\bf a}}$ lowers and $a_{{\bf a}}^{\dag}$ raises $E$ by 1, i.e.
\begin{eqnarray}\label{defanihilation}
Ha_{{\bf a}}|n_1,n_2,n_3,n_0\rangle
(E-1)a_{{\bf a}}|n_1,n_2,n_3,n_0\rangle,
\\
\label{defcreation}
Ha_{{\bf a}}^{\dag}|n_1,n_2,n_3,n_0\rangle\
(E+1)a_{{\bf a}}^{\dag}|n_1,n_2,n_3,n_0\rangle.
\end{eqnarray}
§.§ Construction for space-like photons
From the previous section we learn that (<ref>) and (<ref>) define lowering and raising energy operators respectively and the raising operators will be denoted with a dagger. These operators are also know in the literature as annihilation and creation operators, but at this point this terminology will not be used, since the definition of the ${\bf a}=0$ polarization degree (time-like) may be ambiguous.
It is quite evident that for the polarizations indexed by ${\bf a}=1,2,3$, the raising energy operators create new states. Let us then define vacuum states for these three dimensions as normalized states that are annihilated by lowering energy operators
\begin{eqnarray}\label{vac123}
\textrm{a}_{{\bf{j}}}|0\rangle
\end{eqnarray}
Now let us normalize these space-like states to 1. The state of $n_{{\bf{j}}}$ excitations must be proportional to $n$ raising energy operators acting on ground state
\begin{eqnarray}
\textrm{a}_{{\bf{j}}}^{\dag n}|0\rangle
\end{eqnarray}
Then the scalar product
\begin{eqnarray}
\langle
\rangle
\nonumber\\
\langle 0|
\textrm{a}_{{\bf{i}}}^{n}
\textrm{a}_{{\bf{j}}}^{\dag n}|0\rangle
\delta_{{\bf{j}}}{_{{\bf{i}}}} n
\langle 0|
\textrm{a}_{{\bf{j}}}^{n-1}
\textrm{a}_{{\bf{i}}}^{\dag n-1}|0\rangle
\nonumber\\
\delta_{{\bf{j}}}{_{{\bf{i}}}} n(n-1)
\langle 0|
\textrm{a}_{{\bf{j}}}^{n-2}
\textrm{a}_{{\bf{i}}}^{\dag n-2}|0\rangle=...
\qquad
\end{eqnarray}
Going further with this recurrence, we get
\begin{eqnarray}
\langle
\rangle
\delta_{{\bf{i}}}{_{{\bf{j}}}} n!
\langle 0
\end{eqnarray}
Now we can give a normalized definition of bras and kets
\begin{eqnarray}
\frac{1}{\sqrt{n!}}
\textrm{a}_{{\bf{j}}}^{\dag n}|0\rangle,~~~~{\bf{j}}=1,2,3,
\nonumber\\
\langle n_{{\bf{j}}}|
\frac{1}{\sqrt{n!}}
\langle 0|
\textrm{a}_{{\bf{j}}}^n,~~~~{\bf{j}}=1,2,3.
\end{eqnarray}
This means that the action of raising and lowering operators is defined as follows
\begin{eqnarray}
\textrm{a}_{{\bf{j}}}^{\dagger}| n_{{\bf{j}}}\rangle &=& \sqrt{n+1}| n_{{\bf{j}}}
\\
\textrm{a}_{{\bf{j}}}| n_{{\bf{j}}}\rangle &=& \sqrt{n}| n_{{\bf{j}}}-1\rangle.
\end{eqnarray}
We also define number operators for these three polarizations
\begin{eqnarray}~\label{n123}
\textrm{a}_{{\bf{j}}}^\dagger\textrm{a}_{{\bf{j}}}| n_{{\bf{j}}}\rangle
n_{{\bf{j}}}| n_{{\bf{j}}}\rangle,~~~~{\bf{j}}=1,2,3.
\end{eqnarray}
Furthermore, for the $q$ representation of vacuum (<ref>), we get a differential equation
\begin{eqnarray}\label{}
\frac{1}{\sqrt{2}}
\left(
q_{{\bf{j}}}+\frac{\partial}{\partial q_{{\bf{j}}}}
\right)
\end{eqnarray}
with a simple solution
\begin{eqnarray}\label{}
\right)
\end{eqnarray}
Here $A$ is the normalization constant
\begin{eqnarray}\label{}
\left(
\frac{1}{\pi}
\right)^{\frac{1}{4}}.
\end{eqnarray}
The whole construction for these three polarization degrees of freedom is evident. Now without any confusion we can say that operators that raise the total energy, create particles, so can be called creation operators. Also those that lower the total energy should annihilate the vacuum. Not only the states have positive norms but also the vacuum in the canonical $q$ representation is proper, i.e. satisfies the requirement of going to zero at infinity, making it possible to normalize the function.
§.§ Construction for time-like photons, where lowering energy operators annihilate vacuum
The problem arises for the ${\bf a}=0$ degree. The question for the time-like polarization is: do the lowering energy operators annihilate particles and raising energy operators create ones or maybe do the lowering energy operators create particles and raising energy operators annihilate them.
First, let us assume that the lowering operator annihilates the vacuum, i.e.
\begin{eqnarray}\label{vac0}
\textrm{a}_{0}|0\rangle
\end{eqnarray}
Then, the state of $n_{0}$ excitations is proportional to $n$ raising operators acting on the ground state, so that
\begin{eqnarray}
\textrm{a}_{0}^{\dag n}|0\rangle
\end{eqnarray}
and the scalar product reads
\begin{eqnarray}
\langle
\rangle
\nonumber\\
\langle 0|
\textrm{a}_{0}^{n}
\textrm{a}_{0}^{\dag n}|0\rangle
\langle 0|(-)
\textrm{a}_{0}^{n-1}
\textrm{a}_{0}^{\dag n-1}|0\rangle
\nonumber\\
(-)^2 n(n-1)
\langle 0|
\textrm{a}_{0}^{n-2}
\textrm{a}_{0}^{\dag n-2}|0\rangle=...
\qquad
\end{eqnarray}
Going further with the recurrence we get
\begin{eqnarray}
\langle
\rangle
(-)^n n!
\langle 0
\end{eqnarray}
It looks like the states corresponding to odd values of $n$ give negative norms. Giving a normalized to 1 definition of bras and kets becomes a problem now. If we assume
\begin{eqnarray}
\frac{1}{\sqrt{n!}}
(i \textrm{a}_{0}^{\dag })^n|0\rangle,
\nonumber\\
\langle n_{0}|
\frac{1}{\sqrt{n!}}
\langle 0|
\left(i \textrm{a}_{0}\right)^n,
\end{eqnarray}
then saving the positivity of the scalar product has an effect on the hermitian conjugate operation.
On the other hand we could follow Gupta <cit.> and leave the metric indefinite.
But even then another problem arises, i.e. using the $q$ representation of vacuum, we get a differential equation
\begin{eqnarray}\label{}
\frac{1}{\sqrt{2}}
\left(
q_{0}-\frac{\partial}{\partial q_{0}}
\right)
\end{eqnarray}
with a solution that is divergent at infinity, i.e.
\begin{eqnarray}\label{}
\frac{q_{0}^2}{2}
\right).
\end{eqnarray}
All these conclusions may suggest that another point of view on the time-like polarization is needed.
§.§ Construction for time-like photons, where raising energy operators annihilate vacuum
Now let us assume that the raising operator annihilates the vacuum, which means that the energy spectrum is bounded from the top and to raise the total energy level we need to annihilate a particle, i.e.
\begin{eqnarray}\label{vac0}
\textrm{a}_{0}^{\dagger}|0\rangle
\end{eqnarray}
Such a construction has lots of advantages which will be shown in this section.
Now the state of $n_{0}$ excitations is proportional to $n$ lowering energy operators acting on ground state, so that
\begin{eqnarray}
\textrm{a}_{0}^{n}|0\rangle.
\end{eqnarray}
This means that creating new particles is lowering the total energy. Then the scalar product is
\begin{eqnarray}
\langle
\rangle
\nonumber\\
\langle 0|
\textrm{a}_{0}^{\dag n}\textrm{a}_{0}^{n}
\langle 0|
\textrm{a}_{0}^{\dag n-1}
\textrm{a}_{0}^{n-1}|0\rangle
\nonumber\\
\langle 0|
\textrm{a}_{0}^{\dag n-2}
\textrm{a}_{0}^{n-2}|0\rangle=...,
\end{eqnarray}
so that
\begin{eqnarray}
\langle
\rangle
\langle 0
\end{eqnarray}
Giving a normalized definition of bras and kets is not a problem now
\begin{eqnarray}
\frac{1}{\sqrt{n!}}
\textrm{a}_{0}^{n}|0\rangle,
\nonumber\\
\langle n_{0}|
\frac{1}{\sqrt{n!}}
\langle 0|
\textrm{a}_{0}^{\dag n}.
\end{eqnarray}
Furthermore, this means that for this representation the raising and lowering energy operators are defined as
\begin{eqnarray}
\textrm{a}_{0}| n_{0}\rangle &=& \sqrt{n+1}| n_{0}+1\rangle,
\\
\textrm{a}_{0}^{\dagger}| n_{0}\rangle &=& \sqrt{n}| n_{0}-1\rangle.
\end{eqnarray}
It turns out that in this case we do not have to choose between the positivity of the scalar product and the hermitian conjugate operation. It should be stressed that the number operator for the time-like polarization should be defined carefully
\begin{eqnarray}~\label{n0}
\textrm{a}_{0}\textrm{a}_{0}^\dagger| n_{0}\rangle
n_{0}| n_{ 0}\rangle.
\end{eqnarray}
Now formulating the vacuum state is not problematic. Using the $q$ representation we get a differential equation
\begin{eqnarray}\label{}
\frac{1}{\sqrt{2}}
\left(
q_{0}+\frac{\partial}{\partial q_{0}}
\right)
\end{eqnarray}
with a solution
\begin{eqnarray}\label{}
\frac{-q_{0}^2}{2}
\right).
\end{eqnarray}
Summarizing all the above, from now on we will use the $a_0^{\dag}$ for an operator that annihilates vacuum and $a_0$ for the creation operator of time-like polarization states.
Another interesting fact comes from such interpretation of the time-like photon. To see this we first split the Hamiltonian in two: for the transverse polarizations $H_{12}$ and the unmeasured ones $H_{03}$, such that
\begin{eqnarray}\label{H1203}
\left(
\right)
\end{eqnarray}
\begin{eqnarray}
\label{H12}
\nonumber\\
\nonumber\\
\left(
\right)
\\
\label{H03}
\nonumber\\
\nonumber\\
\left(
\right)
\end{eqnarray}
From this we see that for such a definition of $a_0$, the energy of the ground state comes only from transverse polarizations.
§ FOUR-DIMENSIONAL OSCILLATOR REDUCIBLE REPRESENTATION ALGEBRA
§.§ Motivation for reducible representation algebra proposed by Czachor
In 1925 Heisenberg, Born and Jordan postulated that energies of classical free fields look in Fourier space like oscillator ensembles. It should be stressed that at that time Heisenberg, Born and Jordan did not know the notation of Fock space and may not fully understood the role of eigenvalues of operators. Having to consider oscillators with different frequencies they considered one oscillator for each frequency mode. In such case the ensemble had to be infinite, since the number of modes was infinite.
It is a well known problem that the standard canonical procedures for field quantization result in various infinity problems.
It was shown by Czachor <cit.> that the assumption of having one oscillator for each frequency mode may not be natural. This assumption continued in a series of papers on reducible representation of CCR <cit.> - <cit.>. The main idea for reducible representation algebra is that each oscillator is a wave packet, a superposition of infinitely many different momentum states.
To describe this concept in more detail let us first introduce a spectral decomposition of the frequency operator $\Omega$
\begin{eqnarray}\label{omega}
\Omega
\int
d \Gamma ({\bf k})~
\omega({\bf k})~
|{\bf k}\rangle\langle{\bf k}|,
\end{eqnarray}
so that (<ref>) fulfills the following eigenvalue problem
\begin{eqnarray}
\Omega|{\bf k}\rangle
\int
d \Gamma ({\bf k}')~
\omega({\bf k}')
|{\bf k}'\rangle\langle{\bf k}'|
{\bf k}\rangle
\omega({\bf k})
|{\bf k}\rangle.
\end{eqnarray}
Here $d \Gamma ({\bf k})$ is the Lorentz invariant measure
\begin{equation}\label{Lormeasure}
=\frac{d^3k}{(2\pi)^{3} 2 |{\bf k}|}
\end{equation}
Furthermore, kets of momentum are normalized to
\begin{equation}
\langle{\bf k}|{\bf k}'\rangle
(2\pi)^{3}2|{\bf k}|\delta^{(3)}
({\bf k},{\bf k}')
\delta_{\Gamma}({\bf k},{\bf k}'),
\end{equation}
and the resolution of unity is
\begin{equation}
\int_{R^3}
d \Gamma ({\bf k})
|{\bf k}\rangle \langle {\bf k}|
\end{equation}
The energy for photons, assuming the convention $\hbar=1$, is $E({\bf k})=\omega({\bf k})=|{\bf k}|$. Then the simplest Hamiltonian for one kind of polarization can be written in the form
\begin{eqnarray}
\Omega
\otimes
\left(
\frac{1}{2}
\right)
\nonumber\\
\int
d \Gamma ({\bf k})~
\omega({\bf k})~
|{\bf k}\rangle\langle{\bf k}|
\otimes
\left(
\frac{1}{2}
\right),
\end{eqnarray}
so that
\begin{eqnarray}
|{\bf k}, n
\rangle
\omega({\bf k})
\left(
\right)
|{\bf k}, n
\rangle.
\end{eqnarray}
Here ket $|n\rangle$ is the eigenvector of a “standard theory" number operator
$a^{\dagger}a$, where $[a,a^{\dagger}]=1$.
Now let us introduce an operator that lives in both: the momentum and polarization spaces
\begin{eqnarray}\label{ared}
a({\bf k},1)
|{\bf k}\rangle\langle{\bf k}|
\otimes
\end{eqnarray}
In bracket of $a({\bf k},1)$ in (<ref>), ${\bf k}$ indicates that this representation is reducible and $1$ that it is the $N=1$ oscillator representation. Using the resolution of unity, we can also define an operator within the whole spectrum of frequencies
\begin{eqnarray}
\int
d \Gamma ({\bf k})~
a({\bf k},1)
\otimes
\end{eqnarray}
such that the commutator $[a(1),a^{\dagger}(1)]=I\otimes 1$. Here $1$ in the bracket of $a(1)$ denotes that this is the $N=1$ oscillator representation.
§.§ Lie algerba
The four-dimensional $N=1$ (or 1-oscillator) representation of CCR acts in the Hilbert space
${\cal H}(1)$ spanned by kets of the form
\begin{eqnarray}\label{ket41xy}
&&\hspace{-6pt}|{\bf k},n_1,n_2,n_3,n_0\rangle
\nonumber\\
|{\bf k}\rangle\otimes
\frac{
\quad
\end{eqnarray}
Operators $a_{1}, a_{2}, a_{3}, a_{0}$ satisfy the commutation relations typical for irreducible
representation of CCR (<ref>). In (<ref>) $a_1^{\dag},~a_2^{\dag}$ stand for creation operators for linear polarized photons and $a_0$, $a_3^{\dag}$ are both creation operators for time-like and longitudinal photons respectively.
For the reducible representation we define the ladder operators
\begin{equation}
a_{{\bf a}}({\bf k},1)
|{\bf k}\rangle\langle{\bf k}|\otimes a_{{\bf a}},
~~~~~ {\bf a}=0,1,2,3.
\end{equation}
Then the following CCR algebra holds
\begin{eqnarray}\label{CCR41}
[a_{{\bf a}}({\bf k},1),a_{{\bf {b}}}
({\bf k}',1)^{\dagger}]
-g_{{\bf {ab}}}
\delta_{\Gamma}({\bf k},{\bf k}')
|{\bf k}\rangle\langle{\bf k}|\otimes 1_4
\nonumber\\
g_{{\bf {ab}}}
\delta_{\Gamma}({\bf k},{\bf k}')
I({\bf k} ,1)
\end{eqnarray}
This algebra representation is reducible, since the right-hand side of the commutator (<ref>) is an operator-valued distribution $I({\bf k},1)=|{\bf k}\rangle\langle{\bf k}|\otimes 1_4$ belonging to the center of algebra, i.e.
\begin{equation}
[a_{{\bf a}}({\bf k},1),I({\bf k}',1)]
[a_{{\bf a}}({\bf k},1)^{\dagger},I({\bf k}',1)]
\end{equation}
where operator $I({\bf k},1)$ forms the resolution of unity for
$\mathcal{H}(1)$ Hilbert space
\begin{equation}
\int d \Gamma({\bf k}) ~I({\bf k},1)
\end{equation}
The number operator for the reducible representation algebra in $\mathcal{H}(1)$ Hilbert space will be defined as
\begin{eqnarray}\label{n1}
n_{{\bf j}}({\bf k},1)
|{\bf k}\rangle\langle{\bf k}|\otimes a_{{\bf j}}^{\dagger}a_{{\bf j}},
~~~~~ {\bf j}=1,2,3,
\\
n_{0}({\bf k},1)
|{\bf k}\rangle\langle{\bf k}|\otimes a_{{0}}a_{{0}}^{\dagger}.
\end{eqnarray}
We can also define a number operator within the whole spectrum of frequencies as follows
\begin{eqnarray}\label{n(1)}
n_{{\bf a}}(1)
\int
d\Gamma({\bf k})
n_{\bf a}({\bf k},1),
~~~~~ {\bf a}=0,1,2,3.
\end{eqnarray}
Then the eigenvalue definition of the number operator, i.e. the number of photons of one kind of polarization within the whole frequency spectrum, would be
\begin{eqnarray}
n_{{\bf a}}(1)
|{\bf k},n_1,n_2,n_3,n_0\rangle
n_{{\bf a}}
|{\bf k},n_1,n_2,n_3,n_0\rangle.
\end{eqnarray}
Now the following Lie algebra for the reducible representation holds
\begin{equation}
[a_{{\bf a}}({\bf k},1),
a_{{\bf b}}({\bf k}',1)^{\dagger}]
-g_{{\bf a}{\bf b}}
\delta_{\Gamma}({\bf k},{\bf k}')
I({\bf k},1),
\end{equation}
\begin{equation}
[a_{{\bf a}}({\bf k},1),
n_{{\bf b}}({\bf k}',1)]
-g_{{\bf a}{\bf b}}
\delta_{\Gamma}({\bf k}, {\bf k}')
a_{{\bf b}}({\bf k},1),
\end{equation}
\begin{equation}
[a_{{\bf a}}({\bf k},1)^{\dagger},
n_{{\bf b}}({\bf k}',1)]
g_{{\bf a}{\bf b}}
\delta_{\Gamma}({\bf k}, {\bf k}')
a_{{\bf b}}({\bf k},1)^{\dagger} .
\end{equation}
Furthermore, for the representation within the whole frequency spectrum we have
\begin{equation}
[a_{{\bf a}}(1),
a_{{\bf b}}(1)^{\dagger}]
g_{{\bf a}{\bf b}}
\end{equation}
\begin{equation}
[a_{{\bf a}}(1),
n_{{\bf b}}(1)]
g_{{\bf a}{\bf b}}
a_{\bf b}(1),
\end{equation}
\begin{equation}
[a_{{\bf a}}(1)^{\dagger},
n_{{\bf b}}(1)]
g_{{\bf a}{\bf b}}
a_{{\bf b}}(1)^{\dagger} .
\end{equation}
As we can see, the Lie algebra for the whole frequency spectrum has the “standard theory" structure.
§.§ Hamiltonian
Now, let us introduce a covariant Hamiltonian in the reducible representation algebra
\begin{eqnarray}
\Omega
\otimes
\end{eqnarray}
Here $\Omega$ is the frequency operator (<ref>) and $H$ is defined in (<ref>).
Hamiltonian (<ref>) can be also written in terms of operators $a_{{\bf a}}^{\dagger}$ and $a_{{\bf a}}$ in the form
\begin{eqnarray}
\int
d \Gamma({\bf k})
~|{\bf k}|~
|{\bf k}\rangle \langle{\bf k}|
\otimes
\left(
-a_{{\bf a}}^{\dagger}
a^{{\bf a}}
\right)
\end{eqnarray}
Now the following commutation relations hold for the reducible representation algebra
\begin{equation}
[H(1),a_{{\bf a}}({\bf k},1)]
-|{\bf k}|~a_{{\bf a}}({\bf k},1),
\end{equation}
\begin{equation}
[H(1),a_{{\bf a}}({\bf k},1)^{\dagger}]
|{\bf k}|~a_{{\bf a}}({\bf k},1)^{\dagger}.
\end{equation}
Moreover, within the whole frequency spectrum of the ladder operators, we get
\begin{equation}
[H(1),a_{{\bf a}}(1)]
d \Gamma({\bf k})~
|{\bf k}|~a_{{\bf a}}({\bf k},1)
-\Omega\otimes a_{{\bf a}},
\end{equation}
\begin{equation}
[H(1),a_{{\bf a}}(1)^{\dagger}]
\int
d \Gamma({\bf k})~
|{\bf k}|~a_{{\bf a}}({\bf k},1)^{\dagger}
\Omega\otimes a_{{\bf a}}^{\dag}.
\end{equation}
Let us also assume that the eigenvalue of the covariant Hamiltonian operator (<ref>) acting on the four-dimensional space of 1-oscillator reducible representation is the total energy denoted by $E(1)$
\begin{eqnarray}
&&\hspace{-6pt}H(1)|{\bf k}, n_1,n_2,n_3,n_0\rangle
\nonumber\\
E(1)|{\bf k}, n_1,n_2,n_3,n_0\rangle
\nonumber\\
|{\bf k}|~(n_1+n_2+n_3-n_0+1)|{\bf k}, n_1,n_2,n_3,n_0\rangle.\qquad
\end{eqnarray}
Now it can be shown that indeed $a_{{\bf a}}(1)$ lowers and $a_{{\bf a}}(1)^{\dag}$ raises the total energy by $|{\bf k}|$, i.e.
\begin{eqnarray}\label{defanihilationred}
&&\hspace{-6pt}H(1)a_{{\bf a}}(1)|{\bf k},n_1,n_2,n_3,n_0\rangle
\nonumber\\
(E(1)-|{\bf k}|)a_{{\bf a}}(1)|{\bf k},n_1,n_2,n_3,n_0\rangle,
\end{eqnarray}
\begin{eqnarray}
&&\hspace{-6pt}H(1)a_{{\bf a}}(1)^{\dag}|{\bf k},n_1,n_2,n_3,n_0\rangle\label{defcreationred}
\nonumber\\
(E(1)+|{\bf k}|)a_{{\bf a}}(1)^{\dag}|{\bf k},n_1,n_2,n_3,n_0\rangle.
\end{eqnarray}
§.§ $N$-oscillator space
Now let us discuss an extension of this model to arbitrary $N$ oscillators. The parameter $N$ characterizes reducible representations but is not directly related to the number of photons. The Hilbert space for any $N$ oscillators reads
\begin{equation}
\mathcal{H}(N)
\underbrace{\mathcal{H}(1)\otimes\ldots\otimes\mathcal{H}(1)}_{N}
\mathcal{H}(1)^{\otimes N}.
\end{equation}
So that the $\mathcal{H}(N)$ Hilbert space is spanned by kets of the form
\begin{eqnarray}\label{ket41N}
|{\bf k}_1,n_1^{1},n_2^{1},n_3^{1},n_0^{1}
\rangle\otimes\dots\otimes |{\bf k}_N,n_1^{N},n_2^{N},n_3^{N},n_0^{N}\rangle.\qquad
\end{eqnarray}
Let us also define an operator
\begin{equation}\label{A^(n)}
\underbrace{I\otimes...\otimes I}_{n-1}
\otimes A\otimes\underbrace{I\otimes...\otimes I}_{N-n}.
\end{equation}
The top index $(n)$ shows the “position" of the $A$ operator in $\mathcal H(N)$ space.
Then we construct the Hamiltonian for $N$ oscillators as follows
\begin{eqnarray}\label{covredHN}
\sum_{n=1}^{N}
\end{eqnarray}
A natural extension for the ladder operators to $N$-oscillator reducible representations would be
\begin{eqnarray}\label{aN}
a_{{\bf a}}({\bf k},N)
\frac{1}{\sqrt{N}}
\sum_{n=1}^{N}
a_{{\bf a}}({\bf k},1)^{(n)}.
\end{eqnarray}
Term $\frac{1}{\sqrt{N}}$ is the normalization factor for $N$-oscillator representations. The CCR algebras still hold
\begin{eqnarray}
[a_{{\bf b}}({\bf k},N),
a_{{\bf b}}({\bf k}',N)^{\dagger}]
g_{{\bf a}{\bf b}}
\delta_{\Gamma}({\bf k},{\bf k}')
I({\bf k},N),
\end{eqnarray}
where at the right-hand side there is an operator
\begin{equation}
I({\bf k},N)
\frac{1}{N}
\sum_{n=1}^N
I({\bf k},1)^{(n)}
\end{equation}
which for all $N$ is also in the center of algebra, since
\begin{equation}
[a_{{\bf a}}({\bf k},N),I({\bf k}',N)]
[a_{{\bf a}}({\bf k},N)^{\dagger},I({\bf k}',N)]
\end{equation}
Then $I({\bf k},N)$ forms the resolution of unity for
$\mathcal{H}(N)$ Hilbert space, i.e.
\begin{eqnarray}
\int d \Gamma({\bf k} )~I({\bf k},N)
\underbrace{I(1)\otimes...\otimes I(1)}_{N}
\end{eqnarray}
Again algebra representations within the whole frequency spectrum holds the “standard theory" structure.
Using definition (<ref>) of the number operator for $N=1$, we write
\begin{eqnarray}\label{nIIIN}
n_{{\bf a}}({\bf k},N)
\sum_{n=1}^N n_{{\bf a}}({\bf k}, 1)^{(n)}.
\end{eqnarray}
Furthermore, the following Lie algebras for reducible $N$-oscillator representations hold
\begin{equation}
[a_{{\bf a}}({\bf k},N),
n_{{\bf b}}({\bf k}',N)]
-g_{{\bf a}{\bf b}}
\delta_{\Gamma}({\bf k}, {\bf k}')
a_{{\bf b}}({\bf k},N),
\end{equation}
\begin{equation}
[a_{{\bf a}}({\bf k},N)^{\dagger},
n_{{\bf b}}({\bf k}',N)]
g_{{\bf a}{\bf b}}
\delta_{\Gamma}({\bf k}, {\bf k}')
a_{{\bf b}}({\bf k},N)^{\dagger} .
\end{equation}
§.§ Vacuum and energy of vacuum
The subspace of vacuum states is spanned by vectors of the form
\begin{eqnarray}
|{\bf k}_1,0,0,0,0\rangle\otimes\dots\otimes |{\bf k}_N,0,0,0,0\rangle.
\end{eqnarray}
Vacuum in this representation is any state annihilated by any annihilation operator.
\begin{equation}\label{defvac}
a_{{\bf j}}(1)|O(1)\rangle=0,~~~~{\bf{j}}=1,2,3,
\end{equation}
\begin{equation}
\end{equation}
Therefore, in $N=1$ oscillator representation, we may write
\begin{equation}\label{vacN1}
\int d\Gamma({\bf k}) O({\bf k})
|{\bf k},0,0,0,0\rangle.
\end{equation}
From the normalization condition
\begin{equation}
\langle O(1)
\end{equation}
we get
\begin{equation}\label{sqrvac}
\int d\Gamma({\bf k}) |O({\bf k})|^{2}
\int d\Gamma({\bf k}) Z({\bf k})
\end{equation}
Here the scalar field $Z({\bf k})=|O({\bf k})|^{2} $ represents vacuum probability density. Furthermore, square integrability of (<ref>) implies that $Z({\bf k})$ must decay at infinity. This point is of special importance for reducible representation algebra quantization. It turns out that regularization can be a consequence of employing such special form of scalar field in the definition of vacuum.
Recalling (<ref>), where it is shown that the energy of the vacuum comes only from the two transverse polarization degrees of freedom, we calculate
\begin{eqnarray}\label{vacN1E}
\int d \Gamma({\bf k})
~|{\bf k}|~
O({\bf k})
|{\bf k},0,0,0,0\rangle.
\end{eqnarray}
The extension to $N$-oscillator space is assumed to be a tensor product of $N=1$ vacuum states, i.e.
\begin{equation}
|O(1)\rangle^{\otimes N}
\underbrace{|O(1)\rangle\otimes...\otimes|O(1)\rangle}_{N}.
\end{equation}
Of course, the normalization condition for the $N$-oscillator representations still holds
\begin{equation}
\langle O(N)
\langle O(1)
\end{equation}
Let us also take definition (<ref>) for the Hamiltonian in $N$-oscillator representations and write
\begin{eqnarray}\label{vacNE}
\sum_{n=1}^N
\end{eqnarray}
We see that the expectation value of the vacuum energy does not depend on the $N$ parameter, i.e.
\begin{eqnarray}\label{vacNE}
&&\hspace{-6pt}\langle O(N)|
\nonumber\\
\langle O(1)
\langle O(1)|
\nonumber\\
\int d \Gamma({\bf k})
~|{\bf k}|~
Z({\bf k}).
\end{eqnarray}
This means that the energy of vacuum is not zero and depends only on the vacuum probability density $Z({\bf k})$. For the vacuum energy to be convergent we must demand
\begin{eqnarray}\label{vaccond}
\int d \Gamma({\bf k})
~|{\bf k}|~
Z({\bf k})
\end{eqnarray}
Therefore, the convergence of vacuum is guaranteed by the proper choice of the vacuum probability density function and does not require the $N$ parameter at all. Furthermore, this analysis shows that the $N$ parameter may even be a finite number.
§.§ Vector space corresponding to Maxwell's theory
Let us start by presenting the potential operator in reducible representation algebra for $N=1$ oscillator
\begin{eqnarray}\label{potN1}
i\int d\Gamma({\bf k})
g_{a}{^{\bf a}}({\bf k})a_{{\bf a}}({\bf k},1)
e^{-ik\cdot x}+\rm{H.c.}\qquad
\end{eqnarray}
Here, not just two operators corresponding to the polarizations, but four types are introduced. The same four-dimensional quantization, where in the place of the annihilation operator of the Gupta-Bleuler type potential for the time-like degree stands a creation operator, was already formulated by Czachor and Naudts <cit.> and further by Czachor and Wrzask <cit.>. Let us point out that $a_0$ is indeed a lowering total energy operator which is a creation operator like in <cit.>, only it is denoted here, on the contrary to mentioned papers, without a dagger. Notation in this paper comes straightforward from the analysis in Section <ref> and is convenient for the possibility of writing collective formulas. In (<ref>) $g_{a}{^{\bf a}}(\bf k)$ is a field of Minkowski tetrads. More on the Minkowski and null tetrad in Penrose notation is presented in <ref>. Furthermore, in <ref> we show that the potential operator (<ref>) transforms as a four-vector.
The four-divergence of the potential operator in $N=1$ oscillator representation reads
\begin{eqnarray}
\nonumber\\
\frac{1}{\sqrt{2}}
\int d\Gamma({\bf k})
\left(
a_0({\bf k},1)
a_3({\bf k},1)
\right)
e^{-ik\cdot x}+{\rm H.c.}\qquad\quad
\end{eqnarray}
As we can see this does not correspond to classical Maxwell theory, that is the Lorenz condition does not hold on the four-vector potential operator. Operator $a_0({\bf k},1)-a_3({\bf k},1)$ in Dürr and Rudolph paper <cit.> is called a “bad ghost". This is a very adequate name because it spoils the classical electrodynamics correspondence.
Returning to our problem, it would be appreciable for the theory to eliminate “bad ghosts" and this can be done by solving a weaker, i.e. in averages Lorenz condition
\begin{eqnarray}\label{weeklorenz}
\langle \Psi_{EM}(1)|
\partial^{a}A_a(x,1)
|\Psi_{EM}(1) \rangle
\end{eqnarray}
We will try to impose this condition on the Hilbert space instead of on the operators. Here $\Psi_{EM}(1)$ are vectors that satisfy (<ref>). Later, it will be shown that such vectors give equivalence to standard free-field Maxwell electromagnetism theory. From (<ref>) it follows that
\begin{eqnarray}\label{weeklorenz2}
\langle\Psi_{EM}(1)|
\int d\Gamma({\bf k})
\left(
a_0({\bf k},1)
a_3({\bf k},1)
\right)
|\Psi_{EM}(1) \rangle
\nonumber\\
\end{eqnarray}
Solving (<ref>) we find that $\Psi_{EM}(1)$ has a coherent-like structure.
Therefore, we define a displacement-like operator for time-like and longitudinal photons. We will start from the $N=1$ oscillator representation denoting
\begin{eqnarray}\label{DEMN1}
&&\hspace{-6pt}{\cal D}_{03}(\alpha,1)
\nonumber\\
\exp
\left(
\int d\Gamma({\bf k})
\Big(
\overline{\alpha({\bf k})}
(a_{0}({\bf k},1)
a_{3}({\bf k},1))
{\rm H.c.}
\Big)
\right).
\nonumber\\
\end{eqnarray}
Here $\alpha({\bf k})$ is a function corresponding to the “amount of displacement" and can in general be dependent on ${\bf k}$.
Acting with operator (<ref>) on vacuum we get a vector state of the form
\begin{eqnarray}
|\Psi_{EM}(1) \rangle=
{\cal D}_{03}(\alpha,1)|O(1)\rangle
\nonumber\\
\sum_{n_0,n_3=0}^{\infty}
\int d\Gamma({\bf k})
O({\bf k})
\exp
\left(-
|\alpha({\bf k})|^2
\right)
\nonumber\\
\frac{
(\alpha({\bf k}))^{n_3} (\overline{\alpha({\bf k})})^{n_0}}
|{\bf k},n_1,n_2,n_3,n_0\rangle.
\end{eqnarray}
Operator (<ref>) shifts the ladder operators by $\alpha({\bf k})$
\begin{eqnarray}
{\cal D}_{03}(\alpha,1)^{\dag}a_0({\bf k},1)
{\cal D}_{03}(\alpha,1)
a_0({\bf k},1)
\alpha({\bf k})I({\bf k},1),
\nonumber\\
\\
{\cal D}_{03}(\alpha,1)^{\dag}a_3({\bf k},1)
{\cal D}_{03}(\alpha,1)
a_3({\bf k},1)
\alpha({\bf k})I({\bf k},1).
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
{\cal D}_{03}(\alpha,1)^{\dag}
\left(
a_0({\bf k},1)
a_3({\bf k},1)
\right)
{\cal D}_{03}(\alpha,1)
\nonumber\\
a_0({\bf k},1)
a_3({\bf k},1),
\end{eqnarray}
so that the week Lorenz condition (<ref>) holds. Furthermore, for $\Psi_{EM}(1)$ the number of time-like photons is equal to the number of longitudinal ones
\begin{eqnarray}\label{eqnon31}
\langle\Psi_{EM}(1)|
\left(
n_0({\bf k},1)
n_3({\bf k},1)
\right)
|\Psi_{EM}(1) \rangle
\end{eqnarray}
Therefore, their contribution cancels against each other.
The extension to arbitrary $N$-oscillator representations can be made by
\begin{eqnarray}
\underbrace{|\Psi_{EM}(1)\rangle\otimes...\otimes|\Psi_{EM}(1)\rangle}_{N}
|\Psi_{EM}(1)\rangle^{\otimes N}
\end{eqnarray}
and the week Lorenz condition holds also for $N$-oscillator representations
\begin{eqnarray}\label{weeklorenzN}
\langle\Psi_{EM}(N)|
\int d\Gamma({\bf k})
\left(
a_0({\bf k},N)
a_3({\bf k},N)
\right)
|\Psi_{EM}(N) \rangle
\nonumber\\
\end{eqnarray}
Also for arbitrary $N$ oscillators, the number of longitudinal photons equals to the number of time-like ones, i.e.
\begin{eqnarray}\label{n_0n_3N}
\langle\Psi_{EM}(N)|
\left(
n_0({\bf k},N)
n_3({\bf k},N)
\right)
|\Psi_{EM}(N) \rangle
\end{eqnarray}
Let us note that this is not the usual Gupta-Bleuler condition such that
\begin{eqnarray}\label{guptacon2}
\left(
\right)
|\Psi \rangle
\end{eqnarray}
due to two aspects: a different definition of $a_0$ ladder operator and because it does not hold on the vector states but on the inner product.
§.§ Electromagnetic field operator
The electromagnetic field operator for $N=1$ oscillator representation is by definition
\begin{eqnarray}\label{e-mdef}
\partial_{a}A_{b}(x,1)-\partial_{b}A_{a}(x,1).
\end{eqnarray}
This can be written explicitly as
\begin{eqnarray}\label{e-m1}
\nonumber\\
2\int d\Gamma({\bf k})
k_{[a}({\bf k})
g_{b]}{^{\bf{a}}}({\bf k})
a_{\bf{a}}({\bf k},1)
e^{-ik\cdot x}
+{\rm H.c.}
\nonumber\\
\int d\Gamma({\bf k})
\left[
-k_a({\bf k})
x_{b}({\bf k})
k_{b}({\bf k})
x_{a}({\bf k})
\right]
a_{1}({\bf k},1)
e^{-ik\cdot x}
\nonumber\\
\int d\Gamma({\bf k})
\left[
-k_a({\bf k})
y_{b}({\bf k})
k_{b}({\bf k})
y_{a}({\bf k})
\right]
a_{2}({\bf k},1)
e^{-ik\cdot x}
\nonumber\\
\frac{1}{\sqrt{2}}
\int d\Gamma({\bf k})
\left[
t_a({\bf k})
z_{b}({\bf k})
t_{b}({\bf k})
z_{a}({\bf k})
\right]
\nonumber\\
\left(
a_{0}({\bf k},1)
a_{3}({\bf k},1)
\right)
e^{-ik\cdot x}
+{\rm H.c.}
\end{eqnarray}
Here $k_a({\bf k})$ is the element of the null tetrad introduced in <ref>.
The field tensor (<ref>) can be split into two parts: consisting the transverse photon operators and the “bad ghost" operators corresponding to particles unmeasured in experiments. It can be shown that in $\Psi_{EM}(1)$ the electromagnetic field operator corresponds to standard electromagnetism theory.
Furthermore, let us also check the free Maxwell equations for the electromagnetic field operator
\begin{eqnarray}
\partial_c F_{ab}(x,1)\label{maxwelleq}
\partial_a F_{bc}(x,1)
\partial_b F_{ca}(x,1)
\\
\partial_{a}\label{maxwelleq2}
\nonumber\\
\frac{i}{\sqrt{2}}
\int d\Gamma({\bf k})
k^{b}({\bf k})
\left(
a_{0}({\bf k},1)
a_{3}({\bf k},1)
\right)
e^{-ik\cdot x}+{\rm H.c.}
\nonumber\\
\end{eqnarray}
The second equation (<ref>) consists of a “bad ghost", but in $\Psi_{EM}(1)$ averages
\begin{eqnarray}
\langle \Psi_{EM}(1)|
\partial_{a}
|\Psi_{EM}(1) \rangle
\end{eqnarray}
As in the potential operator, the extension of the electromagnetic field to $N$-oscillator representations is equivalent to the extension for $N$ representations of the creation and annihilation operators in (<ref>).
§ LORENTZ TRANSFORMATION
§.§ Bogoliubov type transformation
In <ref> and <ref> transformation properties of spin-frames and the field of Minkowski tetrads are introduced in detail. In this section we will concentrate on the corresponding transformation on the ladder operators.
At this point only the irreducible representation of CCR will be considered. Let us first define new ladder operators
\begin{equation}
L_{\bf{a}}{^{\bf b}}(\Theta,\phi)a_{\bf{b}}.
\end{equation}
Transformation matrix $L_{\bf{a}}{^{\bf {b}}}(\Theta,\phi)$ can be written explicitly in terms of the Wigner phase $\Theta(\Lambda,{\bf k})$ and parameter $\phi({\bf k})=|\phi({\bf k})|e^{i\xi({\bf k})}$ related to the gauge transformation. This is shown in (<ref>).
$L_{\bf{a}}{^{\bf b}}(\Theta,\phi)$ has the property of leaving the metric invariant, i.e.
\begin{equation}
\end{equation}
so that the new operators satisfy the same CCR
\begin{eqnarray}
L_{\bf{a}}{^{\bf {c}}}(\Theta,\phi)a_{\bf{c}}
L_{\bf{b}}{^{\bf {d}}}(\Theta,\phi)a_{\bf{d}}^{\dagger}
\nonumber\\
-L_{\bf{a}}{^{\bf {c}}}(\Theta,\phi)
L_{\bf{b}}{^{\bf {d}}}(\Theta,\phi)
\end{eqnarray}
Therefore, there must exist an unitary Bogoliubov-type transformation $U(\Theta,\phi) $ satisfying
\begin{eqnarray}\label{bogulubovtrans}
L_{\bf{a}}{^{\bf b}}(\Theta,\phi)
\end{eqnarray}
In the next section an explicit representation of $U(\Theta,\phi)$ will be given.
§.§ Wigner rotations and gauge transformation in four-dimensional polarization space
In 1939 Wigner studied the subgroups of the Lorentz group, whose transformations leave the four-momentum of a given free particle invariant <cit.>. The maximal subgroup of the Lorentz group, which leaves the four momentum invariant is called the little group. This implies that the little group governs the internal space-time symmetries of relativistic particles. Wigner showed in his paper that invariant space-time symmetries are dictated by O(3)-like little groups in the case of massive particles and by E(2)-like little groups in the case of the massless ones. The application for photons has been discussed in many papers, among all <cit.>–<cit.>. It is also known that the Lorentz group is a very natural language for polarized light.
In order to explicitly construct transformation $U(\Theta,\phi)$ in (<ref>), let us first introduce the representation of the Lie algebra generators of rotations $J_{\bf i}$ around an ${\bf i}$-th axis. This first will be done in canonical variables (<ref>) introduced earlier
\begin{equation}
J_{\bf i}=\varepsilon_{\bf {ijk}}q_{\bf j}p_{{\bf k}}.
\end{equation}
Using formulas (<ref>)-(<ref>) we can write the generators of rotations in terms of the ladder operators
\begin{eqnarray}
&J_{\bf i}&=-i\varepsilon_{\bf {ijk}}a_{\bf j}^{\dagger}a_{{\bf k}}.
\end{eqnarray}
Also let us introduce boosts $K_{\bf i}$ along an ${\bf i}$-th axis, first in terms of canonical variables (<ref>)
\begin{equation}
K_{\bf i}=p_{\bf i}q_{0}-p_{0}q_{\bf i}.
\end{equation}
Again using formulas (<ref>)-(<ref>) we write the generators of boosts in terms of the ladder operators of general form
\begin{eqnarray}
&K_{\bf i}&=i (a_{\bf i}^{\dagger}a_{0}
-a_{0}^{\dagger}a_{\bf i}).
\end{eqnarray}
Interestingly, the same form of generators was introduced earlier in <cit.>, although without the knowledge of canonical variables (<ref>). These generators satisfy the following commutation relations
\begin{equation}
[J_{\bf i},J_{\bf j}]
i\varepsilon_{\bf {ijk}}J_{{\bf k}},
\end{equation}
\begin{equation}
[K_{\bf i},K_{\bf j}]
-i\varepsilon_{\bf {ijk}}J_{{\bf k}},
\end{equation}
\begin{equation}
[J_{\bf i},K_{\bf j}]
i\varepsilon_{\bf{ijk}}K_{{\bf k}}.
\end{equation}
Wigner in his paper <cit.> showed that the little group for massless particles moving along $z$ axis is generated by the rotation generators around $z$ axis $J_3$ and two other generators. These other two generators are combinations of $J_{\bf i}$ and $K_{\bf i}$ and form a representation of an Euclidean group E(2), i.e.
\begin{equation}\label{L1L2}
\qquad
\qquad
\end{equation}
with the following commutation relations
\begin{equation}
[L_{1},L_{3}]=-iL_{2}, \quad
[L_{2},L_{3}]=iL_{1}, \quad
\end{equation}
The physical variable associated with $J_3$ is the helicity degree of freedom of massless particles, but it was not clear what is the physical interpretation of generators $L_1$ and $L_2$. In 1971 Janner and Janssen <cit.> showed that those generators generate translations and are responsible for gauge transformations of the four potential. This will be discussed further in Section <ref>.
Moreover, it should be stressed that only $L_3$ annihilates the vacuum states, since it is normally ordered, contrary to generators $L_1$ and $L_2$.
Now the Bogoliubov-type transformation can be written as
\begin{equation}\label{U(irr)}
\exp(i\alpha_{1}L_{1}+i\alpha_{2}L_{2}+i\alpha_{3}L_{3})
\end{equation}
with parameters
\begin{equation}
\alpha_{1}(\Theta,\phi)
\frac{\Theta(\Lambda, {\bf k})}{\sin\Theta(\Lambda, {\bf k})}
|\phi({\bf k})|\sin (\xi({\bf k})+\Theta(\Lambda, {\bf k})),
\end{equation}
\begin{equation}
\alpha_{2}(\Theta,\phi)
\frac{\Theta(\Lambda, {\bf k})}{\sin\Theta(\Lambda, {\bf k})}
|\phi({\bf k})|\cos(\xi({\bf k})+\Theta(\Lambda, {\bf k})),
\end{equation}
\begin{equation}
\alpha_{3}(\Theta)
2\Theta(\Lambda, {\bf k}).
\end{equation}
As we can see the parameter $\alpha_3$ depends only on the Wigner phase.
We can also write formula (<ref>) in following forms
\begin{eqnarray}
\nonumber\\
\exp{\left(-i|\phi| \sin(\xi+2\Theta) L_1-i|\phi| \cos(\xi+2\Theta) L_2\right)}
\nonumber\\
\exp{\left(-2 i \Theta L_3\right)}\label{UGUR}
\\
\exp{\left(-2 i\Theta L_3\right)}\label{URUG}
\nonumber\\
\exp{\left(-i |\phi| \sin \xi L_1
-i |\phi| \cos \xi L_2\right)}
\end{eqnarray}
and define transformations associated with the Lorentz transformation and gauge transformation respectively
\begin{equation}
U(\Lambda,{\bf k})
\exp{\left(-2 i\Theta(\Lambda, {\bf k}) L_3\right)},\label{UR}
\end{equation}
\begin{eqnarray}
U_G({\bf k})\label{UG}
\exp{\left(-i |\phi({\bf k})| \sin \xi({\bf k}) L_1
-i |\phi({\bf k})| \cos \xi({\bf k}) L_2\right)},
\hspace{-1cm}
\nonumber\\
\end{eqnarray}
such that
\begin{equation}
U(\Lambda,{\bf k})^{\dag}
U(\Lambda,{\bf k})
R_{{\bf{a}}}{^{\bf{b}}}(\Lambda,{\bf k})
\end{equation}
\begin{equation}
U_G({\bf k})^{\dag}
U_G({\bf k})
G_{{\bf{a}}}{^{\bf{b}}}({\bf k})
\end{equation}
where $R_{{\bf{a}}}{^{\bf{b}}}(\Lambda,{\bf k})$ and
$G_{{\bf{a}}}{^{\bf{b}}}({\bf k})$ are transformation matrices (<ref>) and (<ref>) introduced in <ref>.
§.§ Lorentz transformation for the reducible representation algebra
To construct a Lorentz transformation for the reducible $N=1$ oscillator
representation the Bogoliubov-type transformation must be written as
\begin{equation}
\int d\Gamma({\bf k})
|{\bf k}
\rangle\langle
\otimes
U(\Lambda,{\bf k}).
\end{equation}
Then, for the hermitian conjugate we can write
\begin{equation}\label{WU}
\int d\Gamma({\bf k})
\rangle\langle
{\bf k}|
\otimes U(\Lambda,{\bf k})^{\dag}.
\end{equation}
Let us denote
\begin{eqnarray}\label{Wop}
\int d\Gamma({\bf k})
|{\bf k}
\rangle\langle
\end{eqnarray}
This operator is not dependent on spin and acts only on momentum ${\bf k}$, i.e.
\begin{eqnarray}
W(\Lambda)|{\bf {k}}\rangle
|{\bf{\Lambda k}}\rangle.
\end{eqnarray}
Operator (<ref>) leaves the inner product invariant and therefore is an unitary operator. The hermitian conjugate of (<ref>) is
\begin{eqnarray}
\int d\Gamma({\bf k})
\rangle\langle
{\bf k}|,
\\
&&\hspace{-6pt} W(\Lambda)^{\dag}|{\bf {k}}\rangle
\end{eqnarray}
For the $N$-oscillator extension we can write
\begin{eqnarray}
U(\Lambda,0,1)^{\otimes N}.
\end{eqnarray}
Then, the transformation rule for the ladder operators in $N$-oscillator reducible representations
\begin{eqnarray}\label{translora}
a_{\bf{a}}({\bf k},N)
\nonumber\\
R_{{\bf{a}}}{^{\bf{b}}}(\Lambda,{\bf k})
\end{eqnarray}
Furthermore, it is shown in <ref> that
\begin{eqnarray}\label{Ulambdalambda'}
&& U(\Lambda,0,1)U(\Lambda',0,1)
\end{eqnarray}
§.§ Transformation properties of vacuum
Let us remind ourselves that the definition of vacuum in this representation is not unique, i.e.
\begin{eqnarray}
\int d\Gamma({\bf k}) O({\bf k})
|{\bf k},0,0,0,0\rangle.
\nonumber
\end{eqnarray}
Then the Lorentz transformation acts on vacuum as follows
\begin{eqnarray}\label{vacuumtrans}
\int d\Gamma({\bf k}) O({\bf{\Lambda^{-1}k}})
|{\bf k},0,0,0,0\rangle
\end{eqnarray}
The transformed vacuum state is again a vacuum state, but the probability of finding ${\bf k}$ is modified by the Doppler effect. As a byproduct we observe that the vacuum field transforms as a
scalar field
\begin{eqnarray}
O({\bf k})
\end{eqnarray}
This also implies the following transformation rule of the vacuum probability density
\begin{eqnarray}
Z({\bf k})
\end{eqnarray}
Of course, the norm of such transformed vacuum, due to the Lorentz invariant measure (<ref>), is invariant
\begin{eqnarray}
&&\hspace{-6pt}\langle O(1)| U(\Lambda,0,1)^{\dagger} U(\Lambda,0,1)
\nonumber\\
\int d\Gamma({\bf k}) |O({\bf{\Lambda^{-1}k}})|^{2} =1.
\qquad
\end{eqnarray}
§.§ Gauge transformation
Quantum field theory is assumed to be gauge invariant. The change of gauge is a change in electromagnetic potential that does not produce any change in physical observables. We will show that for the covariant reducible representation algebra there exists a transformation that corresponds to a gauge transformation of the potential in $\Psi_{EM}(1)$ vector space.
Now let us start from the potential operator (<ref>) and see, how it transforms after transformation
\begin{equation}
\int
d \Gamma ({\bf k})
|{\bf k}\rangle \langle {\bf k}|
\otimes
U_G({\bf k})^{\dag} .
\end{equation}
\begin{eqnarray}\label{potN1gauge3}
\tilde{A}_{a}(x,1)
\nonumber\\
\partial_a \varphi(x,1)
B_a (x,1).
\end{eqnarray}
\begin{eqnarray}\label{potN1gauge4}
\sqrt{2}
\int d\Gamma({\bf k})
|\phi({\bf k})|
\left(
\cos\xi a_1({\bf k},1)\right.
\nonumber\\
\left.
\sin\xi a_2({\bf k},1)
\right)
e^{-ik\cdot x}
\\
B_a (x,1)\label{potN1gauge5}
i\int d\Gamma({\bf k})
\Big(
-x_{a}({\bf k})
|\phi|\cos \xi
y_{a}({\bf k})
|\phi|\sin \xi
\nonumber\\
\left(
z_{a}({\bf k})
t_{a}({\bf k})
\right)
\frac{|\phi|^2}{2}
\Big)
\nonumber\\
\times
\left(a_3({\bf k},1) -a_0({\bf k},1)
\right)
e^{-ik\cdot x} +\rm{H.c.}
\hspace*{-20pt}
\end{eqnarray}
The $B_a(x,1)$ term contains the “bad ghost" operator, but in $\Psi_{EM}(1)$ vector space this contribution vanishes, i.e.
\begin{eqnarray}\label{}
&&\hspace{-6pt}\langle \Psi_{EM}(1)|
\tilde{A}_{a}(x,1)
|\Psi_{EM}(1) \rangle
\nonumber\\
\langle \Psi_{EM}(1)|
\left(
\partial_a \varphi(x,1)
\right)
|\Psi_{EM}(1) \rangle.
\end{eqnarray}
Moreover, the potential operator (<ref>) under transformation (<ref>) holds the weaker Lorenz condition, i.e.
\begin{eqnarray}\label{potN1Lorenzgauge}
\partial^a
\tilde{A}_{a}(x,1)
\partial^a
\end{eqnarray}
so that
\begin{eqnarray}
\langle \Psi_{EM}(1)|
\partial^{a}\tilde{A}_{a}(x,1)
|\Psi_{EM}(1) \rangle
\end{eqnarray}
Earlier, in Section <ref>, it was shown that there exists a space denoted by $\Psi_{EM}(1)$ in which a weaker Lorenz condition (<ref>) holds. This result should be compared with those of Janner and Janssen <cit.> followed by Han, Kim and Son <cit.>. They worked out a similar conclusion that $L_1$ and $L_2$ generators carry gauge transformations for the potential operator $(A_0, A_1, A_2, A_3)$, where $A_0=A_3$. Here the conclusion is the same for equal numbers of longitudinal and time-like photons, i.e. with $n_0=n_3$.
§.§ Invariants in a combined homogeneous Lorentz and gauge transformation
The combined homogeneous Lorentz and gauge transformation $U(\Theta,\phi)$ mixes transverse ladder operators $a_1,~a_2$ with excitations $a_3, ~a_0$.
Let us first take a closer look at the “bad ghost" operator $a_{3}-a_0$, i.e.
\begin{eqnarray}
U(\Theta,\phi)^{\dag} (a_{3}-a_0) U(\Theta,\phi)
a_{3} -a_{0}.
\end{eqnarray}
As we can see, this operator is invariant under transformation (<ref>).
It is also easy to show that the covariant total number of photons does not change due to the combined Lorentz and gauge transformation, i.e.
\begin{eqnarray}
\left(
\right)
\nonumber\\
\end{eqnarray}
Furthermore for the Lorentz transformation we get
\begin{eqnarray}
&&\hspace{-6pt}U(\Lambda,{\bf k})^{\dag}
\left(
\right)
U(\Lambda,{\bf k})
\\
&&\hspace{-6pt}U(\Lambda,{\bf k})^{\dag}
\left(
\right)
U(\Lambda,{\bf k})
\end{eqnarray}
This implies that the Lorentz transformation alone does not mix the two observable in experiments transverse polarizations with the two unobservable ones.
§.§ Four-translations in four-dimensional oscillator representation
Let us begin with the $N=1$ oscillator representation and denote $U({\bf 1},y,1)= e^{iP(1)\cdot y}$.
The generator of four-translations, the four-momentum for the reducible representation reads
\begin{eqnarray}\label{P(1)}
{\textstyle\int} d\Gamma({\bf k})k_{\bf{a}}|{\bf k}\rangle
\langle {\bf k}|\otimes H,
\end{eqnarray}
where $P_0(1)$ is of course the covariant Hamiltonian (<ref>) introduced earlier in section <ref>.
One can immediately verify that
\begin{eqnarray}
e^{iP(1)\cdot y}a_{\bf a}({\bf k},1)e^{-iP(1)\cdot y}
a_{\bf a}({\bf k},1) e^{-iy\cdot k},
\end{eqnarray}
implying the following transformation on the vector potential
\begin{eqnarray}
U({\bf 1},y,1)^{\dagger}A_{a}(x,1)U({\bf 1},y,1)
\end{eqnarray}
Furthermore, the four-momentum for arbitrary $N$-oscillator reads
\begin{eqnarray}\label{P(N)}
\sum_{n=1}^N P_{\bf{a}}(1)^{(n)}
\end{eqnarray}
Vectors (<ref>) are simultaneously the eigenvectors of $P_{\bf{a}}(N)$, i.e.
\begin{eqnarray}
&&\hspace{-6pt}P^{\bf{a}}(N) |{\bf k}_1,\dots,{\bf k}_N,
\nonumber\\
\Big(k^{\bf{a}}_1\big(n_1^{1}+n_2^{1}+n_3^{1}-n_0^{1}+1\big)
\nonumber\\
\nonumber\\
&&\quad\times~|{\bf k}_1,\dots,{\bf k}_N, n_0^{1},\dots,n_3^{N}\rangle.
\end{eqnarray}
Then the following transformation rule for the ladder operators in the $N$-oscillator reducible representation holds
\begin{eqnarray}
e^{iP(N)\cdot y}a_{\bf a}({\bf k},N)e^{-iP(N)\cdot y}
a_{\bf a}({\bf k},N) e^{-iy\cdot k},
\end{eqnarray}
\begin{eqnarray}
U({\bf 1},y,N)^{\dagger}A_{a}(x,N)U({\bf 1},y,N) &=& A_{a}(x-y,N).\quad
\end{eqnarray}
Then the Poincaré group, i.e. the semi-direct product of homogeneous Lorentz transformation and space-time translation groups is
\begin{eqnarray}
U({\bf 1},y,1) U(\Lambda,0,1),
\end{eqnarray}
and the composition law of two successive Poincaré transformations holds
\begin{eqnarray}
U(\Lambda_2,y_2,1) U(\Lambda_1,y_1,1)
\end{eqnarray}
§ SUMMARY AND CONCLUSIONS
In summary, we have proposed a construction for the four-dimensional polarization space coming from a definition of a covariant Hamiltonian (<ref>). Further analysis is presented for such formalism, especially regarding the interpretation of the ladder operators for the time-like polarization. Strong arguments are given in favor of an interpretation in which the operator annihilating vacuum is a raising energy operator. It turns out that assuming for the time-like photons the energy spectrum bounded from the top, we can preserve the positive norms as needed for the probability interpretation of quantum mechanics. Such an interpretation gives a non divergent vacuum representation and saves the hermiticity operation. Analysis presented in this paper is in agreement with the four-dimensional quantization of the potential operator in mentioned <cit.>, <cit.> papers. Further reducible representation algebras for the four-dimensional polarization oscillators are presented. Using the covariant Hamiltonian (<ref>) for $N$-oscillator reducible representation algebras, we find that such formalism is free from vacuum energy divergences. We show that the convergence of vacuum is guaranteed by a proper choice of the vacuum probability density function $Z({\bf k})$.
Next, states $\Psi_{EM}$, which reproduce standard electromagnetic fields (i.e. photons with two transverse polarizations) from the four-dimensional covariant formalism (i.e. with two additional longitudinal and time-like photons) are shown explicitly and discussed. It is interesting that such states have a coherent-like structure.
Finally, a homogeneous Lorentz transformation for the four-dimensional oscillator is introduced. Further, generators of these transformations coming from the canonical variables are shown.
We point out that there exists a transformation that corresponds to a gauge transformation of the potential for $\Psi_{EM}$ vectors. Next, invariants of introduced transformations are shown. Let us stress that the “ghost operator" coming from the two extra degrees of photon polarization is an invariant.
I am grateful to Marek Czachor and Gosia Śmiałek-Telega for discussions and critical comments.
§ PENROSE ABSTRACT INDICES NOTATION FOR TETRADS
Three types of indices were introduced in this paper: the boldface indices
${\bf{a}}$, ${\bf{b}}$ take numerical values $0$, $1$, $2$, $3$ and are related to a concrete choice of basis in Minkowski tetrad. The italics $a$, $b$ are abstract indices and specify types of tensor objects.
Indices that are boldfaced primed ${{\bf a}'}=00',01',10',11'$ are related to a concrete choice of basis in null tetrad.
The abstract index formalism allows to work at a basis independent level, with all the operations on indices we know from the matrix calculus.
However first we will introduce the Penrose spin-frame notation. The symplectic form $\varepsilon_{AB}$ is a skew-symmetric complex bilinear form, i.e. $\varepsilon_{AB}=-\varepsilon_{BA}$,
such that the action of the bilinear form on vectors $\varepsilon_{AB}\psi^A\phi^B$ is a complex number. The element of the dual spin-space is written with a unprimed subscript $\psi_A$.
It is often convenient to use a collective symbol $\varepsilon{_{{\bf A}}}{^A}$ for a spin-frame basis such that
\begin{equation}
\varepsilon_{0}{^{A}}=\omega^{A},
\qquad
\varepsilon_{1}{^{A}}=\pi^{A}.
\end{equation}
Then the components of $\varepsilon_{AB}$ with respect to the spin-frame basis are
\begin{eqnarray}
\varepsilon_{{\bf A} {\bf {B}}}
\varepsilon_{A B}
\varepsilon_{{\bf A}}{^A}
\varepsilon_{{\bf {B}}}{^B}
\left(
\begin{array}{cc}
\end{array}
\right).
\end{eqnarray}
The dual basis denoted as $\varepsilon{_A}{^{{\bf A}}}$ must then satisfy
\begin{eqnarray}
\varepsilon_{{\bf A}}{^A}
\varepsilon_{A}{^{{\bf {B}}}}
\varepsilon_{{\bf A}}{^{ {\bf {B}}}}
\left(
\begin{array}{cc}
\end{array}
\right).
\end{eqnarray}
From this it implies that
\begin{equation}
\varepsilon_{A}{^0}=-\pi_A,~~~~
\varepsilon_{A}{^1}=\omega_A.
\end{equation}
Therefore, the spin-frame and dual spin-frame can be written in a matrix notation, i.e.
\begin{eqnarray}
\varepsilon_{ A}{^{{\bf A}}}\label{epsilon^01}
\left(
\begin{array}{cc}
-\pi_A \\
\omega_A
\end{array}
\right)
\qquad
\varepsilon_{{\bf A}}{^{ A}}
\left(
\begin{array}{cc}
\omega^A\\
\pi^A
\end{array}
\right),
\end{eqnarray}
and any spin-frame satisfies the following
\begin{eqnarray}
\varepsilon_{AB}=\omega_{A}\pi_{B}-\pi_{A}\omega_{B}
\end{eqnarray}
Returning to the tetrads, the Minkowski space has the signature $(+,-,-,-)$ and the metric tensor is denoted by $g_{ab}$, $g^{ab}$. $g_{{\bf {ab}}}$ and $g^{{\bf {ab}}}$ are matrices diag$(+,-,-,-)$.
Minkowski tetrad $g_{a}{^{\bf{a}}}$, indexed by indices that are partly boldfaced and partly italic, consists of four four-vectors $g_{a}{^0}$, $g_{a}{^1}$, $g_{a}{^2}$, $g_{a}^{~3}$. There are two types of tetrads, related to the four-momentum $k^{{\bf{a}}}$ of a massless particles. The momentum independent tetrad $g_{a}^{~{\bf a}}$ satisfies $k^0=|{\bf k}|=k^a g_{a}{^0}$, $k^1=k^a g_{a}{^1}$, $k^2=k^a g_{a}{^2}$, $k^3=k^a g_{a}{^3}$, and defines decomposition into energy and momentum in the Lorentz invariant measure.
The null-vector $k^{a}=k^{a}({\bf k})$ plays the role of a flag pole for spinor field $\pi^{A}({\bf k})$ and can be written
in a spinor notation:
$k^{a}({\bf k})=\pi^{A}({\bf k})\pi^{A'}({\bf k})$,
where $\pi^{A}({\bf k})$ is a spinor field defined by $k^{a}({\bf k})$
up to a phase factor. For any $\pi^{A}({\bf k})$ there is
another spinor field associated $\omega^{A}({\bf k})$
satisfying the spin-frame condition $\omega_{A}({\bf k})\pi^{A}({\bf k})=1$.
We also consider a general field of tetrads defined on the light cone $g_{a}^{~~{\bf a}}({\bf k})$. Here $g_{a}^{~~1}({\bf k})$ and $g_{a}^{~~2}({\bf k})$ can play role of transverse polarization vectors.
Indices $a$ and ${\bf a}$ can be raised and lowered by means of the
Minkowski metric tensor $g_{ab}$, $g^{ab}$, $g_{\bf{ab}}$ and $g^{\bf{ab}}$.
The null tetrad is indexed by indices that are partly boldfaced primed and partly italic
It is important to distinguish between ${\bf a}$ and $\bf{a'}$, and we will employ the convention where ${{\bf a}'}=00',01',10',11'$ and $g_{\bf{a'b'}}=\varepsilon_{\bf{AB}}\varepsilon_{\bf{A'B'}}$,
$g^{\bf{a'b'}}=\varepsilon^{\bf{AB}}\varepsilon^{\bf{ A'B'}}$, where ${{\bf A}}=0,1$ and ${{\bf A'}}=0',1'$. We raise and lower indices
$\bf{AA'}$ by means of $\varepsilon_{\bf{AB}}\varepsilon_{\bf{A'B'}}$ and indices ${\bf a}'$ by means of matrix
\begin{equation}
\left(
\begin{array}{cccc}
\end{array}
\right).
\end{equation}
Therefore the null tetrad associated with spin-frames can be written as
\begin{eqnarray}\label{nulltetrad}
\varepsilon^{A}{_{\bf B}}
\varepsilon^{A'}{_{\bf B'}}
\left(
\begin{array}{c}
\omega^{A}\omega^{A'}\\
\omega^{A}\pi^{A'}\\
\pi^{A}\omega^{A'}\\
\pi^{A}\pi^{A'}
\end{array}
\right)
\left(
\begin{array}{c}
\omega^{a}\\
\bar{m}^{a}\\
\end{array}
\right),\qquad
\nonumber\\
\varepsilon_{A}{^{\bf B}}
\varepsilon_{A'}{^{\bf B'}}
\left(
\begin{array}{c}
\pi_{A}\pi_{A'}\\
\omega_{A}\omega_{A'}
\end{array}
\right)
\left(
\begin{array}{c}
\omega_{a}
\end{array}
\right).\qquad
\end{eqnarray}
Here, $g^{a}{_{01'}}$ and $g^{a}{_{10'}}$ can play the role for circular photon polarization vectors.
There is a relation between Minkowski tetrad, indexed by indices that are
partly boldfaced and partly italic, and null tetrad
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}\label{Mintetrad}
\left(
\begin{array}{c}
\end{array}
\right)
\frac{1}{\sqrt{2}} \left(
\begin{array}{cccc}
\end{array}
\right) \left(
\begin{array}{c}
\omega^{A}\omega^{A'}\\
\omega^{A}\pi^{A'}\\
\pi^{A}\omega^{A'}\\
\pi^{A}\pi^{A'}
\end{array}
\right)
\nonumber\\
\frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
\omega^{a}+k^{a}\\
i m^{a} - i\bar{m}^{a}\\
\omega^{a}-k^{a}
\end{array}
\right)
\left(
\begin{array}{c}
\end{array}
\right),
\end{eqnarray}
and dually we can write
\begin{eqnarray}
\end{eqnarray}
\begin{eqnarray}
\left(
\begin{array}{c}
\end{array}
\right)
\frac{1}{\sqrt{2}} \left(
\begin{array}{cccc}
\end{array}
\right) \left(
\begin{array}{c}
\pi_{A}\pi_{A'}\\
\omega_{A}\omega_{A'}
\end{array}
\right)
\nonumber\\
\frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
i \bar{m}_{a} -i m_{a}\\
\end{array}
\right)
\left(
\begin{array}{c}
\end{array}
\right).
\end{eqnarray}
Here the $g$s with the partly boldfaced and partly boldfaced primed
indices are the Infeld-van der Waerden symbols which can be written in the following matrix forms
\begin{eqnarray}
g_{~~{\bf{b}}'}^{{\bf{a}}} \label{gupadob'}
\frac{1}{\sqrt{2}} \left(
\begin{array}{cccc}
\end{array}
\right),
\end{eqnarray}
\begin{eqnarray}
g_{{\bf{a}}}^{~~{\bf{b}}'} \label{gdoaupb'}
\frac{1}{\sqrt{2}} \left(
\begin{array}{cccc}
\end{array}
\right).
\end{eqnarray}
These Infeld-van der Waerden symbols in their matrix form, in Penrose abstract index formalism, are used to translate formulas into matrix forms.
Furthermore, the relation between Minkowski tetrad and the metric tensor $g_{ab}$ is
\begin{eqnarray}\label{metrictensorM}
g_a{^{{\bf a}}} ({\bf k})g_b{^{{\bf {b}}}} ({\bf k})
g_{{\bf a}{\bf {b}}}
\nonumber\\
-x_{a}({\bf k})x_{b}({\bf k})
-y_{a}({\bf k})y_{b}({\bf k})
-z_{a}({\bf k})z_{b}({\bf k})
+t_{a}({\bf k})t_{b}({\bf k}).
\nonumber\\
\end{eqnarray}
Analogically we get the following relation between null tetrad and the metric tensor $g_{ab}$
\begin{eqnarray}\label{metrictensorN}
g_a{^{{\bf a}'}} ({\bf k})g_b{^{{\bf {b}}'}} ({\bf k})
g_{{\bf a}'{\bf {b}}'}
\nonumber\\
k_{a}({\bf k})\omega_{b}({\bf k})
+\omega_{a}({\bf k})k_{b}({\bf k})
-m_{a}({\bf k})\bar{m}_{b}({\bf k})
-\bar{m}_{a}({\bf k})m_{b}({\bf k}).
\nonumber\\
\end{eqnarray}
§ TRANSFORMATION PROPERTIES OF SPIN-FRAMES
We introduce two symmetries that leave the spin-frame condition
\begin{equation}\label{s-fcondition}
\omega_{A}({\bf k})\pi^{A}({\bf k})=1
\end{equation}
invariant. First, the spinor field transformation associated with the homogeneous Lorentz transformation
\begin{eqnarray}
&&\hspace{-6pt}\pi_{A}({\bf k})~~\mapsto~~ \Lambda\pi_{A}({\bf k})
\Lambda_{A}^{~~B}\pi_{B}({\bf{\Lambda^{-1}k}}),
\nonumber\\
&&\hspace{-6pt}\pi^{A}({\bf k})~~\mapsto~~ \Lambda\pi^{A}({\bf k})
\pi^{B}({\bf{\Lambda^{-1}k}})
\Lambda^{-1}{_{B}}{^{A}}.
\end{eqnarray}
Here $\Lambda_{A}^{~~B}$ is
an unprimed SL(2,C) matrix corresponding to $\Lambda_{a}^{~~b}\in$ SO(1,3).
The null vector $k_a({\bf k})$ that plays the role of a flag-pole for the spinor field $\pi_A({\bf k})$, i.e.
\begin{equation}k_a({\bf k})=\pi_{A}({\bf k})\pi_{A'}({\bf k}),
\end{equation}
must be invariant, so
\begin{equation}
\Lambda\pi_{A}({\bf k})
\Lambda\pi_{A'}({\bf k})
=\pi_{A}({\bf k})\pi_{A'}({\bf k})
\end{equation}
must be satisfied and hence
\begin{equation}\label{lorpitrans}
\Lambda\pi_{A}({\bf k})
e^{-i\Theta(\Lambda,{\bf k})}
\pi_{A}({\bf k}).
\end{equation}
The angle $\Theta(\Lambda,{\bf k})$ is called the Wigner phase. Note that in most literature it is the doubled value $2\Theta(\Lambda,{\bf k})$ which is called the Wigner phase. In analogy
\begin{eqnarray}
&&\hspace{-6pt}\omega_{A}({\bf k})~~\mapsto~~ \Lambda\omega_{A}({\bf k})
\Lambda_{A}^{~~B}\omega_{B}({\bf{\Lambda^{-1}k}}),
\nonumber\\
&&\hspace{-6pt}\omega^{A}({\bf k})~~\mapsto~~ \Lambda\omega^{A}({\bf k})
\omega^{B}({\bf{\Lambda^{-1}k}})
\Lambda^{-1}{_{B}}{^{A}}.
\end{eqnarray}
and the spin-frame condition (<ref>) has to hold. We assume a special case, i.e.
\begin{equation}\label{loromtrans}
\Lambda\omega_{A}({\bf k})
e^{i\Theta(\Lambda,{\bf k})}
\omega_{A}({\bf k}).
\end{equation}
Let us also define another symmetry
\begin{eqnarray}\label{gaomtrans}
\omega_{A}({\bf k})~~&\mapsto&~~
\tilde{\omega}_{A}({\bf k})
\omega_{A}({\bf k})+\phi({\bf k})\pi_{A}({\bf k}),
\\
\pi_{A}({\bf k})~~&\mapsto&~~\label{gapitrans}
\tilde{\pi}_{A}({\bf k})
=\pi_{A}({\bf k}),
\end{eqnarray}
which also keeps the spin-frame condition (<ref>). Here $\phi({\bf k}) = |\phi({\bf k})| e^{i\xi({\bf k})}$
is at this point any complex number. It is interesting that the ambiguity of $\phi({\bf k})$ at the spinor level manifests itself in electrodynamics.
The spin-frame condition
\begin{equation}\label{spinfrcond}
\Lambda\tilde{\omega}_{A}({\bf k})\Lambda\pi^{A}({\bf k})
= 1
\end{equation}
holds for the most general transformation written as
\begin{eqnarray}\label{omegatildelambda}
\Lambda\tilde{\omega}_{A}({\bf k})
e^{i\Theta(\Lambda,{\bf k})}
\left(\omega_{A}({\bf k})+\phi({\bf k})\pi_{A}({\bf k})\right)
e^{i\Theta(\Lambda,{\bf k})}\tilde{\omega}_{A}({\bf k}).
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
&&\hspace{-6pt}\Lambda\tilde{\omega}_{A}({\bf k})
\Lambda_{A}^{~~B}\omega_{B}({\bf{\Lambda^{-1}k}})
\phi({\bf{\Lambda^{-1}k}})\Lambda_{A}^{~~B}\pi_{B}({\bf{\Lambda^{-1}k}})\qquad
\nonumber\\
&& =
e^{i\Theta(\Lambda,{\bf k})}
\left(\omega_{A}({\bf k})+
\phi({\bf{\Lambda^{-1}k}})e^{-2i\Theta(\Lambda,{\bf k})}\pi_{A}({\bf k})\right),
\end{eqnarray}
and from this the Lorentz transformation rule for $\phi({\bf k})$ is
\begin{equation}\label{transphi}
\Lambda\phi({\bf k})
\phi({\bf{\Lambda^{-1}k}})
e^{2i\Theta(\Lambda,{\bf k})}
\phi({\bf k}).
\end{equation}
It should be stressed that there is a difference in interpretation of $\phi({\bf k})$ compared with <cit.> and <cit.>. Here the Lorentz transformation is not parametrized by $\phi({\bf k})$ and this implies a difference in (<ref>) notation, when compared with $\Lambda\omega_{A}({\bf k})
e^{i\Theta(\Lambda,{\bf k})}
\left(\omega_{A}({\bf k})+\phi({\bf k})\pi_{A}({\bf k})\right)$ from <cit.> and <cit.>. This way the succession of these two transformations is emphasised. It is important to stress that the Lorentz transformation and the transformation parametrized by $\phi({\bf k})$ do not commute.
Now let us introduce an SL(2,C) matrix that links two spin-frames
\begin{eqnarray}\label{LAB}
\varepsilon_{\bf{A}}^{~~A}({\bf k})
\Lambda
\tilde{\varepsilon}^{~~{\bf{B}}}_{A}({\bf k}).
\end{eqnarray}
This matrix can explicitly be written in terms of the Wigner phase $\Theta(\Lambda, {\bf k})$ and the $\phi({\bf k})$ parameter, i.e.
\begin{eqnarray}\label{LABmatrix}
\left(
\begin{array}{cc}
\omega_{A}({\bf k})
\Lambda
\pi^{A}({\bf k})&
\omega^{A}({\bf k})
\Lambda
\tilde{\omega}_{A}({\bf k})\\
\pi^{A}({\bf k})
\Lambda
\tilde{\omega}_{A}({\bf k})
\end{array}
\right)
\nonumber\\
\left(
\begin{array}{cc}
e^{-i\Theta(\Lambda,{\bf k})}
& -\phi({\bf k})e^{i\Theta(\Lambda,{\bf k})}\\
0 & e^{i\Theta(\Lambda,{\bf k})}\\
\end{array}
\right).
\end{eqnarray}
Furthermore, this matrix can be split into two SL(2,C) matrices corresponding to gauge and homogeneous Lorentz transformations, i.e.
\begin{eqnarray}\label{LABex}
L_{\bf A}{^{\bf B} }(\Theta,\phi)
\left(
\begin{array}{cc}
& -\phi({\bf k})\\
0 &1\\
\end{array}
\right)
\left(
\begin{array}{cc}
e^{-i\Theta(\Lambda,{\bf k})}
0 & e^{i\Theta(\Lambda,{\bf k})}\\
\end{array}
\right)
\end{eqnarray}
Finally, let us define those two matrices corresponding to gauge transformation and Wigner rotations respectively
\begin{eqnarray}\label{GABex}
G_{\bf A}{^{\bf B}}
({\bf k})
L_{\bf A}{^{\bf B} }(0,\phi({\bf k}))
\varepsilon_{\bf A}^{~~A}({\bf k})
\tilde{\varepsilon}^{~~{\bf B}}_{A}({\bf k})
\nonumber\\
\left(
\begin{array}{cc}
& -\phi({\bf k})\\
0 &1\\
\end{array}
\right),
\end{eqnarray}
\begin{eqnarray}\label{RAB}
R_{\bf A}{^{\bf B} }(\Lambda,{\bf k})
L_{\bf A}{^{\bf B} }(\Theta(\Lambda,{\bf k}),0)
\varepsilon_{\bf A}^{~~A}({\bf k})
\Lambda
\varepsilon^{~~{\bf B}}_{A}({\bf k})
\label{RABex}
\nonumber\\
\left(
\begin{array}{cc}
e^{-i\Theta(\Lambda,{\bf k})}
0 & e^{i\Theta(\Lambda,{\bf k})}\\
\end{array}
\right).
\end{eqnarray}
§ TRANSFORMATION PROPERTIES OF TETRADS
In this appendix we will introduce a transformation on the Minkowski and null tetrads that corresponds to the spin-frame transformations (<ref>), (<ref>), (<ref>) and (<ref>).
Now, for any homogeneous Lorentz transformation $\Lambda_{a}{^{b}}$, parametrized by the Wigner phase $\Theta(\Lambda, {\bf k})$, and any gauge transformation parametrized by some complex number $\phi({\bf k})$, let us define the following
matrix associated with the Minkowski tetrad
\begin{equation}\label{Lab}
g_{~~{\bf{a}}}^{a}({\bf k})
\Lambda_{a}^{~~b}
\tilde{g}^{~~{\bf{b}}}_{b}
({\bf k})\Lambda
\tilde{g}^{~~{\bf{b}}}_{a}
({\bf k}).
\end{equation}
Matrix $L_{\bf{a}}{^{\bf {b}}}$ can be written explicitly parametrized by
$\Theta(\Lambda,{\bf k})$ and $\phi({\bf k})$.
\begin{eqnarray}\label{Labap}
\nonumber\\
\left(
\begin{array}{cccc}
& -|\phi|\cos(\xi+2\Theta)
\\
-|\phi|\cos \xi
& \cos 2\Theta
& -\sin 2\Theta
&|\phi|\cos \xi
\\
|\phi|\sin \xi
&\sin 2\Theta
& \cos 2\Theta
&-|\phi|\sin \xi
\\
\frac{|\phi|^{2}}{2}
& -|\phi|\cos(\xi+2\Theta)
& |\phi|\sin(\xi+2\Theta)
& 1-\frac{|\phi|^{2}}{2}
\end{array}
\right).\label{LLLLL}
\nonumber\\
\end{eqnarray}
Here $\Theta=\Theta(\Lambda,\bf k)$, $\phi=\phi(\bf k)$ and $\xi=\xi(\bf k)$.
This matrix has the property of linking two Minkowski tetrads in a way that
\begin{eqnarray}
g_{c}{^{\bf{a}}}({\bf k})
g_{c}{^{\bf{a}}}({\bf k})
g_{~~{\bf{a}}}^{a}({\bf k})
\Lambda_{a}{^b}
\tilde{g}^{~~{\bf{b}}}_{b}
\nonumber\\
\Lambda_{a}^{~~b}
\tilde{g}^{~~{\bf{b}}}_{b}
\nonumber\\
\Lambda_{c}^{~~b}
\tilde{g}^{~~{\bf{b}}}_{b}
\end{eqnarray}
The metric tensor must be invariant under this combined Lorentz and gauge transformation, therefore
\begin{eqnarray}
&&\hspace{-6pt}g_{a}{^{\bf{a}}}({\bf k})
g_{b}{^{\bf{b}}}({\bf k})
\nonumber\\
g_{a}{^{\bf{c}}}({\bf k})
g_{b}{^{\bf{d}}}({\bf k})
\nonumber\\
g_{a}{^{\bf{c}}}({\bf k})
g_{b}{^{\bf{d}}}({\bf k})
\end{eqnarray}
and this implies
\begin{eqnarray}
\\
\\
\end{eqnarray}
$L_{{\bf{a}}}{^{\bf{b}}}(\Theta,\phi)$ can be written as a product of the Wigner rotation matrix $R_{{\bf{a}}}{^{\bf{b}}}(\Lambda,{\bf k}) $ and a gauge transformation matrix $G_{{\bf{a}}}{^{\bf{b}}}({\bf k})$, i.e.
\begin{eqnarray}
G_{{\bf{a}}}{^{\bf{c}}}({\bf k})
R_{\bf{c}}{^{\bf{b}}}(\Lambda,{\bf k}),
\end{eqnarray}
\begin{eqnarray}\label{Gab}
G_{{\bf{a}}}{^{\bf{b}}}({\bf k})
\left(
\begin{array}{cccc}
& -|\phi|\cos\xi
\sin\xi
\\
-|\phi|\cos \xi
& 1
& 0
&|\phi|\cos \xi
\\
|\phi|\sin \xi
& 1
-|\phi|\sin \xi
\\
\frac{|\phi|^{2}}{2}
& -|\phi|
\cos\xi
& |\phi|
\sin\xi
& 1-\frac{|\phi|^{2}}{2}
\end{array}
\right),
\end{eqnarray}
\begin{eqnarray}\label{Rab}
R_{\bf{a}}{^{\bf{b}}}(\Lambda,{\bf k})=
\left(
\begin{array}{cccc}
\\
& \cos 2\Theta
& -\sin 2\Theta
\\
&\sin 2\Theta
& \cos 2\Theta
\\
& 0
& 0
& 1
\end{array}
\right).
\end{eqnarray}
Of course, those two matrices do not commute, i.e.
\begin{eqnarray}\label{GRRG}
G_{{\bf{a}}}{^{\bf{c}}}({\bf k})
R_{\bf{c}}{^{\bf{b}}}(\Lambda,{\bf k})
R_{{\bf{a}}}{^{\bf{c}}}(\Lambda,{\bf k})
\nonumber\\
\end{eqnarray}
From (<ref>) we can get
\begin{eqnarray}
g_{~~{\bf{a}}}^{a}({\bf k})
\Lambda_{a}^{~~b}
\tilde{g}^{~~{\bf{b}}}_{b}
\nonumber\\
g_{~~{\bf{a}}'}^{a}({\bf k})
\Lambda_{a}^{~~b}
\tilde{g}_{b}^{~~{\bf{b}}'}
\nonumber\\
\end{eqnarray}
$g_{{\bf{a}}}{^{\bf{a}'}}$ and $g{_{\bf{b}'}}^{\bf{b}}$ are the Infeld-van der Waerden symbols introduced in (<ref>)–(<ref>) and matrix $L_{{\bf{a}}'}^{~~{\bf{b}}'}$ links two null tetrads
\begin{eqnarray}
\nonumber\\
g_{~~{\bf{a}}'}^{a}({\bf k})
\Lambda_{a}^{~~b}
\tilde{g}^{~~{\bf{b}}'}_{b}
\nonumber\\
\varepsilon_{\bf{A}}^{~~A}({\bf k})
\varepsilon_{{\bf{A}}'}^{~~A'}({\bf k})
\Lambda_{A}^{~~B}
\Lambda_{A'}^{~~B'}
\tilde{\varepsilon}^{~~{\bf{B}}}_{B}
\tilde{\varepsilon}^{~~{\bf{B}}'}_{B'}
\nonumber\\
\end{eqnarray}
where $L_{\bf{A}}^{~~{\bf{B}}}(\Theta,\phi)$ is the transformation matrix (<ref>) linking two spin-frames.
§ COMPOSITION LAW
From (<ref>) and (<ref>) it is easy to show that $W(\Lambda)$ and $W(\Lambda)^{\dag}$ satisfy the following composition law
\begin{eqnarray}
|{\bf {k}}\rangle
|{\bf {\Lambda \Lambda' k}}\rangle
W(\Lambda \Lambda')
|{\bf k}\rangle ,
\\
|{\bf {k}}\rangle
|{\bf {(\Lambda'\Lambda)^{-1}k}}\rangle
W(\Lambda' \Lambda)^{\dag}
|{\bf k}\rangle
\end{eqnarray}
This means that they are unitary representations of the Lorentz group. We shall now take a closer look at the composition law for the reducible representations of $U(\Lambda,0,1)$ (<ref>)
\begin{eqnarray}\label{Ulambdalambda'}
\nonumber\\
\left(
\int d\Gamma({\bf k})
|{\bf k}
\rangle\langle
{\bf{\Lambda^{-1}k}}|\otimes U(\Lambda,{\bf k})
\right)
\nonumber\\
\left(
\int d\Gamma({\bf k}')
|{\bf k}'
\rangle\langle
{\bf{\Lambda'^{-1}k'}}|\otimes U(\Lambda',{\bf k}')
\right)
\nonumber\\
\int d\Gamma({\bf k})
|{\bf k}\rangle
\langle {\bf{(\Lambda\Lambda')^{-1}k}}|
\otimes U(\Lambda,{\bf k})
\end{eqnarray}
The left-hand side of the above should be equal to
$\int d\Gamma({\bf k})|{\bf k}\rangle\langle {\bf{(\Lambda\Lambda')^{-1}k}}|\otimes U(\Lambda\Lambda',{\bf k})$ and this implies the following condition
\begin{eqnarray}\label{Ucomposition}
U(\Lambda\Lambda',{\bf k})
U(\Lambda,{\bf k})
\end{eqnarray}
For the hermitian conjugates we can write
\begin{eqnarray}
\nonumber\\
\left(
\int d\Gamma({\bf k}')
\langle {\bf k}'|
\otimes U(\Lambda',{\bf k}')^{\dag}
\right)
\nonumber\\
\left(
\int d\Gamma({\bf k})
\langle {\bf k}|
\otimes U(\Lambda,{\bf k})^{\dag}
\right)
\nonumber\\
\int d\Gamma({\bf k})
\langle {\bf k}|
\otimes
U(\Lambda,{\bf k})^{\dag}
\end{eqnarray}
This implies
\begin{eqnarray}
U(\Lambda\Lambda',{\bf k})^{\dag}
U(\Lambda,{\bf k})^{\dag}.
\end{eqnarray}
The reducible representation is unitary thus
\begin{eqnarray}
\nonumber\\
\left(
\int d\Gamma({\bf k})
|{\bf k}\rangle
\langle {\bf{\Lambda^{-1}k}}|
\otimes U(\Lambda,{\bf k})
\right)
\nonumber\\
\left(
\int d\Gamma({\bf k}')
\langle {\bf k}'|
\otimes U(\Lambda,{\bf k}')^{\dag}
\right)
\nonumber\\
\int d\Gamma({\bf k})
|{\bf k}\rangle
\langle {\bf k}|
\otimes U(\Lambda,{\bf k})
U(\Lambda,{\bf k})^{\dag}
\otimes 1_4.
\end{eqnarray}
On the other hand from (<ref>) putting $\Lambda'=\Lambda^{-1}$ we get
\begin{eqnarray}
\nonumber\\
\left(
\int d\Gamma({\bf k})
|{\bf k}
\rangle\langle
{\bf{\Lambda^{-1}k}}|\otimes U(\Lambda,{\bf k})
\right)
\nonumber\\
\left(
\int d\Gamma({\bf k}')
|{\bf k}'
\rangle\langle
{\bf{\Lambda k'}}|\otimes U(\Lambda^{-1},{\bf k}')
\right)
\nonumber\\
\int d\Gamma({\bf k})
|{\bf k}\rangle
\langle {\bf k}|
\otimes U(\Lambda,{\bf k})
\otimes 1_4,\qquad
\end{eqnarray}
and this implies
\begin{eqnarray}
U(\Lambda,{\bf k})^{\dag}
\end{eqnarray}
Condition (<ref>) imposes a composition law for the $R_{{\bf{a}}}^{~~{\bf{b}}}(\Lambda, {\bf k})$ matrix such that
\begin{eqnarray}\label{Labcomposition}
R_{{\bf{a}}}^{~~{\bf{b}}}(\Lambda\Lambda',{\bf k})
R_{{\bf{a}}}^{~~{\bf{c}}}(\Lambda,{\bf k})
\end{eqnarray}
The composition law will be shown here on the spinor level of SL(2,C) matrix
\begin{eqnarray}\label{compRAB}
R_{\bf{A}}{^{\bf{C}}}(\Lambda,{\bf k})
R_{\bf{A}}{^{\bf{B}}}(\Lambda\Lambda', {\bf k}).
\end{eqnarray}
Let us first remind ourselves of formula (<ref>) and write the Wigner Rotation matrix
\begin{eqnarray}\label{RAB}
R_{\bf{A}}^{~~{\bf{B}}}(\Lambda, {\bf k})
\varepsilon_{\bf{A}}{^{A}}({\bf k})
\Lambda\varepsilon_{A}{^{\bf{B}}}({\bf k}).
\end{eqnarray}
We can show that
\begin{eqnarray}
&&\hspace{-6pt}R_{\bf{A}}{^{\bf{C}}}(\Lambda,{\bf k})
\nonumber\\
\varepsilon_{\bf{A}}{^{B}}({\bf k})
\Lambda\varepsilon_{B}{^{\bf{C}}}({\bf k})
\varepsilon_{\bf{C}}{^{A}}({\bf{\Lambda^{-1}k}})
\Lambda'\varepsilon_{A}{^{\bf{B}}}({\bf{\Lambda^{-1}k}})
\nonumber\\
\varepsilon_{\bf{A}}{^{B}}({\bf k})
\Lambda\varepsilon_{B}{^{\bf{C}}}({\bf k})
\Lambda\varepsilon_{\bf{C}}{^{E}}({\bf{k}})\Lambda_{E}{^A}
\Lambda'_{A}{^{C}}
\varepsilon_{C}{^{\bf{B}}}({\bf{(\Lambda\Lambda')^{-1}k}})
\nonumber\\
\varepsilon_{\bf{A}}{^{B}}({\bf k})
\Lambda\varepsilon_{B}{^{\bf{C}}}({\bf k})
\Lambda\varepsilon_{\bf{C}}{^{E}}({\bf{k}})
\Lambda\Lambda'\varepsilon_{D}{^{\bf{B}}}({\bf{k}})
\nonumber\\
\varepsilon_{\bf{A}}{^{B}}({\bf k})
\Lambda\varepsilon_{B}{^{\bf{C}}}({\bf k})
\Lambda\varepsilon_{\bf{C}}{^{A}}({\bf k})
\Lambda\Lambda'\varepsilon_{A}{^{\bf{B}}}({\bf k})
\nonumber\\
\varepsilon_{\bf{A}}{^{B}}({\bf k})
\varepsilon_{B}{^{A}}
\Lambda\Lambda'\varepsilon_{A}{^{\bf{B}}}({\bf k})
\nonumber\\
\varepsilon_{\bf{A}}{^{A}}({\bf k})
\Lambda\Lambda'\varepsilon_{A}{^{\bf{B}}}({\bf k})
R_{\bf{A}}{^{\bf{B}}}(\Lambda\Lambda', {\bf k}).
\end{eqnarray}
Furthermore, let us also prove the composition law of Lorentz transformations on the gauge parameter $\phi({\bf k})$, i.e.
\begin{equation}\label{}
\Lambda\Lambda'\phi({\bf k})
\phi({\bf{(\Lambda\Lambda')^{-1}k}})
e^{2i\Theta(\Lambda\Lambda',{\bf k})}
\phi({\bf k}).
\end{equation}
Proof: First, let us denote
\begin{eqnarray}\label{}
\Lambda\Lambda'\phi({\bf k})
\Lambda'\phi({\bf{\Lambda^{-1}k}})
\phi({\bf{\Lambda'^{-1}(\Lambda^{-1}k}}))
\nonumber\\
\phi({\bf{(\Lambda\Lambda')^{-1}k}}).
\end{eqnarray}
Using such a notation and transformation rule (<ref>) for $\phi({\bf k})$, we can show that
\begin{eqnarray}
\nonumber\\
\Lambda'\phi({\bf{\Lambda^{-1}k}})
\phi({\bf{\Lambda^{-1}k}})
\\
e^{2i\Theta(\Lambda,{\bf k})}
\phi({\bf k})
e^{2i\Theta(\Lambda\Lambda',{\bf k})}
\phi({\bf k}).
\qquad
\end{eqnarray}
The last step of this proof can be shown using composition low (<ref>) in its explicit SL(2,C) matrix form (<ref>).
§ TRANSFORMATION PROPERTIES OF THE POTENTIAL AND ELECTROMAGNETIC FIELD OPERATOR
Let us recall the potential operator for $N=1$ oscillator representation (<ref>)
\begin{eqnarray}
i\int d\Gamma({\bf k})
g_{a}{^{\bf a}}({\bf k})a_{{\bf a}}({\bf k},1)
e^{-ik\cdot x}+\rm{H.c.}
\nonumber
\end{eqnarray}
The potential operator transforms under Lorentz transformation (<ref>) as a four-vector
\begin{eqnarray}
U(\Lambda,0,1)^{\dag} A_{a}(x,1)U(\Lambda,0,1)
\Lambda{_a}{^b}A_{b}(\Lambda^{-1}x,1).\qquad
\end{eqnarray}
\begin{eqnarray}
U(\Lambda,0,1)^{\dag} A_{a}(x,1)U(\Lambda,0,1)
\nonumber\\
i\int d\Gamma({\bf k})
g_{a}{^{\bf{a}}}({\bf k})
R_{\bf{a}}{^{\bf{b}}}(\Lambda,{\bf k})
e^{-ik\cdot x}+\rm{H.c.}
\nonumber\\
i\int d\Gamma({\bf k})
\Lambda g_{a}{^{\bf{b}}}({\bf k})
e^{-ik\cdot x}+\rm{H.c.}
\nonumber\\
i\int d\Gamma({\bf k})
\Lambda{_a}{^b} g_{b}{^{\bf{b}}}({\bf{\Lambda^{-1}k}})
e^{-ik\cdot x}+\rm{H.c.}
\nonumber\\
i\int d\Gamma({\bf k})
\Lambda{_a}{^b} g_{b}{^{\bf{b}}}({\bf k})
a_{\bf{b}}({\bf k},1)
e^{-ik\cdot \Lambda^{-1}x}+\rm{H.c.}
\nonumber\\
\Lambda{_a}{^b}A_{b}(\Lambda^{-1}x,1).
\end{eqnarray}
We can also extend this calculation to any natural number $N$, where the four-potential in $N$-oscillator representation is denoted by $A_a(x,N)$, i.e.
\begin{eqnarray}
U(\Lambda,0,N)^{\dag} A_{a}(x,N)U(\Lambda,0,N)
\Lambda{_a}{^b}A_{b}(\Lambda^{-1}x,N).\qquad
\end{eqnarray}
Recall the electromagnetic field operator (<ref>)
\begin{eqnarray}
\nonumber\\
\int d\Gamma({\bf k})
\left(
k_a({\bf k})
g_{b}{^{\bf{a}}}({\bf k})
k_{b}({\bf k})
g_{a}{^{\bf{a}}}({\bf k})
\right)
a_{\bf{a}}({\bf k},1)
e^{-ik\cdot x}
+{\rm H.c.}
\nonumber
\end{eqnarray}
The electromagnetic field operator transforms under Lorentz transformation (<ref>) like a tensor
\begin{eqnarray}
U(\Lambda,0,1)^{\dagger} F_{ab}(x,1) U(\Lambda,0,1)
\Lambda_{a}{^{c}}
\Lambda_{b}{^{d}}
\end{eqnarray}
\begin{eqnarray}
&&\hspace{-6pt}U(\Lambda,0,1)^{\dagger} F_{ab}(x,1) U(\Lambda,0,1)
\nonumber\\
2\int d\Gamma({\bf k})
k_{[a}({\bf k})
g_{b]}{^{\bf{a}}}({\bf k})
R_{\bf{a}}{^{\bf{b}}}(\Lambda,{\bf k})
e^{-ik\cdot x}
+{\rm H.c.}
\nonumber\\
2\int d\Gamma({\bf k})
\Lambda k_{[a}({\bf k})
\Lambda g_{b]}{^{\bf{a}}}({\bf k})
e^{-ik\cdot x}
+{\rm H.c.}
\nonumber\\
2\int d\Gamma({\bf k})
\Lambda k_{[a}({\bf \Lambda k})
\Lambda g_{b]}{^{\bf{a}}}({\bf \Lambda k})
e^{-i\Lambda k\cdot x}
+{\rm H.c.}
\nonumber\\
\Lambda_{a}{^{c}}
\Lambda_{b}{^{d}}
\int d\Gamma({\bf k})
k_{[c}({\bf k})
g_{d]}^{~~{\bf a}}({\bf k})
a_{\bf{a}}({\bf k},1)
e^{-i k\cdot \Lambda^{-1}x}+{\rm H.c.}~
\nonumber\\
\Lambda_{a}{^{c}}
\Lambda_{b}{^{d}}
\end{eqnarray}
The same can be shown for any $N$-oscillator representation.
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|
1511.00259
|
$^1$Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
$^2$Institut für Theorie der Statistischen Physik, RWTH Aachen University and JARA—Fundamentals of
Future Information Technology, 52056 Aachen, Germany
We study the zero-frequency current noise of the interacting resonant level model for arbitrary
bias voltages using a functional renormalization group approach. For this we extend the existing
nonequilibrium scheme by deriving and solving flow equations for the current-vertex functions.
On-resonance artificial divergences of the latter found in lowest-order
perturbation theory in the two-particle interaction are consistently removed. Away from
resonance they are shifted to higher orders. This allows us to gain a comprehensive picture
of the current noise in the scaling limit. At high bias voltages, the current noise exhibits
a universal power-law decay, whose exponent is, to leading order in the interaction, identical
to that of the current. The effective charge on resonance is analyzed in detail,
employing properties of the vertex correction. We find that it is only modified to second or
higher order in the two-particle interaction.
Valid PACS appear here
§ INTRODUCTION
Remarkable advances in nanotechnology open up the possibility to explore transport
beyond the linear response regime in experimentally well-controlled situations.
Among nanoscale conductors, quantum dot systems have attracted much attention, as they offer a versatile arena in which to study nonequilibrium transport phenomena of interacting
fermions. At sufficiently low temperatures, transport is dominated by quantum mechanics.
In addition, the local two-particle interaction results in fascinating many-body effects,
often accompanied by the emergence of new energy scales.
A generic nonequilibrium setup is given by a quantum dot coupled to several leads with different
chemical potentials.
The interacting resonant level model (IRLM) is a prominent example in which strong correlations play
an essential role.<cit.> It describes a single-level quantum dot dominated
by charge fluctuations which are affected by the local two-particle interaction of dot and
lead fermions. The IRLM was originally introduced as a close relative of the Kondo
model,<cit.> and, since then, many studies have been performed to elucidate
its nonlinear
transport properties.<cit.> These have
shown that universal features appear in nonequilibrium transport if the lead bandwidth
$\Delta$ is much larger than any other energy scale.
In this scaling regime, the current at large bias voltage is suppressed following a power law,
whose exponent depends on the strength of the local interaction.<cit.>
Accumulated knowledge in mesoscopic physics elucidates that the higher order cumulants of
the current are of great importance to characterize the nonequilibrium transport.<cit.>
A vast amount of research on the current noise has shown that it contains information which
cannot be obtained from conductance measurements, e.g., the effective charge.<cit.>
While a unified picture for the current in the IRLM has been established by various methods,
the understanding of its higher cumulants is rather limited for the moment.
A major obstacle is the absence of a general framework to treat the effects of strong
correlations in a nonequilibrium situation.
To compute the current noise, one in general needs
to consider the current-vertex function.<cit.>
A perturbative approach to the current noise of the IRLM based on the
Keldysh technique was put forward in Ref. PhysRevB.76.193307.
Important insights into the noise of the IRLM under on-resonance conditions were gained by
utilizing a special symmetry of this model for a particular interaction strength,
which is known as self-duality.<cit.> This symmetry makes it possible to
map the IRLM to a solvable boundary sine-Gordon model even in the presence of a
driving bias voltage.<cit.>
The effective charge of the quasiparticles of the IRLM at the self-dual point has been
investigated using field-theoretical techniques and the density-matrix renormalization
group method.<cit.> For the relatively
strong interaction at which self-duality is established, the quasiparticles
of the IRLM were found to have effective charge $e^{*}=2e$ by examining the shot
which was confirmed computing the full counting statistics.<cit.>
This has to be contrasted to $e^{*}=e$ in the noninteracting limit. In spite of this remarkable
achievement, the bias voltage dependence of the current noise of the IRLM away from this self-dual
point and off resonance is still an open question. In addition, finite temperature effects were so
far not investigated. Considering this situation, it is strongly desirable to develop a systematic
framework to calculate the current noise and its full counting statistics in general parameter
Recently, a functional renormalization group (FRG) method was developed to describe the
nonequilibrium properties of correlated fermions.<cit.> Logarithmic divergences,
which manifest themselves in plain perturbation theory even in the equilibrium IRLM, are consistently
resummed employing FRG.<cit.> This method also contributed to deepen our
understanding of the nonequilibrium transport properties of the
IRLM.<cit.> The renormalization of the hopping between
the dot level and the leads can be described using a surprisingly simple
approximation.<cit.> In this paper, we utilize the FRG approach to
elucidate the current noise of the nonequilibrium IRLM. In FRG, we can obtain the
vertex function by deriving and solving its flow equation.
For the level that is on resonance, an artificial divergence of the vertex correction obtained in lowest-order
plain perturbation theory in the interaction is removed in the FRG scheme.
For off-resonance conditions, a severe divergence, found in perturbation theory
if the level energy is aligned with one of the leads
chemical potentials, is shifted to higher orders. These achievements allow us to gain a comprehensive picture
of the zero-frequency current noise in the scaling limit.
We show that the current noise is governed by universal power-law scaling in the large bias
voltage regime with an exponent which, to leading order in the interaction, is the same as that
of the current. The effective charge is discussed by relating it to the vertex correction. We
find that $e^{*} = e \left[ 1+ {\mathcal O}(u^2) \right] $ with the dimensionless amplitude
of the interaction $u$.
This paper is organized as follows. We describe the model and outline the FRG scheme to
compute the current noise in Sec. <ref>. The results for the
noise are presented in Sec. <ref>. A summary is given in Sec. <ref>.
§ MODEL AND FORMALISM
§.§ Model
(Color online)
(a) Schematic picture of the IRLM.
(b) The Keldysh contour.
The (two-reservoir) IRLM describes a spinless fermionic level coupled to
delocalized fermions in the left and right
leads. We consider the local repulsion between the fermion in this level and those in both leads.
This situation can be modeled by a three-site dot system, in which a particle occupying the central
site feels the interaction with the adjacent ones. A schematic picture of the system is shown in
Fig. <ref>(a). The hopping amplitudes $t_{L/R}>0$ between the three-site dot region
and the leads are
assumed to be much larger than the intersite ones, such that the lattice sites $1$ and $3$ are
effectively incorporated into the leads.
The single-site model studied in the original paper<cit.> is restored in this limit.<cit.>
The model is described by the action
\begin{align}
=&\sum^{3}_{i,j=1} \int d\tau d\tau' \bar{d}_{i}(\tau) {\bm g}^{-1}_{dij}(\tau,\tau') d_{j}(\tau') \nonumber \\
&+\sum_{\alpha=L,R}\sum_{\bm{k}} \int d\tau d\tau' \bar{c}_{\alpha \bm{k}}(\tau) g^{-1}_{\alpha \bm{k}}(\tau,\tau') c_{\alpha \bm{k}}(\tau') \nonumber \\
&-\frac{1}{\sqrt{N}} \sum_{\bm{k}} \int d\tau
\Bigl[ t_L e^{iA_L(\tau)}\bar{d}_{1}(\tau)c_{L\bm{k}}(\tau) \nonumber \\
&\hspace{40pt} + t_R \bar{d}_{3}(\tau)c_{R\bm{k}}(\tau) + {\rm H.c.} \Bigr] + S_{U},
\label{eq1}
\end{align}
where the isolated lead and dot Green's functions are given by
$ g^{-1}_{\alpha\bm{k}}(\tau,\tau') \equiv \left[ i\frac{d}{d\tau'} - \epsilon_{\alpha \bm{k}} \right] \delta(\tau,\tau')$ and
\begin{align}
{\bm g}^{-1}_{d}(\tau,\tau')=
\left(
\begin{array}{ccc}
i\frac{d}{d\tau'} -(\epsilon_{1} - U_1/2) & -t_{12} & 0 \\
-t_{12} & i\frac{d}{d\tau'} - (\epsilon_{2} -(U_1+U_3)/2) & -t_{23}\\
0 & -t_{23} & i\frac{d}{d\tau'} - (\epsilon_{3} - U_3/2)
\end{array}
\right) \delta(\tau,\tau'),
\end{align}
respectively, and the interaction part is given as
\begin{align}
& \equiv - \int d\tau \left( U_1 \bar{d}_{2}(\tau)d_{2}(\tau)\bar{d}_{1}(\tau)d_{1}(\tau) \right. \nonumber \\
&\hspace{40pt} \left.+ U_3 \bar{d}_{2}(\tau)d_{2}(\tau)\bar{d}_{3}(\tau)d_{3}(\tau) \right).
\end{align}
The argument $\tau$ combines the time $t$ and the Keldysh index $\nu=\mp$
[see Fig. <ref>(b)].
The symbol $\int d\tau $ denotes integration over $t$ and summation over $\nu$.
The Grassmann field $\bar{d}_{i}(\tau)$ $[d_{i}(\tau)]$ creates [annihilates] a spinless
fermion at time $t$, on the Keldysh contour branch with index $\nu$, and on lattice
site i. Similarly
$\bar{c}_{\alpha {\bm k}}(\tau)$ $[c_{\alpha {\bm k}}(\tau)]$ is a Grassmann field for creating [annihilating]
a fermion in lead $\alpha$ with momentum ${\bm k}$. The Green's functions in Eq. (<ref>)
must be understood as $2 \times 2$ matrices in the Keldysh space and the addends contain
matrix products which are left implicit.
The delta function on the Keldysh contour is defined as $\delta(\tau,\tau') \equiv \delta(t-t') \sigma^{\nu \nu'}_{z}$ with the standard Pauli matrix $\sigma_{z}$.
The energy level of each site is denoted by $\epsilon_{i}$ for $i=1,2,3$, and the hopping
amplitude between the sites 1 (3) and 2 by $t_{12}>0$ $(t_{23}>0)$. The energy level $\epsilon_{i}$ is
defined such that $\epsilon_1=\epsilon_2=\epsilon_3=0$ corresponds to the particle-hole
symmetric case. The third term of the action Eq. (<ref>)
describes the hopping between the quantum dot and
the leads with $N$ sites. In our expressions we always take the thermodynamic
limit $N \to \infty$. The auxiliary vector potential $A_L(\tau)$ is incorporated using
the Peierls substitution and later used as a source field to generate the current-vertex
function.<cit.> We choose units with $k_B=1$, $\hbar=1$, and elementary charge
§.§ Generating functional
The noninteracting reservoirs can be integrated out yielding the action
\begin{align}
=&\sum^{3}_{i,j=1} \int d\tau d\tau' \bar{d}_{i}(\tau) {\bm G}^{-1}_{0ij}(\tau,\tau') d_{j}(\tau')
+ S_{U},
\end{align}
where $ {\bm G}^{-1}_{0}(\tau,\tau') \equiv {\bm g}^{-1}_{d}(\tau,\tau') - {\bm \Sigma}^{-1}_{\rm res}(\tau,\tau') $.
The tunneling self-energy is given as
\begin{align}
\label{eq:tunneling self-energy for IRLM}
&\bm{\Sigma}_{\rm res}(\tau,\tau') \nonumber \\
\left(
\begin{array}{ccc}
t^2_Le^{i[A_L(\tau)-A_L(\tau')]} g_{L}(\tau,\tau') & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & t^2_R g_{R}(\tau,\tau')
\end{array}
\right),
\end{align}
with $ g_{\alpha}(\tau,\tau') = \frac{1}{N}\sum_{{\bm k}}g_{\alpha{\bm k}}(\tau,\tau')$.
We consider the system with a time-independent bias voltage $V$, which is applied symmetrically
to the leads, i.e., $\mu_L=V/2$ and $\mu_R=-V/2$. Both of the leads are assumed to be in equilibrium.
Since in its bias-voltage-driven nonequilibrium steady state the system is invariant under
timetranslation, we perform a Fourier transform to energy representation.
As usual in the Keldysh formalism we use the representation with the retarded (r), Keldysh (K), and advanced (a) components instead of the one with $\nu=\mp$ if appropriate.
The component which vanishes in the two-point Green's function due to causality but appears for other vertices (see below) is denoted by $\tilde{\mbox K}$.
If the source field $A_L(\tau)$ is set to be zero, the relevant
parts of the tunneling self-energy are obtained as
\begin{align}
\bm{\Sigma}^{\rm r}_{\rm res}(\omega) &=
\left(
\begin{array}{ccc}
\displaystyle - \frac{i\Delta_L}{2} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & \displaystyle - \frac{i\Delta_R}{2}
\end{array}
\right), \\
\bm{\Sigma}^{\rm K}_{\rm res}(\omega) &=
\left(
\begin{array}{ccc}
i \Delta_L \left(2f_L(\omega)-1\right) & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & i \Delta_R \left( 2f_R(\omega) -1 \right)
\end{array}
\right).
\end{align}
Here, we use the wide-band limit and define the bandwidth, $\Delta_{\alpha} \equiv 2\pi
\rho_{\alpha} t_{\alpha}^2$. For simplicity, we assume $\Delta_L=\Delta_R \equiv \Delta$,
$t_{12}=t_{23}\equiv t$, $\epsilon_1 = \epsilon_3=0$, $\epsilon_2=\epsilon$, and
$U_1=U_3\equiv U \geq 0$. This implies that at particle-hole symmetry $\epsilon=0$
transport is resonant. We thus refer to this case either as the particle-hole symmetric
point or the on-resonance situation.
Following the standard functional integral approach to quantum many-body
physics,<cit.> we introduce the additional source term
\begin{align}
S^{\rm s} &\equiv \sum^{3}_{i=1} \int d\tau \left[
\bar{\eta}_i(\tau)d_i(\tau) +\bar{d}_i(\tau)\eta_i(\tau) \right],
\end{align}
which allows us to generate correlation functions by functional derivatives.
The corresponding generating functional is given by
\begin{align}
W[\eta,\bar{\eta},A] \equiv -i \ln \int {\cal D} \left[ \bar{d},d \right] \exp \left[i (S + S^{\rm s}) \right].
\end{align}
The effective action<cit.> is defined using the Legendre transform as
\begin{align}
\Gamma [ \langle d \rangle^{\rm s}, \langle \bar{d} \rangle^{\rm s}, A]
&\equiv W [\eta,\bar{\eta},A]
- \sum^{3}_{i=1}\int d\tau \left[ \bar{\eta}_{i}(\tau) \langle d_{i} \rangle^{\rm s} (\tau) \right. \nonumber \\
&\hspace{80pt} \left.+ \langle \bar{d}_{i} \rangle^{\rm s} (\tau) \eta_{i}(\tau) \right],
\end{align}
\begin{align}
\langle {\cal O} \rangle ^{\rm s} \equiv \frac{\int {\cal D} \left[ \bar{d}, d \right] {\cal O} \exp \left[i (S+S^{\rm s} ) \right] }{\int {\cal D} \left[ \bar{d}, d \right] \exp \left[i (S+S^{\rm s} ) \right]} .
\end{align}
This effective action acts as the generating functional of one-particle irreducible vertex functions
(e.g., the self-energy). The vertex expansion of the effective action is given as
\begin{align}
\Gamma [ \langle d \rangle^{\rm s}, \langle \bar{d} \rangle^{\rm s}, A]
= \sum^{\infty}_{m,n=0}\frac{(-1)^m}{(m!)^2}\frac{1}{n!}
& \prod^{m}_{j,k=0}\prod^{n}_{l=0} \int d\tau_j d\tau'_k d\tau''_l
\; \gamma^{(2m,n)}_{i'_1\cdots i'_m;i_1\cdots i_m;\alpha_1\cdots \alpha_n}(\tau'_1,\cdots,\tau'_m; \tau_1,\cdots,\tau_m;
\tau''_1,\cdots,\tau''_n)
\nonumber \\ & \times
\langle \bar{d}_{i'_1} \rangle^{\rm s}(\tau'_1) \cdots \langle \bar{d}_{i'_m} \rangle^{\rm s}(\tau'_m)
\langle d_{i_1} \rangle^{\rm s}(\tau_1) \cdots \langle d_{i_m} \rangle^{\rm s}(\tau_m) A_{\alpha_1}(\tau''_1) \cdots A_{\alpha_n}(\tau''_n),
\end{align}
where repeated site indices are summed over.
We note that the auxiliary vector potential $A_{\alpha}(\tau)$ is also defined on the Keldysh contour;
i.e., it has the two components $A^{-}_{\alpha}(t)$ and $A^{+}_{\alpha}(t)$.
Hence, there are in total $2^{2m+n}$ components for the current-vertex function
The doubled degrees of freedom of the vector potential allow one to completely describe both the
dynamical evolution as well as the statistical correlation.<cit.>
We define the source and the physical component of the vector potential as
\begin{align}
A^{\rm s}_{\alpha}(t)\equiv \frac{1}{2}\left[ A^{-}_{\alpha}(t) - A^{+}_{\alpha}(t) \right], \quad
A^{\rm p}_{\alpha}(t)\equiv \frac{1}{2}\left[ A^{-}_{\alpha}(t) + A^{+}_{\alpha}(t) \right],
\end{align}
We utilize $A^{\rm s}(t)$ to derive the expression of the current-vertex functions and
their flow equations.
After deriving all the equations, we take the physically relevant limit $A^{\rm s}(t)\to 0$.
As we introduced the bias voltage via the chemical potentials of the leads and
do not consider any further fields, the physical component of the vector potential
is set to zero as well.
The source and physical components of the current-vertex functions are defined as
\begin{align}
\left(\gamma^{(0,1)}\right)^{\rm s}_{\alpha} (t)
\equiv \left. \frac{\delta \Gamma}{\delta A^{\rm s}_{\alpha}(t)} \right|_{A_\alpha^{\mp}=0}, \quad
\left(\gamma^{(0,1)}\right)^{\rm p}_{\alpha} (t)
\equiv \left. \frac{\delta \Gamma}{\delta A^{\rm p}_{\alpha}(t)} \right|_{A_\alpha^{\mp}=0} ,
\end{align}
The components of the higher order current-vertex functions are defined in the same way.
Then, the current noise can be written as<cit.>
\begin{align}
S_{\alpha\alpha'}(t,t') &\equiv
\langle I_{\alpha}(t)I_{\alpha'}(t') \rangle
+\langle I_{\alpha'}(t')I_{\alpha}(t) \rangle
-2\langle I_{\alpha}(t) \rangle \langle I_{\alpha'}(t') \rangle \nonumber \\
&=S^{0}_{\alpha\alpha'}(t,t') + S^{U}_{\alpha\alpha'}(t,t'), \label{eq:current noise}
\end{align}
where the two terms are defined as
\begin{align}
\label{eq:expression of bubble term}
&\equiv \frac{1}{2} \int dt_1 dt'_1 \bm{G}^{\nu_1\nu'_1}_{i_1 i'_1}(t_1,t'_1) \sigma^{\nu'_1\nu'_1}_{z} \left( \bm{\gamma}^{(2,2)}_{\rm res} \right)^{\nu'_1\nu_1;{\rm ss}}_{i'_1i_1;\alpha\alpha'}(t'_1,t_1;t,t') \sigma^{\nu_1\nu_1}_{z}\nonumber \\
& \hspace{20pt} + \frac{1}{2} \int dt_1 dt_2 dt'_1 dt'_2 \bm{G}^{\nu_2\nu'_1}_{i_2 i'_1}(t_2,t'_1) \sigma^{\nu'_1\nu'_1}_{z} \left( \bm{\gamma}^{(2,1)}_{\rm res} \right)^{\nu'_1\nu_1;{\rm s}}_{i'_1i_1;\alpha}(t'_1,t_1;t) \sigma^{\nu_1\nu_1}_{z} \nonumber \\
& \hspace{120pt} \times \bm{G}^{\nu_1\nu'_2}_{i_1 i'_2} (t_1,t'_2) \sigma^{\nu'_2\nu'_2}_{z} \left( \bm{\gamma}^{(2,1)}_{\rm res} \right)^{\nu'_2\nu_2;{\rm s}}_{i'_2i_2;\alpha'}(t'_2,t_2;t')\sigma^{\nu_2\nu_2}_{z}, \\
\label{eq:expression of vertex correction}
&\equiv \frac{1}{2} \int dt_1 dt_2 dt'_1 dt'_2
\bm{G}^{\nu_2\nu'_1}_{i_2 i'_1}(t_2,t'_1) \sigma^{\nu'_1\nu'_1}_{z} \left( \bar{\bm{\gamma}}^{(2,1)} \right)^{\nu'_1\nu_1;{\rm s}}_{i'_1i_1;\alpha}(t'_1,t_1;t) \sigma^{\nu_1\nu_1}_{z} \nonumber \\
& \hspace{100pt} \times \bm{G}^{\nu_1\nu'_2}_{i_1 i'_2} (t_1,t'_2) \sigma^{\nu'_2\nu'_2}_{z} \left( \bm{\gamma}^{(2,1)}_{\rm res} \right)^{\nu'_2\nu_2;{\rm s}}_{i'_2i_2;\alpha'}(t'_2,t_2;t')\sigma^{\nu_2\nu_2}_{z},
\end{align}
\begin{align}
\left( \bar{\bm{\gamma}}^{(2,1)} \right)^{\nu'_1\nu_1;{\rm s}}_{i'_1i_1;\alpha}(t'_1,t_1;t)
\equiv
\left( \bm{\gamma}^{(2,1)} \right)^{\nu'_1\nu_1;{\rm s}}_{i'_1i_1;\alpha}(t'_1,t_1;t)
-\left( \bm{\gamma}^{(2,1)}_{\rm res} \right)^{\nu'_1\nu_1;{\rm s}}_{i'_1i_1;\alpha}(t'_1,t_1;t).
\label{eq:gammabar}
\end{align}
Here, $\gamma^{(2,n)}_{\rm res}$ (for $n>0$) is the noninteracting part of the $(2+n)$-point
current-vertex function.
Thus $ \bar{\bm{\gamma}}^{(2,1)}$ defined in Eq. (<ref>) is the interaction induced
part of the three-point vertex function.
In Eqs. (<ref>) and (<ref>)
the repeated Keldysh indices $\nu=\mp$ and site indices are summed over.
The first term $S^0$ on the right-hand side
of Eq. (<ref>) is called the bubble term, while the second $S^U$ is the vertex
correction to the noise.
Their diagrammatic representations are shown in Fig. <ref>.
The effect of the repulsive interaction is included in the self-energy of the propagator, ${\bm \Sigma}_{U}= {\bm G}^{-1}_{0} - {\bm \gamma}^{(2,0)}$, and the three-point vertex function, $\gamma^{(2,1)}$, both being determined by the FRG approach.
Diagrammatic representation of the current noise.
The circle represents the three-point vertex function.
The left diagram is called the bubble term, while the right is called the vertex correction.
§.§ Functional renormalization group approach
In setting up the functional renormalization group approach,<cit.> we use a
reservoir cutoff as the flow parameter.<cit.> The Kubo-Martin-Schwinger (KMS)
condition is automatically preserved in this scheme, which is important to perform
calculations consistent with the fluctuation dissipation theorem in the limit $V \rightarrow0$.
The flow parameter is introduced as an additional tunneling self-energy,
\begin{align}
\bm{\Sigma}^{\rm r}_{{\rm aux},\Lambda}(\omega) &= - \frac{i\Lambda }{2} \bm{1}, \\
\bm{\Sigma}^{\rm K}_{{\rm aux},\Lambda}(\omega) &=i \Lambda \left[2f_{\rm aux}(\omega) -1 \right] \bm{1},
\end{align}
where $f_{\rm aux}(\omega)$ is the Fermi-Dirac distribution function of the auxiliary structureless
reservoirs and $\bm{1}$ is the identity matrix of dimension $3$.
It was examined earlier that results for the current are independent of the choice of the
temperature in the auxiliary reservoirs.<cit.>
In this paper, we utilize the auxiliary reservoirs with infinite temperature and thus
$f_{\rm aux}(\omega)=1/2$.
This simplifies the flow equations as the Keldysh component of the auxiliary self-energy
vanishes. The full Green's function is obtained by the Dyson equation
\begin{align}
\left( \bm{G}^{\rm r}_{\Lambda} \right) ^{-1}(\omega)
&= \left( \bm{G}^{\rm r}_{0} \right) ^{-1}(\omega) - \bm{\Sigma}^{\rm r}_{{\rm aux},\Lambda}(\omega) -
\bm{\Sigma}^{\rm r}_{U,\Lambda}(\omega), \\
\left( \bm{G}^{\rm K}_{\Lambda} \right) (\omega)
&= \bm{G}^{\rm r}_{\Lambda} (\omega) \left[\bm{\Sigma}^{\rm K}_{\rm res} (\omega) + \bm{\Sigma}^{\rm K}_{U,\Lambda} (\omega) \right] \bm{G}^{\rm a}_{\Lambda} (\omega) .
\end{align}
The scale-dependent propagator $\bm{S}_{\Lambda}(\tau,\tau')$ appearing in the RG flow equations
(see below) is defined as
\begin{align}
\bm{S}_{\Lambda}(\tau,\tau') \equiv \int d\tau_1 d\tau_2 \bm{G}_{\Lambda}(\tau,\tau_1)
\frac{d \bm{\Sigma}_{{\rm aux},\Lambda}(\tau_1,\tau_2)}{d\Lambda}\bm{G}_{\Lambda}(\tau_2,\tau').
\end{align}
with components
\begin{align}
\bm{S}^{\rm r}_{\Lambda}(\omega)
&= \frac{-i}{2} \bm{G}^{\rm r}_{\Lambda}(\omega) \bm{G}^{\rm r}_{\Lambda}(\omega), \\
\bm{S}^{\rm K}_{\Lambda}(\omega)
&= \frac{-i}{2} \bm{G}^{\rm r}_{\Lambda}(\omega) \bm{G}^{\rm K}_{\Lambda}(\omega)
+ \frac{i}{2} \bm{G}^{\rm K}_{\Lambda}(\omega) \bm{G}^{\rm a}_{\Lambda}(\omega).
\end{align}
We consider the model with $\Lambda_{\rm init} \rightarrow \infty$ as the initial
one of the flow, as all the
vertex functions can be calculated exactly in this limit. The initial conditions of the self-energy
and the vertex functions are summarized in Appendix <ref>. The set of
coupled flow equations has to be integrated down to $\Lambda=0$, at which the auxiliary reservoirs are
decoupled and the cutoff-free problem of interest is restored.
In order to implement numerical calculations, we need to truncate the infinite hierarchy of the flow
equations to a given order. In this paper, we use the lowest order truncation, which is known as
the static approximation,<cit.> to determine the flow equations of the self-energy
and current-vertex functions. Flow equations are truncated at the first order in the interaction,
$U$. The remaining terms are the Hartree-Fock-type diagram for the self-energy – we note that
due to the underlying RG procedure our approximation is not equivalent to the Hartree-Fock approximation – and
the RPA-type diagram for the vertex function. The diagrammatic representation of the flow equations
is given in Fig. <ref>. In spite of this simple treatment, this approximation for the self-energy
is known to describes the rich properties of nonequilibrium transport due to the built-in
renormalization.<cit.> In particular, logarithmic divergences
found in the scaling limit of the IRLM in plain perturbation theory are consistently resummed to
power laws.<cit.> The current-vertex function has not yet been treated
within the present truncated FRG scheme, and its role for the current noise is discussed below.
We note that higher order corrections can be systematically included in principle by incorporating
flow equations of higher order vertices.
Diagrammatic representation of the flow equations of (a) the self-energy and (b) the three-point current-vertex function in the static approximation.
The square represents the two-particle interaction.
The flow equation of the four-point vertex function is ignored in the static approximation, and
its value is replaced by the initial one which is given by the
anti-symmetrized bare two-particle interaction $U_{ik;jl}$ (see Appendix <ref>).
The flow equation of the self-energy is obtained as
(repeated site indices are summed over)
\begin{align}
\frac{d }{d\Lambda}\left({\bm \Sigma}^{\rm r}_{U,\Lambda}\right)_{ij}
\label{eq:flow eq. of the self-energy}
= \frac{iU_{ik;jl}}{2} \int \frac{d\omega}{2\pi} \left(\bm{S}^{\rm K}_{\Lambda}\right)_{lk} (\omega) .
\end{align}
Within the present approximation, the self-energy is frequency independent due to the structure of
the right-hand side. Hence, single-particle Green's functions can be interpreted as effective
noninteracting ones with renormalized parameters.
The flow equation of the retarded component of the interaction-induced
part of the three-point vertex function [see Eq. (<ref>)] is given by
\begin{align}
\frac{d}{d\Lambda} \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm r};{\rm s}}_{ij;L}
\label{eq:flow eq. of ret. three-point vertex func.}
= \frac{iU_{ik;jl}}{2}
\left[
\left(\bm{\Phi}^{(2,1)}_{\Lambda}\right)^{\rm K}_{lk}
+\left(\bm{\Phi}^{(2,1)}_{\Lambda}\right)^{\rm \tilde{K}}_{lk}
\right]
\end{align}
\begin{align}
&\left( \bm{\Phi}^{(2,1)}_{\Lambda} \right)^{\rm K} \equiv \int \frac{d\omega}{2\pi}
\left[
{\bm S}^{\rm r}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm r};{\rm s}}_{;L} {\bm G}^{\rm K}_{\Lambda}(\omega) \right. \nonumber \\
& + {\bm S}^{\rm K}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm a};{\rm s}}_{;L} {\bm G}^{\rm a}_{\Lambda}(\omega)
+ {\bm S}^{\rm r}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm K};{\rm s}}_{;L} {\bm G}^{\rm a}_{\Lambda}(\omega) \nonumber \\
& \left.
+ {\bm S}^{\rm K}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{\tilde{{\rm K}};{\rm s}}_{;L} {\bm G}^{\rm K}_{\Lambda}(\omega)
+({\bm S} \leftrightarrow {\bm G})
\right],
\end{align}
\begin{align}
&\left( \bm{\Phi}^{(2,1)}_{\Lambda} \right)^{\rm {\tilde K}} \nonumber \\
&\equiv \int \frac{d\omega}{2\pi}
\left[
{\bm S}^{\rm a}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{\tilde{{\rm K}};s}_{;L} {\bm G}^{\rm r}_{\Lambda}(\omega)
+({\bm S} \leftrightarrow {\bm G})
\right].
\end{align}
The abbreviation $({\bm S} \leftrightarrow {\bm G})$ denotes the terms which are obtained by
mutually replacing ${\bm S}$ and ${\bm G}$ in the preceding ones in the same parenthesis.
Similarly, the flow equation of the Keldysh component is obtained as
\begin{align}
\frac{d}{d\Lambda} \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm K};{\rm s}}_{ij;L}
\label{eq:flow eq. of Keldysh three-point vertex func.}
= \frac{iU_{ik;jl}}{2}
\left[
\left(\bm{\Phi}^{(2,1)}_{\Lambda}\right)^{\rm r}_{lk}
+\left(\bm{\Phi}^{(2,1)}_{\Lambda}\right)^{\rm a}_{lk}
\right]
\end{align}
\begin{align}
&\left( \bm{\Phi}^{(2,1)}_{\Lambda} \right)^{\rm r}
\equiv \int \frac{d\omega}{2\pi}
\left[
{\bm S}^{\rm r}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm r};{\rm s}}_{;L} {\bm G}^{\rm r}_{\Lambda}(\omega) \right. \nonumber \\
+ {\bm S}^{\rm K}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{\tilde{{\rm K}};{\rm s}}_{;L} {\bm G}^{\rm r}_{\Lambda}(\omega)
+ ({\bm S} \leftrightarrow {\bm G})
\right],
\end{align}
\begin{align}
&\left( \bm{\Phi}^{(2,1)}_{\Lambda} \right)^{\rm a}
\equiv \int \frac{d\omega}{2\pi}
\left[
{\bm S}^{\rm a}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm a};{\rm s}}_{;L} {\bm G}^{\rm a}_{\Lambda}(\omega) \right. \nonumber \\
& \left.
+ {\bm S}^{\rm a}_{\Lambda}(\omega) \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{\tilde{{\rm K}};{\rm s}}_{;L} {\bm G}^{\rm K}_{\Lambda}(\omega)
+ ({\bm S} \leftrightarrow {\bm G})
\right].
\end{align}
Again, repeated site indices are summed over.
The argument of the three-point vertex functions is omitted as these turn out to be independent of
frequency in the static approximation. In contrast to the self-energy, these vertex functions do not
have a simple interpretation in reference to a noninteracting model.
Using the initial condition and the flow equation, we can prove that the three-point current-vertex
functions fulfill the symmetry relations
\begin{align}
\left[ \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm r};{\rm s}}_{ij;L} \right]^{*}
\label{eq:symmetry relation 4 of current-vertex functions}
&= -\left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm a};{\rm s}}_{ji;L}, \\
\left[ \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm K};{\rm s}}_{ij;L} \right]^{*}
\label{eq:symmetry relation 5 of current-vertex functions}
&= \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm K};{\rm s}}_{ji;L}, \\
\left[\left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm \tilde{K}};{\rm s}}_{ij;L}\right]^{*}
\label{eq:symmetry relation 6 of current-vertex functions}
&= \left( {\bm \gamma}^{(2,1)}_{\Lambda} \right)^{{\rm {\tilde{K}}};{\rm s}}_{ji;L}.
\end{align}
In the static approximation, we can derive the additional relations
\begin{align}
\left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm r};{\rm s}}_{ij;L}
&= \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm a};{\rm s}}_{ij;L}, \\
\left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm K};{\rm s}}_{ij;L}
&= \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm {\tilde{K}}};{\rm s}}_{ij;L}.
\end{align}
Hence, it is sufficient to determine the retarded and the Keldysh component of the three-point vertex function.
We determine the self-energy and the three-point current-vertex functions by solving these
flow equations numerically, and use the $\Lambda=0$ functions in the formula of the current
noise given in Eqs. (<ref>)-(<ref>).
Details of the numerical implementation are
given in Appendix <ref>.
§ RESULTS
§.§ A consistency check for the current
There have already been an extensive number of studies on the steady-state current of the IRLM
to elucidate its rich properties.<cit.>
We here focus on the case of the bandwidth $\Delta$ being much larger than the other energy scales of the
problem. It is known as the scaling limit, and universal features of the steady-state
current, such as the power-law behavior at large bias voltages, manifest
In an earlier FRG approach, the current was computed from the self-energy employing the Meir-Wingreen
formula.<cit.> In this subsection, it is discussed that an alternative FRG formulation
can be developed in which a flow equation for the current is derived and solved.
We show that both schemes provide the same results up to linear order in $U$, which is
the one to which our truncation is controlled. As the flowing current-vertex functions derived above enter the flow equation for the current this provides
a nontrivial consistency-check of our formulation later used to study the noise.
(Color online)
The dependence of $\bar{I}$ on the dimensionless interaction $u$ for various $V$.
The unit of energy, $T_{\rm K}$, is introduced in Sec. <ref>.
The parameters are $t / \Delta=0.001$, $\epsilon / T_{\rm K}=0$, and $T / T_{\rm K}=0$.
The explicit expression of the current using the Meir-Wingreen formula is given as
\begin{align}
I^{\rm MW}_{\Lambda} = \frac{1}{2\pi} \int d\omega
T^{\Lambda}_{LR}(\omega)(f_L(\omega)-f_R(\omega)) \right. \nonumber \\
& \left. +T^{\Lambda}_{L {\rm aux}}(\omega)(f_L(\omega)-f_{\rm aux}(\omega))
\right],
\end{align}
\begin{align}
& \Delta_L \Delta_R \left( {\bm G}^{\rm r}_{\Lambda} \right)_{13}(\omega) \left( {\bm G}^{\rm a}_{\Lambda} \right)_{31}(\omega), \\
T^{\Lambda}_{L {\rm aux}}(\omega)=
& \Delta_L \Lambda \left( {\bm G}^{\rm r}_{\Lambda} {\bm G}^{\rm a}_{\Lambda}\right)_{11}(\omega).
\end{align}
This should be equivalent to the current obtained by solving its flow equation, which is denoted
by $I^{\rm flow}_{\Lambda}$.
If we focus on their difference, i.e., $ \bar{I}_{\Lambda} \equiv I^{\rm MW}_{\Lambda}-I^{\rm flow}_{\Lambda}$,
its flow equation is given by
\begin{align}
\frac{d \bar{I}_{\Lambda} }{d\Lambda}
= &\frac{-i}{2} \int \frac{d\omega}{2\pi} \left( {\bm S}_{\Lambda} \right)^{\rm r}_{ij}(\omega) \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm r};{\rm s}}_{ji;L}(\omega;\omega;0) \nonumber \\
& - \frac{i}{2} \int \frac{d\omega}{2\pi} \left( {\bm S}_{\Lambda} \right)^{\rm a}_{ij}(\omega) \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{{\rm a};{\rm s}}_{ji;L}(\omega;\omega;0) \nonumber \\
& - \frac{i}{2} \int \frac{d\omega}{2\pi} \left( {\bm S}_{\Lambda} \right)^{\rm K}_{ij}(\omega) \left( \bar{{\bm \gamma}}^{(2,1)}_{\Lambda} \right)^{\tilde{\rm K};{\rm s}}_{ji;L}(\omega;\omega;0) \nonumber \\
& + \frac{i}{2} \Delta_L \int \frac{d\omega}{2\pi}
\left( (2f_L(\omega)-1)
\left[ \left( {\bm G}^{\rm r}_{\Lambda} \frac{d{\bm \Sigma}^{\rm r}_{U}}{d\Lambda}{\bm G}^{\rm r}_{\Lambda} \right. \right. \right.\nonumber \\
& \left. \left. - \left. {\bm G}^{\rm a}_{\Lambda} \frac{d{\bm \Sigma}^{\rm a}_{U}}{d\Lambda}{\bm G}^{\rm a}_{\Lambda} \right)_{11}(\omega) \right] \left({\bm G}^{\rm r}_{\Lambda} \frac{d{\bm \Sigma}^{\rm r}_{U}}{d\Lambda}{\bm G}^{\rm K}_{\Lambda} \right)_{11}(\omega) \right. \nonumber \\
& \left.
- \left({\bm G}^{\rm K}_{\Lambda} \frac{d{\bm \Sigma}^{\rm a}_{U}}{d\Lambda}{\bm G}^{\rm a}_{\Lambda} \right)_{11}(\omega) \right),
\label{eq:flow eq. of the current}
\end{align}
with initial condition $ \bar{I}_{\Lambda_{\rm init}}=0$.
As mentioned above, it contains the flowing current-vertex function
$\bar{{\bm \gamma}}^{(2,1)}_{\Lambda}$. The right-hand side is zero if the infinite hierarchy
of flow equations is kept, but may become finite if we use approximations, e.g., the
static one.
Equation (<ref>) together with the expression for the self-energy
Eq. (<ref>) as well as the vertex functions
Eqs. (<ref>) and
(<ref>) can be solved numerically.
The resulting value of the relative difference, $\left| \bar{I} \right| / I^{\rm MW}$
as a function of the dimensionless interaction $u\equiv U/\Delta$ is plotted in
Fig. <ref>.
Within the static approximation the difference should be of second order in $u$, i.e.,
$ \bar{I} \equiv \bar{I}_{\Lambda=0} = {\cal O}(u^2)$, which is consistent with the results
shown in Fig. <ref> for
various $V$. This finding indicates that we can consistently
determine the current by solving its flow equation and that the flow of the current-vertex function was
properly implemented. As briefly discussed in the next section, which is mainly on the noise, we
can reproduce all the known results for the current, e.g., power-law scaling with a $U$-dependent
exponent at large voltages from $I^{\rm flow}_{\Lambda}$.
§.§ On-resonance current noise
(Color online)
The dependence of noise and its logarithmic derivative on $V$ for various $u$.
The parameters are $\epsilon / T_{\rm K}=0$, $t / \Delta=0.001$, and $T / T_{\rm K}=0$.
(Color online)
(a) The dependence of the logarithmic derivative of the current and noise on $V$ for various $u$.
(b) The exponents at large bias voltages for various $u$.
The parameters are $\epsilon / T_{\rm K}=0$, $t / \Delta=0.001$, and $T / T_{\rm K}=0$.
It was established by previous works that the low-energy physics of the IRLM is governed by a single
energy scale, $T_{\rm K}$ (see, e.g., Ref. karrasch2010functional). Here this
universal energy scale is introduced as $T_{\rm K} \equiv 8 |\bar{t}^{\rm ren}|^2 / \Delta$ with
the renormalized hopping amplitude $\bar{t}^{\rm ren} \equiv t + \left. \bm{\Sigma}^{\rm r}_{12} \right|_{T=V=\epsilon=0}$
at the end of the RG flow. An alternative definition using the susceptibility is discussed in
Appendix <ref>. The current shows a crossover from the linear response regime to
power-law decay<cit.> at $V \simeq T_{\rm K}$.
Hence, it is natural to expect $T_{\rm K}$ as the characteristic energy scale of the current noise
as well. It is sufficient to focus on the component $ S \equiv S_{LL}(\omega=0)$ due to the charge
conservation. In this subsection we consider the transport on resonance with
The dependence of the temperature $T=0$ zero-frequency current noise obtained by numerically
solving the flow equations (for details of the implementation see Appendix <ref>)
on the bias voltage $V$ is shown in Fig. <ref> for various $u$. The
logarithmic derivative $d \log[S(V)]/d \log(V)$, approximated by centered differences, is shown
in the same figure. If $S(V)$ is governed by power-law behavior, the exponent can be read off from
the plateau value of this quantity. From this, it is evident that $S(V)$ is proportional
to $V^3$ for small $V$. The curves for the current noise collapse into a single one in the
linear response regime ($V\leq T_{\rm K}$) if properly scaled by $T_{\rm K}$. This indicates
that the prefactor of the leading $(V/T_{\rm K})^3$ behavior of the noise is independent of
the two-particle interaction. For $u=0$ the noise computed by FRG agrees with the analytic
expression given in Ref. PhysRevB.82.205414. The latter is shown as the thick
dashed line in Fig. <ref>.
It is well established that the $T=0$ current shows power-law suppression at high bias voltages,
\begin{align}
I / T_{\rm K} & \sim \left( V / T_{\rm K} \right)^{\alpha_I},
\end{align}
with an interaction-dependent exponent which to leading order in $u$ is given by
\begin{align}
\label{eq:exponent of the current}
\alpha_{I} = -\frac{4u}{\pi}.
\end{align}
The constant logarithmic derivative at large bias voltages in Fig. <ref>
indicates that the current noise exhibits power-law behavior in the same regime as well:
\begin{align}
S / T_{\rm K} & \sim \left( V / T_{\rm K} \right)^{\alpha_S} .
\end{align}
The logarithmic derivatives of the current and noise are compared in Fig. <ref>.
The dashed lines in Fig. <ref>(a) and the solid line in Fig. <ref>(b)
indicate the exponent Eq. (<ref>). For the current, this value can be
obtained analytically using FRG.<cit.>
The logarithmic derivative of the noise is found to reach the same value as that of the current at
sufficiently large bias voltages.
Previous works employing FRG showed that the behavior of the current can be understood from an
effective noninteracting model with renormalized parameters, in particular a renormalized
level-lead hopping.<cit.>
As the current-vertex corrections enter the expression for the noise, it is not obvious that
a similar mapping can be used for the noise.
The contribution of the vertex correction to the current noise is discussed in more detail in the
next section.
(Color online)
The dependence of the noise on $V$ for (a) $T<T_{\rm K}$ and (b) $T>T_{\rm K}$ for various $u$ and $T$.
The parameters are $\epsilon / T_{\rm K}=0$ and $t / \Delta=0.001$.
We show plots of the noise as a function of $V$ for various temperatures in
Fig. <ref>. The black dashed lines in Fig. <ref>(a)
are the thermal noise calculated via the fluctuation-dissipation theorem $S_{\rm th}=4 GT$ for each
temperature with the linear conductance defined (and numerically computed) as
$G \equiv d \left. I / dV \right|_{V=0}$.
The excellent agreement confirms that the current noise obeys the fluctuation dissipation relation
in the zero-bias limit, $S(V/ T_{\rm K} \rightarrow 0)=4GT$. The crossover from thermal to shot noise
occurs around voltages which fulfill $S_3V^3 \sim TG$, where $S_3$ is the coefficient of the $V^3$
term in the current noise $S$.
The independence of the coefficient $S_3$ on $u$ in the low-bias regime is discussed later in detail.
The power-law behavior at large voltages is observed for temperatures
sufficiently lower than $T_{\rm K}$. The current noise calculated at temperatures larger than $T_{\rm K}$
is shown in Fig. <ref>(b). The power-law decay at high bias voltages survives
even in this limit if the bias voltage is larger than $T$. In other words, the current noise exhibits
power-law decay at sufficiently large voltages which satisfy $V \gg \max\{T,T_{\rm K}\}$. Due to this
renormalization effect, the value of the current noise at high voltages can become even smaller than the
value in the zero bias limit. We note, however, that the current is suppressed as well with the same exponent.
The dependence of the equilibrium thermal noise on $T$ is shown in Fig. <ref>.
Except for the vertex correction, which is irrelevant at small voltages (see below) the thermal
noise is written as
\begin{align}
S_{\rm th}= \frac{1}{\pi} \int d\omega T_{LR}(\omega)
&\left[ f_{L}(\omega)(1-f_L(\omega)) \right. \nonumber \\
&\left. +f_R(\omega)(1-f_R(\omega))\right],
\end{align}
\begin{align}
\label{eq:Transmission amplitude}
T_{LR}(\omega) \equiv \Delta_L \Delta_R
\left({\bm G}^{\rm r}\right)_{13}(\omega)\left({\bm G}^{\rm a}\right)_{31}(\omega).
\end{align}
Hence, the numerically observed power-law decay at high temperature in Fig. <ref> can be
understood as a renormalization of the transmission amplitude.
As is shown in the figure, the thermal noise can be exactly translated into the linear conductance
via the fluctuation-dissipation relation, $4 G T$. This is owing to the reservoir cutoff scheme, in
which the KMS condition is guaranteed.
(Color online)
The dependence of the equilibrium noise $S$ on $T$ for various $u$.
The parameters are $\epsilon / T_{\rm K}=0$, $V/T_{\rm K}=0$, and $t / \Delta=0.001$.
§.§ The effective charge
(Color online)
The dependence of the ratio between the noise and the backscattering current on $V$ for various $u$.
The parameters are $\epsilon / T_{\rm K}=0$, $t / \Delta=0.001$, and $T / T_{\rm K}=0$.
The ratio between the noise and the current can be interpreted as an effective charge of carriers
when the transport is governed by Poisson statistics.<cit.>
On resonance, the effective charge is defined as the ratio between the noise and the
backscattering current.<cit.>
For $u=0$, the IRLM becomes the resonant level model and can be
solved exactly. In this case, the effective charge $e^*$ is $e$.
Another solvable point is the self-dual one<cit.> reached at relatively
large interaction. At this, field theoretical techniques and the density-matrix renormalization group
approach were utilized to show that the effective charge is $2e$. It is, however, unknown, how $e^*$
crosses over from $e$ to $2e$ when $u$ is increased. In this subsection, we study this issue using
FRG. Since our scheme is based on an expansion in terms of the interaction strength (on the right-hand side
of RG flow equations), we are bound to small-to-intermediate $u$.
The effective charge is defined as
\begin{align}
\label{eq:effective charge}
e^{*} = \lim_{V\rightarrow 0} \frac{S(V)}{2I_{\rm BS}(V)},
\end{align}
with the backscattering current
\begin{align}
I_{\rm BS} \equiv GV - I.
\end{align}
The dependence of the ratio $S/2I_{\rm BS}$ on $V$ is shown in
Fig. <ref> for various $u$.
It is evident that the value of $e^*/e$ can be reliably read off at
$V / T_{\rm K}=10^{-2}$. Obviously $e^*$ does not depend on the interaction in our
approximation, from which we conclude that $e^* /e =1+{\cal O}\left( u^2 \right)$
as all terms to linear order in $u$ are kept in the truncated RG equations.
This numerical observation can be substantiated by analytic considerations.
We first discuss the relation between the effective charge and the
vertex correction. The transmission amplitude Eq. (<ref>) can
be expanded in terms of the bias voltage as
\begin{align}
T_{\rm LR}(\omega)= T^{(0)}_{\rm LR}(\omega) - T^{(2)}_{\rm LR}(\omega) \left(\frac{V}{T_{\rm K}}\right)^2 + \cdots,
\end{align}
in the linear-response regime ($V<T_{\rm K}$).
If the bias voltage is much smaller than the scale of the energy dependence of the
transmission amplitude $(V \ll T_{\rm K})$, the current can be evaluated at the Fermi energy as
\begin{align}
\frac{I}{T_{\rm K}}= \frac{1}{2\pi} \left[ T^{(0)}_{\rm LR} \frac{V}{T_{\rm K}} - T^{(2)}_{\rm LR} \left(\frac{V}{T_{\rm K}}\right)^3 \right].
\end{align}
Since we are considering the on-resonance case, the zeroth-order coefficient $T^{(0)}_{LR}$
is unity. The leading term of the backscattering current is thus
\begin{align}
\frac{I_{\rm BS}}{T_{\rm K}} = \frac{1}{2\pi}T^{(2)}_{\rm LR}(0)\left(\frac{V}{T_{\rm K}}\right)^3.
\end{align}
We expand the (odd) backscattering current as
\begin{align}
\label{eq: def. of Coef. of I_BS}
I_{\rm BS}= G_3 \left(\frac{V}{T_{\rm K}}\right)^3 + G_5 \left(\frac{V}{T_{\rm K}}\right)^5 + {\cal O} \left( \left(\frac{V}{T_{\rm K}}\right)^7 \right).
\end{align}
From field theoretical considerations<cit.> it is known that the
backscattering current is given by
\begin{align}
\label{eq:Coef. of I_BS}
& \frac{2\pi I_{\rm BS} }{ T_{\rm K} }
=\frac{1}{3} \left[ 1 + {\mathcal O}(u^2) \right]
\left( \frac{V}{T_{\rm K}}\right)^3 \nonumber \\ & - \frac{1}{5} \left[ 1- \frac{20}{3} \frac{u}{\pi}
+ {\mathcal O}(u^2) \right] \left( \frac{V}{T_{\rm K}}\right)^5
+ {\mathcal O} \left( \left[ \frac{V}{T_{\rm K}}\right]^7 \right).
\end{align}
This analytical result can also be obtained by FRG (see endnote [52] of
Ref. PhysRevLett.112.216802; for a similar analysis of the current as function
of temperature see Ref. PhysRevB.87.075130). By comparing the coefficients in
Eq. (<ref>) with those in Eq. (<ref>), we find
\begin{align}
\frac{G_3}{T_{\rm K}}=\frac{1}{2\pi}T^{(2)}_{\rm LR}(0) = \frac{1}{6\pi},
\end{align}
independent of $U$. We note in passing that this result can be reproduced by our
numerical calculations as shown in Fig. <ref>(a)
if $T_{\rm K}$ is properly chosen (see Appendix <ref>). This exemplifies that our
numerics gives highly accurate results and that coefficients of expansions in $V/T_{\rm K}$
can be reliably determined.
Considering the above discussion and the definition of $e^*$, the remaining question is whether the
leading $(V / T_{\rm K})^3$ term in the current noise, $S_3$, has an order $u$ correction or not.
We already mentioned above that this does not seem to be the case [see Fig. <ref>].
To analyze this further, we show the dependence of $S_3$ on bias voltage $V$ in
Fig. <ref>(b). The third-order coefficient $S_3$ is independent
of the interaction strength
at low bias voltages, which is consistent with the result of $e^*/e$ shown in
Fig. <ref>.
(Color online)
(a) The dependence of the third order coefficient of the (a) current $G_3$ and (b) noise $S_3$ on $V$ for various $u$.
The parameters are $\epsilon / T_{\rm K}=0$, $t / \Delta=0.001$, and $T / T_{\rm K}=0$.
In the subsequent discussion, we give a microscopic explanation of the $u$ independence of
the third-order coefficient, $S_3$, by dividing the noise into the bubble term and the vertex
correction. If we denote the bubble and vertex correction terms by $S_0 \equiv S^{0}_{LL}(\omega=0)$
and $S_U \equiv S^{U}_{LL}(\omega=0)$, respectively, the current noise can be written as
\begin{align}
\end{align}
At zero-temperature, the bubble term is given as
\begin{align}
S_{0}=\frac{1}{\pi} \int^{V/2}_{-V/2} d\omega\; T_{LR}(\omega)\left[1-T_{LR}(\omega)\right].
\end{align}
If we ignore the frequency dependence of the transmission amplitude, we obtain
\begin{align}
\frac{S_{0}}{T_{\rm K}}= \frac{1}{\pi} \frac{V}{T_{\rm K}} T_{LR}(0)\left[1-T_{LR}(0)\right].
\end{align}
Since we are considering the on-resonance case [$T^{(0)}_{\rm LR}(0)=1$], the
contribution linear in $V$ to the bubble term vanishes. The lowest order contribution in $V$ is thus
\begin{align}
\frac{S_{0} }{T_{\rm K}}
&= \left. \frac{1}{\pi}\left( \frac{V}{T_{\rm K}}\right)^3 T^{(2)}_{\rm LR}(0) \left[ 2T^{(0)}_{\rm LR}(0)-1\right] \right|_{T^{(0)}_{LR}(0)=1} \nonumber \\
&= \frac{1}{\pi}\left( \frac{V}{T_{\rm K}}\right)^3 T^{(2)}_{\rm LR}(0) .
\end{align}
With this, the effective charge is obtained as
\begin{align}
&= \left. \frac{S}{2I_{\rm BS}} \right|_{V=0} \nonumber \\
&= 1 + \left. \frac{S_U}{2I_{\rm BS}} \right|_{V=0} \nonumber \\
&= 1 + \left. \frac{3\pi S_U}{\left( V / T_{\rm K}\right)^{3}} \right|_{V=0}.
\label{eq:effective charge and the vertex correction}
\end{align}
This relation shows that the $U$ dependence of the effective charge is incorporated
via the vertex correction $S_U$ analyzed next.
(Color online)
The vertex correction calculated using a FRG scheme and a plain perturbation theory for various $t$ as a function of $V$.
The parameters are $u=0.1$, $\epsilon / T_{\rm K}=0$, and $T / T_{\rm K}=0$.
The current-vertex functions enter the expression for the vertex correction
$S_U$ [see Eq. (<ref>)]. The three-point vertex function
can be computed in two different ways by either plain perturbation theory or by solving its
flow equations (<ref>) and
(<ref>). It is well established that
the self-energy computed in leading order perturbation theory in $u$ is plagued by a logarithmically divergent
To avoid this known problem in a perturbative computation of $S_U$, depicted
in the right diagram of Fig. <ref>, we dressed the two
propagators by the self-energy computed within FRG. This way we single out possible problems
of a perturbative calculation of the three-point vertex function itself.
The dependence of $S_U$ on $V$ obtained by perturbation theory and FRG is
shown in Fig. <ref>
for different $t/\Delta$. In both computations, the vertex correction scales as $V^4$
for small $V$, which is subleading compared with the bubble term which goes as $V^3$.
A significant difference is that the results obtained by perturbation theory becomes gradually
larger the deeper one goes into the scaling limit $t /\Delta \ll 1$.
The vertex corrections calculated using FRG for the three-point vertex are free of
this problem and collapse into a single curve if rescaled by $T_{\rm K}$. This indicates
that, in analogy to the self-energy, the FRG regularizes the divergences of the vertex correction.
(Color online)
The vertex correction divided by $u^2$ calculated using our FRG scheme for various $u$ as a function of $V$.
The parameters are $\epsilon / T_{\rm K}=0$, $t/\Delta=0.001$, and $T / T_{\rm K}=0$.
The vertex correction fully computed by FRG and divided by $u^2$ is plotted for various $u$ in
Fig. <ref>.
From this figure, it is evident that $S_U$ depends on $u$ and
$V$ as $\left| S_U \right|/ T_{\rm K} \propto u^2 (V/T_{\rm K})^4$.
Because of the $u^2$ prefactor in the linear response regime ($V<T_{\rm K}$), the vertex
correction is not under control within the static approximation which only contains all terms
to order $u$. However, the vertex correction does not contribute to the effective charge because
it is of order $V^4$ while the bubble term scales as $V^3$
[see Eq. (<ref>)].
§.§ Off-resonance current noise
(Color online)
(a) The current noise and (b) the vertex correction calculated from a plain perturbation theory for various $u$ and $t$ away from the particle-hole symmetric point as a function of $V$.
The parameters are $\epsilon / T_{\rm K}=10$ and $T / T_{\rm K}=0$.
In this subsection, we investigate the current noise away from the particle-hole symmetric point
($\epsilon \neq 0$).
We start out by considering the current noise with the three-point vertex functions calculated using plain
perturbation theory, but with all propagators dressed by the FRG self-energy [see above].
The noise as a function of voltage for different $u$ and $t / \Delta$
is shown in Fig. <ref>(a).
If the level energy aligns with one of the leads chemical potentials
($\epsilon \sim \pm V/2$) a peak develops.
The peak exhibits divergent behavior for decreasing $t/\Delta$, that is, when going into the scaling
limit, at fixed $u$. We reemphasize that this divergence originates from the vertex function, as
logarithmic divergences of the self-energy have already been removed by employing the FRG self-energy.
The vertex correction to the noise $S_U$ divided by $u$ is shown in
Fig. <ref>(b).
From this we conclude that the term diverging for $t/\Delta \to 0$ has a prefactor $u$. Plain
perturbation theory can thus not be used to study the current noise away from particle-hole symmetry
in the scaling limit even for very small $u$.
(Color online)
(a) The current noise and (b) the vertex correction divided by $u^2$ calculated using our FRG scheme for various $u$ and $t$ away from the particle-hole symmetric point as a function of $V$.
The parameters are $\epsilon / T_{\rm K}=10$ and $T / T_{\rm K}=0$.
The current noise and its vertex correction determined by our FRG scheme are shown in
Fig. <ref>. The vertex functions are
obtained by solving their flow equations Eqs. (<ref>)
and (<ref>). The divergent behavior of the current noise
observed in Fig. <ref>(a) is essentially
removed for the curves in Fig. <ref>(a).
Further down we comment on the weak features still visible in the regime $\epsilon \sim \pm V/2$.
This indicates that, as for the self-energy (and thus the current when employing the Meir-Wingreen formula),
the RG-based scheme regularizes the leading-order divergences. To further analyze this the vertex
correction to the noise $S_U$ divided by $u^2$ is shown in
Fig. <ref>(b). This figure indicates that
the divergence with prefactor $u$ of first order perturbation theory
(see Fig. <ref>(b)) is pushed to order $u^2$ within FRG.
As our truncation does not contain all terms ${\mathcal O}(u^2)$ we do not
control $S_U$ to this order. This second order divergence in $S_U$ manifests as the artificial dip of
the noise for $u=0.02$ and $t/\Delta=0.001$ and the shoulders for the other parameter sets
found in Fig. <ref>(a).
When next considering larger interactions we thus take $t/\Delta=0.01$ instead of $0.001$ as
before to avoid this order $u^2$ artifact.
The current noise as a function of $V$ is shown in
Fig. <ref> for a variety of $u$.
The current noise is proportional to $V$ in the linear-response regime
(see the plot for $V<T_{\rm K}$ as well as the analytic considerations in
Sec. <ref>).
At large bias voltages, the current crosses overs to a power-law decay with an interaction-dependent exponent.
The exponent agrees with the one found for $\epsilon=0$: $\alpha_S=-4u/\pi$.
This is in accordance with our intuition that
the bias voltage dominates the transport for $V \gg \epsilon$.
The dependence of the current noise on $\epsilon$ at fixed $V$ is shown in
Fig. <ref>.
It is independent of $\epsilon$ for $\epsilon \ll V$ as the level is placed inside the bias window.
The noise starts to decrease when the
level energy is beyond the bias window ($\epsilon \gtrsim V/2$). The weak features found when
the level is aligned with one of the lead chemical potentials were discussed above.
For large $\epsilon \gg T_{\rm K}$, the noise crosses over to a power-law decay as a function
of $\epsilon$ as the renormalization of the hoping amplitude is cut by the level position in this case.
For $u=0$ the exponent is $-2$ and the interacting part of the exponent is found to be twice that
of the $V$ dependence; see the logarithmic derivative shown in
Fig. <ref> for $\epsilon/T_{\rm K} \leq 1$ and
$\epsilon/T_{\rm K} \geq 10$.
(Color online)
The dependence of the current noise on $V$ for various $u$ away from the particle-hole symmetric point.
The parameters are $\epsilon / T_{\rm K}=10$, $t / \Delta=0.01$, and $T / T_{\rm K}=0$.
(Color online)
The dependence of the current noise and its logarithmic derivative on $\epsilon$ for various $u$.
The parameters are $V / T_{\rm K}=5$, $t / \Delta=0.01$, and $T / T_{\rm K}=0$.
§ SUMMARY
In the present paper, we have developed a FRG scheme to describe the current noise of the nonequilibrium IRLM.
The coupled set of flow equations of the current-vertex functions and the self-energy are derived and solved
to determine the current noise within the lowest-order approximation in the two-particle interaction.
The vertex correction of the current noise shows divergent behavior in the scaling limit, if it is calculated
using plain perturbation theory. This divergence is removed in our FRG method at the particle-hole
symmetric point, which makes it possible to perform a reliable analysis in the deep scaling limit.
In this regime, the current noise is found to show a power-law decay at high voltages characterized by
the same exponent as that of the current. This property is robust against temperature.
The effective charge of the IRLM at the particle-hole symmetric point can be reliably extracted
and is found to be interaction independent to linear order. This behavior can be understood from
the properties of the vertex contribution to the noise by combining analytical arguments
and the numerical results.
The current noise away from the particle-hole symmetric point determined by plain perturbation
theory shows a severe leading order divergence, which originates from the current-vertex correction.
We showed that the divergent term which is proportional to $u$ is consistently removed in our scheme
and pushed to order $u^2$; this lies beyond our control. Although the remaining order $u^2$ divergence
is an obstacle to calculate the current noise for the particle-hole asymmetric case in the scaling limit
($t /\Delta \ll 1$), we obtain reliable results down to $t/\Delta=0.01$.
We showed that the current noise shows a power-law decay for $\max\{V,\epsilon\} \gg T_{\rm K}$.
The present paper shows that the FRG method allows one to reliably calculate the current noise
in the scaling regime. A higher order FRG calculation – possible in principle, complicated in practice –
would be desirable to further elucidate the crossover of the effective charge from the noninteracting
case ($e^{*}/e =1$) to the self-dual point ($e^{*}/e =2$) at relatively large $u$.
Furthermore, the higher order contributions need to be taken into account in order to remove a diverging
term of order $u^2$ away from particle-hole symmetry and discuss the current noise in the deep scaling
limit for this case.
Another step for the future would be to extend the FRG treatment
to determine the full counting statistics of interacting fermion systems.
We thank Takeo Kato, Akinori Nishino, Katharina Eissing, and Peter Schmitteckert for very useful discussions.
T.J.S. acknowledges financial support provided by the Advanced Leading Graduate Course
for Photon Science (ALPS). This work was supported by the Deutsche Forschungsgemeinschaf via RTG 1995.
§ INITIAL CONDITIONS
The initial condition of the self-energy for $\Lambda_{\rm init} \to \infty$
is written as
\begin{align}
\left(\bm{\Sigma}^{\rm r}_{U,\Lambda_{\rm init}}\right)_{11}(\omega) &= U_1 n_2,\\
\left(\bm{\Sigma}^{\rm r}_{U,\Lambda_{\rm init}}\right)_{22}(\omega) &= U_1 n_1 + U_3 n_3 ,\\
\left(\bm{\Sigma}^{\rm r}_{U,\Lambda_{\rm init}}\right)_{33}(\omega) &= U_3 n_2,\\
\left(\bm{\Sigma}^{\rm K}_{U,\Lambda_{\rm init}}\right)_{ij}(\omega) &= 0,
\end{align}
where $n_i$ is the occupation of the $i$th site.
By a simple diagrammatic argument,<cit.> current-vertex functions are found to
be identical to those of the noninteracting system in the limit of $\Lambda_{\rm init} \rightarrow \infty$;
\begin{align}
&\left(\bm{\gamma}^{(2,n)}_{\Lambda_{\rm init}}\right)^{\nu'_1\nu_1;{\rm s}\cdots {\rm s}}_{ij;\alpha_1\cdots\alpha_n} (t'_1,t_1;t''_1\cdots t''_n) \nonumber \\
&=\left(\bm{\gamma}^{(2,n)}_{\rm res}\right)^{\nu'_1\nu_1;{\rm s}\cdots {\rm s}}_{ij;\alpha_1\cdots\alpha_n} (t'_1,t_1;t''_1\cdots t''_n) \ \ \ ({\rm for} \ n>0) .
\end{align}
The noninteracting current-vertex functions can be determined using the Ward-Takahashi identity
\begin{align}
&\left(\bm{\gamma}^{(2,1)}_{\rm res}\right)_{11;L} (\tau',\tau;\tau'') \nonumber \\
&= i \left[ \delta(\tau',\tau'') - \delta(\tau,\tau'') \right]
\left(\bm{\Sigma}_{\rm res}\right)_{11} (\tau',\tau).
\end{align}
The other components of the three-point vertex functions are zero because the source
field $A_L(\tau)$ is only included in the (1,1)-component of the tunneling self-energy
Eq. (<ref>).
The initial conditions of the three-point current-vertex functions are obtained as
\begin{align}
\left(\bm{\gamma}^{(2,1)}_{\Lambda_{\rm init}}\right)^{{\rm r};{\rm s}}_{ij;\alpha_1} (\omega_1,\omega_1;0)
&= - \delta_{i1}\delta_{j1}\delta_{\alpha_1 L} \Delta_L (1-2f_L(\omega_1)),\\
\left(\bm{\gamma}^{(2,1)}_{\Lambda_{\rm init}}\right)^{{\rm a};{\rm s}}_{ij;\alpha_1} (\omega_1,\omega_1;0)
&= \delta_{i1}\delta_{j1}\delta_{\alpha_1 L} \Delta_L (1-2f_L(\omega_1)),\\
\left(\bm{\gamma}^{(2,1)}_{\Lambda_{\rm init}}\right)^{{\rm K};{\rm s}}_{ij;\alpha_1} (\omega_1,\omega_1;0)
&= - \delta_{i1}\delta_{j1}\delta_{\alpha_1 L} \Delta_L ,\\
\left(\bm{\gamma}^{(2,1)}_{\Lambda_{\rm init}}\right)^{{\tilde {\rm K}};{\rm s}}_{ij;\alpha_1} (\omega_1,\omega_1;0)
&= \delta_{i1}\delta_{j1}\delta_{\alpha_1 L} \Delta_L.
\end{align}
Here, we show only the case with $\omega_1=\omega'_1$ because we focus on the zero-frequency current
noise in this paper. We note that $\left(\bm{\gamma}^{(2,1)}_{\Lambda_{\rm init}}\right)^{{\tilde {\rm K}};{\rm s}}$
does not need to be zero. The multi-point current vertices are determined by recursively using the Ward-Takahashi
identity, and the initial conditions for four-point current-vertex functions are
\begin{align}
&\left(\bm{\gamma}^{(2,2)}_{\Lambda_{\rm init}}\right)^{{\rm r};{\rm ss}}_{ij;\alpha_1\alpha_2} (\omega_1,\omega_1;0,0)
= 2i \delta_{i1}\delta_{j1}\delta_{\alpha_1 L}\delta_{\alpha_2 L} \Delta_L ,\\
&\left(\bm{\gamma}^{(2,2)}_{\Lambda_{\rm init}}\right)^{{\rm a};{\rm ss}}_{ij;\alpha_1\alpha_2} (\omega_1,\omega_1;0,0)
= -2i \delta_{i1}\delta_{j1}\delta_{\alpha_1 L}\delta_{\alpha_2 L} \Delta_L ,\\
&\left(\bm{\gamma}^{(2,2)}_{\Lambda_{\rm init}}\right)^{{\rm K};{\rm ss}}_{ij;\alpha_1\alpha_2} (\omega_1,\omega_1;0,0) \nonumber \\
&= 2i \delta_{i1}\delta_{j1}\delta_{\alpha_1 L}\delta_{\alpha_2 L} \Delta_L (1-2f_L(\omega_1)),\\
&\left(\bm{\gamma}^{(2,2)}_{\Lambda_{\rm init}}\right)^{{\tilde {\rm K}};{\rm ss}}_{ij;\alpha_1\alpha_2} (\omega_1,\omega_1;0,0) \nonumber \\
&= - 2i \delta_{i1}\delta_{j1}\delta_{\alpha_1 L}\delta_{\alpha_2 L} \Delta_L (1-2f_L(\omega_1)).
\end{align}
The initial conditions of the four-point and higher-point vertex functions are determined by the
bare action. If we denote the antisymmetrized bare two-particle interaction<cit.>
by $U_{ij;kl}$, these vertex functions are written as
\begin{align}
&\left(\bm{\gamma}^{(4,0)}_{\Lambda_{\rm init}}\right)^{\nu'_1\nu'_2;\nu_1\nu_2}_{ij;kl}(\omega'_1,\omega'_2;\omega_1,\omega'_1+\omega'_2-\omega_1) \nonumber \\
\left\{
\begin{array}{l}
\displaystyle -\nu'_1{U_{ij;kl}} \ \;\;\;{\rm if} \ \nu'_1=\nu'_2=\nu_1=\nu_2, \\
0 \ \;\;\;\;\;\;\;\;\;\; {\rm otherwise}.
\end{array}
\right. \\
&\left(\bm{\gamma}^{(4,m)}_{\Lambda_{\rm init}}\right)^{\nu'_1\nu'_2;\nu_1\nu_2;\nu''_1 \cdots \nu''_m}_{ij;kl;\alpha_1\cdots\alpha_m}(\omega'_1,\omega'_2;\omega_1,\omega_2;\omega''_1,\cdots,\omega''_m) \nonumber \\
&=0 \ \ (m>0).
\end{align}
§ NUMERICAL DETAILS
For the plots in the main text, we take the error of solving the set of
ordinary differential equations to be $10^{-6} T_{\rm K}$.
The frequency grid points are determined using geometric sequences with a scale factor $\Delta \omega=10^{-8} T_{\rm K}$.
The number of the grid points is $N_{\rm grid}=4801$, which is sufficient
to produce $N_{\rm grid}$ independent results.
As our main interest lies in the scaling regime, the band width $\Delta$ should be taken to be large enough for the $t/\Delta$ correction to be negligible.
We used $\Delta=10^4$ and $10^6$ for $t/\Delta=0.01$ and $0.001$, respectively.
§ DEFINITION OF $T_{\RM K}$
Several ways exist to define the emergent low-energy scale $T_{\rm K}$. Within the FRG approach the most
natural ones are either by the renormalized hopping amplitude or the susceptibility.
We used the renormalized hopping amplitude in the main text. The other definition utilized in previous FRG
works is
\begin{align}
T^{\rm sus}_{\rm K} \equiv -\frac{2}{\pi} \left( \left. \frac{d \langle n_2 \rangle}{d \epsilon} \right|_{T=V=\epsilon=0} \right)^{-1},
\end{align}
where $\langle n_2 \rangle$ is the occupation of the quantum dot site 2.
Deep in the scaling regime both definitions can equivalently be used when comparing with
field theoretical results obtained for $t/\Delta \to 0$. We found that, for the $t/\Delta$ reachable
by us, results rescaled with the $T_{\rm K}$ derived from the renormalized hopping show weaker
$t/\Delta$ corrections and are thus closer to the field theoretical predictions. For this reason
we used this definition in the main text.
|
1511.00501
|
Institut de Fìsica d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB. 01893 Bellaterra (Barcelona) Spain.
Uncertainties in modeling neutrino-nucleus interactions are a major contribution to systematic errors in Long Base Line neutrino oscillation experiments.
Accurate modeling of neutrino interactions requires additional experimental observables such as the Adler angles which carry information about the polarization of the $\Delta$ resonance and the interference with non-resonant single pion production. The Adler angles were measured with limited statistics in bubble chamber neutrino experiments as well as in electron-proton scattering experiments. We discuss the viability of measuring these angles in neutrino interactions with nuclei.
§ INTRODUCTION
The next generation of Long Base Line (LBL) accelerator neutrino oscillation experiments <cit.> aims at the discovery of CP violation in the lepton sector and the determination of the neutrino mass hierarchy. Systematic errors are likely to limit the accuracy of these measurements because the energy region of this new generation of experiments, ranging from 0.5 to 10 GeV, is dominated by several poorly measured cross-section channels: charged and neutral current quasi-elastic scattering, single-pion production, multi-pion resonant production and collective nuclear responses such as short and long range nuclear correlations<cit.>. Additional challenges are our incomplete understanding of the nuclear effects contributing to the cross-section and inaccuracies in reconstructing the neutrino energy.
In neutrino oscillation experiments, accurate measurements of oscillation parameters demand that uncertainties arising from the errors on neutrino fluxes and from the neutrino interactions themselves be properly factorized. In turn, this requires the correct modeling of neutrino-nucleus cross sections channels.
Neutrino cross-section knowledge suffers from three levels of uncertainties: i) Cross-sections at the nucleon level are not perfectly known. Vector form factors are derived from electron scattering, but axial and pseudoscalar form factors are assumed to be dipolar and constrained by the PCAC hypothesis. ii) Cross-sections are modified by effects due to the nuclear medium through short- and long-range correlations and by uncertainties in nucleon kinematics inside the nucleus. iii) The particles produced in the primary interaction cross the high-density nuclear medium which alters the particle composition of the event. Experimentally, the picture is confused even further by the typically broad neutrino energy spectrum and by beam flux uncertainties.
During the last decade efforts were focused on the description and measurement of quasi-elastic scattering; however we now also need to accurately model interactions occurring at higher energies such as single pion production. These data are sparse and contradictory even at the nucleon level <cit.>. In addition, current and future measurements can be performed only on nuclei.
In this paper, we explore the possibility of measuring Adler angles <cit.> in neutrino charged-current pion production on nuclei. A full description of the theoretical implications of Adler angles measurements can be found in the references <cit.>.
Experimental results from electron scattering on hydrogen have been published by the CLAS <cit.> collaboration and previously by earlier experiments <cit.>. These results were obtained on hydrogen targets, thereby avoiding initial and final state nuclear interactions and with the advantage of knowing the initial electron kinematics.
Here, we discuss the possibility of measuring these angles in modern neutrino experiments, which are typically performed in broad-band beams and using heavy nuclear targets. In these experiments the neutrino energy must be reconstructed from the data, the initial target nucleon is not at rest and the final state particles undergo nuclear re-scattering.
This work is based on the NEUT <cit.> Monte Carlo model, described in the following section. The angular observables are illustrated next, followed by Monte Carlo results and conclusions.
§ MONTE CARLO MODEL
Neutrino interactions are simulated with the NEUT <cit.> program libraries, which include neutral current (NC) and charged current (CC) processes of elastic and quasi-elastic scattering, meson exchange currents, single meson production, single gamma production, coherent pion production and non-resonant inelastic scattering.
NEUT uses the Rein-Sehgal<cit.> model to simulate neutrino-induced single pion production and an ad-hoc model for multiple pion production up to a hadronic invariant mass of 2.0 GeV/c$^2$. NEUT also includes a contribution from non-resonant pion production and the $\Delta$ polarization values measured in deuterium <cit.>.
Final State Interactions (FSI) of hadrons taking place within the nuclear medium are also simulated using a microscopic cascade model. In the case of final state pions, the considered processes are inelastic scattering, pion absorption and charge exchange. The simulated nucleon interactions are elastic scattering as well as single and double $\Delta$ production. FSI interactions alter both the multiplicity of pions in the final state as well as the kinematics of the pions.
For the current evaluation, events are generated with a T2K neutrino spectrum <cit.>. The simulation includes all channels present in the NEUT Monte Carlo. Most current experiments measure neutrino cross-sections on carbon-based detectors, such as plastic scintillator (Polystyrene). To avoid confusion between hydrogen and carbon targets, we decided to study interactions on carbon nuclei. In the case of Polystyrene there will be an additional $\approx$15% of interactions occurring with a free proton, in which the neutrino energy and the Adler angles will be well-determined, except for detector effects.
We consider both the $\Delta^{++}$ and the $\Delta^{+}$ decaying to a $\pi^+$ because they are indistinguishable at the experimental level for most of the final state nucleon momenta.
§ ADLER ANGLES
Definition of the Adler Angles at the nucleon (true) level (a) and the nuclear level (b). The momenta of the particles are defined in the $\vec q = \vec p_{\nu}-\vec p_{\mu}$ rest frame.
The Adler reference system <cit.> describes the p$\pi^+$ final state in the p$\pi^+$ reference system. The two angles are defined as in Fig.<ref>, where $\phi$ and $\theta$ are sensitive to the transverse and longitudinal polarization of the p$\pi^+$ system for interactions mediated by a $\Delta^{++}$, $\Delta^{+}$ and for non-resonant contributions. The two angles are properly defined at the nucleon interaction level but they are altered by the Final State Interactions and theFermi momentum of the target nucleon.
§.§ Adler angles at the level of the nucleus
Modern experiments detect neutrino interactions on targets consisting of relatively heavy nuclei (carbon, oxygen, iron, argon); therefore the definition of the Adler angles needs to be modified. The first modification is mandated by the fact that normally the proton is not detected. In this case, the p$\pi^+$ reference system needs to be redefined based on detector observables, namely the lepton and the $\pi^+$. In addition, we reconstruct the neutrino energy assuming that the target nucleon is at rest, thereby ignoring its intrinsic Fermi momentum, and assuming that the neutrino direction is known. In this scenario, energy-momentum conservation allows to estimate the neutrino energy as :
$$E_{\nu} = { m_p^2 - A_p^2 + | \vec p_{\mu} + \vec p_{\pi} |^2 \over 2 ( A_p + \vec d_{\nu} \cdot ( \vec p_{\mu} + \vec p_{\pi} ) ) } $$
$$ A_p = m_p - E_{bind} - E_{\mu} - E_{\pi} $$
where $(E_{\mu}, \vec p_{\mu})$ and $(E_{\pi}, \vec p_{\pi})$ are the four-momenta of the muon and the pion, $\vec d_{\nu}$ is the neutrino direction, $E_{bind}$ is the target nucleon binding energy ($\approx$ 25 MeV in NEUT for a carbon target) and $m_p$ is the free proton mass. The target nucleon momentum cannot be inferred when the outgoing proton is not detected. The uncertainty introduced by this approximation will be discussed later. This definition of the neutrino energy and the assumption that the target nucleon is at rest allow us to calculate the invariant mass of the p-$\pi$ system and to estimate the values of the observables used in deuterium experiments <cit.>.
We approximate the direction of the final p-$\pi^+$ system by the momentum transfer to the nucleus ($\vec q = \vec p_{\nu}-\vec p_{\mu}$).
The angle between the true p-$\pi^+$ system and the estimated $\vec q$ is shown in Figure <ref>. Under the aforementioned assumptions this approximation causes a bias of about 0.2 rad.
Angle between the $\vec p_{p} + \vec p_{\pi}$ system and the $\vec q$ approximation at the level of the nucleus.
The same observables can be reconstructed for the cases of $\Delta^- \rightarrow n \pi^-$ and $\Delta^0 \rightarrow p \pi^-$ to measure the Delta polarization in anti-neutrino nucleus interactions.
§.§ Fermi momentum versus Final State Interactions
To evaluate the relative contributions to the Adler angles of the Fermi momentum and the FSI, we compute the Adler angles under three assumptions:
i) true: we estimate the parameters using the full kinematic information at the level of the nucleon. These results are experimentally measurable only with a hydrogen target.
ii) pre-FSI: we use the true kinematics of the pion at the level of the nucleon but we ignore the target nucleons momentum. In this case the effect of the Fermi momentum is taken into account but the FSIs are ignored.
iii) post-FSI: we use the information of the pion leaving the nucleus and ignore the kinematic information of the target nucleon. These are the actual experimental observables and they contain the effect of both the Fermi momentum and of the FSI.
§ MONTE CARLO PREDICTIONS
§.§ Selection of events and their categories
Events are identified from the interactions at the nucleon level and from the multiplicity of particles leaving the nucleus. For the first criterion we rely on the Monte Carlo code to tag single pion production events, produced resonantly or not according to the model of Rein and Sehgal <cit.>. In order to assign the multiplicity of the pion final state we look for the number of $\pi^+$, $\pi^-$, $\pi^0$ and $e^{\pm}$ emitted by the nucleus, after the FSI. We define a one-$\pi^+$ topology when one and only one $\pi^+$ is emitted by the nucleus and no other particle from the above list is present. We do not count emitted protons, neutron and nuclear gammas because they are often produced but in current experiments they are detected with low efficiency. Nuclear de-excitation gammas play a negligible role in the description of the neutrino-nucleus interactions discussed here.
The one-$\pi^+$ events are then divided into three categories according to the true type of nucleon interaction: i) single pion production, the signal we want to study ii) deep inelastic scattering iii) other processes: for example, $\eta$ and kaon production. In what follows the last two categories are considered background. The fraction of true charged-current one-$\pi^+$ reactions in which a single $\pi^+$ emerges from the nucleus is $\sim$ 75%. In $\sim$ 43% of them the pion momentum is altered by FSI. The background is estimated to be $\sim$ 18% of single $\pi^+$ events leaving the nucleus. These numbers depend on the error on the actual neutrino flux and on the MC models; they only have indicative value.
Relative spread of the reconstructed neutrino energy $ ( E_{reco} - E_{\nu} ) / E_{\nu} $ as a function of the true neutrino energy. The black circles (squares) represent the mean value of the relative difference post(pre)-FSI. The empty circles (squares) show the RMS of the reconstructed neutrino energy relative error post(pre)-FSI.
§.§ Neutrino energy and hadronic invariant mass reconstruction
The relative spread of the reconstructed neutrino energy is shown in Figure <ref> as a function of the neutrino energy. At low energy, due to the relatively large contribution of the target nucleon Fermi momentum, the bias is large, but it decreases from 10% to almost zero at higher neutrino energies due the reduced contribution of the target´s Fermi momentum to the total interaction energy. The RMS error is also larger at low energies where it reaches 20% while it decreases to 10% for high neutrino energies. Opposite to other observables, the reconstructed pre-FSI energy is biased towards higher values of energy than the post-FSI reconstruction. The FSI compensates, through the pion energy loss inside the nucleus, the effect of the Fermi momentum decreasing the reconstructed energy.
A cut on the invariant mass of the p-$\pi^+$ system was suggested in the original paper <cit.>, and applied by the deuterium experiments <cit.> to constrain the reactions into the region dominated by the $\Delta^{++}(1232)$ resonance. Estimating the neutrino energy allows to obtain the invariant mass of the p-$\pi$ system, taken to be the $\mu-\nu$ invariant mass ($W_{reco}$), see Figure <ref>. The plot shows the contribution of the different backgrounds; the deep inelastic background dominates at high invariant mass values. The invariant mass threshold ($W >$ 2 GeV) that defines the DIS region in NEUT is clearly seen in Figure <ref>. This seems to indicate that the reconstructed $W$ value is a sensitive observable in validating the implementation of the transition from multi-pion production to DIS in the Monte Carlo.
The accuracy of the reconstruction is shown in Figure <ref>. The invariant mass reconstruction shows a small bias ( $<$ 4%) over almost the full range of W and also a relatively small RMS error ($\sim$ 8%). The figure also shows the contribution of the Fermi momentum (black squares) and the combined effect of Fermi momentum and FSI (black circles). The bias (below 1200 MeV) in the reconstructed $W$ is mainly caused by the Fermi momentum of the target nucleon.
Reconstructed hadronic invariant mass. The empty histogram shows the CC one-$\pi$ events. The vertical line histogram shows the invariant mass when the pion momentum is modified by the FSI. The horizontal line histogram shows the CC-DIS contribution to the CC one-$\pi^+$ sample and the mesh-filled histogram shows the remaining backgrounds.
Relative p-$\pi$ invariant mass reconstruction error ($ ( W_{reco} - W ) / W $ ) as a function of the true p-$\pi$ invariant mass . The black circles (squares) represent the mean value of the relative error post(pre)-FSI. The empty circles (squares) show the RMS of the reconstructed invariant mass relative error post(pre)-FSI.
§.§ Reconstructed Adler angles
True (solid line) and reconstructed (dotted line) Adler $\theta$ angle (top panel) and $\phi$ angle (bottom panel). The true CC 1$\pi^+$ distributions include all interactions with the target nucleon and the reconstructed ones for those events with pions leaving the nucleus. The dashed line histogram shows the result when the Fermi momentum is ignored in the reconstruction of the true CC 1$\pi^+$ result. The histogram filled with vertical lines shows the CC 1$\pi^+$ events in which the pion momentum is modified by the FSI. The histogram filled with horizontal lines is the CC-DIS contribution to the CC 1$\pi^+$ sample and the histogram filled with a mesh contains the rest of the backgrounds. The distributions are normalized to unity.
The reconstructed Adler angles are shown in Figure <ref>. The true distributions (solid line) are computed for all interactions at the nucleon level and the reconstructed distributions (dotted line) are shown for all events with a single pion leaving the nucleus.
This first look at the result allows to reach some preliminary conclusions: the transverse coordinates ($\phi$) are almost unaffected by nuclear effects while the longitudinal observable ($\theta$) is modified by the change in momentum of the outgoing pion. Background events tend to accumulate at low values of $\theta$ while $\pi^+$ rescattering events accumulate at high values of $\theta$ as it is shown in Figure <ref>
Reconstructed $\phi$ angle minus the true $\phi$ as a function of the $\phi$ angle. The black circles (squares) show the mean value of the difference post(pre)-FSI . The empty circles(squares) points show the RMS of the difference post(pre)-FSI.
The difference between the reconstructed and the true $\phi$ angle is shown in Figure <ref>. The average bias is close to zero while the maximal RMS error is nearly 1.2 rad. The dependence on the true $\phi$ angle can be explained by the fact that angles around 0 and $\pi$ rad correspond to pions contained in the neutrino-muon reaction plane, see Figure <ref>. These are the cases where the Fermi momentum that was ignored in the event reconstruction will produce the smallest effect because the true motion of the target nucleon should be contained in the reaction plane. The comparison between the RMS due to the Fermi momentum (0.4 rad) and Fermi momentum plus FSI (1.2 rad), Figure <ref>, indicates that the main contribution is the re-scattering of pion on its way out of the nucleus.
The dispersion in the reconstructed values of $\theta$ is shown in Figure <ref>. In this case, the bias goes up to 0.15 rad and the RMS is as large as 0.4 rad. One can see that $\theta$ is very sensitive to the accuracy of the neutrino energy reconstruction and the intranuclear scattering of the charged pion. The FSI are the dominant contribution to the RMS of $\theta$ over its whole range.
Reconstructed $\theta$ angle minus the true $\theta$ angle as a function of $\theta$. The black circles (squares) show the mean value of the difference post(pre)-FSI. The empty circles(squares) show the RMS of the difference post(pre)-FSI. .
§ BIAS IN ASYMMETRY MEASUREMENTS
We estimated the potential bias in the reconstruction of the Adler angles by means of a toy Monte Carlo. We take the dependence found from the deuterium results cited earlier <cit.> and weigh the events according to the $\phi$ angle at the nucleon level with a simple parity-violating function:
$$ weight = ( 1 + \alpha \sin{ \phi } ) $$
where $\alpha$ is derived from the asymmetry calculation in the above paper <cit.> to be 0.083. We fit the same angular dependence for the distributions of the $\phi$ angle at the nucleon and the nucleus level after subtracting the background and for events with W and $W_{reco}$ below 1400 MeV in order to select events dominated by $\Delta^{++}$ and $\Delta^{+}$ resonant contributions; see Figure <ref>. The results are shown in Figure <ref>. The fitted $\alpha$ values, $0.082\pm0.004$ at the nucleon level and $0.053\pm0.005$ at the nucleus level are similar. The errors shown only include the effects of the Monte Carlo statistics. For the reconstructed $\phi$, we have subtracted the non-CC 1$\pi^+$ interactions. Background modeling will introduce an additional source of uncertainty that is not evaluated in this study.
The bias in the determination of the asymmetry was estimated by generating several $\phi$ functional dependencies and adjusting the angular dependence. The results are shown in Table <ref>. Although there is a bias in the determination of the $\alpha$ parameter caused by the smearing of the reconstructed $\phi$ angle, there is still sensitivity to the determination of $\alpha$. An accurate experimental measurement should consider the variation of the polarization with the event kinematics <cit.>.
Table <ref> shows the result when we take into account the effect of the Fermi momentum. The effect of the Fermi momentum seems to be negligible at this level, as expected from Figure <ref>. The same calculation was performed for an angular dependence of the type $(1+\alpha \sin{2\phi})$. The results are shown in Table <ref> . The results are worse than in the previous case due to the convolution with the reconstructed $\phi$ RMS of the faster oscillation frequency.
The values of the angular dependence fits ($\alpha$) for different values of transverse polarization and $\sin{\phi}$ dependence. Results are shown for pions before and after the FSI.
2c post-FSI 2c pre-FSI
$\alpha_{true}$ $\alpha_{reco}$ $ {\alpha_{true}-\alpha_{reco} \over \alpha_{true} }$ $\alpha_{reco}$ $ {\alpha_{true}-\alpha_{reco} \over \alpha_{true} }$
0.02 0.0166 $\pm$ 0.0050 -0.17 0.0248 $\pm$ 0.0055 0.243
0.04 0.0285 $\pm$ 0.0050 -0.29 0.0440 $\pm$ 0.0055 0.101
0.06 0.0404 $\pm$ 0.0050 -0.32 0.0632 $\pm$ 0.0055 0.054
0.10 0.0644 $\pm$ 0.0050 -0.36 0.1015 $\pm$ 0.0054 0.015
0.12 0.0763 $\pm$ 0.0050 -0.36 0.1206 $\pm$ 0.0054 0.005
0.18 0.1121 $\pm$ 0.0050 -0.38 0.1781 $\pm$ 0.0053 -0.011
The values of the angular dependence fits ($\alpha$) for different values of transverse polarization and $\sin{2\phi}$ dependence. Results are shown for pions pre- and post-FSI.
2c post-FSI 2c pre-FSI
$\alpha_{true}$ $\alpha_{reco}$ $ {\alpha_{true}-\alpha_{reco} \over \alpha_{true} }$ $\alpha_{reco}$ $ {\alpha_{true}-\alpha_{reco} \over \alpha_{true} }$
0.02 0.0078 $\pm$ 0.0050 -0.60 0.0129 $\pm$ 0.0055 -0.35
0.04 0.0192 $\pm$ 0.0050 -0.52 0.0308 $\pm$ 0.0055 -0.23
0.06 0.0305 $\pm$ 0.0050 -0.49 0.0487 $\pm$ 0.0055 -0.19
0.10 0.0531 $\pm$ 0.0050 -0.47 0.0845 $\pm$ 0.0054 -0.16
0.12 0.0644 $\pm$ 0.0050 -0.46 0.1024 $\pm$ 0.0054 -0.15
0.18 0.0983 $\pm$ 0.0050 -0.45 0.1561 $\pm$ 0.0054 -0.13
$\phi$ distribution at the nucleon (black dots) and nucleus (black triangles) levels weighted by the angular dependence as described in the text. The solid line is the result of the fit to the function $A(1+\alpha\sin{\phi})$
We have also estimated the bias in the forward-backward asymmetry. Events are not reweighted and the asymmetry is computed as:
$$ A_{FB} = { N_{\cos{\theta}>0} - N_{\cos{\theta}<0} \over N_{\cos{\theta}>0} + N_{\cos{\theta}<0} } $$
for the distributions of $\theta$ both at the nucleon and the nucleus levels, after removing the background. The values of the resulting asymmetries are: $-0.007\pm 0.003$ (as predicted by the NEUT Monte Carlo) and $-0.179\pm0.003$ from the reconstructed observable. The observed bias is produced by the FSI and Fermi momentum within the nucleus because the Adler $\theta$ is very significantly modified, as shown in Figure <ref>. The dependence of the $\theta$ angle on the FSI and Fermi momentum makes it a very useful observable when investigating the nuclear effects on the results of the reaction.
§ CONCLUSIONS
We have shown that it is possible to measure the Adler angles in neutrino-nucleus scattering. The results based on the NEUT Monte Carlo show that one can determine the transverse polarization of the $\Delta$ resonance because the information is reasonably well maintained despite the FSI and the need to reconstruct the energy of the incoming neutrino from the experimental data. The longitudinal polarization is shown to depend strongly on the kinematics of the emerging pion, but it appears to allow investigatingf the effects of pion re-scattering, high mass resonances and deep inelastic processes on the CC one-$\pi^+$ tracks emerging from the nucleus. The results indicate that current high-statistics experiments can explore complex observables like Adler angle as a function of the kinematic parameters of the scattering process such as the energy of the neutrino, the hadronic invariant mass and four-momentum transfer.
The author acknowledges the support received from the Ministerio de Economia y Competitividad under grants FPA2014-59855 and Centro de Excelencia Severo Ochoa SEV-2012-0234, some of which include FEDER funds from the European Union. He would like to to acknowledge support from Y. Hayato on the NEUT Monte Carlo, discussions with J. Nieves on the implications of Adler angle measurements and help from S. Bordoni and M. Cavalli-Sforza in revising this paper.
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1511.00290
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Department of Mathematics, Northwestern University,
2033 Sheridan Road, Evanston, IL 60208, USA
Department of Mathematics, Northwestern University,
2033 Sheridan Road, Evanston, IL 60208, USA
[2000]14D07; 14F10, 14F17
MP was partially supported by the NSF grant DMS-1405516
We prove the weak positivity of the kernels of Kodaira-Spencer-type maps for pure Hodge module
extensions of generically defined variations of Hodge structure.
§.§ Introduction
Let $X$ be a smooth projective complex variety, and $U \subseteq X$ a dense open subset. It is of fundamental importance that (extensions of) variations of Hodge structure on $U$ come with inherent positivity properties. This study was initiated by Griffiths, who proved when $U= X$ that the lowest term in the Hodge filtration has a semi-positive definite metric, so in particular is a nef vector bundle. This fact was extended by Fujita <cit.> (when $X$ is a curve) and Kawamata <cit.> to include the case when $D = X - U$ is a simple normal crossings divisor, and the variation has unipotent monodromy along its components. Generalizations of this result were provided in recent work by Fujino-Fujisawa <cit.> and Fujino-Fujisawa-Saito <cit.>. It is possible, and very useful for applications, to see these results as part of a wider picture involving all kernels of Kodaira-Spencer type maps associated to meromorphic connections with log-poles and unipotent monodromy. This was first considered by Zuo <cit.>, while recently a detailed study has been provided by Brunebarbe <cit.>; see Section <ref>.
In this paper we give a further extension to the setting of Hodge modules. This is very convenient when dealing with arbitrary
families of varieties; see for instance <cit.>, where the present results are used towards proving Viehweg's hyperbolicity conjecture for families of maximal variation, and more generally to put constrains on the spaces on which certain geometrically relevant Hodge modules can exist.
Let $\bV$ be a polarizable variation of Hodge structure on $U$, with quasi-unipotent local monodromies. By a fundamental theorem of M. Saito <cit.>, $\bV$ admits a unique pure Hodge module extension $M$ with strict support $X$; conversely, any pure Hodge module with strict support $X$ is generically a variation of Hodge structure. See Section <ref> for further background.
We consider in particular the filtered left $\Dmod_X$-module $(\Mmod, F_{\bullet})$ underlying $M$. For each $p$, we have a natural Kodaira-Spencer type $\shO_X$-module homomorphism
$$\theta_p: {\rm gr}_p^F \Mmod \longrightarrow {\rm gr}_{p+1}^F \Mmod \otimes \Omega_X^1$$
induced by the $\Dmod$-module structure, and we denote
$$K_p (M) : = {\rm ker}~ \theta_p.$$
If $M$ is a polarizable pure Hodge module with strict support $X$, then the torsion-free sheaf $K_p (M)^\vee$ is weakly positive
for any $p$.
In the case when $D$ is a simple normal crossings divisor, and $\bV$ has unipotent monodromy along its
components, we will see in Section <ref> that there is a close relationship between $K_p (M)$ and $K_p (\cV^{\ge 0})$, where $\cV^{\ge 0}$ is the Deligne canonical extension of $\bV$ and
$K_p (\cV^{\ge 0})$ is defined analogously. For this, the
results of <cit.> and <cit.> on logarithmic connections can be applied to deduce weak positivity, as explained
in Section <ref>. We will then proceed in Sections <ref> and <ref> by successive reductions
to this case, using some of the main results from Saito's theory.
A seemingly different weak positivity result was proved by Schnell and the first author using Kodaira-Saito vanishing, as explained in <cit.> and <cit.>: it states that the lowest non-zero graded piece $F_{{\rm low}} \Mmod$ in the filtration on a Hodge module extending a generic variation of Hodge structure is weakly positive. (This includes Viehweg's result on $f_* \omega_{X/Y}$ for a morphism $f:X \rightarrow Y$ of smooth projective varieties.) Using duality, one can in fact deduce this as a special case of thm:WP; indeed, it is observed in <cit.>, see also the end of <cit.>, that $F_{{\rm low}} \Mmod$ can be related to the dual of ${\rm gr}_F^{{\rm top}} \Mmod(*D)$, the
top non-zero graded piece of the localization of $\Mmod$ along $D$, which in turn coincides with $K_{{\rm top}} (M)$.[This is the analogue of deducing the Fujita-Kawamata semipositivity results as special cases of the result of Brunebarbe and Zuo cited above.] It would be interesting to relate thm:WP to vanishing theorems as well.
§.§ Background material
In this section we review basic terminology and facts regarding weak positivity, filtered $\Dmod$-modules, and Hodge modules.
Weak positivity.
We start by recalling the notion of weak positivity introduced by Viehweg <cit.>; it is a higher rank analogue of the notion of a pseudo-effective line bundle, known to have numerous important applications to birational geometry.
Let $X$ be a smooth quasi-projective variety. A torsion-free coherent sheaf $\shF$ on $X$ is weakly positive over
an open set $U \subseteq X$ if for every integer $\alpha > 0$ and every ample line bundle $H$ on $X$, there exists
an integer $\beta > 0$ such that
$$\widehat{S}^{\alpha \beta} \shF \otimes H^{\otimes \beta}$$
is generated by global sections at each point of $U$. It is simply called weakly positive if such an open set $U$ exists.
Here the notation $\widehat{S}^k \shF$ stands for the reflexive hull of the sheaf $S^k \shF$.
The following basic lemma will be used for detecting weak positivity.
Let $\shF$ and $\shG$ be torsion-free coherent sheaves on $X$. Then the following hold:
(1) If $\shF \rightarrow \shG$ is surjective over $U$, and if $\shF$ is weakly positive over $U$, then $\shG$ is weakly positive over $U$.
(2) If $f: X\rightarrow Y$ is a birational morphism such that $f|_U$ is an isomorphism, and $E$ is a divisor supported on the exceptional locus of $f$ such that $\shF\otimes\shO_X(E)$ weakly positive over $U$, then $f_*\shF$ is weakly positive over $f(U)$.
(3) If $\pi: X' \rightarrow X$ is a finite morphism and $\pi^*\shF$ is weakly positive over $\pi^{-1}(U)$,
then $\shF$ is weakly positive over $U$.
Filtered $\Dmod$-modules and the de Rham complex.
Let $X$ be a complex manifold, or a smooth complex algebraic variety, of dimension $n$. If $(\Mmod, \fddot)$ is a filtered left
$\Dmod_X$-module, then the filtered de Rham complex of $(\Mmod, \fddot)$ is
\[\text{DR}(\Mmod):=[\Mmod \rarr \Mmod \otimes \Omega^1_X \rarr \cdots \rarr \Mmod \otimes \Omega^n_X] [n],\]
with filtration given by
\[F_p\text{DR}(\Mmod):=[F_p \Mmod \rarr F_{p+1} \Mmod \otimes \Omega^1_X \rarr \cdots \rarr F_{p+n}\Mmod \otimes \Omega^n_X][n].\]
The associated graded complexes for this filtration are
\[\gr^F_p\text{DR}(\Mmod):=[\gr^F_p \Mmod \rarr \gr^F_{p+1} \Mmod \otimes \Omega^1_X \rarr \cdots \rarr \gr^F_{p+n}\Mmod \otimes \Omega^n_X][n].\]
These are complexes of coherent $\shO_X$-modules, placed in degrees $-n, \ldots, 0$.
The Kodaira-Spencer kernels of the filtered $\Dmod_X$-module $(\Mmod, \fddot)$ are the coherent sheaves
$$K_p (\Mmod): = {\rm ker} \big( \theta_p: {\rm gr}_p^F \Mmod \longrightarrow {\rm gr}_{p+1}^F \Mmod \otimes \Omega_X^1\big)$$
where $\theta_p$ are the $\shO_X$-module homomorphisms considered above. Equivalently,
\begin{equation}\label{left_coh}
K_p (\Mmod) \simeq \cH^{-n} \gr^F_p {\rm DR}(\Mmod).
\end{equation}
Hodge modules and variations of Hodge structure.
Let $X$ be a smooth complex algebraic variety of dimension $n$, and let
$$\bV = (\mathcal{V}, F_\bullet, \V_{\QQ})$$
be a polarizable variation of $\QQ$-Hodge structure of weight $k$ on an open set $U \subset X$.
Here $\V_{\QQ}$ is a local system of $\QQ$-vector spaces on $U$,
$\mathcal{V} = \V_{\QQ}\otimes_{\QQ} \shO_U$, and $F_p = F_p \mathcal{V}$ an increasing[This convention is adopted in order to match the standard Hodge module terminology; usually one would consider a decreasing filtration where $F^p$ corresponds to our $F_{-p}$.] filtration of
subbundles of $\mathcal{V}$ satisfying Griffiths transversality with respect to the connection associated to $\mathcal{V}$.
In <cit.>, Saito associates to this data a pure Hodge module of weight $n+ k$ on $X$, whose
main constituents are:
(1) A filtered regular holonomic left $\Dmod_X$-module $(\Mmod, F_{\bullet})$, with $F_{\bullet} \Mmod$
a good filtration by $\OX$-coherent subsheaves, whose restriction to $U$ is $\cV$ together with its connection and Hodge filtration.
(2) A $\QQ$-perverse sheaf $P$ on $X$ such that $\DR_X(\Mmod) \simeq P \tensor_{\QQ} \CC$.
Moreover, one of Saito's fundamental results <cit.>, states that there is a unique such extension if we impose the condition that $M$ have strict support $X$, i.e. not have any sub or quotient objects with support strictly smaller than $X$. Its underlying perverse sheaf is $P = {\rm IC}_Z( \V_{\QQ}) = {}^pj_{!*}\V_\Q$, the intersection complex of the given local system, and therefore one sometimes uses the notation
\[M : = j_{!*}\bV.\]
These are the main objects we consider in this paper; we refer to them as the Hodge module extension of the generically defined VHS. In Section <ref> we will give a more concrete description of the Hodge filtration in the case when the complement of $U$ is a simple normal crossings divisor.
Assuming that the complement of $U$ in $X$ is a divisor $D$, we can also consider the filtered $\Dmod_X$-module
$(\cV (*D), F_{\bullet})$, where $\cV(*D)$ is Deligne's meromorphic connection extending $\cV$. This underlies a natural mixed Hodge module extension of $\bV$ introduced in <cit.>, denoted $j^*j^{-1} M$, and sometimes called the localization along $D$. More precisely
\[ j_*j^{-1}M =(\cV(*D), \fddot, j_*\V_\Q),\]
We will also have a more concrete description of this Hodge module in the case when $D$ is a simple normal crossings divisor.
§.§ Weak positivity for logarithmic variations of Hodge structure
In this section we recall background on meromorphic connections with log poles, and the
results of <cit.> and <cit.> that are used in the proof of our main theorem.
Logarithmic connections.
We begin by reviewing the theory of logarithmic connections; see for instance <cit.> and <cit.>.
Let $X$ be a smooth complex variety of dimension $n$, and let $D=\sum D_i$ be a reduced simple normal crossings divisor.
We will call such an $(X, D)$ a smooth log pair. Write $U= X \smallsetminus D$, and denote the inclusion by
\[j: U\hookrightarrow X.\]
Suppose $\cV$ is a holomorphic vector bundle of finite rank on $X$. An integrable logarithmic connection on $\cV$ along $D$ is a $\CC$-linear morphism
\[\nabla:\cV \rarr \cV \otimes \Omega^1_X (\text{log}~D)\]
satisfying the Leibniz rule and $\nabla^2=0$. By analogy with the definition above, the logarithmic de Rham complex is defined as
\[{\rm DR}_D (\cV):=[\cV \otimes \Omega^\bullet_X(\text{log} D)][n].\]
For each $D_i$, composing the Poincaré residue and $\nabla$ induces the residue map
$$\Gamma_i \in \text{End}(\cV|_{D_i}).$$
Now given a $\CC$-local system $\V$ on $U$, $\cV=\V\otimes\shO_U$ is a vector bundle with integrable connection, and this can be extended to an integrable logarithmic connection on $X$. Such an extension is unique if the eigenvalues of $\Gamma_i$ are required to be in the image of a fixed section $\tau$ of the projection $\CC\rarr\CC / \Z$; see <cit.>. If $\tau$ is chosen so that the real parts belong to the interval $[0, 1)$, the corresponding extension of $\cV$ is the Deligne canonical extension, denoted by $\cV^{\geq0}$.
If $E=\sum\alpha_i D_i$ is any Cartier divisor supported on $D$, $\cV (E)$ is also an integrable logarithmic connection, with residue $\Gamma^E_i$ given locally by
\begin{equation}\label{res}
\Gamma^E_i=\Gamma_i-\alpha_i\cdot\text{Id}.
\end{equation}
See for instance <cit.>.
Logarithmic variations of Hodge structure.
Following <cit.>, it will be convenient to consider the following notion which combines
logarithmic connections and variations of Hodge structure:
Let $(X, D)$ be a smooth log pair with $X$ projective, and set $U:=X \smallsetminus D$. A log variation of Hodge structure (log VHS)
along $D$ consists of the following data:
(1) A logarithmic connection $(\cV, \nabla)$ along $D$.
(2) An exhaustive increasing filtration $\fddot$ on $\cV$ by holomorphic subbundles (the Hodge filtration)[Again, to be consistent with conventions for Hodge modules, we use increasing filtrations.], satisfying the Griffiths transversality condition
\[\nabla F_p\subseteq F_{p+1} \otimes \Omega_X^1(\text{log}~D).\]
(3) A $\Q$-local system $\V^U_\Q$ on $U$, such that $(\cV|_{U}, \fddot|_U, \V^U_\Q)$ is a variation of Hodge structure on
A log VHS is polarizable if the variation of Hodge structure defined on $U$ is so.[Polarizability as defined here is slightly different from the notion considered in <cit.>; however they are equivalent when the residues of $\nabla$ are nilpotent.]
Filtered logarithmic de Rham complexes for log variations of Hodge structure are defined just as in the case of filtered
$\Dmod$-modules. We can also consider the associated graded quotients of the filtered de Rham complex,
\[\gr^F_p {\rm DR}_D(\cV) = [\gr^F_{p+\bullet} \cV \otimes \Omega_X^\bullet(\text{log}~D)][n],\]
which are $\shO_X$-linear complexes of holomorphic vector bundles in degrees $-n, \ldots, 0$.
In this context it is known that the duals of the Kodaira-Spencer-type kernels
$$K_p (\cV) : = \cH^{-n} \text{gr}^F_p\text{DR}_D(\cV) \simeq
{\rm ker} \big( {\rm gr}_p^F \cV \overset{\theta_p}{\longrightarrow} {\rm gr}_{p+1}^F \cV \otimes \Omega_X^1 ({\rm log}~D)\big)$$
satisfy a positivity property; this extends the well-known Fujita-Kawamata semi-positivity theorem, and is due to
Zuo <cit.>; see also Brunebarbe <cit.> (or <cit.>) for
an alternate proof.
Let $(X, D)$ be a smooth log pair with $X$ projective, and let $\cV$ be the bundle with logarithmic connection underlying a polarizable log VHS with nilpotent residues along $D$. If $\shF$ is a holomorphic subbundle of $K_p (\cV)$, then
$\shF^{\vee}$ is nef.
For the proof of the next statement, it will be useful to note the following: if
\[g: (Y, E)\rarr (X, D)\]
is a morphism of smooth projective log pairs, where $E=(f^{*}D)_{\text{red}}$, then a simple local calculation shows that
(1) If $\cV$ underlies a (polarizable) log VHS along $D$, then so does $f^*\cV$, along $E$;
(2) If furthermore the residues of $\cV$ along $D$ are nilpotent, then so are those of $f^*\cV$ along $E$.
As a consequence of sp we have the following statement, which in the geometric case is essentially <cit.>.
If $\cV$ underlies a polarizable log VHS with nilpotent residues along $D$, then $K_p (\cV)^\vee$ is weakly positive.
Let us denote for simplicity
$$\shE_p : = {\rm gr}_p^F \cV.$$
If $K_p (\cV)$ is already a subbundle of $\shE_p$, then we are done by sp, as nef vector bundles
are weakly positive. This need not be the case in general; however, a standard resolution of singularities argument
applies to provide a birational morphism $\rho: X' \rightarrow X$ with $X'$ smooth projective, such that
$\rho^* K_p (\cV)$ has a morphism to a subbundle $\shF$ of $\rho^*\shE_p$, which is generically an isomorphism.
On the other hand, we have a commutative diagram
\begin{tikzcd}
\rho^*\shE_p \drar{\theta'_p} \rar{\rho^*\theta_p} &\rho^*\shE_{p+1} \otimes \rho^* \Omega_X^1(\text{log}~D) \dar{\psi} \\
& \rho^*\shE_{p+1} \otimes \Omega_{X'}^1(\text{log}~E)
\end{tikzcd}
where $E=(\rho^*D)_{\text{red}}$ and $\psi$ is induced by the natural map
\[\rho^* \Omega_X^1(\text{log}~D) \rarr \Omega_{X'}^1(\text{log}~E).\]
Using the remark before the statement of the Corollary, $\theta^\prime_p$ again corresponds to a log VHS (induced by the same generic variation of Hodge structure), with a logarithmic connection along $E$. Since $\rho$ is birational, $\theta'_p(\shF)$ is generically 0, and so identically $0$ since it embeds in a vector bundle. Hence $\shF$ is contained in $\text{Ker}(\theta'_p)$, and so $\shF^\vee$ is nef by sp. By wplemma(1) we obtain that
$\rho^* K_p (\cV)^\vee$ is weakly positive as well. Finally, this implies that $ K_p (\cV)^\vee$ itself is weakly positive,
using wplemma(2).
§.§ Normal crossings case
In this section we establish the main result for pure Hodge module extensions of variations of Hodge structure defined on the complement of a simple normal crossings divisor, by reducing to the result for logarithmic connections in the previous section.
Hodge modules associated to variations of Hodge structure.
Let $(X, D)$ be a smooth log pair, with $X$ projective and $\dim X = n$.
Denote $U=X \smallsetminus D$ and $j: U\hookrightarrow X$. We consider a polarizable variation of Hodge structure
\[\bV=(\cV, \fddot, \V_\Q)\]
over $U$, with quasi-unipotent local monodromies along the components $D_i$ of $D$. In particular the eigenvalues of all residues are rational numbers. For $\alpha \in \ZZ$, we
denote by $\cV^{\geq\alpha}$ (resp. $\cV^{>\alpha}$) the Deligne extension with eigenvalues of residues along the $D_i$ in $[\alpha, \alpha+1)$ (resp. $(\alpha, \alpha+1]$). Recall that $\cV^{\geq\alpha}$ is filtered by
\begin{equation}\label{filtr}
F_p\cV^{\geq\alpha}=\cV^{\geq\alpha}\cap j_*F_p\cV,
\end{equation}
while the filtration on $\cV^{>\alpha}$ is defined similarly. The terms in the filtration are
locally free by Schmid's nilpotent orbit theorem <cit.> (see also e.g. <cit.> for the quasi-unipotent case);
we have that $(\cV^{\geq\alpha(>\alpha)}, \fddot, \V_\Q)$ is a polarizable log VHS.
Following Section <ref>, let now $M$ be the pure Hodge module with strict support $X$ uniquely extending
$\bV$. It is proved in <cit.> that
\[M=(\Dmod_X\cV^{>-1}, \fddot, j_{!*}\V_\Q),\]
\[F_p\Dmod_X\cV^{>-1}=\sum_i F_i\Dmod_X \cdot F_{p-i}\cV^{>-1}.\]
We also consider the natural mixed Hodge module extension of $\bV$, namely the localization
\[j_*j^{-1}M =(\cV(*D), \fddot, j_*\V_\Q).\]
See e.g. <cit.>. We recall that $\cV(*D)$ is Deligne's meromorphic connection extending $\cV$, with lattice
$\cV^{>\alpha}$ (or $\cV^{\geq\alpha}$) for any $\alpha\in\QQ$, namely
\[\cV(*D)=\cV^{>\alpha}\otimes \shO_X (*D),\]
with filtration given by
\[F_p\cV(*D)=\sum_i F_i\Dmod_X \cdot F_{p-i}\cV^{\geq-1}.\]
In <cit.>, Saito constructed a filtered quasi-isomorphism that will be used in what follows, namely
\begin{equation}\label{qiso1}
\big([\cV^{\geq0}\otimes \Omega^\bullet_X(\text{log} ~D)][n], \fddot\big) \simeq \big(\text{DR}( j_*j^{-1}M ), \fddot \big).
\end{equation}
Here the notation on the right hand side refers to the filtered de Rham complex of the underlying filtered
Unipotent reduction and proof in the normal crossings case.
In the setting of the previous section, a standard argument using Kawamata's covering construction <cit.> provides a finite flat morphism of smooth projective log pairs
\[f: (Y, E)\rarr (X, D)\]
with $(f^*D)_\text{red}=E$, such that the pull-back $\V_1: = f_1^* \V_\Q$ has unipotent local monodromies along all irreducible components $E_i$ of $E$, where $f_1=f|_{f^{-1}(U)}$.
To perform the reduction to the case of $f^* \bV$ on $(Y, E)$, let us first recall that one calls a
lattice for $\cV (*D)$ any locally free sheaf $\cL$ on $X$ satisfying
$$\cV(*D) \simeq \cL \otimes \shO_X(*D),$$
and preserved by the action of $f\cdot \nabla$, where $\nabla$ is the meromorphic connection on $\cV(*D)$
and $f$ is a local equation of $D$. We also consider the same notions for $E$.
For the argument, first, it is not hard to see that $f^* \cV(*D)$ is a regular meromorphic connection on $Y$ extending $\V_1$. This implies
\[ f^* \cV(*D) \simeq \cV_1(*E),\]
e.g. by Deligne's Riemann-Hilbert correspondence for meromorphic connections with regular singularities
(see <cit.>). Moreover, $f^*\cV^{\geq0}$ is a lattice of $\cV_1(*E)$.
A simple local calculation shows that the eigenvalues of the residue of the connection along each component of $E$ are nonnegative integers. On the other hand,
\[ \cV_1(*E)=\lim_{\longrightarrow}\cV_1^{\geq 0}(-kE)\]
over $k \in \ZZ$. Hence
\[f^*\cV^{\geq 0}\subseteq \cV_1^{\geq 0}(- kE)=\cV_1^{\geq k},\]
for some integer $k$.
We claim that $k \ge 0$, so that in particular
\begin{equation}\label{placement}
f^*\cV^{\geq 0}\subseteq \cV_1^{\geq 0}.
\end{equation}
This is a special case of the following general statement:
Let $Y$ be a smooth complex variety and $E = \sum E_i$ a simple normal crossings divisor on $Y$. Suppose that $\cW$ is a lattice for a regular meromorphic connection with poles along $E$, and with quasi-unipotent local monodromies along all
$E_i$. Denoting by $\Gamma_i$ the residue of $\cW$ along $E_i$ with respect to the connection, if
\[t : = \textup{min}~\{ \textup{eigenvalues of } \Gamma_i \textup{ for all } i \},\]
\[\cW\subseteq \cW^{\geq t},\]
where $\cW^{\geq t}$ is the Deligne extension with eigenvalues of all residues in $[t, t +1)$.
Suppose $k$ is the smallest integer such that $\cW\subseteq \cW^{\geq -k}$. By the Artin-Rees Lemma (see for instance <cit.>), there is an integer $\ell \geq 0$ such that
\[\cW\subseteq \cW^{\geq -k}(-\ell E_i) \,\,\,\, {\rm and} \,\,\,\, \cW\not\subseteq \cW^{\geq -k} \big(-(\ell +1)E_i\big). \]
$$\cW/\cW\cap\cW^{\geq -k} \big(-(\ell +1)E_i \big) \neq 0,$$
and given that $\cW (- E_i) \subseteq \cW^{\geq -k} \big(-(\ell +1)E_i\big)$ we have a short exact sequence
\[0\rightarrow \dfrac{\cW\cap\cW^{\geq -k} \big(-(\ell+1)E_i \big)}{\cW(-E_i)}\rightarrow \cW|_{E_i}\rightarrow \dfrac{\cW}{\cW\cap\cW^{\geq -k}\big(-(\ell+1)E_i \big)}\rightarrow 0. \]
Because of our choice of $k$, and the identification
$$\cW^{\geq -k}(-\ell E_i) \simeq \cW^{\geq 0} ( kE -\ell E_i), $$
the last term in the exact sequence has an induced action of $\Gamma_i + (k - \ell)\cdot {\rm Id}$, with non-negative eigenvalues; see also (<ref>). (Note that since the first term in the sequence comes from an intersection of lattices, it is preserved by the residue action on $\cW|_{E_i}$.) Since this action is induced by the action of $\Gamma_i$ on $E_i$, by definition it follows that
$$ - k + \ell \ge t.$$
We need to show that $-k \ge t$. If on the contrary we assume that $k + t > 0$, it follows that $\ell > 0$ as well. But the exact same argument
can be run for every $E_i$, and so it follows that
$$\cW \subseteq \bigcap_i \cW^{\ge - k} (- E_i) = \cW^{\ge - k + 1},$$
which contradicts the minimality of $k$.
Although not needed for the statement above, it is worth noting that the argument above can be continued inductively
in order to reconstruct the entire minimal polynomial of $\Gamma_i$ acting on $\cW|_{E_i}$. Roughly speaking, denoting
\[\cW_1:=\cW\cap\cW^{\geq -k} \big(-(\ell +1)E_i\big),\]
one can repeat an appropriate procedure for $\cW_1$ instead of $\cW$. The process will eventually stop for dimension reasons.
To deal with the pull-back of the Hodge filtration, we use the following simple lemma (see also <cit.>):
Let $\shE$ be a locally free sheaf on $X$, and let $\shF$ and $\shG$ be two subsheaves of $\shE$ such that $\shG$ is locally free, and of maximal rank at each point as a subsheaf of $\shE$ (i.e. $\shG$ is a subbundle). If $\shF|_U=\shG|_U$ for some nonempty Zariski open subset $U\subseteq X$, then $\shF\subseteq \shG$.
Putting together (<ref>) and Lemma <ref>, we conclude:
\[f^*F_p\cV^{\geq 0}\subseteq F_p\cV_1^{\geq 0}.\]
Recall now that we set
\[K_p (\cV^{\geq 0}) :=\shH^{-n} \text{gr}^F_p\text{DR}_D(\cV^{\geq0})
\simeq
{\rm ker} \big( {\rm gr}_p^F \cV^{\geq 0} \overset{\theta_p}{\longrightarrow} {\rm gr}_{p+1}^F \cV^{\geq 0} \otimes \Omega_X^1 ({\rm log}~D)\big). \]
$K_p (\cV^{\geq 0})^{\vee}$ is weakly positive for any $p$.
As in the proof of WP_log, we consider the natural diagram
\begin{tikzcd}
f^*{\rm gr}_p^F \cV^{\geq 0} \rar{f^*\theta_p} \drar{\theta^\prime_p} & f^*{\rm gr}_{p+1}^F \cV^{\geq 0} \otimes f^* \Omega_X^1({\rm log}~D) \dar \\
& f^*{\rm gr}_{p+1}^F \cV^{\geq 0} \otimes \Omega_{Y}^1({\rm log}~E)
\end{tikzcd}
By hodf we have an inclusion
$$\text{Ker} ~\theta'_p \subseteq K_p (\cV_1^{\geq 0})$$
which is generically an isomorphism. Using WP_log for $\cV_1^{\geq 0}$, together with wplemma(1), it follows that $(\text{Ker} ~\theta'_p)^\vee$ is weakly positive, hence so is
$(\text{Ker} ~f^*\theta_p)^\vee$, again by wplemma(1). Therefore $K_p (\cV^{\geq 0})^\vee$ is also weakly positive because of wplemma(3).
Using the notation $K_p (M)$ for the kernels associated to the underlying filtered $\Dmod$-module, we can now deduce the main result in the setting of this section:
$K_p (M)^{\vee}$ is weakly positive for any $p$.
[We thank both the referee and C. Schnell for suggesting this as a replacement for an earlier
argument that needed more justification.]
The filtered quasi-isomorphism (<ref>) induces an isomorphism
\[K_p (\cV^{\geq 0}) \simeq K_p (j_*j^{-1}M). \]
On the other hand, by definition there is a natural morphism
\[K_p (M) \longrightarrow K_p (j_*j^{-1}M)\]
which is an isomorphism over $U=X \smallsetminus D$.
Passing to duals, by WP_Deligne and wplemma(1) we obtain that the dual of $K_p (M)$
is also weakly positive.
Though not necessary for the argument, it is worth noting, as C. Schnell has pointed out to us, that using the $V$-filtration axioms for Hodge modules one can check (independently of the normal crossings hypothesis)
that $K_p (M) \simeq K_p (j_*j^{-1}M )$ for all $p$ as well. Thus the morphism appearing the proof of wp
is in fact an isomorphism.
§.§ General case
We now assume
\[\bV=(\cV, \fddot, \V_\Q)\]
to be a polarizable variation of Hodge structure with quasi-unipotent local monodromies, defined on open subset $U\subset X$ such that $D=X - U$ is an arbitrary divisor.
As in Section <ref>, we denote the pure Hodge module extension of $\bV$ by $M$.
\[f: (X', E)\rarr (X, D) \]
is a log resolution of the pair $(X, D)$ which is an isomorphism over $U$, with
$E=(f^*D)_{\text{red}}$. We also consider the pure Hodge module $M'$ with strict support $X'$ extending $\bV$, this time seen as a variation of Hodge structure on $X' - E$.
First, by the Stability and Decomposition Theorem for pure Hodge modules <cit.>,
we have that each $\cH^i f_*M'$ is a pure Hodge module on $X$, while the underlying filtered
$\Dmod_X$-modules satisfy
\[f_+ (\Mmod^\prime, F_{\bullet}) = \bigoplus_i \cH^i f_{+} (\Mmod', F_{\bullet}) [-i]\]
in the derived category. (Here $f_+$ is the derived direct image functor for filtered left $\Dmod$-modules.)
Moreover, $M$ is a direct summand of $\cH^0 f_*M'$; it is in fact its component with strict support $X$.
By the commutation of direct images with the de Rham functor <cit.> (see also <cit.>),
we have
\[\derR f_*\text{gr}^F_p\text{DR}(M')\simeq\text{gr}^F_p\text{DR}(f_*M')\simeq\bigoplus_i\text{gr}^F_p\text{DR}(\cH^i f_*M')[-i].\]
To simplify the notation, write
\[\shF:=\cH^{-n} \text{gr}^F_p\text{DR}(M'),\]
\[\shG:=\cH^{-n} \text{gr}^F_p\text{DR}(M).\]
As we are looking at the lowest non-zero cohomology of the complexes in question, an easy argument involving the spectral sequence computing $\derR f_*\text{gr}^F_p\text{DR}(M')$ then shows:
\[f_*\shF \simeq \cH^{-n} \bigoplus_i\text{gr}^F_p\text{DR}(\cH^i f_*M')[-i]=\bigoplus_{i+j=-n}\cH^i \text{gr}^F_p\text{DR}(\cH^j f_*M').\]
and therefore $\shG$ is a direct summand of $f_*\shF$. It suffices then to show that $(f_*\shF)^\vee$ is weakly positive.
Since $\shF^\vee$ is weakly positive on $X^\prime$ by Corollary <ref>, this is a consequence of the general Lemma <ref> below.[We thank the referee for suggesting this lemma and its proof; our original argument was unnecessarily complicated.]
Let $f \colon X^\prime \rightarrow X$ be a birational morphism of smooth projective varieties, and let $\shF$ be a coherent
sheaf on $X^\prime$ such that $\shF^\vee$ is weakly positive. Then $(f_* \shF)^\vee$ is weakly positive as well.
For any other sheaf $\shG$ on $X^\prime$, it is an immediate consequence of the definitions that there is a natural
homomorphism of $\shO_X$-modules
$$f_* \mathcal{H}om_{X^\prime} (\shF, \shG) \longrightarrow \mathcal{H}om_X (f_*\shF, f_*\shG).$$
If in particular we take $\shG = \shO_{X^\prime}$, due to the fact that $f$ is birational morphism of smooth varieties
we have $f_* \shO_{X^\prime} \simeq \shO_X$, and consequently this gives a homomorphism
$$\varphi \colon f_* (\shF^\vee) \longrightarrow (f_* \shF)^\vee.$$
Since $f$ is generically an isomorphism, so is $\varphi$, and so Lemma <ref>(1) says that $(f_* \shF)^\vee$
is weakly positive provided $f_* (\shF^\vee)$ is so. But this holds by Lemma <ref>(2).
We are grateful to Christian Schnell for corrections and comments on a first draft. We also thank Junecue Suh for useful discussions, and to Yohan Brunebarbe for sharing a preprint of his work. Finally, our thanks go to the referee for extremely helpful
corrections and suggestions for improvement.
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|
1511.00554
|
Color-period diagram for M 48
Barnes, Weingrill, Granzer, Spada & Strassmeier
Leibniz Institute for Astrophysics Potsdam (AIP),
An der Sternwarte 16, 14482 Potsdam, Space Science Institute,
4750 Walnut Street, Boulder, CO 80301, USA
Rotation periods are increasingly being used to derive ages for cool single field stars. Such ages are based on an empirical understanding of how cool stars spin down, acquired by constructing color-period diagrams (CPDs) for a series of open clusters.
Our main aims here are to construct a CPD for M 48, to compare this with other clusters of similar age to check for consistency, and to derive a rotational age for M 48 using gyrochronology.
We monitored M 48 photometrically for over 2 months with AIP's STELLA I 1.2 m telescope and the WiFSIP 4K imager in Tenerife. Light curves with 3 mmag precision for bright (V$\sim$14 mag) stars were produced and then analysed to provide rotation periods. A cluster CPD has then been constructed.
We report 62 rotation periods for cool stars in M 48. The CPD displays a clear slow/I-sequence of rotating stars, similar to those seen in the 625 Myr-old Hyades and 590 Myr-old Praesepe clusters, and below both, confirming that M 48 is younger. A similar comparison with the 250 Myr-old M 34 cluster shows that M 48 is older and does not possess any fast/C-sequence G or early K stars like those in M 34, although relatively fast rotators do seem to be present among the late-K and M stars. A more detailed comparison of the CPD with rotational evolution models shows that the cluster stars have a mean age of 450 Myr, and its (rotating) stars can be individually dated to $\pm 117$ Myr (26%). Much of this uncertainty stems from intrinsic astrophysical spread in initial periods, and almost all stars are consistent with a single age of $450$ Myr. The gyro-age of M 48 as a whole is 450$\pm$50 Myr, in agreement with the previously determined isochrone age of 400$\pm$100 Myr.
§ INTRODUCTION
The study of stellar rotation, both among field stars and in open clusters,
was synonymous with $v\,sin\,i$ measurements for several decades. See Kraft
(1970), and references therein, for a review of developments until 1970.
Good starting points for developments into the 1980s and 1990s, especially
those relating to open clusters, are
Stauffer & Hartmann (1987), Soderblom et al. (1993), and Queloz et al. (1998).
More recently the emphasis has shifted to photometric rotation
period measurements, partly because they avoid the $sin\,i$ ambiguity inherent
to spectroscopic measurements.
Such measurements began with the pioneering work of
Van Leeuwen, Alphenaar & Meys (1987)[See also Van Leeuwen & Alphenaar (1982).], who measured rotation periods photometrically for 11 cool stars in the Pleiades open cluster, and were soon followed by the remarkable Hyades
rotation-period measurements[No older cluster was measured successfully until Meibom et al. (2011b) studied the 1 Gyr-old cluster NGC 6811 using the Kepler Space Telescope.] of Radick et al. (1987). The Mt. Wilson sample of stars,
with periods measured from variability in chromospheric emission
(Baliunas et al. 1996) and rooted in earlier spectroscopic work, including the
discovery of stellar chromospheric activity cycles (Wilson 1978), is also
A large body of subsequent work by the astronomical community
[e.g., Bouvier et al. 1993 (T Tauri stars), Barnes et al. 1999 (IC 2602),
Irwin et al. 2007 (NGC 2516), Meibom et al. 2009 (M35), Hartman et al. 2010
has shown that rotation period measurements constitute a distinct new probe of
stellar evolution that provides both similar and new information, as compared
with `classical' methods, such as isochrone fitting of color-magnitude diagrams
(CMDs), from which it is steadily becoming independent.
Such independence is valuable because young clusters contain few (or sometimes
even no) bona fide giant members, making isochrone fitting particularly
Indeed, it is now recognized that color-period diagrams (CPDs) of open clusters
are similar to CMDs, and provide a useful complementary means of characterizing
open clusters, particularly in terms of ranking them by age
(e.g., Barnes 2003, Meibom et al. 2015).
Making the reasonable assumption that stars in both open clusters and the
field spin down in similar ways because spindown is governed by processes
internal to stars allows field star ages and those of any accompanying planets
to be derived using gyrochronology (e.g., Barnes 2007), a valuable ability in
the Kepler (and soon PLATO) era.
Another notable aspect of open cluster CPDs is that they
often display sequences of rotating stars (Barnes 2003).
Clusters like the Hyades (Radick et al. 1987; Delorme et al. 2011) and
NGC 6811 (Meibom et al. 2011b) display a clear slow or I-sequence
(consisting of relatively slowly-rotating stars whose periods increase steadily
as cool star spectral types change from F- to G- to K-type).
These stars have converged onto this sequence over the several-100 Myr ages of
these clusters.
Zero-age main sequence clusters, such as IC 2391 (Patten & Simon 1996) and
IC 2602 (Barnes et al. 1999), display only weak evidence for such a sequence.
(However, the paucity of available stars could play a role in the
difficulty of recognizing it in such cases.)
Certain young clusters, such as M 35 (Meibom et al. 2009), and the
Pleiades (Hartman et al. 2010), display evident fast/C-sequences of
rotating stars (consisting of many G-, K-, and M-type stars with rotation
periods, $P \lesssim$ 1 d),
in addition to the slow/I-sequence that characterizes older clusters.
This fast sequence appears to dissipate rapidly on a timescale that depends
on stellar mass, because these stars spin down to populate the cluster's slow
sequence instead.
This is still a subject of active research. For recent ideas, see
Matt et al. (2015), Gallet & Bouvier (2015), Brown (2014),
Epstein & Pinsonneault (2014), and references therein.
Earlier ideas concerning slow- and fast-rotating stars (with varying emphasis
on the mass dependence of rotation suggested by rotational sequences) can be
found in, e.g., MacGregor & Brenner (1991), Chaboyer et al. (1995), Collier-Cameron et al. (1995), Barnes & Sofia (1996), Bouvier et al. (1997), and Sills et al. (2000).
A significant part of the difficulty in understanding the morphological
changes that occur between the Pleiades-type ($\lesssim$200 Myr) clusters
and the Hyades-type ($\sim$600 Myr) clusters is the lack of appropriate
published rotation-period observations for clusters of intermediate age.
The only one available to date is the M 37 cluster (Hartman et al. 2009),
whose rotational age has been claimed to be younger than its 540 Myr
isochrone age.
With an isochrone age of 400 Myr (Balaguer-Nunez et al. 2005),
M 48 lies squarely in this intermediate age range and could help in
elucidating the transitional rotational behavior.
It is therefore desirable to construct a reliable CPD to characterize the
M 48 cluster and to explore its properties empirically in the context of
other well-studied clusters.
In particular, a CPD would allow a comparison between the age determined from
rotation and the age determined from classical isochrone fitting.
More generally, such studies also increase our basic knowledge
(e.g., photometry and membership) of the often unknown lower main sequence
populations of these clusters, a difficult task in the pre-CCD era.
We have studied the open cluster M 48 (NGC 2548)
in this context. An additional context is provided by the STELLA Open Cluster
Survey, which aims to provide rotation periods for a series of open clusters.
For a related study of the IC 4756 cluster, see Strassmeier et al. (2015).
Despite being a Messier (1781) object[The cluster apparently lies $2.5 \degree$ south of Messier's position, and was identified with NGC 2548 by Oswalt Thomas in 1934.],
M 48 ($\alpha_{2000}$ = 08 13 43, $\delta_{2000}$ = –05 45 00)
has not been the subject of many prior studies, a fact that may be related
to its being located in the southern sky.
Ebbighausen (1939) performed a proper motion study of the upper main
sequence (B and A spectral types) and giants of the cluster within a
$15^{\prime}$ radius, and identified 74 of these stars as probable members.
Another dedicated cluster study was not published for more than 60 years,
until Wu et al. (2002) performed another proper motion study of a
$1.6 \times 1.6 \degree$ region around the cluster and identified 165
stars as probable cluster members. This particular paper did not provide
any photometric information.
However, a related study by Balaguer-Nunez, Jordi & Galadi-Enriquez (2005;
hereafter BJG05) has revealed that the Wu et al. (2002) astrometric cluster
members were brighter than $V \approx 13$. BJG05 focused on providing
multicolor Stromgren photometry for a cluster-centered
$34^{\prime} \times 34^{\prime}$ region to a depth of $V \approx 22$, and using
this photometry they constructed a candidate member list to a depth of $V = 18$.
They also (re-)determined the cluster's basic parameters, including the age,
400${\pm}$100 Myr.
Almost contemporaneously, Rider et al. (2004) provided photometry in the
Sloan filter system for the cluster region to a depth of $g^{\prime}_0 = 16$
and found that a 400 Myr isochrone matched their photometry reasonably
Finally, the most recent work on this cluster by Wu et al. (2006)
provided 13-band photometry in a specialized (BATC) filter set over a
$58 ^{\prime} \times 58 ^{\prime}$ field, and identified 323 stars as
(SED-based) photometric candidate cluster members. An agreement level of
80% was claimed between this membership criterion and earlier
proper motion studies.
Curiously, no photometry in Johnson colors is currently available for this
cluster beyond the bright stars studied photoelectrically by Pesch (1961)
and the ($V, I$) study of Sharma et al. (2006), the latter restricted to stars
within an $8^{\prime}$ radius of the cluster center.
The study presented here builds on these prior ones with an emphasis on the
rotational properties of the cluster's stars.
The rest of this paper is organized as follows.
The observations are discussed in Section 2.
We present the cluster color-magnitude diagram in Johnson B and V
colors to a depth of V$\sim$20 in Section 3.
The variability analysis is presented in Section 4,
leading to the construction of the cluster color-period diagram,
followed by relevant comparisons.
Finally, the conclusions are presented in Section 5.
§ THE OBSERVATIONS
The M 48 open cluster was observed with the WiFSIP 4K CCD imager mounted on
the AIP's STELLA I robotic 1.2 m telescope, located at the IAC in Tenerife,
Spain (Longitude: $16\degree 30^{\prime} 35^{\prime \prime}$ West,
Latitude: $28\degree 18^{\prime} 00^{\prime \prime}$).
The STELLA robotic observatory (also containing a 1.2m spectroscopic telescope,
STELLA II) opens and closes automatically every usable night, guided by
measurements of a number of weather and meteorological parameters.
During this interval, it observes a set of targets that are chosen by a
scheduling program on the basis of user-defined priorities.
Details about the facility and its operation may be found in
Strassmeier et al. (2004), Granzer (2004), and
Strassmeier, Granzer & Weber (2010).
The cluster was observed nightly (as allowed by weather) over a two-month
baseline from 4 March 2014 to 7 May 2014.
A $44^{\prime} \times 44^{\prime}$ field, consisting of a $2 \times 2$ mosaic of
$22^{\prime}$ CCD fields (NW, NE, SW, SE) centered on the cluster, was
Our field is significantly larger than the region covered by
BJG05, as can be seen in Fig. 1.
Having this large a field was fortuitous, because cluster stars and rotators
extend across the entire region monitored. In fact, the cluster very likely
extends significantly beyond even our study field[For instance, the rotators we have identified below extend to the edges of the observed field.].
Generally, the telescope cycled sequentially through the four subfields.
The time-series exposures were acquired in the Johnson V band with
exposure times of 30 s (short) and 300 s (long).
The number of visits (either exposure time) per night per field ranged from
zero to 7, as permitted by other scheduled programs, weather, and telescope
Over the observing period, we acquired a total of 1481 short- and long-exposure
frames for the four fields. Of these, we (conservatively) discarded 992 because
of tracking errors, lunar proximity, bad seeing, or conditions deemed too far
from photometric for our purposes, leaving us with 489 high-quality visits to
the four cluster fields (average = 122/field) for the time series observations.
Small differences in the numbers of images for each sub-field arise from
variations in observing conditions, and because our observing program did not
modify individual field observing priorities after the acquisition of
incomplete nightly observing cycles.
On-sky areal coverage of our photometry (a $2 \times 2$ CCD mosaic covering a $44^{\prime} \times 44^{\prime}$ region and an additional central field) in the region of the M 48 cluster is displayed with solid gray lines (green online). The dashed lines indicate the (smaller) region covered by BJG05. Stars brighter than $V = 12$ are marked, with symbol sizes scaled inversely with their V magnitudes. The locations of the 62 identified rotating stars (discussed later) are also indicated with circular symbols.
In addition, sets of Johnson B and V exposures were obtained for each of
the cluster sub-fields on multiple nights, together with standard star
observations of Landolt (2009) standard fields.
This allowed us to obtain colors for the CPD, and of course,
to construct a CMD for the cluster in Johnson colors.
However, because the number of standard stars observed was small, we decided
to place our final photometry on the photoelectric system of Pesch (1961).
We also specifically acquired photometry for a separate field bore-sighted on
the cluster center, to verify and ensure that the $2 \times 2$ mosaic of
cluster sub-fields are placed on a common photometric system.
As part of the robotic observations, bias and flat field frames are obtained
before and after science observations. Two bracketing master bias frames, each
consisting of 25 individual frames, are used for bias subtraction on the
science frames. No dark current subtraction is performed as the dark current is
below 1e$^{-}$/h at the nominal operation temperature. With the limited time
available for twilight sky flats, it is impossible to flat field all of the
21 filters of WiFSIP on a single night. Instead, each twilight phase is used
to calibrate 2-3 filters with 10 individual flat exposures each, grouped
in two five-exposure sequences at opposite derotator settings to level-out
first order illumination gradients remaining in twilight sky flats for imaging
instruments with large fields of view (Chromey & Hasselbacher, 1996).
The master
flat field for individual science frames is then constructed by averaging at
least ten such flat field blocks, but allowing for up to 100 blocks as long as
a maximum time difference between flat block and science frame of less than
$\pm$25 days is not exceeded. The small variations in image scale across the
field of view lead to each pixel receiving light from differing solid angles,
in turn leading to differing light levels even on perfectly flat-illuminated
fields. The average master flat field is corrected for this geometrical effect
before usage on the science image.
For multi-amplifier readout modes like ours, crosstalk between the
amplifiers can lead to ghost signals on corresponding pixel positions. Although
it is a minor effect of a few tenths of a per cent, raw images are
corrected for crosstalk. Subtle gain variations between the different
amplifiers are compensated for by adjusting the amplifier gain to allow for a
steady transition in illumination level across the amplifier read-out edges.
Again, these variations are a few tenths of a per cent,
compared to the nominal gain factors of $1.568$ and $1.587$, for the two
amplifiers used; however, the precision desirable for our time-series analysis
makes such compensation advisable.
Before performing the final photometry, the astrometric world-coordinate
solution is obtained using a modified version of WCSTools (Mink 2002).
The final solution follows the FITS conventions (Calabretta & Greisen 2002)
for a zenithal polynomial projection (ZPN) to third degree, with the remaining
root-mean square (rms) error in the position on the order of $0.2$ arcsec.
§.§ Photometry
Photometry for each frame was performed with SExtractor
(Bertin & Arnouts, 1996) using the isocorr aperture.
This choice was preferable to others such as fixed aperture or
isoauto because it worked better than those choices in this work,
principally because our point spread function (PSF) is sometimes distorted by
tracking errors and/or less-than-perfect seeing conditions.
Using an adaptive aperture preserves the most frames for photometry,
with minimal impact on the photometric quality.
For each field, the frame with the highest ratio of matched to identified
sources in the PPMXL catalog
was selected as the reference frame.
This selection was also verified manually and showed that only photometric
nights were chosen.
The reference frames for all five fields (2$\times$2 mosaic + central
field) ended up being selected from only a few photometric nights
(28 Feb 2014 to 7 Mar 2014). The particular fields standardized on each
night are listed in Table 1.
Dates and numbers of secondary standard stars
Field Date Number of stars
M 48 BVI NE 2014-02-28 184
M 48 BVI NE 2014-03-01 184
M 48 BVI NW 2014-03-03 192
M 48 BVI C 2014-03-06 211
M 48 BVI SE 2014-03-05 209
M 48 BVI SE 2014-03-02 208
M 48 BVI SW 2014-03-07 205
Up to five frames having the smallest offsets with respect to the reference
frame for stars in the range between 10 to 16 magnitudes were then selected.
These offsets calculated from the mean differences between the stars in the
reference frame and the cross-identified ones in the selected frames were
typically less than 4 millimag, with an rms below 1 millimag.
The mean magnitude for each star was calculated from the five selected frames.
The nightly offset of the reference frame to the Landolt standard fields was
finally added to the derived magnitudes.
Only stars within a 0.15" matching radius were identified as cross-matched.
The whole procedure was performed separately for the B and V frames.
§ COLOR-MAGNITUDE DIAGRAM
B and V frames from the best photometric nights were manually selected for
the CMD. For these frames the successive magnitude
determinations for the best-exposed stars were repeatable at the 3 mmag level.
We decided simply to place the frames on the (photoelectric)
photometric standard system of Pesch (1961).
We matched 36 stars from Pesch (1961) listed on Simbad and Webda with our
dataset. Webda lists 37 stars from Pesch and 14 stars from Oja (1976,
priv. comm.). Since the Oja sample included fainter stars, and proved to be
of similar photometric quality, we included them in our calibration sample.
Four stars out of the initially 40 cross-identified ones were omitted for
calibration due to large differences in $V$ magnitude or $B-V$ color (TYC
4859-28-1, HD 68779, BD-05 2451, BD-05 2452). This might arise from
coordinate mismatching or bad photometry. Eventually 30 stars from Pesch and
6 stars from Oja were retained for calibration.
None of our calibration stars showed saturated pixels in the PSF.
The $V$ magnitude of the calibration stars covered a range from 8.184 mag to
14.272 mag and $0.001 < B-V < 1.441$ in color.
For these 36 stars we achieved an rms of 0.032 mag in
$V_{\rm STELLA}$-$V_{\rm Pesch/Oja}$
and 0.020 mag in $(B-V)_{\rm STELLA}$ – $(B-V)_{\rm Pesch/Oja}$.
The CMD in Johnson $B-V$ color for the entire
$44^{\prime} \times 44^{\prime}$ survey area of our study is displayed in the
upper panel of Fig. 2.
The cluster's main sequence is obvious, and is reasonably distinguishable
visually from the background stars in this direction of the Galaxy.
The cluster sequence begins somewhat brighter than $V = 10$, which is where
the A-type cluster stars are located, and extends diagonally downward to
$V \sim 19$, where the cluster's M-type stars become indistinguishable
from the field population.
A binary sequence, while undoubtedly present, is not visually prominent.
A hint of a white dwarf sequence is visible.
Our scientific interests center on the fainter (V > 13) cool FGK stars.
Photometric information for all stars and cross-identifications with BJG05 are
provided in an accompanying table[This table is only available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/
Upper Panel: Color-magnitude diagram (CMD) for the $44^{\prime} \times 44^{\prime}$ M 48 cluster region of our study in Johnson $B-V$ color. A cluster sequence, beginning brighter than $V=10$ and extending diagonally downward to $V \sim 19$, is seen clearly against the field population.
Lower panel: Large filled gray circles (green online) in the M 48 CMD indicate cluster members identified by BJG05, based on astrometry brighter than $V = 13$, and multi-color Stromgren photometry for fainter stars. Most of these BJG05 members are clearly on the cluster sequence in our photometry. Stars proposed by BJG05 to be non-members are additionally indicated with crosses, while the remaining empty circles have no BJG05 membership information.
It is desirable to understand which of these stars have been determined to be
cluster members by prior studies, and to use this information where possible.
The deepest of these studies is that by BJG05.
Beginning with the Wu et al. (2006) astrometric members, which run out at
a depth of $V \sim 13$, they extended the candidate member list downward to
$V \sim 18$ using multi-color Stromgren photometry to select cluster members.
We have cross-identified the cluster members selected by Balaguer-Nunez et
al. (2005) against our photometry, and have marked these stars in the
color-magnitude diagram displayed in the lower panel of Fig. 2.
(As an aside, we note that BJG05 defined the cluster sequence using an
empirical ZAMS constructed by Crawford (1975) and succeeding authors as
referenced in BJG05.)
We observe that the vast majority of the members identified by
BJG05 indeed lie on the cluster's sequence in our
($B, V$) photometry. It therefore appears that the BJG05 selection is an
inclusive, rather than an exclusive one.
While we are fortunate that this prior membership information from
BJG05 is available for a significant fraction of the
stars of interest in our study, our study area is somewhat larger than theirs
(see Fig. 1), so it will not be a surprise that we are able to propose
additional candidate cluster members, some of them even with determined periods.
Accordingly, a number of our rotational candidate cluster members (see Table 2
below) are classified as “–”, indicating that they do not have a BJG05
designation. However, these stars are both on the cluster's photometric
sequence in our ($B, V$) color-magnitude diagram, and have measured rotation
periods consistent with cluster membership.
(In Table 2, M = BJG05 member, and N = BJG05 non-member.)
Ultimately, the BJG05 cluster member selection and ours
are both photometric below $V = 13$, the primary region of interest for this
work, and therefore neither can be considered definitive in this region.
The relative absence of giant cluster members
makes the photometry sub-optimal for an isochrone fit.
However, it is certainly useful to compare the observed cluster main sequence
with a modern theoretical isochrone.
Accordingly, we have calculated (and display in Fig. 3) three suitable
isochrones, based on the models of Spada et al. (2013)
and of Yi et al. (2001).
CMD for M 48 with solar-metallicity isochrones based on Spada et al. (2013; YaPSI) for ages of 400-, and 500 Myr, and from Yi et al. (2001; YY) for 400 Myr. The distance modulus and reddening used are $(m-M) = 9\fm3$ and $E(B-V) = 0\fm08$ respectively.
One of the goals of Spada et al. (2013) was to update the Yale-Yonsei ($Y^2$)
database of stellar models (Yi et al., 2001), with particular attention to the
input physics relevant for the lower mass regime
(i.e., $M \lesssim 0.6\, M_\odot$; $(B-V)_0 \gtrsim 1.3$).
Most notably, the atmospheric boundary conditions are based on the PHOENIX
model atmospheres (Hauschildt et al., 1999; Allard et al., 2011).
For more details on the micro-physics used in the models, see Spada et al.
The isochrones have been calculated for solar metallicity, with ages of
400 Myr (the nominal cluster age), and 500 Myr.
These are displayed in Fig. 3, assuming a distance modulus of 9.3
($d = 725$ pc), and a reddening of $E(B-V)=0\fm08$, as determined by BJG05.
We see, unsurprisingly, that the
400 Myr and 500 Myr
isochrones are essentially identical redward of the A-type stars.
It is also evident from Fig. 3 that the newer isochrones indeed arguably
follow the cluster's main sequence somewhat better for redder colors than the
earlier one (YY; Yi et al. 2001).
§ VARIABILITY
The rotational variability and associated periods of the cluster stars provide
the principal motivation for our study. This goal requires the sustained
acquisition of high-quality imaging data over a sufficiently long time baseline.
We are particularly fortunate with respect to the baseline because ours is
$\gtrsim$2 months long.
This is four times longer than our conservative expectation of $\sim$15 d for
the longest periods in our color range for a Hyades-aged cluster, allowing
multiple phases to be detected for all rotation periods.
Because our robotic STELLA telescopes are able to acquire data during both
good and suboptimal conditions at no additional cost, it is correspondingly
necessary for us to discard the latter from among the images acquired by the
We have therefore rejected images with bad seeing, those with very elongated
PSFs (usually caused by tracking errors), those affected by
lunar proximity, and finally any frames where fewer than 80% of the stars on
our reference images were detected.
Frames with mean deviations greater than 0.2 mag from these best exposures
were also discarded.
Most of the last category consist of images acquired under non-photometric
conditions, as revealed by our weather monitors and standard star observations.
For this particular observing campaign, the final usable high-quality images
constituted $\sim$40% of the total acquired.
Fortunately, this selection does not introduce any significant gaps in the time
series beyond a lunar proximity issue centered on 11 Mar 2014, and minor
weather-related interruptions, mostly in April 2014.
(The light curve for one of our solar-mass stars, displayed in Fig. 4,
demonstrates this continuity in our observations.)
The aperture photometry from the individual exposures was corrected to a common
system defined using the best images by cross-identifying $>$1000 well-measured
stars with $10 < V < 16$ over all four fields and across all the exposures.
These stars were used as photometric references for making the frame-to-frame
corrections relative to the best exposures, allowing the construction of
individual light curves, which were then constructed for all cross-identified
stars in the field of view of the cluster.
We decided to concentrate our efforts on candidate photometric cluster members.
Consequently, we then extracted the light curves of $\sim$1300 stars along the
cluster sequence for careful analysis.
This is a superset of the cluster members of BJG05 with $B-V > 0.40$, which
is the color range of interest in this work.
Upper panel: Light curve for star No. 485,
a solar-mass star with $(B-V)_0 = 0.664$ in our sample. A periodicity of $6-7$ d is evident from inspection of the light curve. Successive data points show that the photometry for this particular star is clearly repeatable at a level better than 0.01 mag. Our other solar-mass rotators have smaller amplitudes of variability.
Lower panel: Power spectrum for star No. 485, showing an unambiguous periodicity of the signal at $P=6.53d$, identified as the rotation period of the star.
All light curves were subjected to multiple methods of frequency
analysis to identify periodicity –
phase dispersion minimization (PDM; Stellingwerf, 1978),
cleaned Fourier analysis (Roberts et al. 1987),
and generalized Lomb-Scargle periodogram (Zechmeister & Kurster, 2011).
While none of these is completely satisfactory in all cases, we have found that
for our dataset the clean algorithm appears to be the best one overall.
Lomb-Scargle and PDM are affected more often by the 1 d or multiple period
alias. We believe that this behaviour partly comes from our long baseline,
during which there is sometimes significant spot evolution, the onset/decline
of spot activity, and in certain cases multiple spot groups.
clean seems to be less sensitive than the other methods to these
problems, piles up power at the rotational frequency,
and also allows the window function to be taken into account.
However, there seems to be no substitute for manual inspection of the
candidate rotators, taking the raw light curve into account,
the results from all three periodicity indicators discussed above and,
finally, the phased curves.
We found 5$\sigma$ of the noise level in the clean spectra to be a
good threshold for accepting periodicity and have generally adopted this as
the criterion for a high-quality period (Quality flag = 1).
However, it cannot be an automatically adopted criterion, so we made the
ultimate decision manually, and on this basis also listed
certain lower quality periods (Quality flag = 2), where
there might be some chance of the listed period being an alias.
There are therefore some stars with higher peaks that are assigned
Quality flag = 2, while the rest are considered high quality.
On this basis, we have presented 62 rotation periods, of which 8 are listed
with Quality flag = 2. Experiments show that even our Q = 2 periods
have a Scargle false alarm probability, FAP < 0.01, i.e., a confidence level
greater than 99%.
The light curve for a good example of a solar-mass periodic variable is
displayed in the upper panel of Fig. 4.
The locations of successive data points in the light curve show that the
photometry for this particular star is clearly repeatable to a level better
than 0.01 mag.
A periodicity of $6-7$ d is obvious in Fig. 4.
Thanks to the density of our observations, simply plotting the unphased
light curve usually allows us to verify the approximate periodicity by
visual inspection, as the upper panel of Fig. 4 shows.
The corresponding Fourier power spectrum for this star, constructed using
the clean algorithm of Roberts et al. (1987) is displayed in the lower
panel of Fig. 4.
The peak is at $6.53$ d, which is consequently listed as the rotation period
of the star.
(Light curves, power spectra, and phased light curves for all the
periodic candidate cluster rotational variables identified in this study are
displayed in the online Appendix to this paper in Figs. A1-A9.[The online appendix is available at http://www.edpsciences.org.])
We note that few stars actually need to be phased before their periodicity
becomes obvious[Our time baseline is long enough for significant spot evolution. Correspondingly, our phased light curves would look significantly better “cosmetically” had we suppressed epochs of low stellar variability. They would also look better had we rephased the light curves to the PDM values, within the error envelope of the clean values.].
The periods derived range from $1.67$ d to $13.3$ d, the latter well within
the sensitivity of our observational window of $\sim$ 2 months.
These 62 periods are listed in Table 2, along with other relevant
information for each star.
This includes the rotation period, its error
(the HWHM of the clean peak), the variability amplitude (from sinusoidal
fits to the phased light curves), cross-identification with BJG05, the relevant
membership indicator, and relevant notes. The periodic stars considered to be
cluster members are indicated in the CMD displayed in Fig. 5.
Color-magnitude diagram for M 48 with the 62 periodic cluster rotational variables highlighted. Solid symbols indicate the 54 high-quality (Q = 1) periods, and unfilled symbols the 8 lower quality (Q = 2) ones. All the periodic cluster member candidates are located on, or relatively near, the cluster sequence. A 400 Myr YaPSI isochrone and the corresponding equal-mass binary sequence are also displayed.
§.§ Empirical comparisons of the color-period diagram
The rotation periods derived by the methods described above have then been
associated with the photometry discussed earlier in the paper, and the
corresponding CPD has been constructed for the cluster stars,
as displayed in Fig. 6.
This diagram is populated by 62 stars with $(B-V)_0$ color range
0.47 mag–1.47 mag, and the period range 1.67 d–13.1 d.
The field contamination from the background, as seen in the CMD in Fig. 2
suggests that only a couple of these could possibly be non-members.
The measured color-period diagram (CPD) for M 48, showing the 62 periodic stars believed to be cluster rotators, and whose positions in the CMD are displayed in Fig. 5. We observe a relatively distinct sequence of stars, reminiscent of the Hyades sequence, blueward of $B-V = 1$, and a more scattered distribution of stars redward of this color. The solar-mass stars show no evidence of the C/fast sequence that characterizes ZAMS clusters.
The bluer (warmer) half of the M 48 CPD displays a clear
sequence of stars ranging from short-period $\sim 2$ d stars at
$(B-V)_0 \sim 0\fm45$ to $\sim 8$ d periods at $(B-V)_0 \sim 0\fm9$.
This is reminiscent of the situation in the 625 Myr-old Hyades cluster, as
discussed below.
There is no evidence of the C/fast sequence of stars with $P \sim$1 d across
the entire $ 0.5 < (B-V)_0 < 1.5$ color range, characteristically seen in
younger open clusters such as the Pleiades (Hartman et al. 2010) or M 35
(Meibom et al. 2009).
This morphology immediately tells us that M 48 is older than M 35
and the Pleiades, without even having to perform a detailed comparison.
Top panel: Comparison of the M 48 CPD (circles, red online) with that of M 34 (250 Myr old; pentagons, dark blue online). We see that the M 48 sequence is clearly above that of M 34, indicating that M 48 is older. Another indicator of M 48's older age is it's lack of fast-rotating stars,which form a sparsely populated sequence in M 34 (with $P \sim$ 1 d and $ 0.6 < (B-V)_0 < 1$).
Middle panel: Comparison of the M 48 CPD with that of the Hyades cluster (triangles, gray online). While the rotation periods of similar-mass stars are comparable between the two clusters, indicating that their ages are roughly comparable, the average solar-type M 48 star (see text) is 0.99 d below that for the 625 Myr-old Hyades, indicating that M 48 is somewhat younger.
Bottom panel: Comparison of the M 48 CPD with that of the 590 Myr-old Praesepe cluster (diamonds, green online), confirming that M 48 is again younger. The average solar-type M 48 star (see text) is closer (0.44 d), consonant with Praesepe's age being slightly lower than that of the Hyades.
A more detailed empirical idea of how the M 48 cluster fits in with other
cluster observations can be obtained by comparing the constructed M 48 CPD
directly with other open cluster CPDs.
Our first empirical comparison is made with the open cluster M 34, as
displayed in the top panel of Fig. 7.
The filled and unfilled symbols for M 34 represent the rotation periods
determined by Meibom et al. (2011a) and James et al. (2010).
Following the original publications, the M 34 data have been de-reddened by
$E(B-V) = 0\fm07$ (Canterna et al. 1979).
We see that M 48 does not possess the fast G- and K-type rotators
blueward of $(B-V)_0 = 1$ that collectively form an (admittedly ill-defined)
fast/C sequence in M 34.
The M 48 slow/I sequence is clearly above that of M 34,
indicating that M 48 is older than 250 Myr, the nominal age of M 34
(Ianna & Schlemmer, 1993).
The I sequences of the two clusters almost overlap at $(B-V)_0=0.9$,
suggesting that it is already time to move beyond models where the dependence
of rotation period, $P$, on age and mass is separable
[e.g., Barnes (2003), Barnes (2007), Mamajek & Hillenbrand (2008),
Meibom et al. (2011a), Angus et al. (2015)].
The corresponding comparison for the Hyades open cluster is displayed in the
middle panel of Fig. 7, using the rotation period measurements
(filled symbols) of Radick et al. (1987), and the more recent measurements
(unfilled symbols) of Delorme et al. (2011). Taylor (2006) has determined that
the Hyades reddening is $E(B-V) \le 0\fm01$ mag, and consequently the $B-V$
values from neither of these studies have been dereddened.
We see that the rotation periods of similar-mass stars in M 48 and in the
Hyades are comparable, indicating that the ages of the two clusters are
roughly comparable. (In their comprehensive study of the Hyades,
Perryman et al. (1998) derived an age of 625 Myr.)
Closer inspection shows that the average M 48 rotational sequence in the CPD
is clearly below the average of the Hyades sequence, telling us that M 48 is
somewhat younger. The average difference between the two sequences in the
well-sampled $0.5 < (B-V)_0 < 0.9$ interval is 0.99 d.
An empirical confirmation of this result is available using the rotation period
determinations for the Praesepe cluster by Delorme et al. (2011) and Kovacs et
al. (2014). These are plotted in the bottom panel of Fig. 7 using filled and
unfilled symbols respectively.
In accordance with the determination of Taylor (2006), the $B-V$ color values
for Praesepe have been de-reddened by $E(B-V) = 0\fm027$.
Praesepe is known to be of very similar age ($590$ Myr; Fossati et al. 2008)
to the Hyades.
Our comparison of the M 48 and Praesepe CPDs is consonant with this result.
In fact, after comparing the Praesepe and Hyades CPDs,
Delorme et al. (2011) concluded that Praesepe is slightly younger than the
Hyades, perhaps by about 50 Myr.
Our comparison in Fig. 7 seems to confirm this conclusion in that the
Praesepe sequence is indeed slightly closer to the M 48 sequence
than that of the Hyades. The average difference between the two sequences in
the $0.5 < (B-V)_0 < 0.9$ interval is 0.44 d, smaller than the Hyades-M48
difference of 0.99 d. These empirical comparisons tell us that the
rotational age of M 48 is almost certainly in the (250, 590) Myr interval.
§.§ Comparison against models with separable (t, M) dependencies
These new M 48 rotation periods also permit a useful comparison with prior
predictions for the locations of stars of its age in the CPD.
Fig. 8 shows how these data compare with a number of empirical studies that
relied on the rotation period $P$ having separable dependencies on age, $t$,
and stellar mass $M$ (or a suitable mass proxy such as color) –
the original gyrochronology formulation of Barnes (2003), the update in
Barnes (2007), the modification to that formulation proposed by Mamajek &
Hillenbrand (2008), one based largely on the M 34 cluster data of
Meibom et al. (2011a), and the recent one of Angus et al. (2015), the last
based on a very limited cluster dataset, and a small number of field stars
with ages determined from asteroseismology.
The displayed comparisons all use the prior published age for M 48 of 400 Myr.
Moderate changes to the age used make moderate changes to the displayed curves,
but do not change the overall behavior.
Comparison of the M 48 CPD with rotational isochrones for models with separable $(t, M)$ dependencies, constructed for an age of 400 Myr. The M 48 data points have been de-reddened by $0\fm08$. These models do not satisfactorily capture the detailed morphology of the observations (see text for details).
We observe that all the models displayed here capture, to a certain extent,
the overall behavior of the M 48 data blueward of $(B-V)_0 \sim 1$.
However, there is a potentially serious problem redward of this color value.
The measured rotation periods seem to decline for redder (lower-mass) stars.
To a certain extent, this behavior is not unexpected, because these empirical
formulations of gyrochronology explicitly ignored the fast rotators (with the
exception of Barnes (2003), which treated them separately as shown in Fig. 8),
electing to concentrate attention on the slow/I sequence stars, which are the
only ones seen in M 48 blueward of $(B-V)_0 = 1$.
There are also certain differences, obvious in Fig. 8, with respect to the
point where these models intersect the color axis, and with the color range
over which each of these provides a good fit (or not) to the periods.
(We will show a detailed comparison with a subsequent preferred model below.)
Finally, it should be noted that the separable empirical models have slightly
differing dependencies on the age, a fact that is imperceptible in such a
comparison, being overwhelmed by the mass dependence.
We have not been able to identify any red stars (say $(B-V)_0 \gtrsim 1$)
that are very slow rotators ($P >$10 d). While our observations do possess
the time baseline to identify such stars, any such stars are fainter than
$V = 16.5$ (see Fig. 5), where the precision of our photometry declines,
affecting our sensitivity to such periods if their variability amplitudes are
small. Our data therefore do not allow us to state conclusively
whether such stars actually exist or not in M 48.
§.§ Comparison with the B2010 model
Our preferred way of interpreting these M 48 rotation period data is with the
gyrochronology formulation of Barnes (2010), hereafter B10, based on the
rotational evolution ideas described in Barnes & Kim (2010).
A significant part of this preference arises from the simplicity of this model,
where only two dimensionless constants, $k_C$ and $k_I$, are required to
specify the two relevant spindown timescales for all fast and slow cool stars,
and hence of the entire main sequence rotational evolution of cool stars.
The B10 formulation describes the behaviors of both the slow/I-type stars and
the fast/C-type stars in a single model, treating them symmetrically[Brown (2014) has called this the `Symmetric Empirical Model.'].
Furthermore, the rotational isochrones constructed using this model are
better than those from other models in matching the measured color-period
diagram of the 2.5 Gyr-old cluster NGC 6819, studied by Meibom et al. (2015).
And we show below that this model describes the morphology of the M 48
CPD in considerable detail.
This B10 model provides the age of an individual star as an explicit function
of its rotation period (and a mass variable), albeit with more complexity than
the separable empirical models discussed above. One can then take a suitable
average over the cluster stars to derive the corresponding cluster age.
The procedure first requires the calculation of the (global) convective
turnover timescale, $\tau$, for each star from its de-reddened $B-V$ color.
This is simply accomplished by interpolation from Table 1 in Barnes & Kim
(2010). Then one applies equation (32) from B10,
\begin{equation}
t = \frac{\tau}{k_C} {\rm ln}\frac{P}{P_0} + \frac{k_I}{2 \tau} \left( P^2 - P_0^2\right).
\end{equation}
This is an explicit expression for the age $t$ of the star, in terms of its
measured rotation period $P$, and $\tau$, the latter a proxy for its mass or
color. Here, $k_C = 0.646$ Myr/d and $k_I = 452$ d/Myr are two dimensionless
constants whose values are decided by the totality of the open cluster rotation
period data, and the solar datum; we have simply adopted unchanged the values
proposed in B10, particularly in view of the model's ability to reproduce well
the CPD of the 2.5 Gyr-old cluster NGC 6819, as discussed in Meibom et al.
$P_0$ is the initial rotation period, set to $P_0 = 1.1$ d, following
B10, but we will also allow it to vary within certain bounds below,
as allowed by observations, and as described in B10.
(It is also possible to use other turnover timescales, but then one must be
careful to recalibrate the constants $k_C$ and $k_I$ to match the totality
of the open cluster and solar rotational data, as discussed in B10.)
Histograms of gyro-age for the individual M48 stars, calculated using the B10 rotational model and assuming an initial period of 1.1 d. The gray histogram shows all 62 stars, while the black histogram is for a sample restricted to the 46 stars in the particularly well-measured $0.55 < (B-V)_0 < 1.00$ region. The larger sample gives a median age of 451 Myr (mean = 441 Myr, S.D. = 117 Myr = 26%), and the smaller one gives similar values, with a median age of 457 Myr (mean = 447 Myr, S.D. = 70 Myr = 16%.) This large age dispersion mostly arises from having to assume a single initial period, $P_0$ (= 1.1 d) for all stars.
Provided that the measured period is greater than $P_0$, each measured
rotation period $P_i$ results in a corresponding gyro-age $t_i$, which is
plotted in the histograms in Fig. 9 for $P_0 = 1.1$ d.
The gray histogram shows the distribution for all 62 stars with
measured rotation periods.
We obtain a distribution that is sharply peaked at a
median value of 451 Myr, and mean value of 441 Myr
but with relatively wide wings, giving a standard deviation of 117 Myr
(= 26%).
A significant portion of this relatively large dispersion can be traced to a
number of outliers. For instance, the outlier at $t_i = 840$ Myr is No.1456,
the one above all the other stars in the CPD, with $(P, (B-V)_0) = (13.3$ d,
$1.054$). Because it lies on the cluster sequence in the CMD, there is no
good reason to discard it at the present time.
The other outliers are mostly either very blue or very red stars (defined for
this study as lying outside the $0.55 < (B-V)_0 < 1.00$ color range.)
The black histogram in Fig. 9 shows the distribution for the 46
stars in this restricted $0.55 < (B-V)_0 < 1.00$ color range, where the
I sequence is particularly well-defined.
This color region contains the Sun and many well-studied open clusters, and is
known from the rotation period observations for NGC 6819 ($2.5$ Gyr) by
Meibom et al. (2015) to be well-calibrated.
In the corresponding histogram, many of the outliers disappear to reveal a
distribution where $38/46 = 83$% of the stars are confined to the
$t = [350,550]$ Myr region.
An M 48 cluster gyro-age outside this range is essentially ruled out.
The formal mean age for this restricted distribution is
$\overline{t} = 447$ Myr (median = 457 Myr),
but with a smaller standard deviation $\sigma_t =$ 70 Myr (= 16%).
These results enable us to confirm the $400\,\pm100$Myr isochrone age of this
cluster, and to propose a mean rotational (gyrochronology) age of $450$ Myr
for M 48.
Theoretical rotational isochrones for $450$ Myr, constructed following Barnes (2010), are compared with the M 48 CPD. The three curves from top to bottom respectively correspond to initial periods of $3.4$ d, $1.1$ d, and $0.12$ d, representing the range allowed by ZAMS cluster observations. Almost all the data points are consistent with a single rotational age of $450$ Myr.
We now show that much of the above scatter arises from intrinsic astrophysical
variations in the initial rotation periods, so that it is likely that the
uncertainty on the gyro-age of M 48 is actually significantly smaller than
the 70 Myr standard deviation for the best-measured stars.
For open clusters that are significantly younger than the Hyades, it is
well-known, and expected, that the rotation periods of cool stars will not
necessarily have converged to a single sequence.
Indeed, in ZAMS clusters, two distinct sequences are sometimes observed,
as discussed earlier, and as is visible in the CPD for M 34
(see Fig. 7).
In the M 48 CPD, stars with masses greater than $0.9 M_{\odot}$ ($B-V_0 < 1.0$)
have essentially converged to a single sequence, while lower mass stars most
certainly have not.
There is a simple way to understand the dispersion in the age distribution
derived above.
Given an age $t$, and using a given initial period $P_0$, one can solve
equation (1) numerically[For a given $P, P_0$, and age $t$, one can also obtain a solution for $\tau$ analytically, as shown in Barnes (2010), by simply solving a quadratic equation.]
to associate a value $P$ to every $\tau$ value of interest, or equivalently the
value of $(B-V)_0$.
Such a curve is the isochrone for that initial period, since it is the
locus of all such equal-age points.
The 450 Myr isochrone for $P_0 = 1.1$ d is displayed in Fig. 10 with
the thick central line (solid green).
Barnes (2010) found that the lower- and upper envelopes of the initial period
distribution could be reasonably set at 0.12 d and 3.4 d respectively.
Carrying out the corresponding calculations for $P_0 = 0.12$ d and
$P_0 = 3.4$ d with the same age of 450 Myr yields the lower (dotted
green) and upper lines (dashed green) respectively.
(Other intermediate initial period values in this range would provide curves
that are bounded by these.)
This collection of curves can be viewed as representing the total rotational
isochrone for M 48, providing a narrow sequence at the blue/solar-like end,
and a wider sequence for lower mass stars.
We observe that this range of initial conditions explains almost all the
scatter in the rotation period measurements as a function of mass, because
only one of our 62 rotation periods lies significantly outside
the range of these isochrones.
Consequently, with $P_0$ allowed to vary within the astrophysical limits
permitted by ZAMS observations,
almost all our stars are consistent with a single age of 450 Myr.
This fact is consistent with the current belief that open clusters are
simple stellar populations, and that they are describable as single-age
We now ask how well we know the gyro-age of the cluster as a whole.
The histogram of ages (displayed in Fig. 9) shows that moving $\pm 50$ Myr
off the mean age halves the bin occupancies.
Bins beyond these ages are occupied by stars for which 1.1 d is not a good
estimate for the initial period, as Fig. 10 shows.
We therefore construct 400 Myr and 500 Myr rotational isochrones, and
display them in Fig. 11 (dotted blue and dashed red, respectively),
in an analogy with classical isochrones.
Neither of these can be considered a reasonable fit to the M 48 CPD.
In particular, the 400 Myr and 500 Myr isochrones for $P_0 = 1.1$ d
pass below and above the core of the rotation period distribution with
$0.6 < (B-V)_0 < 0.9$.
The $P_0 = 3.4$ d isochrones are not particularly informative.
However, the $P_0 = 0.12$ d isochrones for $400-$ and $500$ Myr also pass
respectively below and above the group of the fastest cluster rotators at
near-Solar color (and maybe even redder values), making them significantly
worse fits to the fast edge of the rotation period distribution.
We therefore consider 50 Myr to be a reasonable estimate of the uncertainty
on the gyro-age of M 48, which we suggest is $450\pm50$ Myr.
CPD for M 48 compared with younger ($400$ Myr; dotted, blue online) and older ($500$ Myr; dashed, red online) rotational isochrones. Neither the $400$ Myr nor the $500$ Myr isochrone provide a good match to the data, passing significantly below or above the core group of stars with $0.6 < (B-V)_0 < 0.9$. The cluster's rotational age is clearly within this (400, 500) Myr interval. (For clarity, the (central) $P_0 = 1.1$ d isochrones for the three separate ages are emphasized.)
Another approach is the following.
Ignoring their individual membership of M 48 and treating all the 62 measured
stars independently gives the histogram displayed in Fig. 9, with its standard
deviation of 117 Myr.
Clearly, the uncertainty on the mean age of the cluster
must be much lower than the uncertainty of an individual star.
The standard error (S.E.) of the mean is formally
$\sigma_t/\sqrt{\rm Number\,of\,stars} = 117/\sqrt{62} \approx 15$ Myr.
But taking this to represent the error in cluster age seems unjustified, in
view of the possibility of residual systematic age errors arising from our
incomplete understanding of rotational stellar evolution.
To estimate possible systematic errors, we constructed equivalent age
distributions for the comparison clusters displayed in Fig. 7, again using
the $ 0.55 < (B-V)_0 < 1.0 $ interval.
These give mean ages of 260, 522, and 584 Myr for M 34, Praesepe, and the
Hyades, respectively, which suggest that if the corresponding isochrone ages
of 250, 590, and 625 Myr, are taken as absolute truth, then systematic errors
in the B10 version of gyrochronology could contribute at the level of
$\sim 40$ Myr to the M 48 age uncertainty. Then adding, in quadrature, the
$15$ Myr internal error of the mean for M 48 gives $44$ Myr.
This suggests that the $\pm 50$ Myr value quoted above is not unreasonable.
In closing, it is worth noting that an analysis like the one above might not
be as easily accomplished for a much younger cluster, where the lower-mass
stars are still on the pre-main sequence (and the demands on rotational
evolution models are more severe). Indeed Cargile et al. (2014) appear to have
experienced some difficulties in adapting the Barnes (2010) formulation for
the $\sim 150$ Myr-old Blanco 1 open cluster, and resorted to the separable
$(t, m)$ formulations for $P = P (t,m)$ to interpret that cluster's
§ CONCLUSIONS
We have performed a two-month-long photometric time series campaign on the
southern open cluster M 48 (NGC 2548) using the AIP's STELLA I robotic
imaging telescope and associated WiFSIP 4K imager, located in Tenerife.
We also acquired photometry in the Johnson B and V bands to a depth of
$V \sim 20$ for the $44^{\prime} \times 44^{\prime}$ region centered on the
A relatively clear cluster sequence is visible in our photometry and largely
coincides with the astrometric and photometric candidate members identified by
prior work on the cluster.
This sequence in the CMD is followed closely by a theoretical isochrone
and is also closely matched by the rotational variables.
We constructed light curves populated with $\sim$120 data points each
and with no serious data gaps over the two-month observing baseline. Our
time-series photometry is repeatable at the 3 mmag level for F-G stars,
with the uncertainty increasing as expected for fainter stars.
We successfully derived rotation periods for 62 cool photometric cluster member
stars, 54 of which are classified as higher-quality, and 8 are lower-quality.
These rotation periods and the associated colors of the stars define the M 48
cluster's CPD, the rotational equivalent of the CMD.
In the CPD, these periods delineate a clear sequence blueward
of $(B-V)_0 = 1.0$; redward of this point, the rotation period distribution has
a significantly larger scatter that is likely astrophysical.
While comparable with the rotational sequences in the 625 Myr-old
Hyades open cluster and the 590 Myr-old Praesepe cluster, this sequence
lies below both, empirically demonstrating that M 48 is younger.
Likewise, direct comparison with the CPD for the 250 Myr-old
M 34 open cluster shows that M 48 is older.
We constructed the distribution of gyro-ages for the cluster stars,
finding one that is sharply peaked at $450$ Myr, but with relatively wide
wings, giving a standard deviation of $117$ Myr (= 26%), and suggesting
that this
precision for comparable field star rotators is reasonable and attainable.
We showed that the known (astrophysical) contribution from initial period
variations on the ZAMS accounts for much of the scatter in the rotational
ages, so that almost all the measured rotation periods are actually
consistent with a single cluster age of $450$ Myr.
For the cluster as a whole, the age uncertainty is about 50 Myr.
We therefore propose a mean cluster age from gyrochronology for M 48 of
$450 \pm 50$ Myr.
In sum, this study has added another open cluster of
intermediate age to the canonical literature.
This work was supported by the German
Deutsche Forschungsgemeinschaft, DFG project
number DFG STR645/7-1.
This work is based on data obtained with the STELLA robotic telescopes
in Tenerife, an AIP facility jointly operated by AIP and IAC.
Detailed comments from an anonymous referee are also appreciated.
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Rotation periods of the M 48 stars.
Id V B–V P Perr amp mem Q BJG # Notes
mag mag days days mag
Id V B–V P Perr amp mem Q BJG # Notes
mag mag days days mag
284 13.577 0.56 3.25 0.08 0.013 – 1 Variable in 2nd half
287 13.597 0.56 2.86 0.10 0.018 M 1 BJG 3157
303 13.726 0.55 1.80 0.02 0.007 M 2 BJG 841 Low Amplitude
332 13.951 0.62 6.12 0.41 0.037 – 1
336 13.970 0.69 7.72 0.56 0.030 – 1
343 14.034 0.62 4.88 0.22 0.015 M 1 BJG 3584
363 14.146 0.66 6.06 0.42 0.017 M 2 BJG 2049 2 spot groups, PDM period
368 14.172 0.63 6.24 0.35 0.022 – 1
386 14.255 0.65 6.42 0.35 0.014 M 1 BJG 2109
407 14.395 0.64 5.33 0.26 0.020 M 1 BJG 3680
418 14.439 0.69 5.85 0.36 0.019 – 1
421 14.440 0.75 8.82 0.64 0.012 M 1 BJG 2037
425 14.459 0.69 6.28 0.44 0.009 M 1 BJG 1614
437 14.525 0.72 7.25 0.72 0.019 M 1 BJG 3583
467 14.640 0.71 7.45 0.57 0.014 – 1
482 14.671 0.71 5.95 0.32 0.029 M 1 BJG 2975
485 14.683 0.74 6.53 0.42 0.028 M 1 BJG 4781
501 14.762 0.75 7.50 0.52 0.018 M 1 BJG 521
507 14.781 0.77 6.21 0.36 0.018 – 1
511 14.792 0.74 6.00 0.23 0.020 M 1 BJG 3846
517 14.812 0.76 7.83 1.73 0.011 M 2 BJG 3927 P$\sim 7$ d possible, DR?
540 14.904 0.83 8.57 0.72 0.012 M 1 BJG 2600
555 14.955 0.76 6.14 0.37 0.023 M 1 BJG 1608
556 14.956 0.75 7.19 0.38 0.018 M 1 BJG 4335
602 15.098 0.81 7.35 0.44 0.032 M 1 BJG 3935
620 15.172 0.82 7.41 0.57 0.026 M 1 BJG 3168
633 15.201 0.81 7.90 0.80 0.019 M 1 BJG 1641
649 15.264 0.84 8.31 0.57 0.012 M 1 BJG 1917
652 15.266 0.79 7.47 0.58 0.016 M 1 BJG 3964
657 15.279 0.85 7.59 0.48 0.036 – 1
699 15.365 0.85 7.60 0.79 0.029 M 1 BJG 2997
713 15.399 0.86 7.96 0.64 0.032 M 1 BJG 2704
752 15.496 0.85 8.23 0.51 0.014 M 2 BJG 3852 PDM breaks ambiguity
772 15.552 0.86 8.75 0.79 0.011 M 2 BJG 3597 LS ambiguous
796 15.609 0.91 8.63 0.61 0.022 – 1
807 15.637 0.99 8.92 0.63 0.018 M 1 BJG 915 2 spot groups
862 15.771 0.96 8.75 0.72 0.040 – 1
864 15.771 0.93 7.82 0.54 0.065 – 1
872 15.782 0.96 7.52 0.45 0.049 – 1
898 15.826 0.90 7.36 0.54 0.036 M 1 BJG 1828
909 15.838 0.97 8.03 0.55 0.019 – 1
920 15.866 0.94 8.65 0.67 0.017 – 1
921 15.868 0.98 7.86 0.47 0.022 – 1
923 15.869 0.95 8.87 0.69 0.022 – 1
931 15.889 0.98 8.75 0.68 0.025 – 1
935 15.897 0.96 9.24 0.75 0.037 – 1
937 15.902 0.96 7.52 0.50 0.035 M 1 BJG 2546
954 15.922 0.97 9.70 0.79 0.013 M 2 BJG 1703 Low Amplitude
969 15.956 0.98 9.55 0.90 0.021 N 1 BJG 2435
974 15.966 0.98 9.27 0.68 0.016 M 1 BJG 2033
975 15.967 1.00 9.58 0.62 0.028 M 1 BJG 578
1227 16.409 1.08 8.83 0.72 0.064 M 1 BJG 1362
1455 16.730 1.17 4.81 0.24 0.021 M 2 BJG 2505 Noisy, PDM picks $\sim 9.6$ d
1456 16.730 1.13 13.31 1.58 0.026 M 1 BJG 4598
1605 16.946 1.22 5.81 0.30 0.083 M 1 BJG 4556
1711 17.091 1.22 10.22 0.84 0.063 M 1 BJG 3794
1744 17.130 1.23 6.79 0.44 0.026 – 2 Noisy, PDM picks $\sim 13.3$ d
2010 17.483 1.25 5.82 0.26 0.056 – 1
2071 17.568 1.35 7.30 0.57 0.088 M 1 BJG 2439
2285 17.805 1.35 3.26 0.10 0.075 N 1 BJG 2458
2346 17.852 1.46 9.68 0.72 0.073 M 1 BJG 771
2632 18.161 1.55 1.67 0.03 0.117 M 1 BJG 3428
Mem = M, N, – respectively indicate BJG05 members, non-members, and stars without BJG05 membership information.
Q = 1, 2 respectively indicate high quality and lower quality periods.
§ ONLINE APPENDIX
Online lightcurves 1 (Q = 2 periods in parentheses, x-units = 10 d, y-units = 0.01 mag)
Online lightcurves 2 (Q = 2 periods in parentheses, x-units = 10 d, y-units = 0.01 mag)
Online lightcurves 3 (Q = 2 periods in parentheses, x-units = 10 d, y-units = 0.01 mag)
Online spectra 1 (Q = 2 periods in parentheses, x-units = 1 d, y-units arbitrary, selected rotation period marked with green line, 5 $\sigma$ level marked with blue line
Online spectra 2 (Q = 2 periods in parentheses, x-units = 1 d, y-units arbitrary, selected rotation period marked with green line, 5 $\sigma$ level marked with blue line)
Online spectra 3 (Q = 2 periods in parentheses, x-units = 1 d, y-units arbitrary, selected rotation period marked with green line, 5 $\sigma$ level marked with blue line)
Online phased curves 1 (Q = 2 periods in parentheses, x-units = 1 d, y-units = 0.01 mag), with two phases displayed for each star
Online phased curves 2 (Q = 2 periods in parentheses, x-units = 1 d, y-units = 0.01 mag), with two phases displayed for each star
Online phased curves 3 (Q = 2 periods in parentheses, x-units = 1 d, y-units = 0.01 mag), with two phases displayed for each star
[]Opacity sources.
\begin{array}{p{0.5\linewidth}l}
\hline
\noalign{\smallskip}
Source & T / {[\mathrm{K}]} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Yorke 1979, Yorke 1980a & \leq 1700^{\mathrm{a}} \\
% Yorke 1979, Yorke 1980a & \leq 1700 \\
Kr\"ugel 1971 & 1700 \leq T \leq 5000 \\
Cox \& Stewart 1969 & 5000 \leq \\
\noalign{\smallskip}
\hline
\end{array}
We will now write down the sign (and therefore stability)
determining parts of the left-hand sides of the inequalities
(<ref>), (<ref>) and (<ref>) and thereby
obtain stability equations of state.
The sign determining part of inequality (<ref>) is
$3\Gamma_1 - 4$ and it reduces to the
criterion for dynamical stability
\begin{equation}
\Gamma_1 > \frac{4}{3}\,\cdot
\end{equation}
Stability of the thermodynamical equilibrium demands
\begin{equation}
\chi^{}_\rho > 0, \;\; c_v > 0\, ,
\end{equation}
\begin{equation}
\chi^{}_T > 0
\end{equation}
holds for a wide range of physical situations.
\begin{eqnarray}
\Gamma_3 - 1 = \frac{P}{\rho T} \frac{\chi^{}_T}{c_v}&>&0\\
\Gamma_1 = \chi_\rho^{} + \chi_T^{} (\Gamma_3 -1)&>&0\\
\nabla_{\mathrm{ad}} = \frac{\Gamma_3 - 1}{\Gamma_1} &>&0
\end{eqnarray}
we find the sign determining terms in inequalities (<ref>)
and (<ref>) respectively and obtain the following form
of the criteria for dynamical, secular and vibrational
stability, respectively:
\begin{eqnarray}
3 \Gamma_1 - 4 =: S_{\mathrm{dyn}} > & 0 & \label{DynSta} \\
\frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T } ( \kappa^{}_T - 4 )
+ \kappa^{}_P + 1 =: S_{\mathrm{sec}} > & 0 & \label{SecSta} \\
4 \nabla_{\mathrm{ad}} - (\nabla_{\mathrm{ad}} \kappa^{}_T
+ \kappa^{}_P)
- \frac{4}{3 \Gamma_1} =: S_{\mathrm{vib}}
> & 0\,.& \label{VibSta}
\end{eqnarray}
The constitutive relations are to be evaluated for the
unperturbed thermodynamic state (say $(\rho_0, T_0)$) of the zone.
We see that the one-zone stability of the layer depends only on
the constitutive relations $\Gamma_1$,
$\nabla_{\mathrm{ad}}$, $\chi_T^{},\,\chi_\rho^{}$,
These depend only on the unperturbed
thermodynamical state of the layer. Therefore the above relations
define the one-zone-stability equations of state
and $S_{\mathrm{vib}}$. See Fig. <ref> for a picture of
$S_{\mathrm{vib}}$. Regions of secular instability are
listed in Table 1.
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
§ CONCLUSIONS
* The conditions for the stability of static, radiative
layers in gas spheres, as described by Baker's (<cit.>)
standard one-zone model, can be expressed as stability
equations of state. These stability equations of state depend
only on the local thermodynamic state of the layer.
* If the constitutive relations – equations of state and
Rosseland mean opacities – are specified, the stability
equations of state can be evaluated without specifying
properties of the layer.
* For solar composition gas the $\kappa$-mechanism is
working in the regions of the ice and dust features
in the opacities, the $\mathrm{H}_2$ dissociation and the
combined H, first He ionization zone, as
indicated by vibrational instability. These regions
of instability are much larger in extent and degree of
instability than the second He ionization zone
that drives the Cepheïd pulsations.
Examples for figures using graphicx
A guide "Using Imported Graphics in LaTeX2e" (Keith Reckdahl)
is available on a lot of LaTeX public servers or ctan mirrors.
The file is : epslatex.pdf
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.
Nonlinear Model Results
HJD $E$ Method#2 Method#3
1 50 $-837$ 970
2 47 877 230
3 31 25 415
4 35 144 2356
5 45 300 556
Nonlinear Model Results
HJD $E$ Method#2 4cMethod#3
1 50 $-837$ 970 65 67 78
2 47 877 230 567 55 78
3 31 25 415 567 55 78
4 35 144 2356 567 55 78
5 45 300 556 567 55 78
Spectral types and photometry for stars in the
Star Spectral type RA(J2000) Dec(J2000)
69 B1 V 09 15 54.046 $-$50 00 26.67
49 B0.7 V *09 15 54.570 $-$50 00 03.90
LS 1267 (86) O8 V 09 15 52.787 11.07
24.6 7.58 1.37 0.20
LS 1262 B0 V 09 15 05.17 11.17
MO 2-119 B0.5 V 09 15 33.7 11.74
LS 1269 O8.5 V 09 15 56.60 10.85
The top panel shows likely members of Pismis 11. The second
panel contains likely members of Alicante 5. The bottom panel
displays stars outside the clusters.
Spectral types and photometry for stars in the
Star Spectral type RA(J2000) Dec(J2000)
69 B1 V 09 15 54.046 $-$50 00 26.67
49 B0.7 V *09 15 54.570 $-$50 00 03.90
LS 1267 (86) O8 V 09 15 52.787 11.07a
24.6 7.581 1.37a 0.20a
LS 1262 B0 V 09 15 05.17 11.17b
MO 2-119 B0.5 V 09 15 33.7 11.74c
LS 1269 O8.5 V 09 15 56.60 10.85d
The top panel shows likely members of Pismis 11. The second
panel contains likely members of Alicante 5. The bottom panel
displays stars outside the clusters.
aPhotometry for MF13, LS 1267 and HD 80077 from
Dupont et al.
bPhotometry for LS 1262, LS 1269 from
Durand et al.
cPhotometry for MO2-119 from
Mathieu et al.
[]List of nearby SNe used in this work.
SN name
(with respect to $B$ maximum)
1981B 0 UBV 1
1986G $-$3, $-$1, 0, 1, 2 BV 2
1989B $-$5, $-$1, 0, 3, 5 UBVRI 3, 4
1990N 2, 7 UBVRI 5
1991M 3 VRI 6
4c SNe 91bg-like
1991bg 1, 2 BVRI 7
1999by $-$5, $-$4, $-$3, 3, 4, 5 UBVRI 8
4c SNe 91T-like
1991T $-$3, 0 UBVRI 9, 10
2000cx $-$3, $-$2, 0, 1, 5 UBVRI 11
(1) <cit.>;
(2) <cit.>; (3) <cit.>; (4) <cit.>;
(5) <cit.>; (6) <cit.>; (7) <cit.>;
(8) <cit.>; (9) <cit.>; (10) <cit.>;
(11) <cit.>.
Summary for ISOCAM sources with mid-IR excess
(YSO candidates).
ISO-L1551 $F_{6.7}$ [mJy] $\alpha_{6.7-14.3}$
YSO type$^{d}$ Status Comments
6cNew YSO candidates
1 1.56 $\pm$ 0.47 – Class II$^{c}$ New Mid
2 0.79: 0.97: Class II ? New
3 4.95 $\pm$ 0.68 3.18 Class II / III New
5 1.44 $\pm$ 0.33 1.88 Class II New
6cPreviously known YSOs
61 0.89 $\pm$ 0.58 1.77 Class I HH 30 Circumstellar disk
96 38.34 $\pm$ 0.71 37.5 Class II MHO 5 Spectral type
Summary for ISOCAM sources with mid-IR excess
(YSO candidates).
ISO-L1551 $F_{6.7}$ [mJy] $\alpha_{6.7-14.3}$
YSO type$^{d}$ Status Comments
6cNew YSO candidates
1 1.56 $\pm$ 0.47 – Class II$^{c}$ New Mid
2 0.79: 0.97: Class II ? New
3 4.95 $\pm$ 0.68 3.18 Class II / III New
5 1.44 $\pm$ 0.33 1.88 Class II New
6cPreviously known YSOs
61 0.89 $\pm$ 0.58 1.77 Class I HH 30 Circumstellar disk
96 38.34 $\pm$ 0.71 37.5 Class II MHO 5 Spectral type
Sample stars with absolute magnitude
Catalogue $M_{V}$ Spectral Distance Mode Count Rate
Catalogue $M_{V}$ Spectral Distance Mode Count Rate
Gl 33 6.37 K2 V 7.46 S 0.043170
Gl 66AB 6.26 K2 V 8.15 S 0.260478
Gl 68 5.87 K1 V 7.47 P 0.026610
H 0.008686
Gl 86
[Source not included in the HRI catalog. See Sect. 5.4.2 for details.]
5.92 K0 V 10.91 S 0.058230
Sample stars with absolute magnitude
Catalogue $M_{V}$ Spectral Distance Mode Count Rate
Catalogue $M_{V}$ Spectral Distance Mode Count Rate
Gl 33 6.37 K2 V 7.46 S 0.043170
Gl 66AB 6.26 K2 V 8.15 S 0.260478
Gl 68 5.87 K1 V 7.47 P 0.026610
H 0.008686
Gl 86
[Source not included in the HRI catalog. See Sect. 5.4.2 for details.]
5.92 K0 V 10.91 S 0.058230
§ BACKGROUND GALAXY NUMBER COUNTS AND SHEAR NOISE-LEVELS
Because the optical images used in this analysis...
Plotted above...
Because the optical images...
These studies, however, have faced...
|
1511.00538
|
Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
We study the nature of entanglement in presence of Deutschian closed timelike curves (D-CTCs) and we observe that qubits traveling along a D-CTC allow unambiguous discrimination of Bell states with Local Operations and Classical Communications (LOCC), that is otherwise known to be impossible. A consequence of this leads us to discover that localized D-CTCs can create entanglement between two parties, using just local operations and classical communication. This contradicts the fundamental definition of entanglement.
03.67.Bg, 03.65.Ud, 04.20.Gz, 04.60.-m
Entanglement and Closed Timelike Curves(CTC) are perhaps the most exclusive features in quantum mechanics and general theory of relativity (GTR) respectively. Interestingly, both theories, advocate nonlocality through them.
While the existence of CTCs <cit.> is still debated upon, there is no reason for them, to not exist according to GTR <cit.>. CTCs come as a solution to Einstein's field equations, which is a classical theory itself. Seminal works due to Deutsch <cit.>, Lloyd et al. <cit.>, and Allen <cit.> have successfully ported these solutions into the framework of quantum mechanics.
The formulation due to Lloyd et. al, through post-selected teleportation (P-CTCs) have been also experimentally verified <cit.>.
The existence of CTCs has been disturbing to some physicists, due to the paradoxes, like the grandfather paradox or the unproven theorem paradox, that arise due to them.
Deutsch resolved such paradoxes by presenting a method for finding self-consistent solutions of CTC interactions. The Deutschian model of CTCs (D-CTCs) impose a boundary condition, in which the density operator of the CTC system that interacts with a chronology respecting (CR) system is the same, both before and after it enters the wormhole. Formally,
$$ \rho_{CTC} = \Phi(\rho_{CTC}) = Tr_{CR} \big( U (\rho_{CR} \otimes \rho_{CTC}) U^ {\dagger} \big) $$
where $\rho_{CR}$ is the density matrix for chronology-respecting system, $\rho_{CTC}$ is the initial density matrix of the qubit traveling along the closed timelike curve, and $U$ is the interaction unitary. Mathematically, this can be seen as nature finding a fixed point solution of the map, $\Phi$, that depends on the chronology respecting system <cit.>.
Although a complete theory of quantum gravity is yet to be formulated, quantum information theorists have been studying the implications of the existence of CTCs and the nature of information with CTC-assisted models of computation. Here, we turn our attention to understand the implications of existence of D-CTCs on entanglement. Recent studies of CTC-assisted models of computation, show them to be extremely powerful and be able to carry out non-trivial tasks, such as distinguish between non-orthogonal states <cit.>, clone unknown quantum states <cit.>, be able to signal superluminally <cit.> and find a solution of any problem in the computational class PSPACE efficiently, in polynomial time (PSPACE=P) <cit.>.
We begin by understanding the problem of Bell state discrimination and ask if it might be possible to distinguish between Bell states, using only local operations and classical communication (LOCC), given only a single copy of the state, from a set of four Bell States. We then try to understand its implications.
It is known, in the conventional model of quantum mechanics, it is possible to distinguish between any two Bell states using LOCC <cit.>, however it is impossible to deterministically discriminate between four or even three Bell states <cit.>. Here we take another look and study the problem of Bell state discrimination with the assumption of the existence of D-CTCs in nature.
Bell state discrimination with LOCC is defined as follows. Suppose a referee, prepares a single copy of a maximally entangled Bell state
$$\Ket{\varphi}_{AB} \in_R \{ \Ket{\Phi^+}_{AB}, \Ket{\Phi^-}_{AB}, \Ket{\Psi^+}_{AB}, \Ket{\Psi^-}_{AB} \}$$
where $\Ket{\Phi^{\pm}} = \frac{1}{\sqrt{2}} \big(\Ket{00} \pm \Ket{11} \big)$ and $\Ket{\Psi^{\pm}} = \frac{1}{\sqrt{2}} \big(\Ket{01} \pm \Ket{10} \big)$, and gives one qubit to Alice ($\Ket{\varphi}_{A}$) and one qubit to Bob ($\Ket{\varphi}_{B}$), who are spatially separated and allowed only local operations and classical communication. Their (Alice and Bob's) objective is to determine which state was given to them.
One strategy Alice and Bob can pick would be the following.
Alice prepares a (known) state $\Ket{\psi} = \alpha \Ket{0} + \beta \Ket{1}$, $0<\alpha \neq \beta<1$ and perform a Bell measurement on her (known) state and her part of the local entangled qubit, $\Ket{\varphi}_{A}$, and classically communicates the measurement outcomes to Bob. Depending on the Bell state Alice and Bob were sharing, the decomposition can be given as follows, for each of the four possible Bell states.
$\Ket{\psi}_{A} \Ket {\Phi^+}_{AB} = \frac{1}{2} \big( \Ket{\Phi^+}_A \Ket{\psi}_B + \Ket{\Phi^-}_A (Z\Ket{\psi}_B) + \Ket{\Psi^+}_A (X\Ket{\psi}_B) + \Ket{\Psi^-}_A (ZX\Ket{\psi}_B) \big)$
$\Ket{\psi}_{A} \Ket {\Phi^-}_{AB} = \frac{1}{2} \big( \Ket{\Phi^+}_A (Z\Ket{\psi}_B) + \Ket{\Phi^-}_A \Ket{\psi}_B + \Ket{\Psi^+}_A (ZX\Ket{\psi}_B) + \Ket{\Psi^-}_A (X\Ket{\psi}_B) \big)$
$\Ket{\psi}_{A} \Ket {\Psi^+}_{AB} = \frac{1}{2} \big( \Ket{\Phi^+}_A (X\Ket{\psi}_B) + \Ket{\Phi^-}_A (ZX\Ket{\psi}_B) + \Ket{\Psi^+}_A \Ket{\psi}_B + \Ket{\Psi^-}_A (Z\Ket{\psi}_B) \big)$
$\Ket{\psi}_{A} \Ket {\Psi^-}_{AB} = \frac{1}{2} \big( \Ket{\Phi^+}_A (ZX \Ket{\psi}_B) + \Ket{\Phi^-}_A (X\Ket{\psi}_B) + \Ket{\Psi^+}_A (Z\Ket{\psi}_B) + \Ket{\Psi^-}_A \Ket{\psi}_B \big)$
Bob performs the necessary unitary operations on his share of the entangled qubit depending on the classical communication from Alice, as follows, (I, X, Y, Z are the Pauli operators),
$ 00 \rightarrow I, \hspace{1em} 01 \rightarrow X, \hspace{1em} 10 \rightarrow Z, \hspace{1em} 11 \rightarrow Y$
In a sense, they `force' the teleportated states to pick up the unitary error associated with Alice's Bell state measurement $\Ket{\Phi^+}_A$.
Now all that remains for Bob is to mark out the unitary error his resultant state contains.
For this, Bob uses a variant of BHW-circuit <cit.> to distinguish between non-orthogonal states $\{ \alpha \Ket{0} + \beta \Ket{1} , \alpha \Ket{0} - \beta \Ket{1},\alpha \Ket{1} + \beta \Ket{0},\alpha \Ket{1} - \beta \Ket{0} \}$, as implemented in Fig.1, where the unitaries are defined as
U_{00} = \begin{bmatrix} \alpha & \beta \\ -\beta & \alpha \end{bmatrix} \otimes \mathbb{I}, \hspace{2em}
U_{01} = (X \otimes X) \circ ( \begin{bmatrix} \beta & \alpha \\ \alpha & -\beta \end{bmatrix} \otimes \mathbb{I} ), $
$U_{10} = (X \otimes \mathbb{I}) \circ ( \begin{bmatrix} \beta & \alpha \\ -\alpha & \beta \end{bmatrix} \otimes \mathbb{I} ), \hspace{2em}
U_{11} = {\begin{bmatrix} \alpha & \beta \\ \beta & -\alpha \end{bmatrix} \otimes X }
The circuit first swaps the CTC system with the CR system. Following that it performs a controlled unitary with the CR systems as the control and CTC systems as the target. Finally, it measures the CR system in the computational basis. The CTC system is a nonlinear system. This is because the outcome of $\rho_{CTC}$, after the desired interactions, depends on the initial $\rho_{CTC}$ (before the interactions) and the CR system $\rho{CR}$. Also, $\rho_{CTC}$ (before the interactions) depends on CR system $\rho{CR}$. The objective here is to harness the non-linearity and exploit the two CTC qubits to effect the following map
$(\alpha \Ket{0} + \beta \Ket{1}) \otimes \Ket{0} \rightarrow \Ket{00} ,$
$(\alpha \Ket{0} - \beta \Ket{1}) \otimes \Ket{0} \rightarrow \Ket{01},$
$(\beta \Ket{0} + \alpha \Ket{1}) \otimes \Ket{0} \rightarrow \Ket{10},$
$(\beta \Ket{0} - \alpha \Ket{1}) \otimes \Ket{0} \rightarrow \Ket{11}.$
It can be seen, that these self-consistent solutions for the CTC qubits are unique, and satisfy Deutsch's criteria.
BHW circuit to distinguish between states $\{ \alpha \Ket{0} \pm \beta \Ket{1}, \alpha \Ket{1} \pm \beta \Ket{0} \}$ using Deutschian formulations of CTC.
Let us understand one instance of what is happening in the circuit. Suppose the teleported state was $\Ket{\psi} = \alpha \Ket{1} + \beta \Ket{0}$. According to the desired interaction, it first swaps the information in the CR system and the CTC system. So, the CTC system now carries $(\alpha \Ket{1} + \beta \Ket{0})\otimes \Ket{0}$. Since the CR system is now carrying $\Ket{1} \otimes \Ket{0}$, which the CTC system was initialized as, before the swap; unitary $U_{10}$ now acts on the CTC system and results in the CTC system to become $\Ket{1} \otimes \Ket{0}$, before it disappears in the wormwhole. Thus Deutsch's criteria for chronology respecting system is met and the qubits traveling along a CTC path remain the same both before and after the interaction.
What is essentially happening here is Alice prepares a known state, $\Ket{\psi}$, and teleports it to Bob. The information of an entangled channel are not stored in the states but in the correlations. By teleporting the state, through the entangled channel, $\Ket{\psi}$ is affected by the correlation. In a sense, the correlation of the entanglement gets downloaded in the state. By studying the change of the teleported state from the prepared state, it becomes possible to understand the nature of correlation in the channel.
The circuit then, by measuring $b_1$ and $b_2$, of the chronology respecting qubits, learns which of the two conjugate eigenstates (through measurement $b_1$) and the eigenvalue ($(-1)^{b_2}$), the teleported state is in.
The distinguishability of non-orthogonal states allows Bob to conclusively determine the Bell state that he shared with Alice. The corresponding Bell states, compared to the state identified by Bob, using the BHW circuit (Fig 1), are shown in Table 1.
Table 1: Corresponding Bell States Alice & Bob share
Measurements State Identified Conclusive
Outcomes, $b_1,b_2$ by Bob Bell State
0,0 $\alpha \Ket{0} + \beta \Ket{1}$ $\Ket{\Phi^+}$
0,1 $\alpha \Ket{0} - \beta \Ket{1}$ $\Ket{\Phi^-}$
1,0 $\alpha \Ket{1} + \beta \Ket{0}$ $\Ket{\Psi^+}$
1,1 $\alpha \Ket{1} - \beta \Ket{0}$ $\Ket{\Psi^-}$
This strategy works efficiently to discriminate between the Bell states, if we could use D-CTCs.
But what does this say about entanglement in general? Consider the Smolin State, a certain four-party unlockable bound-entangled state <cit.>, shared between Alice, Bob, Charlie & Dan,
\begin{eqnarray*}
\rho = & \frac{1}{4}( \Ket{\Phi^+}\Bra{\Phi^+}^{AB} \otimes \Ket{\Phi^+}\Bra{\Phi^+}^{CD} + \Ket{\Phi^-}\Bra{\Phi^-}^{AB} \otimes \Ket{\Phi^-}\Bra{\Phi^-}^{CD} + \\
& \Ket{\Psi^+}\Bra{\Psi^+}^{AB} \otimes \Ket{\Psi^+}\Bra{\Psi^+}^{CD} + \Ket{\Psi^-}\Bra{\Psi^-}^{AB} \otimes \Ket{\Psi^-}\Bra{\Psi^-}^{CD}
\end{eqnarray*}
It can be seen that entanglement between AB and CD is 0, i.e
$$\varepsilon(AB:CD) = 0$$
and the state is invariant under permutation. Thus,
$$\varepsilon(AB:CD) = \varepsilon(AC:BD) = \varepsilon(AD:BC) = 0$$
In other words, $\rho$ is separable across the three bipartite cuts AB : CD, AC : BD and AD : BC <cit.>.
The logarithmic negativity <cit.>, $E_N$, of the state $\rho$, in AC:BD cut, is
$$E_N(\rho) = log_2 || (\rho^{T})^{AC} ||_1 = log_2 ( |1/{\sqrt{2}}|^2 + |1/{\sqrt{2}}|^2) = 0$$
Since the distillable entanglement, $E_D$, is upper bounded by logarithmic negativity <cit.>, we can say
$$ E_D(\rho) \leq E_N(\rho) =0$$
Thus distillable entanglement is exactly zero for a Smolin state.
Now since CTC-assisted computation allow discrimination allow Bell State discrimination, given the Smolin state to Alice, Bob, Charlie and Dan, Alice and Bob can distinguish their Bell state without meeting. Following that, Alice and Bob classically communicate their Bell states to Charlie and Dan respectively, who now have share a maximally entangled Bell state. Hence $1-ebit$ was distilled using only local operations and classical communication, from the Smolin state through a D-CTC assisted computation. This shows, existence of D-CTCs would imply the possibility of creating entanglement using LOCC, which is otherwise impossible according to current formulation of quantum mechanics.
To conclude, our work here raises fundamental questions concerning the nature of entanglement in a world with Deutschian closed timelike curves, that drastically changes our current understanding of quantum mechanics. An intuitive resolution to this might lead to support the chronology protection conjecture <cit.>, which loosely says such closed timelike curves cannot exist in nature. If this were to be indeed true, such contradictions could indeed be evaded. However, it was recently shown that one can also replicate the effects of Deutschian closed timelike curves in quantum states, in chronology respecting open timelike curves <cit.>. So, in a sense, this may not be a problem with the Deutschian formalism, but a problem in nature.
A full theory of quantum gravity, we expect would perhaps resolve such challenges and contradiction between the implications of CTCs and laws of quantum mechanics and hope this work will help motivate further research.
|
1511.00108
|
Let $V$ be a finite set of indices, and let $B_i$, $i=1,\ldots,m$, be subsets of $V$ such that $V=\bigcup_{i=1}^{m}B_i$.
Let $X_i$, $i\in V$, be independent random variables, and let $X_{B_i}=(X_j)_{j\in B_i}$.
In this paper, we propose a recursive computation method to calculate the conditional expectation $E\bigl[\prod_{i=1}^m\chi_i(X_{B_i}) \,|\, N\bigr]$ with $N=\sum_{i\in V}X_i$ given, where $\chi_i$ is an arbitrary function.
Our method is based on the recursive summation/integration technique using the Markov property in statistics.
To extract the Markov property, we define an undirected graph whose cliques are $B_j$, and obtain its chordal extension, from which we present the expressions of the recursive formula.
This methodology works for a class of distributions including the Poisson distribution (that is, the conditional distribution is the multinomial).
This problem is motivated from the evaluation of the multiplicity-adjusted $p$-value of scan statistics in spatial epidemiology.
As an illustration of the approach, we present the real data analyses to detect temporal and spatial clustering.
Keywords and phrases:
change point analysis,
chordal graph,
graphical model,
Markov property,
spatial epidemiology.
§ INTRODUCTION
\[
X_V=(X_i)_{i\in V}=(X_1,\ldots,X_n), \quad V=\{1,\ldots,n\},
\]
be a random vector whose index set is $V$.
Throughout the paper, we use the convention that $X_Z=(X_i)_{i\in Z}$ when $Z$ is a set of indices.
Suppose that $X_i$ are distributed independently according to the Poisson distribution and consider the distribution of $X_V$ when $\sum_{i\in V}X_i=N$ is given.
That is, $X_V$ follows the multinomial distribution with probability $p_V=(p_i)_{i\in V}$ and total sum $N$:
\[
X_V |_{\sum_{i\in V}X_i=N} \,\sim \mathrm{Mult}(N;p_V).
\]
Let $Z_1,\ldots,Z_m$ be subsets of $V$ satisfying $V=\bigcup_{i=1}^m Z_i$.
The main technical result of the paper is to provide an algorithm to evaluate the conditional expectation
\begin{equation}
\label{expectation}
E\Bigl[{\prod}_{i=1}^m \chi_i(X_{Z_i}) \,|\, N \Bigr],
\end{equation}
where $\chi_i(X_{Z_i})$ is an arbitrary function of $X_{Z_i}$.
We also consider a generalization where $X_i$ are distributed as a class of distributions including the multinomial distribution.
This problem arises from the evaluation of the multiplicity-adjusted $p$-value of scan statistics.
We begin with stating the scan statistics in spatial epidemiology, which is a typical example in this framework.
In a certain country, there are $n$ districts.
Let $V=\{1,\ldots,n\}$ be the set of districts.
For each district $i\in V$, we suppose that the number $X_i$ for event we are focusing on (e.g., number of patients with some disease) as well as its expected frequency $\lambda_i$ estimated from historical data are available.
$X_i$ is assumed to be distributed according to the Poisson distribution with parameter $\theta_i\lambda_i$, $\mathrm{Po}(\theta_i\lambda_i)$, independently for $i\in V$.
The parameter $\theta_i$ is known as the standardized mortality ratio (SMR).
Figure <ref> depicts an example of a choropleth map of SMRs.
For 44 districts, we indicate the values of the SMRs with different colors.
A set of adjacent districts with SMRs higher than other areas is called a disease cluster.
The detection of such disease clusters is a major interest in spatial epidemiology.
To detect a disease cluster, we settle a family of subsets
\[
\mathcal{Z} = \{ Z_1,\ldots,Z_m \}, \quad Z_i\subset V,
\]
as candidates of a disease cluster, and define a scan statistic $\varphi_{Z_i}(X_{Z_i})$ for each $Z_i\in\mathcal{Z}$.
$Z_i$ is called the scan window.
The choice of the scan windows is an important research topic in spatial epidemiology
When $\varphi_{Z_i}(X_{Z_i})$ is larger than a threshold, say $c$, we declare that $Z_i$ is a disease cluster.
As such a scan statistic, <cit.> proposed the use of the likelihood ratio test (LRT) statistic $\varphi_Z(X_Z)$ for the null hypothesis
$H_0\,:\,\theta_i \equiv \theta_{0}$ (constant) against the alternative
\[
H_Z : \ \theta_i=\theta_{Z} \ (i\in Z),\ \ \theta_i=\theta_{\overline Z} \ (i\notin Z) \ \ \mbox{such that}\ \theta_{Z} > \theta_{\overline Z}
\]
under the conditional distribution with $N=\sum_{i\in V}X_i$ given.
The conditional inference (inference under the conditional distribution) leads to a similar test independent of the nuisance parameter $\theta_{0}$.
The expression of $\varphi_Z(X_Z)$ is given in Section <ref>.
When the hypothesis $H_Z$ holds, the disease cluster $Z$ is called a hotspot.
This is a typical problem of multiple comparisons.
The $p$-value to assess the significance should be adjusted to incorporate the multiplicity effect.
One method is to define the $p$-value from the distribution of the maximum $\max_{Z \in \mathcal{Z}} \varphi_Z(X_Z)$ under $H_0$:
\begin{equation}
P\Bigl(\max_{Z \in \mathcal{Z}} \varphi_Z(X_Z) \le c \,|\, N \Bigr)
= E\Bigl[{\prod}_{Z\in\mathcal{Z}} \chi_Z(X_Z) \,|\, N \Bigr],
\label{conditional}
\end{equation}
where $\chi_Z(X_Z) = \1{\{ \varphi_Z(X_Z) \le c \}}$.
This is of the form of (<ref>).
Note that when $N$ is given, the distribution of $X_V$ is
$\mathrm{Mult}(N; p_V)$, where $p_i=\lambda_i/\sum_{i\in V}\lambda_i$.
In spatial epidemiology, the $p$-value is usually estimated using Monte Carlo simulation.
Although this is convenient and practical in most cases, when the true $p$-value is very small, it is difficult to obtain a precise value even when the number of random numbers in the Monte Carlo is large.
Therefore, we have good reason to conduct exact computation according to the definition (<ref>) by enumerating all possibilities.
However, this is generally difficult because of the computational complexity.
In the area of multiple testing comparisons, many techniques to reduce computational time have been proposed.
For example, <cit.> demonstrated that in a change point problem,
the computational time for the distribution of the maximum could be reduced using the Markov property among statistics.
See also <cit.>, <cit.> and references therein.
In this paper, we develop a similar computation technique by taking advantage of the Markov structure among scan statistics.
The proposed method is based on the theory of a chordal graph, which is the foundation of the theory of graphical models <cit.>.
The chordal graph theory provides rich tools, in not only statistics but also many fields of mathematical science.
In particular, in numerical analysis, this is a major tool used to conduct large-scale matrix computation (e.g., <cit.>,<cit.>).
We also apply the chordal graph theory to retrieve the Markov structure to reduce the computational time by using the recursive summation/integration technique.
Our technique is then similar to those used in the efficient computation of maximum likelihood estimator of graphical models
(e.g., <cit.>, <cit.>, <cit.>, <cit.>, <cit.>).
The remainder of the paper is organized as follows.
Section <ref> provides the recurrence computational formula in the multinomial distribution under the assumption that the running intersection property holds.
We evaluate the computational complexity,
and show that the recurrence computation methodology works for a class of distributions including the Poisson distribution (that is, the conditional distribution is the multinomial).
Section <ref> proposes a method to detect the Markov property,
and Section <ref> presents illustrative real data analyses to detect temporal and spatial clustering.
Section <ref> briefly summarizes our results and discusses further research.
§ RECURSIVE COMPUTATION OF CONDITIONAL EXPECTATIONS
In this section, we provide an algorithm to compute the expectation
(<ref>) when the sequence $Z_1,\ldots,Z_m$ of subsets of $V$
has a nice property,
which is called the running intersection property given in Definition <ref>.
We will consider the general case in the next section.
In this section, we use the symbol $B_i$ instead of $Z_i$.
§.§ The case where $m=2$
We start with case $m=2$.
For $1<l_1\le l_2<n$, let $B_1=\{1,\ldots,l_2\}$ and $B_2=\{l_1+1,\ldots,n\}$.
Suppose that
\[
X_{B_1\cup B_2} = (X_1,\ldots,X_n) \sim \mathrm{Mult}(N;p_{B_1\cup B_2})
\]
is a random vector distributed according to the multinomial distribution with summation $N$ and probability $p_{B_1\cup B_2}=(p_1,\ldots,p_n)$.
We consider the evaluation of the expectation
\begin{equation}
E[\chi_1(X_{B_1}) \chi_2(X_{B_2})]
\label{naive}
\end{equation}
exactly, where $\chi_1$ and $\chi_2$ are arbitrary functions.
Obviously, $X_{B_1}$ and $X_{B_2}$ are not independent;
there is an overlap $X_{C_1}=(X_{l_{1+1}},\ldots,X_{l_2})$ unless $C_1=B_1\cap B_2$ is empty.
Moreover, there is a restriction that
\[
\sum_{i\in B_1}X_i + \sum_{i\in B_2}X_i - \sum_{i\in C_1}X_i = N.
\]
Instead of the problem of evaluating the expectation, by a change of viewpoint,
we first consider the problem of generating random numbers
$X_{B_1\cup B_2}=(X_{R_1},X_{R_2})$, where $R_1=B_1\setminus C_1$, $R_2=B_2$.
$X_{B_1\cup B_2}$ can be generated according to the following three steps:
\begin{align}
(M_2,M_1)|_N &\sim \mathrm{Mult}\Bigl(N;\Bigl(\sum_{i\in R_2} p_i,\sum_{i\in R_1} p_i\Bigr)\Bigr),
\label{M2M1} \\
X_{R_2}|_{M_2} &\sim \mathrm{Mult}\Bigl(M_2;p_{R_2}/\sum_{i\in R_2} p_i\Bigr),
\label{XR2} \\
X_{R_1}|_{M_1} &\sim \mathrm{Mult}\Bigl(M_1;p_{R_1}/\sum_{i\in R_1} p_i\Bigr),
\label{XR1}
\end{align}
where $X_{R_2}$ and $X_{R_1}$ are independent given $(M_2,M_1)$.
Correspondingly, we divide the expectation in (<ref>) into three parts as
\begin{align}
E[\chi_1(X_{B_1}) \chi_2(X_{B_2})]
&= E^{(M_2,M_1)|N}\bigl[ E^{X_{R_2}|M_2}\bigl[ E^{X_{R_1}|M_1}[\chi_1(X_{B_1}) \chi_2(X_{B_2})] \bigr]\bigr] \nonumber \\
&= E^{(M_2,M_1)|N}\bigl[ E^{X_{R_2}|M_2}\bigl[ \chi_2(X_{B_2}) E^{X_{R_1}|M_1}[\chi_1(X_{B_1}) ] \bigr]\bigr] \nonumber \\
&= E^{(M_2,M_1)|N}\bigl[ E^{X_{R_2}|M_2}[ \chi_2(X_{B_2}) \xi_1(M_1,X_{C_1})] \bigr],
\label{E}
\end{align}
\begin{align}
\xi_1(M_1,X_{C_1})
&= E^{X_{R_1}|M_1}[\chi_1(X_{B_1})] \nonumber \\
&= E^{X_{R_1}|M_1}[\chi_1(X_{C_1},X_{R_1})].
\label{xi}
\end{align}
The procedure for the numerical computation of (<ref>) is as follows.
(i) For possible values of $M_1$ and $X_{C_1}$, compute $\xi_1(M_1,X_{C_1})$ in (<ref>).
Here, the expectation is taken over $X_{R_1}$ according to (<ref>) with $X_{C_1}$ fixed.
The results are stored in memory as a tabulation.
(ii) Compute $E[\chi_1(X_{B_1}) \chi_2(X_{B_2})]$ according to (<ref>).
Here, $M_2$, $M_1$, and $X_{R_2}$ are random variables having distributions (<ref>) and (<ref>).
This technique substantially reduces the computational cost.
To see this, we enumerate the number of required summations in detail.
$M_1$ takes the values $0\le M_1\le N$ and $X_{C_1}$ takes the values of all nonnegative integer vectors whose sum is less than or equal to $M_2=N-M_1$.
In the expectation $E^{X_{R_1|M_1}}[\cdot]$ in (<ref>),
the variable $X_{R_1}$ runs over all nonnegative vectors whose sum is $M_1$.
\[
\sharp\Bigl\{ (x_1,\ldots,x_d)\in\Z^d \mid x_i\ge 0,\ {\sum}_{i=1}^d x_i=n \Bigr\} = {n+d-1 \choose d-1},
\]
\[
\sharp\Bigl\{ (x_1,\ldots,x_d)\in\Z^d \mid x_i\ge 0,\ {\sum}_{i=1}^d x_i\le n \Bigr\} = {n+d \choose d},
\]
we see that the number of summations to prepare the table $\xi_1(M_1,X_{C_1})$ is
\[
\sum_{M_1=0}^N {N-M_1+|C_1| \choose |C_1|} {M_1+|R_1|-1 \choose |R_1|-1}
= {N+|C_1|+|R_1| \choose |C_1|+|R_1|} = {N+|B_1| \choose |B_1|}.
\]
In the expectations $E^{(M_2,M_1)|N}[ E^{X_{R_2}|M_2}[\cdot]]$ in (<ref>), the variable $X_{R_2}$ runs over all nonnegative vectors whose sum is $M_2$, and $M_2+M_1=N$.
Hence, the number of summations for computing (<ref>) is
\[
\sum_{M_2=0}^N {M_2+|R_2|-1 \choose |R_2|-1} = {N+|R_2| \choose |R_2|} = {N+|B_2| \choose |B_2|}.
\]
Therefore, the number of summations is
\[
{N+|B_1| \choose |B_1|} + {N+|B_2| \choose |B_2|} = O(N^{\max(|B_1|,|B_2|)}).
\]
On the other hand, when we do not use this technique, the number of summations is
\[
{N+|B_1\cup B_2|-1 \choose |B_1\cup B_2|-1} = O(N^{|B_1\cup B_2|-1}),
\]
which is of larger order than $O(N^{\max(|B_1|,|B_2|)})$.
§.§ The case of general $m$
This technique for reducing computational time is available in a general setting.
To describe it, we introduce several notions from the theory of chordal graphs and graphical models <cit.>.
A sequence of sets $B_1,\ldots,B_m$ is said to have the running intersection property (RIP) if for each $1\le i\le m-1$, there is a $k(i)>i$ such that
\begin{equation}
\label{rip}
B_i \cap \Bigl( {\bigcup}_{j>i} B_j \Bigr) = B_i \cap B_{k(i)}.
\end{equation}
Throughout this section, we suppose that sequence $B_1,\ldots,B_m$ satisfies the running intersection property.
Note that the indices $i$ of $B_i$ in Definition <ref> are reversely ordered from the conventional definition.
The function $k(\cdot)$ defines a directed graph $(V,E)$,
where $V=\{1,\ldots,m\}$ and $E=\{(i,k(i)) \mid i\in V \}$.
The running intersection property is explained in detail in Section <ref>.
For $i,j\in V$, write $i\preceq j$ iff $j=i$ or
\[
j=\underbrace{k(k(\cdots(k}_{h\,\mathrm{times}}(i))\cdots)) \ \ \mbox{for some $h$}.
\]
Let $R_i=B_i\setminus B_{k(i)}$ ($i<m$) and $R_m=B_m$, where
$R_i$ are residual sets.
The disjoint union is denoted by $\sqcup$.
The proposition below follows from the running intersection property.
\[
V = \bigsqcup_{i=1}^m R_i.
\]
Obviously $V=\bigcup_{i=1}^m B_i=\bigcup_{i=1}^m R_i$.
We prove $R_i\cap R_j=\emptyset$ for $i\ne j$.
Suppose that $R_i\cap R_j\ne\emptyset$ for $i<j$.
Let $x\in R_i\cap R_j$.
Since $x\in B_i,B_j$, $x\in B_i\cap(\bigcup_{j>i} B_j)=B_i\cap B_{k(i)}$.
On the other hand, $x\in R_i$ implies $x\notin B_{k(i)}$.
This is a contradiction.
Write $C_i=B_i\cap B_{k(i)}$ ($i=1,\ldots,m-1$), $C_m=\emptyset$,
$R_i=B_i\setminus C_i$ and $T_i=\bigsqcup_{j\preceq i}R_j$.
$T_i$ is defined recursively as
\[
R_i \sqcup \bigsqcup_{j\in k^{-1}(i)} T_j & (k^{-1}(i)\ne\emptyset), \\
R_i & (k^{-1}(i)=\emptyset).
\end{cases}
\]
We state the proposed summation technique in general form in the theorem below.
Suppose that a sequence of sets $B_1,\ldots,B_m$ satisfies the running intersection property with respect to a function $k(\cdot)$.
Define $C_i$, $R_i$ and $T_i$ as before.
Let $\chi_i(X_{B_i})=\chi_i(X_{C_i},X_{R_i})$ be an arbitrary function of $X_{B_i}=(X_{C_i},X_{R_i})$.
Suppose that
\[
X_V \sim \mathrm{Mult}(N;p_V), \quad V=\bigcup_{i=1}^m B_i,
\]
is a multinomial random vector.
\begin{equation}
\label{xi0}
\xi_i(N_i,X_{C_i}) = E\Bigl[{\prod}_{j\preceq i} \chi_j(X_{B_j}) \mid X_{C_i}, {\sum}_{j\in T_i} X_j = N_i \Bigr].
\end{equation}
In particular, define
\[
\xi_m(N,\emptyset) = E\Bigl[{\prod}_{i=1}^m \chi_i(X_{B_i})\Bigr].
\]
\begin{align}
& \xi_i(N_i,X_{C_i}) \nonumber \\
&= \begin{cases}
\displaystyle
E^{\left(M_i,(N_j)_{j\in k^{-1}(i)}\right)|N_i}\Bigl[E^{X_{R_i}|M_i}\Bigl[\chi_i(X_{C_i},X_{R_i}){\prod}_{j\in k^{-1}(i)}\xi_j(N_j,X_{C_j})\Bigr]\Bigr] \\
\displaystyle
& \hspace*{-20mm}
(k^{-1}(i)\ne\emptyset), %\hspace*{+20mm}
\\
\displaystyle
E^{X_{R_i}|N_i} \bigl[\chi_i(X_{C_i},X_{R_i})\bigr]
& \hspace*{-20mm}
(k^{-1}(i)=\emptyset). \end{cases} \nonumber \\
\label{recursive}
\end{align}
Here, the conditional expectations in (<ref>) are with respect to
\begin{align}
(M_i,(N_j)_{j\in k^{-1}(i)})|N_i &\,\sim\, \mathrm{Mult}\biggr( N_i;\frac{(\sum_{j\in R_i}p_j,(\sum_{l\in T_j}p_l)_{j\in k^{-1}(i)})}{\sum_{j\in T_i} p_j} \biggl), \nonumber \\
X_{R_i}|M_i &\,\sim\,\mathrm{Mult}\biggl( M_i;\frac{p_{R_i}}{\sum_{j\in R_i} p_j} \biggr) \quad \mbox{for } k^{-1}(i)\ne\emptyset,
\label{condprob}
\end{align}
\begin{align*}
X_{R_i}|N_i &\,\sim\,\mathrm{Mult}\biggl( N_i;\frac{p_{R_i}}{\sum_{j\in R_i} p_j} \biggr) \quad \mbox{for } k^{-1}(i)=\emptyset.
\end{align*}
We prove the case $k^{-1}(i)\ne\emptyset$.
The case $k^{-1}(i)=\emptyset$ is similar and easier, and therefore omitted.
Noting that $\prod_{j\preceq i} \chi_j(X_{B_j}) = \chi_i(X_{B_i}) \prod_{j\in k^{-1}(i)} \prod_{l\preceq j} \chi_l(X_{B_l})$, we have
\begin{align}
\xi_i(N_i,X_{C_i}) = E\Bigl[ & \chi_i(X_{B_i}) E\Bigl[ {\prod}_{j\in k^{-1}(i)} {\prod}_{l\preceq j} \chi_l(X_{B_l}) \nonumber \\
& \mid X_{B_i},{\sum}_{l\in T_j} X_l = N_j\ (j\in k^{-1}(i)),X_{C_i},{\sum}_{j\in T_i} X_j = N_i \Bigr] \nonumber \\
& \mid X_{C_i},{\sum}_{j\in T_i} X_j = N_i \Bigr].
\label{xi1}
\end{align}
As $B_i\supset C_i,R_i$ and $T_i = R_i \sqcup \bigsqcup_{j\in k^{-1}(i)} T_j$,
the conditions on $X_{B_i}$ and $\sum_{l\in T_j} X_l\,(=N_j)$ implies the conditions on $X_{C_i}$ and $\sum_{j\in T_i} X_j\,(=N_i)$.
That is,
\[
N_i = M_i + {\sum}_{j\in k^{-1}(i)} N_j, \quad M_i={\sum}_{j\in R_i} X_j.
\]
Therefore, the inner conditional expectation on the right-hand side of (<ref>) is rewritten as
\[
E\Bigl[ {\prod}_{j\in k^{-1}(i)} {\prod}_{l\preceq j} \chi_l(X_{B_l})
\mid X_{B_i},{\sum}_{l\in T_j} X_l = N_j\ (j\in k^{-1}(i)) \Bigr].
\]
Moreover, when ${\sum}_{l\in T_j} X_l\,(=N_j)$ ($j\in k^{-1}(i)$) are given,
$X_{T_j}$ ($j\in k^{-1}(i)$) are independent,
and hence the expression above is rewritten as
\begin{align*}
{\prod}_{j\in k^{-1}(i)} & E\Bigl[ {\prod}_{l\preceq j} \chi_l(X_{B_l})
\mid X_{B_i},{\sum}_{l\in T_j} X_l = N_j \Bigr] \\
&= {\prod}_{j\in k^{-1}(i)} E\Bigl[ {\prod}_{l\preceq j} \chi_l(X_{B_l})
\mid X_{C_j},{\sum}_{l\in T_j} X_l = N_j \Bigr] \\
&= {\prod}_{j\in k^{-1}(i)} \xi_i(N_j,X_{C_j}).
\end{align*}
In the above, the second equality follows because of
$(\bigcup_{l\preceq j} B_l) \cap B_i=C_j$ (which will be proved later) and hence
$\prod_{l\preceq j} \chi_l(X_{B_l})$ is a function on $X_{B_i}$ through $X_{C_j}$.
Now, we have the formula
\begin{align*}
\xi_i(N_i,X_{C_i}) = E\Bigl[ \chi_i(X_{C_i},X_{R_i}) {\prod}_{j\in k^{-1}(i)} \xi_i(N_j,X_{C_j}) \mid X_{C_i},{\sum}_{j\in T_i} X_j = N_i \Bigr].
\end{align*}
Actually this is equivalent to (<ref>), because under the conditional distribution, $(M_i,(N_j)_{j\in k^{-1}(i)})$ ($M_i=\sum_{j\in R_i}X_j$) and
$X_{R_i}$ have the distributions given in (<ref>).
Finally, we prove $(\bigcup_{l\preceq j} B_l) \cap B_i=C_j$ for $i=k(j)$.
$(\bigcup_{l\preceq j} B_l) \cap B_i \supset B_j\cap B_i = B_j\cap B_{k(j)} = C_j$.
It suffices to prove
$B_l \cap B_i \subset C_j$ for $l\preceq j$.
Recall that $l\preceq j$ implies $l<k(l)<k(k(l))<\cdots<k(k(\cdots(k(l))\cdots))=j$.
Also $i=k(j)$ implies $j<i$.
Hence, $B_l \cap B_i\subset B_l\cap (\bigcup_{h>l}B_h)=B_{k(l)}$,
$B_l \cap B_i = (B_l \cap B_i)\cap B_i \subset B_{k(l)}\cap B_i=B_{k(k(l))}$.
By repeatedly applying this manipulation, we reach $B_l \cap B_i\subset B_j$ and
$B_l \cap B_i = (B_l \cap B_i) \cap B_i\subset B_j\cap B_i=C_j$ follows.
Theorem <ref> provides an algorithm to compute the exact expectation
$E\bigl[{\prod}_{i=1}^m \allowbreak \chi_i(X_{B_i})\bigr]$ with $X_V\sim\mathrm{Mult}(N;p_V)$ by updating the tables $\xi_i(N_i,X_{C_i})$, $i=1,\ldots,m$.
We can evaluate the number of required summations as before.
The result is summarized below without proof.
The number of summations required in the algorithm of Theorem <ref> is
\begin{equation}
\sum_{i=1}^{m-1} {N+|B_i|+|k^{-1}(i)| \choose |B_i|+|k^{-1}(i)|} +
{N+|B_m|+|k^{-1}(m)|-1 \choose |B_m|+|k^{-1}(m)|-1} = O(N^{\mathrm{deg}})
\label{order}
\end{equation}
\begin{equation}
\mathrm{deg} = \max\left(\max_{1\le i\le m-1}(|B_i|+|k^{-1}(i)|),|B_m|+|k^{-1}(m)|-1\right).
\label{deg}
\end{equation}
Note that when $i<m$, the value $\xi_i(N_i,X_{C_i})$ is needed for $0\le N_i\le N$.
Whereas, when $i=m$, only the value for $N_i=N$ is needed.
This is the reason why the case $i=m$ is exceptional in (<ref>).
The number of summations (<ref>) is smaller than in the absent of this recursive computation technique:
\[
{N+|V|-1 \choose |V|-1} = O(N^{|V|-1}), \quad V=\bigcup_{i=1}^m B_i.
\]
As shown, the proposed algorithm has an advantage in time complexity.
Whereas, it requires memory space to restore the tables.
Since in (<ref>), $N_i$ and the elements of $X_{C_i}=(X_i)_{i\in C_i}$ are arbitrary nonnegative integers such that $N_i+\sum_{i\in C_i}X_i\le N$, the size of the table for $\xi_i(N_i,X_{C_i})$ is
\[
{N+|C_i|+1 \choose |C_i|+1} = O(N^{|C_i|+1}).
\]
However, in the process of computation of (<ref>) for $i=1,\ldots,m$, the table $\xi_i(N_i,X_{C_i})$ can be deleted (i.e., the memory space can be released) once $\xi_{k(i)}(N_{k(i)},X_{C_{k(i)}})$ is computed.
Therefore, the problem of space complexity does not matter in practice.
§.§ A class of distributions for recursive computation
Although Theorem <ref> is stated for the multinomial distribution,
it also works for a class of distributions including the multinomial distribution.
Let $B_1$ and $B_2$ be index sets such that $V=B_1\cup B_2$,
and let $C_1 = B_1 \cap B_2$, $R_1 = B_1 \setminus C_1$ and $R_2=B_2$ again.
Suppose that $X_i$ is distributed independently for $i\in V$ according to a certain distribution.
Under the conditional distribution where $N=\sum_{i\in V}X_i$ is given,
if we pose additional conditions that $(\sum_{i\in R_2}X_i,\sum_{i\in R_1}X_i)=(M_2,M_1)$ is given, then $X_{R_1}$ and $X_{R_2}$ become independent.
Therefore, the three steps (<ref>)–(<ref>) for generating random numbers,
and the corresponding decomposition of the expectation continue to hold for an arbitrary distribution of $X_i$.
If explicit expressions for probability density functions of the conditional distributions
$X_V|N$, $X_{R_2}|M_2$, $X_{R_1}|M_1$, and $(M_2,M_1)|N$ are available,
we have the the computation formula of the type (<ref>).
In general, if some explicit formula is available for the probability density function of the conditional distribution
\begin{equation}
\label{generalcond}
\Bigl({\sum}_{j\in R_i}{X_j}\Bigr)_{i\ge 1}\,\Big|_{N},
\end{equation}
where the $R_i$ are subsets of $V$ such that $V=\bigsqcup_{i\ge 1}R_i$,
we can construct the recurrence computation formula of the type of Theorem <ref>.
The class of distributions having the explicit conditional density function of (<ref>) includes
the normal distribution, the Gamma distribution, the binomial distribution and the negative binomial distribution.
The conditional distributions of (<ref>) corresponding to the above four distributions are the (degenerate) normal distribution, the Dirichlet distribution, the multivariate hypergeometric distribution, and the Dirichlet-multinomial distribution, respectively.
For these distributions, the recurrence computation formula of Theorem <ref> works by replacing summations with integrations when the distribution is continuous.
§ EXTRACTION OF MARKOV STRUCTURE
As shown in the previous section, when the sequence $Z_1,\ldots,Z_M$ of subsets of $V$ has the running intersection property, the expectation (<ref>) can be evaluated efficiently with the recursive summation/integration technique proved in Theorem <ref>.
However, in general, $Z_1,\ldots,Z_M$ does not have this property.
To apply this technique to general cases, one method is to prepare another sequence $B_1,\ldots,B_m$ having the running intersection property such that each $Z_j$ is included into at least one of $B_1,\ldots,B_m$.
If we have such $B_i$, by defining a function
$\tau:\{1,\ldots,M\}\to\{1,\ldots,m\}$ so that $Z_j \subset B_{\tau(j)}$,
the expectation (<ref>) is written as
\[
E\Bigl[{\prod}_{i=1}^m \chi_i(X_{B_i})\Bigr], \quad
\chi_i(X_{B_i}) := {\prod}_{j\in \tau^{-1}(i)}\chi_j(X_{Z_j}),
\]
which can be dealt with by Theorem <ref>.
Such a sequence $B_i$, $i=1,\ldots,m$, is known to be obtained through a chordal extension in Algorithm <ref>.
Here, we summarize some notions and basic facts on chordal graphs.
Let $G = (V,E)$ be a connected undirected graph, where $V$ is a set of
vertices and $E \subset V \times V$ is a set of edges.
$G$ is called complete if every pair of vertices is joined by an edge.
For a subset $B \subseteq V$, let $G(B)$ denote a subgraph induced by $B$.
That is,
\[
G(B) = (B,E(B)),\quad E(B) = \{(i,j)\in E \mid i,j\in B\}.
\]
When $G(B)$ is complete, $B$ is called a clique.
A clique that is maximal with respect to inclusion is called a maximal clique.
$G$ is called chordal if every cycle with length greater than three has a chord.
Let $B_1,\ldots,B_m$ be a sequence of all maximal cliques of $G$ satisfying the running intersection property (<ref>) in Definition <ref>.
The sequence is called perfect if for every $i=1,\ldots,m$, $B_i \cap B_{k(i)}$ is a clique of $G$.
Denote by ${\cal B}=\{B_1,\ldots,B_m\}$ the set of maximal cliques of $G$.
A vertex $v \in V$ is called simplicial if its adjacent vertices form a clique in $G$.
A perfect elimination ordering $v_1,\ldots,v_n$ of $G$ is an ordering of vertices of $G$ such that for every $i$, $v_i$ is a simplicial in $G(\{v_{i+1},\ldots,v_n\})$.
Let $\mathrm{Simp}(B_i)$ denote the set of simplicial vertices of $G$ in $B_i$.
Then $B_i$ is called a boundary clique if there exists $B_j$, $j\ne i$, such that
\[
B_i \cap (V \setminus \mathrm{Simp}(B_i)) = B_i \cup B_j.
\]
The following proposition on the property of chordal graphs is well known
(e.g., <cit.> and <cit.>).
Let $G$ be an undirected graph.
The four statements below are equivalent.
(i) $G$ is chordal.
(ii) $G$ has a perfect sequence of the maximal cliques.
(iii) $G$ vertices can be ordered to have a perfect elimination ordering.
We generate a perfect sequence $B_1,\ldots,B_m$ from $Z_1,\ldots,Z_M$ according to the following procedure.
Step 0.
Define an undirected graph $G=(V,E)$ with vertices $V=\{1,\ldots,n\}$ and edges
\[
E = \{ (i,j)\in V\times V \mid i,j\in Z \ \mbox{for some}\ Z\in\mathcal{Z} \},
\]
where $\mathcal{Z}=\{ Z_1,\ldots,Z_M \}$.
Step 1.
Add edges of $E_1$ to $G$ so that the extended graph $\widetilde G=(V,E\cup E_1)$ is a chordal graph.
Step 2.
Identify the perfect sequence of the maximal cliques $B_1,\ldots,B_m$ of $\widetilde G$.
This sequence has the running intersection property and $V=\bigcup_{i=1}^m B_i$.
The procedure for Step 1 is referred to as the chordal extension.
Constructing the chordal extension such that the maximum size of the maximal clique is minimum is known to be a NP-hard problem <cit.>.
In this section, we propose a heuristic method to construct an approximately best chordal extension.
For Step 2, we provide a method proved in Theorem <ref>.
The other method for the same purpose based on the maximum cardinality search procedure is also known <cit.>.
We now explain Steps 0–2 in detail using a small example.
§.§.§ Step 0: Defining an undirected graph
Suppose that the vertex set is $V=\{1,2,\ldots,9\,(=n)\}$, and the family of subsets of $V$ is given by
\begin{align}
\mathcal{Z} = \Bigl\{ &
\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}, \{8\}, \{9\},
\nonumber \\
& \{4,5\}, \{7,8\}, \{4,8\}, \{3,7\}, \{4,5,8\}, \{2,4\}, \nonumber \\
& \{1,3\}, \{2,3\}, \{2,4,5\}, \{3,6\}, \{8,9\} \Bigr\}.
\label{Z20}
\end{align}
The associated undirected graph $G$ is shown in Figure <ref> (left).
§.§.§ Step 1-a: Renumbering of vertices
Step 1 is divided into three parts.
In this substep, we renumber the vertices.
Consider the following procedure to make all vertices removed from the graph $G$ sequentially:
First, find a vertex, say $v_1$, that has the minimum number of adjacent edges in the graph $G$.
Then, remove $v_1$ as well as its adjacent edges $E_1 = \{ (v_1,i) \in E \}$ from the graph $G$.
Then, find a vertex, say $v_2$, that has the minimal number of adjacent edges in the graph $(V\setminus\{v_1\},E\setminus E_1)$ with $n-1$ vertices.
Continuing with this procedure, we have a sequence of vertices $v_1,v_2,\ldots,v_n$.
This renumbering is called the minimum degree (MD) ordering <cit.>.
The MATLAB function symamd is available to obtain an MD ordering <cit.>.
In our example,
\[
\begin{pmatrix} i \\ v_i \end{pmatrix} =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
9 & 1 & 6 & 3 & 7 & 2 & 4 & 5 & 8
\end{pmatrix}.
\]
We illustrate the resulting renumbered graph in Figure <ref> (right).
Graph $G$
(left: original, right: after renumbering vertices).
§.§.§ Step 1-b: Triangularization
In this substep, we add additional edges to the graph $G$.
For $i=1,\ldots,n$, if there exist $j,k$ such that $i<j<k$,
$(i,j)\in E$ and $(i,k)\in E$, then add the edge $(j,k)$ to $G$ (if it does not exist).
This procedure is referred to as triangularization.
Let $\widetilde G=(V,\widetilde E)$ be the resulting graph after the triangularization process.
From this construction, for each $i$, the set of vertices
\begin{equation}
\label{simplicial}
V_i = \Bigl\{ j\in V \mid j>i,\ (i,j)\in \widetilde E \Bigr\}\cup\{i\}
\end{equation}
forms a clique.
That is, the vertices are ordered to be a perfect elimination ordering.
Figure <ref> (left) depicts the triangulated graph $\widetilde G$ with added edges in red.
In our example, two edges $(5,6)$ and $(6,9)$ are added.
$V_i$ in (<ref>) are
\begin{align*}
V_1 &=\{1,9\},\ V_2=\{2,4\},\ V_3=\{3,4\},\ V_4 =\{4,5,6\},\\
V_5 &=\{5,6,9\},\ V_6=\{6,7,8,9\},\ V_7=\{7,8,9\},\ V_8=\{8,9\},\ V_9=\{9\}.
\end{align*}
§.§.§ Step 1-c: Finding maximal cliques
Let $V_1,\ldots,V_n$ be the sequence of cliques defined in (<ref>).
From these $n$ cliques, we remove all “non-maximal” $V_i$ such that
\[
V_i \subset V_l \quad\text{for some $l<i$}.
\]
In our example, the maximal cliques are
\begin{align*}
V_1 &=\{1,9\},\ V_2=\{2,4\},\ V_3=\{3,4\},\ V_4 =\{4,5,6\},\\
V_5 &=\{5,6,9\},\ V_6=\{6,7,8,9\}.
\end{align*}
§.§.§ Step 2: Identifying a perfect sequence of the maximal cliques
Suppose that there are $m$ ($\le n$) remaining cliques after removing the non-maximal $V_i$.
Among the set of maximal cliques, we introduce an order defined below.
For two cliques $V=\{v_1,\ldots,v_l\}$ and $V'=\{v'_1,\ldots,v'_{l'}\}$
such that $v_1<\cdots<v_l$ and $v'_1<\cdots<v'_{l'}$, define a lexicographic order:
\begin{equation}
\begin{split}
V<V' \quad \mbox{iff}\quad
& v_l<v_{l'} \\
&\text{or}\ \ v_l=v_{l'},\ v_{l-1}<v_{l'-1} \\
&\text{or}\ \ v_l=v_{l'},\ v_{l-1}=v_{l'-1}, v_{l-2}<v_{l'-2} \\
&\text{or}\ \ \cdots.
\end{split}
\label{lex}
\end{equation}
This comparison procedure should stop properly if one of two is not a proper subset of the other.
According to this order, we order the $m$ maximal cliques $B_1,\ldots,B_m$, which is shown to have the running intersection property by Theorem <ref> below.
The function $k(\cdot)$ is defined as
\begin{equation}
\label{k(i)}
k(i) = \min\Bigl\{ k>i \mid B_i\cap\Bigl({\bigcup}_{j>i} B_j\Bigr) \subset B_k \Bigr\}.
\end{equation}
In our example, we obtain the final results
\begin{align}
B_1 &= \{2,4\},\ B_2=\{3,4\},\ B_3=\{4,5,6\},\ B_4 =\{1,9\}, \nonumber \\
B_5 &= \{5,6,9\},\ B_6=\{6,7,8,9\},
\label{Bi}
\end{align}
\[
\begin{pmatrix} i \\ k(i) \end{pmatrix} =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 3 & 5 & 5 & 6 & -
\end{pmatrix},
\]
\begin{align}
& k^{-1}(1)=\emptyset,\ k^{-1}(2)=\{1\},\ k^{-1}(3)=\{2\},\ k^{-1}(4)=\emptyset, \nonumber \\
& k^{-1}(5)=\{3,4\},\ k^{-1}(6)=\{5\}
\label{ki}
\end{align}
(see Figure <ref> (right)).
Chordal graph $\widetilde G$
(left: added edges are in red, right: maximal cliques are in green).
The required number of summations is estimated by Theorem <ref>.
In our example, $B_i$ and $k^{-1}(i)$ are given in (<ref>) and (<ref>).
For example, when $N=\sum_i X_i = 28$, there are 314,621 summations using the proposed method, and 30,260,340 summations when we do not.
The theorem below validates Step 2 of the proposed algorithm.
The sequence of the maximal cliques $B_1,\ldots,B_m$ identified in Step 2 has the running intersection property.
That is, if the partial order (<ref>) can be embedded into the total order $B_1,\ldots,B_m$, then
$B_i \cap \bigl(\bigcup_{j>i} B_j\bigr) \subset B_{k(i)}$ for $k(i)>i$.
We prove this proposition by induction. The case where $m=1$ is trivial.
Suppose the proposition holds for all chordal graphs with up to $m-1$ maximal cliques.
Assume that there does not exist $k(1)$ in (<ref>).
Then there exists more than one clique $H_1,\ldots,H_t$, $t>1$, such that
\[
C_1 := B_1 \cap (B_2 \cup \cdots \cup B_m)
= B_1 \cap (H_1 \cup \cdots \cup H_t),
\]
\[
B_1 \cap H_i \ne B_1 \cap H_j, \quad i \ne j
\]
\[
B_1 \cap H_i \subset C_1, \quad i = 1,\ldots, t.
\]
Since every vertex of $C_1$ is included in more than one clique, it is not simplicial in $\widetilde G$ (e.g., <cit.>).
In other words, in the subgraph $\widetilde G(B_1 \cup H_1 \cup \cdots \cup H_t)$,
every element of $C_1$ is not simplicial.
This shows that there exists a $H_i$ such that for all $v \in H_i \setminus C_1$ and for all $v' \in C_1$, $v<v'$.
However, this implies $B_1 > H_i$ in the sense of (<ref>), which is a contradiction.
Therefore, there exists $k(1)$ in (<ref>).
This also shows that $B_1$ is a boundary clique.
We then have
\[
\widetilde G(V \setminus \mathrm{Simp}(B_1)) =
\widetilde G(B_2 \cup \cdots \cup B_m)
\]
and $\widetilde G(B_2 \cup \cdots \cup B_m)$ is a connected chordal graph with the set of maximal cliques $\{B_2,\ldots,B_m\}$ and $B_2 < \cdots < B_m$
(e.g., <cit.>).
From the inductive assumption, a sequence s$B_2,\ldots,B_m$ is perfect.
Therefore, $B_1,B_2,\ldots,B_m$ is also perfect.
§ ILLUSTRATIVE DATA ANALYSIS WITH REAL DATA
§.§ <cit.>'s scan statistic
In this section, we provide illustrative data analyses with real data.
As stated in Section <ref>, in spatial epidemiology,
<cit.>'s statistic serves as a standard scan statistic.
Using the notations $X_V=(X_i)_{i\in V}$, $p_V=(p_i)_{i\in V}$, and $Z\subset V$ defined in Section <ref>,
this statistic is written as
\begin{equation}
\varphi_N(X_Z) = \begin{cases}
N \Bigl\{ p\left( \frac{\widehat p}{p}\log \frac{\widehat p}{p} -
\frac{\widehat p}{p} + 1 \right) + (1-p) & \\
\quad \times \left( \frac{1-\widehat p}{1-p}\log \frac{1-\widehat p}{1-p} -
\frac{1-\widehat p}{1-p} + 1 \right) \Bigr\},
& \hspace*{-5mm} \mbox{if } \frac{\widehat p}{p}\ge 1, \\
0,& \hspace*{-5mm} \mbox{otherwise}, \end{cases}
\label{Kulldorff}
\end{equation}
$\widehat p=\sum_{i\in Z} X_i/\sum_{i\in V} X_i$ and $p = \sum_{i\in Z}\lambda_i/\sum_{i\in V}\lambda_i$.
We employ this statistic in this section.
§.§ Implementation
The recursive summation algorithm for Theorem <ref> is implemented in C.
The algorithm from Section <ref> to construct a sequence $B_1,\ldots,B_m$ having the running intersection property is implemented in MATLAB.
For $\mathcal{Z}$ given in (<ref>), if we suppose
\[
\textstyle
(X_i) = (2, 7, 7, 2, 2, 2, 2, 2, 2), \quad N=\sum_i X_i = 28,
\]
and $\lambda_i \equiv 1$, the maximum of the scan statistic is
$\max_{Z\in\mathcal{Z}}\varphi_Z(X_Z) = 5.167364$,
and the maximum is attained at $Z=\{2,3\}$.
The $p$-value is 0.01371293.
The number of summations is 314,621 (coinciding with the value by (<ref>)).
The computational time with C is less than 1 sec (MacBook Air 11-inch Early 2014 1.4GHz Intel Core i5).
Without using the proposed algorithm, the number of summations is 30,260,340,
and the computational time is 130 sec (ibid).
§.§ Monthly frequencies of spontaneous abortions
The method of spatial scan is applicable to detect clustering in the time domain.
Figure <ref> depicts monthly frequencies of trisomy among karyotyped spontaneous abortions of pregnancies.
The data are taken over 24 months from July 1975 to June 1977 in three New York hospitals.
Total count is $N=62$.
The data are tabulated in <cit.>.
Using these, we detect the clustering in the time domain,
that is, the period with a frequency higher than normal.
Monthly frequencies of trisomy among karyotyped spontaneous abortions of pregnancies.
<cit.> pointed out that the maximum frequency per consecutive two months is 14 for the time $t=18,19$.
He considered the conditional distribution of the maximum number of events for any arbitrary two-month period with $N=62$ given under the condition that events occur independently and uniformly over every 24 months, and reported that a value of 14 corresponds the $p$-value of $0.038$.
For the $p$-value formula, see <cit.> and <cit.>.
For the same data, we apply our method to obtain the exact $p$-value.
Let $L$ be the maximum window size, and we consider frequencies during the period less than or equal to $L$.
For example, when $L=3$, the scan window consists of all of one point of time, successive two-, and three-month periods.
That is,
\[
\mathcal{Z} = \bigl\{ \{1\},\ldots,\{24\},\{1,2\},\ldots,\{23,24\},\{1,2,3\},\ldots,\{22,23,24\}\bigr\}.
\]
The chordal graph used here for the calculation is given in Figure <ref>.
Chordal graph for the temporal clustering.
We apply <cit.>'s scan statistic in (<ref>) with the assumption that $\lambda_i$ are constant.
The largest five statistics for $L=5$ are listed in Table <ref>.
The last four columns provide the corresponding $p$-values,
which are also evaluated under the cases where $L$ is less than or equal to five.
Largest five statistics and their $p$-values.
statistic period
$L=5$ $L=4$ $L=3$ $L=2$
5.954 17,18,19 0.0175 0.0151 0.0135 NA
5.847 18,19 0.0217 0.0194 0.0180 0.0140
5.143 18,19,20,21,22 0.0453 NA NA NA
4.507 17,18,19,20 0.0716 0.0695 NA NA
4.507 16,17,18,19 0.0716 0.0695 NA NA
Table <ref> suggests that the period $t=17,18,19$ is detected as a temporal cluster with the smallest $p$-value of 0.0135.
§.§ Gallbladder cancer in Yamagata
Figure <ref> depicts the choropleth map of standardized mortality ratios (SMRs) for gallbladder cancer (among male) analyzed in <cit.> and <cit.>.
The data are for Yamagata Prefecture, Japan, which consists of 44 municipalities (villages, towns, and cities), for the period 1996–2000, based on the age-specific mortality rates from the 1985 national census population.
The total observed number of deaths was $N=147$.
SMRs of gallbladder cancer (among male) in Yamagata Prefecture 1996–2000.
Figure <ref> depicts the scan statistics $\varphi_N(X_Z)$ when $Z$ consists of one district under the same condition as Figure <ref>.
Scan statistic $\varphi_N(X_Z)$ when $Z$ consists of one district.
For these data, we apply two kinds of scan windows.
(i) Each window $Z$ consists of one district (i.e., $|Z|=1$),
and hence the number of windows is equal to the number of districts.
(ii) Each window consists of one district, or two districts adjacent to each other (i.e., $|Z|\le 2$).
The number of scan windows is (i) 44 ($|Z|=1$) and (ii) 154 ($|Z|\le 2$).
Unfortunately, the computation is infeasible for the latter case because the number of summations is more than $10^{14}$ (see Table <ref>).
Therefore, we randomly divided the 154 scan windows into two groups (groups 1 and 2), and calculated the $p$-values for each group and summed them.
This yields conservative $p$-values because this manipulation is nothing more than the Bonferroni correction.
The generated graphs and their chordal extensions are shown in Figure <ref>, and their features are summarized in Table <ref>.
Number of scan windows and generated graphs.
$N$ $M$ $n$ $e$ $\widetilde e-e$ $m$ deg # of summations
whole data 147 154 44 110 47 35 8 141,445,034,516,085
group 1 147 76 44 56 16 33 6 82,837,604,771
group 2 147 78 44 48 10 33 6 35,091,700,432
[t]12cm$N$: # of total events ($=\sum X_i$),
$M$: # of scan windows, $n$: # of vertices,
$e$: # of edges of original graph $G$, $\widetilde e$: # of edges in the chordal graph $\widetilde G$,
$m$: # of cliques of $\widetilde G$, deg: defined in (<ref>).
Table <ref> shows several of the largest scan statistics for the windows $|Z|=1$ and $|Z|\le 2$, and the corresponding $p$-values.
The district of Sakata and Yuza is detected as a spatial disease cluster with the smallest $p$-value of 0.00953.
Several of the largest statistics and their $p$-values.
statistic districts $|Z|\le 2$ $|Z|=1$
7.651 {Sa, Yu} 0.00953 NA
4.578 {Sa} 0.1847 0.0433
4.356 {Sa, Hi} 0.2247 NA
4.247 {Sa, Mi} 0.2541 NA
3.924 {Sa, Am} 0.3444 NA
3.570 {Sa, Ya} 0.4795 NA
3.364 {Ma} 0.5699 0.1739
3.205 {Fu, Ha} 0.6458 NA
3.071 {Yu} 0.7025 0.2065
[t]11cmSa: Sakata (municipality code: 06204), Am: Amarume (06422),
Fu: Fujishima (06423), Ha: Haguro (06424), Mi: Mikawa (06426),
Yu: Yuza (06461), Ya: Yawata (06462), Ma: Matsuyama (06463),
Hi: Hirata (06464).
Generated graphs for the windows $|Z|\le 2$
(left: group 1, right: group 2; circle: municipal capital, black line: edges of $G$, red line: additional edges for the chordal extension).
§ SUMMARY AND ADDITIONAL REMARKS
In this paper, we proposed a recursive summation method to evaluate a class of expectation in multinomial distribution, and applied it to the evaluation of the $p$-values of temporal and spatial scan statistics.
This approach enabled us to evaluate the exact multiplicity-adjusted $p$-values.
Our proposal has an advantage where the true $p$-value is too small and is barely estimated precisely by Monte Carlo simulations.
The proposed algorithm is easily modified to a class of distributions including
the normal distribution, the Dirichlet distribution, the multivariate hypergeometric distribution, and the Dirichlet-multinomial distribution
by replacing recursive summations with recursive numerical integrations if necessary.
On the other hand, our proposed method has a limitation that it only works when the total data count $N$ and the window size are not large.
The limitation is due to $N$, $|B_i|$ (the sizes of the maximum clique), and $|k^{-1}(i)|$.
As shown in Section <ref>, when the size of the scan windows is large, we can divide the whole scan windows into a number of groups, then compute the $p$-value for each group, and sum them.
We have implemented the proposed algorithms.
However, they are still under development.
Our algorithm requires for loop calculations,
where the numbers of the nests and the ranges of running variables depend on the input (scan windows).
The dimensions of the arrays also depend on the data.
These make source coding complicated, and the resulting code inefficient.
One approach to overcome this difficulty may be the use of a preprocessor to generate the C program.
This approach has been successfully used in computer algebra (e.g., page 674 of <cit.>).
§.§.§ Acknowledgments
The authors are grateful to Takashi Tsuchiya, Anthony J. Hayter, Nobuki Takayama, and Yi-Ching Yao for their helpful comments.
This work was supported by JSPS KAKENHI Grant Numbers 21500288 and 24500356.
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|
1511.00005
|
1Department of Astrophysics, American Museum of
Natural History, Central Park West at 79th Street, New York, NY
10024, USA
2Institut für Theoretische Astrophysik, Zentrum
für Astronomie der Universität Heidelberg, 69120 Heidelberg, Germany
3Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007
4Physics Program, The Graduate Center, CUNY, New York, NY 10016
5Kavli Insitute for Theoretical Physics, University of
California, Santa Barbara, Santa Barbara, CA 93106
Accretion disks around supermassive black holes (SMBHs) in active
galactic nuclei contain stars, stellar mass black holes, and other
stellar remnants, which perturb the disk gas
gravitationally. The resulting density perturbations exert
torques on the embedded masses causing them to migrate through the
disk in a manner analogous to planets in
protoplanetary disks. We determine the strength and direction of
these torques using an empirical analytic description dependent on
local disk gradients, applied to two different analytic,
steady-state disk models of SMBH accretion disks. We find that there
are radii in such disks where the gas torque changes sign, trapping
migrating objects. Our analysis shows that major migration traps
generally occur where the disk surface density gradient changes sign
from positive to negative, around 20–300$R_{\rm g}$, where $R_{\rm
g}=2GM/c^{2}$ is the Schwarzschild radius. At these traps,
massive objects in the AGN disk can accumulate, collide, scatter,
and accrete. Intermediate mass black hole formation is likely in
these disk locations, which may lead to preferential gap and cavity
creation at these radii. Our model thus has significant
implications for SMBH growth as well as gravitational wave source
§ INTRODUCTION
At present, the observational evidence for intermediate mass black
holes (IMBHs; $M\sim 10^2$–$10^6$ M$_{\odot}$) is much less
compelling than that for supermassive black
holes (SMBHs; $M>10^{6}$ M$_{\odot}$) or stellar mass black holes
($M\lesssim 40$ M$_{\odot}$). Several IMBH
candidates have been identified, including off-nuclear X-ray sources
such as HLX-1 (likely $\sim 10^3$–$10^5$ )
<cit.> and
optical emission line sources in dwarf galaxies
<cit.>. IMBHs are a missing
link between stellar-mass black holes and SMBHs, and indeed are
good candidates for the seeds of SMBHs <cit.>. IMBH
candidates are hard to confirm, although they are predicted to be
wandering throughout massive galaxy halos
<cit.> or lurking in dwarf
galaxies <cit.>.
An additional potential habitat for IMBHs is the accretion disks
around SMBHs in active galactic nuclei (AGN). Massive objects (stellar
remnants and stars) will exist in these disks
where they can collide, accrete and grow. If a mechanism exists to
efficiently collect compact objects into an orbit where they can
collide, this mass buildup could result in the efficient formation of
IMBHs in AGN disks.
Migration toward trapping orbits may be such a mechanism. Objects
orbiting within differentially rotating disks exchange angular
momentum with the gas around them as they orbit, which results in a
torque, typically causing the objects to migrate. Under the azimuthally
isothermal assumption, masses within disks were shown to migrate only
inwards <cit.>. However,
<cit.> found that in the more realistic case of an
adiabatic midplane, migration
can proceed outwards under some circumstances.
<cit.> used an extensive set of numerical
simulations to empirically define the conditions determining the sign
and strength of migration. Locations where the torque changes sign
from positive to negative have outwardly migrating objects meeting inwardly
migrating objects in an equilibrium, zero-torque orbit, forming a
migration trap. Such traps have been
predicted to exist in protoplanetary disks <cit.>, where
they can lead to rapid growth of giant planet cores
<cit.>. <cit.> pointed out that, by analogy,
IMBH might be able to form efficiently and grow at super-Eddington
rates in SMBH accretion disks, if they contained migration
traps. Eventually, the resulting object may be able to
clear a gap in the disk, which would produce a range of observational
signatures <cit.>.
Here we show that simple, analytic, steady-state models of AGN
disks do indeed predict migration traps, at radii that are independent
of the SMBH mass and the mass ratio between the migrator and the
central SMBH. We further briefly discuss the importance and
observational implications of migration traps in AGN disks.
§ METHODS
In this section we describe the torque model of
<cit.>, and discuss its application to two
different steady-state AGN accretion disk models.
§.§ Torque Model
The torque model is based on simulations performed to study the
behavior of objects in protoplanetary disks, but the physical
processes modeled are no different in optically-thick AGN accretion
disks. We assume that the mass of the migrating object (i.e. a
stellar mass black hole) remains constant, and neglect accretion or
feedback effects on the gas. The torque model includes a linear estimate of the Lindblad (wave) torque plus a simple but nonlinear contribution from adiabatic corotation torques. It is valid for the unsaturated case, where a
temperature gradient is maintained by turbulent and viscous diffusion,
as opposed to the gradient being erased as angular momentum is
transferred between the migrating object and nearby gas.
Saturation can be neglected so long as the diffusion timescale is
short compared to the libration timescale on which the torque acts
We model the torques using the analytical fits of
<cit.> to a broad range of simulations that
included non-isothermal effects and a non-linear model of adiabatic
corotation torques. For the locally isothermal case, the normalized
torque is
\begin{equation}
\Gamma_{\rm{iso}}/\Gamma_0 = -0.85 - \alpha - 0.9\beta,
\end{equation}
while for the purely adiabatic case the normalized torque is
\begin{equation}
\gamma \Gamma_{\rm{ad}}/\Gamma_0 = -0.85 - \alpha - 1.7\beta + 7.9\xi / \gamma.
\end{equation}
The adiabatic index $\gamma = 5/3$, and the variables $\alpha$,
$\beta$, and $\xi$ are the negative gradients of the local density,
temperature, and entropy, with values
\begin{equation}\label{eqn:torques}
\alpha = -\frac{\partial{\rm{ln} \Sigma}}{\partial{\rm{ln} r}}; \beta = -\frac{\partial{\rm{ln} T}}{\partial{\rm{ln} r}}; \xi = \beta - (\gamma - 1)\alpha .
\end{equation}
The torques are normalized by
\begin{equation}\label{eqn:normalize}
\Gamma_0 = (q/h)^2 \Sigma r^4 \Omega^2,
\end{equation}
where $q$ is the mass ratio of the migrator to the
SMBH, $h$ is the aspect ratio of the disk, and $\Omega$ is the
rotational velocity. Interpolating between the isothermal and
adiabatic torque regimes, we obtain
\begin{equation}\label{eqn:torquetotal}
\Gamma = \frac{\Gamma_{\rm ad}\Theta^2 + \Gamma_{\rm iso}}{(\Theta + 1)^2}
\end{equation}
where $\Theta$ is the ratio of the radiative and dynamical timescales
$t_{rad}/t_{dyn}$. <cit.> show that $\Theta$ depends on
the local disk properties as
\begin{equation}\label{eqn:theta}
\Theta = \frac{c_v \Sigma \Omega \tau_{\rm{eff}}}{12 \pi \sigma T^3}
\end{equation}
where c$_v$ is the thermodynamic constant with constant volume,
$\tau_{\rm{eff}}$ is the effective optical depth, and $\sigma$ is the
Stefan-Boltzmann constant. The value of $\tau_{\rm{eff}}$ is taken at
the midplane <cit.> as
\begin{equation}
\tau_{\rm{eff}} = \frac{3\tau}{8} + \frac{\sqrt{3}}{4} + \frac{1}{4\tau}
\end{equation}
where $\tau$ is the true optical depth, calculated by $\tau =
\kappa\Sigma/2$, where $\kappa$ is the opacity.
In summary, each of the torque components depends on the properties of the temperature, density, and entropy gradients. For particular values of these gradients, the torques may cancel, resulting in a region with zero torque, i.e. a migration trap. Our goal is to investigate whether stable traps exist, i.e. whether there are regions where the gradient of the torque is negative.
AGN disks are sufficiently ionized for magnetorotational instability to drive turbulence. The resulting density perturbations produce stochastic torques that can drive diffusive, random walk, migration <cit.>. <cit.> quantify when diffusive migration dominates over advective (type I) migration. Simulations of fully-ionized regions of stratified protoplanetary disks suggest that for interesting ranges of migrator mass and radius, type I migration prevails <cit.>. Such stochastic perturbations were shown by <cit.> to be necessary for multiple objects to reach equilibrium orbits and collide. We defer numerical simulations of the AGN case to future work.
§.§ Disk Models
We examine the torques expected in disks described by two
steady-state, analytic SMBH accretion disk models derived by
<cit.> and <cit.>. These models are derived from different basic
assumptions, but both contain many characteristics we expect in
realistic AGN disks. Neither model includes direct modeling of magnetic fields, nor effects due to general relativity.
SG assume a classical thin Keplerian $\alpha$-disk
<cit.> in a steady state with a constant, high,
accretion rate (Eddington ratio of 0.5). In order to remain stable
and prevent fragmentation (i.e. maintain $Q \gtrsim
1$), SG assume that stars form in the outer disk.
Energetic feedback from the newly formed stars increases the velocity
dispersion and sound speed of the gas, maintaining Q close to unity,
supporting the disk against global gravitational instability and
inhibiting further star formation. This approach is supported by the
existence of nuclear star clusters in the vicinity of SMBHs, which may
have formed in this way
<cit.>. The disk opacity model
of SG is based on <cit.> for high temperatures ($T
\gtrsim 10^4$ K) and <cit.> for lower temperatures.
Models of accretion disks from SG (blue) and TQM (red). From top to bottom, we show temperature, surface density (in g cm$^{-2}$), disk aspect ratio $h/r$, optical depth $\tau$, and Toomre Q vs radius. The top axis represents the translation from gravitational radius to parsecs for a $10^8$ SMBH.
The model of TQM, on the other hand, extrapolates a star-forming galaxy
disk inward to the SMBH.
Angular momentum transport is assumed to
take place due to global gravitational instabilities, such as bars and spiral
inflows, rather than unresolved turbulent viscosity. TQM use a more
up-to-date opacity model based on <cit.>. TQM address
gravitational fragmentation by considering two regimes: one where the external
accretion rate is high enough that the gas fraction of the disk
remains constant, allowing rapid inflow to continue; and another where
the star formation timescale is shorter than the gas advection time,
and thus accretion to the inner regions is more limited, as the gas is
consumed in star formation.
Figure <ref> shows profiles from both models
of the disk temperature $T$, surface density $\Sigma$, aspect ratio $h/r$, and optical depth
$\tau$. Although the profiles are
qualitatively comparable, there are major differences
between the models. For example, the surface
density and optical depth in SG are 2–3 orders of magnitude above
those of TQM in the inner disk. The differences in opacity and the assumed
dynamics of the inflow are
the root cause of these differences. SG assume that a constant
turbulent viscosity drives the inflow;
while TQM assume the inflow speed is a constant fraction of the local
sound speed. In both cases the high Thompson scattering opacity
from electrons produced by the ionization of hydrogen causes the inner
disk to be optically thick. At intermediate radii, where the electron
density drops precipitously, the opacity drops correspondingly,
allowing the disk to cool and become thinner. At larger radii,
where the temperature is low enough for dust grains to survive, dust opacity becomes important in the
disk, so the disk again thickens and cools further. The disk masses (integrated out to 1 pc) are $3.7 \times 10^7$ and $6.5 \times 10^6$ for SG and TQM, respectively.
§ RESULTS
Figure <ref> shows the result of calculating the torques
from Equation (<ref>) in a disk with the profile given
by SG around a $10^8$ SMBH for a migrator of mass 100 . The
figure shows the absolute value of the torque vs radius; black lines
represent negative torque, and thick red lines represent positive
torque. The spikes mark the points where the torque crosses zero. The
direction of the torque is also given by arrows for clarity. These
highlight the two migration traps in this disk model: one at $\log R =
1.39 R_{\rm g}$, and the other at $\log R = 2.52 R_{\rm g}$,
corresponding to 24.5 and 331 $R_{\rm g}$, or 0.0004 and 0.003 pc for
a 10$^8$ SMBH. The Toomre Q parameters at the trap locations are $\sim 10^5$ and 16, respectively, indicating that these regions are quite stable.
The absolute value of the torque $\Gamma$ for the SG model, scaled
by a factor of $10^{49}$ g cm$^2$ s$^{-2}$, vs. gravitational radius
R$_{\rm g}$. Black lines indicate where the torque is negative, and
thick red lines where it is positive. The arrows point in the
direction of the torque, and show that inward- and outward-pointing
torques meet at two of the zero-crossings, forming migration traps.
These estimates are for a fiducial value of $M_{\rm SMBH} =
10^8$ and mass ratio $q = 10^{-6}$.
However, we repeated
our calculations for a range of each value ($5 \times 10^5 < M_{\rm
SMBH} < 5 \times 10^9$ and $0.1 < q < 10^{-6}$) and found no
difference in the radial location of the migration traps in terms of
$R_g$. We should expect this result, since the variables that depend
on the mass ratio $q$ and $M_{\rm SMBH}$ are $\Gamma_0$ and $\Theta$,
as seen in equations (<ref>) and (<ref>). These mass adjustments change the magnitude of the torques but not their radial
position; i.e. the trap locations are not affected. However, the SG model assumes a particular value of $M_{\rm
SMBH}$. As we do not have access to their full set of models, we are
unable to vary the black hole mass self-consistently in our
Figure <ref> shows the results for the same torque
calculation using the TQM model. We find one migration trap, at $\log
R = 2.39 R_{\rm g}$ (245 $R_{\rm g}$, or 0.002 pc for a
10$^8$ SMBH). At this radius, Q = 3.5. This trap occurs precisely at the point where the
disk profiles are vertical, and the derivative is undefined (see
Figure <ref>). To explore the robustness of this result,
we made the profile differentiable by shifting the endpoints of each
vertical section of the profile to vary the slope. In the extreme
case, we adjusted the surface density profile to effectively round off
the sharp peak at $\log R = 2.4 R_{\rm g}$. Regardless of these
changes, the migration trap continues to exist at the point where the
surface density slope changes from positive to negative.
Significantly, migration traps also exist in the SG model at the same
locations—the points where the slope shifts from positive to
negative, indicating that the slope change of the surface density
profile is a key factor in determining where migration traps exist in
these models (see also <cit.>).
The absolute value of the torque $\Gamma$ for the TQM model, scaled by a factor of $10^{49}$ g cm$^2$ s$^{-2}$, vs. normalized radius $R/R_{\rm g}$. Black lines indicate where the torque is negative, and red thick lines where it is positive. The inset shows $\Gamma$ on a linear scale for a small region to better visualize the migration trap.
Note that in Figure <ref> there is a small surface
density discontinuity at $\log R \sim 3.2 R_{\rm g}$; however it does
not yield a migration trap in Figure <ref>. Again we
adjusted the endpoints of the vertical section of the profile to
verify the robustness of this result. We found that the magnitude of
the vertical change in the profile was insufficient to cause the
torque to change sign.
Thus, both a slope change and a large change in magnitude of the
surface density of an AGN disk appear to be needed in order to create
a migration trap.
§ IMPLICATIONS
The occurrence of migration traps in simple models of AGN disks
implies that IMBH may form efficiently and quickly due to stellar
black hole collisions at such locations, by analogy with giant planet
core formation at migration traps in protoplanetary disks
<cit.>. Ignoring migration traps, <cit.> conservatively predict that a $10 M_{\odot}$ black hole around a $10^8 M_{\odot}$ SMBH can double its mass via collisions and gas accretion in 10 Myr. However, including migration traps can boost the collision rate of disk objects by more than a factor of 100. For a migrator at $10^4 R_{\rm g}$ in a migration trap with enhanced surface density of compact objects of $\Sigma_0 = 350$ g cm$^{-2}$, assuming a reasonable distribution of eccentricities, the growth rate can reach over $dM/dt \sim 10^{-5} M_{\odot}$ yr$^{-1}$, which would result in a $10 M_{\odot}$ black hole growing to $\sim 100 M_{\odot}$ in 10 Myr. We also point out that the build-up of the IMBH (i.e. the merging of stellar mass black holes) is detectable in the local universe with LIGO. Nearby quasars such as Mrk 231 are good candidates to model and search for such events.
If this predicted growth occurs, there are observable implications, both in electromagnetic and gravitational radiation. For example, if the IMBH to SMBH mass
ratio becomes large enough ($q \geq 10^{-4}$) a gap can form in the disk at the
migration trap radius, leading to a flux decrement in the optical/UV disk
SED. <cit.>.
If the IMBH migrates into the central SMBH, a robust gravitational wave signal could be detected. Such a scenario is more likely for a lower mass ($M <10^{7.5}M_{\odot}$) primary or closer-in ($200$ $R_{\rm g}$) migrator. A binary system of mass $M_{b}=M_{1}+M_{2}$ will decay via
gravitational wave emission on a timescale <cit.>
\begin{equation}%\label{eqn:tgw}
\tau_{\rm GW} \approx \frac{5}{128}
\frac{c^5}{G^3}\frac{a_b^4}{M_b^2\mu_b}(1-e_b^2)^{7/2},
\end{equation}
where the binary reduced mass $\mu_{b}=M_{1}M_{2}/M_{b}$, the binary
semi-major axis is $a_{b}$, and its eccentricity $e_{b}\approx 0$.
Rewriting in terms of $M_1$,$M_2$, $a_b$ and normalized by
$R_{g1}=2GM_{1}/c^{2}$, the gravitational radius of the primary, yields
\begin{equation}
\tau_{\rm GW} \approx 0.01 \rm{Myr} \left(\frac{M_1}{10^6 M_{\odot}}\right)^2 \left(\frac{M_2}{10^3 M_{\odot}}\right)^{-1} \left( \frac{a_b}{200R_{g1}}\right)^4.
\end{equation}
For a fiducial AGN disk lifetime of $\sim$10 Myr, an IMBH formed at
$200$ $R_{\rm g}$ in a disk around a SMBH with $M <10^{7.5}M_{\odot}$
should merge with the primary within the disk lifetime. If such
mergers are common, detectable gravitational wave events
will be more frequent than previously supposed
<cit.> and can be observed by the planned $LISA$
mission <cit.>, with a
complementary electromagnetic counterpart observable via oscillations
in the FeK$\alpha$ line <cit.>.
On the other hand, IMBHs that form around more massive SMBHs, or more
than twice as far away in the disk will outlast the AGN disk and
survive, potentially until the next accretion episode. Such a binary
system can affect the galactic bulge, scattering stars and altering
the potential well. The IMBH can also itself grow due to gas accretion
<cit.>, changing the mass ratio of the system and
possibly being visible as a mini-quasar with shifting radial
velocities. We will return to some of these consequences in future
§ SUMMARY
Migration traps are equilibrium orbits in disks where regions of outward migration meet regions of inward migration. We study migration of massive objects in AGN accretion disks to determine whether migration traps exist in such environments. We
examine two different steady-state, analytic models of AGN disks, and
find that, despite the different assumptions used, migration traps
occur in both models. These migration traps occur at locations
of significant change in both the magnitude and gradient of the surface density.
In the traps, massive objects, such as stellar mass black holes, can
accumulate and merge, resulting in the formation of IMBHs.
These IMBHs could ultimately clear
out a gap in the accretion disk, producing multiple observable
signatures <cit.>. The buildup of IMBHs could be a significant gravitational wave source for LIGO, and mergers of these IMBHs with their
central SMBHs
would increase the number and strain amplitude of expected
gravitational wave sources detectable by $eLISA$.
Our prediction is based on analytical models that neglect evolution
and make strong simplifying assumptions about the dynamics. Further
studies may need to include effects shown to be important in
the protoplanetary context, including torques due to magnetic fields
<cit.>, and accretion heating feedback from the migrator
<cit.>. Accretion disk dynamics are more
complex than the assumptions of either SG or TQM, as can be
seen from the substantial differences between the models.
Ultimately, migration depends on the detailed physical state of
the disk, including the temperature, density, opacity, and turbulence.
We therefore stress that our results should not be interpreted
literally, but rather as a promising possibility worthy of further
detailed modeling. We also assume that trapped compact objects will have common orbits with low eccentricity, and merge without any dynamical consequences. Further studies must determine whether collisions of migrators will perturb the disk and affect migration, and whether the scattering of compact objects will result in ejections from the disk and prevent our scenario entirely. A full, three-dimensional, time-evolving model
will ultimately be needed in order to make robust predictions of
whether AGN disks can efficiently form IMBHs within migration traps.
Thanks to Alex Hubbard, Yuri Levin, Cole Miller, Shane Davis, and the anonymous referee for
useful discussions. JMB acknowledges generous support from the Helen Gurley
Brown Trust. M-MML acknowledges support from
NSF grant AST-1109395 and the Alexander von Humboldt Foundation.
BM and KESF acknowledge support from NSF PAARE grant
AST-1153335 and NSF grant PHY-1125915.
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|
1511.00279
|
Department of Physics and Astronomy, Appalachian State University,
Boone, NC 26808
Department of Physics and Physical Science, Marshall University, Huntington, WV 25755
Vatican Observatory Research Group, Steward Observatory, Tucson, AZ
Department of Physics and Astronomy, Appalachian State University,
Boone, NC 26808
Department of Physics and Astronomy, Appalachian State University,
Boone, NC 26808
Department of Physics and Physical Science, Marshall University, Huntington, WV 25755
This is the first in a series of papers presenting methods and results from
the Young Solar Analogs Project, which began in 2007. This project monitors
both spectroscopically and photometrically a set of 31 young (300 - 1500 Myr)
solar-type stars with the goal of gaining insight into the space environment
of the Earth during the period when life first appeared. From our
spectroscopic observations we derive the Mount Wilson $S$ chromospheric
activity index ($S_{\rm MW}$), and describe the method we use to transform our
instrumental indices to $S_{\rm MW}$ without the need for a color term. We
introduce three photospheric indices based on strong absorption
features in the blue-violet spectrum – the G-band, the Ca1 resonance
line, and the Hydrogen-$\gamma$ line – with the expectation that
these indices might prove to be useful in detecting variations in the surface
temperatures of active solar-type stars. We also describe our photometric
program, and in particular our “Superstar technique” for differential
photometry which,
instead of relying on a handful of comparison stars, uses the photon flux in
the entire star field in the CCD image to derive the program star magnitude.
This enables photometric errors on the order of 0.005 – 0.007 magnitude.
We present time series plots of our spectroscopic data for all four
indices, and carry out extensive statistical tests on those time series
demonstrating the reality of variations on timescales of years in all four
indices. We also statistically test for and discover correlations and
anti-correlations between the four indices. We discuss the physical basis of
those correlations. As it turns out, the “photospheric” indices
appear to be most strongly affected by emission in the Paschen continuum. We
thus anticipate that these indices may prove to be useful proxies for
emission in the ultraviolet Balmer continuum. Future papers in this
series will discuss variability of the program stars on medium
(days – months) and short (minutes to hours) timescales.
§ INTRODUCTION
The Young Solar Analogs Project is a long-term spectroscopic and
photometric effort to monitor a sample of Young Solar Analogs (YSAs) in
order to gain a deeper understanding of their magnetically related stellar
activity. YSAs give us a window into the conditions in the early solar
system when life was establishing a foothold on the Earth. That early life
had to contend with a hostile space environment, including strong ultraviolet
fluxes from a young active sun (without the benefit of an ozone layer),
an enhanced solar wind, strong and frequent flares, as well
as significant variability in the solar irradiance. By studying solar-type
stars with ages corresponding to this period ($\sim$0.3 – 1.5 Gyr) in the
history of the solar system, we can gain insight not only into the conditions
on the early Earth, but a better understanding of the space environment
experienced by Earth analogs, and the implications that might have for the
development of life on those worlds.
Stellar activity is closely related to the dynamics of the magnetic field of
the star. The existence of the chromosphere and corona and the associated
far-ultraviolet (FUV), extreme-ultraviolet (EUV) and X-ray emissions
of a solar-type star are the result of magnetic heating, and solar and
stellar active regions are associated with strong local enhancements in
the stellar magnetic field. The direct detection of the magnetic fields of
solar-type stars is difficult and direct measurement of FUV (both emission-line
and Balmer continuum), EUV, and X-ray fluxes requires space-based
observations, so the monitoring of
magnetic activity and FUV, EUV, and X-ray fluxes in those stars depends upon
more easily measured proxies such as, traditionally, the chromospheric
flux in the cores of the Ca2 H & K lines. Recent studies have shown
that Ca2 H & K fluxes are correlated in solar-type stars with
both X-ray luminosities <cit.>;
<cit.> and FUV excesses <cit.>. Thus
ground-based monitoring of Ca2 H & K fluxes has played and
continues to
play a vital role in the study of stellar magnetic activity, and serves as
a valuable proxy for the direct measurement of ultraviolet and X-ray fluxes.
Long-term monitoring of the Ca2 H & K fluxes in a sample of F-, G-,
and K-type dwarfs began at Mount Wilson in 1966 <cit.>, and
continued until 2003. That program monitored about 100 stars on a
continuous basis. The stars in the Mount Wilson program range from young
stars with very active chromospheres to old stars with minimal activity.
The program discovered stellar activity cycles similar to that of the
Sun in about 60% of the sample, with a further 25% varying with no
well-defined cycle, and the remainder showing little variation at all.
The Lowell Observatory SSS (solar-stellar
spectrograph) program started in 1988 and continues today
<cit.>. It employs a fiber-fed spectrograph that enables Ca2
H & K
measurements to be carried out on both the Sun and stars with the same
instrument. That program, unlike the Mount Wilson project, focuses closely
on 28 stars that are most like the Sun in terms of spectral type
(F8 – G8, with most in the range G0 – G2). Many in this sample are “solar
twins”, and thus have ages and metallicities similar to that of the Sun,
but a few of the program stars may be described as “young solar analogs” with
activity levels much higher than the Sun. Lowell Observatory, unlike the
Mount Wilson project, carries out near-contemporaneous precision photometric
observations, in the Strömgren $b$ and $y$ bands, of a number of the
SSS-program solar-type stars as well as
others <cit.>. They have found, as might be expected in
analogy with the Sun, that many of the SSS stars are brightest when at the
highest activity levels, but, surprisingly, others are faintest when most
active. It is the most active stars in their sample that show an inverse
correlation between brightness and activity, suggesting that stars, as they
age and decline in activity, flip from inverse- to direct-correlation behaviors.
The “Sun in Time” project <cit.> carried out, over the course 20 years,
multi-wavelength studies of a small sample of solar analogs (G0 - G5) with ages
ranging from $\sim 50$ Myr to 9 Gyr. That project
found that the early Sun was most likely rotating 10 times faster
than at present and
that its coronal X-ray and transition-region/chromospheric EUV and FUV
fluxes were several hundred times higher than the present. This project
as well confirmed that Ca2 H & K observations are useful proxies for
estimating X-ray, EUV, and FUV fluxes and variability.
Spectroscopic features in the optical other than the Ca2 H & K lines
may yield useful stellar activity data. The core of the H$\alpha$ line samples
the chromosphere, but other strong features in the spectrum may be sensitive to
photospheric manifestations of stellar activity. Prime among these in the
blue-violet region of the spectrum are the 4305Å G-band (a molecular
feature arising from the CH molecule), the 4227Å
Ca1 resonance line, and the 4340Å H$\gamma$ line. These three
features are temperature sensitive in late-F, G, and early K-type stars,
with the G-band increasing in strength through the F and G-type stars, coming
to a broad maximum in the late G-type through early K-type stars and then
declining toward later types.
The Ca1 resonance line is negatively correlated with the effective
temperature, and the H$\gamma$ line positively correlated.
Thus these spectral features may be useful in tracking the presence and
areal coverage of sunspots and faculae on the photospheric disk. In addition,
high-resolution images of the solar surface taken in the G-band show bright
points (GBPs) that are strongly correlated with magnetic structures such as
intergranular lanes and extended facular regions <cit.>.
We will discuss in Sections <ref>, <ref>, and <ref>
our definition of spectroscopic indices for the measurement of the G-band,
the Ca1 resonance line, and the H$\gamma$ line. In
5.1 we test the sensitivity of these photospheric indices to temperature
variations, and in 5.4 examine correlations between these indices and with
the Mount Wilson chromospheric activity index. These tests enable us to
evaluate the usefulness of these indices as temperature indicators.
For the purpose of this project, we define a YSA as an
F8 – K2 dwarf with an age between 0.3 and 1.5 Gyr. A sample of 40 candidate
YSAs north of $-10$ were chosen from the NStars project <cit.> on
the basis of the following criteria: 1) Their spectral types should lie
between F8 – K2, as we are interested in solar-type stars, and not late-K
and M-type dwarfs. In addition, within that spectral-type range, the
“photospheric” features we have identified (G-band, Ca1 resonance
line, and the H$\gamma$ line) may be measured with sufficient accuracy. 2) The
stars should be north of $-10^\circ$ declination, and sufficiently bright
($V < 8.0$) that they may be observed at high signal-to-noise (S/N $\ge 100$)
on a routine basis in a reasonable length of time with our equipment (see
<ref>) and 3) they should have
ages approximately between 0.3 and 1.5 Gyr, for the reasons explained above.
Initial ages were estimated on the basis of the “snapshot” Ca2 H & K
activity measures provided by the Nearby Stars project, and the calibration of
<cit.> and, later, when it became available, and we had derived
better average activity measures of our program stars, that of
<cit.>. Some ages were also refined via the determination of
rotational periods <cit.>. The list was thus culled to 31 YSAs
(see Table <ref>). Many of these stars have been monitored
spectroscopically since 2007. We note that this list includes the star
HD 189733, even though that star apparently has an age $> 4$Gyr. The
activity age of HD 189733 is approximately 600 Myr <cit.>, but
this young age is inconsistent with the low X-ray flux of its M-dwarf
companion <cit.>. Its rapid rotation and high activity
presumably derives from the transfer of angular momentum from a close-orbiting
hot jupiter <cit.>. We have
retained this star in our program not only because of its intrinsic interest,
but because insights may come from comparing its activity behavior to young
stars with similar rotation periods and activity levels.
Young Solar Analog Stars
Basic Observational Data
Name SpTa V B-V Duplicityb Programc
HD 166 G8 V 6.10 0.75 s,a
HD 5996 G9 V (k) 7.67 0.75 s
HD 9472 G2+ V 7.63 0.68 s
HD 13531 G7 V 7.36 0.70 s
HD 16673 F8 V 5.78 0.52 s
HD 27685 G4 V 7.84 0.67 s,c
HD 27808 F8 V 7.13 0.52 s,c
HD 27836 G0 V (k) 7.61 0.60 s,c
HD 27859 G0 V (k) 7.80 0.60 s,c
HD 28394 F8 V 7.02 0.50 SB,c
HD 42807 G5 V 6.44 0.66 s SSS
HD 76218 G9- V (k) 7.69 0.77 s
HD 82885 G8+ V 5.41 0.77 V(Bd) MtW,SSS
HD 96064 G8+ V 7.64 0.77 V(B: M0+ Ve)
HD 101501 G8 V 5.32 0.72 s MtW,SSS
HD 113319 G4 V 7.55 0.65 s
HD 117378 F9.5 V 7.64 0.56 s
HD 124694 F8 V 7.19 0.52 cpm
HD 130322 G8.5 V 8.04 0.78 Ex;hj
HD 131511 K0 V 6.01 0.83 SB
HD 138763 F9 V 6.51 0.58 s
HD 149661 K0 V 5.76 0.83 V ? MtW
HD 152391 G8.5 V (k) 6.64 0.76 s MtW
HD 154417 F9 V 6.01 0.58 s MtW
HD 170778 G0- V (k) 7.52 0.59 s
HD 189733 K2 V (k) 7.65 0.93 V,Ex;hj
HD 190771 G2 V 6.17 0.64 V
HD 206860 G0 V 5.94 0.59 V (T2.5e),Ex;j MtW
HD 209393 G5 V (k) 7.97 0.68 s
HD 217813 G1 V 6.64 0.60 s
HD 222143 G3 V (k) 6.58 0.65 s
aSpectral types from <cit.> and <cit.> unless
otherwise indicated.
bKey to duplicity notes: s = single, a = member of association,
c = member of cluster, SB = spectroscopic binary, V = visual binary (along
with spectral types of companions, if known), cpm = common proper motion
companion, Ex = exoplanet host: hj = hot jupiter; j = jupiter-mass planet.
cThe stars indicated are in common with other spectroscopic
activity programs, in particular MtW = Mount Wilson project
<cit.> and the Solar/Stellar spectrograph project <cit.>.
dSimbad lists a spectral type of M5 V for HD 82885B, but gives
no source.
eBrown dwarf companion <cit.>.
The Lowell SSS project has shown the importance and value of contemporaneous
photometry, and so we added a photometric component to our project in 2011.
We monitor our program stars in 5 photometric bands, the Strömgren-$v$
($\lambda_{\rm eff} = 4100$Å), Johnson-Cousins $B$
(4450Å), $V$ (5510Å), and $R$
(6530Å) bands, and a 3 nm-wide passband centered on the
H$\alpha$ line (6563Å). This photometric system is optimized to detect
stellar-activity variations. For instance, it is well-known that late-type
active stars show greater variability at shorter wavelengths; this is
related to a greater contrast between the photosphere and the spots, and
a similar increase in the contrast between the photospheric faculae and
the photosphere at those wavelengths. During flare events, emission in
the Paschen continuum rises sharply with decreasing wavelength. For both
these reasons, it is expected that photometric variability will be more
apparent in the Strömgren-$v$ filter than in the Strömgren-$b$
($\lambda_{\rm eff} = 4670$Å) filter employed by the Lowell SSS project.
Variation in stellar activity,
especially during flare events, should also be apparent in the H$\alpha$
line. We will examine the relationship between these photometric data and
the spectroscopic indices we present in this paper in Paper II of this
§ OBSERVATIONS
§.§ Spectroscopy
Spectroscopic observations for this project have been carried out primarily
with the G/M spectrograph on the Dark Sky Observatory (Appalachian State
University) 0.8-m reflector. Except for early in the endeavor, observations
for this
project on that instrument have been obtained with the 1200 g mm$^{-1}$
grating in the first order. That grating gives a spectral range of
3800 – 4600Å, with a resolution of 1.8Å/2 pixels ($R \sim 2300$). This
spectral range includes the Ca2 H & K lines as well as the
Ca1 resonance line, the G-band, and the H$\gamma$ line. Exposures
have been calculated to give a S/N of at least 100 in the continuum near
the Ca2 H & K lines, which means that the S/N near the G-band
is consistently better than 150. A few early observations were made with
the 600 gmm$^{-1}$ grating (used in the first order), yielding a resolution
of 3.6Å/2 pixels and the 1000 g mm$^{-1}$ grating (used in the second order)
giving a resolution of $\sim 1$Å/2 pixels. Before April 2009, our spectra
were recorded on a thinned, back-illuminated $1024 \times 1024$ pixel
Tektronics CCD operated in the multipinned-phase mode. Since April 2009,
we have been using an Apogee
camera with a $1024 \times 256$ pixel e2v technologies CCD30-11 chip with
enhanced ultraviolet sensitivity. These two chips have very similar pixel
sizes and spectral sensitivities, and we have detected only minor changes
in the instrumental systems (detailed below) in the transition between the two
An Fe-Ar hollow-cathode comparison lamp was observed for wavelength
calibrations, and the spectroscopic data were reduced with
IRAF[IRAF is distributed by the National Optical Astronomy
Observatory, which is operated by the Association of Universities for
Research in Astronomy, Inc. under cooperative agreement with the National
Science Foundation.] using standard techniques.
Since January 2013 the VATTspec spectrograph on the Vatican Advanced
Technology Telescope (VATT; 1.8-m, located on Mount Graham, Arizona) has
also been used for this project, primarily for
high-cadence, high-S/N observations designed to detect flares and other
short-term events on these stars. Those observations will be discussed in a
later paper in this series. For these observations,
the VATTspec is used with a 1200 g mm$^{-1}$ grating which gives a resolution
of 0.75Å/2 pixels in the vicinity of the Ca2 H & K lines, with
a spectral
range of 3640 – 4630Å. The spectra are recorded on a low-noise STA0520A
CCD with 2688$\times$512 pixels (University of Arizona Imaging Technology
serial number 8228). Two hollow cathode lamps, Hg and Ar, were observed
simultaneously for wavelength calibrations, and the spectroscopic data were
again reduced with IRAF using standard techniques.
We have also obtained high-resolution echelle data for six of our stars with
the FIES spectrograph on the Nordic Optical Telescope <cit.>. These
data, which were obtained under the Nordic Optical Telescope Service Observing
Program employed the FIES spectrograph with the high-resolution fiber,
yielding a resolution of 65,000, and a spectral range from 3640 – 7360Å.
Spectra from the FIES spectrograph were reduced with FIEStool.
§.§ Photometry
An important component of the Young Solar Analogs project is concurrent
photometry of our program stars. The analysis of this photometry and how it
relates to our spectroscopic observations will be the subject of Paper II in
this series.
In March 2011 we began obtaining photometric observations in the
Strömgren-$v$, Johnson-Cousins $B$, $V$, $R$ and narrowband H$\alpha$
filter system, described in the previous section, by employing a CCD camera on
a 0.15-m 1300mm focal-length astrograph attached to the 0.8-m Dark Sky
Observatory reflector. The detector is a KAF-8300 monochrome CCD, operated
with on-chip $2 \times 2$ binning
to give an effective pixel size of $10.8 \times 10.8\mu$m. The CCD utilizes
an SBIG “even illumination shutter” which ensures uniform exposures over
the entire field even for very short exposures. Flat fields are obtained
every night with a “Flipflat” luminescent panel which offers more consistent
flats than sky flats. This instrument, which has
a 48 $\times$ 36 field of view, is known as the “Piggy-back”
telescope. It enables us to obtain photometry simultaneously with the
In April 2012 we installed a small robotic dome at the Dark Sky Observatory
containing a clone of the Piggy-back telescope mounted on a German equatorial
mount. This robotic telescope employs the CCDAutopilot5 and Pinpoint
software which, when combined, allow fully automated operation with precise
and consistent centering of the object to within a few arcseconds. This
telescope enables us to obtain photometry on every clear night, as the
YSA project has access to the 0.8-m and Piggy-back telescopes only
$\sim 11 - 12$ nights a month.
Both the Robotic and the Piggy-back telescopes are operated very slightly
out of focus so that the star image is spread over a number of pixels. This
enables more precise photometry. Multiple exposures are obtained for each
target, which are reduced and then combined using the IRAF xregister
Since August 2014 we have also obtained photometry with a wide-field
imager mounted on the Robotic telescope. This wide-field imager consists of
an ST-8300 SBIG CCD, a filter wheel with Johnson $B$, $V$ and $R$ filters, and
a Pentax 150mm f/3.5 camera lens. This setup yields a
$6.9^\circ \times 5.3^\circ$
field of view, and supplements the Robotic telescope data for program stars
which do not have sufficient comparison stars in the 48 $\times$
36 field of view of the main telescope.
§.§.§ Photometric Reduction Technique
Reducing the photometric data from the Piggy-back and Robotic telescopes is
challenging in a number of ways. First, despite the small aperture (0.15-m),
some of our program stars are bright ($V < 6$), which requires short exposures.
To mitigate these difficulties, the telescopes are slightly defocused, and we
obtain multiple exposures which are stacked using IRAF routines which preserve
the stellar flux. None of our fields are crowded, and so photometry is
carried out on the stacked images using the IRAF APPHOT package.
We utilize differential photometry to determine the magnitudes of
our program stars. In most cases, the program star is the brightest in the
field. Suitable comparison and check stars are typically one or two magnitudes
fainter than the program star, so the standard differential photometry
technique leads to unacceptably large photometric errors. To achieve
better photometric
accuracy we have devised an improved method, which we call the “Superstar
technique” (SST). The SST, instead of utilizing a handful of comparison
stars, considers
the photon flux in the entire star field in the image. Thus the SST adds up the
flux from many different sources, both
bright and faint, and constructs from that summed flux a “super” comparison
star that often has comparable flux to the program star. The technique
compares each individual source against the summed flux, thus enabling, in
an interactive fashion, the elimination of variable stars from the final summed
flux. In this way a reference file of comparison stars, often 20 – 50 objects,
(the “reference stars”) is constructed. The individual fluxes in that
reference file are based on
averages over a large number of nights, so the relative fluxes are known to
high precision. To determine the magnitude of the program star
for a given night, the SST identifies as many of the reference stars as
possible on the stacked frame for that night (it is not necessary to identify
all of the reference stars) and uses those identified to construct the
“super” comparison star. The summed flux for that super comparison is
compared to the summed flux of the identified stars in the reference file,
and that ratio enables the calculation of a $\Delta m$ for that
particular observation. That $\Delta m$ is added to the instrumental
magnitude of the program star to give the magnitude for that observation.
The magnitudes so determined are not yet on the standard system, but are
offset by a constant zeropoint shift. If a number of the reference stars
have measured magnitudes on a standard system, they can be used to calculate
that zeropoint shift. However, most of our work can
be carried out in the instrumental system.
The Superstar technique gives best results when the program star is situated
in a rich stellar field, enabling the summation of scores of stellar fluxes
into the single super comparison star. For those stars in our program for
which 20 or more reference stars are available, the typical photometric
errors in the individual Johnson-Cousins $B$, $V$, and $R$
magnitudes are on the order of 0.005 – 0.007 mag. The errors in the
Strömgren-$v$ and H$\alpha$ bands tend to be somewhat higher:
0.007 – 0.010 mag. For the brightest stars in our program and stars with
sparse fields ($ < 20$ reference stars)
the errors are higher, and typically range, on good nights, from 0.010 - 0.015
magnitude, with slightly higher errors in Strömgren-$v$ and H$\alpha$.
These are the stars that will benefit from the photometry obtained with the
wide-field imager that is mounted on the Robotic telescope (see above).
We defer a deeper discussion of the photometric errors until Paper II which
will be devoted to an analysis of the photometric data as well as its
relationship to the spectroscopic data discussed in this paper.
§ BASIC PHYSICAL PARAMETERS
Young Solar Analog Stars
Basic Physical Data
Name $T_{\rm eff}$(K) $\log g$ [M/H] $\xi_t$ $v\sin i$ km s$^{-1}$
km s$^{-1}$
HD 166 5454 4.52 $+0.05$ 1.3 4.5 (0.2) Keck
HD 5996 5463 4.60 $+0.01$ 0.7 0.0 (1.5) Elodie
HD 9472 5705 4.46 $-0.03$ 1.1 3.1 (0.2) Keck
HD 13531 5595 4.54 $-0.02$ 1.1 6.1 (0.1) Keck
HD 16673 6241 4.38 $-0.05$ 1.3 7.3 (0.2) Elodie
HD 27685 5681 4.43 $+0.13$ 1.0 1.6 (1.0) Elodie
HD 27808 6217 4.31 $+0.11$ 1.2 12.7 (0.2) Elodie
HD 27836 5843 4.35
HD 27859 5887 4.36 $+0.06$ 1.2 7.3 (0.2) Keck
HD 28394 6243 4.31 $+0.09$ 1.2 22.0 (1.0) Keck
HD 42807 5722 4.55 $-0.03$ 1.2 5.0 (0.2) Keck
HD 76218 5380 4.56 $+0.07$ 1.0 3.4 (0.2) Keck
HD 82885 5487 4.43 $+0.29$ 1.3 3.2 (0.2) Keck
HD 96064 5402 4.54 $+0.13$ 0.6 2.8 (0.5) Elodie
HD 101501 5535 4.55 $-0.04$ 1.0 2.8 (0.4) Keck
HD 113319 5736 4.53 $-0.05$ 1.1 3.6 (0.2) Keck
HD 117378 6000 4.51 $-0.07$ 1.3 10.2 (0.2) NOT
HD 124694 6195 4.44 $+0.05$ 1.2 17.6 (0.5) NOT
HD 130322 5385 4.53 $+0.05$ 1.0 0.0 (1.5) Elodie
HD 131511 5215 4.51 $+0.07$ 1.2 4.7 (0.2) NOT
HD 138763 6040 4.43
HD 149661 5255 4.57 $-0.01$ 1.0 1.5 (0.2) Paranal
HD 152391 5443 4.53 $+0.02$ 1.2 4.3 (0.2) NOT
HD 154417 6022 4.42 $-0.02$ 1.4 6.8 (0.2) Keck
HD 170778 5925 4.48 $+0.01$ 1.3 7.9 (0.2) NOT
HD 189733 5049 4.59 $+0.04$ 1.1 2.9 (0.2) Keck
HD 190771 5789 4.45 $+0.12$ 1.5 5.4 (0.2) NOT
HD 206860 5986 4.49 $-0.07$ 1.5 10.0 (0.2) Keck
HD 209393 5670 4.58 $-0.10$ 1.0 4.0 (0.2) Keck
HD 217813 5876 4.45 $+0.00$ 1.5 4.4 (0.2) Keck
HD 222143 5787 4.43 $+0.06$ 1.3 3.2 (0.2) Keck
Sun 5774 4.44 $+0.00$ 1.0 1.8 (0.2) NSO
aKeck: The Keck Observatory Archive
https://koa.ipac.caltech.edu/cgi-bin/KOA/nph-KOAlogin; Elodie: The Elodie
Archive http://atlas.obs-hp.fr/elodie/, <cit.>; NOT: Nordic
Optical Telescope Service Observing Proposal P50-410; Paranal: The UVES
Paranal Observatory Project (POP), <cit.>,
http://www.eso.org/sci/observing/tools/uvespop.html; NSO: <cit.>.
Table <ref> presents basic physical data, namely effective
surface gravities ($\log g$), metallicities ([M/H]), microturbulent velocities
($\xi_t$), and projected rotational velocities ($v\sin i$) for the program
stars. The effective temperatures were determined using the
infrared flux method formulae of <cit.>, specifically, those
for $b-y$, $B-V$, and $V-K_s$, where $K_s$ is the 2MASS K-magnitude
<cit.>. The effective temperatures presented are straight means
of the values based on those three indices, except for some of the brighter
stars for which $K_s$ is saturated and thus unreliable. The statistical error
associated with these temperatures is on the order of $\pm 70$K, with an
additional systematic error in the zeropoint of the system of about $15 - 20$K
<cit.>. The gravities were calculated via the absolute
bolometric magnitudes, based on Hipparcos parallaxes as recalculated by
<cit.> and bolometric corrections from <cit.> along
with the mass-luminosity relationship from <cit.>, and have
errors on the order of $\pm 0.10$ in the $\log$. Metallicities,
microturbulent velocities, and projected rotational velocities were calculated
from measurements of high-resolution archival spectra from the HIRES
spectrograph on the Keck 10-m telescope, the Elodie spectrograph on the 193-cm
telescope at the Observatoire de Haute-Provence, the UVES spectrograph on the
ESO VLT provided by the UVES Paranal Observatory Project, as well as new
observations with the FIES spectrograph on the Nordic Optical Telescope.
Projected rotational velocities were calculated with the cross-correlation
method. To do this, we first estimated the line-spread function (LSF) for each
spectrum by measuring the FWHM in ångstroms of a number of telluric lines
in the atmospheric $\alpha$-band of oxygen, centered $\sim 6300$Å or, in
some cases the $\alpha^\prime$ band centered near 5800Å, and
then transformed that FWHM to the echelle orders containing the spectral range
6050 – 6200Å where most of the measurements for calculating $v\sin i$ and
[M/H] were made. Once the LSF was characterized, we computed synthetic
spectra in the 6050 – 6200Å range with the
code of <cit.> and solar-metallicity
ATLAS12 models <cit.> calculated with the effective
temperatures and gravities in Table <ref>. Those synthetic
spectra were then
convolved with the LSF. Cross correlations were obtained between the
synthetic spectrum and the observed spectrum, and the synthetic spectrum and
rotationally broadened versions of itself for a range of rotational velocities.
These cross correlations were normalized at a common point and compared
to derive the rotational velocity of the program star. Our results are in
very good agreement with those of <cit.> who used the
cross-correlation method of <cit.>.
Once the LSF and the $v\sin i$ were known, we used a $\chi^2$ minimization
method comparing the observed and synthetic spectra to determine both the
metallicity and the microturbulent velocity for
each program star. For the synthetic spectra, we used a spectral line list
in the region 6050 – 6200Å with updated $\log(gf)$ values from the
NIST Atomic Spectra Database, version 5.2 <cit.>. Broadening parameters
and $\log(gf)$ values were adjusted, when necessary, by
reference to the Solar Flux Atlas <cit.>. The metallicities and
microturbulent velocities are recorded in Table <ref>. We
estimate errors in
that Table to be $\pm 0.05$ dex for the metallicity, and about
$\pm 0.3$ km s$^{-1}$ for the microturbulent velocity.
The projected rotational velocities will be used in a later paper in
this series to
interpret periodicities observed in our activity and photometric data.
§ SPECTROSCOPIC INDICES FOR STELLAR ACTIVITY
Our project measures four spectroscopic indices from the spectra obtained on
the G/M spectrograph. These are the Ca2 H & K chromospheric
activity index, based
on the Mount Wilson “$S$” index (hereinafter $S_{\rm MW}$), and indices for
the Ca1 4227Åresonance line, the 4305Å G-band, and the 4340Å H$\gamma$ line.
§.§ Ca2 H & K chromospheric activity indices
§.§.§ Definition and Measurement of the Instrumental Indices
<cit.> and <cit.> introduced the Mount Wilson
chromospheric activity index, $S_{\rm MW}$, which recorded the chromospheric flux
in the cores of the Ca2 H & K lines in ratio with flux in the
“continuum” on either side of those lines. Their instrument employed
effective triangular bands with full width at half peak of 1.09Å centered
on the cores of the H & K lines, and continuum bands of 20Å width to the
violet side (3891.067 – 3911.067Å) and the red (3991.067 – 4011.067Å).
The fluxes measured through these bands are ratioed to give the $S_{\rm MW}$
We measure two instrumental indices from the DSO spectra, the $S_2$ index
which measures the flux in the cores of the H & K lines with 2Å-wide
rectangular bands and the $S_4$ index which employs 4Å-wide rectangular
bands in the H & K cores. Both indices utilize the same continuum
bands as the Mount Wilson Project. The indices are calculated (in analogy
with the Mount Wilson index) with the equations
S_2 = 5f_K2 + f_H2/f_v + f_r
S_4 = 5f_K4 + f_H4/f_v + f_r
where the $f$'s are the monochromatic fluxes (i.e. the integrated flux
divided by the bandwidth) through the various bands described above. In
particular, $f_{K2}$ and $f_{K4}$ are the fluxes measured in the core of the
Ca2 K-line using 2 and 4Å bandpasses respectively; $f_{H2}$ and
$f_{H4}$ are the same for the Ca2 H-line, and $f_v$ and $f_r$ are the
fluxes in the two continuum bands. The
DSO spectra do not have sufficient resolution to directly measure 1Å
fluxes in
the cores of the Ca2 H & K lines.
However, the 0.75Å/2 pixel resolution of the VATTspec spectra does
allow direct
measurement of an $S_1$ index, which employs rectangular 1Å passbands in
the cores of the H & K lines. The advantage of the $S_1$ index is that it
is closer to the original instrumental system of the Mount Wilson project
(although that project utilized a triangular passband) and the transformation
from $S_1$ to $S_{\rm MW}$ is linear and does not involve a color ($B-V$) term,
the $S_2 \rightarrow S_{\rm MW}$ and $S_4 \rightarrow S_{\rm MW}$ transformations
are both nonlinear and require a color term (see below).
Steps in the measurement of the $S_1$, $S_2$, and $S_4$ indices include
transforming the stellar spectrum in question to the rest frame of the star,
the rebinning of the spectrum to a uniform spacing of 0.1Å, followed by
the numerical integration of the spectrum in the various passbands. We
employ the raw (non-flux-calibrated) spectrum for these calculations. The
division by the sum of the continuum fluxes ($f_v + f_r$) in the above
equations accounts for changes in the slope of the continuum due to
differing amounts of atmospheric extinction, although for routine observations
we attempt to observe the star as close to the meridian as possible. For
moderately high S/N spectra (S/N $> 100$), all
three indices may be measured to a precision of $\sim 0.001$ in the
§.§.§ Calibration of the Instrumental Indices: Transformation to the
Mount Wilson index
The transformation of $S_4$ to $S_{\rm MW}$, as described in <cit.> is
problematical, as the relationship is highly nonlinear. In addition, it
was not appreciated at the time that there is a small but significant color
term in the transformation. The transformation for $S_2$ is better behaved,
but is still non-linear, and a color term is still required. As stated
above, the $S_1$ indices measured in the VATTspec spectra are linearly
correlated with the Mount Wilson $S_{\rm MW}$, and that transformation does not
involve a color term. To derive that transformation, we have observed
with the VATTspec a number of the chromospheric activity calibration stars
used by <cit.> in their original calibration of $S_{18}$, which is the
same as the $S_4$ index of the present paper.
The relationship between the VATTspec $S_1$ index and the mean $S_{\rm MW}$
recorded for those calibration stars in <cit.> is given by:
S_MW = -0.0011 + 4.6920S_1 σ= 0.0119
and illustrated in Figure <ref>. The goodness of fit is not
improved with a quadratic term, and the residuals show no correlation with
$B-V$. Most of the scatter in that relationship may be traced to the
variability of the calibration stars, especially the more active
calibration stars.
The $S_1 \rightarrow S_{\rm MW}$ (Mount Wilson) transformation for
spectra. The calibration is linear, and has no significant color term.
As mentioned above the $S_2 \rightarrow S_{\rm MW}$ and the $S_4 \rightarrow
S_{\rm MW}$
transformations are both non-linear and require a color term. The non-linear
nature of these transformations is problematical when attempting an
extrapolation of the transformation to very active stars. Because the
resolution of the DSO spectra
is $\sim 1.8$Å/2 pixels, we cannot directly measure a DSO $S_1$ index.
experimentation with the VATTspec spectra suggests a solution. The actual
Ca2 H & K chromospheric emission in main-sequence stars is
intrinsically narrow
(FWHM $\sim 0.5$Å), narrower than even the 1Å passband employed by
the Mount Wilson project.
That flux is entirely contained in the H & K passbands employed in the
$S_1$, $S_2$, and $S_4$ indices, but those passbands involve successively
amounts of photospheric flux. This suggests that it should be possible
to use the $S_2$ and the $S_4$ indices to extrapolate linearly to
an $S_1$ index: $S_1 = 1.5S_2 - 0.5S_4$. That this is feasible can be
demonstrated with the VATTspec spectra. Figure <ref> shows the
correlation between the directly measured
VATTspec $S_1$ index, and $S_1^\prime$ extrapolated from $S_2$ and $S_4$. The
two are linearly related, and $S_1^\prime$ can predict the directly measured
$S_1$ index to better than $\pm$ 1%.
The relationship between the directly measured $S_1$(VATT) activity
index and the extrapolated $S_1^\prime$ index, based on the $S_2$ and
$S_4$ indices.
This provides a way to derive a linear transformation with no color term
between the instrumental DSO system and the Mount Wilson system. An $S_1$
extrapolated index is formed from the $S_2$ and $S_4$ instrumental indices,
and that $S_1$ index is calibrated to the Mount Wilson $S_{\rm MW}$ index via
observations of the chromospheric activity calibration stars of <cit.>.
For most of those calibration stars we have only a few ($< 5$) observations
scattered over the past 15 years. These we refer to as “snapshot”
observations. However, as part of the YSA project we have intensively
observed eight Mount Wilson stars – HD 45067, HD 143761, HD 207978, HD 82885,
HD 101501, HD 152391, HD 154417, and HD 206860. The first three of these
stars are regularly observed “chromospherically stable stars” used to monitor
the stability of our instrumental system (see below), and the
latter five are active G-type stars. For these stars, we can form multi-year
means for the instrumental indices that are much better correlated with
the Mount Wilson means than the snapshot observations of the other calibration
stars. In deriving the calibration, we give the snapshot observations a
weight of 1 and the multi-year means a weight of 5. This yields the
calibration (see Figure <ref>):
S_MW = 0.0323 + 4.8335S_1 σ= 0.0077
The residuals from the calibration show no evidence for a color term. In
addition, as the figure illustrates, extrapolation of this linear relationship
seems to hold for very active stars.
The DSO $S_1 \rightarrow S_{\rm MW}$(Mount Wilson) calibration.
The ordinate
is the mean Mount Wilson $\langle S_{\rm MW} \rangle$ index <cit.>.
small circles represent snapshot (single to a few) observations of the Mount
Wilson calibration stars <cit.>. The large squares represent Mount
Wilson stars that have been regularly observed at DSO since 2007. For
these stars the
8-year mean $S_1$ index (in some cases derived from over a hundred
observations) is used. These stars are given five times the weight of the
snapshot stars in deriving the calibration. Finally, the crosses represent
individual snapshot observations of very active Mount Wilson stars. These
stars were
not used in the derivation of the calibration, but indicate that extrapolation
of the calibration is adequate even for very active stars.
The top panel shows the histogram of the S/N values of our
observations. The S/N values are estimated in the continuum just longwards of
the Ca2 H line. The arrow indicates the average S/N, about 180. The
central panel shows the results of a Monte Carlo simulation of the measurement
error of $S_{\rm MW}$ as a function of S/N. At S/N = 180, the measurement
error is about $\pm 0.0025$. The bottom panel shows a similar simulation
for the G-band index. While the S/N in the continuum at the G-band is
$\sim 1.3$ – $1.4$ times that just longwards of the Ca2 H line,
we have plotted, for simplicity, the G-band errors against the H-line S/N.
At S/N = 180, the measurement error in the G-band index is approximately
$\pm 0.0005$.
The precision of our determinations of $S_{\rm MW}$ depend on the S/N of the
observations. We have attempted to estimate those precisions via a Monte-Carlo
method that begins with a synthetic spectrum of the Ca2 H & K region
smoothed to a resolution of 1.8Å/2 pixels (the resolution of the
DSO spectra).
The Monte-Carlo technique simulates exposing on the spectrum until a certain
S/N is achieved in the continuum just longwards of Ca2 H. That
exposure is processed through our measuring programs in exactly the same
way as the real spectra, including the velocity correction (the synthetic
spectra are given random radial velocity shifts between $-30$ and $+30$ km/s),
measurements of $S_2$, $S_4$, the calculation of $S_1$, and the transformation
to the Mount Wilson system)
enabling a calculation of the error $\Delta S_{\rm MW}$
for a given simulation. Those errors are plotted against S/N in the middle
panel of Figure <ref>. In the top panel of that same Figure is a
histogram of the S/N values of our observations. The average S/N $\sim 180$,
for which a measurement precision of $\pm 0.003$ in the $S_{\rm MW}$ index
is estimated. Indeed this error estimate (which does not include any possible
systematic errors in the transformation of our instrumental system to the
Mount Wilson system) is consistent with our measurements of $S_1$ in
the set of “chromospherically stable” stars (see below). The bottom panel
of the figure shows a similar calculation for the G-band index (see below).
§.§.§ Stability of the Dark Sky Observatory Instrumental System
The seasonal mean residuals in the instrumental $S_1$ index
observed for the three chromospherically “stable” stars, HD 45067
(filled circles), HD 143761 (diamonds), HD 207978 (squares). The outer “error”
bars indicate the standard deviation in the measured index for a given
season. The inner error bars indicate the standard error of the mean.
This diagram and similar ones for the other instrumental indices for the
G-band, Ca1, and H$\gamma$ can be used to assess the stability of
the instrumental system and to derive corrections to apply to the observed
To monitor the stability of the Dark Sky Observatory instrumental system, we
have regularly observed for the past 5 years, every clear night, at least
one chromospherically “stable” star, chosen from a set of stars showing
flat activity on the Mt. Wilson project <cit.>. The stable
stars that we observe are
HD 45067, HD 143761, and HD 207978. During the course of an observing season,
the standard deviations for night-to-night variations of those stars range
from 0.0004 – 0.0012 in $S_1$. The lower figure in that range translates to
a standard deviation in $S_{\rm MW} \sim 0.0019$, in line with our Monte Carlo
estimates for the observational error in that index. To monitor any changes
in the instrumental system, we have adopted the period July 1, 2011
(MJD = JD - 2450000 = 5743) to June 30, 2013 (MJD = 6445) as the
reference zeropoint baseline for the instrumental system. Residuals in the
seasonal means of the instrumental indices relative to that baseline
will then reveal changes in the instrumental system. This is illustrated
in Figure <ref> for the $S_1$ index.
That figure shows that
the instrumental system has remained very stable from the time that we
began regular monitoring of the chromospherically stable stars. However,
beginning September 1, 2013 (MJD = JD - 2450000 = 6536), there was a very
small but abrupt shift in the
instrumental system. That shift can be traced to the return of the CCD to the
manufacturers for repairs because of the failure of the vacuum seal. During
that visit, not only was the vacuum seal repaired, but a new driver was
installed that fixed a very low-level but variable bias pattern. In addition,
improved optical baffling was installed in the interior of the CCD housing
which may have slightly reduced the already very low level of scattered light.
To correct for this shift in the instrumental system, we subtract 0.0007 from
the $S_1$ indices obtained since September 1, 2013. That correction may be
propagated, if required, to the $S_2$ and $S_4$ indices using the relationships
between those indices. We have derived similar very small corrections to
the other observed indices.
Before April 2009, the spectroscopic data for this project were obtained with
a Tektronics CCD (see <ref>) on the same spectrograph. We
have investigated the difference in the instrumental system between the
two CCDs using spectra of inactive F-, G- and K-type stars taken with both
CCDs and find a small systematic difference between the two systems of 0.0019
in the measurement of the $S_1$ index. This correction has been applied to
the earlier data.
§.§ The G-band Index
Band definitions for the Photospheric Indices
Band Name Violet Edge Red Edge
Continuum ($c_1$) 4208.0Å 4214.0Å
Ca1 4226.7Å 4225.7Å 4227.7Å
Continuum ($c_2$) 4239.4Å 4245.4Å
Continuum ($c_3$) 4263.0Å 4266.0Å
G-band 4298.0Å 4312.0Å
Continuum ($c_4$) 4316.0Å 4320.5Å
Continuum ($c_5$) 4329.0Å 4334.0Å
H$\gamma$ 4339.5Å 4341.5Å
Continuum ($c_6$) 4345.0Å 4349.5Å
The variation of the three photospheric indices defined in this
paper as a function of $B-V$ color and spectral type. The G-band (solid
circles) comes to a maximum in the early K-type stars, and then declines.
The Ca1 index (open circles) grows with increasing $B-V$
linearly until the mid K-type stars, after which it appears to saturate. The
H$\gamma$ index (open triangles) decreases linearly with increasing $B-V$.
The stars used for this diagram are the Mount Wilson calibration
stars of <cit.>, and the $B-V$ data are from <cit.>.
At the suggestion of <cit.>, an index has been designed to measure
the G-band molecular feature in the
blue-violet region of the spectrum. This wide, deep feature
arises from the blended Q-branches of the 0-0 and
1-1 vibrational bands of the diatomic CH molecule. The G-band appears
first in the
early F-type stars, strengthens through the F- and G-type stars, comes to a
broad maximum in the early K-type stars on the main sequence, and then
weakens toward later types <cit.>. The G-band index is
measured by numerically integrating
the stellar monochromatic flux in a 14Å rectangular band centered at
4305Å (corresponding closely to the visible extent of the G-band in
low-resolution spectra, and similar to the passband of G-band interference
filters used in observations of the sun) and
ratioing that with “continuum” fluxes measured in two bands on either side
of the G-band (see Table <ref>). The G-band index is defined as:
1 - 1/14Å∫_4298Å^4312Å I(λ)dλ/0.247c_3 + 0.753c_4
where $c_3$ and $c_4$ represent the monochromatic fluxes in the two
continuum bands, respectively. Because the continuum bands are not situated
symmetrically relative to the G-band passband, the weightings in the
denominator are designed to give the “continuum” value at the wavelength
of the center of the G-band passband. The ratio is subtracted from
unity to give
an index that varies between 0 and 1: 0 when the G-band is absent,
1 when the G-band is perfectly black. As expected, the G-band index is a
strong function of $B-V$ (see Figure <ref>) and the spectral type.
The G-band index will also be a function of metallicity and $\log g$
<cit.>. We investigate in <ref> the relationship of the
G-band index to stellar activity.
A Monte Carlo error analysis similar to that described for the $S_{\rm MW}$ index
was carried out for the G-band index. This is illustrated in the lower panel of
Figure <ref>. The typical measurement error for the G-band index
at S/N $= 100$ is $\pm 0.0017$ and at S/N $= 180$ is $\pm 0.0011$. The
Monte Carlo analysis appears to have captured the important sources of measurement
error for the G-band, as may be deduced from Figure <ref>, where the
standard deviations of the seasonal G-band data for all the program stars
and the “chromospherically stable” reference stars are plotted against the
G-band index. The horizontal line in that figure, which corresponds well with
the lower envelope of the points, is the Monte Carlo G-band error for S/N $=180$.
The dispersions that lie above that line presumably arise from actual stellar
variability, a point that will be considered in <ref> below.
Verification of the Monte Carlo error analysis for the G-band index.
This figure plots the seasonal dispersions ($\sigma$) of the G-band indices
for all of the program stars plus the chromospherically stable reference stars
against the G-band index. The horizontal line, which corresponds well with
the lower envelope of the distribution of points, represents the Monte Carlo
error calculation for S/N $=180$, the average S/N of our spectra.
§.§ The Ca1 Index
Another prominent absorption feature in the blue-violet spectrum of G- and
K-type stars is the resonance line of Ca1 at 4226.7Å. This absorption
line grows steadily in strength toward later types, at least up to mid K-type
stars. It is also sensitive to surface gravity, especially in the K-type
stars <cit.>. We have devised an index similar to that of
the previously defined G-band index. The Ca1 index is measured by
integrating over a 2Å-wide band centered on the Ca1 line and ratioing
that with fluxes in two symmetrically placed continuum bands. The formula
used is:
1 - 1/2Å∫_4225.7Å^4227.7Å I(λ)dλ/0.5(c_1 + c_2)
where $c_1$ and $c_2$ are the continuum bands defined in Table <ref>.
As can be seen in Figure <ref>, the Ca1 index behaves as
designed; it grows linearly from the F-type stars into the K-type stars,
only saturating after a spectral type of K3.
A Monte Carlo error analysis similar to that illustrated in Figure
<ref> was carried out for the Ca1 index, giving a
measurement error of $\pm 0.0027$ at S/N $= 180$. This value again corresponds
well with the lower envelope of Ca1 index seasonal dispersions (see
discussion in <ref> above).
§.§ The H$\gamma$ Index
Both the G-band index and the Ca1 index grow with decreasing
temperature (at least up to the early K-type stars), and so it is useful to
another index that decreases with the temperature. The hydrogen lines behave
in exactly this way in the F-, G-, and K-type stars. The best hydrogen line
to use in the spectral range provided by our spectra from the Dark Sky
Observatory is H$\gamma$. An index based on the H$\beta$ line would probably
be preferable, because of the less crowded surroundings, but that line is
outside our spectral range. The H$\gamma$ index is defined similarly to the
Ca1 index, with a 2Å-wide band centered on the H$\gamma$ line and
flanking “continuum” bands (specified in Table <ref>). The formula
used is
1 - 1/2Å∫_4339.5Å^4341.5Å I(λ)dλ/0.4286c_5 + 0.5714c_6
The H$\gamma$ index behaves as designed, declining in strength with declining
temperature (Figure <ref>). However, it appears to have only about
half the temperature sensitivity of the Ca1 index.
A Monte Carlo error analysis similar to that illustrated in Figure
<ref> was carried out for the H$\gamma$ index, giving a
measurement error of $\pm 0.0020$ at S/N $= 180$. The larger errors for the
Ca1 and H$\gamma$ indices relative to the G-band index arise
primarily from the narrower “science” bands.
A montage of Ca2 H & K activity index ($S_{\rm MW}$) time series (upper
panel), and G-band index, Ca1, and H$\gamma$ times series (lower panels)
for our program stars (montage continued in Figures <ref>, <ref>, and <ref>). All the graphs are
scaled identically, with a range of 0.15 in $S_{\rm MW}$, 0.03 in the G-band
index, 0.08 in the Ca1 index, and 0.05 in the H$\gamma$ index so
that amplitudes of variations and seasonal dispersions can be intercompared
directly. The solid lines are Bezier curves drawn through the seasonal means. Typical error bars
for S/N $= 180$ spectra are shown in the upper left-hand corner of the panels for the first star.
Continuation of the montage in Figure <ref>.
Continuation of the montage in Figures <ref> and <ref>.
Continuation of the montage in Figures <ref>, <ref>,
and <ref>.
§ STATISTICAL ANALYSIS OF THE SPECTROSCOPIC RESULTS
Young Solar Analog Stars
Mean Activity Data, Predicted
Rotational Periods in days, and Chromospheric Activity Ages
$\langle S_{\rm MW} \rangle$
$\langle\log(R^\prime_{\rm HK})\rangle$
$P_{\rm rot}(R^\prime_{\rm HK})$
$P_{\rm max}(v\sin i)$
$\langle$G$\rangle$ $\sigma$
$\langle$Ca1$\rangle$ $\sigma$
$\langle$H$\gamma\rangle$ $\sigma$
(error) (error) (Myr)
HD 166 0.429 0.017 -4.393 7.52d (2.79) 10.03d (0.45) 375 0.459 0.002 0.539 0.006 0.384 0.005
HD 5996 0.376 0.019 -4.465 11.30d (2.80) $\infty$ 763 0.463 0.002 0.528 0.009 0.366 0.005
HD 9472 0.322 0.011 -4.495 10.06d (2.19) 16.23d (1.05) 762 0.408 0.002 0.448 0.009 0.405 0.005
HD 13531 0.369 0.013 -4.431 8.06d (2.37) 7.38d (0.12) 486 0.436 0.002 0.488 0.009 0.391 0.005
HD 27685 0.310 0.017 -4.510 10.14d (2.08) 32.71d (20.45) 810 0.425 0.002 0.456 0.010 0.410 0.006
HD 27808 0.255 0.011 -4.541 4.36d (0.80) 5.15d (0.08) 548 0.275 0.002 0.330 0.010 0.472 0.004
HD 27836 0.346 0.011 -4.397 4.03d (1.46) 210 0.343 0.003 0.387 0.011 0.431 0.008
HD 27859 0.312 0.009 -4.460 5.80d (1.47) 8.14d (0.22) 390 0.360 0.002 0.391 0.011 0.436 0.005
HD 28394 0.259 0.007 -4.520 3.46d (0.69) 3.01d (0.14) 876 0.241 0.002 0.307 0.011 0.479 0.006
HD 42807 0.339 0.013 -4.451 7.58d (2.01) 8.91d (0.36) 494 0.416 0.002 0.450 0.008 0.402 0.005
HD 76218 0.392 0.019 -4.458 11.37d (2.91) 12.54d (0.74) 753 0.470 0.003 0.570 0.010 0.362 0.006
HD 82885 0.287 0.018 -4.632 20.77d (2.91) 16.00d (1.00) 2184 0.479 0.002 0.551 0.007 0.389 0.003
HD 96064 0.476 0.028 -4.355 5.81d (1.82) 15.61d (2.79) 230 0.453 0.003 0.544 0.012 0.368 0.008
HD 101501 0.315 0.019 -4.540 13.90d (2.56) 15.55d (2.22) 1185 0.460 0.002 0.518 0.007 0.380 0.004
HD 113319 0.303 0.020 -4.511 9.37d (1.92) 12.77d (0.71) 743 0.408 0.002 0.439 0.008 0.406 0.007
HD 117378 0.302 0.010 -4.452 4.20d (1.11) 4.73d (0.09) 297 0.331 0.003 0.368 0.011 0.433 0.007
HD 124694 0.282 0.010 -4.473 3.35d (0.80) 3.08d (0.09) 344 0.281 0.002 0.331 0.008 0.453 0.006
HD 130322 0.253 0.030 -4.720 26.31d (3.01) $\infty$ 3202 0.484 0.004 0.564 0.011 0.363 0.004
HD 131511 0.445 0.021 -4.455 12.57d (3.27) 9.49d (0.40) 797 0.496 0.002 0.611 0.006 0.357 0.005
HD 138763 0.316 0.012 -4.436 4.46d (1.28) 283 0.322 0.002 0.366 0.009 0.440 0.008
HD 149661 0.303 0.019 -4.651 24.23d (3.24) 27.73d (3.70) 2581 0.506 0.002 0.623 0.007 0.352 0.004
HD 152391 0.391 0.022 -4.445 10.28d (2.81) 10.35d (0.48) 651 0.467 0.002 0.538 0.007 0.371 0.005
HD 154417 0.263 0.010 -4.550 6.92d (1.23) 8.12d (0.24) 654 0.330 0.002 0.373 0.007 0.448 0.005
HD 170778 0.315 0.014 -4.444 4.97d (1.37) 6.35d (0.16) 322 0.344 0.003 0.392 0.008 0.430 0.007
HD 189733 0.510 0.021 -4.503 16.91d (3.57) 13.58d (0.94) 1167 0.507 0.002 0.685 0.007 0.325 0.007
HD 190771 0.326 0.011 -4.462 7.37d (1.85) 9.61d (0.36) 492 0.401 0.002 0.453 0.007 0.421 0.005
HD 206860 0.317 0.014 -4.438 4.73d (1.34) 5.01d (0.05) 305 0.339 0.002 0.384 0.007 0.433 0.004
HD 209393 0.360 0.013 -4.429 7.38d (2.19) 10.61d (0.53) 440 0.419 0.002 0.464 0.010 0.387 0.005
HD 217813 0.310 0.012 -4.462 5.82d (1.47) 11.89d (0.54) 397 0.353 0.002 0.430 0.008 0.429 0.004
HD 222143 0.310 0.015 -4.497 8.87d (1.92) 16.55d (1.03) 675 0.389 0.002 0.452 0.008 0.424 0.005
Index-Variation Kolmogorov-Smirnov Significance Tests
Probability of the Null Hypothesis
Star ID $S_{\rm MW}$ G-band
Ca1 H$\gamma$ $N_{\rm seasons}$
HD 166 $< 10^{-5}$ 0.020 4
HD 5996 $< 10^{-5}$ 0.028 0.0023 5
HD 9472 0.00075 $< 10^{-5}$ 5
HD 13531 $< 10^{-5}$ 0.030 0.024 5
HD 27685 $< 10^{-5}$ 0.00002 6
HD 27808 $< 10^{-5}$ 5
HD 27836 5
HD 27859 0.011 5
HD 28394 0.014 5
HD 42807 $< 10^{-5}$ 0.014 $< 10^{-5}$ 0.0153 6
HD 76218 $< 10^{-5}$ 0.00033 7
HD 82885 $< 10^{-5}$ 0.00059 0.0011 0.00026 7
HD 96064 $< 10^{-5}$ 0.029 0.00019 8
HD 101501 $< 10^{-5}$ 0.00033 0.0045 0.013 7
HD 113319 $< 10^{-5}$ 0.00046 0.0014 0.00045 6
HD 117378 4
HD 124694 $< 10^{-5}$ 0.00069 0.00016 5
HD 130322 $< 10^{-5}$ 0.033 5
HD 131511 0.025 0.038 7
HD 138763 0.00076 0.023 6
HD 149661 0.00043 4
HD 152391 4
HD 154417 0.00075 5
HD 170778 0.00022 0.0011 5
HD 189733 0.00097 0.015 $< 10^{-5}$ 8
HD 190771 0.0005 0.024 0.049 0.010 6
HD 206860 $< 10^{-5}$ 0.0031 6
HD 209393 0.00036 0.048 5
HD 217813 $< 10^{-5}$ $< 10^{-5}$ 5
HD 222143 $< 10^{-5}$ 0.0011 5
Pearson Statistical Tests of Index-to-Index Correlations
Star ID $S_{\rm MW} - G$ $S_{\rm MW} - Ca~I$ $S_{\rm MW} - H\gamma$
$G - Ca~I$ $G - H\gamma$ $Ca~I - H\gamma$
HD 166 $+0.216,0.117$ $-0.137,0.323$ $-0.091,0.511$ $-0.066,0.634$ $+0.039,0.779$ $-0.234,0.089$
HD 5996 $-0.190,0.081$ $-0.102,0.351$ $-0.236,0.030$ $\mathbf{ +0.479,0.000}$ $\mathbf{ +0.379,0.000}$ $+0.226,0.038$
HD 9472 $-0.025,0.843$ $-0.170,0.175$ $-0.066,0.603$ $\mathbf{ +0.330,0.007}$ $+0.214,0.086$ $+0.222,0.075$
HD 13531 $-0.106,0.338$ $+0.055,0.621$ $-0.093,0.400$ $+0.257,0.018$ $\mathbf{ +0.489,0.000}$ $+0.178,0.106$
HD 27685 $\mathbf{ -0.391,0.001}$ $+0.267,0.033$ $-0.205,0.104$ $+0.133,0.295$ $\mathbf{ +0.330,0.008}$ $\mathbf{+0.360,0.004}$
HD 27808 $-0.215,0.102$ $-0.146,0.268$ $-0.188,0.153$ $+0.021,0.875$ $+0.091,0.494$ $-0.104,0.435$
HD 27836 $+0.050,0.767$ $-0.024,0.887$ $-0.111,0.514$ $+0.363,0.027$ $\mathbf{+0.558,0.000}$ $\mathbf{+0.499,0.002}$
HD 27859 $+0.074,0.682$ $+0.044,0.809$ $-0.083,0.647$ $+0.062,0.731$ $+0.158,0.381$ $+0.258,0.147$
HD 28394 $-0.185,0.240$ $-0.208,0.186$ $\mathbf{-0.506,0.001}$ $-0.057,0.722$ $+0.363,0.018$ $+0.221,0.160$
HD 42807 $\mathbf{-0.291,0.004}$ $+0.044,0.675$ $-0.245,0.017$ $+0.151,0.144$ $\mathbf{+0.390,0.000}$ $+0.208,0.043$
HD 76218 $\mathbf{-0.323,0.000}$ $+0.019,0.837$ $-0.170,0.066$ $\mathbf{+0.496,0.000}$ $\mathbf{+0.579,0.000}$ $\mathbf{+0.570,0.000}$
HD 82885 $\mathbf{-0.464,0.000}$ $\mathbf{-0.292,0.002}$ $+0.113,0.241$ $+0.155,0.108$ $\mathbf{+0.249,0.009}$ $+0.064,0.507$
HD 96064 $-0.164,0.142$ $-0.030,0.787$ $\mathbf{-0.296,0.007}$ $\mathbf{+0.425,0.000}$ $\mathbf{+0.404,0.000}$ $\mathbf{+0.279,0.011}$
HD 101501 $\mathbf{-0.351,0.000}$ $-0.135,0.168$ $-0.021,0.830$ $-0.017,0.866$ $\mathbf{+0.388,0.000}$ $-0.016,0.869$
HD 113319 $\mathbf{-0.508,0.000}$ $-0.038,0.731$ $\mathbf{-0.493,0.000}$ $+0.053,0.627$ $\mathbf{+0.503,0.000}$ $+0.135,0.212$
HD 117378 $+0.045,0.756$ $-0.187,0.193$ $\mathbf{-0.501,0.000}$ $-0.073,0.615$ $-0.075,0.606$ $\mathbf{+0.345,0.014}$
HD 124694 $-0.266,0.032$ $-0.053,0.677$ $\mathbf{-0.457,0.000}$ $+0.133,0.292$ $+0.103,0.413$ $+0.190,0.129$
HD 130322 $\mathbf{-0.429,0.003}$ $+0.046,0.762$ $-0.010,0.943$ $\mathbf{+0.390,0.007}$ $+0.258,0.082$ $\mathbf{+0.481,0.001}$
HD 131511 $\mathbf{-0.307,0.010}$ $-0.165,0.176$ $-0.115,0.347$ $+0.166,0.172$ $\mathbf{+0.320,0.007}$ $+0.181,0.138$
HD 138763 $+0.055,0.730$ $-0.253,0.106$ $-0.350,0.023$ $\mathbf{+0.500,0.001}$ $\mathbf{+0.490,0.001}$ $\mathbf{+0.520,0.000}$
HD 149661 $\mathbf{-0.442,0.005}$ $\mathbf-0.346,0.031$ $+0.102,0.536$ $+0.364,0.023$ $+0.317,0.049$ $-0.130,0.430$
HD 152391 $-0.414,0.026$ $-0.068,0.726$ $-0.281,0.140$ $+0.194,0.314$ $+0.129,0.505$ $-0.085,0.662$
HD 154417 $-0.090,0.635$ $-0.041,0.831$ $+0.072,0.706$ $+0.187,0.323$ $-0.005,0.980$ $-0.106,0.578$
HD 170778 $-0.152,0.277$ $\mathbf{-0.524,0.000}$ $\mathbf{-0.385,0.004}$ $+0.223,0.108$ $+0.301,0.029$ $\mathbf{+0.361,0.008}$
HD 189733 $-0.116,0.245$ $-0.067,0.502$ $-0.017,0.867$ $\mathbf{+0.341,0.000}$ $+0.210,0.033$ $\mathbf{+0.262,0.007}$
HD 190771 $\mathbf{-0.307,0.003}$ $\mathbf{-0.285,0.006}$ $-0.240,0.022$ $\mathbf{+0.289,0.006}$ $+0.177,0.095$ $\mathbf{+0.266,0.011}$
HD 206860 $-0.073,0.466$ $\mathbf{-0.470,0.000}$ $\mathbf{-0.445,0.000}$ $+0.164,0.102$ $+0.066,0.513$ $+0.177,0.076$
HD 209393 $\mathbf{-0.270,0.011}$ $+0.078,0.466$ $-0.208,0.051$ $\mathbf{+0.473,0.000}$ $+0.220,0.038$ $+0.189,0.077$
HD 217813 $-0.228,0.038$ $-0.192,0.081$ $\mathbf{-0.284,0.009}$ $+0.200,0.070$ $+0.191,0.083$ $-0.049,0.662$
HD 222143 $-0.089,0.411$ $-0.006,0.959$ $-0.211,0.050$ $-0.053,0.623$ $+0.163,0.131$ $+0.097,0.372$
Figures <ref>, <ref>, <ref>, and <ref>
show montages (in order of HD number) of time series of the Ca2 H & K
index (transformed to the Mount Wilson system index $S_{\rm MW}$), the G-band
index, the Ca1 index, and the H$\gamma$ index for our program stars.
Table <ref> lists mean values for the $S_{\rm MW}$ index and
$\log(R^\prime_{\rm HK})$ from our observations of the program stars.
The $S_{\rm MW}$ index measures
the flux in the cores of the Ca2 H & K lines, but that flux includes
contributions from both the chromosphere and the photosphere. A quantity
$R^\prime_{\rm HK}$, which is a useful measure of the chromospheric flux only,
may be derived from $S_{\rm MW}$ using a method outlined in <cit.>.
The $\log(R^\prime_{\rm HK})$ index
may be calibrated against age and rotation period <cit.>. We
have included columns in Table <ref> listing the expected rotation
period (in days), $P_{\rm rot}(R^\prime_{\rm HK})$, based on the calibration of
<cit.>, as well as the upper limit to the rotation period derived
from our $v\sin i$
values listed in Table <ref> ($P_{\rm max}(v\sin i)$), along with
associated errors. Note that, with the possible exception of HD 82885, the
rotation periods derived from the activity levels are consistent,
within the errors, with the rotation period upper limits deduced from the
projected rotational velocities. Chromospheric “activity ages” for our
program stars, based on the calibration of <cit.>, are also included
in Table <ref>. As it turns out, all but three of our stars (HD 82885,
HD 130322, and HD 149661) lie within the target age limits for this project,
0.3 – 1.5 Gyr. However, the discrepancy between $P_{\rm rot}(R^\prime_{\rm HK})$ and
$P_{\rm max}(v\sin i)$ for HD 82885 suggests that the chromospheric activity
age for that star may not be accurate. For instance, <cit.> quote
a rotation period for HD 82885 of 18.6 days. This gives a gyrochronological
age, using the calibration of <cit.>, of 1.6 Gyr.
§.§ The Sensitivity of the Photospheric Indices to
Temperature Variations
It was hypothesized in the Introduction that the three “photospheric” indices
defined in <ref> – <ref> will be primarily sensitive to
temperature, and thus might be
useful in measuring integrated temperature changes on the stellar surface
arising from spots and/or photospheric faculae. To determine the usefulness
of these indices for that purpose, we need to assess their sensitivity to
these changes. Figure <ref> displays plots of these
indices (using the Mount Wilson calibration stars from Table 5 of
<cit.>) versus
$B-V$. That Figure shows that in the realm of the late F-type
stars to the early K-type stars all three indices vary approximately linearly
with $B-V$. The following equations are straight-line fits to the linear
portions of those curves:
\begin{align*}
B-V =\, & 0.259 + 0.966\, {\rm G}\\
B-V =\, & 0.212 + 1.010\,{\rm Ca\, I}\\
B-V =\, & 1.68 - 2.539\, {\rm H}\gamma
\end{align*}
where G, Ca1, and H$\gamma$ refer to their respective indices, and
$B-V$ refers to the Johnson $B-V$ index. Both the G-band index and the
Ca1 index have slopes of nearly unity with respect to $B-V$, and so
changes in those indices should translate directly into changes in $B-V$.
The H$\gamma$ index has a sensitivity that is smaller by about a factor of
We will report in Paper II that many of our stars vary $\le 0.03$ – 0.07 magnitude
in the Johnson $V$-band, and in the instances where we can measure color ($B-V$)
changes, those changes are generally $\le 0.01$ mag. This is roughly what we
might expect if variations in brightness (due to sunspots and faculae) move
the star parallel to the main sequence. If the observed changes in the
photospheric indices arise solely from temperature effects, we might therefore expect
to observe variations in the G-band and Ca1 up to 0.01 in the index,
and by a factor of about 2.5 smaller in H$\gamma$. Such changes should be
detectable in at least the G-band and Ca1 indices, as the measurement
errors in those indices are on the order of 0.001 – 0.003. Indeed, because
of those measurement errors, these indices are potentially more useful in
measuring temperature changes than photometric colors where the errors are
larger. Interestingly, the data
in Table 4 do indeed indicate variations in Ca1 of about the expected
magnitude ($\le 0.01$), but the observed variations in the G-band are smaller
by a factor of two or more ($\le 0.004$). Hence, while it is plausible that
the observed variations in Ca1 are temperature related, it is clear
that the variations in the G-band may have a different or more complex origin.
The observed variations in H$\gamma$ are smaller than those observed in
Ca1, but not by the factor we would expect if those variations are
governed by temperature alone. We will examine these questions in more
detail in 5.4 below.
§.§ Statistical Tests for Season-to-Season Variability
The $S_{\rm MW}$ plots are the traditional tool for detecting and characterizing
activity cycles in stars. The detection and characterization of
activity cycles in active stars requires time series observations that exceed,
preferably by a factor of two or more, the period or characteristic timescale
of the star in question. That normally requires observations over decades,
and so, except for stars that our program has in common with other long-term
surveys, such as the Mt. Wilson program, we are limited in what we can say
on that subject. What can be done at the current stage of the project is to
1) evaluate the reality of the variations in the seasonal means and/or
variances of the four “activity” indices – the $S_{\rm MW}$, G-band,
Ca1, and H$\gamma$ indices – that are suggested by the time series
montages and 2) to examine and try to understand the existence of correlations between
those indices.
To assess the significance of the variations in the four indices on a
year-to-year basis (variations within a given observing season will be examined
in Paper II of this series where we will evaluate rotation periods for our
stars), we have employed the Kolmogorov-Smirnov (KS)
statistical test. KS tests are used in judging the significance of whether
or not two experimental or observational distributions of a certain variable
differ; the difference may arise from either a difference in the means or in
the variances of the two distributions. We may consider each set of seasonal
data (the “clumps” in Figures <ref>, <ref>,
<ref>, and <ref>) as independent samples of the index in
question, and compare those samples for a given star on a pair-wise basis
using the KS test to ascertain whether significant variation in the mean
value and/or variance of the index has occured over the period we have
observed the star. The way we perform the tests is as follows. Let us
suppose we have observed the star for four years (four observing seasons),
seasons 1, 2, 3, and 4. We then carry out KS tests on each of the following
six pair-wise comparisons: $1 \leftrightarrow 2$, $1 \leftrightarrow 3$,
$1 \leftrightarrow 4$, $2 \leftrightarrow 3$, $2 \leftrightarrow 4$, and
$3 \leftrightarrow 4$. The KS test yields a $p$-statistic for each comparison.
Smaller values of $p$ indicate higher significance. For instance, $p = 0.01$
indicates that the null hypothesis (no variation in the mean value or
variance of the index) may be rejected with a confidence of 99%. But the fact
that we need to estimate the significance of variations in a time series rather
than simply between two seasons complicates the analysis. For instance, let
us suppose we have five seasons of observations. This results in a set
of 10 pair-wise comparisons. If only one of those comparisons results in
$p = 0.01$, that does not rise to the level of significance because we would
expect, on the average, in a set of 10 comparisons, to encounter $p \le 0.01$
10% of the time – a significance of only 90%. However, if a given set
contains multiple comparisons with small $p$, we may then combine those
probabilities in assessing the significance of the observed variation.
We use a Monte Carlo technique to evaluate these probabilities. A random number
generator was used to generate multiple gaussian distributions of an observational
variable, all with the same mean and variance (and thus for these artificial data
the null hypothesis is true). In total we generated 100,000 sets of 4-season data,
each involving 6 pair-wise comparisons, for a total of 600,000 comparisons, and
evaluated each comparison with KS statistics. We did the same for sets of 5-season, 6-, 7-, and
8-season data, the latter involving 2.8 million pair-wise comparisons. We were
able to verify, for instance, that comparisons with $p \le 0.01$ were encountered
with the expected frequency. We then used these artificial data sets to evaluate
the significance of variations in our observational data. To take a real example, in
one of our 5-season data sets (HD 9472), we had, for the $S_{\rm MW}$ index, the following
values of $p$: 0.0116, 0.0132, 0.0188, 0.0281, and 0.0315. The remaining five
comparisons had $p > 0.05$. We then used the 5-season Monte Carlo data to
ask “What proportion of 5-season sets have $p_{\rm min} \le 0.0116$ and four
other comparisons with $p \le 0.0315$?” The result yields an overall $p = 0.00075$.
We have listed in Table 5 all the time series for which the overall $p \le 0.05$,
indicating “significant” variability.
Spurious significant $p$ values can
be created by outliers in the dataset. We have reduced to a minimum the
number of outliers in the dataset by rejecting all spectra with S/N $< 50$ and
by examining each spectrum to eliminate those with obvious defects (such as
cosmic rays) in the wavelength bands used for the calculation of the indices.
The remaining outliers cannot be rejected on a statistical basis (and may
indeed represent true excursions of the star) and so are included in the
statistical tests.
It is clear from Table 5 that almost all of the program stars show significant
season-to-season variations in $S_{\rm MW}$. The ones that do not have only 4 or 5
seasons of data, so it is entirely possible that with a few more seasons of data
all will show significant variation. 50% show significant variations in the G-band
index, 40% in the Ca1 index, and 37% in the H$\gamma$ index. We
expect that continuing the project for a few more years will increase those
proportions as well. We emphasize that a lack of significant variation in
the seasonal means and variances does not imply that the star is constant
within a given season. For instance, it is well known, and we will further
demonstrate in Paper II, that the “scatter” (at least for the $S_{\rm MW}$ index)
within a given season can arise from rotational modulation in the index.
§.§ Comments on the Nature of the Observed Variability
A number of our stars that have significant season-to-season variations
appear to be showing very short-term periodic or “pseudo periodic”
behavior in the $S_{\rm MW}$ index. Examples include HD 9472, HD 13531,
HD 27685 (superimposed on a secular rise in activity), HD 217813,
as well as some others. Despite the shortness of the datasets, the
above-mentioned stars show significant periods in the range of 2 – 4 years
with a
Lomb-Scargle analysis. To judge the reality of such short periods, which are
considerably shorter than the periods found in <cit.> we may refer
to other similar datasets. For example, a number of stars in the
<cit.> dataset appear to show very similar behavior (see the stars
HD 39587, HD 131156, HD 152391, HD 115404, HD 201092 for some possible
examples). This short-term variation appears to come
and go and is often superimposed on longer timescale variations. Analysis of
the Lockwood et al. or similar datasets will be required to evaluate the reality of
these variations.
Stars in our dataset show a variety of behaviors associated with the dispersion
in the $S_{\rm MW}$ activity index within a given season. For instance, the
stars HD 27859 ($\langle\sigma_{S_{\rm MW}}\rangle = 0.007$), HD 124694 (0.008),
HD 154417 (0.007), HD 217813 (0.008), and HD 222143 (0.006)
all show very tight activity dispersions within a given season. On the other
hand, HD 130322 (0.014), HD 131511 (0.018), and HD 189733 (0.018) show average
seasonal dispersions greater by a factor of two or more within a given season.
This distinction appears to be intrinsic, as we are careful to achieve
adequate S/N for all of our observations, and there are bright and
“faint” stars in both sets. We note, however, that those stars
that have particularly low seasonal dispersions are F and early G-type stars,
while the three with the higher dispersions are all K-type stars. If the seasonal
dispersions arise from rotational modulation, as active regions
rotate across the stellar disk, then this suggests that the late-type stars
mentioned above may be dominated by one or a small number of active regions, whereas
for the F- and early G-type stars in the project sample, active regions are smaller and more
dispersed across the stellar disk. One caution should be noted: the activity
behavior of HD 189733 may not be typical of young active K-type stars, as it
has apparently been spun up by angular momentum transfer from its hot-jupiter
companion (see Introduction). HD 131511 does, however, behave in quite a
similar way to HD 189733. Even though HD 131511 is a spectroscopic binary, its
stellar companion is in a much wider orbit
<cit.>, and
probably has not yet had an important influence on the angular momentum of the
primary. We also note that in a recent Nordic Optical Telescope FIES spectrum
of HD 131511, the emission in the cores of the Ca2 H & K lines appear
symmetrical, and so we see no evidence for emission from the companion. This
will need to be verified by further spectra at different phases of the
companion's orbit.
The variation in the seasonal dispersion of $S_{\rm MW}$ with time for
three stars, HD 76218 (solid line), HD 131511 (dashed line), and HD 189733
(dotted line).
Interestingly, some of our stars appear to show variations in their seasonal
dispersion behavior. Whether that variation in dispersion is cyclical can
only be determined with longer time series. The F-test is the appropriate
test for determining the statistical significance of season-to-season
differences in the variance of an index. HD 131511, for instance, appears
to alternate between seasons with high and moderate dispersions in the
$S_{\rm MW}$ index. Examination of the $S_{\rm MW}$ plot for HD 131511 (Figure
<ref>) shows four seasons with relatively high dispersions and
three with moderate dispersions. F-tests carried out on the
21 pair-wise comparisons between the seven seasons show highly significant
variations, with an overall $p = 0.0011$ (calculated using the same Monte
Carlo technique employed to evaluate the KS tests).
HD 189733 may be behaving in a similar way, although the statistics are
of lower significance ($p = 0.045$). HD 76218 apparently also varies in seasonal
dispersion ($p = 0.016$). The variation in HD 76218 is unusual in the sense
that when the seasonal dispersion is the highest, the activity level is at
or near a minimum. This is opposite to the sun which shows the greatest dispersion
in the Ca2 flux at activity maximum <cit.>. Figure <ref>
shows the variation in seasonal dispersion with time for HD 76218, HD 131511,
and HD 189733. A possible interpretation of this
behavior is that these stars vary between a state in which the active regions
are relatively small, numerous, and dispersed (low-to-moderate seasonal
dispersion) and a state which is dominated by one or a few large active
regions (high seasonal dispersion). This variation in seasonal dispersion
may represent a novel type of activity cycle in stars, or it may be evidence
for a flipflop cycle <cit.> and/or active longitudes. More observations
will be required to fully characterize this behavior.
§.§ Correlations Between Indices
A further question to address is whether or not significant correlations
exist between the four “activity” indices measured in this paper. In the Introduction
we gave the rational for the three “photospheric” indices defined in this paper –
the G-band, Ca1, and H$\gamma$ indices – and suggested that these three
indices might vary in step with activity variations largely through related temperature
changes in the photosphere connected with changes in spots and photospheric faculae.
How the photospheric indices would vary in relation to $S_{\rm MW}$ would then depend on
whether cool spots or hot photospheric faculae dominate. If the photospheric indices
vary primarily on the basis of temperature, we would expect the G-band index to vary
directly with the Ca1 index, and inversely with respect to the H$\gamma$ index.
We will see below whether this is indeed the case.
Table 6 shows the results of Pearson's r-tests for linear correlations between the
four indices. These comparisons are made with the original observations, and not with
the seasonal means. Since all of these indices are measured in a single spectrum, we
do not have to worry about time differences between the observations of the different
indices. The first column in Table 6 is the stellar ID, the second tabulates the results of the
Pearson r-test for correlations between $S_{\rm MW}$ and the G-band index, the third the
same for $S_{\rm MW}$ and Ca1, the fourth for $S_{\rm MW}$ and H$\gamma$, the fifth
for the G-band and Ca1, the sixth for the G-band and H$\gamma$, and the seventh
for Ca1 and H$\gamma$. Each comparison consists of two numbers, Pearson's
linear correlation coefficient $r$, and the $p$-statistic, from which the probability of
the null hypothesis (zero correlation) may be calculated. Small $p$ indicates a
signficant correlation. Correlations with $p \le 0.015$ are indicated with bold type
in Table 6. We have adopted $p \le 0.015$ as a useful standard for judging the
significance of these correlations because, for a given index, and 30 tabulated stars,
we should expect at that significance level only 0.5 spurious correlations.
A glance at Table 6 shows the presence of multiple significant, in many cases highly significant,
correlations, although none of those correlations are particularly strong ($r < 0.6$). We have
examined each of these correlations graphically to assure ourselves that none are caused by one
or a few “outliers”. Could these correlations arise from instrumental effects? We reject
that for a number of reasons: 1) We have not included in these tests data from the earlier
Photometrics CCD, and so that means that all of the observations involved in these tests
have been carried out with the same CCD on the same spectrograph on the same telescope, and
all have been reduced identically. 2) The passbands used in defining these indices do not overlap,
and so a spectral defect (cosmic ray, etc.) that affects one index will not affect another. 3) While
some stars show highly significant correlations, others do not. Instrumental effects would
lead to significant correlations (or not) in all stars, not just a limited number.
Let us now examine the nature of those correlations. For the $S_{\rm MW}$ – G-band comparison,
11 out of the 30 stars show significant ($p \le 0.015$) correlations, and all of those are negative correlations,
meaning that in those stars $S_{\rm MW}$ and the G-band vary oppositely; when one increases, the other
decreases. Note that the correlations that do not rise to our level of significance are as well almost
all negative. For the
$S_{\rm MW}$ – Ca1 comparison, only 4 of the 30 stars show a significant correlation,
but again all of those are negative. For $S_{\rm MW}$ – H$\gamma$, 8 of the 30 stars show
significant correlations, and again all of those correlations are negative.
So, a strong conclusion is that where significant correlations are present, the
“photospheric” indices are all negatively correlated with $S_{\rm MW}$.
What about correlations between the photospheric indices? Examination of Table 6 shows the presence
of many highly significant correlations between these indices, all of which are positive. So,
the tendency is, when the G-band weakens, so too do Ca1 and H$\gamma$.
This behavior is not consistent with the hypothesis that the photospheric indices are primarily
affected by temperature changes in the photosphere arising from changes in spots
and photospheric faculae. What then are the possible physical causes behind the observed behaviors?
One possibility that we must consider is whether the Ca1 and H$\gamma$ indices are
affected by changes in CH opacity. The G-band is a molecular feature, arising from the CH molecule,
but CH absorption lines are ubiquitous in the region of the spectrum containing
the Ca1 4227Å resonance line and H$\gamma$. To test this hypothesis we calculated a
number of synthetic spectra for late F, mid-G, and early K-type stars, all identical except for
differences in CH absorption strength (appropriate for the size of the variations we observe in
the G-band index), and then measured the resulting Ca1 and H$\gamma$ indices. Those indices showed
very small changes compared to the resulting changes in the G-band index, and in the opposite sense,
which would yield negative correlations instead of the positive ones observed.
The negative correlation between $S_{\rm MW}$ and the H$\gamma$ index might be understood on the basis
of line emission. In the spectrum of the solar chromosphere, both Ca2 H & K and H$\gamma$ (as
well as, of course, H$\alpha$ and H$\beta$) are seen in emission, and so it is reasonable to
expect that $S_{\rm MW}$ and H$\gamma$ would be negatively correlated on this basis, as emission
fills in the H$\gamma$ line, resulting in a index smaller than for a purely photospheric line,
while chromospheric emission yields an increase in $S_{\rm MW}$ over what would be measured for
pure absorption in Ca2 H & K. Neither the G-band nor Ca1 show
up in any significant way in the chromospheric spectrum, so this mechanism does not help to explain
the negative correlations of those indices with $S_{\rm MW}$ or their positive correlations with H$\gamma$.
As noted above, the existence of direct correlations between all three photospheric indices
is difficult to understand on the basis of temperature differences. This
suggests that the physical cause underlying those direct correlations does not
depend on temperature. Mechanisms that may be relevant here were noted
by <cit.> who observed that the equivalent widths of metallic lines (especially
low-excitation lines) in
the blue-violet part of the spectrum were reduced in
certain active stars, apparently due either to continuum emission arising in the
chromosphere or upper photosphere leading to the phenomenon of “veiling” or to nonradiative heating in the
upper layers of the photosphere in plage regions resulting in weaker line cores
Indeed <cit.> noted a similar phenomenon in the spectra of active K-type dwarfs, particularly
in the vicinity of the Ca1 line. Interestingly, they noted that some
active K dwarfs show this phenomena, and other equally active dwarfs do not.
Both of these mechanisms can help to explain not only the direct correlations between
the G-band, the Ca1 line, and H$\gamma$, but also
are consistent with the negative correlations between those indices and $S_{\rm MW}$
because as stellar activity increases, both the veiling and/or core-weakening and the emission in
Ca2 H & K would presumably increase together. Furthermore, a closer
look at Table 6 reveals
that the most significant G-band anti-correlations with $S_{\rm MW}$ occur at
spectral types where the G-band is near its maximum strength, and most of the
significant H$\gamma$ anti-correlations appear in the late-F and early G-type stars
where H$\gamma$ is still a strong feature, exactly what one would expect if
the mechanisms suggested by <cit.> were active.
We might then ask why temperature effects, hypothesized at the beginning of this paper
to be the primary drivers of changes in the “photospheric” indices do not appear to be important?
This question requires further investigation, but it may be that for the indices
considered in this paper, temperature effects arising from changes in both photospheric
faculae and spots – which would tend to cancel – sufficiently balance out so that
temperature variations become only a secondary cause in driving changes in these indices.
While it may be disappointing that the purpose for which we
designed these
indices has not been realized, it does appear that these indices can be used
to monitor the emission flux in the Paschen continuum arising from stellar
activity. This suggests that these three indices may also prove to be useful
proxies for monitoring emission in the ultraviolet Balmer continuum,
which is largely inaccessible from the ground. If that proves to be the case,
these indices would be of direct utility in achieving the original goals of
this project.
§ CONCLUSIONS
This paper reports on initial results from the Young Solar Analogs project,
which began in 2007 and which is monitoring the stellar activity of 31
late F-, G-, and early K-type stars with ages between 300 million and 1.5
billion years. We have detailed the transformation between our
instrumental Ca2 activity indices and the Mount Wilson $S$ activity
index. In addition, we have defined three new photospheric indices based
on the G-band, the Ca1 resonance line in the blue-violet, and the
H$\gamma$ line, and have examined, on a detailed statistical basis, how those
indices vary and how they are related. All four indices show strong evidence
for variability on a multi-year timescale in our data. The anti-correlations
between $S_{\rm MW}$ and the photospheric indices and the positive correlations
between the photospheric indices suggest the presence of varying continuum emission
and/or non-radiative heating of the upper layers of the photosphere
in at least some of the program stars. Further observations and modelling will
be required to better understand these physical mechanisms and
to evaluate the utility of the “photospheric” indices as proxies for
ultraviolet continuum emission. Subsequent papers in this series
will examine medium-term variations in these indices and the multi-band photometry,
as well as short-term variations.
The authors would like to thank an anonymous referee for careful and detailed
comments that resulted in a considerably improved paper.
The authors would also like to thank Lee Hawkins, Dark Sky Observatory engineer,
for expert and enthusiastic technical assistance in the construction and
maintenance of the Robotic dome. We are also pleased to acknowledge the
assistance of Mike Hughes (Electronics technician), Dana Greene, Machinist,
and David Sitar, all at Appalachian State University. This project has been
supported by NSF grant AST-1109158. We are also grateful for funding
provided by The Fund for Astrophysical Research during an early stage of
this project. This research has made use of the Keck Observatory Archive
(KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet
Science Institute (NExScI), under contract with the National Aeronautics and
Space Administration. This research has also made use of the Elodie Archive
(http://atlas.obs-hp.fr/elodie/). We also acknowledge use of archival spectra
from the UVES Paranal Observatory Project (ESO DDT Program ID 266.D-5655).
It is also a pleasure to acknowledge the service observing program at the
Nordic Optical Telescope. This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France. We acknowledge with pleasure the Veusz
software package[http://home.gna.org/veusz/] which was used for the
figures in this paper.
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|
1511.00603
|
Department of Physics, Eastern Mediterranean University, Gazimağusa,
We present three parameters exact solutions with possible black holes in $%
2+1-$dimensional $f\left( R\right) =R^{n}$ modified gravity coupled
minimally to a cloud of strings. These three parameters are $n,$ the cloud
of string coupling constant $\xi $ and an integration constant $C$. Although
in general one has to consider each set of parameters separately; for $n$
an even integer greater than one we give a unified picture providing black
holes. For $n\geq 1$ we analyze null / timelike geodesic within the context
of particle confinement.
§ INTRODUCTION
The advantages of working in lower-dimensional gravity, specifically in $%
2+1- $dimensions has been highlighted extensively during the recent decades.
The interest started all with the discovery of $2+1-$dimensional black hole
solution by Banados, Teitelboin and Zanelli (BTZ) <cit.>. The physical
source of the BTZ black hole was a cosmological constant which was
subsequently extended with the presence of different sources <cit.>.
Physically how significant are these sources?. Addition of scalar <cit.> and electromagnetic, both linear and non-linear fields has almost
been a routine while exotic and phantom fields also found room of
applications in the problem. A source that is less familiar is a cloud of
strings <cit.> which was considered in Einstein's general relativity.
Within this context in $3+1-$dimensions the importance of a string cloud has
been attributed to the action-at-a distance interaction between particles.
For a detailed geometrical description of a string cloud we refer to <cit.>. In this study we extend such a source to the $f\left( R\right) =R^{n}$
gravity which is a modified version of general relativity <cit.>. In $D-$
dimensional spacetime the energy-momentum tensor for a string cloud is
represented by $T_{\mu }^{\nu }=\frac{\xi }{r^{D-2}}diag\left(
1,1,0,0,...,0\right) $ where $\xi $ is a positive constant. In $3-$
dimensions, which will be our concern here this amounts to $%
T_{0}^{0}=T_{1}^{1}=\frac{\xi }{r},$ with $T_{2}^{2}=0$ where our labeling
of coordinates is $x^{\mu }=\left( t,r,\theta \right) .$ Compared with the
energy-momentum of the scalar and electromagnetic sources which are of the
order $\frac{1}{r^{2}}$ in $2+1-$dimensions the order $\frac{1}{r}$ for a
string cloud may play an important role. Briefly the singularity at $r=0$ is
weaker in comparison with different sources. This makes the motivation for
us to conduct the present study. We note that our cloud of strings is
reminiscent of the wormhole "fur coat" model in $5D$ Reissner-Nordström
black hole <cit.>.
Such a cloud of $1-$dimensional string play the role of particles in analogy
with the $1-$dimensional gas atoms. The spatial geometry is confined to the
polar plane with the cyclic angular coordinate. With the addition of time
variable the sheet description of the string becomes more evident. The
strings are open, originating at the singular origin and extending to
infinity, vanishing with $r\rightarrow \infty .$ Thus, for $r\rightarrow
\infty $ our model reduces to the source-free (vacuum) $f\left( R\right) $
model, which derives its power from the curvature of geometry. For $\xi =0$
in $f\left( R\right) =R$ model the spacetime is automatically flat unless
supplemented by other sources. In $f\left( R\right) $ gravity, on the other
hand even though we can take $\xi =0$ we have still room for a non-flat
metric albeit this may not be a black hole.
We investigate the field equations of $f\left( R\right) $ gravity in the
presence of a string cloud. In general, these are highly non-linear
differential equations but owing to the simplicity of our source and the $%
2+1-$dimensions we are fortunate to obtain a large class of exact solutions.
For particular choice of the parameter $n$ our solution can be interpreted
as black holes, while for other choices it corresponds to cosmological
§ STRING CLOUD SOURCE IN $F\LEFT( R\RIGHT) =R^{N}$ GRAVITY
Let us start with the action of the $2+1-$dimensional $f\left( R\right) -$
gravity coupled to the cloud of strings
\begin{equation}
I=\frac{1}{16\pi G}\int d^{3}x\sqrt{-g}f\left( R\right) +I_{S}
\end{equation}
in which $f\left( R\right) =R^{n}$ is a function of Ricci scalar $R,$ $n$ is
a real constant and
\begin{equation}
I_{S}=\int_{\Sigma }m\sqrt{h}d\lambda ^{0}d\lambda ^{1}
\end{equation}
where $\lambda ^{A}$, $(A=0,1)$ are the string parameters. The world-sheet
bivector is defined by
\begin{equation}
\sum\nolimits^{\mu \nu }=\epsilon ^{AB}\frac{\partial x^{\mu }}{\partial
h^{A}}\frac{\partial x^{\nu }}{\partial h^{B}}
\end{equation}
in which $\epsilon ^{01}=1=-\epsilon ^{10}$ is the $2-$dimensional
Levi-Civita symbol. Note that the string metric has the line element
\begin{equation}
\end{equation}
Accordingly the corresponding energy-momentum tensor of the string-cloud is
\begin{equation}
T^{\mu \nu }=\rho \frac{\sum\nolimits^{\mu \alpha }\sum\nolimits_{\alpha
}^{\nu }}{\sqrt{-h}}
\end{equation}
where $\rho $ is the energy-density and $h=\det \left( h_{AB}\right) $
refers to the world sheet of the string. The latter expression for $T^{\mu
\nu }$ is meaningful as long as $h<0.$
Variation of the action $I$ with respect to $g_{\mu \nu }$ provides the
field equations (in metric formalism)
\begin{equation}
\frac{df}{dR}R_{\mu }^{\nu }-\frac{1}{2}f\delta _{\mu }^{\nu }-\nabla ^{\nu
}\nabla _{\mu }\frac{df}{dR}+\delta _{\mu }^{\nu }\square \frac{df}{dR}%
=T_{\mu }^{\nu }
\end{equation}
in which $\square $ is the covariant Laplacian and
\begin{equation}
T_{\mu }^{\nu }=\frac{\xi }{r}diag.\left[ 1,1,0\right]
\end{equation}
is the energy momentum tensor of the cloud of string in which $\xi $ is a
real constant. The static and circular symmetric line element is set to be
\begin{equation}
ds^{2}=-U\left( r\right) dt^{2}+\frac{1}{V\left( r\right) }%
dr^{2}+r^{2}d\theta ^{2}
\end{equation}
with two unknown functions $U\left( r\right) $ and $V\left( r\right) .$ The
field equations explicitly written are
\begin{multline}
\frac{df}{dR}R_{t}^{t}-\frac{f}{2}+V\left( R^{\prime 2}\frac{d^{3}f}{dR^{3}}+%
\frac{d^{2}f}{dR^{2}}R^{\prime \prime }\right) + \\
\left( \frac{V^{\prime }}{2}+\frac{V}{r}\right) R^{\prime }\frac{d^{2}f}{%
dR^{2}}=\frac{\xi }{r},
\end{multline}
\begin{equation}
\frac{df}{dR}R_{r}^{r}-\frac{f}{2}+\left( \frac{VU^{\prime }}{2U}+\frac{V}{r}%
\right) R^{\prime }\frac{d^{2}f}{dR^{2}}=\frac{\xi }{r}
\end{equation}
\begin{multline}
\frac{df}{dR}R_{\theta }^{\theta }-\frac{f}{2}+V\left( R^{\prime 2}\frac{%
d^{3}f}{dR^{3}}+\frac{d^{2}f}{dR^{2}}R^{\prime \prime }\right) + \\
\left( \frac{VU^{\prime }}{2U}+\frac{V^{\prime }}{2}\right) R^{\prime }\frac{%
\end{multline}
in which
\begin{equation}
R_{t}^{t}=-\frac{\left[ V^{\prime }U^{\prime }Ur+2VU^{\prime \prime
}Ur-VU^{\prime 2}r+2VU^{\prime }U\right] }{4rU^{2}},
\end{equation}
\begin{equation}
R_{r}^{r}=-\frac{\left[ V^{\prime }U^{\prime }Ur+2VU^{\prime \prime
}Ur-VU^{\prime 2}r+2V^{\prime }U^{2}\right] }{4rU^{2}}
\end{equation}
\begin{equation}
R_{\theta }^{\theta }=-\frac{\left( rVU^{\prime }+rUV^{\prime }\right) }{%
\end{equation}
Note that, a prime stands for the derivative with respect to $r$. A set of
solutions to the above field equations are given by
\begin{equation}
U\left( r\right) =r^{2\left( n-1\right) }\left( C-\frac{2n^{2}\xi r^{\frac{%
1-2n+2n^{2}}{n}}}{\alpha \left( 1-2n+2n^{2}\right) \left( 4n^{2}-6n+1\right)
\end{equation}
\begin{equation}
V\left( r\right) =r^{\frac{2\left( 1-n\right) \left( 2n-1\right) }{n}%
}U\left( r\right) ,
\end{equation}
in which $C$ is an integration constant and
\begin{equation}
\alpha ^{n}=n^{n}\left( \frac{4n^{2}-6n+1}{4\left( 2n-1\right) \xi }\right)
\end{equation}
To complete the set of solutions we provide also the explicit form of the
Ricci scalar $R$ which reads as
\begin{equation}
R=\frac{4\xi n\left( 2n-1\right) }{\left( 4n^{2}-6n+1\right) \alpha r^{\frac{%
\end{equation}
Let's note that $n\neq 1/2$ is a real constant which must exclude also the
values $\frac{3\pm \sqrt{5}}{4}$ since they are the roots of the denominator
of $R$.
We would like to add also that, due to an arbitrary value for $n$ and $\xi $
our solution is a three parameter solution. For any specific value of $n$
one has to choose an appropriate sign for $\xi $ which in cases both sign
may be acceptable. Setting $n$ and the sign of $\xi $ give us an equation
for $\alpha $ given by (17). Depending on the number of real solutions this
equation may admit, we will get different metric functions. For instance
let's consider $n=2.$ In this case one finds $\alpha ^{2}=4\left( \frac{%
12\xi }{5}\right) $ which imposes $\xi >0$ and consequently there are two
solutions for $\alpha $ given by $\alpha =\pm 2\sqrt{\frac{12\xi }{5}}.$ For
positive/negative $\alpha $ one finds
\begin{equation}
U_{\pm }\left( r\right) =r^{2}\left( C\mp \frac{2\sqrt{15\xi }r^{\frac{5}{2}}%
}{75}\right) .
\end{equation}
Here clearly $U_{-}\left( r\right) $ is a black hole solution considering $%
C<0$ and therefore we may rewrite the metric as
\begin{equation}
U_{-}\left( r\right) =r^{2}\left\vert C\right\vert \left( \left( \frac{r}{%
r_{h}}\right) ^{\frac{5}{2}}-1\right)
\end{equation}
in which the horizon is located at
\begin{equation}
r_{h}=\left( \frac{75\left\vert C\right\vert }{2\sqrt{15\xi }}\right) ^{%
\frac{2}{5}}.
\end{equation}
In general, for $n\geq 2$ an even integer number the picture is the same as $%
n=2$ i.e., $\alpha =\pm n\left( \frac{4n^{2}-6n+1}{4\left( 2n-1\right) \xi }%
\right) ^{\frac{1-n}{n}}$ with $\xi >0.$ The general solution hence can be
cast as a black hole solution provided $C<0$ and negative $\alpha $ such that
\begin{equation}
U_{-}\left( r\right) =r^{2\left( n-1\right) }\left\vert C\right\vert \left(
\left( \frac{r}{r_{h}}\right) ^{\frac{1-2n+2n^{2}}{n}}-1\right)
\end{equation}
in which the location of the horizon is given by
\begin{equation}
r_{h}=\left( \frac{2\left( 1-2n+2n^{2}\right) \left( 2n-1\right) \left\vert
C\right\vert }{n\left( \frac{4\left( 2n-1\right) }{4n^{2}-6n+1}\xi \right) ^{%
\frac{1}{n}}}\right) ^{\frac{n}{1-2n+2n^{2}}}.
\end{equation}
For odd integers and other real numbers as we mentioned above, one must
carefully go through any individual case and find the final form of the
spacetime. Ultimately it gives either a black hole solution as we mentioned
above or a particle solution. For instance, in the case of $n=2$ the
positive branch i.e., $U_{+}\left( r\right) $ with positive $C$ the solution
represents a particle solution <cit.>.
§.§ $f\left( R\right) =R$ with geodesics
In the second example we consider $n=1$ which simply gives the Einstein's $R$
-gravity coupled to the cloud of string minimally. Let's note that this
solution was found first by Bose, Dadhich and Kar in <cit.>. The
solution becomes
\begin{equation}
U\left( r\right) =V\left( r\right) =-M+2\xi r
\end{equation}
\begin{equation}
R=-\frac{4\xi }{r}.
\end{equation}
As one observes, there is no restriction on the sign of $\xi $ and therefore
the solution represents either a singular black hole or a naked singularity.
In the case of the black hole the singularity is hidden behind an event
horizon located at
\begin{equation}
r_{h}=\frac{M}{2\xi }.
\end{equation}
§.§.§ Geodesics confinement for $f\left( R\right) =R$
The black hole solution in $R$-gravity given by (24) is rather interesting
if we assume $M,\xi >0$. In this section we investigate the null and
timelike geodesics of this solution. Let's start with the Lagrangian
\begin{equation}
L=\frac{1}{2}g_{\mu \nu }\frac{dx^{\mu }}{d\lambda }\frac{dx^{\nu }}{%
d\lambda }
\end{equation}
in which $\lambda $ is an arbitrary parameter. We also introduce the null ($%
\epsilon =0$) and timelike ($\epsilon =1$) geodesics by
\begin{equation}
g_{\mu \nu }\frac{dx^{\mu }}{d\lambda }\frac{dx^{\nu }}{d\lambda }=-\epsilon
\end{equation}
The line element
\begin{equation}
ds^{2}=-Udt^{2}+\frac{dr^{2}}{U}+r^{2}d\theta ^{2}
\end{equation}
in which $U=-M+2\xi r$ admits two Killing vectors, i.e., $\partial _{t}$ and
$\partial _{\theta }$ corresponding to the conserved energy and angular
momentum given by
\begin{equation}
E=-U\frac{dt}{d\lambda }
\end{equation}
\begin{equation}
\ell =r^{2}\frac{d\theta }{d\lambda }.
\end{equation}
Considering these one finds the only equation to be solved as
\begin{equation}
\left( \frac{dr}{d\lambda }\right) ^{2}+U\left( \epsilon +\frac{\ell ^{2}}{%
r^{2}}\right) =E^{2}.
\end{equation}
For the null radial geodesics with $\epsilon =\ell ^{2}=0$ one finds
\begin{equation}
r_{\pm }=r_{0}e^{\pm 2\xi \left( t-t_{0}\right) }+r_{h}\left( 1-e^{\pm 2\xi
\left( t-t_{0}\right) }\right)
\end{equation}
in which $r_{0}$ and $t_{0}$ are the initial location and time of the null
particle with respect to a distant observer. The sign $\pm $ stands for the
direction of the initial velocity with positive for outward and negative for
inward motion. As one can see
\begin{equation}
\frac{dr_{\pm }}{dt}=\pm 2\xi \left( r_{\pm }-r_{h}\right)
\end{equation}
\begin{eqnarray}
\lim_{t\rightarrow \infty }r_{+} &=&\infty \\
\lim_{t\rightarrow \infty }\frac{dr_{+}}{dt} &=&\infty \notag
\end{eqnarray}
\begin{eqnarray}
\lim_{t\rightarrow \infty }r_{-} &=&r_{h}. \\
\lim_{t\rightarrow \infty }\frac{dr_{-}}{dt} &=&0. \notag
\end{eqnarray}
In the case of timelike radial geodesics with $\epsilon =1,\ell ^{2}=0$ one
finds (for $\lambda =\tau $)
\begin{equation}
r=r_{0}-\frac{1}{2}\xi \tau ^{2}
\end{equation}
in which $\tau $ is the proper time and $r_{0}$ is the initial position of
the particle. To reach the horizon, the proper time needed by the particle
is just $\tau _{h}=\sqrt{\frac{2\left( r_{0}-r_{h}\right) }{\xi }}$In terms
of the coordinate time $t$ the equation of motion becomes
\begin{equation}
\left( \frac{dr}{dt}\right) ^{2}=U^{2}\left( 1-\frac{U}{E^{2}}\right) .
\end{equation}
If we consider the particle is at rest at $t=t_{0}$ we find $E^{2}=M\left(
\frac{r_{0}}{r_{h}}-1\right) $ that consequently implies
\begin{equation}
\left( \frac{dr}{dt}\right) ^{2}=4\xi ^{2}\left( r-r_{h}\right) ^{2}\left(
\frac{r_{0}-r}{r_{0}-r_{h}}\right)
\end{equation}
which admits the solution
\begin{equation}
r=r_{0}-\left( r_{0}-r_{h}\right) \tanh ^{2}\left[ \xi \left( t-t_{0}\right) %
\right] .
\end{equation}
Clearly the particle is attracted toward the black hole and when $%
t\rightarrow \infty $ the limit goes to $r=r_{h}.$ This time is not
comparable with the proper time interval needed for the particle to cross
the horizon.
In circular motion for a null particle one finds the only unstable orbit is
located at $r_{c}=2r_{h}$ (photon circle's radius) at which for the null
particle one finds
\begin{equation}
\frac{E^{2}}{\ell ^{2}}=\frac{\xi }{2r_{h}}.
\end{equation}
This can easily be justified from Eq. (32), by considering a potential of
the particle given by
\begin{equation}
V\left( r\right) =U\left( r\right) \left( \epsilon +\frac{\ell ^{2}}{r^{2}}%
\right) -E^{2}
\end{equation}
whose derivative satisfies $\frac{dV}{dr}=0$, at $r=r_{c},$ and $\epsilon =0$
for the null geodesics. For the timelike geodesics the similar analysis
applies with the substitution $\epsilon =1.$
Furthermore, for a timelike particle we find the stable orbit at $r=r_{c}$
where the angular momentum and the energy of the particle are given by
\begin{equation}
\ell ^{2}=\frac{r_{c}^{3}}{r_{c}-2r_{h}}
\end{equation}
\begin{equation}
E^{2}=4\xi \frac{\left( r_{c}-r_{h}\right) r_{h}}{\left( r_{c}-2r_{h}\right)
\end{equation}
We see clearly that $r_{c}>2r_{h}$ which is outside a photon circle around
the black hole.
§.§.§ Generalization to $f\left( R\right) =R^{n}$ with $n$ an even
integer and $n\geq 2$
The geodesic analysis given in the previous part can be generalized with $%
f\left( R\right) =R^{n}.$ As we are interested in the black hole geodesics,
we impose $n$ to be even integer bigger than one i.e., $n\geq 2.$ In order
to keep the mathematical expression analytic we only consider the radial,
null-geodesics. Accordingly one finds from the Euler-Lagrange equations
\begin{equation}
\pm \frac{dr}{d\lambda }=Er^{\frac{\left( 1-n\right) \left( 2n-1\right) }{n}}
\end{equation}
\begin{equation}
\pm \frac{dt}{d\lambda }=\frac{E}{U}
\end{equation}
in which $E$ is the energy of the particle and $\lambda $ is an affine
parameter. Combining these two equations we find
\begin{equation}
\frac{dr}{dt}=\pm r^{\frac{\left( 1-n\right) \left( 2n-1\right) }{n}}U
\end{equation}
with its integral expression in the form
\begin{equation}
\int_{r_{0}}^{r}\frac{r^{\frac{\left( n-1\right) \left( 2n-1\right) }{n}}dr}{%
U}=\pm \left( t-t_{0}\right)
\end{equation}
where $t_{0}$ is the initial time when the particle is located at $%
r_{0}>r_{h}$ and $r_{h}$ is the event horizon. Plugging (22) into the latter
equation we reexpress
\begin{equation}
\int_{r_{0}}^{r}\frac{r^{\frac{1-n}{n}}dr}{\left\vert C\right\vert \left(
\left( \frac{r}{r_{h}}\right) ^{\frac{1-2n+2n^{2}}{n}}-1\right) }=\pm \left(
t-t_{0}\right) .
\end{equation}
Obviously an exact integral of this expression is out of our reach. Even for
particular values of $n$ we have to appeal to the asymptotic behaviors.
Considering $r$ to be large i.e., $\frac{r_{h}}{r}\ll 1,$ and $n>0$ one
approximately finds
\begin{equation}
r^{2\left( 1-n\right) }-r_{0}^{2\left( 1-n\right) }\simeq \mp \omega \left(
\end{equation}
in which the constant $\omega $ is given by
\begin{equation}
\omega =2\left( n-1\right) \left\vert C\right\vert \left( \frac{1}{r_{h}}%
\right) ^{\frac{1-2n+2n^{2}}{n}}.
\end{equation}
Let's add that considering $\frac{r_{h}}{r}\ll 1$ in (49) implicitly implies
that the initial location of the particle is far from the horizon i.e., $%
\frac{r_{h}}{r_{0}}\ll 1$ too. Latter equation implies that at least for
this specific choices the motion is confined.
§ CONCLUSION
In search of alternative black holes to the BTZ in $2+1-$dimensions a
particular case was considered in Einstein's theory of $f\left( R\right) =R$
in which the source is a cloud of string <cit.>. Projected in the polar
plane the string can be considered as radial lines originating at the origin
and extending in radial in/out direction. This excludes the possibility of
closed string in such a geometry. The advantage of such a choice of source
becomes evident when substituted into the complicated $f\left( R\right)
=R^{n}$ gravity which we consider here. In other words our geometry is
powered by such a cloud of string in the $f\left( R\right) =R^{n}$ gravity
with the energy-momentum tensor $T_{0}^{0}=T_{1}^{1}=\frac{\xi }{r}$ and $%
T_{2}^{2}=0$ with $\xi =$constant. Obviously this satisfies the null energy
conditions. Although $n$ can be arbitrary in principle there are
restrictions on the choice of $n$ for a meaningful expression. For instance,
$n=\frac{1}{2}$ and $n=\frac{3\pm \sqrt{5}}{4}$ are to be excluded in the
class of metric solutions. Depending on the other values of $n$ we obtain an
infinite class of possible metrics that describe black holes / naked
singularities in $2+1-$dimensional $f\left( R\right) =R^{n}$ gravity theory.
An interesting physical conclusion to be drawn from these solutions is the
role that the power $n$ plays in the confinement of (especially) null
geodesics. Although the most general analytic integration of the geodesics
is lacking we obtain an approximate connection between the parameter $n$ and
the confinement of the null geodesics for $n$ an even integer greater than
one. Whether a similar relation occurs in higher dimensions $(D>3)$ for a
cloud of strings as source remains to be seen.
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|
1511.00255
|
Understanding how the brain stores and processes information is central to mathematical neuroscience. Neural data is often represented as a neural code: a set of binary firing patterns $\C\subset\{0,1\}^n$. We have previously introduced the neural ring, an algebraic object which encodes combinatorial information, in order to analyze the structure of neural codes. We now relate maps between neural codes to notions of homomorphism between the corresponding neural rings. Using three natural operations on neural codes (permutation, inclusion, deletion) as motivation, we search for a restricted class of homomorphisms which correspond to these natural operations. We choose the framework of linear-monomial module homomorphisms, and find that the class of associated code maps neatly captures these three operations, and necessarily includes two others - repetition and adding trivial neurons - which are also meaningful in a neural coding context.
§ INTRODUCTION AND SUMMARY OF RESULTS
§.§ Introduction
A major challenge of mathematical neuroscience is to determine how the brain processes and stores information. By recording the spiking from a population of neurons, we obtain data about their relative activity. Much can be learned by considering this neural data in the form of a neural code: a set of binary firing patterns recorded from brain activity. In this context, a neural code on $n$ neurons is a subset $\C\subset\{0,1\}^n$, with each binary vector in $\C$ representing a pattern of activity.
This type of neural code is referred to in the literature as a combinatorial neural code <cit.>
as it contains only the combinatorial information of which neurons fire together, without precise spike times or firing rates. These codes can be analyzed to determine important features of the neural data, using tools from coding theory <cit.> or topology <cit.>. In <cit.>, we introduced the neural ideal and the corresponding neural ring, algebraic objects associated to a neural code that store its combinatorial information.
Thus far, work involving the neural ring has been primarily concerned with using the algebraic properties of the ring, such as the canonical form for the neural ideal, to extract structural information about the code <cit.>. However, a neural code $\C$ is not an isolated object, but rather a member of a family of similar codes which may be obtained from $\C$ through certain natural operations, or maps. We define a code map from a code $\C$ to another code $\D$ to be any well-defined function $q:\C\rightarrow \D$. Considering the relationships between codes which are described by these maps allows us to determine precisely how information changes from one code to another. We are interested primarily in a set of `nice' code maps which reflect a meaningful relationship between the corresponding neural data. Our primary motivating examples of `nice' maps are listed below.
* Permutation. If $\sigma:[n]\rightarrow [n]$ is a permutation of $[n]$, then $\sigma$ induces a permutation map $q$ on any code $\C\subset\{0,1\}^n$, defined by $q( (c_1,...,c_n) ) = (c_{\sigma(1)},...,c_{\sigma(n)})$ for any codeword $c=(c_1,...,c_n)\in \C$. Applying a permutation to a code results in a code which is structurally identical to the original. That is, the labeling of the neurons is not important to the combinatorial properties of the code.
* Deleting neurons. Let $\alpha\subset[n]$ be a subset of neurons which we want to ignore. The deletion of these neurons is equivalent to projection onto the remaining neurons. That is, order $([n]\backslash \alpha) = \{i_1,...,i_k\}$ so $i_1<\cdots<i_k$; the code map deleting neurons in $\alpha$ is then given by $q((c_1,...,c_n)) = (c_{i_1},...,c_{i_k})$ for any codeword $c=(c_1,...,c_n) \in \C$.
* Inclusion: When two codes $\C$ and $\D$ on the same set of neurons are related by containment, then they are related by an inclusion map. That is, when $\C\subseteq \D$, the inclusion map is defined so $q(c) =c$ for any codeword $c \in \C$.
One type of code operation which we would not consider to be meaningful in the neural context is bit-reversal, where we map each binary vector to the vector with complementary support (that is, the map defined by $q((c_1,...,c_n)) = (1-c_1,...,1-c_n)$). As an operation on neural data, it makes little sense to associate the activity of a subset of neurons to the activity of the complementary set.
Understanding the relationships between codes, like those described above, requires an understanding of how combinatorial structure changes when these maps are applied. Since the neural ring stores structural information, we here consider how operations on neural codes relate to algebraic operations between neural rings. Our question, therefore, is the following:
What types of maps on neural rings correspond to `nice' maps between codes?
The class of `nice' maps certainly includes the three maps described above. In determining what other maps are in this class, we will in part allow the algebra to guide us. That is, the algebraic structure we choose for the best notion of neural ring homomorphism in order to capture these three maps may unavoidably capture other code maps. We are willing to consider those code maps `nice' as well, provided they have a meaningful interpretation as an operation on neural data.
Describing relationships between maps on varieties and maps on their associated rings is not an uncommon consideration in algebraic geometry; see for example <cit.>. However, our goal is twofold: we wish to not only relate maps between codes to a notion of homomorphism between rings, but also to restrict algebraically to a special class of homomorphisms such that the associated code maps are similar to the maps described above. Finally, since permutation maps preserve all combinatorial structure of the code, we aim to define isomorphism between neural rings in some way which captures permutation maps.
The organization of this paper is as follows. In the remainder of Section 1, we outline our main results and explain our final choice for a notion of neural ring homomorphism which meets our stated restrictions. In Section 2, we provide proofs of the initial results about the correspondence between code maps and ring homomorphisms, and examine the structure of the neural rings as modules. In Section 3 we describe our choice of neural ring homomorphism which captures the desired features, and present the relevant proofs. Finally, in Section 4, we discuss the effect of our chosen code maps on the canonical form of the neural ideal, and exhibit an iterative algorithm for adapting the canonical form when a codeword is added.
§.§ Neural rings and the pullback map
First, we briefly review the definition of a neural code and its associated neural ring, and present the most immediate relationship between the set of functions from one code to another, and the set of homomorphisms between the corresponding neural rings.
A neural code on $n$ neurons is a set of binary firing patterns $\C \subset \{0,1\}^n$. If $\C\subset\{0,1\}^n$ and $\D\subset\{0,1\}^m$ are neural codes, a code map is any assignment $q:\C\rightarrow \D$ sending each codeword $c\in \C$ to an image $q(c) \in \D$.
Given a neural code $\C\subset \{0,1\}^n$, we define the associated ideal $I_\C\subset\F_2[x_1,...,x_n]$ as follows:
$$I_\C \stackrel{\text{def}}{=} \{f\in \F_2[x_1,...,x_n] \ | \ f(c) = 0 \ \text{ for all } \ c\in \C\}.$$
The neural ring $R_\C$ is then defined to be $R_\C = \F_2[x_1,...,x_n]/I_\C$, the ring of functions $\C\rightarrow \{0,1\}$.
Since the ideal $I_\C$ is precisely the set of polynomials which vanish on $\C$, we can make use of the ideal-variety correspondence and obtain a relationship between the code maps and ring homomorphisms immediately by using the pullback. Given a code map $q:\C\rightarrow \D$, each $f\in R_\D$ is a function $f:\D\rightarrow \{0,1\}$, and therefore we may “pull back" $f$ by $q$ to a function $f\circ q:\C\rightarrow \{0,1\}$, which is an element of $R_\C$. Hence for any $q:\C\rightarrow \D$ we may define the pullback map $q^*:R_\D\rightarrow R_\C$, where $q^*(f) = f\circ q$. That is, for every $f\in R_\D$, the following diagram commutes:
\[
\xymatrix{ \C \ar[dr]_{q^*f = f\circ q} \ar[r]^q & \D\ar[d]^f\\ & \{0,1\}}
\]
In fact, the pullback provides a bijection between code maps and ring homomorphisms, as the following theorem states.
There is a 1-1 correspondence between code maps $q:\C\rightarrow \D$ and ring homomorphisms $\phi:R_\D\rightarrow R_\C$, given by the pullback map. That is, given a code map $q:\C\rightarrow \D$, its pullback $q^*:R_\D\rightarrow R_\C$ is a ring homomorphism; conversely, given a ring homomorphism $\phi:R_\D\rightarrow R_\C$ there is a unique code map $q_\phi:\C\rightarrow \D$ such that $q_\phi^* = \phi$.
Theorem <ref> is proven in Section 2, and is a special case of Proposition 8, p.243 in <cit.>. The bijective correspondence described by the pullback is unsatisfying in its generality: any code map we can define has a corresponding ring homomorphism. Additionally, using the pullback we can show that two neural rings are isomorphic if and only if their corresponding codes have the same number of codewords (Lemma <ref>); no other similarity of structure is reflected by isomorphism. Therefore, we will go beyond the ring structure to determine an algebraic framework by which the activity of the neurons (and thus the combinatorial structure of the code) can be tracked.
§.§ Neural rings as modules
The neural ring is constructed by associating the activity of neuron $i$ to the behavior of the variable $x_i$. When these rings are viewed abstractly without the details of their presentation (as with the pullback map), this relationship is lost. To ensure that we retain the presentation of the ring, we view the neural rings as modules under a ring on the variables $x_i$ with no code-related relationships between the variables. This allows us to track each variable $x_i$ via the module action, and hence to track the activity of neuron $i$.
For a neural ring $R_\C$ where $\C$ is a code on $n$ neurons, we consider $R_\C$ as an $R[n]$-module, where $R[n] = \F_2[x_1,...,x_n]/\langle x_i^2-x_i\ | \ i=1,...,n\rangle$ is the neural ring on the full code $\{0,1\}^n$. The module action is standard multiplication of functions; see Section 2.1 for a precise definition. In particular, we note that $R_\C$ contains a basis of characteristic functions $\rho_c$, where
$$\rho_c(v) \stackrel{\text{def}}{=} \left\{\begin{array}{ll} 1 & \text{if }v=c\\ 0 & \text{if }v\neq c\end{array}\right.$$
The module action is then more easily described by noting that each variable $x_i\in R[n]$ either preserves a basis element or neutralizes it, depending on the associated codeword:
$$x_i\cdot \rho_c = \left\{\begin{array}{ll} \rho_c & \text{if } c_i = 1\\ 0 & \text{if }c_i =0\end{array}\right. = c_i \rho_c$$
For more about this module action, see Section 2.1. Most importantly, this module structure is not only a property of neural rings, but can be used to characterize them completely.
Suppose $M$ is an $R[n]$-module. Then $M$ is isomorphic to a neural ring $R_\C$ as an $R[n]$-module if and only if $M$ has an $\F_2$-basis $\rho_1,...\rho_d$ such that the following hold:
* For all $i\in [n]$ and $j\in [d]$, $x_i\cdot \rho_j\in \{\rho_j,0\}$.
* For all $j,k \in [d]$ there exists at least one $i\in [n]$ such that exactly one of $x_i\cdot \rho_j$ and $x_i\cdot \rho_k$ is $0$
For such an $R[n]$-module, we can determine $\C$ from the module action.
Theorem <ref> is proven in Section 2.1. This theorem shows that by considering the action of the variables $x_i$ on a neural ring $R$, we can determine the unique code $\C \subset\{0,1\}^n$ for which $R = R_\C$. Having described a module framework which retains the information about the presentation of the neural ring, we consider how module homomorphisms relate to code maps. However, we need to do a bit of work first, since if we take our module structure as described above with no modification, we would be unable to consider any relationship between neural rings on different numbers of neurons, as they are modules under different rings. In fact, the only code maps this structure would cover are inclusion maps between codes of the same length; this is the substance of Lemma <ref>. To address this issue, we turn to some well-known algebraic machinery that allows us to extend the notion of module homomorphisms to modules under different rings.
§.§ Compatible ring homomorphisms
Suppose $R,S$ are rings with $\tau:R\rightarrow S$ a ring homomorphism. Any $S$-module $M$ may then be viewed as an $R$-module via the homomorphism $\tau$, using the action $r\cdot m = \tau(r)\cdot m$ for any $r\in R, m\in M$. In the neural ring framework, this says that given a ring homomorphism $\tau:R[m]\rightarrow R[n]$, we can consider the $R[n]$-module $R_\C$ as an $R[m]$-module. This motivates the following definition:
Given an $R$-module $M$, an $S$-module $N$, and a group homomorphism $\phi:M\rightarrow N$ (so $\phi(x+y) = \phi(x) + \phi(y)$), we say that a ring homomorphism $\tau:R\rightarrow S$ is compatible with $\phi$ if $\phi$ is an $R$-module homomorphism, where $N$ is viewed as an $R$-module via $\tau$. That is, $\tau$ is compatible with $\phi$ if $\phi(r \cdot x ) = \tau(r) \cdot \phi(x)$ for all $r\in R, x\in M$. Equivalently, $\tau$ is compatible with $\phi$ if for every $r\in R$, the following diagram commutes:
\[
\xymatrix{ M\ar[d]^\phi \ar[r]^{r\cdot} & M \ar[d]^\phi \ar[d]^\phi \\ N \ar[r]^{\tau(r)\cdot} & N }
\]
Having defined compatible ring homomorphisms, we now need to determine when compatible homomorphisms can be found. In fact, for any ring homomorphism $\phi$ between neural rings it is possible to find at least one compatible homomorphism. Moreover, any compatible homomorphism will act as an extension of the map $\phi$ to a broader class of functions, as the following theorem states.
Suppose $\C$ and $\D$ are neural codes on $n$ and $m$ neurons, respectively.
* If $\phi:R_\D\rightarrow R_\C$ is a ring homomorphism, then there exists a ring homomorphism $\tau:R[m]\rightarrow R[n]$ which is compatible with $\phi$, and thus $\phi$ is an $R[m]$-module homomorphism.
* For any ring homomorphism $\phi:R_\D\rightarrow R_\C$, the set of compatible ring homomorphisms is exactly the set of ring homomorphisms $\tau:R[m]\rightarrow R[n]$ which are extensions of $\phi$, in the sense for all $f\in R_\D$, $(\phi(f))^{-1}(1) \subseteq (\tau(f))^{-1}(1)$.
Theorem <ref> is proven in Section 2.2. It shows that compatibility is a powerful descriptive tool. Most notably, since the rings $R[n]$ and $R[m]$ are themselves neural rings, any homomorphism between them is associated to a code map via Theorem <ref>. In Corollary <ref> we use the second part of Theorem <ref> to prove that that for a homomorphism $\tau$ to be compatible with a homomorphism $\phi$, the associated code map $q_\tau$ between the full codes $\{0,1\}^n$ and $\{0,1\}^m$ must be an extension of the code map $q_\phi$ between the smaller codes $\C$ and $\D$.
However, Theorem <ref> also indicates that by using compatibility to widen our possible allowed module homomorphisms, we obtain again the general class of code maps, similarly to when we considered only the pullback map. That is, any code map corresponds to a ring homomorphism by Theorem <ref> and thus induces a module map with a compatible ring homomorphism. Hence, every possible code map can be described by a module homomorphism, contrary to our goal of obtaining only a restricted class of homomorphisms. We have, however, made one small improvement over our original ring homomorphisms: we are now able to track the changes in the activity of neuron $i$ by considering the effect of the image $\tau(x_i)$ of the indicator variable. To restrict to on a class of code maps which contains natural operations but not every possible map, we control the activity of neuron $i$ across the map by placing restrictions on the activity of these indicators under the compatible homomorphisms $\tau$.
The most obvious restriction would be to require that every variable be sent to itself: that $\tau(x_i) = x_i$ for all $i$ for which it is possible, and $0$ otherwise. However, this notion is quite restrictive. Most notably, permutation is not an allowable map under this restriction, since the activity of neuron $i$ is forced to remain the same. The only maps we can capture are projection onto the first $k$ neurons, inclusion, and adding 0's to the end of codewords (and compositions thereof). We are not permitted even to delete, say, the second neuron of three, because that would involve renaming neuron $3$ as the `new' neuron 2. See Proposition <ref> for the formal description of the corresponding code maps.
Since this notion is too restrictive to even capture permutation, we will instead consider a slight relaxation of this restriction which will allow a broader class of code maps.
§.§ Neural ring homomorphism
Instead of restricting to maps $\tau$ where $\tau(x_i) = x_i$, we make a slight generalization: we consider linear-monomial maps: maps where $\tau(x_i) \in \{x_j, 0, 1\}$. Allowing $\tau(x_i) = x_j$ opens the door for permutation maps. There are other possibilities, as we can map multiple variables to the same, or to constants $0$ or $1$, so we get more maps than we had under our previous, very restrictive notions. We therefore make the following definition:
Let $\C$ and $\D$ be neural codes on $n$ and $m$ neurons, respectively. A ring homomorphism $\phi:R_\D\rightarrow R_\C$ is a neural ring homomorphism if there is a compatible linear-monomial ring homomorphism $\tau:R[m]\rightarrow R[n]$.
This is our final choice for a `best' notion of homomorphism between neural rings, which captures all our desired code maps, and a few extra maps which are still meaningful. Our main result is Theorem <ref>, which shows that neural ring homomorphisms correspond to compositions of the following operations between neural codes.
$\phi$ is a neural ring homomorphism if and only if $q_\phi$ is a composition of the following code maps:
* Permutation of labels
* Adding a codeword (inclusion)
* Deleting the last neuron
* Repeating a neuron
* Adding a trivial neuron (always 0 or 1) to the end of each word.
We then define a neural ring isomorphism to be an isomorphism $\phi: R_\D\rightarrow R_\D$ which has a compatible linear-monomial isomorphism $\tau:R[m]\rightarrow R[n]$, and find that neural ring isomorphisms are exactly permutation maps.
Neural ring isomorphisms correspond exactly to those code maps which permute the labels on neurons.
The proofs of Theorem <ref> and Corollary <ref> are found in Section 3.1. The set of possible maps described in this result is a restricted class of code maps including our three primary examples of permutation, deletion, and inclusion. Although for deletion the list includes only deletion of the last neuron, note that by composing this operation with permutation we can choose to delete any subset, and project onto the remaining neurons. See Section 3.2 for a detailed description of this, as well as a few examples of other natural maps which these compositions produce. Furthermore, the notion of neural ring isomorphism is a more specialized version of general ring isomorphisms, and which exactly captures permutations. We can repeatedly add codewords to include one code into another. Finally, not every possible code map is captured by neural ring homomorphisms; the first example in Section 1.6 gives an example of a code map which does not correspond to a neural ring homomorphism.
§.§ Examples
We here provide a few examples of code maps, the associated ring homomorphisms and module actions, and determination of whether or not these are neural ring homomorphisms or isomorphisms. Throughout, $\B_n = \langle x_i(1-x_i) \ | \ i\in [n] \rangle$ is the ideal of Boolean relationships.
* Let $\C = \{00,11\}$ and $\D = \{111,000\}$, and suppose the code map $q$ is a bijection with $q(00) = 111$ and $q(11) = 000$. The pullback map $\phi_q:R_\D\rightarrow R_\C$ sends $\rho_{111}$ to $\rho_{00}$ and $\rho_{000}$ to $\rho_{11}$.
The module action on $R_\D$ is given by $x_i\cdot \rho_c = c_i \rho_c$ and extending by linearity. In particular, $x_i \cdot\rho_{000} = 0$ for all $i$ and $x_i\cdot \rho_{111} = \rho_{111}$ for all $i$, and thus for example , $(x_1+x_2)\cdot \rho_{111} = x_1\cdot \rho_{111} + x_2\cdot \rho_{111} = 2\rho_{111} = 0$. Similarly, the module action on $R_\C$ is given by $x_i\cdot \rho_{c} = c_i \rho_c$ and extending by linearity.
$\phi_q$ is a ring homomorphism; in fact, it is an isomorphism as $q$ is a bijection. However, $\phi_q$ is is not a neural ring homomorphism. Let $\tau:R[3]\rightarrow R[2]$ be a compatible ring homomorphism. Then, we have $\phi_q(x_1\cdot \rho_{111}) = \phi_q(\rho_{111}) = \rho_{00}$, but also $\phi_q(x_1)\cdot \rho_{111}) = \tau(x_1)\cdot \rho_{00}$, so $\tau(x_1)$ cannot be $x_1, x_2,$ or $0$. To be a neural ring homomorphism, we require $\tau(x_1)$ to be linear monomial; the only remaining option is $\tau(x_1) = 1$. But then, $\phi_q(x_1\cdot \rho_{000}) = \phi_q(0) = 0$, but on the other hand $\phi_q(x_1\cdot \rho_{000}) = \tau(x_1)\cdot \phi_q(\rho_{000}) = 1\cdot \rho_{11} =\rho_{11}$ and these are not equal. So there is no compatible $\tau$ which is also a linear monomial function on the variables, and hence $\phi_q$ is not a neural ring homomorphism.
* Let $\C = \{000, 100, 101, 110\}$ and $\D = \{00, 10, 11, 01\}$. Suppose the code map $q$ is given by $q(000) = 00$, $q(100) = q(101) = 10$ and $q(110) = 11$. The pullback map $\phi_q:R_\D\rightarrow R_\C$ sends $\rho_{00}$ to $\rho_{000}$, $\rho_{10}$ to $\rho_{100} + \rho_{101}$, $\rho_{11}$ to $\rho_{110}$ and $\rho_{01}$ to zero.
The module action on $R_\D$ is given by $x_i \cdot\rho_{c} = c_i\rho_c$ and extending by linearity; for example,
$$(x_1x_2)\cdot (\rho_{10} + \rho_{11}) = x_1\cdot(x_2\cdot \rho_{10}) + x_1\cdot(x_2\cdot\rho_{11}) = x_1\cdot 0 + x_1\cdot \rho_{11} = \rho_{11}$$
Similarly, the module action on $R_\C$ is given by $x_i\cdot \rho_c = c_i\rho_c$ and extending by linearity.
$\phi_q$ is a ring homomorphism, though not a ring isomorphism; its kernel contains $\rho_{01}$ which is nonzero. It is also a neural ring homomorphism. In fact, there is exactly one possible compatible linear-monomial homomorphism $\tau:R[2]\rightarrow R[3]$, given by $\tau(x_1) = x_1$ and $\tau(x_2) = x_2$. We also note that this code map is projection onto the first 2 coordinates (or, deleting the the 3rd neuron) composed with adding the codeword $01$.
* Let $\C = \{0000, 0011, 1000, 1011\}$ and $\D = \{000, 100, 010, 110\}$, and let the code map $q$ be given by $q(0000) = 000$, $q(0011) = 100$, $q(1000) = 010$ and $q(1011) = 110$. The pullback map $\phi_q$ sends $\rho_{000}$ to $\rho_{0000}$, $\rho_{100}$ to $\rho_{0011}$, $\rho_{010}$ to $\rho_{1000}$ and $\rho_{110}$ to $\rho_{1011}$.
$\phi_q$ is a ring homomorphism, and as $q$ is a bijection, we see that $R_\C$ and $R_\D$ are actually isomorphic as rings. $\phi_q$ is also a neural ring homomorphism; in fact, there are several compatible linear-monomial functions $\tau:R[3]\rightarrow R[4]$. One possibility is $\tau_1$ where $\tau_1(x_1) = x_3$, $\tau_1(x_2) = x_1$ and $\tau_1(x_3) = x_2$; this corresponds to the idea that $q$ could be described as deleting neuron 4, and then permuting neurons 1, 2, and 3. Alternatively, we could use $\tau_2$ where $\tau_2(x_1) = x_4, \tau_2(x_2) = x_1$, and $\tau_2(x_3) = 0$; this corresponds to the idea that $q$ could be described as permuting the neurons to the order 4,1,2,3; deleting the last two neurons (2 and 3), and adding a trivial neuron which is always $0$. However, while $R_\C$ and $R_\D$ are isomorphic and $\phi_q$ is a neural ring homomorphism, there is no compatible linear-monomial map $\tau$ which is itself an isomorphism (as $R[3]$ and $R[4]$ are not isomorphic) so $\phi_q$ is not a neural ring isomorphism.
Finally, now that we have seen how examples work at the level of the rings and the code maps, we consider the practical effect of the basic maps given in Theorem <ref> on the canonical form, one of the major objects of study from our previous work <cit.>.
§.§ Effect on the canonical form
In our original paper <cit.>
we focused our attention on structural information about the code which could be obtained from the ideal $I_\C$, rather than the neural ring $R_\C$. In particular, we split the ideal $I_\C$ into two pieces: $I_\C = \B+J_\C$, where $\B$ covered the standard Boolean relations $\B = \langle x_i^2-x_i \ | \ i=1,...,n\rangle$. $J_\C$, on the other hand, covered the relationships unique to the code: $J_\C = \langle \rho_v \ | \ v \in \{0,1\}^n \backslash \C \rangle$. The polynomials which generate $J_\C$ are examples of pseudo-monomials: polynomials of the form $\prod_{i\in \sigma} x_i \prod_{j\in \tau} (1-x_j)$ for $\sigma\cap \tau = \emptyset$. A pseudo-monomial $f\in J_\C$ is minimal if there is no other pseudo monomial $g\in J_\C$ of smaller degree so $f=gh$ for some $h\in \F_2[x_1,...,x_n]$. If we take the set of all minimal pseudo-monomials, we obtain the canonical form of $J_\C$, written:
$$CF(J_\C) = \{ f \in \F_2[x_1,...,x_n] \ | \ f \text{ is a minimal pseudo-monomial of } J_\C
\}$$
The canonical form provides not only a condensed set of generators for $J_\C$, but also a minimal combinatorial description of the code itself. It is thus an important and useful object, and so we examine the effect of our basic natural code maps on the canonical form. It's important to note that the effects we examine here, unlike the homomorphisms above, are not contravariant. That is, for a code map $q:\C\rightarrow \D$, we describe at how to change $CF(J_\C)$ to obtain $CF(J_\D)$. The basic code maps have the following effects:
* Permutation: permuting the neurons requires that we change the canonical form via the same permutation. For example, if neuron 1 is relabeled as neuron 3, then all relationships involving $x_1$ must now be changed to involve $x_3$ instead. That is, we perform the same permutation on the labels of the variables.
Suppose $\C = \{000,010, 001, 110, 011\}$. Then $CF(J_\C) = \{x_1(1-x_2), x_1x_3\}$. If we permute the neurons under the permutation $(123)$, then the new code is $\D = \{000, 100, 001, 011, 101\}$, and the new canonical form is $CF(J_\D) = \{x_2(1-x_3), x_1x_2\}$.
* Adding codewords: this is the most difficult operation to describe, because the effect of this map can vary greatly depending on the relationship of the codewords being added to the codewords already present. We have, however, written an algorithm which adapts the canonical form iteratively by codeword. In short, all elements of the canonical form which still vanish on the new code are preserved, whereas those that do not are multiplied by a linear term and included if not redundant. See Section 4 for a more detailed description of this algorithm.
Let $\C = \{000,010, 001, 110, 011\}$, so $CF(J_\C) = \{x_1(1-x_2), x_1x_3\}$. Suppose we add the codeword $111$, so $\D = \{000, 010,001, 110, 011, 111\}$. Note that $x_1(1-x_2)$ still vanishes on $\D$, but $x_1x_3$ no longer does, so we must fix it. The 3 linear terms that vanish on $111$ are $(1-x_1), (1-x_2)$, and $(1-x_3)$; multiplying $x_1x_3$ by either $(1-x_1)$ or $(1-x_3)$ won't result in a pseudomonomial, so the only fix that works is $x_1x_3(1-x_2)$. This, however, is redundant, since it is a multiple of $x_1(1-x_2)$. So our new canonical form is simply $CF(J_D) = \{x_1(1-x_2)\}$.
* Deleting the last neuron: To drop neuron $n$ from the end of each word, we change the canonical form by merely removing all elements of $CF(J_\C)$ which involve variable $x_n$ in any way.
Let $\C = \{000,010, 001,110, 011\}$, so $CF(J_\C) = \{x_1(1-x_2), x_1x_3\}$. Suppose we drop neuron 3 from the end of each codeword, and project onto the other two neurons. Then $\D = \{00, 01, 11\}$, and its associated canonical form is $CF(J_\D) = \{x_1(1-x_2)\}$.
* Repeating a neuron: to repeat the behavior of neuron $i$ with a new neuron $x_{n+1}$, we duplicate all elements of $CF(J_\C)$ which involve $x_i$, replacing the variable $x_i$ with $x_{n+1}$. Except in the rare instance where neuron $i$ is trivial (i.e. when either $x_i$ or $(1-x_i)$ is in $J_\C)$, we must also add the relations $x_i(1-x_{n+1})$ and $x_{n+1}(1-x_i)$ to the canonical form.
Let $\C = \{000,010, 001,110, 011\}$, so $CF(J_\C) = \{x_1(1-x_2), x_1x_3\}$. Suppose we add a new neuron, neuron 4, which duplicates neuron 2. Then $\D = \{0000, 0101, 0010, 1101, 0111\}$, and its associated canonical form is $CF(J_\D) = \{x_1(1-x_2), x_1(1-x_4), x_1x_3, x_2(1-x_4), x_4(1-x_2)\}$.
* Adding a trivial neuron: to add a trivial $n+1$st neuron which is always $0$ (or always 1) we add the element $x_{n+1}$ (respectively, $1-x_{n+1})$ to the canonical form.
Let $\C = \{000,010, 001,110, 011\}$, so $CF(J_\C) = \{x_1(1-x_2), x_1x_3\}$. Suppose we add a new neuron, neuron 4, which is always 1. Then $\D = \{0001, 0101, 0011, 1101, 0111\}$, and its associated canonical form is $CF(J_\D) = \{x_1(1-x_2),x_1x_3, (1-x_4)\}$.
Knowing how these maps affect the canonical form allows us to transform structural information about the original code quickly and algorithmically into structural information about the resulting image code. Primarily useful here is the algorithm for changing the canonical form in the event of adding a codeword, because it gives us an iterative algorithm for finding the canonical form by building up the code one word at a time, a substantial improvement in efficiency and transparency over our original algorithm from <cit.>. This algorithm and a proof of its efficacy are found in Section 4.
§ HOMOMORPHISMS AND CODE MAPS
In this section, we give a detailed description of the pullback correspondence between code maps $\C\rightarrow \D$ and ring homomorphisms $R_\D\rightarrow R_\C$, including the method for determining one from the other.
Recall that the neural ring $R_\C$ for a given code $\C$ is exactly the ring of functions $f:\C\rightarrow \{0,1\}$, and as such is an $\F_2$-vector space. Elements of neural rings may be denoted in different ways: they can be written as polynomials, where it is understood that the polynomial is a representative of its equivalence class mod $I_\C$. Alternatively, using the vector space structure, an element of $R_\C$ can be written as a function which, since each function maps into $\F_2$, is defined completely by the codewords which support it. We will make use of the latter idea frequently, and find it helpful to identify a canonical basis of characteristic functions $\{\rho_c\,|\,c\in \C\}$, where
$$\rho_c(v) = \left \{\begin{array}{ll} 1 & v=c \\ 0 & \text{else}\end{array}\right.$$
In polynomial notation, $\rho_c$ is represented by the polynomial $\prod_{c_i = 1} x_i \prod_{c_j = 0} (1-x_j)$. The characteristic functions $\rho_c$ form a basis for $R_\C$ as an $\F_2$-vector space, and they have several useful properties; those most important for our proofs are described briefly here.
* Each element of $R_\C$ corresponds to the formal sum of basis elements for the codewords in its support: $\displaystyle f=\sum_{f(c) = 1} \rho_c$.
* The product of two basis elements is 0 unless they are identical: $\rho_c\rho_d = \left\{\begin{array}{ll} \rho_c & c=d\\ 0 & \text{ else }\end{array}\right.$.
* If $1_\C$ is the identity of $R_\C$, then $\displaystyle 1_\C = \sum_{c\in \C} \rho_c$.
These basis elements prove essential to making our proofs intuitive. Recall that given a code map $q:\C\rightarrow \D$ the pullback map $q^*:R_\D\rightarrow R_\C$ is given by $q^*(f) = f\circ q$; we now show briefly that $q^*$ is a ring homomorphism.
For any code map $q:\C\rightarrow \D$, the pullback $q^*:R_\D\rightarrow R_\C$ is a ring homomorphism.
First, note that $q^*(f) = f\circ q$ is a well-defined function $\C\rightarrow \{0,1\}$ and any such map is an element of $R_\C$, the ring of all such functions. Now we need to prove $q^*:R_\D\rightarrow R_\C$ is a ring homomorphism. To show equality in $R_\C$, we need only show that two functions evaluate the same on any codeword in $\C$. Since elements of $R_\C$ and $R_\D$ are polynomial mappings, we see that both addition and multiplication are preserved:
$$q^*(f+g) (c) = (f+g)(q(c)) = f(q(c))+ g(q(c)) = q^*f(c) + q^*g(c)$$
$$q^*(fg)(c) = (fg)(q(c)) = f(q(c))g(q(c)) = q^*(f)(c) q^*(g)(c).$$
Thus $q^*$ is a ring homomorphism.
This immediately raises the question: which ring homomorphisms can be obtained as the pullback of code maps? First we make the following observation:
For any ring homomorphism $\phi:R_\D \rightarrow R_\C$, and any element $c\in\C$, there is a unique $d\in \D$ such that $\phi(\rho_{d})(c) = 1$.
To prove existence, note that $\sum_{c\in \C} \rho_c = 1_{\C} = \phi(1_{\D}) = \phi(\sum_{d\in \D} \rho_d) = \sum_{d\in \D} \phi(\rho_d)$. For each $c\in \C$, $1 = \rho_c(c) = \left(\sum_{c'\in \C} \rho_{c'}\right)(c) = \left(\sum_{d\in \D} \phi(\rho_d)\right)(c)$, and thus
$\phi(\rho_d)(c)=1$ for at least one $d\in \D$.
To prove uniqueness, suppose there exist distinct $d,d'\in \D$ such that $\phi(\rho_d)(c) = \phi(\rho_{d'})(c) = 1$. Then we would have $1= (\phi(\rho_d)\phi(\rho_{d'}))(c) = \phi(\rho_d\rho_{d'})(c) = \phi(0)(c) = 0$, but this is a contradiction. Thus such a $d$ must be unique.
This allows us to define a unique code map corresponding to any ring homomorphism.
Given a ring homomorphism $\phi:R_\D\rightarrow R_\C$, we define the associated code map $q_\phi:\C\rightarrow \D$ as follows:
$$q_\phi(c) = d_c$$
where $d_c$ is the unique element of $\D$ such that $\phi(\rho_{d_c})(c) =1$.
Let $\C = \{110, 111, 010, 001\}$ and $\D = \{00, 10, 11\}$. Let $\phi:R_\D\rightarrow R_\C$ be defined by $\phi(\rho_{11} )= \rho_{110} + \rho_{111} + \rho_{010}$, $\phi(\rho_{00}) = \rho_{001}$, and $\phi(\rho_{10}) = 0$. Then the corresponding code map $q_\phi$ will have $q_\phi(110) = q_\phi(111) = q_\phi(010) = 11$, and $q_\phi(001) = 00$. Note that there is no element $c\in \C$ with $q_\phi(c) = \rho_{10}$ so $q_\phi$ is not surjective.
We can now prove Theorem <ref>: that code maps and ring homomorphisms are in bijection via the pullback map.
This is a special case of a proposition from <cit.>. We include our own proof for completeness, and also because it shows constructively how to obtain $q_\phi$ from $\phi$.
Lemma <ref> shows that the pullback $q^*$ is a ring homomorphism, so we now prove that any homomorphism can be obtained as the pullback of a code map. Given a ring homomorphism $\phi:R_\D \rightarrow R_\C$, define $q_\phi$ as above. We must show that the $q_\phi^* = \phi$, and moreover that $q_\phi$ is the only code map with this property.
The fact that $q_\phi^* = \phi$ holds essentially by construction: let $f\in R_\D$, so $f=\sum_{f(d) = 1} \rho_d$. Then, for any $c\in \C$,
$$q_\phi^*(f)(c) = f(q_\phi(c)) = \sum_{f(d) = 1} \rho_d(q_\phi(c)) = \sum_{f(d)=1}\rho_d(d_c) = \left\{\begin{array}{ll} 1 & \text{ if } f(d_c) = 1\\ 0 & \text{ if }f(d_c) = 0 \end{array}\right.$$
whereas, remembering from above that there is exactly one $d\in \D$ such that $\phi(\rho_d)(c)=1$ and that this $d$ may or may not be in the support of $f$, we have
$$\phi(f)(c) = \sum_{f(d)=1} \phi(\rho_d)(c)= \left\{\begin{array}{ll} 1 & \text{ if }d_c\in f^{-1}(1) \\ 0 & \text{ if } d_c\notin f^{-1}(1) \end{array} \right. = \left\{\begin{array}{ll} 1 & \text{ if }f(d_c) = 1 \\ 0 & \text{ if }f(d_c) = 0 \end{array} \right. . $$
Thus, $\phi = q_\phi^*$.
Finally, to see that $q_\phi$ is the only code map with this property, suppose we have a different map $q\neq q_\phi$. Then there is some $c\in \C$ with $q(c) \neq q_\phi(c)$; let $d_c = q_\phi(c)$, so $q(c)\neq d_c$. Then $\phi(\rho_{d_c})(c) = 1$ by definition, but $q^*(\rho_{d_c})(c) = \rho_{d_c}(q(c)) =0$ as $q(c)\neq d_c$. So $q^*$ does not agree with $\phi$ and hence $\phi$ is not the pullback of $q$, so $q_\phi$ is the unique code map with pullback $\phi$.
While Theorem <ref> gives a clean bijective correspondence between code maps and ring homomorphisms, it is too general to capture only a class of natural code maps. As the theorem shows, any well-defined code map has a corresponding neural ring homomorphism, even a random assignment $q:\C\rightarrow \D$. Our question from this point, therefore, is the following:
What algebraic constraints can be put on homomorphisms between neural rings to capture some meaningful restricted class of code maps?
Since we have determined that ring homomorphism alone is insufficiently restrictive, the next natural idea is to examine ring isomorphism which should be more restrictive. This is true, but the standard notion of isomorphism in fact captures very little actual similarity between codes beyond the number of codewords, as the following lemma shows.
A ring homomorphism $\phi:R_\D\rightarrow R_\C$ is an isomorphism if and only if $q_\phi:\C\rightarrow \D$ is a bijection.
Note that $R_\C\cong \F_2^{|\C|}$ and $R_\D \cong \F_2^{|\D|}$, and $\F_2^{|\C|} \cong \F_2^{|\D|}$ if and only if $|\C| = |\D|$. Suppose $\phi$ is an isomorphism; then we must have $|\C| = |\D|$. If $q_\phi$ is not injective, then there is some $d\in \D$ such that $\phi(\rho_d)(c) = 0$ for all $c\in \C$. But then $\phi(\rho_d) = 0$, which is a contradiction since $\phi$ is an isomorphism so $\phi^{-1}(0) = \{0\}$. Thus $q_\phi$ is injective, and since $|\C| = |\D|$, this means $q_\phi$ is a bijection.
On the other hand, suppose $q_\phi:\C\rightarrow \D$ is a bijection. Then $|\C| = |\D|$, so $R_\C\cong R_\D$, and as both are finite, $|R_\C|= |R_\D|$. Consider an arbitrary element $f\in R_\C$. For each $c\in f^{-1}(1)$, there is a unique $d\in \D$ so $\phi(\rho_d) = c$; furthermore as $q_\phi$ is a bijection, all these $d$ are distinct. . Then
$$\phi\big(\sum_{\substack{d=q_\phi(c),\\ c\in f^{-1}(1)}} \rho_d\big) = \sum_{\substack{d=q_\phi(c)\\c\in f^{-1}(1)}} \phi(\rho_d) = \sum_{c\in f^{-1}(1)} \rho_c = f. $$
Hence $\phi$ is surjective, and since $|R_\C| = |R_\D|$, we know $\phi$ is also bijective and hence an isomorphism.
This shows us that not only would two permutation-equivalent codes have isomorphic neural rings, but in fact any two codes with the same number of codewords have isomorphic neural rings. Lemma <ref> highlights the main difficulty with using ring homomorphism and isomorphism alone: the neural rings are rings of functions from $\C$ to $\{0,1\}$, and the abstract structure of such a ring depends solely upon the size of the code, $|\C|$. By considering such rings abstractly without their presentation, we can reflect no other structure, not even such basic information as the number of bits per codeword. In particular, we cannot track the behavior of the variables $x_i$ which represent individual neurons, since once we move to an abstract view of the ring the presentation is lost, and it is merely an $\F_2$-vector space determined completely by the number of codewords.
In order to track the activity of neurons, we turn next to modules, where we can consider the neural ring with its presentation, tracking the variables which represent the neurons via their action on elements of the neural ring. We will consider each neural ring as a carefully designed module under the ring for the full code on the same number of neurons, which will allow us to keep track of the structure.
§.§ Neural rings as modules
Given a code $\C$ on $n$ neurons, we consider $R_\C$ as a module under $R[n]$ under the module action
$$R[n]\times R_\C\rightarrow R_\C$$
given by multiplication - that is, for a codeword $c\in \C$, we have
$$(r\cdot f ) (c) = r(c) f(c).$$
In particular, for any basis element $\rho_c$, we have
$$x_i\cdot \rho_c = \left\{\begin{array}{ll} \rho_c & \text{if } c_i = 1\\ 0 & \text{if }c_i =0\end{array}\right. .$$
and we can determine the rest of the action by extension. In particular, for any $r\in R[n]$ and $f\in R_\C$, we will have $(r\cdot f)^{-1}(1) = (r^{-1}(1)\cap f^{-1}(1))$.
This module action is equivalent to multiplication of polynomials, with the result considered as an element of $R_\C$. In particular, we always impose the relationships $x_i(1-x_i) = 0$, which also means $x_i^2 = x_i$ and $(1-x_i)^2 = (1-x_i)$. The use of these relationships can be seen more clearly in the following example:
Consider again the code $\C = \{000,100,101\}$. $R_\C$ is a module under $R[3] = R_{\{0,1\}^3}$.
* Consider the element $1-x_1$ of $R[3]$. Then
$$(1-x_1)\cdot (1-x_1)(1-x_2)(1-x_3) = (1-x_1)(1-x_2)(1-x_3).$$
$$(1-x_1)\cdot x_1(1-x_2)(1-x_3) = 0.$$
* Consider the element $x_1x_2$ of $R[3]$. Although $x_1x_2$ is a nonzero element in $R[3]$, it evaluates to $0$ for all codewords in $\C$, so for any element $f\in R_\C$, we have $x_1x_2\cdot f = 0$.
As another way to look at this action, we can consider each neural ring using the basis of characteristic functions and perform the same calculations. $R[3]$ is itself a neural ring, and so has basis elements $\{\rho_c\,| \, c\in \{0,1\}^3\}$. We can look at the action in terms of these basis elements (using the same examples as above)
* The element $1-x_1 \in R[3]$ is written $\rho_{000} + \rho_{010} + \rho_{001} + \rho_{011}$. In $R_\C$, we have $(1-x_1)(1-x_2)(1-x_3) = \rho_{000}$ and $x_1(1-x_2)(1-x_3) = \rho_{100}$. Then
$$(\rho_{000} + \rho_{010} + \rho_{001} + \rho_{011}) \cdot \rho_{000} = \rho_{000}$$
$$(\rho_{000} + \rho_{010} + \rho_{001} + \rho_{011})\cdot \rho_{011} = 0.$$
* In $R[3]$, $x_1x_2 = \rho_{110} + \rho_{111}$. Although this is a nonzero element in $R[3]$, $110$ and $111$ are not in $\C$ and thus $R_\C$ has no basis elements for them. Thus, for any element $f\in R_\C$, we have $(\rho_{110} + \rho_{111})\cdot f = 0$.
We have already observed that a neural ring is an $\F_2$-vector space with basis $\{\rho_c\,|\, c\in \C\}$. As an $R[n]$-module, the basis elements are the unique elements of $R_\C$ (besides $0$) such that $x_i\cdot f\in \{f,0\}$ for all $i\in [n]$. We also see that any two basis elements can be distinguished by the action of some $x_i$; that is, for $\rho_c\neq \rho_{c'}$, there is some $i$ so $c_i\neq c'_i$, and hence exactly one of $x_i\cdot \rho_c$ and $ x_i\cdot \rho_{c'}$ will be $0$. In fact, these characteristics are sufficient to characterize the neural rings as $R[n]$-modules. This is the substance of Theorem <ref>, which we now prove.
Let $M$ be an $R[n]$-module with $\F_2$-basis $\rho_1,...,\rho_d$ with the given properties. First, we identify the basis elements $\rho_j$, if not already known. We can do this by considering the action $x_i\cdot m$ on every element of $M$. Note that if $m$ is neither a basis element $\rho_j$, nor $0_M$, then it is a formal sum of at least two basis elements; by property 2, we can distinguish between these two elements with some $x_i$, so $x_i\cdot m$ will be neither $m$ nor $0$. Hence, the set $\{m\in M \,|\, x_i\cdot m \in \{m,0\}\text{ for all }i\in [n], m\neq 0_M\}$ will be exactly the set $\{\rho_1,...,\rho_d\}$. It is possible that a basis element $\rho_j$ may be such that $x_i\cdot \rho_j =0$ for all $i\in [n]$, but this basis element is not the same as the identity $0_M$ and we distinguish them if necessary by considering the action of $1-x_i$.
Once we have the set of basis elements $\rho_1,...,\rho_d$, we associate a related code $\C\subset\{0,1\}^n$. To each basis element $\rho_j$ we associate a unique word $c^j$ by taking $c^j_i = \left\{\begin{array}{ll}1 & x_i\cdot \rho_j = \rho_j\\ 0 & x_i\cdot \rho_j = 0 \end{array}\right.$ This assignment is well-defined by property 1, and unique by property 2. We then take $\C = \{c^j\,|\,j\in [d]\}$. Then we make the obvious choice of module isomorphism: $\phi: M\rightarrow R_\C$ given by $\phi(\rho_j) = \rho_{c^j}$. By our choice of associated codewords, we have ensured that the module action is preserved across $\phi$: that is, $\phi(x_i \cdot\rho_j) = x_i\cdot \rho_{c^j}$, and as both modules are $\F_2$-vector spaces, they are isomorphic as $R[n]$ modules.
On the other hand, suppose $M$ is an $R[n]$-module which is isomorphic to a neural ring $R_\C$; let $\phi:R[n]\rightarrow M$ be a module isomorphism. Since $R_\C$ is a finite $\F_2$-vector space, we know $M$ is an $\F_2$-vector space with basis $\{ \phi(\rho_c)\ | \ c\in \C\}$; the relevant properties of the basis follow immediately by properties of module isomorphism.
Suppose $M$ is an $R[3]$-module with 3 basis elements $\rho_1,\rho_2,\rho_3$. Suppose, for example, that $x_1\cdot \rho_1 = \rho_1$ and $x_2\cdot \rho_1 = x_3\cdot \rho_1 = 0$. We thus know $\rho_1$ corresponds to the word $100$ and hence $\rho_1\leftrightarrow \rho_{100}$. Similarly, suppose we then find that $\rho_2\leftrightarrow \rho_{000}$ and $\rho_3\leftrightarrow \rho_{101}$; then $M \cong R_\C$ as $R[n]$-modules, where $\C = \{000, 100, 101\}$.
Now that we have a framework to consider a neural ring $R_\C$ on $n$ neurons as an $R[n]$-module that preserves the code structure, we consider module homomorphisms between neural rings as an alternative to the too-general ring homomorphisms
Let $R$ be a ring and suppose $M,N$ are $R$-modules. Then a map $\phi:M\rightarrow N$ is an $R$-module homomorphism if for any $m\in M, n\in N$, and $r,s\in R$, we have
$$f(r\cdot m + s\cdot n) = r\cdot f(m) + s\cdot f(n).$$
A module homomorphism $\phi$ is an $R$-module isomorphism if $\phi$ is a bijection.
For neural rings, this is an incredibly restrictive notion, because we can only consider module homomorphisms between $R_\C$ and $R_\D$ when the codes $\C$ and $\D$ have the same length $n$ and are thus both $R[n]$-modules. Under this framework, the code maps we are allowed are only inclusion.
Given a code map $q:\C\rightarrow \D$, where $\C,\D$ are both subsets of $\{0,1\}^n$, the induced ring homomorphism $\phi_q$ is a module homomorphism $R_\D\rightarrow R_\C$ if and only if $q$ is an inclusion map $q:\C\hookrightarrow \D$, and it is a module isomorphism if and only if $q$ is the identity, i.e. $\C=\D$ and $q(c) = c$ for all $c$.
Because we have an $\F_2$-vector space structure, $\phi_q$ is a module homomorphism if and only if for every $i\in[n]$ and $d\in \D$, we have
$$x_i\cdot \phi_q(\rho_d) = \phi_q(x_i\cdot \rho_d).$$
These are equal if and only if they evaluate the same on every element $c\in \C$. Computing, we see
$$x_i\cdot \phi_q(\rho_d )(c) = \left\{\begin{array}{ll} 1 &\text{if } c_i=1 \text{ and }q(c) = d \\ 0 &\text{if }c_i = 0 \text{ or } q(c)\neq d \end{array}\right.$$
$$\phi_q(x_i\cdot \rho_d)(c) = \left\{\begin{array}{ll} 1 &\text{if } d_i =1 \text{ and }q(c)=d \\ 0 & \text{if } d_i=0 \text{ or }q(c)\neq d\end{array}\right.$$
and these are equal on every $c\in \C$ if and only if whenever $q(c) = d$, we have $c_i=d_i$ for all $i \in [n]$. This is only true if $q$ is an inclusion map.
Finally, we have an isomorphism if and only if $\phi_q$ is a bijection, but this happens if and only if $q$ is itself a bijection, as we saw with ring homomorphisms. Thus $\phi_q$ is a module isomorphism if and only if $q$ is the identity map.
Clearly, this notion is far too restrictive; in particular, code maps between two codes of different lengths are not permitted at all; nor are permutations possible. In the next section, we show how to extend our list of possible maps further by allowing module homomorphisms between $R_\D$ and $R_\C$ under different rings $R[n]$ and $R[m]$.
§.§ Modules under different rings
As we discussed in Section 1.3, we may consider an $R[n]$-module $R_\C$ as an $R[m]$ module, provided we have a ring homomorphism $\tau:R[m]\rightarrow R[n]$.
For any given $n,m$ there is a standard map $\tau:R[m]\rightarrow R[n]$:
$$\tau(x_i) = \left\{\begin{array}{ll} x_i &\text{if } i\leq n \\ 0 & \text{if } i>n\end{array}\right.$$
Now, our question becomes: which code maps induce $R[m]$-module homomorphisms under this construction? Unfortunately, we still find the class of allowable code maps to be very restrictive, and dependent almost entirely on the relative sizes of $m$ and $n$:
A code map $q:\C\rightarrow \D$ where $\C\subset\{0,1\}^n$ and $\D\subset\{0,1\}^m$ induces an $R[m]$-module homomorphism $\phi_q:R_\D\rightarrow R_\C$ under the map $\tau$ described above if and only if
* $[n>m]$ $q$ is the deletion of the last $n-m$ neurons, possibly composed with an inclusion map.
* $[n=m]$ $q$ is an inclusion map.
* $[n<m]$ $q$ is given by adding $m-n$ trivial neurons (never firing, always 0) to the end each word, possibly composed with an inclusion map.
All parts proceed very like the proof of Lemma <ref>.
* $\phi_q$ is a module homomorphism if and only if for every $i\in[m]$ and $d\in \D$, we have
$$\tau(x_i)\cdot \phi_q(\rho_d) = \phi_q(x_i\cdot \rho_d).$$
These are equal if and only if they evaluate the same on every element $c\in \C$. For all $i\in[m]$,
$$\tau(x_i)\cdot \phi_q(\rho_d )(c) = x_i\cdot \phi_q(\rho_d)(c) = \left\{\begin{array}{ll} 1 &\text{if } c_i=1 \text{ and }q(c) = d \\ 0 &\text{if }c_i = 0 \text{ or } q(c)\neq d \end{array}\right.$$
$$\phi_q(x_i\cdot \rho_d)(c) = \left\{\begin{array}{ll} 1 &\text{if } d_i =1 \text{ and }q(c)=d \\ 0 & \text{if } d_i=0 \text{ or }q(c)\neq d\end{array}\right.$$
and these are equal on every $c\in \C$ if and only if whenever $q(c) = d$, we have $c_i=d_i$ for all $i \in [m]$. This exactly determines the map given by deleting the last $n-m$ neurons, as $\C\subset\{0,1\}^n$, $\D\subset\{0,1\}^m$, and $c$ agrees with $q(c)$ on the first $m$ neurons. However, since we place no restriction on $\phi_q$ being a bijection, inclusion may also be permitted; i.e., it is possible that $q(\C)\subsetneq \D$.
* If $n=m$, then we are exactly in the case of Lemma <ref> so the only possible map is the inclusion map $q:\C\hookrightarrow \D$.
* $\phi_q$ is a module homomorphism if and only if for every $i\in[m]$ and $d\in \D$, we have
$$\tau(x_i)\cdot \phi_q(\rho_d) = \phi_q(x_i\cdot \rho_d).$$
These are equal if and only if they evaluate the same on every element $c\in \C$. Computing for $i\leq n$, we have
$$\tau(x_i)\cdot \phi_q(\rho_d )(c) = x_i\cdot \phi-q(\rho_d)(c) = \left\{\begin{array}{ll} 1 &\text{if } c_i=1 \text{ and }q(c) = d \\ 0 &\text{if }c_i = 0 \text{ or } q(c)\neq d \end{array}\right.$$
$$\phi_q(x_i\cdot \rho_d)(c) = \left\{\begin{array}{ll} 1 &\text{if } d_i =1 \text{ and }q(c)=d \\ 0 & \text{if } d_i=0 \text{ or }q(c)\neq d\end{array}\right..$$
and these are equal on every $c\in \C$ and if and only if whenever $q(c) = d$, we have $c_i=d_i$ for all $i \in [n]$.
For $i>n$, we know
$$\tau(x_i)\cdot \phi_q(\rho_d )(c) = 0 \cdot \phi_q(\rho_d)(c) = 0$$
$$\phi_q(x_i\cdot \rho_d)(c) = \left\{\begin{array}{ll} 1 &\text{if } d_i =1 \text{ and }q(c)=d \\ 0 & \text{if } d_i=0 \text{ or }q(c)\neq d\end{array}\right..$$
so these are equal on every $c\in \C$ if and only if whenever $q(c) = d$ we have $d_i=0$ for all $i>n$.
These two requirements combine to describe the map given by adding $m-n$ zeros to the end of each codeword $c\in \C$. As in (1), we also must allow inclusion since we are not requiring that $\phi_q$ to be a bijection and it is possible that $q(\C)\subsetneq\D$.
Hence, these are the only 3 possible types of maps under the canonical map $\tau$.
Once again, this set of allowable code maps is far too restrictive; in particular, permutation maps are still omitted. However, we have obtained the ability to consider module homomorphisms between neural rings of different length, so this construction is a good step towards our goal. We therefore consider situations where this map $\tau$ is allowed to be more general, not just the canonical map described above.
As a code map $q:\C\rightarrow \D$ determines the related map $\phi_q$ between the modules $R_\D$ and $R_\C$, the code map $q$ places certain restrictions on which ring homomorphisms $\tau$ may be used between $R[m]$ and $R[n]$. Since we have a preconceived idea of which code maps are desirable (permutation, dropping a neuron, inclusion), it is natural to let the code maps drive the choice of ring homomorphism. The ring homomorphisms for which a certain map $\phi_q$ is still a module homomorphism are the compatible homomorphisms, which we formally define in Section 1.4.
It is worth noting that not every possible function between two neural rings has a compatible ring homomorphism.
Consider the codes $\C = \{000,100,101\}$ and $\D = \{00,10,11\}$ , and let the map $\phi:R_\D\rightarrow R_\C$ be given by $\phi(\rho_{00}) = \rho_{000} + \rho_{100}$, $\phi(\rho_{10}) = \rho_{100}$, and $\phi(\rho_{11})= 0$. Extending by linearity to all elements of $R_\D$ gives us a group homomorphism, which is easy to check, although not a ring homomorphism (also easy to check).
As a polynomial map, this is the group homomorphism given by: $x_1\rightarrow y_1(1-y_2)(1-y_3)$, $x_2\rightarrow 0$. There is, however, no compatible ring homomorphism $\tau:R[2]\rightarrow R[3]$ so that $\phi$ is an $R[2]$-module homomorphism. To see this, note that any such homomorphism $\tau$ would require $\phi(\rho_{00}\cdot \rho_{00}) = \tau(\rho_{00})\phi(\rho_{00}) = \tau(\rho_{00}) \cdot[\rho_{000} + \rho_{100}]$. But as $\rho_{00} \cdot \rho_{00} = \rho_{00}$, this must equal $\rho_{000} + \rho_{100}$. So $\tau(\rho_{00})$ must preserve $\rho_{000}$ and $\rho_{100}$. Similarly, $\tau(\rho_{10})$ must preserve $\rho_{100}$. So $\tau(\rho_{10}) \tau(\rho_{00})$ must preserve $\rho_{100}$ at least, so $\tau(\rho_{10})\tau(\rho_{00})\neq 0$.
Note $\tau(\rho_{00}\rho_{10}) = \tau(0)=0$, but as $\tau$ is a ring homomorphism, we also have $\tau(\rho_{00}\rho_{10}) = \tau(\rho_{00})\tau(\rho_{10}) \neq 0$. So no such $\tau$ can exist; there is no compatible ring homomorphism for $\phi$.
However, ring homomorphisms between two neural rings are guaranteed to have at least one compatible ring homomorphism $\tau$; this is the substance of the first part of Theorem <ref>. For this result, we use the idea that elements of $R_\D$ can also be thought of as elements of the ambient ring $R[m]$. For example, each basis element $\rho_d$ of $R_\D$ is the function which detects only the codeword $d$; however, since $d\in \{0,1\}^m$, we know $R[m]$ has a basis element $\rho_d$ which detects only $d$ as well, and we associate these two basis elements. Likewise, any function $f\in R_\D$ corresponds the subset $f^{-1}(1)\subset \D$ which it detects, so we can consider $f$ as a function in $R[m]$ which detects the same set of codewords. Theorem <ref> not only states that any ring homomorphism $\phi_q$ between neural rings has at least one compatible homomorphism, but also states that the set of compatible ring homomorphisms is given those maps which agree as far as possible with the original $\phi_q$. We now prove Theorem <ref>.
Let $\phi:R_\D\rightarrow R_\C$ be a ring homomorphism. To construct a compatible ring homomorphism $\tau:R[m]\rightarrow R[n]$, first select one basis element $\rho_d$ of $R_\D$. Note that $\rho_d$ (as the function which detects exactly the codeword $\{d\}$) is also a basis element of $R[m]$, and define $\tau(\rho_d ) = \phi(\rho_d) + \sum_{v\in \{0,1\}^n \backslash \C}\rho_v$; that is, $\tau(\rho_d)$ will detect all the same codewords as $\phi(\rho_d)$, but also all the codewords of $\{0,1\}^n$ which are not part of $\C$. For all other $d\in \D$, define $\tau(\rho_d) = \phi(\rho_d)$, and for all $v\in \{0,1\}^m\backslash \D$, define $\tau(\rho_v ) = 0$. Extend $\tau$ to all elements of $R[m]$ by linearity; that is, if $f = \sum \rho_c$, then $\tau(f) = \sum \tau(\rho_c)$. This gives a ring homomorphism.
Now, we will show that the property of compatibility is equivalent to the property of extensions. Let $\tau:R[m]\rightarrow R[n]$ be a ring homomorphism, and let $f=\sum_{d\in f^{-1}(1)} \rho_d \in R_\D$. As $f\cdot f = f$, then $\tau$ is compatible with $\phi$ if and only if we have $\phi(f)=\phi(f\cdot f) = \tau(f)\cdot \phi(f)$, which occurs if and only if $\tau(f)$ detects at least the same codewords as $\phi(f)$, which happens if and only if $\tau(f)^{-1}(1)\supseteq \phi(f)^{-1}(1)$.
Since our ambient rings $R[n]$ and $R[m]$ are themselves neural rings, we note that that each map $\tau:R[m]\rightarrow R[n]$ is also a ring homomorphism between the neural rings for the complete codes. We have shown that code maps correspond to ring homomorphisms, and thus each ring homomorphism $\tau$ corresponds to a unique code map $q_\tau:\{0,1\}^m \rightarrow \{0,1\}^n$ between the complete codes. Furthermore, Theorem <ref> shows that $\tau$ is compatible with $\phi$ if and only if it is an extension of $\phi$, in that $\phi(f)^{-1}(1)\subseteq \tau(f)^{-1}(1)$. We also know from Theorem <ref> that given a ring homomorphism $\phi:R_\D\rightarrow R_\C$, we can always find a ring homomorphism $\tau:R[m]\rightarrow R[n]$ which is compatible with $\phi$. We now confirm that idea by considering the code maps and prove the following Lemma, which shows we can generate a compatible $\tau$ at the level of the codes. We do this by taking any code map $q':\{0,1\}^n\rightarrow \{0,1\}^m$ which extends $q$ (so $q'(c)=q(c)$ for all $c\in \C$) and using the corresponding ring homomorphism $\tau_q$. Then $\tau_{q'}$ will take each function to its pullback by $q'$, and it will be compatible with $\phi$.
The ring homomorphism $\tau:R[m]\rightarrow R[n]$ is compatible with the ring homomorphism $\phi:R_\D\rightarrow R_\C$ if and only if $q_\phi = q_\tau\big|_\C$.
Suppose $\tau$ is compatible with $\phi$. Note that $q_\tau\big|_\C =q_\phi$ if and only if $q_\tau(c) = q_\phi(c)$ for all $c\in \C$. So, suppose by way of contradiction that $q_\tau(c) \neq q_\phi(c)$ for some $c\in \C$. Let $d=q_\phi(c)$. Then, $\phi(\rho_d)(c) =1$. But $\phi(\rho_d) = \phi(\rho_d\cdot \rho_d) = \tau(\rho_d) \cdot \phi(\rho_d)$, and we know $\tau(\rho_d)(c) = 0$, not $1$, so $\tau$ and $\phi$ cannot be compatible. This is a contradiction.
Now, suppose $q_\phi = q_\tau\big|_\C$. Suppose $\phi(\rho_d)(c) = (1)$. By our code map-homomorphism correspondence, this means that $q_\phi(c) = d$. So then $q_\tau(c) = d$ also, and thus again by the correspondence, $\tau(\rho_d)(c)(1)$. Thus, for each $d\in \D$, we have $\tau(\rho_d)^{-1}(1) \supseteq \phi(\rho_d)^{-1}(1)$, and by Theorem <ref>, $\tau$ is compatible with $\phi$.
Consider again the codes $\C = \{000,100,101\}$ and $\D = \{00,10,11\}$. Let the map $\phi:R_\D\rightarrow R_\C$ be given by $\phi(\rho_{00}) = \rho_{000}$, $\phi(\rho_{10} )= \rho_{100} + \rho_{101}$, and $\phi(\rho_{11}) = 0$, and extend by linearity to all other elements. It is easy to check that $\phi$ is a ring homomorphism.
Consider $\tau_1:R[2]\rightarrow R[3]$ given by $\tau_1(\rho_{00}) = \rho_{000} + \rho_{001}+\rho_{010}+\rho_{110} + \rho_{011}+ \rho_{111}, \tau_1(\rho_{10}) = \rho_{100} + \rho_{101}$, and $\tau_1(\rho_{11}) = 0$. Extend again by linearity. Then $\tau_1$ is a compatible ring homomorphism, which is again quick to check.
Now, consider $\tau_2:R[2]\rightarrow R[3]$ given by $\tau_2(\rho_{00}) = \rho_{000} + \rho_{001}+\rho_{010}+\rho_{110} + \rho_{011}+ \rho_{111}, \tau_2(\rho_{10}) = \rho_{100}$, and $\tau_2(\rho_{11}) = \rho_{101}$. $\tau_2$ is not a compatible ring homomorphism, as if it were, we would have
$$\rho_{100} + \rho_{101} = \phi(\rho_{10}) = \phi(\rho_{10}\rho_{10}) = \tau_2(\rho_{10})\phi(\rho_{10}) =\rho_{100}\cdot(\rho_{100} + \rho_{101}) = \rho_{100}$$
which is a contradiction.
Theorem <ref>, along with Corollary <ref> show that compatibility alone is too general a notion. While restricting our $\tau$ to the standard map $x_i\mapsto x_i$ was far too restrictive, allowing $\tau$ to be any compatible map once again gives us all possible code maps. In Theorem <ref> it was established that ring homomorphisms $\phi:R_\D\rightarrow R_\C$ between two neural rings are in correspondence with the set of possible functions $q:\C\rightarrow \D$. Theorem <ref> makes it clear that even considering module homomorphisms does not capture the restricted class of code maps we desire, since not every code map preserves structure, but every code map generates a related ring homomorphism and therefore a related module homomorphism. Using module properties we can extract code structure and track its behavior across the homomorphism, but we we have not yet ensured that structure is preserved across maps.
§ LINEAR-MONOMIAL MODULE HOMOMORPHISMS & NEURAL RING HOMOMORPHISMS
We have seen from Proposition <ref> that if we require indeterminates map to themselves if possible, and $0$ if not, we obtain a very restricted class of code maps. In the rest of Section 2, we proved that if we make no requirements on the indeterminates and take module homomorphisms under any compatible homomorphism, we can get any possible code map. What we are working towards is something between these ideas - a set of code maps where neurons preserve their basic activity, but we might only look at a subset of the neurons, or we might allow their names to change.
Therefore, we now place make a slight relaxation on the model from Proposition <ref> and consider linear monomial homomorphisms - where $\tau(x_i)\in \{x_j,0,1\}$, as discussed in Section 1.5. The code maps which have a possible compatible homomorphism which is also linear-monomial we will choose as our neural ring homomorphisms. Not all ring homomorphisms are linear-monomial, as shown by the following example:
Consider the map $\tau:R[1]\rightarrow R[2]$ given by $\tau(\rho_1) = \rho_{01} + \rho_{10}$ and $\tau(\rho_0) = \rho_{00} + \rho_{11}$. Here $\tau(x_1) = x_1+x_2$, and $\tau(x_i)\notin \{x_1,x_2,0,1\}$ as would be required.
Observe that a linear-monomial ring homomorphism $\tau:R[m]\rightarrow R[n]$ is defined by a vector $S=(s_1,...,s_m)$, where $S_i\in [n]\cup\{0,u\}$, so that
$$\tau(x_i) = \left\{\begin{array}{ll} x_j & \text{if }s_i = j \\ 0 & \text{if }s_i = 0\\ 1 & \text{if }s_i=u\end{array}\right. .$$
$S$ is a vector which stores the pertinent information about $\tau$, and each possible $S$ with $s_i \in [n]\cup \{0,u\}$ defines a possible neuron-preserving $\tau$. We refer to the $\tau$ defined by $S$ as $\tau_S$.
We have defined linear-monomial homomorphism at the level of maps between the ambient rings $R[n]$ only, and not between neural rings in general, even though the indeterminate $x_i$ does have meaning in a general neural ring. This allows us to have a linear-monomial homomorphism $\tau$ for any given $S$ without accidentally contradicting given relationships; that is, we don't need to consider relationships amongst the $x_i$'s in $R[n]$, since there are no relationships in $R[n]$ besides the Boolean relationships $x_i(1-x_i) = 0$. Thus, any choice of the vector $S$ will give a valid ring homomorphism.
The composition of two linear-monomial homomorphisms is also linear-monomial.
Suppose $S=(s_1,...,s_n)$ with $s_i\in [m]\cup\{0,u\}$ and $T=(t_1,...,t_m)$ with $t_i \in [\ell]\cup\{0,u\}$ are given as above, with $\tau_{S}:R[n]\rightarrow R[m]$ and $\tau_{T}: R[m]\rightarrow R[\ell]$. To prove the lemma, we need to find $W = (w_1,...,w_n)$ with $w_i\in [\ell]\cup\{0,u\}$ so $\tau_{W} = \tau_T\circ \tau_S:R[n]\rightarrow R[\ell]$.
Define the vector $W$ by
$$w_i =\left\{\begin{array}{ll} t_{s_i} & \text{if }s_i \in [m]\\ 0 & \text{if }s_i = 0 \\ u &\text{if } s_i = u \end{array}\right.$$
We use variables $z_i$ for $R[n]$, $y_i$ for $R[m]$, $x_i$ for $R[\ell]$ to make it clear at all times what ring we are working in. Then, unraveling the definitions,
$$\tau_W(z_i) = \left\{\begin{array}{ll} x_j &\text{if } t_{s_i} = j \\ 0 &\text{if } t_{s_i} = 0 \\ & \text{or if }s_i = 0 \\ 1 & \text{if }t_{s_i} = u\\ & \text{or if } s_i = u\end{array}\right.
= \left\{\begin{array}{ll} x_j &\text{if } s_i = k \text{ and } t_k=j \\ 0 & \text{if }s_i = k \text{ and } t_k = 0\\ & \text{or if }s_i = 0 \\ 1 &\text{if } s_i = k \text{ and } t_k = u \\ & \text{or if }s_i = u\end{array}\right. \hspace{1in}$$
\hspace{1in} = \left\{\begin{array}{ll} x_j &\text{if } \tau_S(z_i) = y_k \text{ and } \tau_T(y_k ) = x_j \\ 0 &\text{if } \tau_S(z_i) = y_k \text{ and } \tau_T(y_k) = 0\\ &\text{or if } \tau_S(z_i) = 0 \\ 1 & \text{if }\tau_S(z_i) = y_k \text{ and } \tau_T(y_k) = 1 \\ & \text{or if }\tau_S(z_i) =1 \end{array}\right. = \left\{\begin{array}{ll} x_j & \text{if }\tau_T\circ \tau_S(z_i) = x_j \\ 0 & \text{if }\tau_T\circ\tau_S(z_i) = 0 \\ 1 & \text{if }\tau_T\circ \tau_S(z_i) = 1 \end{array}\right. .$$
Given such a vector $S=(s_1,...,s_m)$, where $s_i\in [n]\cup\{0,u\}$, define the code map $q_S:\{0,1\}^n\rightarrow\{0,1\}^m$ to be given by $q_S(c) = d$ if and only if
$$d_i = \left\{\begin{array}{ll} c_j &\text{if } s_i = j \\ 0 &\text{if } s_i = 0\\ 1 & \text{if }s_i = u \end{array}\right. .$$
Let $\C = \{0,1\}^n$ and $\D = \{0,1\}^m$, and suppose $q_S:\C\rightarrow \D$ is defined by $S$ as above. Then $\phi_{q_S} = \tau_S$, and thus $q_S= q_{\tau_S}$ by Corollary <ref>.
Write $q_S= q$ and $\tau_S = \tau$. Note $\phi_q(y_i) = q^*y_i$, so for any $c\in \C$,
$$q^*y_i(c) = y_i(q(c)) = d_i = \left\{\begin{array}{ll} c_j &\text{if } s_i = j \\ 0 & \text{if }s_i = 0 \\ 1 & \text{if }s_i = u\end{array}\right. .$$
On the other hand, we also have
$$\tau(y_i)(c) = \left\{\begin{array}{ll} x_j(c) &\text{if } \tau(y_i) = x_j \\ 0 & \text{if }\tau(y_i) = 0 \\ 1 & \text{if }\tau(y_i) = 1 \end{array}\right. = \left\{\begin{array}{ll} x_j(c) & \text{if }s_i = j\\ 0 &\text{if } s_i = 0 \\ 1 & \text{if }s_i = u \end{array}\right. = \left\{\begin{array}{ll} c_j &\text{if } s_i = j \\ 0 & \text{if }s_i = 0 \\ 1 &\text{if } s_i = 1 \end{array}\right..$$
Thus $\tau$ and $\phi_q$ are identical on $\{y_i\}$, and hence are identical everywhere.
Here we show how our natural motivating code maps, as well as some other types, are defined by $S$-vectors. Throughout, let $\C$ be the full code on $n$ neurons and $c=(c_1,...,c_n)$ an element of $\C$. $\D$ will be the full code on $m$ neurons, but the relationship of $m$ and $n$ will vary for each example.
* Permuting the labels: Let $\sigma\in \mathcal S_n$ be a permutation. To relabel the code so neuron $i$ is relabeled $\sigma(i)$, we use $S = (\sigma(1),...,\sigma(n))$. Then $q_S(c) = d$, where $d=(c_{\sigma(1)},...,c_\sigma(n))$. We require $q_S(\C) = \D$.
* Adding a codeword: Let $S = (1,2,...,n)$. This defines an inclusion map, so $q(c) = c$. We may use this anytime we have $q(\C)\subseteq \D$. Then all codewords in $\D\setminus q(\C)$ are “added." Again this is exactly the same as our inclusion maps defined under the canonical $\tau$ with $n=m$.
* Deleting the last neuron: Let $S=(1,2,...,n-1)$. Then $q_S(c) = d$, where $d=(c_1,...,c_{n-1})$. We require $q_S(\C) = \D$. In particular, this is the map given by the canonical $\tau$, where $m=n-1$.
* Adding a neuron which repeats neuron $i$ to the end of each word: $S = (1,2,...,n,i)$. Then $q_S(c) = d$, where $d = (c_1,...,c_n, c_i)$. We require $q_S(\C) = \D$.
* Adding a 1 (respectively, 0) to the end of each codeword: $S= (1,2,...,n,u)$ (respectively $S=(1,2,...,n,0)$ ). Then $q_S(c) = d$ where $d=(c_1,...,c_n,1)$ [respectively $d=(c_1,...,c_n,0)$]. This is an extension of our canonical map which added $m-n$ zeros to the end of each codeword; we are now allowed to add a $0$ to the end of each codeword, or a 1 to the end of each codeword.
We now define a class of homomorphisms between neural rings which has this property of preserving neuron indicators. This will, it turns out, be exactly the right notion for capturing the maps that we discussed in the introduction.
§.§ Neural ring homomorphisms & proof of Theorem <ref>
The examples listed above are powerful evidence that linear-monomial homomorphisms capture precisely the maps we hoped to be natural code maps. We therefore define neural ring homomorphisms and neural ring isomorphisms to be the homomorphisms between neural rings which have compatible linear-monomial homomorphisms; this definition formally appears in Section 1.5. In our main result, Theorem <ref> characterizes the code maps which correspond to neural ring homomorphisms, while Corollary <ref> does the same for neural ring isomorphisms. We find that code maps which correspond to neural ring homomorphisms are exactly those which are compositions of permutation, dropping the last neuron, adding a codeword, repeating a neuron, or adding a trivial neuron. All are natural code maps for neural data, and the list includes our original motivating maps. We further find that neural ring isomorphisms correspond exactly to permutation code maps, matching our idea that these are the only code maps that completely preserve combinatorial structure. We now prove these results, with help from the following lemmas.
$\phi:R_\C\rightarrow R_\D$ is a neural ring homomorphism if and only if there is some $S$ such that $q_\phi = q_S\big|_\C$.
Suppose $\C$ is a code on $n$ neurons and $\D$ a code on $m$ neurons.
If $\phi:R_\D\rightarrow R_\C$ is a neural ring homomorphism then there exists some pure $\tau:R[m]\rightarrow R[n]$ compatible with $\phi$. Let $\tau=\tau_S$, and let $q_S:\{0,1\}^n\rightarrow \{0,1\}^m$ be as above. Then $q_{\tau_S} = q_S$ by Lemma <ref> and $q_\phi = q_{\tau_S}\big|_\C$ by Corollary <ref> so $q_\phi = q_S\big|_\C$. On the other hand, if $q_\phi = q_S\big|_\C$ for some $S$, then $\phi$ is compatible with $\tau_S$ by Corollary <ref>, so as $\tau_S$ is pure, $\phi$ is a neural ring homomorphism.
If $q:\C\rightarrow \D$ is a restriction of $q_T$, and $p:\D\rightarrow \E$ is a restriction of $q_S$, then the composition $p\circ q$ is a restriction of $q_W$ for some $W$.
Let $q_T:\C\rightarrow \D$ and $q_S:\D \rightarrow \E$ be linear-monomial. Suppose $q_T(c) = d$ and $q_S(d) = e$. Let $W$ be defined so $w_i = t_{s_i}$ (where $t_0 = 0$ and $t_u = u$) Then,
$$e_i = \left\{\begin{array}{ll} d_j & \text{if }s_i = j\\ 0 &\text{if } s_i = 0\\ 1 & \text{if }s_i = u \\ \end{array}\right.
= \left\{\begin{array}{ll} c_k &\text{or if } s_i = j \text{ and } t_j = k \\ 0 & \text{if }s_i = j \text{ and } t_j = 0\\ &\text{or if } s_i = 0 \\ 1 &\text{if } s_i = j \text{ and } t_j = u\\ & s_i = u\end{array}\right. $$
$$ = \left\{\begin{array}{ll} c_k &\text{if } t_{s_i} = k \\ 0 &\text{if } t_{s_i} = 0 \\ 1 &\text{if } t_{s_i} = u\\\end{array}\right.
= \left\{\begin{array}{ll} c_k &\text{if } w_i = k \\ 0 & \text{if }w_i = 0 \\ 1 &\text{if } w_i = u\end{array}\right.$$
Thus, $q_S(q_T(c)) = q_W(c)$, and so $q_S\circ q_T$ is defined by $W$.
Lemma <ref> gives that $\phi$ is a neural ring homomorphism if and only if $q_\phi = q_S\big|_\C$ for some $S$. So we need only show that $q_\phi = q_S\big|_\C$ for some $S$ if and only if $q_\phi$ is a composition of the listed maps. On one hand, each of the above maps has a related $S$ vector, so by Lemma <ref> their composition also has one, so if $q_\phi$ is a composition of the above maps then there is certainly an $S$ such that $q_\phi=q_S\big|_\C$.
On the other hand, we show that if $q_\phi = q_S\big|_\C$, then we can construct a composition $q$ of the above maps so $q=q_\phi$.
Let $\C$ be a code on $n$ neurons and $\D$ a code on $m$ neurons. Suppose $q_\phi:\C\rightarrow \D$ is given by $q_S\big|_\C$. code map. For $i=1,...,m$, define the function $f_i\in \F_2[x_1,...,x_n]$ such that $f_i = \left\{\begin{array}{ll} x_j & \text{ if } s_i = j\\ 1 & \text{ if }s_i = u \\ 0 & \text{ if }s_i=0\end{array}\right. $
First we define some intermediate codes: let $\C_0=\C$. For $i=1,...,m$, let $\C_i = \{ (c_1,...,c_n,d_1,...,d_i) \mid c\in \C, d=q(c)\}\subset\{0,1\}^{n+i}$. For $j=1,...,n$, let $\C_{m+j} = \{ (d_1,...,d_m,c_1,...,c_{n-j+1}) \mid c\in \C, d=q(c)\}\subset\{0,1\}^{m+n-j+1}$. Finally, define $\C_{m+n+1} = q(\C)\subset \D$.
Now, for $i=1,...,m$, let the code map $q_i:\C_{i-1}\rightarrow \C_i$ be defined by $q_i(v) = (v_1,...,v_{n+i-1}, f_i(v))\in \C_i$. Note that if $v=(c_1,...,c_n, d_1,...,d_{i-1})$, then $f_i(v)= f_i(c)$, as only the first $n$ places matter. Thus, if $v=(c_1,...,c_n,d_1,...,d_{i-1})$ with $d=q(c)$, then $q_i(v) = (c_1,...,c_n, d_1,...,d_i)$. Neuron by neuron, we add the digits of $q(c)$ on to $c$. Note that $q_i = q_{S_i}\big|_{\C_{i-1}}$ where $S_i=(1,...,n+i-1, s_i)$, so $q_i$ is either repeating a neuron, or adding a trivial neuron, depending on whether $s_i = j$, or one of $u,0$.
Next, take the permutation map given by $\sigma = (n+1,...,n+m,1,...,n)$, so all the newly added neurons are at the beginning and all the originals are at the end. That is, define $q_\sigma:\C_m\rightarrow \C_{m+1}$ so if $v=(v_1,...,v_{n+m}),$ then $q_\sigma(v) = (v_{n+1},...,v_{n+m},v_1,...,v_n)$.
We then delete the neurons $m+1$ through $n+m$ one by one in $n$ code maps. That is, for $j=1,...,n$ define $q_{m+j}:\C_{m+j}\rightarrow \C_{m+j+1}$ by $q_{m+j}(v) = (v_1,...,v_{m+n-j})$.
Lastly, if $q(\C)\subsetneq \D$, then add one last inclusion code map $q_a:q(\C)\hookrightarrow \D$ to add the remaining codewords of $\D$.
Thus, given $c=(c_1,...,c_n)$ with $q(c) = d =(d_1,...,d_m)$, the first $m$ steps give us $q_m\circ\cdots\circ q_1(c) = (c_1,...,c_n,d_1,...,d_m) = x$. The permutation then gives us $q_\sigma(x) = (d_1,...,d_m,c_1,...,c_n) = y$, and then we compose $q_{m+n}\circ\cdots\circ q_{m+1}(y) = (d_1,...,d_n) = d = q(c)$. Finally, if $q(\C)\subsetneq \D$, we do our inclusion map, but as $q_a(d) = d$, the overall composition is a map $\C\rightarrow \D$ takes $c$ to $q_S(c)=d$ as desired. At each step, the map we use is from our approved list.
Suppose $\tau = \tau_S:R[n]\rightarrow R[m]$ is an isomorphism. Then since $R[n]$ and $R[m]$ are finite, they must have the same size. So we must have $2^{2^n} = |R[n]| = |R[m]| = 2^{2^m}$, and thus $n=m$. Thus we can write $\tau:R[n]\rightarrow R[n]$.
As $\tau$ is an isomorphism, we know $\ker \tau = \{0\}$. So we cannot have $\tau(x_i) = 0$ since then $x_i \in \ker \tau_S$, and similarly we cannot have $\tau(x_i) = 1$, since then $1-x_i \in \ker\tau$. As $\tau$ is linear-monomial, this means that $\tau(x_i) \in \{x_1,...,x_n\}$ for all $i=1,...,n$; as $\tau=\tau_S$ for some $S$, this means that $s_i\in [n]$ for all $i$.
If we had $\tau(x_i) = \tau(x_j) = x_k$ for $i\neq j$, then $x_i-x_j \in \ker\tau$, which is a contradiction as $x_i-x_j\neq 0$. So $\tau$ induces a bijection on the set of variables $\{x_1,...,x_n\}$; i.e., $S$ contains each index in $[n]$ exactly once.
Now, consider the corresponding code map $q_\tau$. Let $c\in \{0,1\}^n$, and $q_\tau(c) = d$. We must have $\tau(f)(c) = f(d)$. In particular, we must have $x_j(c) = x_i(d)$, or rather, $c_j = d_i$. So $q_\tau$ takes each codeword $c$ to its permutation where $j\rightarrow i$ iff $\tau(x_i) =x_j$.
Now, we know that if $\tau$ is compatible with $\phi$, then $\phi$ is merely a restriction of the map $\tau$, and so $q_\phi(c) = q_\tau(c)$ for all $c\in \C$. Finally, as $\phi$ is an isomorphism we know $q_\phi$ is a bijection, so every codeword in $\D$ is the image of some $c\in \C$; thus, $q_\phi$ is a permutation map on $\C$, and no codewords are added.
§.§ Examples of neural ring homomorphisms
Beyond these basic code maps, there are many interesting code maps we can build as compositions. Here are some particular examples.
[Deleting an arbitrary set of neurons:]
While our basic code map above is only defined to delete the final neuron, we can in fact delete any set of neurons. To do this, we first apply a permutation map so the set of neurons to be deleted appear at the end of the world. We then compose as many deletion maps as necessary to remove these neurons and project onto the other coordinates. This composition corresponds to a neural ring homomorphism as it is the composition of basic maps.
[Inserting a trivial pattern:] By composing permutation maps with maps which add a trivial neuron (1 or 0), we can transform a code to a longer code which has trivial neurons dispersed throughout.
Localization involves restricting to a piece of the code where a particular activity pattern is present, and looking at the restricted code among the other neurons in this segment. As a code map, this is not well-defined, because there is nowhere to send the codewords which do not exhibit the desired activity pattern. We can, however, consider a kind of reverse localization, where we map the localization back into the code whence it came. This, then, is a composition of adding a trivial pattern (see above), and then an inclusion map. However, not just any inclusion will do - we assume the additional restriction that $\D$ may not contain any codewords exhibiting this activity pattern which aren't in $q(\C)$.
§ NEURAL RING HOMOMORPHISM AND THE CANONICAL FORM
In our previous paper, we focused much of our attention on the canonical form for the neural ideal. The canonical form is a distinguished set of minimal generators for the neural ideal which give a condensed yet complete set of combinatorial information about the code. We had previously developed an algorithm for computing the canonical form using the primary decomposition; however, we now present an algorithm which is iterative.
First, we give a few definitions. First note that the neural ideal $I_\C = \{f\in \F_2[x_1,...,x_n] \ | \ f(c) = 0$ for all $c\in \C\}$ can be decomposed into two pieces:
$$I_\C = \B + J_\C$$
where $\B = \langle x_i^2-x_i \,|\, i\in [n]\rangle$ and $J_\C = \langle \prod_{v_i = 1} x_i \prod_{v_i = 0} (1-x_i)\,|\, v\in \{0,1\}^n\backslash \C\rangle$. Polynomials of the form $\prod_{i\in \sigma} x_i \prod_{j\in \tau} (1-x_j)$, where $\sigma\cap \tau = \emptyset$, are referred to as pseudo-monomials, and the ideal $J_\C$, termed the neural ideal, can be generated by a set of minimal pseudo-monomials which we call the canonical form. For more on this canonical form $CF(J_\C)$ see <cit.>.
Consider the code $\C = \{000,100,010, 001, 110, 011\}$. $J_\C $ is generated by $\langle x_1x_2x_3, x_1(1-x_2)x_3\rangle$, but the canonical form is given by $CF(J_\C) = \{x_1x_3\}$, which also generates $J_\C$.
For the most part, it is simple to describe the effects of our basic code maps on the canonical form. Two proofs (repeating a neuron, adding a codeword) are presented in full detail; the others follow a similar structure and the intuition is briefly sketched. Throughout, $\C$ is a code on $n$ neurons with canonical form $CF(J_\C)$, and $z_i \in \{x_i, 1-x_i\}$ represents one of the two possible non constant linear terms involving $x_i$. Because $J_\C$ is an ideal of $\F_2[x_1,...,x_n]$ we revert back to polynomial notation entirely in this section.
* Permutation: as this map simply permutes the labels on the variables, the canonical form stays nearly the same, but the labels are permuted using the reverse permutation $\sigma^{-1}$. That is, let $\D$ be the code obtained by applying the permutation $\sigma\in \mathcal S_n$ to $\C$. Then $f=z_{i_1}\cdots z_{i_k} \in CF(J_\C)$ if and only if $f_\sigma = z_{\sigma^{-1}(i_1)}\cdots z_{\sigma^{-1}(i_k)} \in CF(J_\D)$. That is, all relationships amongst the variables stay the same; only the labels change to reflect the permutation.
* Adding a codeword: This is the most complex operation on the canonical form. The algorithmic process for obtaining $CF(\C\cup\{v\})$ from $CF(\C)$ is described in detail in the following section, and forms the basis for our iterative algorithm. Essentially, we retain those generators which still apply under the new codeword; those which do not are adjusted by multiplication with an appropriate linear term.
* Deleting a neuron: Let $\C$ be a code on $n$ neurons, and $\D$ the code on $n-1$ neurons obtained by deleting the $n$th neuron. Then $CF(J_\D) = CF(J_\C)\setminus\{ f \mid f=g z_n, g $ a pseudo-monomial$\}$. That is, we simply remove all pseudo-monomials which involved the variable $x_n$; however, the relationships amongst the others are unchanged. We never add new relations, because any relation that holds without neuron $n$ held before, and would have appeared.
* Adding a trivial neuron which is always 1 (always 0): Let $\D$ be the code on $n+1$ neurons obtained by adding a 1 (respectively 0) to the end of each codeword in $\C$. Then $CF(J_\D) = CF(J_\C)\cup \{1-x_{n+1}\}$ (respectively $CF(J_\C) \cup \{x_{n+1}\}$). Here, we add a single generator to indicate that this new neuron is uniformly 1 (or 0) and the relationships amongst the others are unchanged.
* Adding a new neuron which repeats another neuron: Let $\D$ be the code on $n+1$ neurons obtained from $\C$ by adding a new neuron which repeats neuron $i$ for all codewords.
Let $F = \{f\in CF(J_\C)\,|\, f=z_i\cdot g$ for $ g\text{ a pseudo monomial} \}$; let $H$ be the set formed by replacing $x_i$ with $x_{n+1}$ for all $f\in F$. Then in most cases, $CF(J_\D) = CF(J_\C) \cup\{x_i(1-x_{n+1}), x_{n+1}(1-x_i)\} \cup H$. The exception is when $z_i\in CF(J_\C)$; then $CF(J_\D)$ is simply $CF(J_\C)\cup H = CF(J_\C) \cup \{z_{n+1}\}$.
In the first case, suppose $z_n \in CF(J_\C)$. Then we know the $n$th bit is either always 1 or always 0, so by repeating it we are really adding a new neuron which is always 1 or always 0. We can then follow the changes described above, which means we merely add the appropriate $z_{n+1}$ term.
In the second case, suppose $z_n\notin CF(J_\C)$. First, note that $CF(J_\C)\subseteq J_\D$, as a polynomial in $CF(J_\C)$ involves the first $n$ variables only, and the first $n$ bits of any codeword in $\D$ are a copy of a word in $\C$. Note also that any word in $\D$ has the same $n+1$st digit as $n$th digit, so if we take a polynomial from $CF(J_\C)$ which has a $z_n$ term and replace it with the corresponding $z_{n+1}$ term it will still be $0$ on all elements of $\D$; thus $H\subseteq J_\D$. Lastly, since each polynomial has the same $n$ and $n+1$ digits, we have $x_n(1-x_{n+1})$ and $x_{n+1}(1-x_n)\in J_\D$. Note also that any polynomial in $J_\D$ which doesn't involve $x_{n+1}$ is also a polynomial in $J_\C$.
Suppose $f$ is a minimal pseudo-monomial in $CF(J_\D)$. Then we have three cases:
* $z_n$ and $z_{n+1}$ both divide $f$. If both linear terms are the same, then removing one of them also must give something in $J_\D$ (because those two digits are always the same) so they're not really minimal. If they're different, then $f$ is a multiple of either $x_n(1-x_{n+1})$ or $x_{n+1}(1-x_n)$ and since these are minimal it must be one of these.
* $z_{n+1}\big| f$, but $z_n$ does not. Then, changing $x_{n+1}$ to $x_n$ gives a pseudo-monomial $g$ which appears in $J_\C$. It must be minimal, or else its divisor would be a divisor of $f$ (after changing $x_{n+1}$ back to $x_n$ if necessary). Thus $g\in CF(J_\C)$ and hence $f\in H$.
* $z_{n+1}\not | f$. Then $f\in J_\C$. But then it must also be minimal in $J_\C$, or else its divisor would also be in $J_\D$ which would contradict the stated minimality. So $f\in CF(J_\C)$.
Together, this shows that $CF(J_\D)\subseteq CF(J_\C)\cup H \cup \{x_n(1-x_{n+1}), x_{n+1}(1-x_n)\}$.
By similar arguments, we get reverse containment. If $f\in CF(J_\C)$, then it is minimal in $J_\D$ as well, as any divisor of it would also be in $J_\C$. If $f\in H$, then it must be minimal, because the polynomial $g$ obtained by switching $x_n$ back in for $x_{n+1}$ was minimal, so $f\in CF(J_\D)$. And both $x_n(1-x_{n+1})$ and $x_{n+1}(1-x_n)$ are minimal, as we don't have $z_n\in J_\C$ so the $n$th bit sometimes takes value 0 and sometimes value 1. Thus we can't have $z_{n+1}$ in $J_\D$ because the $n+1$st bit also varies.
§.§ Iterative algorithm for computing $CF(J_\C)$
We now describe the algorithm for computing $CF(J_\C)$, starting with the canonical form for a single codeword and iteratively applying the map which adds a single codeword until the whole code $\C$ is present.
The code for this algorithm is available online <cit.>.
Input: $\C\subset\{0,1\}^n$, a code consisting of $m$ codewords. Order codewords arbitrarily $c^1,...,c^m$.
Output: The canonical form for $J_\C$.
Description: Let $\C^k$ be the restriction of $\C$ to the first $k$ codewords. $CF(J_{\C^1})$ is simple to describe:
$$CF(J_{\C^1}) = \{x_i-c^1_i \ | \ i\in [n]\}.$$
We create $CF(J_{\C^k})$ from $CF(J_{\C^{k-1}})$ for $2\leq k\leq m$, by the process described in Algorithm 1, described below. After $m-1$ iterations, we obtain $CF(J_{\C^m}) = CF(J_\C)$. If desired, one also could store the successive iterations and output the canonical form for all the nested codes.
Algorithm 1 describes the process to compute $CF(J_{\C^k})$ from $CF(J_{\C^{k-1}})$ and $c^k$. To do this, we note that $\C^k = \C^{k-1}\cup \{c^k\}$; hence we can use Lemma <ref>, which gives a process to find the canonical form for the union of two codes if we know the canonical forms of each respective ideal. We have $CF(J_{\C^{k-1}}) = \{ f_1,...,f_r\}$, and $CF(J_{\{c^k\}}) =\{x_i - c^k_i \ | \ i\in [n] \}$. By the lemma, we obtain $CF(J_{\C^k})$ by taking all possible products $f_i(x_j-c^k_j)$ and reducing the $0$s and the multiples.
However, since one of our canonical forms consists of only linear terms, we can take a few shortcuts along the way, as described in the following algorithm.
§.§ Algorithm 1: Computing $CF(J_{\C^k})$ from $CF(J_{\C^{k-1}})$ and $c^k$:
Input: $CF(J_{\C^{k-1}}) = \{ f_1,...,f_r\}$ and $CF(J_{\{c^k\}}) = \{x_i - c^k_i \ | \ i\in [n] \}$.
Output: $CF(J_{\C^k})$, where $\C^k = \C^{k-1}\cup\{c^k\}$.
Step 1: Let $M = \{f_i \in CF(J_{\C^{k-1}}) \,|\, f_i(c^k) = 0\}$, and $N = CF(J_{\C^{k-1}}) \backslash M$. Initialize a third set, $L$, as an empty set.
Step 2: There are $n|N|$ possible products of polynomials $f_j$ in $N$ and linear terms $x_i-c^k_i$ in $CF(J_{\{c^k\}})$. Order them arbitrarily.
Step 3: Consider these products one by one. For each product $g=f_j(x_i-c^k_i)$, there are 3 possibilities:
* If $x_i-c^k_i-1$ divides $f_j$, discard $g$ and move on to the next product.
* If a) does not hold, but $g$ is a multiple of an element of $M$, discard $g$ and move on to the next product.
* If neither a) nor b) holds, add $g$ to $L$ and move on to the next product.
Repeat until all products have been considered.
Step 4: Output $M\cup L$.
The process described in Algorithm 1 gives the canonical form for $CF(J_\C^k)$ from $CF(J_\C^{k-1})$ and $CF(J_{c^k})$.
§.§ Proof of Algorithm 1
Throughout, $\C$ and $\D$ are neural codes on the same number of neurons; so, $\C,\D \subseteq \{0,1\}^n$. All polynomials are elements of $\F_2[x_1,...,x_n]$. We will use the following useful facts about $J_\C$:
* If $f$ is a pseudo-monomial and $f(c) = 0$ for all $c\in \C$, then $f\in J_\C$.
* If $f \in J_\C$ is a pseudo-monomial, then $f=gh$ for some $h\in CF(J_\C)$ and $g$ a pseudo-monomial.
* If $\C\subseteq \D$, then $J_\C\supseteq J_\D$.
We also make use of the following definitions: a monomial $x^\alpha$ is square-free if $\alpha_i \in \{1,0\}$ for all $i=1,...,n$. A polynomial is square-free if it can be written as the sum of square-free monomials. For example: $x_1x_2+x_4+x_1x_3x_2$ is square-free. There is a unique square free representative of every equivalence class of $\F_2[x_1,...,x_n]/\B$. To see this, recall that $\F_2[x_1,...,x_n]/\B$ is the set of functions from $\{0,1\}^n\rightarrow \{0,1\}$. There are $2^{2^n}$ such functions; thus there are $2^{2^n}$ equivalence classes. Any polynomial representative of one of these classes can be reduced to a square-free polynomial representative by applying the relations $x_i = x_i^2$ from $\B$. As there are $2^n$ square-free monomials in $\F_2[x_1,...,x_n]$, there are exactly $2^{2^n}$ square-free polynomials total; thus each equivalence class must have a unique representative.
For $h\in \F_2[x_1,..,x_n]$, let $h_R$ denote the unique square-free representative of the equivalence class of $h$. This is similar to choosing the equivalence class of $h$ mod $\B$, but we select a very particular representative of that class.
Then, for $CF(J_\C) = \{f_1,...,f_r\}$ and $CF(J_\D) = \{g_1,...,g_s\}$, we define the set of reduced products
$$P(\C,\D) \stackrel{\text{def}}{=} \{(f_ig_j)_R\,|\, i\in [r], j\in [s]\}$$
and the minimal reduced products as
$$MP(\C,\D) \stackrel{\text{def}}{=} \{h\in P(\C,\D)\,|\, h\neq 0\text{ and } h\neq fg\text{ for any } f\in P(\C,\D), g\neq 1\}.$$
If $\C,\D\subset\{0,1\}^n$, then the canonical form of their union is given by the set of reduced products from their canonical forms: $CF(J_{\C\cup \D}) = MP(\C,\D)_R$.
Let $h=(f_ig_j)_R\in MP(\C,\D)$. Then $h$ is a pseudo-monomial, and note that $h\in J_\C$ as it is a multiple of $f_i$, and $h\in J_\D$ as it is a multiple of $g_j$. Thus $h(c) = 0$ for all $c\in \C\cup \D$, so $h\in J_{\C\cup \D}$.
Now, let $h\in CF(J_{\C\cup \D})$. Then as $J_{\C\cup \D}\subset J_{\C}$, there is some $f_i\in CF(J_{\C})$ so that $h=h_1f_i$, and likewise there is some $g_j\in CF(J_{\D})$ so $h=h_2g_j$ where $h_1, h_2$ are also pseudo monomials. Thus $h$ is a multiple of $(f_ig_j)_R$ and hence is a multiple of some element of $MP(\C,\D)$.
But as every element of $MP(\C,\D)$ is an element of $J_{\C\cup \D}$, and $h\in CF(J_{\C\cup \D})$, this means $h$ must actually be in $MP(\C,\D)$.
Thus, $CF(J_{\C\cup\D}) \subseteq MP(\C,\D)$.
Finally, to see that $MP(\C,\D)\subseteq CF(J_{\C\cup\D})$, if $h\in MP(\C,\D)$, then $h$ is in $J_{\C\cup\D}$. It is thus the multiple of some $f\in CF(J_{\C\cup\D})$. But we have shown that $f\in MP(\C,\D)$, which contains no multiples. So $h=f$ is in $CF(J_{\C\cup\D})$.
(Proof of Proposition <ref>) Since we are considering products between the canonical form for a code $\C^{k-1}$ and that for a single codeword $c^k$, we will dispense with the excess indices and use $\C$ and $c$, respectively. Note that if $c\in \C$, then $M=CF(J_\C)$, so the algorithm ends immediately and outputs $CF(J_\C)$; we will generally assume $c\notin \C$. To show that the algorithm produces the correct canonical form, we apply Lemma <ref>, so it suffices to show that the set $M\cup L$ is exactly $MP(\C, \{c\})$. This requires that all products are considered, and that we remove exactly those which are multiples or zeros.
To see that all products are considered: Let $g\in M$. Since $g(c) = 0$, we know $(g(x_i-c_i))_R=g$ for at least one $i$. So $g\in MP(\C,\{c\})$. Any other product $(gz_j)_R$ will either be $g$, or will be a multiple of $g$, and hence will not appear in $MP(\C,\{c\})$. Thus, all products of linear terms with elements of $M$ are considered, and all multiples or zeros are removed. It is impossible for elements of $M$ to be multiples of one another, as $M\subset CF(\C)$.
We also consider all products of elements of $N$ with the linear elements of $CF(J_c)$. We discard them if their reduction would be 0 (case (a)) or if they are a multiple of anything in $M$ (case (b)). If neither holds, we keep them in $L$. So it only remains to show that no element of $L$ can be a multiple of any other element in $L$, and no element of $M$ can be a multiple of anything in $L$, and thus that we have taken care of every possibility.
First, no element of $M$ may be a multiple of an element of $L$, since if $g\in M$, $fz_i\in L$, and $fz_i\cdot p = g$ for some pseudo-monomial $p$, then $f\big| g$. But this is impossible as $f,g$ are both in $CF(J_\C)$.
Now, suppose $fz_i = h\cdot gz_j$ for $f,g\in CF(J_\C)$ and $fz_i, gz_j\in L$, and $h$ a nontrivial pseudo-monomial. Then as $f\not|g$ and $g\not | f$, we have $i\neq j$, and so $z_j\big|f$. But this means $fz_j=f$ and therefore $f\in M$, which is a contradiction. So no elements of $L$ may be multiples of one another.
|
1511.00362
|
Department of mathematics,
The Ohio State University,
Columbus, OH 43210, USA
EHCCBP]Explicit Realization of Hopf Cyclic Cohomology Classes of Bicrossed Product Hopf Algebras
The author would like to give special thanks to Henri Moscovici.
We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$. We give an explicit way to construct Hopf cyclic cohomology classes of our Hopf algebra $\cH$ and then realize these classes in terms of explicit representative cocycles in the cyclic cohomology of the convolution algebra $\cA$.
§ INTRODUCTION
The geometric Hopf algebras and their cyclic cohomology first appeared as a new geometric tool in the work <cit.> of Connes and Moscovici on the local index formula for transversely hypoelliptic operators on foliations. They constructed a canonical isomorphism <cit.> from the Gelfand-Fuks cohomology of the Lie algebra of formal vector fields on $\Rb_n$ (<cit.>) to the Hopf cyclic cohomology of the Hopf algebra $\cH(n)$ associated to the pseudogroup of local diffeomorphisms of $\Rb_n$. In this way they proved that the local index formula computes characteristic classes of foliations (<cit.>). A detailed account of the above isomorphism, based on the bicrossed product decomposition of $\cH(n)$, was provided by Moscovici and Rangipour (<cit.>), see the diagram from <cit.>:
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ C^{\bullet}_{\ttop}(\Fa_n)& C_{\cw}^{\bullet,\bullet} ( \Fg^{\ast}, \cF) & C_{\Bott}^{\bullet} \big( \Omega_{\bullet} ( G ), \Gamma \big) \\
& C^{\bullet}(\cH(n), \Cb_{\delta}) & C^{\bullet}(C_c^{\ify}(G)\rtimes \Gamma ) \\ };
\path[transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-1) edge node[above] {$ \mathcal{E} $} (m-1-2) (m-1-2) edge node[above] {$ \Theta $} (m-1-3)
(m-1-3) edge node[left] {$ \Phi_{}$} (m-2-3) (m-2-2) edge (m-1-2) ;
\path[bend left,->]
(m-1-1) edge node [above] {$ \mathcal{D} $} (m-1-3);
\end{tikzpicture}
\end{equation*}
where $G=\Rb^{n} \rtimes GL(n)$; $\Gamma$ is a subgroup of $\text{Diff}(n)$, diffeomorphisms group of $\Rb^{n}$.
Recently, Moscovici gave an explicit description of a basis of Hopf cyclic characteristic classes of $\cH(n)$ (<cit.>) in the spirit of Chern-Weil theory. He showed that the image of $\mathcal{D}$ forms a subcomplex of the Bott complex $C_{\Bott}^{\bullet} \big( \Omega_{\bullet} (G), \Gamma \big) $ from diagram above, which is quasi-isomorphic to Hopf cyclic via a restriction of Connes' $\Phi$ map. That is
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\small\matrix (m) [matrix of math nodes, row sep=2.2em, column sep=1.8em,
text height=0.8ex, text depth=0.2ex]
{ C^{\bullet}_{\ttop}(\Fa_n)& C_{\cw}^{\bullet,\bullet} ( \Fg^{\ast}, \cF) & Im (\mathcal{D}) &\\
& C^{\bullet}(\cH(n), \Cb_{\delta}) & Im ( \lambda^{} ) &C^{\bullet}(C_c^{\ify}(G)\rtimes \text{Diff}(n)) & C^{\bullet}(C_c^{\ify}(G)\rtimes \Gamma) \\ };
\path[transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-1) edge node[above] {$ \mathcal{E} $} (m-1-2) (m-1-2) edge node[above] {$ \Theta $} (m-1-3)
(m-1-3) edge node[left] {$ \Phi_{d}$} (m-2-3) (m-2-2) edge node[below] {$ \cong $} node [above]{$ \lambda^{}$} (m-2-3) (m-2-4) edge (m-2-5);
\path[ultra thick,bend left,->,transform canvas={yshift=0.5ex},font=\scriptsize]
(m-1-1) edge node [above] {$ \mathcal{D} $} (m-1-3);
\path[right hook->,font=\scriptsize]
(m-2-3) edge (m-2-4);
\end{tikzpicture}
\end{equation*}
This completes the description of relationship between the Lie algebra cohomology of $\Fa_n$, the Hopf cyclic cohomology of the Hopf algebra $\cH(n)$ and the cyclic cohomology of $C_c^{\ify}(G)\rtimes \text{Diff}(n)$ and $C_c^{\ify}( G )\rtimes \Gamma$. Note that the cohomology classes of the Lie algebra of formal vector fields are sophisticated, the calculation above is carried out by $\mathcal{D} $ map using simplicial connection, instead of calculating $ \mathcal{E} $ and $\Theta$ explicitly.
In this paper we are implementing a similar procedure for a bicrossed product Hopf algebra associated to a matched pair of Lie groups. For a matched pair of Lie groups $G_1, G_2$, with corresponding bicrossed product Lie algebra $\Fg_1 \bowtie \Fg_2$, one can also construct a bicrossed product Hopf algebra $\cR(G_2) \acl \cU(\Fg_1) $. Rangipour and Sütlü (<cit.>) showed that the Hopf cyclic cohomology of the Hopf algebra $\cR(G_2) \acl \cU(\Fg_1) $ is canonically isomorphic via a van Est type quasi-isomorphism to the Lie algebra cohomology of $\Fg_1 \bowtie \Fg_2$ relative to a certain Lie subalgebra $\Fh_2$, as in the following diagram (both arrows are quasi-isomorphisms):
\begin{equation} \label{diagram1}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ C(\Fg_1 \bowtie \Fg_2, \Fh_2)& C^{\bullet,\bullet}_{}( \Fg_1^{\ast} , \cR_2) \\
& C^{\bullet}\big(\cR(G_2) \acl \cU(\Fg_1), ^{\sigma^{-1}}\hspace{-3pt} \Cb_{\delta} \big) \\ };
\path[transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-2) edge node[above] {$ \nu $} (m-1-1) ;
\path[transform canvas={xshift=0ex},->,font=\scriptsize]
(m-2-2) edge node[right] {$ \mathcal{J} $} (m-1-2) ;
\end{tikzpicture}
\end{equation}
In their work, a quasi-isomorphism from the complex $C^{\bullet}\big(\cR(G_2) \acl \cU(\Fg_1), ^{\sigma^{-1}}\hspace{-3pt} \Cb_{\delta} \big)$ of $\cR(G_2) \acl \cU(\Fg_1)$ with coefficient in some module comodule $ ^{\sigma^{-1}}\hspace{-3pt} \Cb_{\delta}$ to the Lie algebra cohomology complex of $(\Fg_1 \bowtie \Fg_2,\Fh_2,M)$ is constructed. However, in order to write down the Hopf cyclic classes explicitly, one would need the precise expression of the map from Lie algebra cohomology complex to the cyclic complex of Hopf algebra.
We invert the map $\nu$, following the integration along simplices in $G/K$ map from the Lie algebra complex to the continuous group complex as in <cit.> and <cit.>.
Instead of inverting $\mathcal{J}$, we find a Hopf action (<ref>), with an invariant trace, of it on the convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$ ($\delta$ for the discrete topology), as well as the convolution algebra $\cA_{\Gamma}=C_c^{\ify}(G_1)\rtimes \Gamma$ for any discrete subgroup $\Gamma$ of $ G_2$. Next we use Connes' $\Phi$ map to go directly to the cyclic cohomology of the algebra. In this case the image of $\Phi$ sits inside the range of the characteristic map from the Hopf cyclic cohomology.
Compared with the work of Moscovici's (<cit.>), unlike the Lie algebra of formal vector fields whose cohomology classes are sophisticated, we take the advantage that for finite dimensional Lie algebra one can represent the cohomology classes by invariant forms explicitly. This way we can use $ \mathcal{E} $ and $\Theta$ directly and write down the explicit Hopf cyclic classes of our Hopf algebra $\cH=\big(\cR(G_2) \acl \cU(\Fg_1)\big)^{\cop}$ in terms of representative cocycles on the convolution algebra, as illustrated by the following diagram:
\begin{equation} \label{diagram2}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\small\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.25ex]
{ C(\Fg_1 \bowtie \Fg_2, \Fh_2)& C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}) & D^{p,q} \\
& C^{\bullet}(\cH, ^{\sigma}\Cb_{\delta}) & Im ( \lambda^{} ) &C^{\bullet}(\cA_{}) & C^{\bullet}(\cA_{\Gamma}) \\ };
\path[ultra thick,transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-1) edge node[above] {$ \mathcal{E} $} (m-1-2) (m-1-2) edge node[above] {$ \Theta $} (m-1-3) ;
\path[transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-3) edge node[left] {$ \Phi_{D}$} (m-2-3) (m-2-2) edge node[below] {$ \cong $} node [above]{$ \lambda^{}$} (m-2-3) (m-2-4) edge (m-2-5);
\path[bend left,->]
(m-1-1) edge node [above] {$ \mathcal{D} $} (m-1-3);
\path[right hook->]
(m-2-3) edge (m-2-4);
\end{tikzpicture}
\end{equation}
This amounts to an indirect way of inverting maps in diagram <ref>, and be used to give a explicit description of Hopf cyclic classes of $\cH=\big(\cR(G_2) \acl \cU(\Fg_1)\big)^{\cop}$. Let us known that the Hopf algebra $\big(\cR(G_2) \acl \cU(\Fg_1)\big)^{\cop}$ actually acts on $\cA_{\Gamma}=C_c^{\ify}(G_1)\rtimes \Gamma$ as well for any discrete subgroup $\Gamma \subset G_2$, so we can complete the diagram with the very bottom right corner.
The paper is organized as follows. In <ref> we introduce some background material.
In <ref> we construct the bicrossed product Hopf algebra
$\cR(G_2) \acl \cU(\Fg_1) $. We show that the induced morphism on (Theorem <ref>) Hopf action of $\big(\cR(G_2) \acl \cU(\Fg_1)\big)^{\cop}$ on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$.
We introduce differentiable and representative cohomologies of action groupoid and the complexes that calculate them in <ref>. We also explain the construction of $D^{p,q} $ as a smaller subcomplex of the previous complexes. At the end of <ref> we give an account of Connes' $\Phi$ map in <ref>, whose restriction $\Phi_D$ on $D^{p,q} $ is going to be the quasi-isomorphism we need.
At the beginning of <ref> we review the maps appear in the work of Rangipour and Sütlü (<cit.>) as listed in diagram <ref> under our setting ($^{\sigma^{-1}}\hspace{-3pt}M_{\delta}=^{\sigma^{-1}}\hspace{-3pt}\Cb_{\delta}$). Then we describe the maps $\mathcal{E}$, $\mathcal{D}$, $\Theta$ in the top row of the diagram above and show that all of them are quasi-isomorphisms in the second section. At the end of <ref> we connect the maps from <ref> to <ref> and prove theorem <ref> that gives the same result of Rangipour and Sütlü (<cit.>) but from a different direction with explicit formulas.
Finally we give an example calculation of transition from Lie algebra cohomology classes as explicit invariant forms on a 4 dimensional Lie group to cyclic cochains on the corresponding convolution algebra $\cA$ in <ref>.
§ PRELIMINARIES AND NOTATIONS
In this chapter we provide background material that will be needed. We first recall the definitions of matched pair of Lie groups and Lie algebras, and then the bicrossed product Hopf algebra constructions. Most of the material is taken from <cit.>.
§.§ Matched pair of Lie groups and Lie algebras
First, we would like to define a matched pair of Lie groups and a matched pair of Lie algebras. The definitions are given in Takeuchi's paper <cit.>, Majid's paper <cit.> and book <cit.>:
Two Lie groups $(G_1,G_2)$ are a matched pair if they act on each other and the left action $\triangleright$ of $G_2$ on $G_1$, the right action $\triangleleft$ of $G_1$ on $G_2$, obey the conditions:
\begin{align}
\begin{aligned}
\forall \ \psi_{1}, \psi_{2}\in G_2,\quad \vp_1, \vp_2 &\in G_1,\qquad \psi_1 \triangleright e = e,\quad e \triangleleft \vp_1=e,\\
\psi_1 \triangleright (\vp_1 \vp_2)&=(\psi_1 \triangleright \vp_1)\big((\psi_1 \triangleleft \vp_1)\triangleright \vp_2\big),\\
(\psi_1 \psi_2) \triangleleft \vp_1&=\big(\psi_1 \triangleleft(\psi_2 \triangleright \vp_1)\big)(\psi_2 \triangleleft \vp_1).
\end{aligned}
\end{align}
One example would be a matched pair of Lie subgroups from a decomposition of a Lie group. Let $G=G_1G_2$ as sets, $G_1\cap G_2={e}$, one defines the actions to be:
\begin{align}
\psi \vp = (\psi \triangleright \vp) (\psi \triangleleft \vp), \qquad \text{for}\ \psi \in G_2 \ \text{and} \ \vp \in G_1,
\end{align}
and check that $\triangleright$ and $\triangleleft$ are matched actions.
Differentiate these Lie groups we get a infinitesimal version: matched pair of Lie algebras:
Two Lie algebras $(\Fg_1,\Fg_2)$ are a matched pair if they act on each other and the left action $\triangleright$ of $\Fg_2$ on $\Fg_1$, the right action $\triangleleft$ of $\Fg_1$ on $\Fg_2$, obey the conditions:
\begin{align}\label{lraction}
\begin{aligned}
&[X_1,X_2] \trt Y =X_1 \trt ( X_2 \trt Y ) - X_2 \trt ( X_1 \trt Y ), \\
&X \tlt [Y_1, Y_2] =(X \tlt Y_1) \tlt Y_2 - (X \tlt Y_2) \tlt Y_1 , \\
& X \trt [Y_1, Y_2] = [X \tlt Y_1, Y_2] + [Y_1, X \trt Y_2] +( X \tlt Y_1 )\trt Y_2 - ( X \tlt Y_2 )\trt Y_1, \\
&[X_1,X_2] \tlt Y =[X_1 \tlt Y,X_2] + [X_1,X_2 \tlt Y] + X_1 \tlt (X_2 \trt Y) - X_2 \tlt (X_1 \trt Y).
\end{aligned}
\end{align}
Similar to the group decomposition example, one would get a matched pair of Lie subalgebras from a decomposition of a Lie algebra. Let $\Fg=\Fg_1 \oplus \Fg_2$ as vector spaces, one defines the actions to be:
\begin{align}
[Y,X] = Y \trt X + Y \tlt X, \qquad \text{for}\ Y \in \Fg_2 \ \text{and} \ X \in \Fg_1,
\end{align}
and check that $\triangleright$ and $\triangleleft$ are matched actions.
One can also assemble a match pair of Lie groups or Lie algebras to one Lie group or Lie algebra. We will introduce this when needed (cf. <ref>).
§.§ Bicrossed product Hopf algebras
We refer the complete argument to <cit.>.
We will use Sweedler notation <cit.> to denote a comultiplication by $\Delta (c) = c_{(1)} \otimes c_{(2)}$, omitting the summation.
Let $C$ be a coalgebra. A left $C$-comodule is a pair $(N, \Delta_{N})$ where $N$ is a vector space and $\Delta_{N}) :N \to C \otimes N$ is a linear map, called the coaction of $C$ on $N$, such that the following diagrams commute:
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ N & C \otimes N & \Cb \otimes N & C \otimes N \\
C \otimes N & C \otimes C \otimes N & & N\\ };
\path[transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-1) edge node[above] {$ \Delta_{N} $} (m-1-2) (m-1-4) edge node[above] {$ \varepsilon \otimes Id $} (m-1-3)
(m-2-1) edge node[above] {$\Delta \otimes Id$} (m-2-2) (m-2-4) edge node[below] {$ \cong $} (m-1-3) (m-2-4) edge node[right] {$ \Delta_{N} $} (m-1-4) ;
\path[transform canvas={xshift=0ex},->,font=\scriptsize]
(m-1-1) edge node[right]{$\Delta_{N}$} (m-2-1) (m-1-2) edge node[right]{$Id \otimes \Delta_{N} $} (m-2-2) ;
\end{tikzpicture}
\end{equation*}
We would also use Sweedler notation to denote a left coaction by $\Delta_{N} (c) = c_{<-1>} \otimes c_{<0>}$. Similarly we will denote right coaction of a coalgebra on a comodule $N$ by $\Delta_{N} (c) = c_{<0>} \otimes c_{<1>}$. One shall take advantage of the signs in this notation as $c_{<0>}$ always stay in the same vector space $N$ while $c_{<-1>},c_{<1>}$ are in the coalgebra $C$, $-$,$+$ signs tell the left/right direction of the coaction.
More detailed information on coalgebras, Hopf algebras and their comodules can be found in <cit.>
When a Hopf algebra $\cH$ acts on an algebra $A$ from the left, we say that $A$ is a left $\cH-$module algebra if
\begin{align*}
& h \trt (ab) = (h_{(1)}\trt a)(h_{(2)} \trt b), \\
&h \trt (1_{A}) =\varepsilon(h ) 1_{A}.
\end{align*}
Then we can form a left cross product algebra $A \al \cH $ built on vector space $A \otimes \cH $ with the product:
\begin{align*}
&(a \al h)(b \al g)= a (h_{(1)} \trt b)\otimes h_{(2)} g, \qquad a,b \in A, \quad h,g \in \cH\\
&1_{A \alsub \cH } =1_{A} \otimes 1_{\cH}.
\end{align*}
Let $\cH$ be a Hopf algebra. Suppose a coalgebra $C$ is a right $\cH-$comodule with coaction
\begin{align*}
\Db (c) = c_{<0>} \otimes c_{<1>},
\end{align*}
then $C$ is a right $\cH-$comodule coalgebra if the coaction commutes with the coproduct and counit, i.e., the following conditions are satisfied:
\begin{align*}
&c_{<0>(1)} \otimes c_{<0>(2)} \otimes c_{<1>} = c_{(1)<0>} \otimes c_{(2)<0>} \otimes c_{(1)<1>}c_{(2)<1>} \\
&\varepsilon(c_{<0>}) c_{<1>} =\varepsilon(c ) 1_{\cH}.
\end{align*}
Then we can form a right cross coproduct coalgebra $\cH \cl C$ built on vector space $\cH \otimes C$ with the coalgebra structure:
\begin{align*}
&\Delta(h \cl c)= h_{(1)} \cl c_{(1)<0>} \otimes h_{(2)} c_{(1)<1>} \cl c_{(2)}, \\
&\varepsilon (h \cl c )=\varepsilon(h) \varepsilon (c),
\end{align*}
for $h \in \cH$ and $c \in C$.
Let $A$, $\cH$ be Hopf algebras, let $A$ be a left $\cH-$module algebra and $\cH$ be a right $A-$comodule coalgebra such that the compatibility conditions
\begin{align*}
&\varepsilon(h \trt a)=\varepsilon(h)\varepsilon(a), \quad \Delta(h \trt a )=h_{(1)<0>} \trt a_{(1)} \otimes h_{(1)<1>} (h_{(2)} \trt a_{(2)} ), \\
&1_{<0>} \otimes 1_{<1>} = 1 \otimes 1, \\
& (gh)_{<0>} \otimes (gh)_{<1>}=g_{(1)<0>} h_{<0>} \otimes g_{(1)<1>} (g_{(2)} \trt h_{<1>}), \\
&h_{(2)<0>} \otimes (h_{(1)} \trt a ) h_{(2)<1>} = h_{(1)<0>} \otimes h_{(1)<1>} ( h_{(2)} \trt a),
\end{align*}
are satisfied for all $a,b \in A$ and $g,h \in \cH$. Then the algebra $A \al \cH$ and coalgebra $A \cl \cH$ form a left-right bicrossed product Hopf algebra $A \acl \cH$ with antipode
\begin{align*}
S_{\aclsub} (a \acl h)= \big(1 \acl S(h_{<0>}) \big) \big( S(a h_{<1>})\acl 1 \big)
\end{align*}
§.§ Additional notation
In this paper $G$ will denote a Lie group, $G^{\delta}$ means same group but with discrete topology, $H$ and $L$ are closed subgroups; $\Fg$, $\Fh$, and $\Fl$ will denote the corresponding Lie algebras. $X$,$Y$,$Z$ are vectors of Lie algebras. $\displaystyle \widetilde{X}$,$\displaystyle \widetilde{Y}$,$\displaystyle \widetilde{Z}$ means the corresponding left invariant vector field. $\cH$ is Hopf algebra, $\cA$ is convolution algebra. $\Db$ is for coaction and $\acl$ is for left-right bicrossed product. $\cU$ is universal enveloping algebra, $\cR$ is algebra of representative functions.
§ HOPF ALGEBRA H AND ITS STANDARD ACTION
§.§ Hopf algebra H
In this section, our goal is to associate a bicrossed product Hopf algebra and a convolution algebra on which the Hopf algebra acts to any matched pair of Lie groups. We prove theorem <ref> and show that the characteristic map is faithful.
Given a matched pair of Lie groups $(G_1,G_2)$, we view $G_2$ as discrete group and denote as $G_2^{\delta}$. Consider the action groupoid $\cG = G_1 \rtimes G_2^{\delta} $ and its convolution algebra $C_c^{\ify}(\cG)$, which is equivalent to the cross product algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$ (see <cit.>). The elements of $\cA$ are finite sums of symbols of the form
\begin{align*}\label{}
fU_{\psi}^{\ast},\qquad \text{where} \quad f \in C_{c}^{\infty}(G_1),\quad U_{\psi}^{\ast}\, \text{ stands for } \widetilde{\psi^{-1}} \in G_2,
\end{align*}
with multiplication:
\begin{align}\label{algebramultiplication}
fU_{\psi_1}^{\ast} \Conv gU_{\psi_2}^{\ast}=f (g\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1} ,
\end{align}
where $\widetilde{\psi_1} \in \Diff (G_1)$ is left action by $\psi_1$. We will also use $fU_{\psi}:= fU_{\psi^{-1}}^{\ast}$ to simplify notation afterwards.
Denote the Lie algebra of $G_1$ by $\Fg_1$ and its basis element by $Z_i,1 \le i \le \dim (G_1)$, the corresponding left-invariant vector field on $G_1$ by $\widetilde{Z}_i$. These vector fields act on $\cA$ by
\begin{align}
\widetilde{Z}_i (fU_{\psi}^{\ast} )=\widetilde{Z}_i (f)U_{\psi}^{\ast}, \qquad \widetilde{Z}_i (f)U_{\psi}^{\ast}\vert_{\vp_0} = \dt f \big(\vp_0 \exp(tZ_i) \big) U_{\psi}^{\ast}.
\end{align}
Now we calculate $L_{\psi^\ast} \widetilde{Z}_i$ for $ \psi \in G_2$:
\begin{align}
\begin{aligned}
\widetilde{Z}_i(f\circ \widetilde{\psi})\vert_{\vp_0} &=\dt (f\circ \widetilde{\psi}) \big(\vp_0 \exp(tZ_i) \big)\\
&=\dt f \Big( \psi \trt \big(\vp_0 \exp(tZ_i) \big) \Big)\\
&=\dt f \Big(( \psi \trt \vp_0)\big((\psi \tlt \vp_0)\trt \exp(tZ_i)\big) \Big)\\
&=\dt f \Big(( \psi \trt \vp_0)\big((\psi^{-1}\tlt (\psi \trt \vp_0))^{-1}\trt \exp(tZ_i) \big) \Big)\\
&=\sum_{j} \big(\Gamma_{i}^j(\psi^{-1})\widetilde{Z}_j(f) \big)\circ \widetilde{\psi} |_{\vp_0},
\end{aligned}
\end{align}
or in short
\begin{align}
\label{zpsi}
\widetilde{Z}_i U_{\psi}^{\ast} &= \sum_{j} U_{\psi}^{\ast} \Gamma_{i}^j(\psi^{-1})\widetilde{Z}_j ,
\end{align}
\begin{align}
\label{gamma}
\sum_{j} \Gamma_{i}^j(\psi)(\vp_0)Z_j:=(\psi^{}\tlt \vp_0)^{-1} \trt Z_i , \quad \text{or}\quad
\Gamma_{i}^j(\psi)(\vp_0):=\big\lag(\psi\tlt \vp_0)^{-1} \trt Z_i,\omega_j\big\rag ,
\end{align}
and $\{\omega_j,1 \le j \le \dim (G_1)\}$ is the dual basis and the action $\trt$ of $G_2$ on $\Fg_1$ is given by the differentiation of $\trt$ on $G_1$ at $e$.
We notice that $\Gamma$ satisfies:
\begin{align}
\label{1-gamma}
\begin{aligned}
&\Gamma_i^j (\psi_1\psi_2)= \sum_{k} (\Gamma_{i}^{k} (\psi_1) \circ \widetilde{\psi_2} )\Gamma_{k}^{j} (\psi_2) , \\
&\Gamma_i^j (\psi^{-1}) \circ \widetilde{\psi} =(\Gamma^{-1} )_{i}^{j} (\psi).
\end{aligned}
\end{align}
From <ref> the definition of $\Gamma_{i}^{j} $, we can see that $\Gamma_{i}^{j} $ is determined only by its value at $e$:
\begin{align}
\label{ytltx}
\Gamma_{i}^{j} (\psi)(\vp_0)=\Gamma_{i}^{j} (\psi \tlt \vp_0)(e),
\end{align}
we will denote
\begin{align}
\gamma_i^j(\psi):=\Gamma _{i}^{j} (\psi)(e) ,
\qquad \text{or}\qquad \label{coact}
\gamma_{i}^j(\psi):=\lag \psi^{-1} \trt Z_i,\omega_j \rag.
\end{align}
Hence similarly we can show $\gamma$ is a homomorphism:
\begin{align}
\label{one-cocycle condition}
\begin{aligned}
&\gamma_i^j(\psi_1 \psi_2)= \gamma_i^k(\psi_1) \gamma_k^j(\psi_2) , \\
&\gamma_i^j (\psi^{-1}) =(\gamma^{-1} )_{i}^{j} (\psi)
\end{aligned}
\end{align}
This property of $\gamma_i^j$ suggests that $\gamma_i^j$ falls into a special kind of functions on $G_2$ called representative functions. We will give the definition originated from <cit.>.
Fix the base field to be $\mathbb{C}$, let $G$ be a Lie group and $\rho : G \to Aut(V)$ a finite dimensional smooth representation. Topologize $End(V)$ so that every linear functional is continuous and topologize $Aut(V)$ by the induced topology. Then, the composition of $\rho$ with a linear functional $\tau \in Aut(V)^{\ast}$ is called a representative function of $G$. Denote $\cR(G)$ the algebra of representative functions when we run over all pairs of finite dimensional representation and linear functional.
It is well known that the representative functions $\cR(G)$ form a commutative Hopf algebra:
\begin{align}
\begin{aligned}
&F_1F_2(\psi)=F_1(\psi) F_2(\psi), \quad \Delta(F)(\psi_1,\psi_2)=F(\psi_1 \psi_2) ,\\
&1(\psi)=1, \quad \varepsilon (F)= F(e), \quad S (F)(\psi)= F(\psi^{-1}).
\end{aligned}
\end{align}
From <ref> we can uniquely express
\begin{align}
U_{\psi}^{\ast} \widetilde{Z}_i U_{\psi}^{} = \Gamma_{i}^j(\psi)\widetilde{Z}_j ,\qquad \text{or } \qquad\widetilde{(\psi^{-1})_{\ast} } (\widetilde{Z}_i) = \Gamma_{i}^j(\psi)\widetilde{Z}_j.
\end{align}
This gives a left action of $G_2$ on $\Fg_1$:
\begin{align} \label{laction}
\psi \trt ({Z}_i)=\widetilde{(\psi^{})_{\ast} } (\widetilde{Z}_i)\vert _{e} =\Gamma_{i}^j(\psi^{-1})(e) {Z}_j =\gamma_{i}^j(\psi^{-1}){Z}_j =S(\gamma_i^j )(\psi) {Z}_j,
\end{align}
which can be dualized using the identification
\begin{align}
({Z}_j)_{<0>} ({Z}_j)_{<1>} (\psi):=\psi \trt ({Z}_i)
\end{align}
to a right coaction of $\cR(G_2)$ on $\Fg_1$:
\begin{align}
\label{rightcoaction}
\blacktriangledown (Z_i)= (Z_i )_{<0 >} \otimes (Z_i) _{<1 >} := Z_j \otimes S(\gamma_i^j ).
\end{align}
The transpose of the left action <ref> gives a right action of $G_2$ on $\Fg_1^{\ast}$:
\begin{align}\label{raction}
(\omega_i) \trt^t \psi=S(\gamma_i^j )(\psi) {\omega}_j,
\end{align}
dualize it we can have a left coaction of $\cR(G_2)$ on $\Fg_1^{\ast}$:
\begin{align}
\label{leftcoaction2}
\blacktriangledown (\omega_i):= S(\gamma_i^j ) \otimes \omega_j .
\end{align}
We also have a left action of $\Fg_1$ on $\cR(G_2)$ from differentiating the left action of $G_1$ on $C^{\ify}(G_2)$ (which comes from the right action of $G_1$ on $G_2$):
\begin{align}
\label{leftaction}
(Z_i \triangleright F )(\psi):=\dt F \big(\psi \triangleleft \exp (t Z_i) \big) ,
\end{align}
For any $Z_i \in \Fg_1$ and any $F \in \cR(G_2)$, we have $(Z_i \triangleright F ) \in \cR(G_2)$.
Both coaction and action can be extended from $\Fg_1$ to $\cU(\Fg_1)$ and we have the following result.
<cit.>by the left action <ref> and the right coaction <ref>, the pair $(\cU(\Fg_1),\cR(G_2))$ is a matched pair of Hopf algebras. i.e., $\cR(G_2)$ is a left $\cU(\Fg_1)-$module algebra, $\cU(\Fg_1)$ is a right $\cR(G_2)-$comodule coalgebra, and they satisfy the compatibility conditions
\begin{align*}
&\varepsilon(u \triangleright F ) =\varepsilon (u) \varepsilon (F), \qquad \blacktriangledown (1)=1 \otimes 1,\\
&\Delta(u \trt F) = u_{(1)_{< 0 > }} \triangleright F_{(1)} \otimes u_{(1)_{< 1 > }} (u_{(2)} \triangleright F_{(2)} ), \\
&\blacktriangledown (uv)= u_{(1)_{< 0 > }} v_{<0>} \otimes u_{(1)_{< 1 > }} (u_{(2)} \triangleright v_{<1>} ) ,\\
& u_{(2)_{< 0 > }} \otimes ( u_{(1)} \triangleright F ) u_{(2)_{< 1 > }} =u_{(1)_{< 0 > }} \otimes u_{(1)_{< 1 > }} ( u_{(2)} \trt F ).
\end{align*}
As a result of this proposition, for a matched pair of Hopf algebras $(\cU(\Fg_1),\cR(G_2))$ we can form the left-right bicrossed product Hopf algebra $\cR(G_2) \acl \cU(\Fg_1)$ in a canonical way.
The algebra structure is given by $\cR(G_2) \al \cU(\Fg_1)$:
\begin{align}\label{product1}
(F \acl u)(G \acl v)=(F (u_{(1)}\trt G) \acl u_{(2)}v)
\end{align}
with unit $(1 \acl 1)$ and the coalgebra structure is given by $\cR(G_2) \cl \cU(\Fg_1)$:
\begin{align}
\Delta_{\aclsub} (F \acl u)=(F_{(1)} \acl u_{(1)_{<0>}}) \otimes (F_{(2)} u_{(1)_{<1>}} \acl u_{(2)})
\end{align}
with counit $\varepsilon (F \acl u)=\varepsilon (F) \varepsilon (u)$, and finally the antipode is given by
\begin{align}
S_{\aclsub} (F \acl u)= \big(1 \acl S(u_{<0>}) \big) \big(S(F u_{<1>})\acl 1 \big)
\end{align}
Fix an order of the basis of $\Fg_1$ and Let $I=(i_1,\dots, i_p)$, $p= \dim{G_1}$ be multi-indeces, ordered lexicographically. We then have $Z_{I}=Z_{1}^{i_1}\cdots Z_{p}^{i_p}$ as a PBW basis of $\cU(\Fg_1)$.
Every element $\sum_{j}F_{j} \acl u_{j}$ in $\cR(G_2) \acl \cU(\Fg_1)$ can be written uniquely as
\begin{align}
\begin{aligned}\label{basis2}
\sum_{j}F_{j} \acl u_{j}=\sum_{I} F_{I} \acl Z_{I},\quad I=(i_1,\dots, i_p).
\end{aligned}
\end{align}
Therefore, $\{F\acl Z_{i}, F \in \cR(G_2) , 1 \le i \le \dim{G_1}\}$ is a generating set of $\cR(G_2) \acl \cU(\Fg_1)$.
For the specific Hopf algebra $\cR(G_2) \acl \cU(\Fg_1)$ we can show that $S_{\aclsub}$ is invertible and $S_{\aclsub}^{-1}$ is given by the formula
\begin{align}
S_{\aclsub}^{-1} (F \acl Z)=\big( S( Z_{<1>}) \acl S(Z_{<0>}) \big) \big( S(F )\acl 1 \big)
\end{align}
on the generators and is extended as an anti-algebra morphism.
The non-trivial check will be on $1 \acl Z_{i}$:
First we observe that $S(Z)=-Z$ for $Z \in \Fg_1$ and
\begin{align}
(-Z)_{<0 >} \otimes (-Z )_{<1 >} = \blacktriangledown (-Z)= -\blacktriangledown (Z)= -(Z )_{<0 >} \otimes (Z) _{<1 >}, \quad Z \in \Fg_1
\end{align}
\begin{align}
\big( S(Z) \big)_{<0>}\otimes \big( S(Z) \big)_{<1>}=S \big( (Z)_{<0>} \big) \otimes (Z)_{<1>}, \quad Z \in \Fg_1
\end{align}
note that the above equality is only true for $Z \in \Fg_1$ not $u \in \cU(\Fg_1)$. The above equality allow us to interchange coaction and antipode on generators.
Now we have
\begin{align}
\begin{aligned}
S_{\aclsub}^{-1} S_{\aclsub}^{} (1 \acl Z)=&S_{\aclsub}^{-1}\Big( (1 \acl S(Z _{<0>}))(S( Z _{<1>})\acl 1)\Big)\\
=&S_{\aclsub}^{-1}\Big(S( Z _{<1>})\acl 1\Big) S_{\aclsub}^{-1}\Big( 1 \acl S(Z _{<0>})\Big)\\
=&\Big( Z _{<1>}\acl 1 \Big) \Big(S( (S(Z _{<0>}))_{<1>}) \acl S((S(Z _{<0>}))_{<0>})\Big) \\
=&\Big( Z _{<1>}\acl 1 \Big) \Big(S( (Z _{<0>})_{<1>}) \acl Z _{<0><0>}\Big) \\
=&\Big(Z _{<1>(2)} S(Z _{<1>(1)}) \acl Z _{<0>} \Big)\\
=&\varepsilon (Z_{<1>}) \acl Z _{<0>}\\
=&1 \acl Z
\end{aligned}\\
\begin{aligned}
S_{\aclsub}^{} S_{\aclsub}^{-1} (1 \acl Z)=&S_{\aclsub}^{}\big(S( Z _{<1>})\acl S(Z _{<0>})\big)\\
=& \Big(1 \acl S( (S(Z _{<0>}))_{<0>}) \Big)\Big( S((S(Z _{<0>}))_{<1>}) Z _{<1>}\acl 1 \Big)\\
=& \Big(1 \acl Z _{<0><0>} \Big)\Big( S((Z _{<0>})_{<1>}) Z _{<1>}\acl 1 \Big)\\
=& \Big(1 \acl Z _{<0>} \Big)\Big( S (Z _{<1>(1)})Z _{<1>(2)} \acl 1 \Big)\\
=& \Big(1 \acl Z _{<0>} \Big)\Big( \varepsilon (Z_{<1>}) \acl 1 \Big)\\
=&1 \acl Z
\end{aligned}
\end{align}
Opposite coalgebra. For any coalgebra $(C,\Delta,\varepsilon)$ we set
\begin{align*}
\Delta^{\rm{op}}=\tau_{C,C} \circ \Delta
\end{align*}
where $\tau_{C,C}$ is the twist map.
Then $(C,\Delta^{\rm{op}},\varepsilon)$ is a coalgebra which we call the opposite coalgebra and denote by $C^{\cop}$.
The opposite coalgebra $\cH=\big(\cR(G_2) \acl \cU(\Fg_1)\big)^{\cop}$ is a Hopf algebra with coproduct $\Delta=\Delta_{\aclsub}^{\rm{op}}$ and antipode $S=S_{\aclsub}^{-1}$.
It was proved in <cit.> that if the antipode of a Hopf algebra has an inverse, the opposite coalgebra is also a Hopf algebra with opposite coproduct and inverse antipode.
Let two Lie groups $(G_1,G_2)$ be a matched pair, define $\cH$ to be the opposite coalgebra $\big(\cR(G_2) \acl \cU(\Fg_1)\big)^{\cop}$, with product given by the same of $\big(\cR(G_2) \acl \cU(\Fg_1)\big)$ (<ref>), coproduct given by $\Delta_{\aclsub}^{\rm{op}}$ and antipode $S_{\aclsub}^{-1}$.
From now on, we will focus on the opposite coalgebra $\cH$, which we will use to construct a Hopf action later.
§.§ Standard action
Next we would like to construct a Hopf action of $\cH$ on $\cA$:
Again take a PBW basis of $\cU(\Fg_1)$, $\{Z_{I}=Z_{1}^{i_1}\cdots Z_{p}^{i_p},I=(i_1,\dots, i_p)\}$.
Every $Z_{I} \in \cU(\Fg_1)$ acts on $\cA$ by
\begin{align}
&\widetilde{Z_{I}}(fU_{\psi}^{\ast})=\widetilde{Z_{1}}^{i_1}\cdots \widetilde{Z_{p}}^{i_p}(f)U_{\psi}^{\ast}, \\
&\widetilde{Z_{I}} (f)U_{\psi}^{\ast}\vert_{\vp_0} = \underbrace{\left.\frac{d}{dt_1}\right|_{_{t_1=0}}}_{\text{$i_1$ times}} \cdots \underbrace{\left.\frac{d}{dt_p}\right|_{_{t_p=0}}}_{\text{$i_p$ times}} f \big(\vp_0 \underbrace{\exp(t_1Z_{1})\dots}_{\text{$i_1$ times}} \cdots \underbrace{\exp(t_pZ_{p})\dots}_{\text{$i_p$ times}} \big) U_{\psi}^{\ast},
\end{align}
on each $fU_{\psi}^{\ast}$ and extend to the finite sums.
$F \in \cR(G_2) $ act on $\cA$ by
\begin{align}
\widetilde{F}(fU_{\psi}^{\ast})(\vp)=F(\psi \tlt \vp)f(\vp)U_{\psi}^{\ast},
\end{align}
on each $fU_{\psi}^{\ast}$ and extend to the finite sums.
Therefore we would like $F \acl Z_{I} $ to act on $\cA$ by
\begin{align}
\widetilde{F \acl Z_{I} }(fU_{\psi}^{\ast})(\vp)=F(\psi \tlt \vp) \widetilde{Z_{1}}^{i_1}\cdots \widetilde{Z_{p}}^{i_p}(f)(\vp)U_{\psi}^{\ast},
\end{align}
or equivalently,
\begin{align}
\widetilde{F \acl Z_{I} }(fU_{\psi}^{\ast})(\vp)=\vp \trt F(\psi^{} )\widetilde{Z_{1}}^{i_1}\cdots \widetilde{Z_{p}}^{i_p}(f)(\vp)U_{\psi}^{\ast}.
\end{align}
The formula
\begin{align}
\widetilde{F \acl Z_{I} }(fU_{\psi}^{\ast})(\vp)=F(\psi \tlt \vp) \widetilde{Z_{1}}^{i_1}\cdots \widetilde{Z_{p}}^{i_p}(f)(\vp)U_{\psi}^{\ast}
\end{align}
defines a Hopf action.
We need to check that the action is well-defined, i.e., commutes with product and coproduct.
First, we want to show it commutes with product. Because of the formula above, it is trivial that
\begin{align}
\begin{aligned}
&(\widetilde{F \acl 1 } )(\widetilde{1 \acl Z_{I} } )=\widetilde{F \acl Z_{I} }, \\
&(\widetilde{F \acl 1 } )(\widetilde{G \acl 1})=\widetilde{(F G) \acl 1 } ,\\
&(\widetilde{1 \acl Z_{I} } )(\widetilde{1 \acl Z_{J} } )=\widetilde{1 \acl Z_{I}Z_{J} }.
\end{aligned}
\end{align}
Therefore we just need to show that
\begin{align}\label{zif}
\begin{aligned}
\widetilde{1 \acl Z_{I} } \widetilde{F \acl 1} =\widetilde{(1 \acl Z_{I} )(F \acl 1)}.
\end{aligned}
\end{align}
This is true because
\begin{align}
\begin{aligned}
&\widetilde{Z_I} \widetilde{F} (fU_{\psi}^{\ast} )(\vp)\\
=&\widetilde{Z_I} (F(\psi \tlt \vp)f(\vp)U_{\psi}^{\ast}) \\
=&F(\psi \tlt \vp)\big(\widetilde{(Z_{I})_{(1)}} f)(\vp)U_{\psi}^{\ast} + \dt F(\psi \tlt (\vp \exp t (Z_{I})_{(2)}))f(\vp)U_{\psi}^{\ast} \\
=&F(\psi \tlt \vp)\big(\widetilde{(Z_{I})_{(1)}} f)(\vp)U_{\psi}^{\ast} + \dt F((\psi \tlt \vp )(\exp t (Z_{I})_{(2)}))f(\vp)U_{\psi}^{\ast} \\
=&F(\psi \tlt \vp)\big(\widetilde{(Z_{I})_{(1)}} f)(\vp)U_{\psi}^{\ast} + ((Z_{I})_{(2)}\trt F)(\psi \tlt \vp)f(\vp)U_{\psi}^{\ast} \\
=&\widetilde{F} \widetilde{(Z_{I})_{(1)}} (fU_{\psi}^{\ast} )(\vp) + \widetilde{(Z_{I})_{(2)}\trt F} (fU_{\psi}^{\ast} )(\vp).
\end{aligned}
\end{align}
Next, we verify on generators that the action commutes with coproduct,
\begin{align}
\begin{aligned}
&\widetilde{F}(fU_{\psi_1}^{\ast} \Conv gU_{\psi_2}^{\ast})(\vp)\\
=&\widetilde{F}(f (g\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1})(\vp)\\
=&F(( \psi_2 \psi_1 )\tlt \vp)f (\vp) (g\circ \widetilde{\psi_1} (\vp))U^\ast_{\psi_2 \psi_1}\\
=&F((( \psi_2 \tlt (\psi_1 \trt \vp)\psi_1\tlt \vp)f (\vp) (g\circ \widetilde{\psi_1} (\vp))U^\ast_{\psi_2 \psi_1}\\
=&F_{(1)}( \psi_2 \tlt (\psi_1 \trt \vp)) F_{(2)}(\psi_1\tlt \vp)f (\vp) (g\circ \widetilde{\psi_1} (\vp))U^\ast_{\psi_2 \psi_1}\\
=&(F_{(1)}( \psi_2 \tlt \vp) \circ \widetilde{\psi_1} )F_{(2)}(\psi_1\tlt \vp)f (\vp) (g\circ \widetilde{\psi_1} (\vp))U^\ast_{\psi_2 \psi_1}\\
=&\widetilde{F}_{(2)}(fU_{\psi_1}^{\ast}) \Conv \widetilde{F}_{(1)}(gU_{\psi_2}^{\ast})(\vp)\\
=&\widetilde{\Delta_{\aclsub}^{\rm{op}}(F \acl 1)} (fU_{\psi_1}^{\ast} \otimes gU_{\psi_2}^{\ast}) (\vp),
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
&\widetilde{Z_i}(fU^\ast_{\psi_1} \Conv gU^\ast_{\psi_2})\\
=&\widetilde{Z_i}(f (g\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1}) \\
=&\widetilde{Z_i}(f) (g\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1}+f\widetilde{Z_i} (g\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1} \\
=&\widetilde{Z_i}(f) (g\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1}+f\Gamma_{i}^j(\psi_1)(\widetilde{Z_j}(g)\circ \widetilde{\psi_1}) U^\ast_{\psi_2 \psi_1} \\
=&\widetilde{Z_i}(fU^\ast_{\psi_1} )\Conv gU^\ast_{\psi_2} +(\Gamma_{i}^j(\psi_1) f)U^\ast_{\psi_1} \Conv \widetilde{Z_j}(gU^\ast_{\psi_2})\\
=&\widetilde{Z_i}(fU^\ast_{\psi_1} )\Conv gU^\ast_{\psi_2} +\widetilde{\gamma_{i}^j}(fU^\ast_{\psi_1} )\Conv \widetilde{Z_j}(gU^\ast_{\psi_2})\\
=&\widetilde{\Delta_{\aclsub}^{\rm{op}}(1 \acl Z_i )} (fU_{\psi_1}^{\ast} \otimes gU_{\psi_2}^{\ast}) (\vp).
\end{aligned}
\end{align}
The formula also defines a Hopf action on $\cA_{\Gamma}=C_c^{\ify}(G_1)\rtimes \Gamma$ as well for any discrete subgroup $\Gamma < G_2^{\delta}$.
In order to do Hopf cyclic theory, we want to show that this Hopf action is equipped a $\delta$-invariant $\sigma$-trace.
We introduce a functional on the discrete crossed product $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$:
\begin{align}\label{tracetau}
\begin{aligned}
\tau (fU^\ast_{\psi}) = \begin{cases} 0,\quad \text{if}\quad \psi \ne 1,\\
\displaystyle \int_{G_1} f \varpi.
\end{cases}
\end{aligned}
\end{align}
Here $\varpi$ is the volume form attached to the dual basis $\widetilde{\omega}_i$ of $\widetilde{Z}_i$ on $G_1$
\begin{align*}
\varpi =\bigwedge \widetilde{\omega}_i ,
\end{align*}
and is in fact not always invariant under $G_2$ action, i.e.,
\begin{equation}\label{volumeform1}
\begin{aligned}
&\widetilde{\psi}_{\ast}(\widetilde{Z_i})=U_{\psi}\widetilde{Z_i}U_{\psi}^{\ast }=\Gamma_i^j(\psi^{-1})\widetilde{Z_j} , \\
&\widetilde{\psi}^{\ast}(\widetilde{\omega_i})=(\Gamma_i^j(\psi^{-1}) \circ \widetilde{\psi^{}} ) \widetilde{\omega}_j ,\\
&\widetilde{\psi}^{\ast}(\varpi)=\bigwedge (\Gamma_i^j(\psi^{-1}) \circ \widetilde{\psi^{}} ) \widetilde{\omega}_j=(\det(\Gamma(\psi^{-1})) \circ \widetilde{\psi^{}} )\varpi .
\end{aligned}
\end{equation}
Evaluating at identity gives
\begin{equation}
\begin{aligned}
\widetilde{\psi}^{\ast}(\bigwedge {\omega}_i)=\det(\gamma)(\psi^{-1}) \bigwedge {\omega}_i .
\end{aligned}
\end{equation}
We see a representative function $\sigma:=\det(\gamma)$ here. It is easy to see that $\sigma$ is a group-like element in $\cR(G_2)$, i.e., $\sigma(\psi_1\psi_2)=\sigma(\psi_1)\sigma(\psi_2)$.
we also have the module function on $\Fg_1$, which is the trace of the adjoint representation $\Fg_1$ on itself,
\begin{align*}
\delta(Z)=\textrm{Tr} ( ad_Z),
\end{align*}
and we can extend it to a character on $\cU(\Fg_1)$ and then $\cH$ by
\begin{align*}
\delta(F \acl Z_I)= \varepsilon (F) \acl \delta(Z_I),
\end{align*}
so we have a group-like element $\sigma \acl 1$ in $\cH$ and a character $\delta$ on $\cH$.
Let us recall the definition of modular pairs in involution as follows:
Let $\cH$ be a Hopf algebra, $\delta : \cH \to \Cb $ be a character and $\sigma \in \cH$ be a group-like element. The pair $(\delta, \sigma)$ is called a modular pair in involution (MPI) if
\begin{align} \label{mpi_condition}
\delta(\sigma)=1,\quad S_{\delta}^{2}(h)=\sigma h \sigma ^{-1},
\end{align}
for all $h \in \cH$, where $S_{\delta}$ is the convolution $\delta \Conv S$:
\begin{align*}
\end{align*}
A modular pair in involution is the easiest coefficient that we can put for Hopf cyclic cohomology. The generalized coefficients are called stable anti-Yetter-Drinfeld (SAYD) module over $\cH$.
Assume $\cH$ is a Hopf algebra with invertible antipode, we can define Yetter-Drinfeld module over $\cH$ as in <cit.>. It is shown in <cit.> that if we replace $S$ with $S^{-1}$, $S^{-1}$ with $S$ in the definition of Yetter-Drinfeld module we can have compatible action and coaction. The new definition gives the so called anti-Yetter-Drinfeld module over $\cH$.
Let $\cH$ be a Hopf algebra with invertible antipode. A right $\cH-$module, left $\cH-$comodule $M$ is called a right-left anti-Yetter-Drinfeld module over $\cH$ if
\begin{align} \label{sayd1}
\Db ( m \tlt h ) =S(h_{(3)}) m_{<-1>} h_{(1)} \otimes m_{<0>} \tlt h_{(2)},\qquad \forall\, m\in M, \, h \in \cH,
\end{align}
moreover, $M$ is called stable if
\begin{align}
m_{<0>} \tlt m_{<-1>} = m.
\end{align}
Let the ground field $\bC$ be a right module over $\cH$ via a character $\delta$ and a left comodule over H via a group-like $\sigma$. Then $^\sigma \bC _ \delta$ is a stable right-left anti-Yetter-Drinfeld module if and only if $(\delta,\sigma)$ is a modular pair in involution.
It's a Lemma in <cit.>. However we would like to give an illustration here:
The second (stability) condition is obvious, so we will just show the first one. On one side, if
\begin{align*}
S_{\delta}^{2}(h)=\sigma h \sigma ^{-1},
\end{align*}
we calculate
\begin{align}
\begin{aligned}
S(h_{(3)})1_{<-1>}h_{(1)} \otimes 1_{<0>}\cdot h_{(2)}&=S(h_{(3)})\sigma h_{(1)} \otimes \delta( h_{(2)})\\
&=S_{\delta}(h_{(2)})\sigma h_{(1)} \sigma^{-1} \sigma \otimes 1\\
&=S_{\delta}(h_{(2)}) S_{\delta}^2( h_{(1)} ) \sigma \otimes 1\\
&=S_{\delta}( S_{\delta} (h_{(1)})h_{(2)}) \sigma \otimes 1\\
&=S_{\delta}( \delta (h_{(1)}) S(h_{(2)}) h_{(3)}) \sigma \otimes 1\\
&=S_{\delta}( \delta (h)\eta) \sigma \otimes 1\\
&= \sigma \otimes \delta(h)\\
&=\blacktriangledown(1\cdot h).
\end{aligned}
\end{align}
On the other side, if
\begin{align*}
\sigma \otimes \delta(h)=S(h_{(3)})\sigma h_{(1)} \otimes \delta( h_{(2)}),
\end{align*}
\begin{align*}
\delta(h)=\sigma^{-1}S_{\delta}(h_{(2)})\sigma h_{(1)},
\end{align*}
we calculate
\begin{align}
\begin{aligned}
h&=\delta(h_{(1)} S(h_{(2)}))h_{(3)}\\
&=\delta(h_{(1)})\sigma^{-1}S_{\delta}((S(h_{(2)}))_{(2)})\sigma (S(h_{(2)}))_{(1)} h_{(3)} \\
&=\delta(h_{(1)})\sigma^{-1}S_{\delta}(S(h_{(2)(1)}))\sigma (S(h_{(2)(2)}))h_{(3)} \\
&=\sigma^{-1}S_{\delta}( \delta(h_{(1)}) S(h_{(2)}))\sigma \varepsilon( h_{(3)}) \eta \\
&=\sigma^{-1}S_{\delta}^2 (h)\sigma .
\end{aligned}
\end{align}
We also have
If $^\sigma \bC _ \delta$ is a right-left stable anti-Yetter-Drinfeld module over $\cH$, then $^{S^{-1}(\sigma)} \bC _ \delta$ is a right-left stable anti-Yetter-Drinfeld module over $\cH^{\cop}$.
straightforward calculation. Apply $S^{-1}$ to equation <ref>.
Now we are ready to show that the pair we constructed is a modular pair in involution.
The pair $(\delta,\sigma^{})$ is a modular pair in involution for the Hopf algebra $\cH$.
<cit.> shows that $(\delta,S\big(\det(\gamma) \big)=\sigma^{-1})$ is a modular pair in involution for the Hopf algebra $\cR(G_2) \acl \cU(\Fg_1)$(we have used antipode in our right coaction so their $\sigma$ is our $\sigma^{-1})$). By <ref>, $^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} $ is a SAYD over $\cR(G_2) \acl \cU(\Fg_1)$. Use previous lemma, $^{\sigma} \bC _{\delta}$ is a right-left SAYD over $\cR(G_2) \acl \cU(\Fg_1)^{\cop}$ and $(\delta,\sigma^{})$ is a modular pair in involution for the Hopf algebra $\cH$.
Let us now calculate $S_{\delta}$ on these generators for future use.
\begin{align}
\begin{aligned}
S_{\delta}(1 \acl Z_i)&=\mu (\delta \otimes S)\Delta(1 \acl Z_i)\\
&=\mu (\delta \otimes S_{\aclsub}^{-1})\Delta_{\aclsub}^{\rm{op}}(1 \acl Z_i) \\
&=\mu (\delta \otimes S_{\aclsub}^{-1})(1 \acl Z_i \otimes 1 \acl 1 + S(\gamma_i^j) \acl 1 \otimes 1 \acl Z_j)\\
&=\delta(Z_i) 1 \acl 1 +\delta_i^j S_{\aclsub}^{-1} (1 \acl Z_j)\\
&=\delta(Z_i) 1 \acl 1 + S_{\aclsub}^{-1} (1 \acl Z_i)\\
&=\delta(Z_i) 1 \acl 1 + \gamma_i^j\acl S(Z_j),
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
S_{\delta}(F \acl 1)&=\mu (\delta \otimes S)\Delta(F \acl 1)\\
&=\mu (\delta \otimes S_{\aclsub}^{-1})\Delta_{\aclsub}^{\rm{op}}(F \acl 1) \\
&=\mu (\delta \otimes S_{\aclsub}^{-1})(F_{(2)}\acl 1)\otimes (F_{(1)}\acl 1)\\
&=\varepsilon (F_{(2)}) S(F_{(1)}) \acl 1\\
&=S(F_{}) \acl 1.
\end{aligned}
\end{align}
With a module pair in involution in hand we can associate a cyclic structure to $\cH$ as suggested in <cit.>. Specifically, we set $C^n(\cH)=\cH^{\otimes n}$ for $n \ge 1$, and $C^0(\cH)=\bC$. The face maps $\delta_i:C^{n-1}(\cH) \to C^{n}(\cH)$, $0 \le i \le n$, are:
\begin{align}
\begin{aligned}\label{face}
\delta_{0}(h^1 \otimes \cdots \otimes h^{n-1} )&=1\otimes h^1 \otimes \cdots \otimes h^{n-1}, \\
\delta_{i}(h^1 \otimes \cdots \otimes h^{n-1} )&= h^1 \otimes \cdots\otimes \Delta (h^i) \otimes \cdots \otimes h^{n-1}, \quad 1 \le i \le n-1,\\
\delta_{n}(h^1 \otimes \cdots \otimes h^{n-1} )&=h^1 \otimes \cdots \otimes h^{n-1} \otimes \sigma^{} ,
\end{aligned}
\end{align}
for $n > 1$, and
\begin{align*}
\delta_{0}(1)&=1 ,\qquad \delta_{1}(1)=\sigma^{},
\end{align*}
for $n=1$.
The degeneracy maps $\sigma_i:C^{n+1}(\cH) \to C^{n}(\cH)$, $0 \le i \le n$, are:
\begin{align}\label{degeneracy}
\sigma_{i}(h^1 \otimes \cdots \otimes h^{n+1} )= h^1 \otimes \cdots\otimes \varepsilon (h^{i+1}) \otimes \cdots \otimes h^{n+1},
\end{align}
for $n > 0$, and
\begin{align*}
\sigma_{0}(h)=\varepsilon(h),
\end{align*}
for $n=0$.
The cyclic operator $\tau_{n}:C^{n}(\cH) \to C^{n}(\cH)$, is defined as:
\begin{align}
\begin{aligned}
\tau_{n}(h^1 \otimes \cdots \otimes h^{n} )=&\big( \Delta^{n-1} S_{\delta}(h^1) \big) \cdot h^2 \otimes \cdots\otimes h^{n} \otimes \sigma^{}, \label{cyclic}\\
=&\sum S(h^1_{(n)})h^2\otimes \cdots \otimes S(h^1_{(2)})h^n \otimes S_{\delta}(h^1_{(1)})\sigma^{}.
\end{aligned}
\end{align}
The periodic Hopf cyclic cohomology $HP^{\bullet}(\cH;^\sigma \bC _ \delta)$ of $\cH$ with coefficients in the modular pair $(\delta,\sigma)$ is, by definition (cf. <cit.>), the $\Zb_2$-graded cohomology of the total complex $CC^{\bullet}(\cH;^\sigma \bC _ \delta)$ associated to the bicomplex $\{CC^{\bullet}(\cH;^\sigma \bC _ \delta), b, B\}$, where
\begin{align}
\begin{aligned}
&CC^{p,q}(\cH;^\sigma \bC _ \delta)=\begin{cases} 0,\quad \text{if}\quad p < q,\\
\displaystyle C^{p-q}(\cH;^\sigma \bC _ \delta),\quad \text{if}\quad q \ge p,
\end{cases} \\
&b: C^{n}(\cH;^\sigma \bC _ \delta) \to C^{n+1}(\cH;^\sigma \bC _ \delta)\\
&b=\sum_{i=0}^{n} (-1)^{i}\delta_{i}, \\
&B: C^{n}(\cH;^\sigma \bC _ \delta) \to C^{n-1}(\cH;^\sigma \bC _ \delta)\\
&B= \big(\sum_{i=0}^{n-1} (-1)^{(n-1)i} \tau^{i}_{n-1} \big) \sigma_{n-1}\tau_{n}(1-(-1)^{n}\tau_{n}) .
\end{aligned}
\end{align}
Let $\cH$ be a Hopf algebra endowed with a modular pair in involution $(\delta,\sigma^{})$. Then $\cH^{\natural}_{(\delta,\sigma^{})}=\{C^n(\cH)\}_{n \ge 0}$ equipped with the operators given by <ref> to <ref> is a module over the cyclic category $\Lambda$.
See <cit.> for proof. Note it is the MPI condition <ref> that ensured the cyclicity.
Now we will take a deeper look at the Hopf action and the trace we just defined.
For any $a,b \in \cA$ and $h \in \cH$ one has
\begin{align}
\begin{aligned}
&\tau(ab)=\tau(b \sigma^{}(a)),\\
&\tau(h(a)b)=\tau(a S_{\delta}(h)(b)),
\end{aligned}
\end{align}
hence $\tau$ is a $\delta$-invariant $\sigma^{}$-trace under the Hopf action.
It suffices to verify the identities for non trivial cases. i.e., we assume $a=fU_{\psi}^{\ast}$ and $b=gU_{\psi}$ (recall from our convention <ref>, $U_{\psi}=U_{\psi^{-1}}^{\ast}$) for the first and third identities and $a=fU_{1}^{\ast}$ for the second one.
\begin{align}
\begin{aligned}
\tau(fU_{\psi}^{\ast} \ast gU_{\psi})&=\int_{G_1} f (g\circ \widetilde{\psi}) \ \varpi\\
&=\int_{G_1} (f\circ \widetilde{\psi^{-1}}) g \ \widetilde{\psi^{-1}}^{\ast} (\varpi)\\
&=\int_{G_1} (f\circ \widetilde{\psi^{-1}}) g \ (\det(\Gamma)(\psi^{})\circ \widetilde{\psi^{-1}}) \varpi\\
&=\int_{G_1} (f\circ \widetilde{\psi^{-1}})(\vp) g (\vp)\ \det(\Gamma)(\psi^{})( \psi^{-1} \trt \vp) \varpi (\vp)\\
&=\int_{G_1} (f\circ \widetilde{\psi^{-1}})(\vp) g (\vp)\ \det(\gamma)(\psi^{} \tlt ( \psi^{-1} \trt \vp)) \varpi (\vp)\\
&=\tau(gU_{\psi} \ast \widetilde {\det (\gamma)^{}} (fU_{\psi}^{\ast})).
\end{aligned}
\end{align}
For the second identity, because of obvious multiplicativity, we verify on generators $1 \acl Z_i $:
\begin{align}
\begin{aligned}
\int_{G_1} \widetilde{Z_i}(f)\ \varpi&= \int_{G_1} \dt f( \vp \exp(tZ_i))\ \varpi(\vp) \\
&=\dt \int_{G_1} f( \vp \exp(tZ_i))\ \varpi (\vp) \\
&=\dt \int_{G_1} f( \vp)\ \varpi (\vp \exp(-tZ_i)) \\
&=\dt \Delta(tZ_i) \int_{G_1} f( \vp)\ \varpi (\vp) \\
&=\delta(Z_i) \int_{G_1} f\ \varpi,
\end{aligned}
\end{align}
and on $F\acl 1 $:
\begin{align}
\begin{aligned}
\int_{G_1} F((1 \tlt \vp )^{}) f (\vp)\ \varpi (\vp)= F(1) \int_{G_1} f\ \varpi.
\end{aligned}
\end{align}
For the third identity, because of anti-multiplicativity of $S_{\delta}$, we verify on generators $F \acl 1 $:
\begin{align}
\begin{aligned}
\tau (\widetilde{F \acl 1}(fU_{\psi}^{\ast}) \Conv gU_{\psi}^{} )
&=\int_{G_1} F((\psi \tlt \vp )^{}) f (\vp) g (\psi \trt \vp) \ \varpi (\vp) \\
&=\int_{G_1} F((\psi^{-1} \tlt (\psi \trt \vp ) )^{-1}) f (\vp) g (\psi \trt \vp) \ \varpi (\vp) \\
&=\int_{G_1} f (\vp) g (\psi \trt \vp)F(( \psi^{-1} \tlt (\psi \trt \vp ) )^{-1}) \ \varpi (\vp) \\
&=\int_{G_1} f (\vp) g (\psi \trt \vp) S(F)(( \psi^{-1} \tlt (\psi \trt \vp) )^{}) \ \varpi (\vp) \\
&=\tau \big(fU_{\psi}^{\ast} \Conv \widetilde{S_{\delta}( F \acl 1)}( gU_{\psi}^{} ) \big).
\end{aligned}
\end{align}
and on $1\acl Z_i$:
\begin{align}
\begin{aligned}
\tau ( \widetilde{ 1 \acl Z_i } (fU_{\psi}^{\ast} \Conv gU_{\psi}^{}))
=&\int_{G_1} \widetilde{Z_i}(f)(g\circ \widetilde{\psi})\ \varpi\\
=& \int_{G_1} \dt f( \vp \exp(tZ_i)) g (\psi \trt \vp)\ \varpi(\vp) \\
=&\dt \int_{G_1} f( \vp) g (\psi \trt (\vp \exp(-tZ_i)))\ \varpi(\vp \exp(-tZ_i)) \\
=&\dt \int_{G_1} f( \vp) g (\psi \trt (\vp \exp(-tZ_i)))\ \varpi(\vp ) \\
&+\dt \int_{G_1} f( \vp) g (\psi \trt \vp )\ \varpi(\vp \exp(-tZ_i)) \\
=& \int_{G_1} f( \vp)\ (\Gamma_{i}^j(\psi^{-1} )\circ \widetilde{\psi})(\vp)(-\widetilde{Z}_j(g)\circ \widetilde{\psi}) ( \vp )\ \varpi(\vp ) \\
&+\dt \Delta(tZ_i) \int_{G_1} f( \vp) g (\psi \trt \vp )\ \varpi(\vp) \\
=& \int_{G_1} f( \vp)\ \gamma_{i}^j(\psi^{-1} \tlt (\psi \trt \vp ) )(-\widetilde{Z}_j(g)\circ \widetilde{\psi}) ( \vp )\ \varpi(\vp ) \\
&+\delta(Z_i)\int_{G_1} f( \vp) g (\psi \trt \vp )\ \varpi(\vp) \\
=& \tau ( fU_{\psi}^{\ast} \Conv ( ( \widetilde{ 1 \acl \gamma _{i}^j } )(\widetilde{ S(Z_j) \acl 1 })(gU_{\psi}^{})))\\
&+\delta(Z_i)\tau(fU_{\psi}^{\ast} \Conv gU_{\psi})\\
=& \tau ( fU_{\psi}^{\ast} \Conv ( ( \widetilde{ 1 \acl \gamma _{i}^j } )(\widetilde{ S(Z_j) \acl 1 })(gU_{\psi}^{})))\\
&+\tau(fU_{\psi}^{\ast} \Conv \widetilde {\delta(Z_i)}(gU_{\psi}))\\
=& \tau (fU_{\psi}^{\ast} \Conv \widetilde{ S_{\delta}( Z_i \acl 1)}(gU_{\psi}^{})),
\end{aligned}
\end{align}
Let $\tau: \cA \to \bC$ be a $\delta$-invariant $\sigma^{}$-trace under the Hopf action of $\cH$ on $\cA$. Then the assignment
\begin{align}
\lambda(h^1 \otimes \cdots \otimes h^n)(a^0,\dots,a^n)=\tau(a^0 h^1(a^1)\cdots h^n(a^n) )
\end{align}
defines a map of $\Lambda$-modules $\lambda^{\natural}: \cH^{\natural}_{(\delta,\sigma^{})} \to \cA^{\natural}$, which induces characteristic homomorphisms in cyclic cohomology:
\begin{align*}
\lambda^{\ast}_{\tau}:HC^{\ast}_{(\delta,\sigma^{})} (\cH) \to HC^{\ast} (\cA).
\end{align*}
For any discrete subgroup $\Gamma < G_2^{\delta}$, $\tau: \cA_{\Gamma} \to \bC$ is also a $\delta$-invariant $\sigma^{}$-trace under the Hopf action of $\cH$ on $\cA_{\Gamma}$. The assignment above also defines a map of $\Lambda$-modules $\lambda^{\natural}: \cH^{\natural}_{(\delta,\sigma^{})} \to \cA_{\Gamma}^{\natural}$, which induces characteristic homomorphisms in cyclic cohomology:
\begin{align*}
\lambda^{\ast}_{\tau}:HC^{\ast}_{(\delta,\sigma^{})} (\cH) \to HC^{\ast} (\cA) \to HC^{\ast} (\cA_{\Gamma}).
\end{align*}
Let ($G_1$, $G_2$) be a matched pair of Lie groups, $\cA$ be the convolution algebra $C_c^{\ify}(G_1)\rtimes G_2^{\delta}$. $\tau: \cA \to \bC$ be a $\delta$-invariant $\sigma$-trace, as defined in this section, under the Hopf action of $\cH=(\cU(\Fg_1) \acr \cR(G_2))^{\cop}$ on $\cA$. Then the assignment
\begin{align}
\lambda(h^1 \otimes \cdots \otimes h^n)(a^0,\dots,a^n)=\tau(a^0 h^1(a^1)\cdots h^n(a^n) ) ,\qquad h^i \in \cH, \quad a^i \in \cA,
\end{align}
defines an injective homomorphism of $\Lambda$-modules $\lambda^{\natural}: \cH^{\natural}_{(\delta,\sigma^{})} \to \cA^{\natural}$.
Before we go to the proof we list two useful lemmas.
the $\sigma^{}-$trace $\tau$ is faithful, i.e., that
\begin{align*}
\tau (ab)=0 ,\quad \forall a \in \cA, \quad \text{implies}\ b=0.
\end{align*}
We just need to show that if $\int_{G_1} f (g\circ \widetilde{\psi}) \ \varpi=0$ for all $g$, then $f=0$, which is the same that if $\int_{G_1} f g \ \varpi=0$ for all $g$, then $f=0$.
Now if we assume $\int_{G_1} f g \ \varpi=0$ for all $g$, but $f(p)\ne 0$ at some $p \in G_1$. Then there exits some small neighborhood $U$ of $p$ such that $f$ is strictly positive or negative, say $f > c >0$ on $U$. We can have a compact set $V_1\subset U$ and a smaller compact set $V_2\subsetneqq V_1 $. Take $g $ to be a smooth cutoff function that is identity on $V_2$ and vanish off $V_1$, then $\int_{G_1} f g \ \varpi =\int_{V_1} f g \ \varpi > \int_{V_2} f g \ \varpi > \int_{V_2} c \ \varpi >0 $, hence a contradiction.
\begin{align*}
h(a)(e)=0 ,\quad \forall a \in \cA , \quad \text{implies}\ h=0.
\end{align*}
Recall <ref> and <ref>, since the $Z_I$ ’s form a PBW basis of $\cU(\Fg_1)$, and that every element $\sum_{j}F_{j} \acl u_{j}$ in $\cR(G_2) \acl \cU(\Fg_1)$ can be written uniquely as
\begin{align}
\begin{aligned}
\sum_{j}F_{j} \acl u_{j}=\sum_{I} F_{I} \acl Z_{I},\quad I=(i_1,\dots, i_p),
\end{aligned}
\end{align}
we just need to prove that if
\begin{align*}
\sum \widetilde{F_{I}} \widetilde{Z_{I}} (fU_{\psi}^{\ast})(e)=0, \qquad \forall fU_{\psi}^{\ast} \in \cA,
\end{align*}
then $F_{I}=0$ for any $I$.
However, for every $\psi \in G_2$ we will have
\begin{align*}
\sum F_{I}(\psi \tlt e) \widetilde{Z_{I}} (f)(e)=0, \qquad \forall f \in C_c^{\ify}(G_1),
\end{align*}
Because $Z_{I}$ form a PBW basis of $\cU(\Fg_1)$, we order the sum by the order of the PBW basis assume that the coefficient $F_{I_n}(\psi )$ is non-zero for $I_n$ with top degree. There exits a function $g_n \in C_c^{\ify}(G_1)$ defined on a neighborhood of $e$ such that $\widetilde{Z_{I_n}} (g_n)(e)=1$ and $\widetilde{Z_{I_s}} (g_n)(e)=0$ for all lower $s < n$. Therefore we have $F_{I_n}(\psi)=0$, a contradiction. Therefore $F_{I}(\psi)=0$ for any $I$. Since this is true for every $\psi \in G_2$, we then have our desired conclusion.
Now we are ready to prove theorem <ref>.
Assume $\tau \big( \sum_{j} a^0 h_j^1(a^1)\cdots h_j^n(a^n) \big)=0$ for any $a^0,,\dots,a^n$, by lemma <ref>, we conclude
\begin{align}
\sum_{j} h_j^1(a^1)\cdots h_j^n(a^n)=0,
\end{align}
when evaluating at the identity, the above equation becomes
\begin{align}
\sum_{j} h_j^1(a^1)\cdots h_j^n(a^n)(e)=0,
\end{align}
from where we want to show
\begin{align*}
\sum_{j} h_j^1\otimes \cdots \otimes h_j^n=0.
\end{align*}
We use induction: When $n=1$, lemma <ref> applies. Now assume the statement is true of $n-1$, i.e., if
\begin{align*}
\sum_{j} h_j^1(a^1)\cdots h_j^{n-1}(a^{n-1})(e)=0,
\end{align*}
we have
\begin{align}
\sum_{j} h_j^1\otimes \cdots \otimes h_j^{n-1}=0.
\end{align}
Denote $a^k=f_kU_{\psi_k}^{\ast}$. If we use <ref> and the PBW basis of $\cU(\Fg_1)$ and write each $h_j^k$ as $F_{j,I(j,k)}^k \acl Z_{j,I(j,k)}^k $. Since $e$ is fixed by every $\psi \in G_2$, when evaluating at the identity the above equality becomes
\begin{equation*}
\sum_{j,I} F_{j,I(j,1)}^1 ((\psi_1 \tlt e)^{}) \widetilde{Z_{j,I(j,1)}^1} (f_1)(e) \cdots F_{j,I(j,n)}^n((\psi_n \tlt e)^{}) \widetilde{Z_{j,I(j,n)}^n} (f_n)(e)=0.
\end{equation*}
We partition the sum by the order of $Z_{j,I(j,n)}^n$ and index with $(1\le s \le N,Q(s))$ and use induction again on $s$. after reordering the above equality becomes
\begin{equation*}
\sum_{I_1 \le I_s \le I_N} F_{I_s}^n(\psi_n^{} ) \widetilde{Z_{I_s}^n} (f_n)(e)( \sum_{Q(s)} F_{Q}^1 (\psi_1^{} ) \widetilde{Z_{Q}^1} (f_1)(e) \cdots F_{Q}^{n-1}(\psi_{n-1} ^{}) \widetilde{Z_{Q}^{n-1}} (f_{n-1})(e))=0.
\end{equation*}
Now if we look at the top ${Z_{I_N}^n}$ with nonzero $F_{I_N}$, because of the PBW basis, there exits a function $g_n \in C_c^{\ify}(G_1)$ defined on a neighborhood of $e$ such that $\widetilde{Z_{I_N}^n} (g_n)(e)=1$ and $\widetilde{Z_{I_s}^n} (g_n)(e)=0$ for all lower $s < N$. Since we assume $F_{I_N}$ is nonzero, there also exist some $\psi_n^{+}$ such that $F_{I_N}(\psi_n^{+}) \ne 0$.
Now if we fix $a^n=g_nU_{\psi_n^{+}}^{\ast}$, the above equality becomes
\begin{equation*}
\sum_{Q(N)} F_{Q}^1 (\psi_1 ^{}) \widetilde{Z_{Q}^1} (f_1)(e) \cdots F_{Q}^{n-1}(\psi_{n-1} ^{}) \widetilde{Z_{Q}^{n-1}} (f_{n-1})(e)=0 \quad \forall f_kU_{\psi_k}^{\ast}, \quad1\le k \le n-1,
\end{equation*}
this is the induction hypothesis for $n-1$ of the first induction, hence we have
\begin{align}
\sum_{Q(N)} h_Q^1\otimes \cdots \otimes h_Q^{n-1}=0.
\end{align}
\begin{align*}
(\sum_{Q(N)} h_Q^1\otimes \cdots \otimes h_Q^{n-1}) \otimes h_N^n=0.
\end{align*}
This means the top degree part, with the new order $(1\le s \le N,Q(s))$, of the original sum
\begin{align*}
\sum_{j} h_j^1\otimes \cdots \otimes h_j^n
\end{align*}
is $0$. By the reversed induction on the degree, the whole term is $0$ and $\lambda$ is injective.
An n-cochain $\vp $ on the algebra $C_c^{\ify}(G_1)\rtimes G_2^{\delta}$ is representative iff it is in the range of the above monomorphism $\lambda$. We denote the representative cochain space as $\text{Im}(\lambda^{\natural})$.
Under the assumption of the above theorem, $\lambda$ is an isomorphism of $\Lambda$-modules $\lambda^{\natural}: \cH^{\natural}_{(\delta,\sigma^{})} \to \text{Im}(\lambda^{\natural})$.
From now on we do not distinct between $\cH^{\natural}_{(\delta,\sigma^{})}$ and $\text{Im}(\lambda^{\natural})$.
§ DIFFERENTIABLE AND REPRESENTATIVE COHOMOLOGIES OF ACTION GROUPOIDS
In this section, we will first review Haefliger's differentiable cohomology of a action groupoid <cit.>, in particular we take our $G_1 \rtimes G_2^{\delta}$ as example. Then we refine the complex that calculate the differentiable cohomology and define representative cohomology of a action groupoid. The differentiable and representative cohomology works in general for any action groupoid. However, for our example $G_1 \rtimes G_2^{\delta}$ with a matched pair of Lie groups $(G_1,G_2)$, There is an even smaller subcomplex which we called $D^{p,q}$ that satisfies some strong covariance property. We will explain in detail the description of $D^{p,q}$ and its relation (as appeared in the last column of diagram <ref>) between the representative cochain space $\text{Im}(\lambda^{\natural})$ from previous section via Conne's $\Phi$ map.
§.§ Decomposition of Lie groups
Recall from <cit.> that a nucleus of a Lie group $G$ is a simply connected solvable closed normal subgroup $L$ of $G$ such that $G/L$ is reductive, in the sense that $G/L$ has a faithful representation and every finite dimensional analytic representation of $G/L$ is semisimple. In this case one proves (<cit.>) that $G = H \ltimes L$, where $H\cong G/L$ is the complementary closed subgroup which is reductive. We will assume all groups are connected through this paper. $H$ acts on $L$ from the right by
\begin{align}
\begin{aligned}
l \tlt h= h^{-1} l h
\end{aligned}
\end{align}
which coincide with the right multiplication of $H$ on $H \backslash G$
\begin{align}
\begin{aligned}
(Hl )h=H(h h^{-1} l h)=H (l \tlt h)
\end{aligned}
\end{align}
Let, in addition, $\Fh,\Fl \subset \Fg$ be Lie algebras of $H,L$ and $G$ respectively.
Parallel decomposition of Linear algebraic group is called the Levi decomposition (<cit.> ) where $G = H \ltimes L $, $H$ the levi factor and $L=G_{u}$ the unipotent radical.
For a matched pair of Lie groups $G_1$ and $G_2$ we can form the bicrossed product group $G_1\bowtie G_2$ with group structure given by
\begin{align*}
(\vp_1\bowtie\psi_1)(\vp_2\bowtie\psi_2)=(\vp_1 \psi_1 \trt \vp_2)\bowtie(\psi_1 \tlt \vp_2 \psi_2).
\end{align*}
For a matched pair of Lie algebras $\Fg_1$ and $\Fg_2$ we can also form the bicrossed product Lie algebra $\Fg_1\bowtie \Fg_2$ with underlying vector space $\Fg_1\oplus \Fg_2$ and bracket relation:
\begin{equation*}
[X_1 \oplus Y_1, X_2 \oplus Y_2] = ([X_1,x_2]+ Y_1 \trt X_2 -Y_2 \trt X_1) \oplus ( [Y_1,Y_2]+Y_1 \tlt X_2 -Y_2 \tlt X_1 ).
\end{equation*}
Now let $G_2 = H_2 \ltimes L_2$ be such decomposition of $G_2$. In general, the simply-connected nucleus $L_2$ of a Lie group is differ from nilpotent by a vector group, to simplify the situation we assume further $L_2$ is nilpotent with Lie algebra $\Fl_2$. In the algebraic case $L_2$ as the unipotent radical is nilpotent. Under our assumption, for both case we have $L_2=\exp (\Fl_2)$ and $\Fl_2=\mathfrak{nil}(\Fg_2)$.
§.§ Differentiable cohomology of action groupoid
Let $\cG=G_1 \rtimes G_2^{\delta}$ be our action groupoid, $\Omega_{G_1}^{\ast}$ be the complex of differential forms on $G_1$. We introduce the following double complex $\label{bigcomplex}C^{p,q},d_1,d_2$. $C^{p,q}= \{ 0 \} $ unless $p \ge 0$ and $0 \le q \le \text{ dim }G_1 $, $C^{p,q}$ be the space of totally antisymmetric maps $\alpha:G_2^{p+1}\to \Omega^{q}(G_1)$ that satisfy the $G_2$-equivariant condition
\begin{align}
\alpha(\psi_0 \psi,\dots,\psi_p \psi )=(\psi^{-1}\trt)^{\ast} \alpha(\psi_0,\dots,\psi_p), \quad \forall \psi,\psi_i \in G_2.
\end{align}
The coboundary $d_1:C^{p,q} \to C^{p+1,q}$ is given by
\begin{align}
(d_1\alpha)(\psi_0,\dots,\psi_{p+1})=\sum_{j=0}^{p+1}(-1)^{q+j}\alpha(\psi_0,\dots,\check{\psi_{j}} , \dots,\psi_{p+1}),
\end{align}
The coboundary $d_2:C^{p,q} \to C^{n,q+1}$ is given by de Rham boundary:
\begin{align}
(d_2\alpha)(\psi_0,\dots,\psi_{p+1})=d \alpha(\psi_0,\dots,\psi_{p+1}).
\end{align}
The complex $C^{p,q} $ has a subcomplex $\label{diffcomplex}C_{d}^{p,q} $, which consists of forms $\alpha(\psi_0 ,\dots,\psi_p )$ that depend on $\psi_i$ smoothly, i.e., we can write $\alpha \in C_{d}^{p,q}$ as
\begin{align}
\alpha(\psi_0,\dots,\psi_p)=\sum c_j \rho_j ,
\end{align}
here $\rho_j$ form a basis of left $G_1$ invariant forms on $G_1$, and $c_j$ are smooth functions on $G_2$ .
Let $\cG=G_1 \rtimes G_2^{\delta}$. The cohomology of the action groupoid $H^{\bullet} ( \cG;\Rb)$ is the cohomology of the simple complex associated to the above double complex $C^{p,q}$ and the differentiable cohomology $H^{\bullet}_{d}( \cG;\Rb)$ is the cohomology of the simple complex associated to the subcomplex $C_{d}^{p,q}=C^{p}_{d}( G_2^{\delta};\Omega_{G_1}^{q})$ whose elements are forms depending smoothly on $p$ elements of $G_2$ as in <ref>.
Let $\cG=G_1 \rtimes G_2^{\delta}$. $H^{\bullet}_{d}( \cG;\Rb)$ is isomorphic to the cohomology of $G_2$-invariant forms on $G_2/K_2 \times G_1$, where $K_2$ is a maximal compact subgroup of $G_2$, and $G_2$ acts diagonally.
§.§ Representative cohomology of action groupoid
We refine Haefliger's differentiable cohomology of groupoid by defining a subcomplex, $C^{p}_{\cR}( G_2^{\delta};\Omega_{G_1}^{q})$ of $C^{p}_{d}( G_2^{\delta};\Omega_{G_1}^{q})$. The elements of $C_{\cR}^{p,q} =C^{p}_{\cR}( G_2^{\delta};\Omega_{G_1}^{q})$ are forms depending representatively on $p+1$ elements of $G_2$. i.e., the coefficient functions are products of $p+1$ representative functions on $G_2$, we can write $\alpha \in C_{\cR}^{p,q}$ as
\begin{align}
\alpha(\psi_0,\dots,\psi_p)=\sum c_j \rho_j ,
\end{align}
where $c_j$ are representative functions on $G_2$.
Since representative functions are smooth, we can get a subcomplex and define the representative cohomology $H^{\bullet}_{\cR}( \cG;\Rb)$:
Let $\cG=G_1 \rtimes G_2^{\delta}$. The representative cohomology $H^{\bullet}_{\cR}( \cG;\Rb)$ is the cohomology of the simple complex associated to the double complex $C^{p}_{\cR}( G_2^{\delta} ;\Omega_{G_1}^{q})$.
Let $\cG=G_1 \rtimes G_2^{\delta}$. The representative cohomology $H^{\bullet}_{\cR}( \cG;\Rb)$ is isomorphic to the cohomology of $G_2$-invariant forms on $L_2 \times G_1$, where $L_2$ is the nucleus of $G_2$, and $G_2$ acts diagonally.
Step 1:
Consider the first inclusion
\begin{align*}
\varepsilon_1 : C^{p}_{\cR}( G_2^{\delta};\Omega_{G_1}^{q}) \to C^{p}_{\cR}( G_2^{\delta} ;\Omega_{(L_2\times G_1)}^{q})
\end{align*}
induced by the projection $\pi: L_2 \times G_1 \to G_1$ induces an isomorphism of the cohomology of the double complexes.
By a familiar spectral sequence argument (<cit.>), we only need to show that the inclusion above induces an isomorphism of the vertical deRham cohomology. This is true because the fiber of $\pi$ is smoothly contractible.
Step 2:
Follow the usual double complex argument as in <cit.>. The augmented sequence of the horizontal complex is exact for every $q$:
\begin{align*}
0 \to (\Omega_{(L_2\times G_1)}^{q})^{G} \xrightarrow{\varepsilon_2} C^{0}_{\cR}( G_2^{\delta} ;\Omega_{(L_2\times G_1)}^{q}) \xrightarrow{d_1} C^{1}_{\cR}( G_2^{\delta} ;\Omega_{(L_2\times G_1)}^{q}) \xrightarrow{d_1} \cdots
\end{align*}
where $(\Omega_{(L_2\times G_1)}^{q})^{G_2}$ are $G_2$-invariant forms on $L_2 \times G_1$, $\varepsilon_2$ is another inclusion. Because of this exactness we can use the usual diagram chasing argument for double complex with exact rows to get the fact that $\varepsilon_2$ induces an isomorphism from the cohomology of augmented column complex to the cohomology of double complex $C^{p}_{\cR}( G_2^{\delta} ;\Omega_{(L_2\times G_1)}^{q})$.
Step 3:
Finally the exactness of the above sequence is provided by the given homotopy (cf., <cit.>):
\begin{align}
\begin{aligned}
&H:\ C^{p}_{\cR}( G_2^{\delta} ;\Omega_{(L_2\times G_1)}^{q})\to \ C^{p-1}_{\cR}( G_2^{\delta} ;\Omega_{(L_2\times G_1)}^{q})\\
&H(\alpha)(\psi_1,\dots,\psi_{p})(l\times m)=(-1)^q \pi(f_{(\psi_1,\dots,\psi_{p})}^\alpha),
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
f_{(\psi_1,\dots,\psi_{p})}^\alpha (h):=\alpha(hl,\psi_1,\dots,\psi_{p})(l\times m),
\end{aligned}
\end{align}
and $\pi$ the projection to its $H_2$ invariant part, since $H_2$ is reductive and every $H_2$ module is completely reducible.
We have
\begin{align}
\begin{aligned}
&H d_1(\alpha)(\psi_0,\dots,\psi_{p})(l\times m)\\
=&(-1)^q \pi( f_{(\psi_0,\psi_1,\dots,\psi_{p})}^{ d_1 \alpha}) \\
=& \alpha(\psi_0,\dots,\psi_{p})(l\times m) - \pi( f_{(\psi_1,\dots,\psi_{p})}^{ \alpha}) +\sum_{j=1}^{p}(-1)^{j+1}\pi( f_{(\psi_0,\dots \check{\psi_j} \dots,\psi_{p})}^{ \alpha}) ,
\end{aligned}
\end{align}
\begin{align*}
d_1 H (\alpha)(\psi_0,\dots,\psi_{p})(l\times m)=\sum_{j=0}^{p}(-1)^{j}\pi( f_{(\psi_0,\dots \check{\psi_j} \dots,\psi_{p})}^{ \alpha}) .
\end{align*}
\begin{align}
H d_1 + d_1 H =I.
\end{align}
§.§ A smaller subcomplex Dpd and Connes' Phi map
The complex $C_{\cR}^{p,q} $ has a subcomplex $D ^{p,q} $, which consists of forms $\alpha(\psi_0 ,\dots,\psi_p )$ that can be written as
\begin{align}
\alpha(\psi_0,\dots,\psi_p)=\sum c_j \rho_j ,
\end{align}
where $c_j$ are smooth functions on $G_1$ which are finite linear combinations of finite products of the following functions,
\begin{align*}
\vp \in G_1 \to F_i \big( (\psi_i \tlt \vp) \big),\quad F_i \in \cR(G_2) ,
\end{align*}
more precisely,
\begin{align}
c_j(\vp)=\sum_{\text{finite}} \prod_{i=0}^{p}F_i \big( (\psi_i \tlt \vp)\big),\quad F_i \in \cR(G_2).
\end{align}
The important fact about $\alpha$ is that it is not only $G_2$-equivariant but also $G$-equivariant, i.e.,
\begin{align}
\alpha(\psi_0 \tlt \phi,\dots,\psi_p \tlt \phi )=(\phi^{-1} \trt)^{\ast} \alpha(\psi_0,\dots,\psi_p), \quad \forall \psi_i \in G_2,\phi=\vp\psi \in G_1G_2=G.
\end{align}
We use Connes' $\Phi$ map from bicomplex $(C^{p,q},d_1,d_2)$ to the $(b,B)$ bicomplex of the algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$ and we show that the resulting cochains are representative as in <ref>, i.e., in the range of $\lambda$. Hence we have $\Phi$ from $(C^{p,q},d_1,d_2)$ to the $(b,B)$ bicomplex of $\cH$.
Let us recall the construction of $\Phi$. As in <cit.>, we let $\mathcal{B}$ be the tensor product,
\begin{align*}
\mathcal{B}=A^\ast(G_1) \otimes \Lambda^{\ast} (\bC (G_2')),
\end{align*}
where $A^\ast(G_1)$ is the algebra of smooth forms with compact support on $G_1$, the exterior algebra $\Lambda^{\ast} (\bC (G_2'))$ of the linear space $\bC (G_2')$ with basis the $\delta_{\psi}, \psi \in G_2$, with $\delta_e=0$. We take the crossed product
\begin{align*}
\mathcal{C}=\mathcal{B} \rtimes G_2.
\end{align*}
The action of $G_2$ on $\mathcal{B}$ by automorphisms is defined as
\begin{align}
\begin{aligned}
U^{\ast}_{\psi} \omega U^{}_{\psi} = \widetilde{\psi}^{\ast} \omega = \omega \circ \widetilde{\psi} \qquad \forall \omega \in A^\ast(G_1) , \\
U^{\ast}_{\psi} \delta_{\psi_2} U^{}_{\psi} =\delta_{\psi_2 \psi} -\delta_{\psi} \qquad \forall \psi_2 \in G_2.
\end{aligned}
\end{align}
We write the elements of $\mathcal{C}$ as a finite sum
\begin{align*}
c=\sum_{\psi} b U^{\ast}_{\psi}, \qquad b \in \mathcal{B},
\end{align*}
then we can write the differential $d$ in $\mathcal{C}$ as
\begin{align}
d (b U^{\ast}_{\psi})= d b U^{\ast}_{\psi} - (-1)^{\partial b}b \delta_{\psi} U^{\ast}_{\psi},
\end{align}
where the first term is only the exterior differential in $A^\ast(G_1)$.
A cochain $\gamma$ in $C^{p,q}$ determines a form $\widetilde{\gamma}$ on $\mathcal{C}$ by
\begin{align}
\widetilde{\gamma}(\omega\otimes\delta_{\psi_1}\cdots \delta_{\psi_p} )=\int_{G_1}\omega \wedge \gamma(e,\psi_1,\dots,\psi_p),\quad \widetilde{\gamma}(b U^{\ast}_{\psi})=0 \quad \text{if}\ \psi \ne e.
\end{align}
we define $\Phi(\gamma)$ by
\begin{align}
\begin{aligned}
\Phi ( \gamma ) (a^0,\dots,a^{l}) &=\frac{p!}{(l+1)!} \sum_{j=0}^{l} (-1)^{j(l-j)}\widetilde{\gamma}(da^{j+1}\cdots da^{l} a^0 da^{1}\cdots da^{j}) \\
\text{where}\quad l&=p+\text{dim}(G_1)-q,\qquad a^0,\dots,a^{l} \in \cA.
\end{aligned}
\end{align}
We have the inclusions:
\begin{align}
\begin{aligned}
D^{p,q} \subset C^{p,q}_{\cR} \subset C^{p,q}_{d} \subset C^{p,q}
\end{aligned}
\end{align}
and we use $\Phi_{d}$, $\Phi_{\cR}$, $\Phi_{D}$ to denote the restriction of $\Phi$ on these above subcomplexes, they are still chain maps because $(b,B)$ maps will not change the type of the cochains.
Images of $\Phi_{D}$ are representative cochains as defined in <ref>.
we follow exactly the same argument as in <cit.>. When we need to rearrange the terms we use the coproduct of $\cR(G_2)$, when we need to permute functions and $U^{\ast}_{\psi}$ we use the fact that the left and right translates of representative function is still a representative function.
$\Phi_{D}$ is a chain map from the bicomplex $\big( D^{p,q} ,d_1,d_2 \big)$ to the $(b,B)-$complex of the Hopf algebra $\cH$.
It is proved in <cit.> that $\Phi$ is a chain map from the bicomplex $\big( C^{p,q} ,d_1,d_2 \big)$ to the $(b,B)-$complex of the algebra $\cA$. From the lemma above, the restriction $\Phi_{D}$ maps $D^{p,q}$ to representative cochain space, i.e., $Im(\lambda^{\natural})$, which is identified with $\cH^{\natural}_{(\delta,\sigma^{})}$ as in Corollary <ref>.
§ EXPLICIT CONSTRUCTION OF HOPF CYCLIC CLASSES
As in diagram <ref>, the work of Rangipour and Sütlü (<cit.>) can be split into two steps: a vertical map from the Hopf cyclic complex of $\cR(G_2) \acl \cU(\Fg_1) $ to the complex $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$ and a horizontal map $\nu$ from $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$ to the relative Lie algebra complex. The first process involves a number of quasi-isomorphic complexes and maps, hence will be recalled at the beginning of this section. We then construct an isomorphism $\Theta$ that link $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$ with $D^{p,q}$ from previous section. In the middle of the section we construct an one-sided inverse of $\nu$ map, which we will call $\mathcal{E}$. To the end we prove the main theorem of our paper: $ \mathcal{E} \circ \Theta \circ \mathcal{D} $ is a quasi-isomorphism, which goes along the opposite direction comparing to the maps of Rangipour and Sütlü, that can be used to transit cohomology classes from Lie algebra cohomology complex to representative cocycles on the convolution algebra explicitly.
§.§ Bicomplexes that calculate Hopf cyclic cohomology of H
In this section we will review a few quasi-isomorphic bicomplex that are all quasi-isomorphic to the cyclic complex of $\cH$. We refer to <cit.> for the detailed discussion. The first step is taken from <cit.>, but with our setting; while the last three steps are extra steps from <cit.>, also with our setting, that help us to link to $D^{p,q}$.
step 1
From $CC^{\bullet}(\cH,^{\sigma^{}} \bC _ \delta)$ to $CC^{\bullet}(\cH^{\cop},^{\sigma^{-1}}\hspace{-3pt} \bC _ \delta)=CC^{\bullet}(\cR(G_2) \acl \cU(\Fg_1),^{\sigma^{-1}}\hspace{-3pt} \bC _ \delta)$.
It was shown in <cit.> that
\begin{align*}
&\mathcal{T}:CC^{\bullet}(\cH,^{\sigma^{}} \bC _ \delta) \to \ CC^{\bullet}(\cH^{\cop},^{\sigma^{-1}}\hspace{-3pt} \bC _ \delta),\\
&\mathcal{T}(1\otimes h^0 \otimes h^1 \times \cdots \otimes h^n)=1\otimes {\sigma^{-1}} h^0 \otimes h^n \otimes \cdots \otimes h^1
\end{align*}
defines an isomorphism of mixed complexes.
Recall that we have used antipode in our right coaction so the element $\sigma$ in <cit.> is our $\sigma^{-1}$. It is proved in <cit.> that our $CC^{\bullet}(\cR(G_2) \acl \cU(\Fg_1),^{\sigma^{-1}}\hspace{-3pt} \bC _ \delta)$ is quasi-isomorphic to a double complex
\begin{equation}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \vdots & \vdots & \vdots \\
\wedge^{2} \Fg_1^{\ast} & \wedge^{2} \Fg_1^{\ast} \otimes \cR & \wedge^{2} \Fg_1^{\ast} \otimes \cR^{\otimes 2} & \cdots \\
\Fg_1^{\ast} & \Fg_1^{\ast} \otimes \cR & \Fg_1^{\ast} \otimes \cR^{\otimes 2} & \cdots \\
\bC & \bC \otimes \cR & \bC \otimes \cR^{\otimes 2} & \cdots \\ };
\path[transform canvas={yshift=0.6ex},->,font=\scriptsize]
(m-2-1) edge node[above] {$ b_{\cR} $} (m-2-2) (m-2-2) edge node[above] {$ b_{\cR} $} (m-2-3) (m-2-3) edge node[above] {$ b_{\cR} $} (m-2-4)
(m-3-1) edge node[above] {$ b_{\cR} $} (m-3-2) (m-3-2) edge node[above] {$ b_{\cR} $} (m-3-3) (m-3-3) edge node[above] {$ b_{\cR} $} (m-3-4)
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(m-2-2) edge node[below] {$ B_{\cR} $} (m-2-1) (m-2-3) edge node[below] {$ B_{\cR} $} (m-2-2) (m-2-4) edge node[below] {$ B_{\cR} $} (m-2-3)
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\end{tikzpicture}
\end{equation}
the coboundary $\partial_{\Fg}$ is the Lie algebra cohomology coboundary of $\Fg_1$ with coefficients in $\cR ^{\otimes \ast}$ with right $\Fg_1$ action given by
\begin{align}
&( F^1 \otimes \cdots \otimes F^p) \tlt Z=- Z \trt (F^1 \otimes \cdots \otimes F^p),
\end{align}
here the left action of $\Fg_1$ on $\cR ^{\otimes p}$ is extended from <ref> but is different from the usual diagonal action. Because the coproduct of $\Fg_1$ we used here is not from $\cU(\Fg_1)$ but from $\cR(G_2) \acl \cU(\Fg_1)$.
The formula is given by
\begin{align*}
&Z \trt (F^1 \otimes \cdots \otimes F^p)= \\
&Z_{(1)_{<0>}} \trt F^1 \otimes Z_{(1)_{<1>}} (Z_{(2)_{<0>}} \trt F^2) \otimes \cdots \otimes Z_{(1)_{<p-1>}}\cdots Z_{(p-1)_{<1>}}(Z_{(p) }\trt F^p).
\end{align*}
The coboundary $b_{\cR}$ involve the coaction $\blacktriangledown$. It is the $b$ operator for coalgebra $\cR$ with coefficients in $\wedge^{\bullet} \Fg_1^{\ast}$. The left coaction is given by <ref>
\begin{align*}
\blacktriangledown (\omega ^i)= S(\gamma^i_j) \otimes \omega^j ,
\end{align*}
and then extend to $\wedge^{\bullet} \Fg_1^{\ast}$.
It has the explicit expression
\begin{align}
\begin{aligned}
& b_{\cR} ( \alpha \otimes F^1 \otimes \cdots \otimes F^p)= \alpha \otimes 1 \otimes F^1 \otimes \cdots \otimes F^p \\
&+ \sum\limits_{1\le i \le p}^{} (-1)^{i} \alpha \otimes F^1 \otimes \cdots \otimes \Delta (F^i) \otimes \cdots \otimes F^p \\
& +(-1)^{p+1} \alpha_{<0>} \otimes F^1 \otimes \cdots \otimes F^p \otimes \alpha_{<-1>} ;
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
& B_{\cR} =(\sum_{i=0}^{p-1}(-1)^{(p-1)i}\tau_{\cR}^{i})\sigma_{\cR}\tau_{\cR} (1-(-1)^{p} \tau_{\cR}),\qquad \text{with}\\
& \tau_{\cR}( \alpha \otimes F^1 \otimes \cdots \otimes F^p)=\alpha_{<0>} \otimes S(F^1)\cdot (F^2 \otimes \cdots \otimes F^p \otimes \alpha_{<-1>})\\
& \sigma_{\cR}( \alpha \otimes F^1 \otimes \cdots \otimes F^p)=\varepsilon (F^p)\alpha \otimes F^1 \otimes \cdots \otimes F^{p-1}.
\end{aligned}
\end{align}
Here we see no $\sigma^{-1}$ or $\delta$, because during the transition we have identified the module $^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} $ with $\wedge^{\text{\tiny dim} (\Fg_1)} \Fg_1^{\ast}$ via Poincaré isomorphism.
In <cit.>, the authors stopped at this complex and derived the desired quasi-isomorphism between relative Lie algebra cohomology and the Hopf cyclic cohomology that we begin from. However, we want to go to $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$ and $D^{p,q}$. Therefore we pass from the previous complex to its homogeneous version and then to the equivariant cochains.
step 2
homogeneous version
We define $C^{p,q}_{\cR}(\wedge \Fg_1^{\ast},\otimes \cR(G_2)):=(\wedge^{q} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p+1})^{\cR(G_2)}$. An element $\sum \alpha \otimes F^0 \otimes \cdots \otimes F^p$ is in $(\wedge^{p} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p+1})^{\cR(G_2)}$ if it satisfies the $\cR-$coinvariance condition:
\begin{equation*}
\sum \alpha_{<0>} \otimes F^0 \otimes \dots \otimes F^p \otimes \alpha_{<-1>}
= \sum \alpha \otimes F^0_{\ (1)} \otimes F^1_{\ (1)} \otimes \dots \otimes F^p_{\ (1)} \otimes F^0_{\ (2)}\cdots F^p_{\ (2)}.
\end{equation*}
The two complexes are isomorphic via
\begin{align}
\begin{aligned}
&\mathcal{I}: \wedge^{q} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p} \to (\wedge^{q} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p+1})^{\cR(G_2)}, \\
&\mathcal{I}(\alpha \otimes F^1 \otimes \cdots \otimes F^p)= \\
&\alpha_{<0>} \otimes F^1_{\ (1)} \otimes S(F^1_{\ (2)} ) F^2_{\ (1)} \otimes \dots \otimes S(F^{p-1}_{\ (2)} ) F^p_{\ (1)} \otimes S(F^p_{\ (2)}) \alpha_{<-1>} ;
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
&\mathcal{I}^{-1}: (\wedge^{q} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p+1})^{\cR(G_2)} \to \wedge^{q} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p}, \\
&\mathcal{I}^{-1}(\alpha \otimes F^0 \otimes \cdots \otimes F^p)= \\
&\alpha \otimes F^0_{\ (1)} \otimes F^0_{\ (2)} F^1_{\ (1)} \otimes F^0_{\ (3)} F^1_{\ (2)} F^2_{\ (1)} \otimes \dots \otimes F^{0}_{\ (p)} \cdots F^{p-2}_{\ (2)} F^{p-1} \varepsilon (F^{p}).
\end{aligned}
\end{align}
\begin{equation}
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\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \vdots & \vdots & \vdots \\
\wedge^{2} \Fg_1^{\ast} & (\wedge^{2} \Fg_1^{\ast}\otimes \cR^{\otimes 2})^{\cR} & (\wedge^{2} \Fg_1^{\ast}\otimes \cR^{\otimes 3})^{\cR} & \cdots \\
\Fg_1^{\ast} & (\wedge^{1} \Fg_1^{\ast}\otimes \cR^{\otimes 2})^{\cR} & (\wedge^{1} \Fg_1^{\ast}\otimes \cR^{\otimes 3})^{\cR} & \cdots \\
\bC & ( \bC \otimes \cR^{\otimes 2})^{\cR} & ( \bC \otimes \cR^{\otimes 3})^{\cR} & \cdots \\ };
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\end{tikzpicture}
\end{equation}
We transform these boundary operators via $\mathcal{I}$ and get
the coboundary $\partial_{\Fg_1}$ is the Lie algebra cohomology coboundary of $\Fg_1$ with coefficients in $ \cR ^{\otimes \bullet}$ with right $\Fg_1$ action now given by the usual diagonal action:
\begin{align*}
&(1\otimes F^1 \otimes \cdots \otimes F^p) \tlt Z\\
&= -1\otimes (\sum_{i=1}^{p} F^1 \otimes \cdots \otimes Z \trt F^i \otimes \cdots \otimes F^p)
\end{align*}
while $b_{\cR}$ has a simple expression,
\begin{align}
\begin{aligned}
b_{\cR} ( \alpha \otimes F^0 \otimes \dots \otimes F^p)= \sum_{i=0}^{p+1} (-1)^i \alpha \otimes F^0 \otimes \dots \otimes F^{i-1} \otimes 1 \otimes F^{i} \otimes
\dots \otimes F^{p}.
\end{aligned}
\end{align}
$B_{\cR}$ also has a simple expression,
\begin{align}
\begin{aligned}
B_{\cR} =(\sum_{i=0}^{p-1}(-1)^{(p-1)i}\tau_{\cR}^{i})\sigma_{\cR}\tau_{\cR} (1-(-1)^{p} \tau_{\cR}) ,
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
& \tau_{\cR}( \alpha \otimes F^0 \otimes \cdots \otimes F^p)=\alpha \otimes F^1 \otimes \cdots \otimes F^p \otimes F^0\\
& \sigma_{\cR}( \alpha \otimes F^0 \otimes \cdots \otimes F^p)=\alpha \otimes F^0 \otimes \cdots \otimes F^{p-1}F^p.
\end{aligned}
\end{align}
step 3
Usually we will look at the complex of totally antisymmetric cochains $C^{p,q}_{\cR}(\wedge \Fg_1^{\ast},\wedge \cR(G_2)):=(\wedge^{q} \Fg_1^{\ast}\otimes \wedge^{p+1} \cR(G_2))^{\cR(G_2)}$
It is quasi-isomorphic to $C^{p,q}_{\cR}(\wedge \Fg_1^{\ast},\otimes \cR(G_2))$ via the antisymmetrization map $\alpha_{\cR}$
\begin{align*}
\alpha_{\cR} (\alpha \otimes F^0 \wedge \cdots \wedge F^p)=
\frac{1}{(p+1)!} \sum_{\sigma \in S_{p+1}} (-1)^{\sigma} \alpha \otimes F^{\sigma(0)} \otimes \cdots \otimes F^{\sigma(p)}.
\end{align*}
The quasi-isomorphism is the same as in the case of group cohomology. We refer to <cit.>.
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \vdots & \vdots & \vdots \\
\wedge^{2} \Fg_1^{\ast} & (\wedge^{2} \Fg_1^{\ast}\otimes \wedge^{2} \cR)^{\cR} & (\wedge^{2} \Fg_1^{\ast}\otimes \wedge^{3} \cR)^{\cR} & \cdots \\
\Fg_1^{\ast} & (\wedge^{1} \Fg_1^{\ast}\otimes \wedge^{2} \cR)^{\cR} & (\wedge^{1} \Fg_1^{\ast}\otimes \wedge^{3} \cR)^{\cR} & \cdots \\
\bC & (\bC \otimes \wedge^{2} \cR)^{\cR} & (\bC \otimes \wedge^{3} \cR)^{\cR} & \cdots \\ };
\path[transform canvas={yshift=0.6ex},->,font=\scriptsize]
(m-2-1) edge node[above] {$ b_{\cR} $} (m-2-2) (m-2-2) edge node[above] {$ b_{\cR} $} (m-2-3) (m-2-3) edge node[above] {$ b_{\cR} $} (m-2-4)
(m-3-1) edge node[above] {$ b_{\cR} $} (m-3-2) (m-3-2) edge node[above] {$ b_{\cR} $} (m-3-3) (m-3-3) edge node[above] {$ b_{\cR} $} (m-3-4)
(m-4-1) edge node[above] {$ b_{\cR} $} (m-4-2) (m-4-2) edge node[above] {$ b_{\cR} $} (m-4-3) (m-4-3) edge node[above] {$ b_{\cR} $} (m-4-4) ;
\path[transform canvas={xshift=0ex},->,font=\scriptsize]
(m-4-1) edge node[left]{$\partial_{\Fg}$} (m-3-1) (m-4-2) edge node[left]{$\partial_{\Fg}$} (m-3-2) (m-4-3) edge node[left]{$\partial_{\Fg}$} (m-3-3)
(m-3-1) edge node[left]{$\partial_{\Fg}$} (m-2-1) (m-3-2) edge node[left]{$\partial_{\Fg}$} (m-2-2) (m-3-3) edge node[left]{$\partial_{\Fg}$} (m-2-3)
(m-2-1) edge node[left]{$\partial_{\Fg}$} (m-1-1) (m-2-2) edge node[left]{$\partial_{\Fg}$} (m-1-2) (m-2-3) edge node[left]{$\partial_{\Fg}$} (m-1-3) ;
\end{tikzpicture}
\end{equation*}
The boundary operators have simple expression when restricted to such complex.
\begin{align}
\begin{aligned}
&b_{\cR} ( \alpha \otimes F^0 \wedge \dots \wedge F^p)= \alpha \otimes 1 \wedge F^0 \wedge \dots \wedge F^p, \\
&\partial_{\Fg} (\alpha \otimes F^0 \wedge \dots \wedge F^p)=\partial \alpha \otimes F^0 \wedge \dots \wedge F^p \\
&\qquad \qquad \qquad \qquad \qquad -\sum_{i,j}^{}\omega^i \wedge \alpha \otimes F^0 \wedge \dots \wedge Z_i \trt F^j \wedge \dots \wedge F^p,
\end{aligned}
\end{align}
and because
\begin{align*}
\tau_{\cR}( \alpha \otimes F^0 \wedge \dots \wedge F^p)= \alpha \otimes F^1 \wedge \dots \wedge F^p \wedge F^0=(-1)^{p} \alpha \otimes F^0 \wedge \dots \wedge F^p,
\end{align*}
we have
\begin{equation*}
(1-(-1)^{p} \tau_{\cR})( \alpha \otimes F^0 \wedge \dots \wedge F^p)= 0,
\end{equation*}
\begin{equation*}
B_{\cR}( \alpha \otimes F^0 \wedge \dots \wedge F^p)= 0.
\end{equation*}
We notice that $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$ is quasi-isomorphic to $C^{p,q}_{\cR}(\wedge \Fg_1^{\ast},\wedge \cR(G_2))$ via the map:
\begin{equation*}
\begin{aligned}
\cJ (\alpha \otimes F^0 \wedge \cdots \wedge F^p) (\psi_0, \dots, \psi_p)=
\frac{1}{(p+1)!} \sum_{\sigma \in S_{p+1}} (-1)^{\sigma} \alpha F^{\sigma(0)} (\psi_0) \cdots F^{\sigma(p)} (\psi_p).
\end{aligned}
\end{equation*}
\begin{equation}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \vdots & \vdots & \vdots \\
\wedge^{2} \Fg_1^{\ast} & C^{1}_{\cR}(G_2,\wedge ^{2} \Fg_1^{\ast}) & C^{2}_{\cR}(G_2,\wedge ^{2} \Fg_1^{\ast}) & \cdots \\
\Fg_1^{\ast} & C^{1}_{\cR}(G_2,\wedge ^{1} \Fg_1^{\ast}) & C^{2}_{\cR}(G_2,\wedge ^{1} \Fg_1^{\ast}) & \cdots \\
\bC & C^{1}_{\cR}(G_2,\bC) & C^{2}_{\cR}(G_2,\bC) & \cdots \\ };
\path[transform canvas={yshift=0.6ex},->,font=\scriptsize]
(m-2-1) edge node[above] {$ b_{\cR} $} (m-2-2) (m-2-2) edge node[above] {$ b_{\cR} $} (m-2-3) (m-2-3) edge node[above] {$ b_{\cR} $} (m-2-4)
(m-3-1) edge node[above] {$ b_{\cR} $} (m-3-2) (m-3-2) edge node[above] {$ b_{\cR} $} (m-3-3) (m-3-3) edge node[above] {$ b_{\cR} $} (m-3-4)
(m-4-1) edge node[above] {$ b_{\cR} $} (m-4-2) (m-4-2) edge node[above] {$ b_{\cR} $} (m-4-3) (m-4-3) edge node[above] {$ b_{\cR} $} (m-4-4) ;
\path[transform canvas={xshift=0ex},->,font=\scriptsize]
(m-4-1) edge node[left]{$\partial_{\Fg}$} (m-3-1) (m-4-2) edge node[left]{$\partial_{\Fg}$} (m-3-2) (m-4-3) edge node[left]{$\partial_{\Fg}$} (m-3-3)
(m-3-1) edge node[left]{$\partial_{\Fg}$} (m-2-1) (m-3-2) edge node[left]{$\partial_{\Fg}$} (m-2-2) (m-3-3) edge node[left]{$\partial_{\Fg}$} (m-2-3)
(m-2-1) edge node[left]{$\partial_{\Fg}$} (m-1-1) (m-2-2) edge node[left]{$\partial_{\Fg}$} (m-1-2) (m-2-3) edge node[left]{$\partial_{\Fg}$} (m-1-3) ;
\end{tikzpicture}
\end{equation}
The boundary operators are
\begin{align*}
& b_{\cR} (c)( \psi_0, \dots, \psi_{p+1})= \sum_{i=0}^{p+1}(-1)^i c (\psi_0, \dots, \hat{\psi_{i}}, \dots, \psi_{p+1}) \\
&\partial_{\Fg} (c)(\psi_0, \dots, \psi_{p})= \partial (c( \psi_0, \dots, \psi_{p})) - \sum_{i}^{}\omega^i \wedge ( Z_i \trt c ) (\psi_0, \dots, \psi_{p}).
\end{align*}
step 4
we define a map $ \Theta:C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}) \to D^{p,q}$
\begin{align}\label{thetamap}
\Theta(\alpha)(\psi_0,\dots, \psi_p) \vert _{\vp} =L_{\vp}^{\ast}(\alpha) (\psi_0 \tlt \vp,\dots,\psi_{p}\tlt \vp)
\end{align}
$\Theta$ is an isomorphism between complexes.
use the strong covariance property of $D^{p,q}$, cochains in $D^{p,q}$ depend only on its value at $e_1 \in G_1$.
§.§ Construction from Lie algebra cohomology complex to Dpq
In this section, we would like to carry out a parallel construction of a chain map from Lie algebra cohomology complex of the pair $(\Fg_1\bowtie \Fg_2 , \Fh_2)$ to the complex $D^{p,q}$, comparing with the infinite dimensional case as in <cit.>, and <cit.>.
Assume $H_2$ is invariant under $G_1$ action, we can have $H_2 \backslash(G_1\bowtie G_2) \cong G_1 \bowtie (H_2 \backslash G_2)$ and $\Fh_2 \backslash (\Fg_1\bowtie \Fg_2 ) \cong \Fg_1 \bowtie ( \Fh_2 \backslash\Fg_2 )$. We can also write down right $\Fh_2$-action on $\Fh_2 \backslash (\Fg_1\bowtie \Fg_2 ) \cong \Fg_1 \bowtie ( \Fh_2 \backslash\Fg_2 )$ by the induced adjoint action $(-\trt) \oplus ad$. In order to have a linear action we need to assume that the action $\trt$ of $\Fh_2$ on $\Fg_1$ is given by derivations, this is the case when our matched pair of Lie groups are from decomposition of Lie group. Under above assumptions we can talk about $\Fh_{2}-$basic forms on the bicrossed product group $G_1 \bowtie G_2$.
For $\psi_0,\dots,\psi_{p} \in G_2$, we let $\Delta(\psi_0,\dots,\psi_{p})$ be the affine simplex with vertices $\pi_{L}(\psi_i)$ in the affine coordinates on $L_2$. i.e.,
\begin{align*}
\Delta(\psi_0,\dots,\psi_{p})=\exp \Big(\sum_{0}^{p} t_i \log \big(\pi_{L}(\psi_i)\big)\Big); \qquad 0 \le t_i \le 1,\sum_{0}^{p} t_i=1
\end{align*}
where $\pi_L$ is the projection $G_2 \to H_2 \backslash G_2$.
We check that the right multiplication of $G_2$ is affine on the exponential coordinates of $L_2$, i.e., assume $\psi=hl$,
\begin{align}
\begin{aligned}
\Delta(\psi_0 \psi,\dots,\psi_{p} \psi)=&\exp \Big(\sum_{0}^{p} t_i \log \big(\pi_{L}(\psi_i \psi)\big)\Big)\\
=&\exp \Big(\sum_{0}^{p} t_i \log \big((\pi_{L}(\psi_i) \tlt h )l)\big)\Big)\\
=&( \Delta(\psi_0,\dots,\psi_{p}) \tlt h) l
\end{aligned}
\end{align}
where we identify $L_2$ with $H_2 \backslash G_2$ to make sense of the right $G_2$ action.
We assume that the right $G_1$ action is affine on the exponential coordinates, i.e.,
\begin{align}
\begin{aligned}
\Delta(\psi_0 \tlt \vp,\dots,\psi_{p} \tlt \vp)=&\exp \Big(\sum_{0}^{p} t_i \log \big(\pi_{L}(\psi_i \tlt \vp)\big)\Big)\\
=&\exp \Big(\sum_{0}^{p} t_i \log \big(\pi_{L}(\psi_i)\tlt \vp)\big)\Big)\\
=& \Delta(\psi_0,\dots,\psi_{p}) \tlt \vp
\end{aligned}
\end{align}
We define $\mathcal{D}$ as discussed in <cit.> and <cit.>.
\begin{align}
&\mathcal{D} :(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}} \to C^{p,q} \\
&\langle \mathcal{D} (\omega)(\psi_0,\dots,\psi_{p}), \eta \rangle = (-1)^{\frac{q(q+1)}{2}} \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} \pi_1^{\ast} \eta \wedge \tilde{\omega}
\end{align}
The group cochain $\mathcal{D} (\omega)$ satisfies the strong covariance property:
\begin{align}
\begin{aligned}
\mathcal{D} (\omega) (\psi_0 \tlt \phi,\dots,\psi_{p} \tlt \phi ) =(\phi^{-1} \trt )^{\ast} \mathcal{D} (\omega)
\end{aligned}
\end{align}
where $\phi=\vp\psi \in G_1G_2=G$.
\begin{align*}
\langle \mathcal{D} (\omega)(\psi_0\phi,\dots,\psi_{p}\phi), \eta \rangle &=\int_{(G_1 \times \Delta (\psi_0\phi,\dots,\psi_{p}\phi))^{-1}} \pi_1^{\ast} \eta \wedge \tilde{\omega}\\
&=\int_{L_{\phi^{-1}}(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} \pi_1^{\ast} \eta \wedge \tilde{\omega}\\
&=\int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} (\widetilde{\phi^{}})^{\ast} (\pi_1^{\ast} \eta \wedge \tilde{\omega})\\
&=\int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} \pi_1^{\ast} ((\phi \trt)^{\ast} \eta) \wedge \tilde{\omega}\\
&=\langle \mathcal{D} (\omega)(\psi_0,\dots,\psi_{p}), (\phi \trt)^{\ast} \eta \rangle \\
&=\langle (\phi^{-1} \trt)^{\ast} \mathcal{D} (\omega)(\psi_0,\dots,\psi_{p}), \eta \rangle
\end{align*}
$\mathcal{D} $ is a map of complexes
The covariance property is checked above. Now we check
\begin{align}
\begin{aligned}
\mathcal{D} (d\omega) =(d_1+d_2) \mathcal{D} (\omega)
\end{aligned}
\end{align}
we have
\begin{align}
\begin{aligned}
&\langle \mathcal{D} (d\omega)(\psi_0,\dots,\psi_{p}), \eta \rangle \\
=& (-1)^{\frac{q(q+1)}{2}} \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} \pi_1^{\ast} \eta \wedge d\tilde{\omega} \\
=& (-1)^{\frac{q(q+1)}{2}} (-1)^{q} ( \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} d(\pi_1^{\ast} \eta \wedge \tilde{\omega} ) \\
& - \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} \pi_1^{\ast} (d\eta) \wedge \tilde{\omega} ) \\
=& (-1)^{q} (-1)^{\frac{q(q+1)}{2}} ( \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} d(\pi_1^{\ast} \eta \wedge \tilde{\omega} ) ) \\
& +(-1)^{\frac{(q+1)(q+2)}{2}} ( \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} \pi_1^{\ast} (d\eta) \wedge \tilde{\omega} ) \\
=& (-1)^{q} (-1)^{\frac{q(q+1)}{2}} ( \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} d(\pi_1^{\ast} \eta \wedge \tilde{\omega} ) ) + \langle d_2 \mathcal{D} (\omega), \eta \rangle.
\end{aligned}
\end{align}
Apply Stokes, the first integral becomes the group coboundary:
\begin{align}
\begin{aligned}
&(-1)^{q} (-1)^{\frac{q(q+1)}{2}} ( \int_{(G_1 \times \Delta (\psi_0,\dots,\psi_{p}))^{-1}} d(\pi_1^{\ast} \eta \wedge \tilde{\omega} ) ) \\
=& (-1)^{q} (-1)^{\frac{q(q+1)}{2}} (\sum_{i=0}^{p+1} (-1)^{i} \int_{(G_1 \times \Delta (\psi_0,\dots,\hat{\psi_{i}},\dots, \psi_{p}))^{-1}} \pi_1^{\ast} \eta \wedge \tilde{\omega} ) \\
=&\sum_{i=0}^{p+1} (-1)^{q+i} \langle \mathcal{D} (\omega)(\psi_0,\dots,\hat{\psi_{i}},\dots, \psi_{p}), \eta \rangle \\
=&\langle d_1 \mathcal{D} (\omega) ,\eta \rangle.
\end{aligned}
\end{align}
Because of the strong covariance property, $D$ is completely determined by its value at $e_1 \in G_1$, consider group cochains with values in $\wedge ^{\bullet} \Fg_{1}^{\ast}$:
\begin{align}
\begin{aligned}
\mathcal{E}(\omega)(\psi_0,\dots,\psi_{p}):=\mathcal{D}(\omega)(\psi_0,\dots,\psi_{p}) \vert _{e_1}.
\end{aligned}
\end{align}
$\mathcal{E}$ is a chain map from $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$ to $C^{p}(G_2,\wedge ^{q} \Fg_1^{\ast}) $
The covariance property is preserved by the evaluation.
Use Prop. <ref>, the evaluation of the second integral in its proof at $e_1$ is clearly group coboundary, while the first integral, when evaluate at $e_1$, gives Lie algebra coboundary with coefficients.
As discussed in <cit.> we can also verify the explicit description of $\mathcal{E}$:
Fix a basis $Z_i$ of $\Fg_1$, denote by $\tilde{Z_i}$ the corresponding left invariant vector fields, $\omega_i$ the dual basis, $\tilde{\omega}_i$ the corresponding left invariant forms fields. Use the formula in <cit.>, we define $\nu(\vp,\psi)=\vp(\psi \tlt \vp)^{-1}$. Let $\imath:L_2 \to L_2$ be the inversion map, $\iota$ be the contraction. Let $\omega$ be in $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$ and $\tilde{\omega}$ the corresponding left invariant $\Fh_2-$basic form on $(G_1\bowtie G_2)/H_2$. It is the key that we can have explicit expression of such invariant form in order to make use of this explicit $\mathcal{E}$ map. Denote
\begin{align}
\begin{aligned}\label{mumap}
&\mu : (\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}} \to \sum_{p+q=n} (\Omega^{p}(L_2) )^{\Fh_{2}} \otimes \wedge^{q} \Fg_1^{\ast}\\
&\mu_q(\omega)=\sum_{|I|=q} \imath^\ast(\iota_{\tilde{Z}_{I}(e)}\nu^\ast(\tilde{\omega})) \otimes \omega_{I}, \\
& I=(i_1 < \dots < i_q)\quad \text{and} \quad \omega_{I}=\omega_{i_1}\wedge \cdots \wedge \omega_{i_q}
\end{aligned}
\end{align}
Use the formula in <cit.> We then write a map $\mathcal{E}=\int\limits_{\Delta} \circ \mu $ from $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$ to $ \sum_{p+q=n} C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$:
\begin{align} \label{mapE}
\mathcal{E} (\omega) (\psi_0,\dots,\psi_{p})= \int\limits_{\Delta(\psi_0,\dots,\psi_{p})} \mu_q(\omega)
\end{align}
one can also compare this map with the integration along simplex map as used by Dupont in <cit.>.
Then we can have
\begin{align}
\mathcal{E} (\omega) (\psi_0,\dots,\psi_{p})= \mathcal{D} (\omega) (\psi_0,\dots,\psi_{p}) \vert _{e_1}
\end{align}
$\mathcal{E} (\omega)$ are representative functions of $\psi_i$s.
Because $L_2=\exp(\Fl_2)$ is nilpotent, representative functions are exactly the polynomial functions. Use exponential coordinates on $L_2$ and we can represent the projection of $ \nu^\ast(\tilde{\omega})$ on $L_2$ as forms with polynomial coefficients. Now apply the contractions $\iota_{\tilde{Z}_{e}^{I}}$, the only new type of coefficients that appear are of the form $Z_I \trt P$ where $P$ is some polynomial function of $L_2$. By Lemma <ref> $Z_I \trt P$ is again representative hence polynomial. Therefore $ \imath^\ast(\iota_{\tilde{Z}_{I}(e)}\nu^\ast(\tilde{\omega})) $ are polynomial differential forms on $L_2$ and $\mathcal{E} (\omega)$ are representative functions on $L_2$.
Furthermore they are $H_2$ invariant, every $H_2$ invariant $L_2$ representative function is representative on $G_2$ because of the semi direct product decomposition $G_2=H_2 \ltimes L_2$. More precisely if we extend any $F \in \cR(L_2)$ trivially to a function on $G_2$ by $F(hl)=F(l)$, then
\begin{align*}
F \big( (h_1l_1)(h_2l_2) \big)&=F \big( (h_1h_2)(l_1 \tlt h_2 l_2 ) \big) =F(l_1 \tlt h_2 l_2 )=F_{(1)}(l_1 \tlt h_2) F_{(2)}(l_2) \\
&=F_{(1)}( l_1) F_{(2)}(l_2) =F_{(1)}( h_1l_1) F_{(2)}(h_2l_2),
\end{align*}
hence $F \in \cR(G_2) $ and $\pi_L^{\ast}$ preserves representative functions.
Therefore $\mathcal{E} (\omega)$ are representative functions on $G_2$.
$\mathcal{E}$ is a chain map from $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$ to $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}) $
Use previous proposition and corollary <ref>.
$\mathcal{D}$ is a chain map from $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$ to $D^{p,q} \subset C^{p,q}$
The map $\mathcal{E}$ can be extended by the isomorphism $\Theta:C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}) \to D^{p,q}$ defined in <ref>
\begin{align*}
\Theta(\alpha)(\psi_0,\dots, \psi_p) \vert _{\vp} =L_{\vp}^{\ast}(\alpha) (\psi_0 \tlt \vp,\dots,\psi_{p}\tlt \vp)
\end{align*}
back to map $\mathcal{D}=\Theta \circ \mathcal{E}$:
\begin{align*}
\mathcal{D}(\omega) (\psi_0,\dots,\psi_{p})\vert_{\vp}=L_{\vp}^{\ast}(\mathcal{E} (\omega)) (\psi_0 \tlt \vp,\dots,\psi_{p}\tlt \vp)
\end{align*}
We can also have the identification below in this case:
$(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}} \cong \bigoplus_{p+q=n}(\wedge^{q} \Fg_1^{\ast}\otimes \wedge^{p} (\Fl_2^{\ast})^{\Fh_2})$
follow the proof in <cit.> the isomorphism is given by
\begin{align}
\begin{aligned}
&\natural: (\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}} \to \bigoplus_{p+q=n}(\wedge^{q} \Fg_1^{\ast}\otimes \wedge^{p} (\Fl_2^{\ast})^{\Fh_2})\\
&\natural (\omega)(X^1,\dots, X^q \vert \xi_1,\dots \xi_p)= \omega (X^1\oplus 0, \dots , X^q \oplus 0, 0 \oplus \xi_1, \dots , 0 \oplus \xi_p)
\end{aligned}
\end{align}
with its inverse
\begin{align}
\begin{aligned}
&\natural^{-1} (\mu \otimes \nu)(X^1 \oplus \xi_1 , \dots , X^{p+q} \oplus \xi_{p+q})\\
=&\sum_{\sigma \in Sh(p,q)}(-1)^{\sigma} \mu (X^{\sigma(1)},\dots,X^{\sigma(p)}) \nu (\xi _{\sigma(p+1)},\dots, \xi _{\sigma(p+q)} )
\end{aligned}
\end{align}
Use a slight modification of <cit.> we can show that $\mu \vert _{e_2} $ coincides with $\natural$.
For $\xi_1,\dots \xi_p \in \Fl_2$, denote
\begin{equation*}
\begin{aligned}
&\Psi(\xi_1,\dots \xi_p)=\\
\end{aligned}
\end{equation*}
Define a map $j:C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}) \to \wedge^{q} \Fg_1^{\ast}\otimes \wedge^{p} (\Fl_2^{\ast})^{\Fh_2}$
\begin{align*}
&j(\alpha)(X^1,\dots, X^q \vert \xi_1,\dots \xi_p) \\
=&\sum_{\sigma \in S_{p}} (-1)^{\sigma} \left.\frac{d}{ds_1}\right|_{_{s_1=0}} \cdots \left.\frac{d}{ds_p}\right|_{_{s_p=0}} \alpha(\Psi(\xi_1,\dots \xi_p))(X^1,\dots, X^q)
\end{align*}
then The map $\mathcal{E}$ has a quasi-inverse $ \natural ^{-1}\circ j$.
$( \natural ^{-1}\circ j) \circ \mathcal{E} $ is identity.
since $\mu \vert _{e_2}$ implement the $\natural$ map, we just need to show that $j\circ \int\limits_{\Delta}$ is the evaluation at $e_2$ .
we write $\xi_{k}=\xi_k^{j}\frac{\partial}{\partial x_{j}}$ and
write any $p$ form as $f_{i_1\dots i_p} dx^{i_1} \wedge \cdots \wedge dx^{i_p}$ in the exponential coordinates.
\begin{equation*}
\begin{aligned}
x^{i_k}=&t_1 ( s_{\sigma(1)} \xi_{\sigma(1)}^{i_k}) +t_2 (s_{\sigma(1)}\xi_{\sigma(1)}^{i_k}+s_{\sigma(2)}\xi_{\sigma(2)}^{i_k} +p_2(s_{\sigma(1)},s_{\sigma(2)})) +\cdots \\
&+ t_k (s_{\sigma(1)}\xi_{\sigma(1)}^{i_k}+ \cdots + s_{\sigma(k)}\xi_{\sigma(k)}^{i_k} +p_{\sigma(k)}(s_{\sigma(1)},s_{\sigma(2)},\dots s_{\sigma(k)})) , \quad 1 \le k \le p
\end{aligned}
\end{equation*}
where $p_2,\dots, p_k$ are $\xi-$valued polynomials with degree higher than $2$.
\begin{equation*}
\begin{aligned}
d(x^{i_k})=& ( s_{\sigma(1)} \xi_{\sigma(1)}^{i_k}) d t_1 + (s_{\sigma(1)}\xi_{\sigma(1)}^{i_k}+s_{\sigma(2)}\xi_{\sigma(2)}^{i_k} +p_2(s_{\sigma(1)},s_{\sigma(2)})) d t_2 +\cdots \\
&+ (s_{\sigma(1)}\xi_{\sigma(1)}^{i_k}+ \cdots + s_{\sigma(k)}\xi_{\sigma(k)}^{i_k} +p_{\sigma(k)}(s_{\sigma(1)},s_{\sigma(2)},\dots s_{\sigma(k)})) d t_k , \quad 1 \le k \le p
\end{aligned}
\end{equation*}
therefore if we calculate the wedge product of $dx^{i_1} \wedge \cdots \wedge dx^{i_p}$, its degree will be at least $p$, viewed as $\xi dt^{1} \wedge \cdots \wedge dt^{p}-$valued polynomials in $s_j$. Since we want to take derivative from $s_1$ to $s_p$, there is exact one term that gives non-zero value:
\begin{align*}
&\sum_{\sigma \in S_{p}} (-1)^{\sigma} \left.\frac{d}{ds_1}\right|_{_{s_1=0}} \cdots \left.\frac{d}{ds_p}\right|_{_{s_p=0}} \\
&\int\limits_{\Delta(\Psi(\xi_1,\dots \xi_p))} \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} dx^{i_1} \wedge \cdots \wedge dx^{i_p} \\
=&\sum_{\sigma \in S_{p}} (-1)^{\sigma} \left.\frac{d}{ds_1}\right|_{_{s_1=0}} \cdots \left.\frac{d}{ds_p}\right|_{_{s_p=0}} \\
&\int\limits_{\Delta^p} \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} s_{\sigma(1)} \xi_{\sigma(1)}^{i_1}dt_1\wedge \cdots \wedge s_{\sigma(p)}\xi_{\sigma(p)}^{i_p} dt_{p} \\
=&\sum_{\sigma \in S_{p}} (-1)^{\sigma} \int\limits_{\Delta^p} \sum_{i_1 < \dots < i_p} f_{i_1\dots i_p} (e_2) \xi_{\sigma(1)}^{i_1}(e_2)dt_1\wedge \cdots \wedge \xi_{\sigma(p)}^{i_p} (e_2) dt_{p} \\
=&\sum_{\sigma \in S_{p}} (-1)^{\sigma} \frac{1}{p! } \sum_{i_1 < \dots < i_p}( f_{i_1\dots i_p} \xi_{\sigma(1)}^{i_1} \cdots \xi_{\sigma(p)}^{i_p} )(e_2) \\
=&(f_{i_1\dots i_p} dx^{i_1} \wedge \cdots \wedge dx^{i_p})(\xi_1^{j}\frac{\partial}{\partial x_{j}}, \dots , \xi_p^{j}\frac{\partial}{\partial x_{j}}) \vert_{e_2}
\end{align*}
$\mathcal{E}$ is a quasi-isomorphism.
In <cit.> the authors showed that $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$ is quasi-isomorphic to the double complex $C^{q,p}(\wedge \Fg_1^{\ast},\otimes \cR(G_2))$ (<ref>). The later is also quasi-isomorphic to $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$. Therefore $\mathcal{E}$ as a chain map with one side inverse between two quasi-isomorphic complexes is a quasi-isomorphism.
we can also give a direct proof here to show that $C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast})$ is quasi-isomorphic to $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$.
we will take a look at the triple complex $C^{s}_{\cR}(G_2,\Omega_{L_2}^{t} \otimes \wedge ^{q} \Fg_1^{\ast})$. During the first step we will fix $q$ and vary $s$, $t$ and view it as a double complex for each fixed $q$.
we put an augmented row
\begin{align*}
C^{s}_{\cR}(G_2,\bC \otimes \wedge ^{q} \Fg_1^{\ast})
\end{align*}
there is an inclusion
\begin{align*}
0 \to C^{s}_{\cR}(G_2,\bC \otimes \wedge ^{q} \Fg_1^{\ast}) \xrightarrow{\varepsilon_1} C^{s}_{\cR}(G_2,\Omega_{L_2}^{0} \otimes \wedge ^{q} \Fg_1^{\ast}) \to C^{s}_{\cR}(G_2,\Omega_{L_2}^{1} \otimes \wedge ^{q} \Fg_1^{\ast}) \to \cdots
\end{align*}
and an augmented column
\begin{align*}
(\Omega_{L_2}^{t})^{G_2} \otimes \wedge ^{q} \Fg_1^{\ast}
\end{align*}
there is another inclusion
\begin{align*}
0 \to (\Omega_{L_2}^{t})^{G_2} \otimes \wedge ^{q} \Fg_1^{\ast} \xrightarrow{\varepsilon_2} C^{0}_{\cR}(G_2,\Omega_{L_2}^{t} \otimes \wedge ^{q} \Fg_1^{\ast}) \to C^{1}_{\cR}(G_2,\Omega_{L_2}^{t} \otimes \wedge ^{q} \Fg_1^{\ast}) \to \cdots
\end{align*}
the columns are exact for $s \ge 0$ since $L$ is smoothly contractible.
the rows are exact for $t \ge 0$ because of the homotopy below (cf., <cit.>):
\begin{align*}
&H:\ C^{s}_{\cR}(G_2,\Omega_{L_2}^{t} \otimes \wedge ^{q} \Fg_1^{\ast}) \to \ C^{s-1}_{\cR}(G_2,\Omega_{L_2}^{t} \otimes \wedge ^{q} \Fg_1^{\ast}) \\
\end{align*}
\begin{align*}
f_{(\psi_1,\dots,\psi_{p})}^c (h):=c(hl,\psi_1,\dots,\psi_{p}) (l) \qquad l\,\in L_2, h\,\in H_2
\end{align*}
and $\pi$ the projection to its $H_2$ invariant part, since $H_2$ is reductive and any $H_2$ module is completely reducible.
By the standard diagram chasing of double complex we have that $\varepsilon_1$ and $\varepsilon_2$ both induces isomorphisms to the cohomology of the double complex. Therefore these augmented complexes are quasi-isomorphic, while the augmented column is isomorphic to the complex $(\wedge^{n} (\Fg_{1} \oplus \Fl_{2})^{\ast})^{\Fh_{2}}$.
§.§ Main theorem
The map $\Phi_{D} \circ \Theta$ from $(\sum_{p+q=n}C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}),d_1)$ to the Hochschild complex of $\cH$ is a quasi-isomorphism.
Recall from Corollary <ref>, we shall identify $\cH^{\natural}_{(\delta,\sigma^{})}$ and $\text{Im}(\lambda^{\natural})$. In order to prove the statement we need to understand the Hochschild cohomology of the algebra $\cA$ with coefficients in the module $\bC$ given by the augmentation $\varepsilon$ on $\cA$.
We use the abstract version of the Hochschild-Serre spectral sequence.
The following is from <cit.>
A subalgebra $\cA_1 \subset \cA$ of an augmented algebra $\cA$ is called normal if the left ideal $J$ generated by $\text{Ker}\, \varepsilon_1$, where $\varepsilon_1=\varepsilon \vert _{\cA_1}$, is also a right ideal.
we let
\begin{align}
\cA_{2}=\cA / J,\qquad \text{or equivalently}\quad \cA_{2}=\bC \otimes_{\cA_1} \cA,
\end{align}
and then we have a spectral sequence converging to the Hochschild cohomology $H^{\ast}_{\cA}(\bC)$ of augmented algebra $\cA$ with coefficients in $\bC$ and with $E_2$ term given by
\begin{align*}
\end{align*}
In our case we let $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$ and $\cA_1=C_c^{\ify}(G_1)$, with augmentation
\begin{align*}
\varepsilon(fU^{\ast}_{\psi})=f(1)\qquad \forall f \in C_c^{\ify}(G_1),\ \psi \in G_2.
\end{align*}
Thus the ideal $J$ is generated by
\begin{align*}
g U^{\ast}_{\psi},\qquad g(1)=0.
\end{align*}
It is a two sided ideal because $\forall \psi \in G_2$, $\tilde{\psi}(1)=1$ and
\begin{align*}
g(1)=0\qquad\text{iff}\quad (g \circ \tilde{\psi})(1)=0
\end{align*}
therefore the algebra $\cA_{2}=\bC \otimes_{\cA_1} \cA$ is the group ring of $G_2$.
Now since the original version of Hochschild-Serre spectral sequence is for unital algebra, in order to apply the spectral sequence we need one more preparation, i.e., we would like to extend the result to our algebra.
A algebra $\cA$ is said to have local units if for every finite family of elements $a_i \in \cA $ there is an element $u \in \cA$ such that $ua_i=a_iu=a_i$ for all $i$.
If an algebra has local units, it is excisive for Hochschild cohomology. <cit.><cit.>.
Now we let $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$ and $\cA_1=C_c^{\ify}(G_1)$. It is obvious that both our algebras $\cA_1$ and $\cA$ have local units (just take $\text{Id} \vert _{\bigcup \text{supp}(f_i)} U^{\ast}_{e}$ and $\text{Id} \vert _{\bigcup \text{supp}(f_i)}$ ). Hence we add a unit to $\cA_1$ as usual:
\begin{align*}
0 \to \bC \to \tilde{\cA_1} \to \cA_1\to 0
\end{align*}
where $\tilde{\cA_1}=\bC \oplus \cA_1$ with multiplication $(\lambda,u)(\mu,v)=(\lambda\mu,\lambda v+u \mu + uv)$ and unit $(1,0)$,
and add a unit to $\cA$:
\begin{align*}
0 \to \bC \cA_2 \to \tilde{\cA} \to \cA\to 0
\end{align*}
where $\tilde{\cA}=\bC \cA_2 \oplus \cA$ with multiplication $(\lambda,u)(\mu,v)=(\lambda\mu,\lambda v+u \mu + uv)$ and unit $(1e,0)$.
Thus we can have a diagram as below:
\begin{equation*}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \dots & H^{p}_{\cA_2}(H^{q}_{\cA_1}(\bC)) & H^{p}_{\cA_2}(H^{q}_{\tilde{\cA_1}}(\bC)) & H^{p}_{\cA_2}(H^{q}_{\bC}(\bC)) & \dots \\
\dots & H^{p+q}_{\cA}(\bC) & H^{p+q}_{\tilde{\cA}}(\bC) & H^{p+q}_{\cA_2}(\bC) & \dots \\ };
\path[transform canvas={yshift=0ex},->,font=\scriptsize]
(m-1-1) edge (m-1-2) (m-1-2) edge (m-1-3) (m-1-3) edge (m-1-4) (m-1-4) edge (m-1-5)
(m-2-1) edge (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge (m-2-4) (m-2-4) edge (m-2-5);
\path[transform canvas={xshift=0ex},->,font=\scriptsize]
(m-1-2) edge (m-2-2) (m-1-3) edge (m-2-3) (m-1-4) edge (m-2-4) ;
\end{tikzpicture}
\end{equation*}
the first row is due to the excision property of $\cA_1$ and the second row is due to the excision property of $\cA$. Now we apply the abstract Hochschild-Serre spectral sequence for the middle and right columns and conclude that $H^{p}_{\cA_2}(H^{q}_{\cA_1}(\bC))$ also converge to $H^{p+q}_{\cA}(\bC)$. This proved the Hochschild-Serre spectral sequence for our algebra $\cA$.
Now $\text{Im}(\lambda^{\natural})$ can be teated as $\cA$ with the weak topology given by the range of $\lambda$, we can have a corresponding spectral sequence which converges to the Hochschild cohomology of $\cH$ and whose $E_2$ term is given by the representative group cohomology of $G_2$ with coefficients in $H^{\ast}_{\cA_1}(\bC)$, which is given by $\wedge^{q} \Fg_{1}^{\ast}$. On $E_2$ level, the map $\Phi_{D} \circ \Theta$ realize the representative group cochains of $G_2$ with coefficients in $\wedge^{q} \Fg_{1}^{\ast}$.
We now have the main result:
Let ($G_1$, $G_2$) be a matched pair of Lie groups, and $L_2$ be a nucleus of $G_2$ that is nilpotent. Let $\Fh_2$, $\Fg_1$, $\Fg_2$ and $\Fl_2$ denote the Lie algebras of $H_2:= G_2/L_2$, $G_1$, $G_2$ and $L_2$ respectively. Let us also assume that $H_2$ is $G_1-$invariant, the action $\trt$ of $\Fh_2$ on $\Fg_1$ is given by derivations, and the right $G_1$ action on $L_2$ is affine in its exponential coordinates. Then we have
$ \Phi_{D} \circ \mathcal{D} $ induces an isomorphism from the relative Lie cohomology of the pair $(\Fg_{1} \bowtie \Fg_{2}, \Fh_{2})$ to the periodic cyclic cohomology of $\cH=(\cU(\Fg_1) \acr \cR(G_2))^{\cop}$ with coefficient $^\sigma \bC _ \delta$, shifted by $\text{dim}(G_1)$. We have
\begin{align}
\begin{aligned}
&\bigoplus_{i\,\text{\tiny even} } H^{i} (\Fg_{1} \bowtie \Fg_{2}, \Fh_{2},\Cb) \cong HP^{\text{\tiny even}} (\cH,^\sigma \bC _ \delta) \\
&\bigoplus_{i\,\text{\tiny odd} } H^{i} (\Fg_{1} \bowtie \Fg_{2}, \Fh_{2},\Cb) \cong HP^{\text{\tiny odd}} (\cH,^\sigma \bC _ \delta)
\end{aligned}
\end{align}
if $\text{dim}(G_1)$ is even, and
\begin{align}
\begin{aligned}
&\bigoplus_{i\,\text{\tiny even} } H^{i} (\Fg_{1} \bowtie \Fg_{2}, \Fh_{2},\Cb) \cong HP^{\text{\tiny odd}} (\cH,^\sigma \bC _ \delta) \\
&\bigoplus_{i\,\text{\tiny odd} } H^{i} (\Fg_{1} \bowtie \Fg_{2}, \Fh_{2},\Cb) \cong HP^{\text{\tiny even}} (\cH,^\sigma \bC _ \delta)
\end{aligned}
\end{align}
if $\text{dim}(G_1)$ is odd.
$ \Phi_{D} \circ \Theta $ is a map between complexes $\{ \sum_{}C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}),d_1,d_2 \}$ and $\{CC^{\bullet}(\cH;^\sigma \bC _ \delta), b, B\}$. By Lemma <ref>, The map $ \Phi_{D} \circ \Theta $ induces an isomorphism on Hochschild cohomology, therefore it also induces an isomorphism on cyclic cohomology. By Proposition <ref>, $\mathcal{E}$ induces isomorphism between relative Lie cohomology of the pair $(\Fg_{1} \bowtie \Fg_{2}, \Fh_{2})$ and the cohomology of $(\sum_{p+q=n}C^{p}_{\cR}(G_2,\wedge ^{q} \Fg_1^{\ast}),d_1,d_2)$. Therefore the map $ \Phi_{D} \circ \mathcal{D} = \Phi_{D} \circ \Theta \circ \mathcal{E}$ induces an isomorphism from the relative Lie cohomology of the pair $(\Fg_{1} \bowtie \Fg_{2}, \Fh_{2})$ to the periodic cyclic cohomology of $\cH=(\cU(\Fg_1) \acr \cR(G_2))^{\cop}$ with coefficient $^\sigma \bC _ \delta$.
The shift by the number $\text{dim}(G_1)$ is included in the formulation of $\Phi_{D}$ map.
§ EXAMPLE CONSTRUCTION AND CALCULATION
§.§ Real diamond group D4
The so called real diamond Lie algebra is the 4-dimensional solvable Lie algebra $\Fd$ with basis $T$, $X$, $Y$, $Z$ satisfying the following commutation relations, see <cit.>:
\begin{align*}
\end{align*}
this real diamond Lie algebra $\Fd=\Rb \ltimes \Fh_3$ is an extension of the one-dimensional Lie algebra $\Rb T$ by the Heisenberg algebra $\Fh_3$ with basis $X$, $Y$, $Z$.
We look at the corresponding Lie group $\Rb \ltimes H_{3}$. A group element is represented as $(\theta,x+\mathbf{i}y,z)$, while the group multiplication is given by :
\begin{equation*}
\begin{aligned}
&(\theta_1,x_1+\mathbf{i}y_1,z_1)\cdot(\theta_2,x_2+\mathbf{i}y_2,z_2) =\\
&\big(\theta_1+\theta_2,e^{-\mathbf{i}\theta_2}(x_1+\mathbf{i}y_1)+(x_2+\mathbf{i}y_2),z_1+z_2-\frac{1}{2}Im((x_1+\mathbf{i}y_1) e^{-\mathbf{i}\theta_2} (x_2-\mathbf{i}y_2) )\big)
\end{aligned}
\end{equation*}
and inverse is given by
\begin{align}
\end{align}
when $\theta=0$ we have the usual $H_{3}$. Denote $G_1=(\theta,0,0)$, $G_2=(0,x+\mathbf{i}y,z)$, we have
\begin{align}
(\theta,x+\mathbf{i}y,z)=(\theta,0,0)\cdot(0,x+\mathbf{i}y,z),\qquad \text{i.e.,} \quad G=G_1G_2.
\end{align}
\begin{align}
(0,x+\mathbf{i}y,z_1) \cdot (\theta,0,0)= (\theta, e^{-\mathbf{i}\theta} (x+\mathbf{i}y) , z )=(\theta,0,0)\cdot(0,e^{-\mathbf{i}\theta} (x+\mathbf{i}y),z)
\end{align}
we have
\begin{align}
\begin{aligned}
&G_2 \trt G_1 :(0,x+\mathbf{i}y,z) \trt (\theta,0,0) = (\theta,0,0), \\
&G_2 \tlt G_1 : (0,x+\mathbf{i}y,z) \tlt (\theta,0,0) = (0,e^{-\mathbf{i}\theta} (x+\mathbf{i}y),z)
\end{aligned}
\end{align}
Now we look at the Lie algebra:
\begin{align*}
\Fd \, = \,\{ T,X,Y,Z \}= \,\{ e_1 ,e_2, e_3 ,e_4 \} \qquad
\Fg_2= \, \{ X,Y,Z\} \qquad
\Fg_1= \, \{ T \}.
\end{align*}
with Lie bracket given by
\begin{align*}
\end{align*}
and actions:
\begin{align}
\Fg_2 \trt \Fg_1 : \text{trivial action} \qquad \Fg_2 \tlt \Fg_1: X \tlt T =Y, Y \tlt T =-X.
\end{align}
The cohomology ring of $H^{\ast}( \Fd )$ is generated by $\theta_1$, $\theta_2\wedge \theta_3\wedge \theta_4$ and $\theta_1\wedge \theta_2\wedge \theta_3\wedge \theta_4$: $H^{0}$ is one dimensional, generated by $\{1\}$, $H^{1}$ is one dimensional, generated by $\{\theta_1\}$, $H^{3}$ is one dimensional, generated by $\{ \theta_2\wedge \theta_3\wedge \theta_4 \}$, $H^{4}$ is one dimensional, generated by $\{ \theta_1\wedge \theta_2\wedge \theta_3\wedge \theta_4 \}$.
On $G_1$ we let $\theta_1=d\theta$.
On $G_2$ we use the global exponential coordinate system $\left[ \begin {array}{ccc} 1&x&z+\frac{1}{2}\,xy\\ \noalign{\medskip}0&1&y
\\ \noalign{\medskip}0&0&1\end {array} \right]
$ of $G_2$ to write $\theta_2=dx,\theta_3=dy,\theta_4=\frac{y}{2}dx-\frac{x}{2}dy+dz$.
Let $\vp=(\theta,0,0)$, $\psi=(0,x+\mathbf{i}y,z)$, we calculate
\begin{align}
\nu(\vp,\psi)=\vp(\psi \tlt \vp)^{-1}&=\big(\theta,-e^{-\mathbf{i}\theta}(x+\mathbf{i}y),-z \big) \\
&=\big(\theta,(-\cos\theta x - \sin \theta y)+i(- \cos \theta y + \sin \theta x),-z \big)
\end{align}
Now use <ref> we calculate
\begin{align*}
\nu^\ast(\theta_1)=d\theta, \qquad \text {therefore,}\qquad \mu (\theta_1)=1 \otimes \theta_1
\end{align*}
\begin{align*}
\nu^\ast(\theta_2\wedge \theta_3\wedge \theta_4)=&\nu^\ast(dx\wedge dy \wedge dz) \\
=&d (-\cos\theta x - \sin \theta y) \wedge d (- \cos \theta y + \sin \theta x) \wedge dz\\
=&(\cos^2\theta+\sin^2\theta)dx \wedge dy \wedge dz \\
+ &(\sin^2 \theta x -\sin \theta \cos \theta y+ \cos \theta \sin \theta y + \cos^2 \theta x) d \theta \wedge dx \wedge dz \\
+&(-\sin \theta \cos \theta x + \cos^2 \theta y+ \sin^2 \theta y + \cos \theta \sin \theta x) d \theta \wedge dy \wedge dz\\
=&dx \wedge dy \wedge dz + x d \theta \wedge dx \wedge dz + y d \theta \wedge dy \wedge dz
\end{align*}
\begin{align*}
\mu_0 (\theta_2\wedge \theta_3\wedge \theta_4)=& dx \wedge dy \wedge dz \otimes 1 \\
\mu_1(\theta_2\wedge \theta_3\wedge \theta_4)=&(x dx \wedge dz + y dy \wedge dz )\otimes \theta_1
\end{align*}
\begin{align*}
\nu^\ast(\theta_1 \wedge \theta_2\wedge \theta_3\wedge \theta_4)=&\nu^\ast(d\theta \wedge dx\wedge dy \wedge dz) \\
=&d\theta \wedge d (-\cos\theta x - \sin \theta y) \wedge d (- \cos \theta y + \sin \theta x) \wedge dz\\
=&(\cos^2\theta+\sin^2\theta)d\theta \wedge dx \wedge dy \wedge dz
\end{align*}
\begin{align*}
\mu (\theta_1 \wedge \theta_2\wedge \theta_3\wedge \theta_4)=d \theta \wedge dx \wedge dy \wedge dz \otimes 1 \end{align*}
Therefore the images of $\theta_1$ and $\theta_2\wedge \theta_3\wedge \theta_4$ give two even cocycles and the image of $1$,$\theta_1\wedge \theta_2\wedge \theta_3\wedge \theta_4$ give two odd cocycles. We take $\theta_2\wedge \theta_3\wedge \theta_4$ as an example.
\begin{align*}
\mathcal{E} (\theta_2\wedge \theta_3\wedge \theta_4) (\psi_0,\dots,\psi_{3})=1 \cdot \int\limits_{\Delta(\psi_0,\dots,\psi_{3})} dx \wedge dy \wedge dz
\end{align*}
\begin{align*}
&dx=(x_1-x_0) dt_1 +(x_2-x_0) dt_2 +(x_3-x_0) dt_3, \\
&dy=(y_1-y_0) dt_1 +(y_2-y_0) dt_2 +(y_3-y_0) dt_3, \\
&dz=(z_1-z_0) dt_1 +(z_2-z_0) dt_2 +(z_3-z_0) dt_3.
\end{align*}
therefore the $D^{3,0}$ part of $\theta_2\wedge \theta_3\wedge \theta_4$ is
\begin{align*}
\mathcal{E} (\theta_2\wedge \theta_3\wedge \theta_4) (\psi_0,\dots,\psi_{3})=\frac{1}{2}\sum_{\sigma \in S_3} (-1)^{\sigma} (x_{\sigma(1)}-x_0) (y_{\sigma(2)}-y_0) (z_{\sigma(3)}-z_0)
\end{align*}
Now for the other component we have
\begin{align*}
\mathcal{E} (\theta_2\wedge \theta_3\wedge \theta_4) (\psi_0,\dots,\psi_{2})=\theta_1 \cdot \int\limits_{\Delta(\psi_0,\dots,\psi_{2})} y \cdot dy \wedge dz x \cdot dx \wedge dz
\end{align*}
\begin{align*}
&dx=(x_1-x_0) dt_1 +(x_2-x_0) dt_2, \\
&dy=(y_1-y_0) dt_1 +(y_2-y_0) dt_2, \\
&dz=(z_1-z_0) dt_1 +(z_2-z_0) dt_2.
\end{align*}
therefore the $D^{2,1}$ part of $\theta_2\wedge \theta_3\wedge \theta_4$ is
\begin{align*}
&\mathcal{E} (\theta_2\wedge \theta_3\wedge \theta_4) (\psi_0,\dots,\psi_{2})\\
=&\frac{1}{6}\theta_1 \cdot ((y_0^2+x_0^2)(z_1-z_2)+(y_1^2+x_1^2)(z_2-z_0)+(y_2^2+x_2^2)(z_0-z_1))
\end{align*}
Use $\Phi$ to go direct from these two parts to the representative cochain on algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$, we have $\Phi(\theta_2 \wedge \theta_3 \wedge \theta_4) \in C^{4}(\cA) \cup C^{2}(\cA)$.
As an example, the $C^{2}(\cA)$ part is:
\begin{align*}
&\Phi(\theta_2 \wedge \theta_3 \wedge \theta_4) (f_{0} U^{\ast}_{\psi_{0} },f_{1} U^{\ast}_{\psi_{1} },f_{2} U^{\ast}_{\psi_{2} })\\
=&\big((y_0^2+x_0^2)(z_1-z_2)+(y_1^2+x_1^2)(z_2-z_0)+(y_2^2+x_2^2)(z_0-z_1) \big) \int_{\Rb}f_0 f_{1}f_{2} \theta_1
\end{align*}
§.§ Other examples
Other examples like higher diamond group can also be calculated explicitly. We can use mathematical software like Maple™ to initiate example Lie groups and Lie algebras and calculate all the Lie algebra cohomology classes we want, and then implement $\mathcal{E}$, $\Theta$ and $\Phi$ maps as in the paper.
§ APPENDIX
§.§ detail of step 1
We give a detailed account of the transition from $CC^{\bullet}(\cR(G_2) \acl \cU(\Fg_1),^{\sigma^{-1}}\hspace{-3pt} \bC _ \delta)$ to $C^{\bullet,\bullet}(\wedge \Fg_1^{\ast},\otimes \cR(G_2))$ in step 1 in <ref>. We just restate what is proved in <cit.>, under our setting.
Define a bi-cyclic complex $C^{\bullet,\bullet}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$, where
\begin{equation*}
C^{p,q}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta}):=^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cR^{p } \otimes \cU^{q}, \qquad p,q \ge 0
\end{equation*}
\begin{equation}
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em,
text height=1.5ex, text depth=0.25ex]
{ \vdots & \vdots & \vdots \\
^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{\otimes 2}& ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{\otimes 2} \otimes \cR& ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{\otimes 2} \otimes \cR^{\otimes 2}& \cdots \\
^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{} & ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{} \otimes \cR& ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{} \otimes \cR^{\otimes 2}& \cdots \\
^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} & ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cR^{}& ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cR^{\otimes 2}& \cdots \\ };
\path[transform canvas={yshift=0.6ex},->,font=\scriptsize]
(m-2-1) edge node[above] {$ b_{\cR} $} (m-2-2) (m-2-2) edge node[above] {$ b_{\cR} $} (m-2-3) (m-2-3) edge node[above] {$ b_{\cR} $} (m-2-4)
(m-3-1) edge node[above] {$ b_{\cR} $} (m-3-2) (m-3-2) edge node[above] {$ b_{\cR} $} (m-3-3) (m-3-3) edge node[above] {$ b_{\cR} $} (m-3-4)
(m-4-1) edge node[above] {$ b_{\cR} $} (m-4-2) (m-4-2) edge node[above] {$ b_{\cR} $} (m-4-3) (m-4-3) edge node[above] {$ b_{\cR} $} (m-4-4) ;
\path[transform canvas={xshift=-0.6ex},->,font=\scriptsize]
(m-4-1) edge node[left]{$b_{\cU}$} (m-3-1) (m-4-2) edge node[left]{$b_{\cU}$} (m-3-2) (m-4-3) edge node[left]{$b_{\cU}$} (m-3-3)
(m-3-1) edge node[left]{$b_{\cU}$} (m-2-1) (m-3-2) edge node[left]{$b_{\cU}$} (m-2-2) (m-3-3) edge node[left]{$b_{\cU}$} (m-2-3)
(m-2-1) edge node[left]{$b_{\cU}$} (m-1-1) (m-2-2) edge node[left]{$b_{\cU}$} (m-1-2) (m-2-3) edge node[left]{$b_{\cU}$} (m-1-3) ;
\path[transform canvas={yshift=-0.6ex},->,font=\scriptsize]
(m-2-2) edge node[below] {$ B_{\cR} $} (m-2-1) (m-2-3) edge node[below] {$ B_{\cR} $} (m-2-2) (m-2-4) edge node[below] {$ B_{\cR} $} (m-2-3)
(m-3-2) edge node[below] {$ B_{\cR} $} (m-3-1) (m-3-3) edge node[below] {$ B_{\cR} $} (m-3-2) (m-3-4) edge node[below] {$ B_{\cR} $} (m-3-3)
(m-4-2) edge node[below] {$ B_{\cR} $} (m-4-1) (m-4-3) edge node[below] {$ B_{\cR} $} (m-4-2) (m-4-4) edge node[below] {$ B_{\cR} $} (m-4-3) ;
\path[transform canvas={xshift=0.6ex},->,font=\scriptsize]
(m-1-1) edge node[right]{$B_{\cU}$} (m-2-1) (m-1-2) edge node[right]{$B_{\cU}$} (m-2-2) (m-1-3) edge node[right]{$B_{\cU}$} (m-2-3)
(m-2-1) edge node[right]{$B_{\cU}$} (m-3-1) (m-2-2) edge node[right]{$B_{\cU}$} (m-3-2) (m-2-3) edge node[right]{$B_{\cU}$} (m-3-3)
(m-3-1) edge node[right]{$B_{\cU}$} (m-4-1) (m-3-2) edge node[right]{$B_{\cU}$} (m-4-2) (m-3-3) edge node[right]{$B_{\cU}$} (m-4-3) ;
\end{tikzpicture}
\end{equation}
We refer to <cit.> for the detail of the construction of the above complex and the proof of its bi-cyclicity.
Next, we identify the diagonal of the above bi-cyclic complex $D^{\bullet}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$ with the standard Hopf cyclic module $CC^{\bullet}(\cR(G_2) \acl \cU(\Fg_1) ,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$ by the map, which is similar to the one defined in <cit.> and <cit.>, $\Psi_{\aclsub}:CC^{\bullet}(\cR(G_2) \acl \cU(\Fg_1),^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta}) \to D^{\bullet}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta}) $:
\begin{equation*}
\begin{aligned}
\Psi_{\aclsub}^{}&(1 \otimes F^1 \acl u^1\otimes \cdots \otimes F^n \acl u^n) \\
=&1 \otimes F^1 \otimes F^2 S(u^1_{<n-1>})\otimes F^3 S(u^1_{<n-2>})S(u^2_{<n-2>})\otimes \cdots \\
&\otimes F^nS(u^1_{<1>})S(u^2_{<1>})\cdots S(u^{n-1}_{<1>}) \otimes u^1_{<0>}\otimes \cdots \otimes u^{n-1}_{<0>}\otimes u^{n},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Psi_{\aclsub}^{-1}&(1 \otimes F^1\otimes \cdots \otimes F^n\otimes u^1 \otimes \cdots \otimes u^n) \\
&=1 \otimes F^1 \acl u^1_{<0>} \otimes F^2 u^1_{<1>} \acl u^2_{<0>} \otimes \cdots \otimes F^n u^1_{<n-1>} \cdots u^{n-1}_{<1>} \acl u^n.
\end{aligned}
\end{equation*}
We relate $\text{Tot} C^{\bullet,\bullet}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$ with the diagonal $D^{\bullet}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$ by Alexander-Whitney map and shuffle map (cf. <cit.>):
\begin{align}
\begin{aligned}
&\qquad AW=\bigoplus_{p+q=n} AW_{p,q},\\
&AW_{p,q}:^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{q} \otimes \cR^{p} \to ^{\sigma^{-1}}\hspace{-3pt} \bC _{\delta} \otimes \cU^{p+q} \otimes \cR^{p+q} \\
&AW_{p,q}=(-1)^{p+q} \uparrow\partial_{0}\cdots \uparrow\partial_{0} \overrightarrow{\partial}_{p+q}\cdots \overrightarrow{\partial}_{p+1},\\
&Sh=\sum_{\sigma \in Sh(p,q)} (-1)^{\sigma} \uparrow s_{\sigma(1)}\cdots \uparrow s_{\sigma(p)}\otimes \overrightarrow s_{\sigma(p+1)}\cdots \overrightarrow s_{\sigma(p+q)}.
\end{aligned}
\end{align}
We note that both Alexander-Whitney map and shuffle map are only chain maps between Hochschild complexes. The lack of explicit cyclic Alexander-Whitney map is the reason that the vertical map in diagram <ref> can not be inverted.
Follow the treatment in <cit.>, we can go from the complex $C^{p,q}(\cU,\cR,^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$ to the quasi-isomorphic complex
$C^{p,q}(\wedge \Fg_1^{},\otimes \cR(G_2),^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta}):= ^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta} \otimes\wedge^{q} \Fg_1^{}\otimes \cR(G_2)^{\otimes p}$ through antisymmetrization map. We can continue from the complex $C^{\bullet,\bullet}(\wedge \Fg_1^{},\otimes \cR(G_2),^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta})$ to the quasi-isomorphic complex $C^{\bullet,\bullet}(\wedge \Fg_1^{\ast},\otimes \cR(G_2)):= \wedge^{\bullet} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes \bullet}$ by Poincaré isomorphism:
\begin{align}
\begin{aligned}
\mathfrak{D}&: \wedge^{\text{\tiny dim}(\Fg_1)} \Fg_1^{\ast} \otimes \wedge^{\text{\tiny dim}(\Fg_1)-q} \Fg_1 \otimes \cR(G_2)^{\otimes p} \to \wedge^{q} \Fg_1^{\ast}\otimes \cR(G_2)^{\otimes p} \\
\mathfrak{D}& (\varpi \otimes \eta \otimes F^1 \otimes \cdots \otimes F^p) = \iota ( \eta) \varpi \otimes F^1 \otimes \cdots \otimes F^p,
\end{aligned}
\end{align}
where $\varpi $ is a volume form and $\iota ( \eta) $ is the contraction by $\eta$.
As in <cit.> and <cit.>, we have the right coadjoint action of $\Fg_1$ on $\varpi $
\begin{align}
\text{ad}^{\ast}(Z) \varpi = \delta(Z) \varpi, \qquad \forall Z \in \Fg_1,
\end{align}
and identify $\wedge^{\text{\tiny dim}(\Fg_1)} \Fg_1^{\ast} $ with $\bC_{\delta}$ as right $\Fg_1$-modules.
use <ref> and <ref> we have the left coaction of $\cR(G_2)$ on $\varpi ^{\ast}$
\begin{align}
\blacktriangledown (\varpi ) = \sigma^{-1} \otimes \varpi ,
\end{align}
and identify $\wedge^{\text{\tiny dim}(\Fg_1)} \Fg_1^{\ast} $ with $^{\sigma^{-1}}\hspace{-3pt}\bC _{\delta}$ as right $\Fg_1$-module and left $\cR(G_2)$-comodule, or, at the same time, as right-left SAYD over $\cR(G_2) \acl \cU(\Fg_1)$.
|
1511.00058
|
Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany
ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany
DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany
Department of Physics, University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany
Free-electron lasers (FELs) produce short and very intense light pulses in the XUV and x-ray regimes.
We investigate the possibility to drive Rabi oscillations in xenon with an intense FEL pulse by using the unusually large dipole strength of the giant-dipole resonance (GDR).
The GDR decays within less than 30 as due to its position, which is above the $4d$ ionization threshold.
We find that intensities around 10$^{18}$ W/cm$^2$ are required to induce Rabi oscillations with a period comparable to the lifetime.
The pulse duration should not exceed 100 as because xenon will be fully ionized within a few lifetimes.
Rabi oscillations reveal themselves also in the photoelectron spectrum in form of Autler-Townes splittings extending over several tens of electronvolt.
§ INTRODUCTION
In the last decade, with the emergence of free-electron lasers (FELs) <cit.>, the door has been opened for studying multiphoton processes in the XUV and x-ray regimes.
Complex ionization dynamics, resulting from a non-trivial interplay of photoabsorption and inner-shell processes such as Auger decay, have been experimentally found when exposing atoms <cit.>, molecules <cit.>, and clusters <cit.> to these high-fluence and high-intensity FEL pulses.
From the extensive works in the last 40 years <cit.>, it is well known that intense optical pulses with frequencies resonant with internal transitions cause the system to behave in completely new ways, resulting in novel effects such as Rabi oscillations, electromagnetically-induced transparency (EIT), lasing without inversion, and population trapping.
These processes have been primarily studied in the outer-valence shells of atomic systems, which are accessible with optical frequencies and the physical realization is quite close to ideal isolated two-, three-, and multi-level systems <cit.>.
With the arrival of FELs, these processes have been extended to the XUV and x-ray regimes: namely stimulated emission <cit.>, Rabi oscillations <cit.>, and processes similar to EIT <cit.>.
At XUV and x-ray photon energies ($\omega > 10$ eV), atomic bound-bound transitions must involve inner-valence and core shells.
Once an electron is removed from a core orbital, the system wants to 'relax' by filling this hole via spontaneous emission (more likely for heavy atoms) or Auger decay (more likely for light atoms) <cit.>.
In order to compete with the relaxation processes of the system, the time scale of the light-driven processes has to be comparable or ideally faster than the time scale of the relaxation.
If this is the case, the light-driven process will not just dominate over the relaxation processes but it also significantly alters the relaxation processes themselves.
The influence of light-driven processes on x-ray fluorescence <cit.> and Auger decay <cit.> has been theoretically studied in recent years.
In this work, we investigate the possibility to induce Rabi oscillations involving the giant-dipole resonance (GDR) in xenon.
The GDR is located at around 100 eV above the ground state and, therefore, also above the $4d$ ionization threshold.
Even though the electron is ultimately ionized at this energy, it is bound for a very short time in the vicinity of the atom leading to an enhanced dipole transition strength which gave the GDR its name <cit.>.
The unusually large dipole strength is beneficial in two ways.
First, it induces fast Rabi oscillations that can compete with the short lifetimes ($<30$ as) of the GDR states.
Second, it ensures that other ionization pathways (out of other sub-shells), which do not involve the GDR, are much weaker and are not of high relevance.
In high-harmonic generation, this large dipole moment can be used to significantly boost the high-harmonic yield around 100 eV <cit.>.
Theoretical studies <cit.> have found that the GDR consists of two sub-resonances.
Recently, an experiment <cit.> has seen first indications of this sub-structure in the XUV two-photon above-threshold-ionization (ATI) spectrum of xenon.
Another goal of this study is, therefore, to investigate whether Rabi oscillations can uncover this sub-structure as well.
In the following, we present in Sec. <ref> our theoretical model.
In Sec. <ref> we estimate which pulses are needed to induce Rabi oscillations, study the population dynamics, and investigate how Rabi oscillations affect the $4d$ hole population and the photoelectron spectrum.
If not noted otherwise, atomic units are used throughout the paper.
§ THEORY
We use our time-dependent configuration interaction singles (TDCIS) approach <cit.>, which we have successfully applied in the strong-field <cit.>, XUV <cit.>, and x-ray regimes <cit.>.
The $N$-body wavefunction ansatz for TDCIS reads
\begin{align}
\label{eq:tdcis}
\ket{\Psi(t)}
\alpha_0(t) \sket{\Phi_0}
\sum_{ai} \alpha^a_i(t) \sket{\Phi^a_i}
\end{align}
where $\Phi_0$ is the Hartree-Fock (HF) ground state, and $\Phi^a_i$ is a one-particle-one-hole excitation where one electron is excited from orbital $i$ into orbital $a$.
The Hamiltonian is the exact non-relativistic $N$-body Hamiltonian,
\begin{align}
\label{eq:ham}
\hat H(t)
\hat H_0 + \hat H_1 + A(t)\,\hat p - E_\text{HF}
\end{align}
which is partitioned into four parts: (i) the Fock operator, $\hat H_0$, describing non-interacting electrons in the HF mean-field potential plus a complex-absorbing potential, (ii) the residual Coulomb interaction, $\hat H_1$, capturing the electron-electron interactions that go beyond the HF mean-field picture, (iii) the light-matter interaction, $A(t)\,\hat p$, in the velocity form of the dipole approximation with $A(t)$ being the vector potential and $\hat p$ being the momentum operator, and (iv) the Hartree-Fock energy, $E_\text{HF}$, which shifts the spectrum such that the HF ground state has the energy $0$.
The residual Coulomb interactions, $\hat H_1$, can be grouped into two classes: intrachannel and interchannel coupling.
Intrachannel coupling, $\sbra{\Phi^a_i} \hat H_1 \sket{\Phi^b_i}$, corrects the mean-field potential due to the missing electron in the atom, and leads to a long-range, $-1/r$, potential for the photoelectron.
Interchannel coupling, $\sbra{\Phi^a_i} \hat H_1 \sket{\Phi^b_j}$ ($i\neq j$), describes the interaction where the excited electron changes the ionic states $i$.
This interaction leads correlated electron dynamics that can significantly change the overall response of the system <cit.> and has large effects on coherence properties <cit.>.
Combining Eqs. (<ref>) and (<ref>), we find the equation of motion for the time-dependent CIS coefficients,
\begin{align}
\label{eq:eom.1}
i\partial_t \, \alpha_0(t)
A(t)\, \sum_{a,i} \sbra{\Phi_0} \hat p \sket{\Phi^a_i}
\alpha^a_i(t)
\\\nonumber
\label{eq:eom.2}
i\partial_t \, \alpha^a_i(t)
(\varepsilon_a-\varepsilon_i) \, \alpha^a_i(t)
\sum_{b,j}
\sbra{\Phi^a_i} \hat H_1 \sket{\Phi^b_j}
\alpha^b_j(t)
\\ &
A(t)\, \Big(
\alpha_0(t)
\sbra{\Phi^a_i} \hat p \sket{\Phi_0}
\! + \!
\sum_{jb} \sbra{\Phi^a_i} \hat p \sket{\Phi^b_j}
\alpha^b_j(t)
\Big)
\end{align}
where $\sket{\Phi^a_i}$ and $\sbra{\Phi^a_i}$ are right and left eigenstates of the non-hermitian Fock operator $\hat H_0$, respectively (see Ref. <cit.>).
The energies of the occupied and virtual orbitals are given by $\varepsilon_i$ and $\varepsilon_a$, respectively.
The existence of the GDR in xenon can be understood in a single-particle picture with a central model potential <cit.>.
CIS intrachannel already improves the description of the GDR in terms of energy position and spectral width <cit.>.
However, many-body effects have to be taken into account in order to reproduce the correct position and width of the GDR.
Interchannel interactions, $\sbra{\Phi^a_i} \hat H_1 \sket{\Phi^b_j}$, within CIS improve greatly the description of the GDR in comparison to intrachannel CIS <cit.>.
To obtain an even better description, electronic correlations of higher order, mostly double excitations, are needed <cit.>.
Recently, it has been shown that interchannel interactions lead to the emergence of a second dipole-allowed resonance state within the GDR <cit.>.
This second resonance is centered at 112 eV and lives only for 11 as ($\Gamma=58.2$ eV).
It is primarily responsible for the spectrally broad GDR feature in the photoionization cross section.
In comparison, the first GDR resonance, which emerges already from a one-particle picture due to a shape resonance, is centered at 73 eV and has a lifetime of 27 as ($\Gamma=24.7$ eV).
§ RESULTS
We use the xcid program [S. Pabst, L. Greenman, A. Karamatskou, Y.-J. Chen, A. Sytcheva, O. Geffert, R. Santra–xcid program package for multichannel ionization dynamics, DESY, Hamburg, Germany, 2015, Rev. 1790] (with the following numerical parameters in [
A pseudo-spectral grid with a radial box size of 100 $a_0$, 500 grid points, and a mapping parameter of $\zeta=0.5$ are used.
The complex absorbing potential starts at 70 $a_0$ and has a strength of $\eta=0.002$.
The maximum angular momentum is 6 and Hartree-Fock orbitals up to an energy of 200 $E_h$ are considered.
The propagation method is Runge-Kutta 4 with a time step $dt=0.005$ a.u.
All $4d, 5s$, and $5p$ orbitals are active in the calculations.
to verify that we can induce Rabi oscillations using the GDR resonances.
With TDCIS, we do not simplify the problem to a two-level system, and we explicitly include all other ionization mechanisms (that lead to singly ionized xenon).
Furthermore, the GDR is properly described as a result of discrete states in the continuum <cit.> which can be modified by the intense XUV pulse itself.
This may lead to trends that deviate from the weak-field behavior.
§.§ Population dynamics
(color online) The population of (a) the ground state, (b) a hole in the $4d$ shell, and (c) a hole in the $5s$ or $5p$ shell in atomic xenon.
The instantaneous peak intensity of the 36 as long (FWHM) Gaussian pulse with a center photon energy of 109 eV is varied from $2.3\cdot 10^{17}$ W/cm$^2$ to $1.2\cdot 10^{18}$ W/cm$^2$.
First, we have a look at the dynamics of the ground state population (cf. Fig. <ref>a) and of the hole populations of different sub-shells of xenon (cf. Fig. <ref>b-c) as it is exposed to a pulse with a center photon energy of 109 eV ($=4$ a.u.) and a FWHM duration (with respect to the intensity profile) of 36 as ($=1.5$ a.u.).
With intensities above $10^{17}$ W/cm$^2$, we basically fully ionize xenon.
As we increase the intensity, the ground state population and the individual hole populations start to show oscillatory behavior and do not monotonically increase as in the “low” intensity limit (cf. $2.3\cdot10^{17}$ W/cm$^2$ results in Fig. <ref>).
These are clear indications that we successfully induce Rabi oscillations.
Rabi oscillations are strongly damped due to the short lifetime of the GDR states leading to a large amount of irreversible ionization per Rabi cycle.
Note that for relatively low intensities the hole populations grow monotonically with time.
In Fig. <ref>, we see that mainly the $4d$ shell is depopulated as expected from the character of the GDR, which mainly corresponds to a configuration where a $4d$ electron is excited into the $l=3$ continuum.
The ionization of the $5s$ and $5p$ shells is ten times smaller but with 10% ionization probability it is large enough that it should be taken into account.
§.§ Final hole population
Unfortunately, it is very hard to monitor the $4d$ hole population as a function of time in an experiment.
With attosecond transient absorption spectroscopy <cit.>, it is in principle possible to do so.
The pump-probe delay between the pulses can already be controlled on a few attosecond scale, but the challenge lies in the generation of a probe pulse with a duration of a few attoseconds to achieve the required time resolution.
This comes on top of the already challenging requirements for the XUV pump pulse.
It is, therefore, easier to probe Rabi oscillations by measuring the final $4d$ hole population or the photoelectron spectrum (see Sec. <ref>).
(color online) The final $4d$ hole population as a function of pulse intensity for different center photon energies $\omega$ (a-c), and two pulse durations, 36 as (blue solid) and 73 as (red dashed).
The $4d$ hole itself is not stable and will decay mainly via Auger decay.
The lifetime of the $4d$ hole is around 6 fs.
Therefore, we can safely neglect the hole decay during the few attoseconds the XUV pulse drives Rabi oscillations.
The hole will, however, eventually decay.
The Auger electron yield is, therefore, directly related to the final $4d$ hole population after the XUV pulse.
Also the Auger electron spectrum should not be affected by the XUV pulse as the $4d$ hole decays predominantly after the pulse under field-free conditions.
In Fig. <ref>, the final $4d$ hole population (after the pulse) is shown as a function of the pulse intensity for the XUV photon energies, (a) $\omega=82$ eV [$=3$ a.u.], (b) $\omega=109$ eV [$=4$ a.u.], and (c) $\omega=136$ eV [$=5$ a.u.], as well as for the pulse durations, (solid blue line) 36 as [$=1.5$ a.u.] and (dashed red line) 73 as [$=3.0$ a.u.].
We see, especially for the 36 as pulse, that the population has oscillatory behavior and does not monotonically grow with the pulse intensity—a clear indication of Rabi oscillations.
The more rapid modulations in the final population for longer pulses is also consistent with Rabi oscillations.
The probability to be in an excited state after the pulse is given by $\sin^2(\Omega_\text{eff}T)$, where $\Omega_\text{eff}=T^{-1}\int_{-\infty}^\infty\!\! dt\ d\,{\cal E}(t)=d\,{\cal E}_\text{max} \sqrt{\pi/(2\ln(2))}$ is the pulse-averaged Rabi frequency, ${\cal E}(t)$ is the pulse envelope, $d$ is the dipole transition strength, and $T$ is the duration of the pulse.
Even though the variations are faster for longer pulses, the visibility decreases with pulse length.
Due to the short lifetimes of the GDR states, which are below 30 as, almost all electrons are irreversibly ionized after 30 as, and the hole population is always close to unity whether or not Rabi oscillations were induced.
The hole creation is most dominant at $\omega=109$ eV where the photoionization cross-section is largest (for the three photon energies shown).
At $\omega=136$ eV, the hole creation increases almost monotonically with the electric field.
Almost no oscillatory behavior is visible indicating that at this photon energy we do not hit a resonance and ionization is predominantly irreversible.
At $\omega=82$ eV, the first peak in the hole population appears quite early followed by a relatively long plateau extending up to $10^{17}$ W/cm$^2$.
This trend is not fully consistent with the Rabi-oscillation picture of a two-level system.
But xenon at these XUV photon energies cannot be described as a two-level system.
(color online) The final $4d$ hole population as a function of pulse intensity and center pulse photon energy.
The pulse duration is (a) 36 as and (b) 73 as.
As we see in Fig. <ref>, where the final $4d$ hole population is shown as a function of intensity and driving photon energy, the photon energy where the $4d$ shell is most strongly ionized shifts to higher energies as the intensity increases.
This explains why for $\omega=82$ eV in Fig. <ref>a) a plateau appears.
At low intensities, it probes the lower end of the GDR.
At higher intensities, the GDR moves to higher energies and the dipole transition strength drops, counter-balancing the increase in field strength.
The energy shift of the GDR is a result of the very intense XUV pulse, which dresses the excited and the continuum states.
The polarizability of a flat continuum is given by the ponderomotive potential $U_p=\frac{E^2}{4\omega^2}$ and shifts the continuum to higher energies <cit.>.
At $E=2$ a.u. and $\omega=109$ eV, this yields an energy shift of less than 2 eV.
The observed energy shift of around 20 eV is much larger.
Furthermore, the energy shifts depend rather linearly than quadratically on $E$.
As we will see in Sec. <ref>, the kinetic energy of the photoelectron even increases or decreases with intensity depending on whether the photon energy is above or below the GDR, respectively.
This clearly shows that the continuum in xenon is structured around the GDR and is not flat.
Furthermore, when $z\,E\gg\omega$ linear energy shifts are expected <cit.>.
In Fig. <ref>, we find the linear energy shifts especially at intensities above 10$^{18}$ W/cm$^2$ where $z\,E>\omega$.
For the short 36 as pulse, the effect of Rabi oscillations is nicely visible.
At $\omega\sim 100$–150 eV, the $4d$ hole population shows a local minimum around $7-11\cdot 10^{17}$ W/cm$^2$ and an island of enhanced ionization emerges at lower intensities.
For the longer pulse, more Rabi oscillation occur as the intensity increases and, therefore, two islands of enhanced ionization occur.
Again, the visibility is greatly reduced for longer pulses as explained above.
At photon energies below 100 eV, no signs of Rabi oscillations can be seen.
Even though the dipole strength falls off quite symmetrically around 100 eV in the weak-field regime, the dressing of the continuum states creates a clear preference for higher energies as the field strength increases.
Furthermore, we see no indication of the two GDR subresonances.
Rabi oscillations seem to be only sensitive to the overall dipole strength.
Note that at each photon energies the ground state is coupled resonantly to an excited/continuum state in contrast to a two-level system where only one excited state at a specific energy exists.
Another general trend that we can observe in Fig. <ref> is the broadening of the spectral feature with intensity.
This indicates, on the one hand, that the lifetimes of the excited states are decreasing.
On the other hand, the spectral broadening is also a direct consequence of saturation of ionization.
We already saw in Figs. <ref> and <ref> that at these intensities we basically fully ionize the system.
Also new multiphoton ionization pathways start to emerge for photon energies below the $4d$ ionization threshold and at intensities above $2\cdot10^{17}$ W/cm$^2$ ($E\geq 1.5$ a.u.).
The ionization is enhanced for specific combinations of field strengths and photon energies.
The photon energy for the ionization enhancement increases as the field strength increases indicating field-dressing effects are involved in these new pathways.
The origin might be, therefore, quite similar to Freeman resonances known from ATI <cit.>, where intermediate states are shifted into resonance by the intense pulse.
§.§ Photoelectron spectrum
Rabi oscillations should be also visible in the photoelectron spectrum.
In a time domain picture, the field-driven coupling between two states leads to Rabi oscillations.
In an energy domain picture, the excited state (also the ground state) splits into two states, known as the Autler-Townes doublet, which are separated by the Rabi frequency $\Omega$ <cit.>.
In our case, the excited state is in the continuum, and the energy splitting should be directly imprinted in the kinetic energy of the photoelectron.
(color online) The photoelectron spectrum as a function of the pulse intensity.
The photon energy is centered at 109 eV and the pulse duration is (a) 36 as and (b) 73 as.
In Fig. <ref>, the total photoelectron spectrum is shown as a function of intensity for (a) a 36 as and (b) a 73 as pulse with $\omega=109$ eV.
We clearly see the Autler-Townes doublet for both pulse durations.
The lower energy branch of the Autler-Townes doublet approaches 0 eV and does not survive at high intensities as the kinetic energy of the electron has to be positive.
At higher intensities, the ionization dynamics become less periodic and additional peaks in the photoelectron spectrum occur <cit.>.
(color online) The photoelectron spectrum as a function of the pulse intensity.
The pulse has a duration of 73 as and the photon energy is centered (a) at 136 eV and (b) at 82 eV.
Once the driving photon energy moves away from $110$ eV, no indication of Rabi oscillations are seen in the $4d$ hole population (see Fig. <ref>).
The same is true for the photoelectron spectrum.
In Fig. <ref>, the photoelectron spectrum is shown for (a) $\omega=136$ eV and (b) $\omega=82$ eV as the pulse intensity is varied.
For $\omega=136$ eV, the kinetic energy of the electron slightly increases with intensity.
The coupling to the lower lying GDR resonance pushes the continuum states above the GDR to lower energies.
At $\omega=163$ eV ($=6$ a.u.)—these results are not shown—the energy separation to the GDR is large and the coupling can be neglected and the kinetic energy of the photoelectron does not significantly change with intensity.
For photon energies below the GDR (see Fig. <ref>b), the trend is reversed and the kinetic energy of the electron decreases with intensity.
This clearly shows that the polarization of the continuum at these frequencies is strongly affected by the GDR and cannot be considered flat.
For a flat continuum, the kinetic energy of the photoelectron spectrum would always decrease by the ponderomotive potential ($\propto\omega^{-2}$) and would never increase as $\omega$ increase.
§ CONCLUSION
We have investigated to which extent intense FEL pulses could be used to drive Rabi oscillations between the neutral ground state and the GDR in xenon.
We found that intensities around $10^{18}$ W/cm$^2$ are needed to see an impact of Rabi oscillations on the $4d$ hole population.
We could find indications that Rabi oscillation can be used to uncover the substructure of the GDR, i.e., the two dipole-allowed resonances at 73 eV and 112 eV <cit.>.
There are two ways how Rabi oscillations can be observed: (1) by measuring the $4d$ hole population via the Auger electron that gets emitted when the $4d$ hole decays, or (2) via the photoelectron spectrum in the form of Autler-Townes splitting.
To see Rabi oscillations in the $4d$ hole population, a short pulse duration comparable to the lifetimes of the GDR states is needed.
For the photoelectron spectrum, a longer pulse seems to be beneficial as the Autler-Townes splitting is spectrally better visible.
The energy shifts of the photoelectron show clearly that the continuum around the GDR is structured and highly polarizable and cannot be assumed flat.
With seeded FELs, pulse intensities and pulse durations are in reach to induce Rabi oscillations that are driven by XUV light.
S.P. is funded by the Alexander von Humboldt Foundation and by the NSF through a grant to ITAMP.
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1511.00432
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Optimal control of steady second grade fluids with Dirichlet boundary conditions
Nadir Arada[Centro de Matemática e Aplicações, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal. E-mail: naar'100fct.unl.pt]
We consider optimal control problems governed by systems describing the flow of an incompressible second grade fluid with Dirichlet boundary conditions. We prove the existence of an optimal solution, derive the corresponding necessary optimality conditions and analyze its asymptotic behavior when the viscoelastic parameter tends to zero.
Key words. Optimal control, Navier-Stokes equations, second grade fluid, Dirichlet boundary conditions, necessary optimality conditions, vanishing viscoelastic parameter.
AMS Subject Classification. $49$K$20$, $76$D$55$, $76$A$05$.
§ INTRODUCTION
One of the important feature of complex non-Newtonian fluids is their ability to exhibit normal stress differences in simple shear flows, leading to characteristic phenomena like rod-climbing or die-swell. The second grade fluid model forms a subclass of differential type fluids of complexity 2, and is one of the simplest constitutive models for flows of non-Newtonian fluids that can predict normal stress differences (cf. <cit.> or <cit.>). The corresponding stress is just a function of the pressure, the velocity gradient and some number of its higher material time derivatives (the Rivlin-Ericksen tensors). As a consequence, only an infinitesimal part of the history of the deformation gradient has an influence on the stress and, while they are good at predicting creep, these models cannot capture stress relaxation. Nevertheless, due to their relative mathematical simplicity, there has been a great deal of interest on these models in recent years as they have been used successfully to predict slow steady motions of slurry flows, food rheology or flow of a water solution of polymers, where relaxation effects frequently seem to be rather insignificant.
The corresponding equations of motion have the form
\begin{equation}\label{equation_sg}
\begin{array}{lll}\partial_t\left(\mathbf y-\alpha_1\Delta \mathbf y\right)-\nu \Delta \mathbf y\hspace{-3mm}&+
\mathbf{curl}\left( \mathbf y-\left(2\alpha_1+\alpha_2\right)\Delta \mathbf y
\right)\times \mathbf y\vspace{2mm}\\
2\mathbf y\cdot \nabla \left(\Delta \mathbf y\right)-\Delta\left(\mathbf y\cdot \nabla \mathbf y\right)\right)
+\nabla \pi= \mathbf u& \mbox{in} \ \Omega,\end{array}
\end{equation}
where $\mathbf y$ is the velocity field, $\alpha_1$ and $\alpha_2$ are viscoelastic parameters (normal stress moduli), $\nu$ is the viscosity of the fluid, $\pi$ is the hydrodynamic pressure, $\mathbf u$ is a given body force and $\Omega\subset \mathbb{R}^2$ is a bounded domain with boundary $\Gamma$. As this equation is set in dimension two, the vector $\mathbf y$ is written in the form $\mathbf y=(y\equiv (y_1,y_2),0)$ in order to define the vector product and the curl. Recall that in two dimensions, $ \mathrm{curl} \, y=\tfrac{\partial y_2}{\partial x_1}-\tfrac{\partial y_1}{\partial x_2}$ and thus $\mathbf{curl} \, \mathbf y=(0,0, \mathrm{curl} \, y)$.
According to <cit.>, if the fluid modelled by equation (<ref>) is to be compatible with thermodynamics in the sense that all motions of the fluid meet the Clausius-Duhem inequality and the assumption that the specific Helmholtz free energy of the fluid is a minimum in equilibrium, then
$$\nu\geq 0, \qquad
\alpha_1\geq 0,\qquad \alpha_1+
\alpha_2=0.$$
We refer to <cit.> for a critical and extensive historical review of second-order fluid models and, in particular, for a discussion on the sign of the normal stress moduli. Here we will restraint to the simplified case $\alpha_1+\alpha_2=0$, with $\alpha_1\geq 0$ and $\nu>0$. Setting $\alpha_1=\alpha$, we can see that the problem of determining the velocity field $\mathbf{y}$ and the associated pressure $\pi$ satisfying the equations governing the flow of an incompressible second grade fluid reduces to
\left\{ \begin{array}{ll}\partial_t\left(\mathbf y-\alpha\Delta \mathbf y\right)-\nu \Delta \mathbf y+
\mathbf{curl}\left( \mathbf y-\alpha\Delta \mathbf y
\right)\times \mathbf y+\nabla \pi= \mathbf u& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf y=0& \mbox{in} \ \Omega.\end{array}\right.
In the inviscid case ($\nu=0$), the second-grade fluid equations are called $\alpha$-Euler equations. Initially proposed as a regularization of the incompressible Euler equations, they are geometrically significant and have been interpreted as a model of turbulence (cf. <cit.> and <cit.>). They also inspired another variant, called the $\alpha$-Navier-Stokes equations that turned out to be very relevant in turbulence modeling (cf. <cit.>, <cit.> and the references therein). These equations contain the regularizing term $-\nu\Delta\left(\mathbf y-\alpha \Delta\mathbf y\right)$ instead of $\nu\Delta\mathbf y$, making the dissipation stronger and the problem much easier to solve than in the case of second-grade fluids. When $\alpha=0$, the $\alpha$-Navier-Stokes and the second grade fluid equations are equivalent to the Navier-Stokes equation.
Since the nonlinear term involves derivatives with higher order than the ones appearing in the viscous term, solving this problem is very challenging. The two dimensional case has been systematically studied for the first time in <cit.> and <cit.> for both steady and unsteady cases with homogeneous Dirichlet boundary conditions. A Galerkin's method in the basis of the eigenfunctions of the operator $\mathbf{curl}(\mathbf{curl}(\mathbf y-\alpha\Delta \mathbf y))$ was especially designed to decompose the problem into a mixed elliptic-hyperbolic type, looking for the velocity $\mathbf y$ as a solution of a Stokes-like system coupled to a transport equation satisfied by $\mathbf{curl}\left( \mathbf y-\alpha\Delta \mathbf y
\right)$. Under minimal restrictions on the data, this approach allows the authors to establish the existence of solutions (and automatically recover $H^3$ regularity) in the steady case, and to prove that the time-dependent version admits
a unique global solution in the two dimensional case. This problem received a lot of attention since these pioneering results and, without ambition for completeness, we refer to <cit.> where existence of a solution in the three dimensional steady case was established under a restriction on the size of the data. We also cite the extensions in <cit.> and <cit.>, where the three dimensional unsteady case was considered: global in time existence for small data was established, the former work using a Schauder fixed point argument while the latter considers the decomposition method on the system of Galerkin equations previously mentionned.
This paper deals with the mathematical analysis of an optimal control problem associated with a steady viscous, incompressible second grade fluid. Control is effected through a distributed mechanical force and the objective is to match the velocity field to a given target field. More precisely, the controls and states are constrained to satisfy the following system of partial differential equations
\begin{equation}\label{equation_etat}
\left\{ \begin{array}{ll}-\nu \Delta \mathbf y+
\mathbf{curl}\left( \mathbf y-\alpha\Delta \mathbf y
\right)\times \mathbf y+\nabla \pi= \mathbf u& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf y=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf y=0&
\mbox{on} \ \Gamma\end{array}\right.\end{equation}
and the optimal control problem reads as
$$(P_\alpha) \
\left\{\begin{array}{ll}\mbox{minimize} & \displaystyle J(u,y)=
\tfrac{1}{2}\int_\Omega\left|y-y_d\right|^2\,dx+\tfrac{\lambda}{2}
\int_\Omega\left|u\right|^2\,dx\vspace{1mm}\\
\mbox{subject to} & (u,y)\in U_{ad}\times H^3(\Omega) \ \mbox{satisfies} \
(\ref{equation_etat}) \ \mbox{for some} \ \pi\in L^2(\Omega),\end{array}\right.$$
where $\lambda\geq 0$, $y_d$ is some desired velocity field in $L^2(\Omega)$ and $U_{ad}$, the set of admissible controls, is a nonempty closed convex subset of
$H(\mathrm{curl};\Omega)=\left\{v\in L^2(\Omega)\mid \mathrm{curl}\, u\in L^2(\Omega) \right\}$.
Deriving the optimality conditions for problems governed by highly nonlinear equations is not an easy task (cf. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). The main difficulties are encountered when studying the solvability of the corresponding linearized and adjoint equations and are closely related with the regularity of the coefficients in the main part of the associated differential operators. As already mentioned, the choice of the special Galerkin basis used to study the state equation is optimal in the sense that it allows us to prove the existence of regular solutions with minimal assumptions on the data. However, the direct application of this approach to study the linearized and adjoint equations does not seem appropriate and does come at a cost. The main disadvantage is that it automatically imposes the derivation of a $H^3$ estimate and this may be achieved only if high order derivatives of the state variable are well defined and if we impose additional restriction on their size. This in turn is only guaranteed if we consider regular, size constrained controls. To overcome this difficulty, our idea is to consider an approximate optimal control problem governed by a state equation involving regularized controls. More precisely, if $(\bar u,\bar y)$ is a solution of $(P_\alpha)$ and $\varepsilon$ is a positive parameter, we consider the control problem
$$(P_\alpha^\varepsilon) \ \left\{\begin{array}{ll} \mbox{minimize} & I(u,y^\varepsilon)=\displaystyle J(u,y^\varepsilon)+\tfrac{1}{2}\int_\Omega
\left|u-\bar u\right|^2\,dx+\tfrac{1}{2}\int_\Omega
\left|{\rm curl}\left(u-\bar u\right)\right|^2\,dx
\vspace{2mm}\\
\mbox{subject to} & (u,y^\varepsilon)\in U_{ad}\times
H^3(\Omega) \ \mbox{such that}
\vspace{2mm}\\
&\left\{ \begin{array}{ll}-\nu \Delta \mathbf y^\varepsilon+
\mathbf{curl}\left(\mathbf y^\varepsilon-\alpha\, \Delta\mathbf y^\varepsilon\right)
\times \mathbf y^\varepsilon+\nabla \pi^\varepsilon= \varrho_\varepsilon(\mathbf u)& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf y^\varepsilon=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf y^\varepsilon=0&
\mbox{on} \ \Gamma,\end{array}\right.
\end{array}
\right.$$
where $\varrho_\varepsilon$ denotes a Friedrich mollifier. A careful analysis enables us to handle the issues mentioned above and to derive the corresponding approximate optimality conditions under natural restriction on the control variable. By passing to the limit in the regularization parameter $\varepsilon$, we recover the optimality conditions for $(P_\alpha)$.
In this paper, we are also interested in the asymptotic behavior of the solutions of $(P_{\alpha})$, when the viscoelastic parameter $\alpha$ tends
to zero. We will prove in particular that
\begin{equation}\label{stability}\lim_{\alpha\rightarrow 0^+}\min(P_\alpha)=\min(P_0),\end{equation}
where $(P_0)$ is the optimal control problem governed by the steady Navier-Stokes equations and defined by
$$(P_0) \ \left\{\begin{array}{ll} \mbox{minimize} & J(u,y)
\vspace{2mm}\\
\mbox{subject to} & (u,y)\in U_{ad}\times H^1(\Omega) \ \mbox{such that}
\vspace{2mm}\\
&\left\{\begin{array}{ll}-\nu \Delta y+
y\cdot \nabla y+\nabla \pi=u& \quad
\mbox{in} \ \Omega,\vspace{2mm}\\
\mathrm{div}\, y=0& \quad
\mbox{in} \ \Omega,\vspace{2mm}\\
y=0& \quad
\mbox{on} \ \Gamma.\end{array}\right.\end{array}
\right.$$
To obtain such a result, we first establish that the sequence of solutions $(y_\alpha)_\alpha$ of (<ref>) converges to $y$, a solution of the Navier-Stokes equation, when $\alpha$ tends to zero. Next we prove that if $(\bar u_\alpha,\bar y_\alpha)$ is a solution to the problem $(P_\alpha)$ then the sequence $(\bar u_\alpha,\bar y_\alpha)_\alpha$ converges to a solution $(\bar u_0,\bar y_0)$ of $(P_0)$. Another aspect concerns the necessary optimality conditions. To study the asymptotic behavior of these conditions, we analyze the adjoint equations for $(P_\alpha)$ and prove that the sequence of adjoint solutions converges to the solution of the adjoint equation for $(P_0)$. The optimality conditions for $(P_0)$ are then obtained by passing to the limit in the optimality conditions for $(P_\alpha)$
The plan of the present paper is as follows. The main results are stated in Section 2. Notation and preliminary results related with the nonlinear terms are given in Section 3. Section 4 is devoted to the existence and uniqueness results for the state and the linearized state equation and to the derivation of the corresponding estimates. In Section 5, we analyze the Lipschitz continuity and the Gâteaux differentiability of the control-to-state mapping and we consider the solvability of the adjoint equation in Section 6. Finally, the proof of the main results are given in Section 7.
§ STATEMENT OF THE MAIN RESULTS
We first establish the existence of optimal solutions for problem $(P_\alpha)$.
Assume that $U_{ad}$ is bounded in $H(\mathrm{curl};\Omega)$.
Then problem $(P_\alpha)$ admits at least a solution.
To derive the corresponding necessary optimality conditions (stated in the next result), we need to restrain the optimal control size. Such a restriction, well known and widely used when dealing with optimal control problems governed by the steady Navier-Stokes equations, should be set within the natural functional framework of $H({\rm curl};\Omega)$, without requiring additional regularity on the control. Besides the difficulties inherent to the highly nonlinear nature of the state equation, and its implications on the linearized and adjoint equations, this is one of the main issues we must overcome.
Let $(\bar{ u}_\alpha,\bar{y}_\alpha)$ be a solution of $(P_\alpha)$. There exists a positive constant
$\bar\kappa$, depending only on $\Omega$, such that if the following condition holds
\begin{equation}\label{control_constraint}
\bar\kappa \left(\left\|\bar u_\alpha\right\|_2+\alpha
\left\|\mathrm{curl}\, \bar u_\alpha\right\|_2\right)<\nu^2\end{equation}
then there exists $\bar{p}_\alpha\in H^1(\Omega)$ weak solution of
\begin{equation}\label{adj_opt_eq_alpha}
\left\{ \begin{array}{ll}-\nu \Delta \bar{\mathbf p}_\alpha-
\mathbf{curl}\,\sigma\left(\bar{\mathbf y}_\alpha\right)\times
\bar{\mathbf p}_\alpha+\mathbf{curl}
\left(\sigma\left(\bar{\mathbf y}_\alpha\times \bar{\mathbf p}_\alpha\right)\right)+\nabla \pi= \bar{\mathbf y}_\alpha-\mathbf y_d& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \bar{\mathbf p}_\alpha=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\bar{\mathbf p}_\alpha=0& \mbox{on} \ \Gamma,\end{array}\right.\end{equation}
and satisfying
\begin{equation}\label{opt_control_alpha}\left(\bar{ p}_\alpha+\lambda\bar{ u}_\alpha,v-\bar{ u}_\alpha\right)\geq 0 \qquad
\mbox{for all} \ v\in U_{ad}.\end{equation}
Finally, we consider the asymptotic analysis of the optimal control $(P_\alpha)$. We first prove that if $(\bar u_\alpha,\bar y_\alpha)$ is a solution of $(P_\alpha)$, then a cluster point (for an appropriate topology) is a solution of problem $(P_0)$ and the stability property (<ref>) holds. Moreorer, if $(\bar u_\alpha,\bar y_\alpha,\bar p_\alpha)$ is defined as in Theorem $\ref{main_1}$, then $\bar p_\alpha$ converges to some $\bar p_0$ satisfying the optimality conditions of problem $(P_0)$. More precisely, we have the following result.
Let $(\bar u_\alpha,\bar y_\alpha,\bar p_\alpha)$ defined as in Theorem $\ref{main_1}$. Then
$i)$ $(\bar u_\alpha,\bar y_\alpha)$ strongly converges in $L^2(\Omega)\times V$ $($up to a subsequence when $\alpha$ tends to zero$)$ to a limit point $(\bar u_0,\bar y_0)$ solution o $(P_0)$.
$ii)$ $\bar p_\alpha$ converges $($up to a subsequence when $\alpha$ tends to zero$)$ for the weak topology of $V$ to $\bar p_0$, weak solution of the adjoint equation
\begin{equation}\label{adjoint_limit}
\left\{\begin{array}{ll}-\nu \Delta \bar p_0-\bar y_0\cdot
\nabla \bar p_0+\left(\nabla \bar y_0\right)^\top \bar p_0+\nabla \pi=\bar y_0-y_d& \quad
\mbox{in} \ \Omega,\vspace{2mm}\\
\mathrm{div}\, \bar p_0=0& \quad
\mbox{in} \ \Omega,\vspace{2mm}\\
\bar p_0=0& \quad
\mbox{on} \ \Gamma\end{array}\right.
\end{equation}
and satisfying the optimality condition
$$\left(\bar p_0+\lambda\bar u_0,v-\bar u_0\right)\geq 0 \qquad
\mbox{for all} \ v\in U_{ad}.$$
Taking into account Remark $\ref{remark1}$ below, it follows that if $\bar u_\alpha$ satisfies $(\ref{control_constraint})$, then the limit $\bar u_0$ satisfies
$S_4^2 S_2\left\|\bar u_0\right\|_2<\nu^2$,
which implies the uniqueness of $\bar y_0$ and $\bar p_0$.
Unlike the two dimensional case where existence of at least a weak solution for the state equation can be established without restriction on the size of the data, the existence of such a solution is only guaranteed for small data in the three dimensional case $($see e.g. $\cite{B99}$$)$. As a consequence, the results stated in Theorem $\ref{main_existence}$, Theorem $\ref{main_1}$ and Theorem $\ref{assympt_1}$ may be extended to the three dimensional case, under additional restrictions on the whole set of admissible controls.
§ NOTATION, ASSUMPTIONS AND PRELIMINARY RESULTS
§.§ Functional setting
Throughout the paper $\Omega$ is a bounded, simply connected domain in $\mathbb{R}^2$. The boundary of $\Omega$ is denoted by $\Gamma$ and is of class $C^{2,1}$. The standard Sobolev spaces are denoted by $W^{k,p}(\Omega)$ ($k\in \mathbb{N}$ and $1<p<\infty$), and their norms by $\|\cdot\|_{k,p}$. We set $W^{k,2}(\Omega)\equiv H^k(\Omega)$ and $\|\cdot\|_{k,2}\equiv \|\cdot\|_{H^k}$. In order to simplify the presentation, we will use the notation
$$\sigma(v)=v-\alpha \Delta v, \qquad v\in H^2(\Omega)$$
in all the sequel. We will also frequently use the scalar product in $L^2(\Omega)$
$$\left(u,v\right)=\displaystyle\int_\Omega u(x)\cdot v(x)\,dx,$$
the semi-norm of $H^1(\Omega)$
$$\left|v\right|_{H^1}=\left\|\nabla v\right\|_2$$
and in order to eliminate the pressure in the different variational formulations, we will work in divergence-free spaces and consider the following Hilbert space
$$V=\left\{v\in H^1_0(\Omega)\mid \mathrm{div} \, v=0 \
\mbox{in} \, \Omega\right\}.$$
Recall also the Poincaré and Sobolev inequalities, respectively given by
$$\left\|v\right\|_2\leq S_{2} \left|v\right|_{H^1}
\qquad \mbox{for all} \ v\in V,$$
$$\left\|v\right\|_4\leq S_{4} \left|v\right|_{H^1}
\qquad \mbox{for all} \ y\in V.$$
We introduce the space
$$V_2=\left\{v\in V\mid {\rm curl}\, \sigma(v)\in L^2(\Omega)\right\}$$
equipped with the scalar product
$$\left(u,v\right)_{V_2}=\left(u,v\right)+\alpha \left(\nabla u,\nabla v\right)+\left({\rm curl}\, \sigma(u),{\rm curl}\, \sigma(v)\right)$$
and associated semi-norm
$$\left|v\right|_{V_ 2}=\left\|{\rm curl}\, \sigma(v)\right\|_2.$$
We finally introduce the space (of controls)
$$H({\rm curl};\Omega)=\left\{v\in L^2(\Omega)\mid {\rm curl} \, v\in L^2(\Omega)\right\}$$
equipped with the scalar product
$$\left(u,v\right)_{H({\rm curl};\Omega)}=
\left(u,v\right)+\left({\rm curl} \, u,{\rm curl} \, v\right)$$
and which is a Hilbert space for the associated norm
$$\left\|v\right\|_{H({\rm curl};\Omega)}=\left(v,v\right)_{H({\rm curl};\Omega)}^{\frac{1}{2}}.$$
§.§ Auxiliary results
The aim of this section is to present some results that will be used throughout the paper. We first recall that the space $V_ 2$, particularly well adapted to handle the partial differential equations we are considering, is continuously embedded in $H^3(\Omega)$ (see e.g. <cit.>).
Any $y\in V_2$ belongs to $H^3(\Omega)$ and there exists a constant $c(\alpha)$ such that
$$\left\|y\right\|_{H^3}\leq c(\alpha) \left|y\right|_{V_2}.$$
The second lemma will be useful when dealing with a priori estimates for the linearized state and adjoint state equations.
Let $y\in V_2$. Then, the following estimate holds
$$\left\|y\right\|_{\infty}\leq \tfrac{c}{\alpha^\frac{1}{3}}
\left|y\right|_{H^1}^{\frac{2}{3}}
\left|y\right|_{V_2}^{\frac{1}{3}},$$
where $c$ is a positive constant only depending on $\Omega$.
Proof. Since $\mathrm{curl}\,\sigma(y)\in L^2(\Omega)$
and $\nabla\cdot \left(\mathrm{curl}\,\sigma(y)\right)=0$, there exists a unique vector-potential $\psi\in H^1(\Omega)$ such that
\psi=\mathrm{curl}\,\sigma(y) &\quad \mbox{in} \ \Omega, \vspace{2mm}\\
\nabla \cdot \psi=0 & \quad\mbox{in} \ \Omega,\vspace{2mm}\\
\psi\cdot n=0 & \quad\mbox{on} \ \Gamma\end{array}\right.$$
\begin{equation}\label{sigma_phi}
\left\|\psi\right\|_{H^1}\leq
It follows that
$${\rm curl} \left(y-\alpha
\Delta y-\psi\right)=0$$
and the fact that $\Omega$ is simply connected implies that
there exists $\pi\in L^2(\Omega)$ such that
\Delta y-\psi+\nabla \pi=0.$$
(For the proof of such a result, see Theorem 2.9, Chapter 1 in <cit.>.)
Hence $y$ is the solution of the Stokes system
$$-\Delta y+\nabla\left(\tfrac{\pi}{\alpha}\right)=\tfrac{1}{\alpha}
\left(\psi-y\right)$$
and satisfies
\begin{equation}\label{y_phi}\left\|y\right\|_{H^2}
\leq \tfrac{c}{\alpha}
\left\|\psi-y\right\|_2.\end{equation}
Observing that
\left(y-\alpha \Delta y+\nabla \pi,y\right)=
\left\|y\right\|_2^2+
\alpha \left|y\right|_{H^1}^2,$$
we obtain
\left\|\psi\right\|_2^2-
\left\|y\right\|_2^2-2\alpha
\left|y\right|_{H^1}^2\leq
\left\|\psi\right\|_2^2.$$
Combining (<ref>) and (<ref>), we deduce that
\begin{equation}\label{yh_curl_sigma} \left\|y\right\|_{H^2}\leq
\tfrac{c}{\alpha}\left|y\right|_{V_2}.
\end{equation}
Finally, the interpolation inequalities
$$\left\|y\right\|_\infty\leq c
\left\|y\right\|_2^{\frac{1}{3}}
\left\|y\right\|_{1,4}^{\frac{2}{3}}\leq c
\left\|y\right\|_2^{\frac{1}{3}}
\left(\left\|y\right\|_{H^1}^\frac{1}{2}\left\|y\right\|_{H^2}^\frac{1}{2}\right)^{\frac{2}{3}},$$
together with the Poincaré inequality and (<ref>) yield
\left\|y\right\|_{H^2}^{\frac{1}{3}}
\leq \tfrac{c}{\alpha^\frac{1}{3}}
\left|y\right|_{H^1}^{\frac{2}{3}}\,
\left|y\right|_{V_2}^{\frac{1}{3}}$$
and the claimed result is proved. $\hfill \Box$
The first identity in the next result is standard and relates the nonlinear term in (<ref>), and similar terms appearing in the linearized and adjoint state equations, to the classical trilinear form used in the Euler and Navier-Stokes equations and defined by
$$b(\phi,z,y)=\left(\phi\cdot \nabla z,y\right).$$
The second identity deals with another term only appearing in the adjoint state equation.
Let $y,z \in V_2$ and $\phi\in V$. Then
$$\left( \mathbf{curl}\, \sigma(\mathbf y)
\times \mathbf z, \boldsymbol\phi\right)=
b\left(\phi,z, \sigma(y)\right)
Let $y, z$ and $\phi$ be in $V_2$. Then
$$\left( \mathbf{curl}\, \sigma\left(\mathbf y\times \mathbf z\right), \boldsymbol\phi\right)
=b\left(z,y, \sigma(\phi)\right)-b\left(y,z,\sigma(\phi)\right).$$
Proof. By using a standard integration by parts, we can easily prove that for every $ y, z\in V_2$
and every $\phi\in V$, we have
$$\begin{array}{ll}\left( \mathbf{curl}\, \sigma(\mathbf y)
\times \mathbf z, \boldsymbol\phi\right)&=\left( \mathbf{curl}\, \sigma(\mathbf y)
, \mathbf z\times \boldsymbol\phi\right)=\left( \sigma(\mathbf y)
, \mathbf{curl}\left(\mathbf z\times \boldsymbol\phi\right)\right)\vspace{2mm}\\
&=\left( \sigma(\mathbf y)
,\boldsymbol\phi\cdot \nabla \mathbf z-\mathbf z\cdot \nabla \boldsymbol\phi\right)=b\left(\phi,z, \sigma(y)\right)
and the first identity is proved. Similarly, for $y, z$ and $\phi$ be in $V_2$ we have
$$\begin{array}{ll}\left( \mathbf{curl}\, \sigma\left(\mathbf y\times \mathbf z\right), \boldsymbol\phi\right)&=
\left( \mathbf{curl}\, \left(\mathbf y\times \mathbf z\right), \boldsymbol\phi\right)-\alpha
\left( \mathbf{curl}\, \Delta\left(\mathbf y\times \mathbf z\right), \boldsymbol\phi\right)\vspace{2mm}\\
&=\displaystyle \left(\mathbf z\cdot \nabla \mathbf y-\mathbf y\cdot \nabla \mathbf z,\boldsymbol\phi\right)-\alpha\left(\Delta\left(\mathbf y\times \mathbf z\right),\mathbf{curl}\, \boldsymbol\phi\right)\vspace{2mm}\\
\alpha\left(\mathbf{curl}\left(\mathbf{curl}\left(\mathbf y\times \mathbf z\right)\right)
-\nabla\left( \mathrm{div}\left(\mathbf y\times \mathbf z\right)\right),\mathbf{curl}\, \boldsymbol\phi\right)\vspace{2mm}\\
&=b\left(z,y,\phi\right)-b\left(y,z,\phi\right)+\alpha\left(\mathbf{curl}\left(\mathbf{curl}\left(\mathbf y\times \mathbf z\right)\right),\mathbf{curl}\,
\boldsymbol\phi\right)\vspace{2mm}\\
&=b\left(z,y,\phi\right)-b\left(y,z,\phi\right)+\alpha \left(\mathbf{curl}\left(\mathbf y\times \mathbf z\right),\mathbf{curl}\left(\mathbf{curl}\, \boldsymbol\phi\right)\right)\vspace{2mm}\\
&=b\left(z,y,\phi\right)-b\left(y,z,\phi\right)-\alpha\left(\mathbf{curl}\left(\mathbf y\times \mathbf z\right),\Delta \boldsymbol\phi-\nabla\left( \mathrm{div}\, \boldsymbol\phi\right)\right)\vspace{2mm}\\
&=b\left(z,y,\phi\right)-b\left(y,z,\phi\right)-\alpha\left(b(\mathbf z,\mathbf y,\Delta\boldsymbol\phi)-b(\mathbf y,\mathbf z,\Delta \boldsymbol\phi)\right)\vspace{2mm}\\
&=b\left(z,y, \sigma(\phi)\right)-b\left(y,z,\sigma(\phi)\right)
\end{array}$$
and the second identity is proved.$\hfill \Box$
As will be seen in the sequel, the first identity in Lemma <ref> enables us to give an adequate variational setting for the state and linearized state equations. Based on the corresponding definitions, we can derive $H^1$ and $H^3$ a priori estimates and establish existence results.
Similarly, combining the two identities in Lemma <ref>, we can propose a variational formulation for the adjoint equation and establish a $H^1$ estimate of the corresponding solution.
This section concludes with a result that will be used to establish a uniqueness
result for the state equation and to derive $H^1$ a priori estimates for the linearized state equation and the adjoint equation.
Let $ y,z \in V_2$. Then
\left|\left( \mathbf{curl}\, \sigma(\mathbf z) \times \mathbf y, \mathbf z\right)\right|
\leq \left( S_4^2\left|y\right|_{H^1}+ \kappa
\alpha\left\|y\right\|_{H^3}\right)\left|z\right|_{H^1}^2,$$
where $\kappa$ is a positive constant only depending on $\Omega$.
Proof. Lemma <ref> together with classical arguments show that
\left( \mathbf{curl}\,\sigma(\mathbf z)\times
\mathbf y,\mathbf z\right)&=b\left(z, y, \sigma(z)\right)
-b\left(y,z, \sigma(z)\right) \vspace{2mm}\\
\left(z\cdot \nabla y-y\cdot \nabla z,\Delta z\right)\vspace{2mm}\\
+\alpha\left(b\left(\mathbf z, \mathbf{curl}\, \mathbf y,\mathbf{curl}\, \mathbf z\right)
-b\left(\mathbf y, \mathbf{curl}\, \mathbf z,\mathbf{curl}\, \mathbf z\right)\right)\vspace{2mm}\\
&\displaystyle +2\alpha\sum_{k=1}^3\left(\nabla \mathbf z_k\times \nabla \mathbf y_k,\mathbf{curl}\,\mathbf z\right)\vspace{-1mm}\\
&\displaystyle =b(z,y,z)
+\alpha b\left(\mathbf z, \mathbf{curl}\, \mathbf y,\mathbf{curl}\, \mathbf z\right)
+2\alpha\sum_{k=1}^2\left(\nabla \mathbf z_k\times \nabla \mathbf y_k,\mathbf{curl}\,\mathbf z\right).
\end{array}$$
$$\begin{array}{ll}&\left|\left( \mathbf{curl}\,\sigma(\mathbf z)\times \mathbf y,\mathbf z\right)\right|
\vspace{2mm}\\
&\displaystyle\leq \|z\|_4^2 \left|y\right|_{H^1}+
\alpha\left(\|z\|_4 \left\|\nabla \mathbf{curl}\,\mathbf y\right\|_{4} \left\|\mathbf{curl}\, \mathbf z\right\|_2+
2\sum_{k=1}^2\left\|\nabla \mathbf z_k\right\|_2
\left\|\nabla \mathbf y_k\right\|_\infty\left\|\mathbf{curl}\,
\mathbf z\right\|_2\right) \vspace{2mm}\\
&\leq \left( S_4^2\left|y\right|_{H^1}+\kappa\alpha
\left\|y\right\|_{H^3}\right) \|\nabla z\|_2^2
\end{array}$$
and the claimed result is proved.$\hfill\Box$
§ STATE EQUATION
§.§ Existence and uniqueness results for the state equation
The state equation can be written in a variational form by taking its scalar product with a test function in $ V$.
Let $ u\in L^2(\Omega)$. A function $ y\in V_2$ is a solution of $(\ref{equation_etat})$ if
\begin{equation}\label{var_form_state}\nu\left(\nabla y,\nabla \phi\right)+\left( \mathbf{curl}\, \sigma(\mathbf y)
\times \mathbf y, \boldsymbol\phi\right)=\left( u, \phi\right) \qquad \mbox{for all} \ \phi\in V.\end{equation}
Due to Lemma <ref>, the nonlinear term in the previous definition can be understood in the following sense
$$\begin{array}{ll}\left( \mathbf{curl}\, \sigma(\mathbf y)\times \mathbf y, \boldsymbol\phi\right)&=
b\left( \phi, y, \sigma(y)\right)-b\left( y,
\phi, \sigma(y)\right)
\vspace{2mm}\\&=b\left( y, y, \phi\right)
-\alpha\left(b\left( \phi, y,\Delta y\right)-b\left( y,
\phi,\Delta y\right)\right).\end{array}$$
Equation (<ref>) was first studied by Cioranescu and Ouazar (<cit.>, <cit.>) in the case of Dirichlet boundary conditions and simply connected domains. These authors proved existence and uniqueness of solutions by using Galerkin's method in the basis of the eigenfunctions of the operator $\mathrm{curl}\left(\mathrm{curl}\, \sigma(y)\right)$. More precisely, by using the fact that the imbedding of $V_2\subset V$ is compact, they prove the existence of a sequence of eigenfunctions $\left(e_j\right)_j\subset V_2$ corresponding to a sequence of eigenvalues $\left(\lambda_j\right)_j$ such that
\begin{equation}\label{eigenfunc}
\left(e_j,\phi\right)_{V_2}=
\lambda_j\left(
\left( e_j,\phi\right)
+\alpha\left(\nabla e_j,\nabla\phi\right)\right),
\qquad \mbox{for all} \ \phi\in V_2\end{equation}
$$0<\lambda_1<\cdots<\lambda_k<\cdots \longrightarrow +\infty.$$
The functions $ e_j$ form an orthonormal basis in $V$ and an orthogonal basis in $V_2$. Moreover,
$$ e_j\in H^4(\Omega), \quad
{\rm curl} \, \sigma(e_j)\in H^1(\Omega)$$
\begin{equation}\label{g_vs_curlg}\left({\rm curl}\, g,
{\rm curl}\, \sigma(e_j)\right)=\lambda_j\left(g, e_j\right)\qquad \mbox{for all} \ g\in H({\rm curl};\Omega).\end{equation}
This method, designed to decompose the problem into a Stokes-like system for the velocity $y$ and a transport equation for $\mathrm{curl}\,\sigma(y)$, allows to establish the existence of global solutions with $H^3$ regularity in the two dimensional case, and uniqueness and local existence in the three dimensional case. It has been extented by Cioranescu and Girault <cit.> to prove global existence in time in the three dimensional case and by Busuioc and Ratiu <cit.> to study the case of Navier-slip boundary conditions.
The following result deals with existence of a solution and is well known (see e.g. <cit.>). For the convenience of the reader, the corresponding estimates are derived herafter.
Let $ u\in H( \mathrm{curl};\Omega)$. Then problem $(\ref{equation_etat})$ admits at least one solution $ y\in V_2$ and this solution satisfies the following estimates
\begin{equation}\label{state_est1}
\left|y\right|_{H^1}\leq \tfrac{S_2}{\nu}
\left\| u\right\|_2,\end{equation}
\begin{equation}\label{state_est2}
\left|y\right|_{V_2}\leq \tfrac{1}{\nu}
\left(S_2\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right),\end{equation}
\begin{equation}\label{state_est3}
\left\| y\right\|_{H^3}\leq
\tfrac{\kappa}{\alpha\nu}\left(S_2\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right),
\end{equation}
where $\kappa$ is a positive constant depending only on $\Omega$.
Proof. Setting $\phi=y$ in (<ref>) and using the Poincaré inequality, we obtain
\left(u,y\right)-
\left( \mathbf{curl}\, \sigma\mathbf (\mathbf y)\times \mathbf y, \mathbf y\right)
\|u\|_2\|y\|_2\leq S_{2}\|u\|_2\left|y\right|_{H^1}.
\end{array}$$
which gives (<ref>). On the other hand, by applying the curl to
$(\ref{equation_etat})$, we obtain
$$-\nu \Delta \left(\mathrm{curl} \, y\right)+y\cdot \nabla
\mathrm{curl} \,\sigma(y)=\mathrm{curl} \, u$$
\begin{equation}\label{transport_state}
\mathrm{curl} \, \sigma(y)+\tfrac{\alpha}{\nu}\,
y\cdot \nabla\left(\mathrm{curl} \,\sigma(y)\right)
\mathrm{curl} \, u+\mathrm{curl} \, y.\end{equation}
Multiplying by $\mathrm{curl} \, \sigma(y)$ and integrating, we get
$$\begin{array}{ll}\left|y\right|_{V_2}^2&=\left\|\mathrm{curl}\, \sigma(y)\right\|_{2}^2\vspace{2mm}\\
&=-\tfrac{\alpha}{\nu}\left(y\cdot \nabla\left(\mathrm{curl} \,\sigma(y)\right),\mathrm{curl}\, \sigma(y)\right)+\left(\tfrac{\alpha}{\nu}\,\mathrm{curl} \, u+\mathrm{curl} \, y,\mathrm{curl}\,
\sigma(y)\right)\vspace{2mm}\\
&=\left(\tfrac{\alpha}{\nu}\,\mathrm{curl} \, u+\mathrm{curl} \, y,\mathrm{curl}\,
\sigma(y)\right) \vspace{2mm}\\
&\leq\left(\tfrac{\alpha}{\nu}\left\|\mathrm{curl} \, u\right\|_2+\left\|\mathrm{curl} \, y\right\|_2\right)
\left|y\right|_{V_2}\vspace{2mm}\\
&\leq\left(\tfrac{\alpha}{\nu}\left\|\mathrm{curl} \, u\right\|_2+\left|y\right|_{H^1}\right)
\left|y\right|_{V_2}
\end{array}$$
and thus
$$\left|y\right|_{V_2}\leq \left|y\right|_{H^1}
\left\| \mathrm{curl} \, u\right\|_2.$$
This estimate together with (<ref>) gives (<ref>).
Finally, since $\mathrm{curl}\,\Delta y\in L^2(\Omega)$ and $\nabla\cdot \left(\mathrm{curl}\,\Delta y\right)=0$, by arguing as in the proof of Lemma <ref>, we can establish the existence of a unique function $\psi\in H^1(\Omega)$ such that
$$-\Delta y -\psi+\nabla \pi=0$$
\begin{equation}\label{y2_sigma} \left\|y\right\|_{H^3}
\leq c\left\|\psi\right\|_{H^1}
\leq \kappa\left\|\mathrm{curl}\,\Delta y\right\|_{2},\end{equation}
where $\kappa$ is a positive constant only depending on $\Omega$.
Combining (<ref>) and $(\ref{y2_sigma})$, we obtain
\kappa^2\left\|\mathrm{curl} \, \Delta y\right\|_2^2\vspace{2mm}\\
&=\left(\tfrac{\kappa}{\alpha}\right)^2\left\|\mathrm{curl} \,
y-\mathrm{curl} \, \sigma(y)\right\|_2^2\vspace{2mm}\\
\left\|\mathrm{curl} \, y\right\|_2^2
\left(\mathrm{curl} \, y,\mathrm{curl} \, \sigma(y)\right)
\right)\vspace{2mm}\\
\left\|\mathrm{curl} \, y\right\|_2^2
\left(\mathrm{curl} \, u,\mathrm{curl} \, \sigma(y)\right)
\right)\vspace{2mm}\\
&\leq \left(\tfrac{\kappa}{\alpha}\right)^2\left(
\left\|\mathrm{curl} \, y\right\|_2^2
\left\|\mathrm{curl} \, u\right\|_2
\right)^2\right)\vspace{2mm}\\
&\leq \left(\tfrac{\kappa}{\alpha}\right)^2\left(
\left|y\right|_{H^1}^2
\left\|\mathrm{curl} \, u\right\|_2
\right)^2\right)\end{array}$$
and thus
$$\left\|y\right\|_{H^3}\leq \tfrac{\kappa}{\alpha}
\left(\left|y\right|_{H^1}+\tfrac{\alpha}{\nu}
\left\|\mathrm{curl} \, u\right\|_2\right).$$
Estimate (<ref>) is then a direct consequence of (<ref>).$\hfill\Box$
As in the case of Navier-Stokes equations, uniqueness of the solution is guaranteed under a restriction on the data. Additional regularity of the solution is obtained under the same restriction for more regular data.
Assume that $ u\in H(\mathrm{curl};\Omega)$. There exists a positive constant $ \bar\kappa$, depending only on $\Omega$, such that if $ u$ satisfies
\begin{equation}\label{uniqueness_condition}
\bar\kappa\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)<\nu^2,\end{equation}
then equation $(\ref{equation_etat})$ admits a unique solution $y$. Moreover, if ${\rm curl} \, u \in H^1(\Omega)$ then $y\in H^4(\Omega)$ and the following estimate holds
\left\|u\right\|_2+
\alpha\left\|{\rm curl} \,u\right\|_2\right)\right)
\left|{\rm curl}\, \sigma(y) \right|_{H^1}
\leq \tfrac{\kappa}{\alpha\nu}
\left(\left\|u\right\|_2+
\alpha\left\|{\rm curl} \,u\right\|_2+\alpha^2\left|
{\rm curl} \, u \right|_{H^1}\right).$$
Proof. Assume that $ y_1$ and $ y_2$ are two solutions of (<ref>) corresponding to $ u$ and denote by $ y$ the difference $ y_1- y_2$. By setting $\phi=y$ in the variational formulation (<ref>), we deduce that
$$\nu\left|y\right|_{H^1}^2+\left( \mathbf{curl} \,\sigma(\mathbf y_1)
\times \mathbf y_1- \mathbf{curl}\, \sigma(\mathbf y_2)
\times \mathbf y_2,\mathbf y\right)=0.$$
Observing that
$$\mathbf{curl} \,\sigma(\mathbf y_1)
\times \mathbf y_1- \mathbf{curl}\, \sigma(\mathbf y_2)
\times \mathbf y_2=\mathbf{curl} \,\sigma(\mathbf y_1)
\times \mathbf y+\mathbf{curl} \,\sigma(\mathbf y) \times \mathbf y_2,$$
and taking into account Lemma <ref>, we deduce that
$$\nu\left|y\right|_{H^1}^2+\left( \mathbf{curl} \,\sigma(\mathbf y)
\times \mathbf y_2,\mathbf y\right)=0.$$
Due to Lemma <ref>, (<ref>) and
(<ref>), it follows that
&\leq \tfrac{1}{\nu}\left( S_4^2\left|y_2\right|_{H^1}+
\kappa\alpha\left\| y_2
\right\|_{H^3}\right)\left|y\right|_{H^1}^2\vspace{2mm}\\
&\leq \tfrac{1}{\nu^2}\left( S_4^2 S_2\left\|u\right\|_2+
\kappa\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)
\left|y\right|_{H^1}^2\vspace{2mm}\\
&\leq \tfrac{ \bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\left|y\right|_{H^1}^2
\end{array}$$
implying that $ y_1= y_2$ if condition (<ref>) is satisfied. This proves the uniqueness result. The regularity result can be similarly established by using classical arguments on the transport equation. For the convenience of the reader, we will give a sketch of the proof and only derive the estimate that shall be applied to the solution of a Galerkin approximation of the problem.
By taking the gradient in (<ref>), we can see that $\varphi =\nabla\left(\mathrm{curl}\, \sigma(y)\right) $ is the solution of the following transport equation
$$\varphi +\tfrac{\alpha}{\nu}\,
y \cdot \nabla \varphi +\tfrac{\alpha}{\nu}
\left(\nabla y\right)^\top \cdot \varphi=
\nabla\left(\tfrac{\alpha}{\nu}\,{\rm curl} \,
u +{\rm curl} \, y \right).$$
$$\begin{array}{ll}\left\|\varphi \right\|_2^2&
=\left(\nabla\left(\tfrac{\alpha}{\nu}\,{\rm curl} \,
u +{\rm curl} \, y \right),
\varphi \right)
-\left(\tfrac{\alpha}{\nu}\left(\nabla y
\right)^\top \cdot
\varphi,\varphi\right)\vspace{1mm}\\
&\leq \left(\tfrac{\alpha}{\nu} \left\|{\rm curl} \, u
\right\|_{H^1}+\left\|{\rm curl} \, y
\right\|_{H^1}\right)
\left\|\varphi \right\|_2+
\tfrac{\alpha}{\nu}\left\|\nabla y \right\|_\infty
\left\|\varphi \right\|_2^2\vspace{2mm}\\
&\leq \left(\tfrac{\alpha}{\nu} \left\|{\rm curl} \, u
\right\|_{H^1}+c\left\|y
\right\|_{H^2}\right)
\left\|\varphi \right\|_2+
\tfrac{c\alpha}{\nu}\left\|y \right\|_{H^3}
\left\|\varphi \right\|_2^2\vspace{2mm}\\
&\leq \left(\tfrac{\alpha}{\nu} \left\|{\rm curl} \, u
\right\|_{H^1}+\tfrac{c}{\alpha}
\left\|{\rm curl} \, \sigma(y) \right\|_{2}\right)
\left\|\varphi \right\|_2+
\tfrac{ \bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\left\|\varphi \right\|_2^2\vspace{2mm}\\
&\leq \tfrac{\kappa}{\alpha\nu}
\left(\left\|u\right\|_2+
\alpha\left\|{\rm curl} \,u\right\|_2+\alpha^2\left\|\nabla\left({\rm curl} \, u \right)\right\|_2\right)
\left\|\varphi \right\|_2+
\tfrac{ \bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\left\|\varphi \right\|_2^2.
\end{array}$$
This gives the estimate and shows that $\mathrm{curl}\, \sigma(y)$ belongs to $H^1(\Omega)$. Arguing as in the proof of (<ref>) and (<ref>), it follows that $y\in H^4(\Omega)$. $\hfill\Box$
Notice that
$ \bar\kappa >S_4^2 S_2$. This
implies that if $u$ satisfies the condition stated in the previous proposition, then the corresponding Navier-Stokes equation has a unique weak solution.
§.§ Linearized state equation
The aim of this section is to study the solvability, in an adequate setting, of the linearized equation associated to the nonlinear state equation. Its solution is involved in the definition of the directional derivative of the control-to-state mapping and is related, through a suitable Green formula, to the adjoint state.
Let $u\in H( \mathrm{curl};\Omega)$, let $ y\in V_2$ be a corresponding solution of $(\ref{equation_etat})$ and consider the linear equation
\begin{equation}\label{linearized}\left\{
\begin{array}{ll}
-\nu\Delta \mathbf z+ \mathbf{curl}\,\sigma(\mathbf z)\times \mathbf y+
\mathbf{curl}\,\sigma(\mathbf y)\times \mathbf z+\nabla \pi=\mathbf w&\quad\mbox{in} \ \Omega,\vspace{2mm}\\
\mathrm{div} \, \mathbf z=0&\quad\mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf z=0&\quad\mbox{on}\ \Gamma,
\end{array}
\right .\end{equation}
where $w\in L^2(\Omega)$.
A function $ z\in V_2$ is a solution of $(\ref{linearized})$ if
\begin{equation}\label{var_lin}
\nu\left(\nabla z,\nabla \phi\right)+
\left(\mathbf{curl}\, \sigma(\mathbf z)\times
\mathbf y+\mathbf{curl}\, \sigma(\mathbf y)\times \mathbf z,
\boldsymbol\phi\right)=
\left( w, \phi\right) \qquad \mbox{for all} \
\phi\in V.\end{equation}
In analogy to the state equation, by taking into account Lemma
<ref>, we can rewrite the previous variational formulation as follows:
$$\nu\left(\nabla z,\nabla \phi\right)+
b\left( \phi, y, \sigma(z) \right)-b\left( y, \phi, \sigma(z) \right)+
b\left( \phi, z, \sigma(y) \right)-b\left( z, \phi, \sigma(y) \right)=
\left( w, \phi\right)$$
for all $\phi\in V$.
As already mentioned, the special Galerkin basis used to study the state equation (<ref>) is particularly well adapted and allows to prove existence of regular solutions with minimal assumptions on the data.
Seeming appropriate, the application of the same arguments to study the solvability of the linearized equation (<ref>) leads, however, to additional, yet expectable, issues. Indeed, after deriving the $H^1$ a priori estimate, this technique will naturally imposes the derivation of a $L^2$ estimate for
$\mathrm{curl}\, \sigma(z)$ (and thus $H^3$ for $z$). This term should satisfy the transport equation
$$\mathrm{curl}\, \sigma(z)+\tfrac{\alpha}{\nu} \,y\cdot \nabla
\left(\mathrm{curl}\, \sigma(z)\right)
+\tfrac{\alpha}{\nu}\,z\cdot \nabla
\left(\mathrm{curl}\, \sigma(y)\right)=\tfrac{\alpha}{\nu}
\,\mathrm{curl}\, w+\mathrm{curl}\, z$$
and in order to obtain the desired estimate, we need to guarantee that the coefficient $\mathrm{curl}\,\sigma(y)$ appearing in the linearized operator belongs to $H^1(\Omega)$. Following Proposition <ref>, this can be achieved if we consider more regular data in the state equation and impose additional restrictions on their size.
On the other hand, let us observe that the variational formulation stated above is well defined for $\sigma(z)\in L^2(\Omega)$ (and thus for $z\in H^2(\Omega)$) and that this regularity would be sufficient to carry out our analysis and derive the necessary optimality conditions. We might consider less restrictive choices for the Galerkin basis, but technical difficulties inherent to Dirichlet boundary conditions need to be managed.
Formally, the natural way to obtain the $H^2$ a priori estimates would be to multiply (<ref>) by $\sigma(z)$
and to integrate. The main difficulty is then to deal with the pressure term
$$\left(\nabla \pi,\sigma(z)\right)=-\left(\pi,\mathrm{div}\, \sigma(z)\right)+\int_\Gamma \pi n\cdot \sigma(z)=\int_\Gamma \pi n\cdot \sigma(z)$$
that does not vanish, unless $\sigma(z)$ is tangent to the boundary, and that we do no know how to adequately estimate.
The next result deals with existence of a regular solutions of the linearized equation when the corresponding data are sufficiently regular.
Let $u\in H( \mathrm{curl};\Omega)$ satisfying condition $(\ref{uniqueness_condition})$ and such that ${\rm curl} \, u \in H^1(\Omega)$, and let $y\in V_2\cap H^4(\Omega)$ be the corresponding solution of $(\ref{equation_etat})$. Then equation $(\ref{linearized})$ admits a unique solution $ z\in V_2$. Moreover, the following estimates hold
\begin{equation}\label{est_H1_z}
\left(1-\tfrac{\bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)\left|z\right|_{H^1}
\leq \tfrac{S_2}{\nu} \left\| w\right\|_2,\end{equation}
\begin{equation}\label{est_V2_z}\left|z\right|_{V_2}
\leq \tfrac{2\alpha}{\nu}\left\|{\rm curl}\, w\right\|_2+
\kappa\left(\tfrac{\alpha}{\nu^\frac{3}{2}}
\left|\mathrm{curl}\,\sigma(y)\right|_{H^1}^\frac{3}{2}+
where $\kappa$ is a positive constant depending only on $\Omega$.
Proof. The proof of Proposition <ref> is split into three steps. We first establish the existence of an approximate solution and a first estimate in $ H^1(\Omega)$. Next, we derive an estimate in $H^3(\Omega)$ and then we pass to the limit.
The solution of $(\ref{linearized})$ is constructed by means of Galerkin's discretization, by expanding the linearized state $z$ in the basis introduced in the previous section.
The approximate problem is defined by
\begin{equation}\label{faedo_galerkin_lin}
\left\{\begin{array}{ll}\mbox{Find} \
z_m=\displaystyle\sum_{j=1}^m \zeta_{j}
e_j \ \mbox{solution}, \ \mbox{for} \ 1\leq j\leq m, \
\mbox{of}\vspace{3mm}\\
\nu\left(\nabla z_m,\nabla e_j\right)+
\left( \mathbf{curl}\,\sigma(\mathbf z_m)\times \mathbf y+
\mathbf{curl}\,\sigma(\mathbf y)\times
\mathbf z_m, \mathbf e_j\right)=
\left(w, e_j\right).\end{array}\right.
\end{equation}
Step 1. Existence of the discretized solution and a priori $H^1$ estimate. We prove that the $H^1$ estimate can be derived if $u$ satisfies the condition $(\ref{uniqueness_condition})$. Let $m$ be fixed and consider $P: \ \mathbb{R}^m \longrightarrow \mathbb{R}^m$ defined by
$$\left(P \zeta\right)_j=\nu\left(\nabla z_m,\nabla e_j\right)+\left(\mathbf{curl}\,\sigma(\mathbf z_m)\times \mathbf y+
\mathbf{curl}\, \sigma(\mathbf y)\times \mathbf z_m, \mathbf e_j\right)-
\left( w, e_j\right), $$
where $ z_m=\sum_{j=1}^m\zeta_j e_j$. The mapping $P$ is obviously continuous. Let us prove that $P(\zeta)\cdot \zeta>0$ if $|\zeta|$ is sufficiently large. Classical arguments together with Lemma <ref> yields
\begin{align}\label{P_zeta}{P}(\zeta)\cdot \zeta&=
\nu \left|z_m\right|_{H^1}^2+\left(\mathbf{curl}\,\sigma(\mathbf z_m)\times \mathbf y, \mathbf z_m\right)-
\left( w, z_m\right)\\
&\geq \left(\nu-\left( S_4^2\left|y\right|_{H^1}+
\kappa\alpha\| y\|_{H^3}\right)\right)
\left|z_m\right|_{H^1}^2
-\left\| w\right\|_2\left\|z_m\right\|_2\nonumber\\
&\geq \left(\nu-\left( S_4^2\left|y\right|_{H^1}+
\kappa\alpha\| y\|_{H^3}\right)\right)
\left|\zeta\right|^2-\left\| w\right\|_2
\left|\zeta\right|\nonumber\\
&\geq \left(\nu-\tfrac{\bar\kappa}{\nu}
\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\right)\left|\zeta\right|^2-
\left\| w\right\|_2\left|\zeta\right|\nonumber\\
&\longrightarrow +\infty \qquad \mbox{when} \
|\zeta|\rightarrow +\infty.\nonumber\end{align}
Due to the Brouwer theorem, we deduce that there exists $ \zeta^\ast\in \mathbb{R}^m$ such that ${P}\left( \zeta^\ast\right)=0$ and thus $ z_m=\sum_{j=1}^m\zeta_j^\ast e_j$ is a solution of problem (<ref>). Due to $(\ref{P_zeta})$ and Lemma <ref>, it follows that
$$\begin{array}{ll}\nu \left|z_m\right|_{H^1}^2&=\left( \mathbf{curl}\,\sigma(\mathbf z_m)\times \mathbf y, \mathbf z_m\right)-
\left( w, z_m\right)\vspace{2mm}\\
&\leq \left( S_4^2\left|y\right|_{H^1}+
\kappa\alpha
\left\| y\right\|_{H^3}\right)
\left|z_m\right|_{H^1}^2+
S_2\left\| w\right\|_2
\left|z_m\right|_{H^1}\vspace{2mm}\\
&\leq\tfrac{\bar\kappa}{\nu}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\left|z_m\right|_{H^1}^2+
S_2\left\| w\right\|_2
\left|z_m\right|_{H^1}
\end{array}$$
which gives
\begin{equation}\label{est_H1_zm}\left(
1-\tfrac{ \bar\kappa}{\nu^2}
\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)\left|z_m\right|_{H^1}
\leq \tfrac{S_2}{\nu} \left\| w\right\|_2.\end{equation}
Step 2. A priori $H^3$ estimate. By taking into account (<ref>) and (<ref>) we have
$$\begin{array}{ll}\left|z_m\right|_{V_2}^2&=\left\|\mathrm{curl} \,\sigma(z_m)\right\|_2^2\vspace{2mm}\\
&=\displaystyle\sum_{j=1}^m \zeta_j^\ast\left(\mathrm{curl} \,\sigma(z_m),{\rm curl}\, \sigma(e_j)\right)\vspace{1mm}\\
&=\displaystyle\sum_{j=1}^m \zeta_j^\ast\left(\lambda_j-1\right)
\left(\left(z_m, e_ j \right)+\alpha
\left(\nabla z_m,\nabla e_ j \right)\right)
\vspace{1mm}\\
&=\displaystyle\sum_{j=1}^m \zeta_j^\ast\left(\lambda_j-1\right)
\left(\mathbf z_m-\tfrac{\alpha}{\nu}\left(\mathbf{curl} \,\sigma(\mathbf z_m)\times \mathbf y+
\mathbf{curl}\, \sigma\left(\mathbf y\right)\times \mathbf z_m-\mathbf w\right),
\mathbf e_ j \right)\vspace{1mm}\\
&=\displaystyle\sum_{j=1}^m \zeta_j^\ast
\left({\rm curl}\,z_m+\tfrac{\alpha}{\nu}\, {\rm curl}\,w,
{\rm curl}\, \sigma(e_ j)\right)\vspace{1mm}\\
&-\tfrac{\alpha}{\nu}\displaystyle\sum_{j=1}^m \zeta_j^\ast
\left(\mathbf{curl}\left(\mathbf{curl} \,\sigma(\mathbf z_m)\times \mathbf y+
\mathbf{curl}\, \sigma\left(\mathbf y\right)\times \mathbf z_m\right),
\mathbf{ curl}\, \sigma(\mathbf e_j)\right)
\end{array}$$
&=\left({\rm curl}\,z_m+\tfrac{\alpha}{\nu}\,
{\rm curl}\,w,
{\rm curl}\, \sigma(z_m)\right)\vspace{3mm}\\
\left(\mathbf{curl}\left(\mathbf{curl} \,\sigma(\mathbf z_m)\times \mathbf y+
\mathbf{curl}\, \sigma\left(\mathbf y\right)\times \mathbf z_m\right),
\mathbf{ curl}\, \sigma(\mathbf z_m)\right)\vspace{2mm}\\
\left({\rm curl}\,z_m+\tfrac{\alpha}{\nu}\,
{\rm curl}\,w, {\rm curl}\, \sigma(z_m)\right)\vspace{2mm}\\
&-\tfrac{\alpha}{\nu}\left(b\left(y,{\rm curl}\, \sigma(z_m),
{\rm curl}\, \sigma(z_m)\right)
+b\left(z_m,{\rm curl}\, \sigma(y),{\rm curl}\,
\sigma(z_m)\right)\right)\vspace{2mm}\\
&=\left({\rm curl}\,z_m+\tfrac{\alpha}{\nu}\,
{\rm curl}\,w, {\rm curl}\, \sigma(z_m)\right)
-\tfrac{\alpha}{\nu}\,b\left(z_m,{\rm curl}\,
\sigma(y),{\rm curl}\, \sigma(z_m)\right).
\end{array}$$
Due to Lemma <ref>, we deduce that
\leq\left\|{\rm curl}\,z_m\right\|_2+
\tfrac{\alpha}{\nu}
\left\|{\rm curl}\, w\right\|_2+
\tfrac{\alpha}{\nu}\left\|z_m\right\|_\infty
\left|\mathrm{curl}\,\sigma(y)\right|_{H^1}
\vspace{2mm}\\
\left\|{\rm curl}\,z_m\right\|_2+
\tfrac{\alpha}{\nu}
\left\|{\rm curl}\, w\right\|_2+
\tfrac{c\alpha^\frac{2}{3}}{\nu}\left|z_m\right|_{H^1}^{\frac{2}{3}}\left|z_m\right|_{V_2}^{\frac{1}{3}}
\left|{\rm curl}\, \sigma(y)\right|_{H^1}\end{array}$$
and by using the Young inequality we finally obtain
\begin{equation}\label{est_V2_zm}
\left|z_m\right|_{V_2}
\leq \tfrac{2\alpha}{\nu}\left\|{\rm curl}\, w\right\|_2+
\kappa\left(\tfrac{\alpha}{\nu^\frac{3}{2}}
\left|\mathrm{curl}\,\sigma(y)\right|_{H^1}^\frac{3}{2}+
Step 3. Passing to the limit. It remains to pass to the limit with respect to $m$. From estimates (<ref>) and (<ref>), it follows that if $u$ satisfies condition $(\ref{uniqueness_condition})$ then there exists a subsequence, still indexed by $m$, and function $ z\in V_2$ such that
$$ z_m \longrightarrow z \qquad \mbox{weakly in} \ V_2.$$
By passing to the limit in (<ref>), we obtain for every $j\geq 1$
$$\nu\left(\nabla z,\nabla e_j\right)+
b\left( e_j, y, \sigma(z)\right)-b\left( y, e_j,
\sigma(z)\right)+b\left( e_j, z, \sigma(y) \right)
-b\left( z, e_j, \sigma(y)\right)=
\left( w, e_j\right) $$
and by density we prove that $ z$ satisfies the variational formulation. Moreover, $z$ satisfies estimates (<ref>) and (<ref>). Finally, since (<ref>) is linear, the uniquess result is direct consequence of estimate (<ref>).$\hfill\Box$
§ ANALYSIS OF THE CONTROL-TO-STATE MAPPING
§.§ Sequential and Lipschitz continuity
We are first concerned with continuity properties of the control-to-state mapping in adequate topologies.
Let $U$ be a bounded closed subset of $H(\mathrm{curl};\Omega)$. Then the control-to-state mapping is sequentially continuous from $U$, endowed with its weak topology, into
Proof. Let $(u_k)_k \subset U$ be a sequence converging to $u$ in the weak topology of $H(\mathrm{curl};\Omega)$ and let $y_k$ be a solution of $(\ref{equation_etat})$ corresponding to $u_k$.
Due to estimates (<ref>) and (<ref>), we have
$$\left|y_k\right|_{V_2}\leq \tfrac{1}{\nu}
\left(S_2\left\|u_k\right\|_2+\alpha
\left\|\mathrm{curl}\, u_k\right\|_2\right),$$
$$\left\|y_k\right\|_{H^3}\leq \tfrac{\kappa}{\alpha \nu}
\left(S_2\left\|u_k\right\|_2+\alpha
\left\|\mathrm{curl}\, u_k\right\|_2\right)$$
and since $(u_k)_k$ is uniformly bounded in $H(\mathrm{curl};\Omega)$, we deduce that the sequence $(y_k)_k$ is bounded in $V_2$. Then there exists a subsequence, still indexed by $k$, and $y\in V_2$, such that $\left(y_k\right)_{k}$ weakly converges to $y$ in $H^3(\Omega)$ and (by using compactness results on Sobolev spaces) strongly in $H^2(\Omega)$.
By passing to the limit in the variational formulation corresponding to $y_k$, we obtain
$$\nu \left(\nabla y,\nabla \phi\right)+
b\left( \phi, y, \sigma(y)\right)-b\left(y,
\phi, \sigma(y)\right)=(u,\phi) \qquad \mbox{for all} \
\phi \in V$$
implying that $y$ is a solution of (<ref>) corresponding to $u$, and the claimed result is proved.$\hfill\Box$
Next, we analyze the local Lipschitz continuity of the state with respect to the control variable. More precisely, if $u_1$, $u_2$ are two controls in $H({\rm curl};\Omega)$ and if $y_1$, $y_2$ are two corresponding states then, by assuming that one of the control variables satisfies the restriction $(\ref{uniqueness_condition})$, we estimate $\left|y_1-y_2\right|_{H^1}$ with respect to $\left\|u_1-u_2\right\|_2$. Under the additional assumption that this control variable is regular enough, we can also estimate $\left|y_1-y_2\right|_{V_2}$ with respect to $\left\|u_1-u_2\right\|_{H({\rm curl};\Omega)}$.
Let $ u_1, u_2\in H(\mathrm{curl};\Omega)$ with $u_2$ satisfying condition $(\ref{uniqueness_condition})$, and let $ y_1, y_2 \in V_2$ be corresponding solutions of $(\ref{equation_etat})$. Then the following estimate holds
\begin{equation}\label{lipschitz_est_H1}
\left(1-\tfrac{ \bar\kappa}{\nu^2}\left(\left\| u_2\right\|_2+\alpha
\left\| \mathrm{curl} \, u_2\right\|_2\right)\right)
\left|y_1- y_2\right|_{H^1}\leq \tfrac{S_2}{\nu} \left\| u_1- u_2\right\|_2.
\end{equation}
Moreover, if ${\rm curl}\,u_2$ belongs to $H^1(\Omega)$ then
\begin{equation}\label{lipschitz_est_V2}
\left|y_1-y_2\right|_{V_2}
\leq \tfrac{2\alpha}{\nu}\left\|{\rm curl}\, (u_1-u_2)\right\|_2
\left|{\rm curl}\, \sigma(y_2)\right|_{H^1}^\frac{3}{2}+
\end{equation}
where $\kappa$ is a positive constant only depending on $\Omega$.
Proof. The proof is split into two steps.
Step 1. A priori $H^1$ estimate.
It is easy to see that $y=y_1-y_2$ satisfies
\begin{equation}\label{y1-y2}\left\{
\begin{array}{ll}
-\nu\Delta \mathbf y+ \mathbf{curl}\, \sigma(\mathbf y)\times \mathbf y_2+
\mathbf{curl}\, \sigma(\mathbf y_1)\times \mathbf y+\nabla \pi= \mathbf u&\quad\mbox{in} \ \Omega,\vspace{2mm}\\
\mathrm{div} \, \mathbf y=0&\quad\mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf y=0&\quad\mbox{on}\ \Gamma,
\end{array}
\right.\end{equation}
where $u=u_1-u_2$. By setting $\phi=y$ in the corresponding variational formulation, we obtain
\left( \mathbf{curl}\, \sigma(\mathbf y)\times \mathbf y_2, \mathbf y\right)=\left( u, y\right).$$
Due to Lemma <ref>, (<ref>) and
(<ref>), it follows that
&\leq \left\| u\|_2\| y\right\|_2+
\left(S_4^2\left|y_2\right|_{H^1}+\kappa
\alpha\left\|y_2\right\|_{H^3}\right)
\left|y\right|_{H^1}^2\vspace{2mm}\\
&\leq S_2\left\|u\right\|_2\left|y\right|_{H^1} +
\tfrac{\bar\kappa}{\nu} \left(\left\|u_2\right\|_2
+\alpha \left\|{\rm curl}\, u_2\right\|_2\right)\left|y\right|_{H^1}^2
\end{array}$$
and (<ref>) holds.
Step 2. A priori $H^3$ estimate. To prove (<ref>), let us first recall that if ${\rm curl}\,u_2\in H^1(\Omega)$, then ${\rm curl}\, \sigma(y_2)\in H^1(\Omega)$ (cf. Proposition <ref>). Using (<ref>), we can see that $\tau={\rm curl}\, \sigma(y_2)-{\rm curl}\, \sigma(y_1)={\rm curl}\, \sigma(y)$ is the solution of the following transport equation
$$\tau+\tfrac{\alpha}{\nu} \,y_1\cdot \nabla \tau
+\tfrac{\alpha}{\nu} \, y\cdot
\nabla \left({\rm curl}\, \sigma(y_2)\right)
={\rm curl}\, y
+\tfrac{\alpha}{\nu}\,{\rm curl} \, u$$
and satisfies
\left( y\cdot
\nabla \left({\rm curl}\, \sigma(y_2)\right),\tau\right)=\left({\rm curl}\, y
+\tfrac{\alpha}{\nu}\,{\rm curl} \, u,\tau\right).$$
By taking into account Lemma <ref> and using the Young inequality, we obtain
\left\|\tau\right\|_2&\leq \left\|{\rm curl}\,y\right\|_2
\left\|{\rm curl} \, u\right\|_2+\tfrac{\alpha}{\nu}
\left\|y\cdot
\nabla \left({\rm curl}\, \sigma(y_2)\right)
\right\|_2\vspace{2mm}\\
&\leq \left\|{\rm curl}\,y\right\|_2
\left\|{\rm curl} \, u\right\|_2+\tfrac{\alpha}{\nu} \left\|y\right\|_\infty \left|{\rm curl}\, \sigma(y_2)\right|_{H^1} \vspace{2mm}\\
&\leq \left\|{\rm curl}\,y\right\|_2
\left\|{\rm curl} \, u\right\|_2+\tfrac{c\alpha^\frac{2}{3}}{\nu}\left|y\right|_{H^1}^{\frac{2}{3}}
\left|{\rm curl}\, \sigma(y_2)\right|_{H^1}
\left\|\tau\right\|_2^{\frac{1}{3}}\vspace{2mm}\\
&\leq \left\|{\rm curl}\,y\right\|_2
\left\|{\rm curl} \, u\right\|_2+
\tfrac{1}{2} \left\|\tau\right\|_2+
\tfrac{c\alpha}{\nu^\frac{3}{2}}
\left|y\right|_{H^1}
\left|{\rm curl}\, \sigma(y_2)\right|_{H^1}^\frac{3}{2}.\end{array}$$
\leq \tfrac{2\alpha}{\nu}\left\|{\rm curl}\, u\right\|_2
\left|{\rm curl}\, \sigma(y_2)\right|_{H^1}^\frac{3}{2}+
which gives the result.$\hfill\Box$
§.§ Gâteaux differentiability
At this stage, we are able to study the Gâteaux-differentiability of the control-to-state mapping.
Let $u, w\in H(\mathrm{curl};\Omega)$ and assume in addition that
${\rm curl}\, u\in H^1(\Omega)$ satisfies condition $(\ref{uniqueness_condition})$. For
$0<\rho<1$, set $ u_\rho= u+\rho w$, and let $y$ and $ y_{\rho}$ be solutions of
$(\ref{equation_etat})$ corresponding to $u$ and $u_\rho$, respectively. Then we have
$$ y_{\rho}= y+\rho
z+\rho r_\rho \qquad
\mbox{with} \ \lim_{\rho\rightarrow 0}
\left|r_\rho\right|_{H^1}=0,$$
where $ z\in V_2$ is a solution of $(\ref{linearized})$ corresponding to $(y,w)$.
Proof. Easy calculation shows that $z_{\rho}=\frac{ y_{\rho}- y}{\rho}$ satisfies
$$-\nu \Delta \mathbf z_\rho+ \mathbf{curl}\,
\sigma\left(\mathbf z_\rho\right)\times \mathbf y+ \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)\times \mathbf z_\rho+\nabla\pi_\rho= \mathbf w.$$
Let $ z\in V_2$ be the solution of (<ref>).
Then $r_\rho= z_\rho- z$ satisfies
$$-\nu \Delta \mathbf r_\rho+
\mathbf{curl}\,\sigma\left(\mathbf r_\rho\right)\times \mathbf y
+ \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)\times
\mathbf r_\rho
+ \mathbf{curl}\,\sigma\left(\mathbf y_\rho-\mathbf y\right)
\times \mathbf z
Multiplying this equation by $r_\rho$, we obtain
\begin{equation}\label{energy_r}\nu \left|r_\rho\right|_{H^1}^2+
\left(\mathbf{curl}\,\sigma\left(\mathbf r_\rho\right)\times \mathbf y+ \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)
\times \mathbf r_\rho+\mathbf{curl}\,\sigma\left(\mathbf y_\rho-\mathbf y\right)
\times \mathbf z,\mathbf r_\rho\right)=0.\end{equation}
It is easy to verify that
\begin{equation}\label{energy_r2}\left( \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)
\times \mathbf r_\rho,\mathbf r_\rho\right)=
\end{equation}
Moreover, by taking into account Lemma <ref> and estimates (<ref>)-(<ref>), we get
\begin{equation}\label{energy_r3}\left|\left(\mathbf{curl}\,
\sigma\left(\mathbf r_\rho\right)\times \mathbf y,\mathbf
\leq \tfrac{\bar\kappa}{\nu}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\left|r_\rho\right|_{H^1}^2.
\end{equation}
Combining (<ref>)-(<ref>), we deduce that
\left(1-\tfrac{ \bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)
\left|r_\rho\right|_{H^1}^2&\leq \tfrac{1}{\nu}
\left|\left(\mathbf{curl}\,\sigma\left(\mathbf y_\rho-\mathbf y\right)
\times \mathbf z,\mathbf r_\rho\right)\right|\vspace{2mm}\\
&\leq \tfrac{1}{\nu}
\left|y_\rho-y\right|_{V_2}
\left\|z\right\|_\infty
\left\|r_\rho\right\|_2\vspace{2mm}\\
&\leq \tfrac{S_2}{\nu}
\left|y_\rho-y\right|_{V_2}
\left\|z\right\|_\infty
\left|r_\rho\right|_{H^1}
\end{array}$$
and thus
$$\left(1-\tfrac{ \bar\kappa}{\nu^2}
\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)
\left|r_\rho\right|_{H^1}\leq \tfrac{S_2}{\nu}
\left|y_\rho-y\right|_{V_2}
\left\|z\right\|_\infty.$$
The conclusion follows by observing that the term on the right-hand side of the previous inequality tends to zero when $\rho$ tends to zero. Indeed, due to (<ref>) and (<ref>),
we have
\left|y_\rho-y\right|_{V_2}&
\leq \tfrac{2\alpha}{\nu}\left\|{\rm curl}\, (u_\rho-u)\right\|_2
\left|{\rm curl}\, \sigma(y)\right|_{H^1}^\frac{3}{2}+
&\leq \left(\tfrac{2\alpha}{\nu}\left\|{\rm curl}\, w\right\|_2
\left|{\rm curl}\, \sigma(y)\right|_{H^1}^\frac{3}{2}+
1\right) \tfrac{S_2\nu}{\nu^2-\bar\kappa
\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2
\right)} \left\|w\right\|_2\right)\rho\vspace{2mm}\\
&\longrightarrow 0 \qquad \mbox{when} \ \rho \rightarrow 0\end{array}$$
and the claimed result is proved. $\hfill\Box$
§ ADJOINT EQUATION
Let $u\in H( \mathrm{curl};\Omega)$ and let $ y\in V_2$ be a corresponding solution of (<ref>). The aim of this section is to study the existence of a weak solution for the adjoint state equation defined by
\begin{equation}\label{adjoint}\left\{
\begin{array}{ll}
-\nu\Delta \mathbf p- \mathbf{curl}\,\sigma(\mathbf y)\times \mathbf p +\mathbf{curl}\left(\sigma\left(\mathbf y\times \mathbf p\right)\right)+\nabla \pi=\mathbf f&\quad\mbox{in} \ \Omega,\vspace{2mm}\\
\mathrm{div} \, \mathbf p=0&\quad\mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf p=0 &\quad\mbox{on}\ \Gamma,
\end{array}
\right .\end{equation}
where $f\in L^2(\Omega)$.
The two identities in Lemma <ref> motivates the following variational formulation.
A function $p\in V$ is a weak solution of $(\ref{adjoint})$ if
\begin{equation}\label{form_var_lin_adj}
\nu\left(\nabla p,\nabla \phi\right)+
b\left( p,\phi, \sigma(y) \right)-b\left( \phi,p, \sigma(y) \right)+
b\left(p, y, \sigma(\phi) \right)-b\left( y,p, \sigma(\phi) \right)=
\left(f, \phi\right)\end{equation}
for all $\phi\in V \cap H^2(\Omega)$.
This formulation allows us to relate the adjoint state to the solution of the linearized equation and is particularly suited to derive the necessary optimality conditions.
As will be seen below, existence of a Galerkin approximate solution
can be established by taking into account the formulation stated in Definition <ref>. A corresponding a priori $H^1$ estimate can be derived and is sufficient to pass to the limit and prove the existence of a weak solution for the adjoint equation. Establishing a uniqueness result is much more challenging and requires higher regularity of the solutions. In this context, the observations raised in Section <ref>, concerning the most appropriate choice for the Galerkin basis, would similarly apply but deriving a $V_2$ estimate for the approximate solution of (<ref>) is far more difficult than in the case of the linearized equation. In order to illustrate our point, we can adapt the decomposition method and easily see that the term ${\rm curl}\, \sigma(p)$ should (formally) satisfy
$${\rm curl}\, \sigma(p)-\tfrac{\alpha}{\nu}\,
p\cdot \nabla\left({\rm curl}\, \sigma(y)\right)+
\tfrac{\alpha}{\nu}\, {\rm curl}
\left({\rm curl}\left(\sigma\left( y\times p\right)\right)\right)=
\tfrac{\alpha}{\nu}\,{\rm curl}\, f+{\rm curl}\, p.
Solving this equation is not an easy task: in addition to high order derivatives of $p$ that we need to manage, the coefficients in ${\rm curl}\left({\rm curl}\left(\sigma\left( y\times \cdot\right)\right)\right)$ also involve high order derivatives of the state variable $y$. Following the ideas developed in Section
<ref>, we may prove that for every integer $k\geq 0$, if $\Gamma$ is of class $C^{k+2,1}$ then the semi-norm
$\left|{\rm curl}\, \sigma(\cdot)\right|_{H^k}$ is equivalent to the norm
$\left\|\cdot\right\|_{H^{k+2}}$. Recalling that ${\rm curl}\, \sigma(y)$ satisfies (<ref>) and in view of the classical regularity results for transport equations (generally based on fixed point arguments) the high order derivatives of the state variable are well defined if we assume that the control is accordingly regular and if we impose an additional restriction on the size of $y$ (and consequently on the corresponding control). Unlike the linearized equation, where the condition on the size of the data is set on the natural space $H({\rm curl};\Omega)$ and also guarantees uniqueness of the solution for the state equation and Gâteaux differentiability of the control-to-state variable, the condition we need to impose here is set on higher-order Sobolev spaces and is much more restrictive.
An other aspect reinforces the idea that the effort in obtaining such regularity results for the adjoint state may not be necessarily compensated. Keeping in mind that our objective is to derive first-order optimality conditions and that the natural framework for the controls is $H({\rm curl};\Omega)$, we should not require a priori additional regularity on this variable (and on the corresponding state). On the other hand,
the results obtained in the previous sections concerning the solvability of the linearized state equation and the differentiability of the
control-to-state mapping are only available in the case of regular data. To overcome this difficulty, our idea is to consider an approximate optimal control problem governed by a state equation involving regularized controls. The results stated in Sections
<ref> and <ref> are then valid and we can derive the corresponding approximate optimality conditions. In order to pass to the limit, when the regularization parameter tends to zero, we only need a uniform estimate for the regularized adjoint state in $V$.
Let $ u\in H(\mathrm{curl};\Omega)$ satisfying condition $(\ref{uniqueness_condition})$ and let $ y\in V_2$ be the corresponding solution of $(\ref{equation_etat})$. Then equation $(\ref{adjoint})$ admits at least a weak solution $p\in V$. Moreover, the following etimate holds
\begin{equation}\label{est_H1_p}
\left(1-\tfrac{ \bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)\left|p\right|_{H^1}
\leq \tfrac{S_2}{\nu} \left\|f\right\|_2.\end{equation}
Proof. We first establish the existence of an approximate solution and derive a corresponding apriori estimate in $H^1(\Omega)$. We next pass to the limit and prove our statement.
Step 1. Existence of an approximate solution and a priori $H^1$ estimate. Consider the approximate problem defined by
\begin{equation}\label{faedo_galerkin_adj}
\left\{\begin{array}{ll}\mbox{Find} \
p_m=\displaystyle\sum_{j=1}^m \zeta_{j}
e_j \ \mbox{solution}, \ \mbox{for} \ 1\leq j\leq m, \
\mbox{of}\vspace{3mm}\\
\nu\left(\nabla p_m,\nabla e_j\right)-\left( \mathbf{curl}\,\sigma(\mathbf y)\times \mathbf p_m, e_j\right)+\left(\mathbf{curl}\left(\sigma\left(\mathbf y\times \mathbf p_m\right)\right), e_j\right)=
\left(f, e_j\right).\end{array}\right.
\end{equation}
where $( e_j)_j\subset H^4(\Omega)$ is the set of the eigenfunctions, solutions of (<ref>).
Due to Lemma <ref>, we have
\begin{align}\label{form_adj_vw}
-\left( \mathbf{curl}\,\sigma(\mathbf y)\times \mathbf p_m, e_j\right)
+\left(\mathbf{curl}\left(\sigma\left(\mathbf y\times \mathbf p_m\right)\right), e_j\right)&
=b\left( p_m, e_j, \sigma(y) \right)-b\left( e_j,p_m, \sigma(y) \right)\nonumber\\
b\left(p_m, y, \sigma( e_j) \right)-b\left( y,p_m, \sigma( e_j) \right).\end{align}
Let then $m$ be fixed and consider $Q: \ \mathbb{R}^m \longrightarrow \mathbb{R}^m$ defined by
$$\begin{array}{ll}\left(Q \zeta\right)_i=&\nu\left(\nabla p_m,\nabla e_j\right)+
b\left( p_m, e_j, \sigma(y) \right)-b\left( e_j,p_m, \sigma(y) \right)\vspace{2mm}\\
b\left(p_m, y, \sigma( e_j) \right)-b\left( y,p_m, \sigma( e_j) \right)-
\left(f, e_j\right), \end{array}$$
where $ p_m=\sum_{i=1}^m\zeta_i e_i$. The mapping $Q$ is obviously continuous. Let us prove that $Q(\zeta)\cdot \zeta>0$ if $|\zeta|$ is sufficiently large. Arguing as in the proof of Proposition <ref>, we may prove that
\begin{align}\label{Q_zeta}{Q}(\zeta)\cdot \zeta&=
\nu \left|p_m\right|_{H^1}^2+b\left(p_m, y, \sigma(p_m) \right)-b\left( y,p_m, \sigma(p_m) \right)-
\left(f, p_m\right)\\
&\geq \left(\nu-\tfrac{\bar\kappa}{\nu} \left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\right)\left|\zeta\right|^2-
\left\|f\right\|_2\left|\zeta\right|\nonumber\\
&\longrightarrow +\infty \qquad \mbox{when} \
|\zeta|\rightarrow +\infty.\nonumber\end{align}
Due to the Brouwer theorem, we deduce that there exists $ \zeta^\ast\in \mathbb{R}^k$ such that ${Q}\left( \zeta^\ast\right)=0$ and thus $ p_m=\sum_{i=1}^k\zeta_i^\ast e_i$ is a solution of problem (<ref>). Taking into account $(\ref{Q_zeta})$ and Lemma <ref>, we deduce that
$$\begin{array}{ll}\nu \left|p_m\right|_{H^1}^2&=
\left( \mathbf{curl}\,\sigma(\mathbf p_m)\times \mathbf y, \mathbf p_m\right)-
\left(f, p_m\right)\vspace{2mm}\\
&\leq\tfrac{ \bar\kappa}{\nu}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)
\left|p_m\right|_{H^1}^2+
\left|p_m\right|_{H^1}
\end{array}$$
which gives
\begin{equation}\label{est_H1_pm}\left(
1-\tfrac{ \bar\kappa}{\nu^2}\left(\left\| u\right\|_2+\alpha
\left\| \mathrm{curl} \, u\right\|_2\right)\right)\left\|\nabla p_m\right\|_2
\leq \tfrac{S_2}{\nu} \left\|f\right\|_2.\end{equation}
Step 2. Passing to the limit. It remains to pass to the limit with respect to $m$. From estimate (<ref>), it follows that if
$\bar u$ satisfies condition $(\ref{uniqueness_condition})$, then there exists a subsequence, still indexed by $m$, and function $ p\in V_2$ such that
$$ p_m \longrightarrow p \qquad \mbox{weakly in} \ V.$$
By taking into account (<ref>) and passing to the limit in (<ref>), we obtain for every $j\geq 1$
$$\nu\left(\nabla p,\nabla e_j\right)+b\left( p, e_j, \sigma(y) \right)-b\left( e_j,p, \sigma(y) \right)+
b\left(p, y, \sigma( e_j) \right)-b\left( y,p, \sigma( e_j) \right))=
\left(f, e_j\right) $$
and by density we prove that $ p$ satisfies the variational formulation (<ref>). Moreover, $p$ satisfies (<ref>). $\hfill\Box$
§ PROOF OF THE MAIN RESULTS
Unless necessary, and in order to simplify the redaction, the index $\alpha$ will be dropped.
§.§ Proof of the existence of an optimal control for $(P_\alpha)$
We first prove Theorem <ref>. Let $(u_k,y_k)_k\subset U_{ad}\times V_2$ be a minimizing sequence. Since $(u_k)_k$ is uniformly bounded in the closed convex set $U_{ad}$, we may extract a subsequence, still indexed by $k$, weakly convergent to some $u\in U_{ad}$. Applying Proposition <ref> with $U=U_{ad}$, it follows that $(y_k)_k$ converges to $y$, solution of (<ref>) corresponding to $u$, in $H^2(\Omega)$. The convexity and continuity of $J$ imply the lower semicontinuity of $J$ in the weak topology and
$$J(u,y)\leq \liminf_k J(u_k,y_k)=\inf(P_\alpha),$$
showing that $(u,y)$ is a solution for $(P_\alpha)$.$\hfill\Box$
§.§ Proof of the necessary optimality conditions
for $(P_\alpha)$
§.§.§ Approximate optimal control problem
For $\varepsilon>0$, we denote by $\varrho_\varepsilon$ a Friedrichs mollifier, i.e. the convolution operator defined by
$$\varrho_\varepsilon(u)=\varrho_\varepsilon \ast u,$$
where $\varrho_\varepsilon(x)=\varepsilon^{-2} \varrho\left(\tfrac{x}{\varepsilon}\right)$ and $\varrho$ is a positive radial compactly supported smooth function whose integral is equal $1$. Let us recall some usefull properties of these mollifiers.
* $\varrho_\varepsilon$ is selfadjoint for the $L^2$ scalar product, i.e.
\left(\varrho_\varepsilon(u),v\right)=\left(u,\varrho_\varepsilon(v)\right)
\qquad \mbox{for all} \ u,v\in L^2(\Omega).$$
* $\varrho_\varepsilon$ commutes with derivatives.
* For $m\in \mathbb{N}$ and $u\in H^m(\Omega)$, we have
\left\|\varrho_\varepsilon(u)\right\|_{H^m}\leq
\left\|u\right\|_{H^m} \quad \mbox{and} \quad
\lim_{\varepsilon \rightarrow 0^+}
\left\|u-\varrho_\varepsilon(u)\right\|_{H^m}=0.$$
Due to the first and second properties, we have
\begin{align}\label{auto_adj_molli}
\left(\varrho_\varepsilon \left(u\right),v
\right)_{H({\rm curl};\Omega)}&=
\left(\varrho_\varepsilon \left(u\right),v\right)+
\left({\rm curl}\, \varrho_\varepsilon \left(u\right),
{\rm curl}\,v\right)\nonumber\\
&=\left(\varrho_\varepsilon \left(u\right),v\right)+
\left(\varrho_\varepsilon \left({\rm curl}\, u\right),
{\rm curl}\,v\right)\nonumber\\
&=\left(u,\varrho_\varepsilon \left(v\right)\right)+
\left({\rm curl}\, u,
\varrho_\varepsilon \left({\rm curl}\,v\right)\right)\nonumber\\
&=\left(u,\varrho_\varepsilon \left(v\right)\right)+
\left({\rm curl}\, u,
{\rm curl}\, \varrho_\varepsilon \left(v\right)\right)\nonumber\\
&=\left(u,\varrho_\varepsilon \left(v\right)
\right)_{H({\rm curl};\Omega)}
\qquad \mbox{for all} \ u,v\in H({\rm curl};\Omega).\end{align}
Let $(\bar u,\bar y)$ be a solution of $(P_\alpha)$ with $\bar u$ satisfying condition (<ref>) and consider the control problem $(P_\alpha^\varepsilon)$ defined in Section <ref>. We first prove the existence of an optimal control for $(P_\alpha^\varepsilon)$. The proof combines the standard arguments already used to establish Theorem <ref> with the properties of the mollifiers.
Assume that $U_{ad}$ is bounded in $H(\mathrm{curl};\Omega)$. Then problem $(P_\alpha^\varepsilon)$ admits a solution.
Proof. Let $(u_{k}^\varepsilon,y_{k}^\varepsilon)_k\subset U_{ad}\times V_2$ be a minimizing sequence for $(P_\alpha^\varepsilon)$. Then there exists a subsequence, still indexed by $k$, and $u^\varepsilon\in U_{ad}$ such that $(u_k^\varepsilon)_k$ weakly converges to $u^\varepsilon$ in $H(\mathrm{curl};\Omega)$. Since $\varrho_\varepsilon$ is linear and continuous, it is weakly continuous and thus $\left(\varrho_\varepsilon \left(u_k^\varepsilon\right)\right)_k$ weakly converges to $\varrho_\varepsilon \left(u^\varepsilon\right)$
in $H({\rm curl};\Omega)$.
By taking into account Proposition <ref> (with $U=\varrho_\varepsilon(U_{ad})$), we deduce that $\left(y_{k}^\varepsilon\right)_{k}$ converges in $H^2(\Omega)$ to
$y^\varepsilon$, a solution of (<ref>) corresponding to $\varrho_\varepsilon \left(u^\varepsilon\right)$. This implies that $(y^\varepsilon,u^\varepsilon)$ is admissible for $(P_\alpha^\varepsilon)$ and by using the convexity and continuity of $I$, we obtain
$$I(u^\varepsilon,y^\varepsilon)\leq \liminf_k I(u_k^\varepsilon,y_k^\varepsilon)
showing that $(u^\varepsilon,y^\varepsilon)$ is a solution for $(P_\alpha^\varepsilon)$.$\hfill\Box$
The next result deals with the necessary optimality conditions for the approximate problem $(P_\alpha^{\varepsilon})$.
Let $(\bar u^\varepsilon,\bar y^\varepsilon)$ be a solution of $(P_\alpha^\varepsilon)$ and assume that $\bar u^\varepsilon$ satisfies condition $(\ref{uniqueness_condition})$. Then there exists $\bar{p}^\varepsilon\in V$ such that
\begin{equation}\label{state_regularized}
\left\{ \begin{array}{ll}-\nu \Delta \mathbf{\bar y}^\varepsilon+
\mathbf{curl}\,\sigma\left(\mathbf{\bar y}^\varepsilon \right)\times \mathbf{\bar y}^\varepsilon+\nabla \pi^\varepsilon= \varrho_\varepsilon(\mathbf{\bar u}^\varepsilon)& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf{\bar y}^\varepsilon=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf{\bar y}^\varepsilon=0&
\mbox{on} \ \Gamma,\end{array}\right.
\end{equation}
\begin{equation}\label{adjoint_regularized}\left\{ \begin{array}{ll}-\nu \Delta \mathbf{\bar p}^\varepsilon-
\mathbf{curl}\,\sigma\left(\mathbf{\bar y}^\varepsilon\right)
\times \mathbf{\bar p}^\varepsilon+\mathbf{curl}\,\sigma\left(\mathbf{\bar y}^\varepsilon\times \mathbf{\bar p}^\varepsilon\right)
+\nabla \tilde\pi^\varepsilon= \mathbf{\bar y}^\varepsilon-\mathbf{y_d}& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf{\bar p}^\varepsilon=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf{\bar p}^\varepsilon=0&
\mbox{on} \ \Gamma,\end{array}\right.\end{equation}
\begin{equation}\label{op_control}
\left(\varrho_\varepsilon\left(\bar{p}^{\varepsilon}\right)+
\lambda\bar{u}^{\varepsilon},
\left(\bar u^\varepsilon-\bar u,v-\bar u^\varepsilon\right)_{H({\rm curl};\Omega)}\geq 0\qquad \mbox{for all} \
v\in U_{ad}.\end{equation}
Proof. Let first notice that if $\bar{u}^{\varepsilon}$ satisfies condition $(\ref{uniqueness_condition})$, then $\varrho_\varepsilon\left(\bar{u}^{\varepsilon}\right)$ satisfies the same condition.
Taking into account Proposition <ref>, we deduce that (<ref>) admits a unique solution ${\bar y}^\varepsilon$ and that this solution belongs to $H^4(\Omega)$. Due to Proposition <ref>, it follows that for every $v\in H(\mathrm{curl};\Omega)$, the linearized equation
\begin{equation}\label{linearized_regularized}
\left\{ \begin{array}{ll}-\nu \Delta \mathbf z+
\mathbf{curl}\,\sigma\left(\mathbf z\right)\times \mathbf{\bar y}^\varepsilon+\mathbf{curl}\,\sigma\left(\mathbf{\bar y}^\varepsilon\right)\times \mathbf z+\nabla \pi^\varepsilon= \varrho_\varepsilon(\mathbf v)& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf z=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf z=0&
\mbox{on} \ \Gamma,\end{array}\right.
\end{equation}
admits a unique solution $\bar z^\varepsilon(v)\in H^3(\Omega)$. Moreover, due Proposition <ref> and Proposition <ref>, the control-to-state mapping $u\mapsto y^\varepsilon(u)$ is Gâteaux differentiable at $\bar u^\varepsilon$ and equation (<ref>) admits at least a solution ${\bar p}^\varepsilon\in V$.
For $\rho\in ]0,1[$ and $v\in U_{ad}$, let $u_{\rho}^\varepsilon=\bar u^\varepsilon+\rho (v-\bar u^\varepsilon)$, $y_\rho^\varepsilon$ be the solution of (<ref>) corresponding to $\varrho_\varepsilon\left(u_{\rho}^\varepsilon\right)$ and $z_{\rho}^\varepsilon=\tfrac{y_{\rho}^\varepsilon-\bar y^\varepsilon}{\rho}$. Since $(\bar u^\varepsilon,\bar y^\varepsilon)$ is an optimal solution for $(P_\alpha^\varepsilon)$ and $(u_{\rho}^\varepsilon,y_{\rho}^\varepsilon)$ is admissible for this problem, we have
$$\displaystyle\lim_{\rho\rightarrow 0}
\tfrac{I(u_{\rho}^\varepsilon,y_{\rho}^\varepsilon)-
I(\bar u^\varepsilon,\bar y^\varepsilon)}{\rho}\geq 0$$
which yields
\begin{equation}\label{zv}\left(\bar z^\varepsilon(v-\bar u^\varepsilon),\bar y^\varepsilon-y_d\right)+
\lambda\left(\bar u^\varepsilon,v-\bar u^\varepsilon\right)+
\left(\bar u^\varepsilon-\bar u,v-\bar u^\varepsilon\right)_{H({\rm curl};\Omega)}\geq 0.\end{equation}
Setting $\phi=\bar z^\varepsilon(v-\bar u^\varepsilon)$ in the variational formulation
(<ref>) corresponding to $\bar p^\varepsilon$ and taking into account the variational formulation (<ref>), we obtain
\begin{align}\left(\bar{y}^\varepsilon-y_d,\bar z^\varepsilon(v-\bar u^\varepsilon)\right)
&=\nu\left(\nabla \bar p^\varepsilon,\nabla \bar z^\varepsilon(v-\bar u^\varepsilon)\right)+
b\left(\bar p^\varepsilon,\bar z^\varepsilon(v-\bar u^\varepsilon), \sigma\left(\bar y^\varepsilon\right) \right)
-b\left(\bar z^\varepsilon(v-\bar u^\varepsilon),{\bar p}^\varepsilon, \sigma\left(\bar y^\varepsilon\right) \right)\nonumber\\
& \ +
b\left(\bar p^\varepsilon, \bar y^\varepsilon, \sigma\left(\bar z^\varepsilon(v-\bar u^\varepsilon)\right) \right)
-b\left(\bar y^\varepsilon,\bar p^\varepsilon, \sigma\left(\bar z^\varepsilon(v-\bar u^\varepsilon)\right) \right)\nonumber\\
\end{align}
The result follows by combining (<ref>) and (<ref>).$\hfill\Box$
Let $({\bar u}^\varepsilon,{\bar y}^\varepsilon)$ be a solution for $(P_\alpha^\varepsilon)$. There exists a subsequence $(\varepsilon_k)_k$ converging to zero, such that
$$\lim_{k\rightarrow +\infty}\left\|{\bar u}^{\varepsilon_k}-\bar u\right\|_{H({\rm curl};\Omega)}=0$$
$$\lim_{k\rightarrow +\infty}\left\|{\bar y}^{\varepsilon_k}-\bar y\right\|_{H^3}=0$$
$$\lim_{k\rightarrow +\infty}I\left({\bar u}^{\varepsilon_k},{\bar y}^{\varepsilon_k}\right)=J(\bar u,\bar y)$$
Proof. Since $(\bar u^\varepsilon)_\varepsilon$ is bounded in $U_{ad}$, there exists a subsequence $(\varepsilon_k)_k$ converging to zero and $u\in U_{ad}$ such that $(\bar u^{\varepsilon_k})_k$ converges to $u$ weakly in $H(\mathrm{curl};\Omega)$. Due to (<ref>)
we have
\displaystyle\lim_{k\rightarrow +\infty}\left(\varrho_{\varepsilon_k}
\left(\bar u^{\varepsilon_k}\right),
\phi\right)_{H({\rm curl};\Omega)}&=\displaystyle
\lim_{k\rightarrow +\infty}\left(\bar u^{\varepsilon_k},
\varrho_{\varepsilon_k}\left(\phi\right)\right)_{H({\rm curl};\Omega)}
\vspace{2mm}\\
&=\displaystyle\left(u,\phi\right)_{H({\rm curl};\Omega)}
\qquad \mbox{for all} \ \phi\in H(\mathrm{curl};\Omega)\end{array}$$
implying that $\left(\varrho_{\varepsilon_k}
\left(\bar u^{\varepsilon_k}\right)\right)_k$ also weakly converges to $u$ in $H(\mathrm{curl};\Omega)$. Due to Proposition
<ref>, we deduce that $(\bar y^{\varepsilon_k})_k$
converges in $H^2(\Omega)$ to $y$, solution of (<ref>) corresponding to $u$. On the other hand, let $y^{\varepsilon_k}_{\bar u}$ be a solution of (<ref>) corresponding to $\varrho_{\epsilon_k}(\bar u)$. Since $(\varrho_{\epsilon_k}(\bar u))_k$ strongly converges to $\bar u$ in $H(\mathrm{curl};\Omega)$, it follows that $\left(y^{\varepsilon_k}_{\bar u}\right)_k$ converges to $\bar y$ in $H^3(\Omega)$. Using the lower semicontinuity of $I$ and the admissibility of $(\bar u,y_{\bar u}^{\varepsilon_k})$ for $(P_\alpha^{\varepsilon_k})$, we obtain
\left\|u\right\|^2_2+\tfrac{1}{2}\left\|u-\bar{u}\right\|^2_{H({\rm curl};\Omega)}&\leq\displaystyle\liminf_{k}
I(\bar{u}^{\varepsilon_k},\bar y^{\varepsilon_k})\vspace{2mm}\\
&\displaystyle\leq\limsup_{k} I(\bar{u}^{\varepsilon_k},\bar y^{\varepsilon_k})\vspace{1mm}\\
&\leq\displaystyle\lim_{k} I(\bar u,y_{\bar u}^{\varepsilon_k})
=\tfrac{1}{2}\left\|\bar y-y_d\right\|^2_2+\tfrac{\lambda}{2}\left\|\bar{u}\right\|^2_2\end{array}$$
and consequently
$$J(u,y)+\tfrac{1}{2}\left\|u-\bar{u}\right\|^2_{H({\rm curl};\Omega)}\leq J(\bar{u},\bar y).$$
Since $(\bar{u},\bar y)$ is solution of $(P_\alpha)$, we have $J(\bar{u},\bar y)
\leq J(u,y)$ and thus $u=\bar{u}$. Recalling that $\bar u$ satisfies condition (<ref>), we deduce that $y=\bar y$ and thus
$$ \lim_{k\rightarrow +\infty}I(\bar{u}^{\varepsilon_k},\bar y^{\varepsilon_k})=J(\bar{u},\bar y).$$
Finally, observing that
$$\begin{array}{ll}\displaystyle\tfrac{1}{2}\limsup_{k}\left\|\bar{u}^{\varepsilon_k}-\bar{u}\right\|^2_{H({\rm curl};\Omega)}
&=\displaystyle\limsup_{k}\left(I(\bar{u}^{\varepsilon_k},\bar y^{\varepsilon_k})-\tfrac{1}{2}
\|\bar y^{\varepsilon_k}-y_d\|^2_2-\tfrac{\lambda}{2}\left\|\bar u^{\varepsilon_k}
\right\|^2_2\right)\vspace{1mm}\\
&\displaystyle\leq J(\bar{u},\bar y)-\tfrac{1}{2}\|\bar{y}-y_d\|^2_2-\tfrac{\lambda}{2}\liminf_{k}
\|\bar u^{\varepsilon_k}\|^2_2\vspace{2mm}\\
\liminf_{k}\left\|\bar{u}^{\varepsilon_k}\right\|^2_2\leq 0\end{array}$$
we conclude that $(\bar{u}^{\varepsilon_k})_k$ converges to $\bar{u}$ strongly in $H({\rm curl};\Omega)$.$\hfill\Box$
§.§.§ Proof of Theorem <ref>
Let $(\bar u^{\varepsilon_k},\bar y^{\varepsilon_k})$ be the solution of $(P^{\varepsilon_k}_\alpha)$ given in Proposition <ref>. Since $\bar u$ satisfies condition $(\ref{uniqueness_condition})$, we deduce that there exists $k_1\in \NN$ such that $\bar u^{\varepsilon_k}$ also satisfies condition $(\ref{uniqueness_condition})$ for every $k> k_1$. Due Proposition <ref>, there exists $\bar{p}^{\varepsilon_k}\in V$ such that
\begin{equation}\label{adjoint_regularized_k}\left\{ \begin{array}{ll}-\nu \Delta \mathbf{\bar p}^{\varepsilon_k}-
\mathbf{curl}\,\sigma\left(\mathbf{\bar y}^{\varepsilon_k}\right)
\times \mathbf{\bar p}^{\varepsilon_k}+\mathbf{curl}\,\sigma\left(\mathbf{\bar y}^{\varepsilon_k}\times \mathbf{\bar p}^{\varepsilon_k}\right)
+\nabla \tilde\pi^{\varepsilon_k}= \mathbf{\bar y}^{\varepsilon_k}-\mathbf{y_d}& \mbox{in} \ \Omega,\vspace{2mm} \\
\mathrm{div} \, \mathbf{\bar p}^\varepsilon_k=0& \mbox{in} \ \Omega,\vspace{2mm}\\
\mathbf{\bar p}^{\varepsilon_k}=0&
\mbox{on} \ \Gamma,\end{array}\right.\end{equation}
\begin{equation}\label{op_control_k}\left(\varrho_{\varepsilon_k}\left(\bar{p}^{\varepsilon_k}\right)+
v-\bar{u}^{\varepsilon_k}\right)\geq 0\qquad \mbox{for all} \
v\in U_{ad}.\end{equation}
Moreover, due to (<ref>), we have the following estimate
\begin{equation}\label{est_H1_p_k}
\left|p^{\varepsilon_k}\right|_{H^1}
\leq \tfrac{S_2\nu}{\nu^2-\bar\kappa
\left(\left\| u^{\varepsilon_k}\right\|_2+\alpha
\left\| \mathrm{curl} \, u^{\varepsilon_k}\right\|_2\right)} \left\|\bar y^{\varepsilon_k}-y_d\right\|_2.\end{equation}
Using once again the strong convergence of $(u^{\varepsilon_k},y^{\varepsilon_k})_k$ in $H({\rm curl; \Omega})\times L^2(\Omega)$, we deduce that the sequence $(p^{\varepsilon_k})_k$
is bounded in $V$. There then exist a subsequence, still indexed by $k$,
and $\bar p$ such that $(\bar p^{\varepsilon_k})_{k}$ weakly converges to $\bar p$ in $V$ and, by using compactness results on Sobolev spaces, $(\bar p^{\varepsilon_k})_{k}$ strongly converges to $\bar p$ in $L^{2}(\Omega)$. Therefore, by passing into the limit in the variational formulation corresponding to $\bar p^{\varepsilon_k}$:
$$ \begin{array}{ll}
&\nu\left(\nabla \bar p^{\varepsilon_k},\nabla \phi\right)+
b\left(\bar p^{\varepsilon_k},\phi,
\sigma\left(\bar y^{\varepsilon_k}\right) \right)
-b\left( \phi,\bar p^{\varepsilon_k}, \sigma\left(\bar y^{\varepsilon_k}\right) \right)+
b\left(\bar p^{\varepsilon_k}, \bar y^{\varepsilon_k}, \sigma(\phi) \right)-b\left(\bar y^{\varepsilon_k},\bar p^{\varepsilon_k}, \sigma(\phi) \right)\vspace{2mm}\\
\left(\bar y^{\varepsilon_k}-y_d, \phi\right)\end{array}$$
we obtain
$$\nu\left(\nabla \bar p,\nabla \phi\right)+
b\left(\bar p,\phi,
\sigma\left(\bar y\right) \right)
-b\left( \phi,\bar p, \sigma\left(\bar y\right) \right)+
b\left(\bar p, \bar y, \sigma(\phi) \right)-b\left(\bar y,\bar p, \sigma(\phi) \right)=
\left(\bar y-y_d, \phi\right)$$
for all $\phi\in V \cap H^2(\Omega)$, that is $\bar p$ is a weak solution of (<ref>). Finally, observing that
\left(\bar p^{\varepsilon_k}\right)-\bar p\right\|_2&\leq
\left\|\varrho_{\varepsilon_k}
\left(\bar p^{\varepsilon_k}-\bar p\right)\right\|_2+
\left\|\varrho_{\varepsilon_k}
\left(\bar p\right)-\bar p\right\|_2\leq
\left\|\bar p^{\varepsilon_k}-\bar p\right\|_2+
\left\|\varrho_{\varepsilon_k}
\left(\bar p\right)-\bar p\right\|_2\vspace{2mm}\\
&\longrightarrow 0 \quad \mbox{when} \ k\rightarrow \infty,\end{array}$$
we obtain (<ref>) by passing into the limit in (<ref>).$\hfill\Box$
§.§ Asymptotic analysis when $\alpha$ tends to zero
The proof of Theorem <ref> is split into three steps. First, we prove that if $(u_\alpha,y_\alpha)$ is admissible for $(P_\alpha)$, then it converges in the weak-$H(curl;\Omega)\times V$ topology to an admissible point $(u_0,y_0)$ for problem $(P_0)$. Next, we prove that if $(\bar u_\alpha,\bar y_\alpha)$ is an optimal solution of $(P_\alpha)$, then the limit point $(\bar u_0,\bar y_0)$ is an optimal solution of $(P_0)$ and the convergence of $\bar u_\alpha$ to $\bar u_0$ is strong in the topology of $L^2(\Omega)$. Finally, we pass to the limit in the adjoint equation and prove that the limit point $\bar p_0$ satisfies an adjoint equation and optimality condition associated with $(P_0)$.
Step 1. Convergence of admissible points. Let $(u_\alpha,y_\alpha)$ be an admissible point for $(P_\alpha)$.
By taking into account (<ref>) and (<ref>), we have
$$\left|y_\alpha\right|_2\leq \tfrac{S_2}{\nu} \left\|u_\alpha\right\|_2, $$
$$ \left|y_\alpha\right|_{V_2}\leq \tfrac{1}{\nu}\left(
S_2\left\|u_\alpha\right\|_2+\alpha \left\|\mathrm{curl}\,u_\alpha\right\|_2\right),$$
and thus $\left(y_\alpha\right)_\alpha$ and
$\left(\mathrm{curl}\, \sigma(y_\alpha)\right)_\alpha$ are bounded independently of $\alpha$. There then exists a subsequence, still indexed by $\alpha$, $u_0\in U_{ad}$, $y_0\in V$ and $\omega_0\in L^2(\Omega)$ such that
u_0 \qquad
\mbox{weakly in} \ L^2(\Omega). $$
$$y_\alpha \longrightarrow y_0 \qquad
\mbox{weakly in} \ H^1(\Omega) \
\mbox{and strongly in} \ L^2(\Omega),$$
$$\mathrm{curl}\, \sigma(y_\alpha)\longrightarrow
\omega_0 \qquad
\mbox{weakly in} \ L^2(\Omega). $$
By taking into account $(\ref{var_form_state})$ and $(\ref{transport_state})$, we have
\begin{equation}\label{vf_yalpha}\nu\left(\nabla y_\alpha,\nabla\phi\right)+\left(\mathbf{curl}\, \sigma(\mathbf{y_\alpha})\times \mathbf{y_\alpha},\boldsymbol\phi\right)=
\left(u_\alpha,\phi\right) \qquad \mbox{for all} \
\phi\in V\end{equation}
\begin{equation}\label{vf_sigma_alpha}\left(\mathrm{curl}\, \sigma(y_\alpha),\phi\right)-
\tfrac{\alpha}{\nu}\, b(y_\alpha,\phi,\mathrm{curl}\, \sigma(y_\alpha))=\left(\tfrac{\alpha}{\nu}\,\mathrm{curl}\, u_\alpha+
\mathrm{curl}\, y_\alpha,\phi\right)\qquad
\mbox{for all} \
\phi\in {\cal D}(\Omega).\end{equation}
The previous convergence results yield
$$\lim_{\alpha\rightarrow 0^+}\left(\mathbf{curl}\, \sigma(\mathbf{y_\alpha})\times \mathbf{y_\alpha},\boldsymbol\phi\right)=\left(\boldsymbol{\omega}_0\times \mathbf{y}_0,\boldsymbol\phi\right)\qquad \mbox{for all} \
\phi\in V$$
$$\lim_{\alpha\rightarrow 0^+}
b(y_\alpha,\phi,\mathrm{curl}\, \sigma(y_\alpha))=
b(y_0,\phi, \omega_0)\qquad
\mbox{for all} \
\phi\in {\cal D}(\Omega).$$
Therefore, by passing to the limit in the previous identities, we obtain
$$\nu\left(\nabla y_0,\nabla \phi\right)+\left(\boldsymbol{\omega}_0\times
\mathbf{y}_0,\boldsymbol\phi\right)=
\left(u_0,\phi\right) \qquad \mbox{for all} \
\phi\in V$$
\mbox{for all} \
\phi\in {\cal D}(\Omega)$$
showing that $\omega_0=\mathrm{curl}\,
y_0$ and that $y_0$ satisfies
$$\nu\left(\nabla y_0,\nabla \phi\right)
\left(u_0,\phi\right) \qquad \mbox{for all} \
\phi\in V.$$
that is, $(u_0,y_0)$ is admissible for $(P_0)$. Let us now prove that the convergence of $y_\alpha$ to $y_0$ is strong. Taking into account the variational formulations corresponding to $y_\alpha$ and $y_0$, we easily see that that $y_\alpha-y_0$ satisfies
$$\begin{array}{ll}\nu \left|y_\alpha-y_0\right|_{H^1}^2
&=\left(u_\alpha-u_0,y_\alpha- y_0\right)-
\left(\mathbf{curl}\, \sigma(\mathbf y_\alpha)\times \mathbf y_0
-\mathrm{curl}\, \mathbf y_0\times \mathbf y_0,
\mathbf y_\alpha-\mathbf y_0\right)\vspace{2mm}\\
&\longrightarrow 0 \qquad \mbox{when} \ \alpha \rightarrow 0^+.
\end{array}$$
Step 2. Convergence to an optimal solution of $(P_0)$. Let us now prove that the limit point $(\bar u_0,\bar y_0)$ of a solution $(\bar u_\alpha,\bar y_\alpha)$ of $(P_\alpha)$ is a solution of $(P_0)$. By taking into account the convergence results established in the first step and the lower semicontinuity of $J$, we obtain
$$\min(P_0)\leq J(\bar u_0,\bar y_0)\leq \liminf_{\alpha\rightarrow 0^+}J(\bar u_\alpha,\bar y_\alpha)=\liminf_{\alpha\rightarrow 0^+}\min(P_\alpha).$$
On the other hand, let $(\hat u,\hat y)$ be a solution of problem $(P_0)$ and let $\hat y_\alpha$ be the solution of $(\ref{equation_etat})$ corresponding to $\hat u$. Then $(\hat u,\hat y_\alpha)$ is admissible for $(P_\alpha)$ and
\begin{equation}\label{semi_con_1}
\min(P_\alpha)\leq J(\hat u,\hat y_\alpha).\end{equation}
Arguing as in the first step, we can establish the convergence of $\hat y_\alpha$ to
$\hat y$ in $V$ and thus
\begin{equation}\label{semi_con_2}\lim_{\alpha\rightarrow 0^+}\min(P_\alpha)\leq \lim_{\alpha\rightarrow 0^+}J(\hat u,\hat y_\alpha)=J(\hat u,\hat y)=\min(P_0).\end{equation}
Combining (<ref>) and (<ref>), we deduce that
\begin{equation}\label{stability_alpha}
\lim_{\alpha\rightarrow 0^+}\min(P_\alpha)=\min(P_0).\end{equation}
To prove the strong convergence of $\bar u_\alpha$ to $\bar u_0$ in $L^2(\Omega)$, observe that
\left\|\bar u_\alpha-\bar u_0\right\|_2^2&=
\left\|\bar u_\alpha\right\|_2^2
-\left\|\bar u_0\right\|_2^2-
2\left(\bar u_\alpha-\bar u_0,\bar u_0\right)
\vspace{2mm}\\
J(\bar u_\alpha,\bar y_\alpha)-
J(\bar u_0,\bar y_0)\right)-\tfrac{1}{\lambda}\left(
\left\|\bar y_\alpha-y_d\right\|_2^2-
\left\|\bar y_0-y_d\right\|_2^2\right)
-2\left(\bar u_\alpha-\bar u_0,\bar u_0\right)\vspace{2mm}\\
\min(P_\alpha)-\min(P_0)\right)-\tfrac{1}{\lambda}\left(
\left\|\bar y_\alpha-y_d\right\|_2^2-
\left\|\bar y_0-y_d\right\|_2^2\right)
-2\left(\bar u_\alpha-\bar u_0,\bar u_0\right).\end{array}$$
Therefore, by taking into account the convergence results of Step 1 and (<ref>), it follows that
\begin{equation}\label{L2_con_alpha}\lim_{\alpha\rightarrow 0^+}
\left\|\bar u_\alpha-\bar u_0\right\|_2=0.\end{equation}
Step 3. Convergence of $\bar p_\alpha$.
Due to (<ref>), we have
$$\left(1-\tfrac{ \bar\kappa}{\nu^2}\left(
\left\|\bar u_\alpha\right\|_2+\alpha\left\|{\rm curl}\,\bar u_\alpha\right\|_2\right)
\right) \left|\bar p_\alpha\right|_{H^1}\leq
\tfrac{\kappa}{\nu}\left\|\bar y_\alpha-y_d\right\|_2,$$
and by taking into account (<ref>), we deduce that
$\left(\bar p_\alpha\right)_\alpha$ is also bounded independently of $\alpha$. There then exists a subsequence, still indexed by $\alpha$ and $\bar p_0\in V$ such that
$$\bar p_\alpha \longrightarrow \bar{p}_0 \qquad
\mbox{weakly in} \ H^1(\Omega) \
\mbox{and strongly in} \ L^2(\Omega).$$
By taking into account the convergence results established in the first step, we deduce that
$$\begin{array}{ll}\displaystyle\lim_{\alpha\rightarrow 0^+}\left(b\left( \bar p_\alpha,\phi, \sigma(\bar y_\alpha) \right)-b\left( \phi,\bar p_\alpha, \sigma(\bar y_\alpha) \right)\right)&=\displaystyle\lim_{\alpha\rightarrow 0^+}\left(\mathbf{curl}\,\sigma(\bar{\mathbf y}_\alpha) \times \bar{\mathbf p}_\alpha,\boldsymbol\phi\right)\vspace{2mm}\\
&=\displaystyle \left(\mathbf{curl}\,\bar {\mathbf y}_0\times \bar{\mathbf p}_0,\boldsymbol\phi\right)\vspace{2mm}\\
&=b(\bar p_0,\bar y_0,\phi)-b(\phi,\bar y_0,\bar p_0)
\end{array}$$
$$\lim_{\alpha\rightarrow 0^+}\left(b\left(\bar p_\alpha, \bar y_\alpha, \sigma(\phi) \right)-b\left( \bar y_\alpha,\bar p_\alpha, \sigma(\phi) \right)\right)=b\left(\bar p_0, \bar y_0,\phi \right)-b\left(\bar y_0,\bar p_0,\phi\right)$$
for all $\phi\in V$. Passing then to the limit in the variational formulation corresponding to $\bar p_\alpha$ yields
\nu\left(\nabla\bar p_0,\nabla\phi\right)+b(\phi,\bar y_0,\bar p_0)-b\left(\bar y_0,\bar p_0,\phi\right)
=\left(\bar y_0-y_d, \phi\right)$$
for all $\phi\in V$ and thus $\bar p_0$ is the unique weak solution of
(<ref>). The optimality condition for the control follows then by passing to the limit in $(\ref{opt_control_alpha})$.$\hfill\Box$
Acknowledgment. This work was partially supported by FCT project PEst-OE/MAT/UI4032/2011.
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|
1511.00344
|
1Department of Geophysics, Tohoku University, Sendai, Japan.
2Max Planck Institute for Solar System Research,
Göttingen, Germany.
3Institute of Astrophysics, Georg-August University,
Göttingen, Germany.
4Department of Physics and Astronomy,
George Mason University, Fairfax, Virginia, USA.
Global characteristics of the small-scale gravity wave (GW) field in the Martian
atmosphere obtained from a high-resolution general circulation model (GCM) are presented
for the first time. The simulated GW-induced temperature variances are in a good agreement
with available radio occultation data in the lower atmosphere between 10 and 30 km. The
model reveals a latitudinal asymmetry with stronger wave generation in the winter
hemisphere, and two distinctive sources of GWs: mountainous regions and the meandering
winter polar jet. Orographic GWs are filtered while propagating upward, and the mesosphere
is primarily dominated by harmonics with faster horizontal phase velocities. Wave fluxes
are directed mainly against the local wind. GW dissipation in the upper mesosphere
generates body forces of tens of m s$^{-1}$ sol$^{-1}$, which tend to close the simulated
jets. The results represent a realistic surrogate for missing observations, which can be
used for constraining GW parameterizations and validating GCM simulations.
§ INTRODUCTION
The dynamical importance of small-scale gravity waves (GWs) has been well recognized in the
terrestrial middle atmosphere <cit.>
and upper atmosphere <cit.>. On Mars, GWs
are generated by flow over much rougher than on Earth topography, by strong convection, and
volatile instabilities of weather systems. Amplitudes of Martian GWs are, generally, larger
than those in the lower atmosphere of Earth <cit.> and in the
thermosphere <cit.>. Upward propagating and ultimately dissipating GWs deposit
a substantial amount of momentum and produce heating and cooling in the Martian middle
atmosphere (50–100 km) and thermosphere (above 100 km) <cit.>. Using the Mars Global
Surveyor (MGS) radio occultation data, <cit.> have demonstrated that spectral
amplitudes of small-scale GWs below $\sim$40 km drop off with respect to their vertical
wavenumbers according to the theoretical saturation and power law dependence of $-3$ slope,
which implies a transfer of wave energy and momentum to the mean flow. Based on the MGS
accelerometer data, <cit.> have found significant body forcing by GWs in the lower
The Martian atmosphere is approximately 100 times less dense than the terrestrial
one. Accordingly, molecular viscosity is to the same degree larger on Mars, and damping by
molecular diffusion and thermal conduction must be taken into account when GW propagation is
considered, as in Earth's thermosphere. GWs of interest have horizontal wavelengths usually
smaller than the conventional resolution of general circulation models (GCMs), and, thus,
their effects have to be parameterized. <cit.> applied the nonlinear
spectral parameterization of small-scale GWs of <cit.> to the output of the
Mars Climate Database <cit.> and demonstrated that dynamical effects of these waves in
the Martian lower thermosphere are very large and, therefore, cannot be ignored. This
parameterization was specifically developed for “whole atmosphere" GCMs, and was
extensively utilized in numerous GW studies in the context of Earth's middle atmosphere and
thermosphere <cit.>. With the parameterization interactively implemented into the Max Planck
Institute Martian GCM (MGCM) <cit.>, <cit.> have shown
that GWs play a very important role in the dynamics of the middle and upper atmosphere of
Mars. They close, and even reverse, the zonal jets, enhance the meridional circulation and
middle atmosphere polar warmings, facilitate a formation of CO$_2$ ice clouds
<cit.>, and modulate the upper atmospheric response to dust storms
<cit.>. GW-induced cooling is as strong in the mesosphere and thermosphere
as the major radiative cooling mechanism – the radiative transfer in the IR bands of CO$_2$
molecules <cit.>, – and can explain the observed temperatures in the lower
thermosphere <cit.>.
GW parameterizations assume a spectrum of wave harmonics at a certain source level in the
lower atmosphere in order to represent GW generation and activity. Accurate estimates of GW
momentum fluxes have, therefore, been recognized as an essential task in the Earth climate
studies. However, with the concerted efforts and numerous observational campaigns
<cit.>, the global picture of GWs is still beyond our reach even on Earth. On
Mars, this goal is even farther away. The progress with numerical modeling has allowed to
circumvent this problem to a certain degree by utilizing high-resolution (GW-resolving)
GCMs. They are now being increasingly used in Earth studies for the interpretation and
validation of observations and constraining parameterizations
<cit.>. This approach is based on the assumption that
comprehensive GCMs can capture a significant portion of GW sources and the details of wave
propagation. Thus, they provide a realistic surrogate for observations.
The first high-resolution GCM for Mars has been reported by <cit.>, however, GWs have
not been considered explicitly at that time. The only other high-resolution MGCM has been
presented by <cit.>. They performed simulations with a horizontal resolution of
$2\times 2$ degrees, and analyzed spatio-temporal spectra of the resolved fields. The major
finding of their work was an enhancement of wave energy for harmonics with zonal wavenumbers
$s$ up to 30 at tidal frequencies at heights where diurnal and semidiurnal tides are large.
Our paper further addresses the lack of knowledge of GW fields in the Martian atmosphere
with the new high-resolution ($\sim$1.1 degrees in horizontal) MGCM, and directly focuses on
smaller-scale ($s>60$) harmonics, which usually have to be parameterized.
The paper is structured as follows. The high-resolution MGCM is described in
Section <ref>. GW variations in the lower atmosphere (10–30 km) are presented and
compared with observations in Section <ref>. Vertical propagation of GWs is
discussed in Section <ref>, while horizontal distributions of their characteristics
are given in Section <ref>.
§ GRAVITY-WAVE RESOLVING MARTIAN GENERAL CIRCULATION MODEL
The high-resolution MGCM used in this study is based on the atmospheric
component of the MIROC (Model for Interdisciplinary Research On Climate)
terrestrial GCM developed collaboratively by the Atmosphere and Ocean Research
Institute (AORI), The University of Tokyo, the National Institute of
Environmental Studies (NIES), and the Japan Agency for Marine-Earth Science
and Technology (JAMSTEC) in Japan <cit.>. It utilizes a spectral
solver for the three-dimensional primitive equations, and has a set of physical
parameterizations appropriate for the Martian atmosphere as described in the
works by <cit.>. The MGCM accounts, among others, for radiative
effects of gaseous carbon dioxide and airborne dust, and interactively
simulates condensation and sublimation of the atmospheric CO$_2$, formation of
CO$_{2}$ ice clouds, snowfalls and seasonal ice cap in the polar atmosphere.
The lower-resolution version of the MGCM has been validated against the
observed zonal mean climatology <cit.>, and extensively been used for
studies of baroclinic planetary waves <cit.>, zonal-mean variability
in the middle- and high-latitudes <cit.>, equatorial semiannual
oscillations <cit.>, winter polar warmings during global dust storms
<cit.>, and CO$_2$ snowfalls in the northern winter polar atmosphere
<cit.>. Recently, this model received the name DRAMATIC (Dynamics,
RAdiation, MAterial Transport and their mutual InteraCtions) MGCM, and has
been used to validate the retrieved temperature in the southern polar night
from the MGS radio occultation measurements <cit.>.
In this study, the MGCM was run at the T106 spectral truncation, which
corresponds approximately to a 1.1$^{\circ}$ $\times$ 1.1$^{\circ}$ (or
$\sim$60 km) horizontal resolution. In the vertical direction, the model
domain extends from the surface to $\sim$80–100 km, and is represented by
49 $\sigma$-levels. Such setup allows for realistically capturing generation
and propagation of GWs with horizontal wavelengths of $3\Delta x\sim$180 km
and longer and, to some extent, their vertical attenuation due to nonlinear
processes. These waves are subgrid-scale in conventional GCMs, and
the dynamical and thermal importance in the Martian atmosphere of the
harmonics of these scales has been demonstrated in the works of
<cit.> and <cit.>, correspondingly.
The local thermodynamic equilibrium was assumed for the radiative effects of
CO$_2$ gas at all heights.
§ GRAVITY WAVE VARIATIONS IN THE LOWER (10-30 KM) ATMOSPHERE
(a) Kinetic $E_k$ and (b) potential energy $E_p$ per unit mass
(in J kg$^{-1}$) of resolved gravity waves with the total wavenumbers greater
than 60 (horizontal wavelengths of less than $\sim$350 km), and (c) the ratio
$E_k/E_p$, averaged between 10 and 100 Pa for 20 sols starting at
$L_{s}$=270$^{\circ}$. White contours on each plot denote the Martian
The results shown here are for the Northern Hemisphere winter solstice, i. e., when Mars is
at perihelion, and the dynamical processes in the atmosphere are most active. All the
figures are based on 20-sol averaged fields centered at the solar longitude $L_s=270^\circ$,
with the dust opacity of $\sim$1.0 in the visible wavelength (a “low dust" condition).
We designate the shortest horizontal-scale fluctuations with the total wavenumber $n>61$
(horizontal wavelengths less than $\sim$350 km) as wave disturbances $\varphi^\prime$. This
choice allows for explicitly considering harmonics, which are known to significantly
contribute to dynamical and thermal forcing of large-scale atmospheric flows, and which are
usually parameterized in GCMs. Correspondingly, the larger-scale ($n\le60$) fields here
represent the “mean" $\bar{\varphi}$ such that
$\varphi=\bar{\varphi}+\varphi^\prime$. Similar definition is applicable to disturbance
covariances. For instance, $\overline{\varphi^\prime \psi^\prime}$ is the product of
shorter-scale fields $\varphi$ and $\psi$ on the globe, of which only the lower-$n$ portion
is taken. Effectively, averaging denoted by overbars is a horizontal spatial averaging, or a
Direct measures of activity of fluctuating fields, which we believe are composed mainly of
gravity waves, are their kinetic and potential energy (per unit mass) $E_k$ and $E_p$,
\begin{equation}
E_k={1\over 2} \biggl( \overline{u^{\prime 2}} +
\overline{v^{\prime 2}} \biggr),
\qquad
E_p={1\over 2} \Bigl(\frac{g}{N}\Bigr)^2
\frac{\overline{T^{\prime 2}}}{\overline{T}^2},
\label{eq:energy}
\end{equation}
where $u^\prime$ and $v^\prime$ are the wind fluctuations in the zonal and
meridional directions, respectively, $g$ is the acceleration of gravity, and
$N$ is the Brunt-Väisälä frequency. The quantities $E_k$ and $E_p$
averaged between 10 and 30 km are shown in Figures <ref>a and
<ref>b. This representation allows for a direct comparison with the
measurements of GW temperature fluctuations derived from MGS occultation data
for the same season <cit.>. Their observations show a
gradual increase of $E_p$ in the Southern Hemisphere from $<$2 J kg$^{-1}$ at
high-latitudes to 10–15 J kg$^{-1}$ and larger over the equator, which is in
an excellent agreement with our simulations in Figure <ref>b.
Measurements are missing for latitudes higher than 20$^\circ$N, where
simulations predict an increase of GW activity, and reaches its maximum (of
greater than 30 J kg$^{-1}$) over the core of the westerly polar night jet (at
$\sim$60$^\circ$). Another observational constraint have been presented
by <cit.>, who derived temperature fluctuations from the Mars Climate
Sounder (MCS) data. Although they were obtained for spatial scales longer than
in our simulations, the magnitudes of variations are in a very good agreement
between 100 and 10 Pa (several K) <cit.>. Observations
also show an enhancement of temperature fluctuations in the Northern
high-latitudes. In the Southern Hemisphere, our simulations do not reproduce
large temperature fluctuations. In addition, <cit.> showed that the
equatorial region has larger $E_p$ than any other latitude region, whereas
our simulations display that this peak is shifted to the middle latitudes of
the Northern Hemisphere.
The distribution of $E_k$ in Figure <ref>a is similar to that of $E_p$.
It also demonstrates the latitudinal asymmetry of gravity wave activity in the
lower atmosphere with the maximum in the winter hemisphere. There is an
equipartition of kinetic energy between the zonal and meridional components of
small-scale wind variations. The results in Figure <ref>a indicate that
the magnitudes of wind fluctuations increase from $\sim$1 m s$^{-1}$ in high
latitudes of the Southern Hemisphere to $\sim$6 m s$^{-1}$ in the middle- and
high latitudes of the Northern Hemisphere. These distributions of $E_p$ and
$E_k$ clearly reflect GW sources in the lower atmosphere.
One property of the small horizontal-scale wave field can immediately be
found by comparing $E_k$ and $E_p$: the kinetic component of energy exceeds
that of potential energy. <cit.> have derived
the relation between the $E_k/E_p$ ratio and the intrinsic frequency of
gravity wave $\hat{\omega}$:
\begin{equation}
\frac{E_k}{E_p} = \frac{1+\biggl({f \over \hat{\omega}} \biggr)^2 }
{1-\biggl({f \over \hat{\omega}} \biggr)^2 },
\label{eq:ratio}
\end{equation}
where $f$ is the Coriolis frequency. It follows from (<ref>) that
smaller-$\hat{\omega}$ (longer-period in the frame of reference moving with
the local wind) GW harmonics have larger $E_k/E_p$ ratios, while the latter
asymptotically approaches unity for high-frequency harmonics. The calculated
ratio $E_k/E_p$ plotted in Figure <ref> points out the
interhemispheric asymmetry in the distribution of the dominant intrinsic
frequencies $\hat{\omega}$ of resolved small-scale waves: they are
a factor of two or more smaller in the winter
hemisphere. Given that their horizontal scales are approximately equal
throughout the globe, this implies smaller intrinsic horizontal phase
velocities $c-\bar{u}$ of GWs in the Northern Hemisphere. These waves are
generated by the meandering strong winter polar jet (large $\bar{u}$), which
means that their observed horizontal phase velocities $c$ (measured with
respect to the surface) are, on the contrary, large. A closer consideration
of small-scale GW-induced fields, for instance, of the horizontal wind
divergence $\partial u^\prime/\partial x +\partial v^\prime/\partial y$
(see Movie S1 in the Supporting Information), confirms that wave packets move
eastward much faster in the winter polar jet region, although somewhat lag
the mean zonal winds. This illustrates the bias in the
horizontal phase velocities of small-scale GWs in the source region first
pointed out in the work by <cit.> and utilized in the prescribed source
spectrum in the GW parameterization studies for Earth <cit.>
and Mars <cit.>. In the mountainous regions, $E_k/E_p$ is, on the
contrary, small (blue shades in Figure <ref>c), which indicates large
intrinsic/small observed horizontal phase velocities. This means that
topographically-induced GWs dominate there, and that the wave packets are
“tied up" to the relief features. Movie S1 clearly demonstrates this
§ VERTICAL PROPAGATION OF GRAVITY WAVES
The latitude-altitude cross-sections of zonal-mean quantities due
to resolved GWs with the total wavenumber of larger than 60 (shaded):
(a) vertical flux of zonal wave momentum $\rho\overline{u'w'}$ (in mPa),
(b) vertical flux of meridional momentum $\rho\overline{v'w'}$,
(c) kinetic wave energy $\rho E_k$ (in mJ m$^{-3}$),
(d) potential wave energy $\rho E_p$.
Black contours in (a) represent the zonal wind (in m s$^{-1}$),
and the meridional wind in (b).
The components of divergences of gravity wave momentum fluxes
(shaded, in m s$^{-1}$ sol$^{-1}$) and the mean wind (contours, in m s$^{-1}$): full (horizontal and vertical)
divergences of (a) zonal and (b) meridional momentum fluxes;
only vertical divergence of (c) zonal and (d) meridional momentum fluxes.
Black contours denote (a, c) the mean zonal, and (b, d) meridional wind.
Having considered GWs in the lower atmosphere, we now turn to their upward
propagation. Vertical fluxes of the zonal and meridional momentum,
$\rho\overline{u^\prime w^\prime}$ and $\rho\overline{v^\prime w^\prime}$, respectively, are
important quantities for examining this. Zonally averaged distributions of the calculated
$\rho\overline{u^\prime w^\prime}$ and $\rho\overline{v^\prime w^\prime}$ are plotted with
color shades in Figures <ref>a and b, respectively, and the mean zonal and meridional
winds are superimposed with contour lines. The fluxes are vector quantities, which are
conserved if no sources and sinks are present. For a given GW harmonic, the momentum flux is
proportional to the intrinsic phase velocities in the corresponding direction, and
characterizes wave propagation with respect to the mean flow. Only in the absence of the
latter, the signs indicate the direction of wave propagation with respect to the surface,
that is, in the east-west or north-south. These results suggest that, in the lower
atmosphere, the fluxes are, generally, directed against the local winds. This means that the
spectra of GWs are dominated by harmonics with observed phase velocities $c$ that are slower
than the local wind (“lagging" the flow), or having opposite signs (moving against the
flow): $c<\bar{u}$ if $\bar{u}>0$, and $c>\bar{u}$ if $\bar{u}<0$. Over the course of
vertical propagation, harmonics are selectively dissipated and/or obliterated due to
breaking or filtering by the mean wind. The net wave momentum flux is determined by a
delicate balance of contributions of “surviving" harmonics from the initial spectrum. Thus,
magnitudes and even the sign of the net flux can vary with height. For instance, the
apparent increase of the magnitude in low latitudes between 100 and 10 Pa in
Figure <ref>a does not necessarily indicate in situ generation of waves with positive
$\rho\overline{u^\prime w^\prime}$. Harmonics with $\rho\overline{u^\prime w^\prime}<0$ in
the incident spectrum are filtered by the easterly wind $\bar{u}<0$, while waves carrying
positive fluxes progressively contribute more, because a) their amplitudes grow with height,
and b) $c-\bar{u}$ and the associated momentum flux increase. Above $\sim$10 Pa, the
opposite occurs. Harmonics with positive flux partly dissipate and deposit their momentum to
the mean flow, as we shall discuss below, and partly their contribution decreases (due to
the mean zonal wind $\bar{u}$ weakening) along with the increase of the contribution of
waves with negative fluxes. Same can be applied to the local maximum of positive meridional
fluxes $\rho\overline{v^\prime w^\prime}>0$ over $\sim$60$^\circ$N in Figure <ref>b.
Since wave momentum fluxes are vector quantities they are not fully suitable for
characterizing the net field, because harmonics with opposite signs may offset and even
cancel contributions of each other. Wave variances provide another proxy for wave activity,
which is devoid of this limitation. Figures <ref>c and d show their zonal mean
latitude-altitude distributions in the form of kinetic and potential energy, $E_k$ and $E_p$
from (<ref>), multiplied by the mean density. $\rho E_k$ exceeds $\rho E_p$
everywhere in the atmosphere, as it does at lower altitudes in Figure <ref>. The
maximum of wave energy is in the lower atmosphere, where these waves are mainly excited, and
decreases with height in each vertical column. However, a clear asymmetry between the
Northern and Southern Hemispheres is seen. Wave activity is stronger, and GWs penetrate
higher in the winter hemisphere. Partially, this may be explained by the asymmetry of
sources in the lower atmosphere, but refractive properties of the atmosphere associated with
the mean winds are likely to play a role as well. Spectra of generated waves in average, are
dominated by harmonics with slower phase velocities, as otherwise would cause an
“ultraviolet catastrophe" (integral of energy over spectrum diverges). These waves are less
affected by strong winds in the core of the westerly jet, and, therefore, are being focused
into it. One more reason for the asymmetry can be related to the oblique propagation: wave
packets composed of harmonics with slower phase velocities can cover significant horizontal
distances upon their vertical propagation. We cannot diagnose the degree of obliqueness
directly from the GCM output, and a ray tracing model is required for that. Most likely, all
three factors contribute to the obtained distributions of GW activity in the middle
atmosphere. Here we simply state that the simulated asymmetry awaits a validation with
observations, and that any successful parameterization of subgrid-scale GWs must reproduce
Figures <ref>a and b show that momentum fluxes ultimately decrease with
height. Divergence of the momentum fluxes quantifies the rate of wave
obliteration, and the amount of momentum transferred to the mean (larger-scale
in our study) flow. Depending on the sign, waves can produce acceleration or
deceleration of the latter.
Figures <ref>a and b present thus calculated forcing along the
corresponding axes:
\begin{equation}
a_x=-\nabla\cdot\overline{{\bf v'}u'}, \qquad
a_y=-\nabla\cdot\overline{{\bf v'}v'},
\label{eq:acc}
\end{equation}
where ${\bf v'}=(u',v',w')$ are the components of velocity fluctuations, and
$\nabla =(\partial/\partial x, \partial/\partial y, \rho^{-1} \partial \rho /
\partial z)$.
As can be seen, $a_x$ and $a_y$ created by the resolved small-scale motions are significant
in the middle atmosphere (tens of m s$^{-1}$ sol$^{-1}$), and directed mainly against the
mean wind. This result is consistent with the estimates of GW drag obtained using the
extended spectral parameterization of <cit.> applied to the distributions of
wind and temperature from the Mars Climate Database <cit.>, and
interactively coupled with the Martian GCM <cit.>. One may notice that it is
significantly smaller than the estimates of <cit.> ($\sim$1000 of m s$^{-1}$
sol$^{-1}$), but it is because we present zonal and time averaged quantities, while their
results are based on individual measurements. Instantaneously, $a_x$ and $a_y$ in our
simulations can reach over 10 000 m s$^{-1}$ sol$^{-1}$. The response of the mean zonal
winds to this forcing is also seen – the jets show the tendency to decrease and close in
the upper portion of the domain. This cannot be achieved in simulations with conventional
(low) resolution without parameterized subgrid-scale GWs, unless an artificial sponge layer
is applied near the top. Thus, our GW-resolving simulations represent a direct confirmation
of the predictions on the dynamical importance and effects of small-scale GWs in the Martian
The plotted divergences further illustrate GW propagation in the equatorial region. They
show weak negative $a_x$ below 10 Pa created by the absorption of harmonics with negative
fluxes by the easterly mean wind $\bar{u}<0$, as is discussed above. Around 10 Pa, strong
dissipation of harmonics with $\overline{u^\prime w^\prime}>0$ produces positive $a_x$
decelerating the mean wind. Above 10 Pa, the remaining harmonics with negative fluxes
deposit the negative momentum upon their dissipation, which results in the acceleration of
the negative flow. The latter seems paradoxial as all the waves with negative fluxes should
have apparently been filtered below by the negative background wind. An in depth explanation
of such phenomenon was given in the paper of <cit.>, and is related to the fact that the projection of the wind on the
direction of wave propagation (that affects the latter) can significantly differ from the
zonal wind alone.
The vertically alternating patches of positive and negative $a_x$ in the equatorial region
tend to enhance the semiannual oscillation of the zonal wind, as discussed in the work of
<cit.>. The meridional component of the GW-induced torque, $a_y$, also plays an
important role in the middle atmosphere. It decelerates the cross-equatorial south-to-north
meridional transport in low and middle latitudes at $\sim$10 Pa induced mainly by thermal
tides, accelerates it somewhat higher (at $\sim$1 Pa), and extends to high latitudes of the
winter (Northern) hemisphere. This leads to the intensification of the downward branch of
the meridional transport cell over the North Pole, which results in the increase of the
adiabatic heating and enhancement of the middle atmosphere polar warming <cit.>.
Similarly, small-scale GWs decelerate the northward meridional flow in the upper mesosphere,
and weaken the the meridional pole-to-pole cell.
Next, we estimate the contributions of the vertical component of the momentum flux
divergence to the net $a_x$ and $a_y$ by plotting
$-\rho^{-1}d\rho \overline{u^\prime w^\prime}/dz$ and
$-\rho^{-1}d\rho\overline{v^\prime w^\prime}/dz$ in Figures <ref>c and d. They are
very close to those in Figures <ref>a and b. This indicates that a) horizontal
propagation of GWs plays a secondary role in forcing the mean flow, and b) GW
parameterizations accounting for only vertical propagation can successfully capture the
major part of subgrid-scale GW effects in GCMs.
§ HORIZONTAL DISTRIBUTIONS OF WAVE FLUXES
The distribution of horizontal flux of zonal wave momentum $\overline{u'w'}$ at (a)
260 Pa and (c) 0.1 Pa pressure levels (shaded, in J kg$^{-1}$). Black contours represent
the topography of Mars. (b) The same as in (a) except for $\overline{v'w'}$. (d) The
divergence of meridional wave momentum (in m s$^{-1}$ sol$^{-1}$) at 0.1 Pa pressure
level. Black contours in (a) and (b) represent the topographical features. Purple contours
in (a), (c) and (d) denote the mean zonal wind velocity $\bar{u}$ (in m s$^{-1}$), while
in (b) denote the mean meridional wind velocity $\bar{v}$.
Many parameterizations use wave momentum fluxes at a certain level in the lower atmosphere
for the specification of sources. Therefore, we plotted the longitude-latitude distributions
of $\overline{u^\prime w^\prime}$ and $\overline{v^\prime w^\prime}$ at $p=260$ Pa in
Figures <ref>a–b. They are shown with color shades, and the corresponding
large-scale winds $\bar{u}$ and $\bar{v}$ are superimposed with contours. Although all
quantities are 20-day averaged, fluxes are seen to be very patchy, which demonstrates that
sources are extremely localized both in space and time. Peak values of the fluxes with
alternating signs occur in the mountainous regions. They are evidently associated with waves
generated by flow over topography. Nevertheless, a clear asymmetry can be seen: fluxes
predominantly have signs opposite to the mean local wind. In the middle- to high-latitudes
of the Northern Hemisphere, the distribution of the zonal flux is significantly
smoother. These GWs are excited within the curvatures of the winter westerly jet, which, in
large, are associated with Kelvin waves moving eastward with time. The meridional fluxes are
negative and directed against the mean meridional wind between the Equator and 45$^\circ$N,
and have alternating direction in other regions, where the mean wind is weak.
For comparison, the zonal momentum fluxes created by harmonics penetrating to
the mesosphere ($p=0.1$ Pa) are shown in Figure <ref>c.
Their distribution is significantly more horizontally homogeneous.
Most orographic GWs (with small with respect to the surface phase
speeds) are filtered out by the wind in the course of their vertical
propagation, and create only a marginal enhancement over the mountainous
regions. This confirms the fact well-known from Earth studies that GWs with
progressively faster horizontal phase speeds dominate at high altitudes
<cit.>. The region with negative (but large)
horizontal wave momentum fluxes in the mesosphere is confined to Northern
high-latitudes, which reflects the favorable propagation conditions for the
corresponding harmonics, and which is in line with our finding using the GW
parameterization <cit.>. The magnitudes of fluxes in
the mesosphere significantly exceed those in the lower atmosphere, which
merely reflects the wave amplitude growth due to exponential density drop
with height.
Finally, we show the calculated vertical divergence of momentum fluxes (wave
drag), $a_x=-\rho^{-1} d(\rho\overline{u^\prime w^\prime})/dz$, in the
mesosphere (Figure <ref>d). It is consistent with the zonal mean
cross-section in Figure <ref>d, but shows a high degree of horizontal
inhomogeneity. Locally, $a_x$ exceeds 200 m s$^{-1}$ sol$^{-1}$ at $p=0.1$ Pa,
but almost nowhere is less than several tens of m s$^{-1}$ sol$^{-1}$.
Obviously, such strong effects of small-scale waves cannot be ignored in the
dynamics of the Martian mesosphere. Note that the values and distributions of
both wave fluxes and acceleration/deceleration obtained in this high-resolution
simulation can and should be served for validation and tuning of GW
§ CONCLUSIONS
We presented the first results of simulations with a new high-resolution
Martian general circulation model (triangle spectral truncation T106) that
resolves (in a 3$\Delta x$ sense) harmonics with horizontal scales down
to $\sim$180 km. In this paper, we concentrated on the Northern winter
solstice (around the solar longitude $L_s=270^\circ$) and GW harmonics
shorter than 350 km. This consideration leaves aside shorter-scale (few tens
of km) harmonics generated by convection, and which can be important in the
upper atmosphere. The main inferences of this first study of its kind are
listed below.
* Magnitudes of temperature variances due to small-scale GWs (or available
potential wave energy $E_p$) between 10 and 30 km are in a good agreement
with those obtained by <cit.> from Mars Global Surveyor radio
occultation data. In addition, simulations show a gradual and steep
latitudinal increase of $E_p$ from South to North with the maximum in the
winter hemisphere, where the observational data are missing.
* Variances of wave-induced horizontal wind fluctuations exhibit a similar
behavior, however, with a steeper growth – the ratio of the wave kinetic
and potential energy, $E_k/E_p$, increases from $\sim$1.5 in the Southern
Hemisphere to about 3 in the Northern one.
* Two major sources of GWs can be identified in the lower atmosphere: the
mountainous regions generating slow, or even non-moving with respect to
the surface wave packets, and the meandering winter westerly jet exciting
faster GW harmonics traveling mainly eastward.
* The majority of generated GWs move slower than the background
wind, and the associated vertical fluxes of horizontal wave momentum
are directed against it.
* Most of GWs are produced in the lower atmosphere, and their fluxes and
energy decay with height.
* Upon vertical propagation and dissipation, these waves deposit their
momentum directed mainly against the local wind, and, thus, provide a wave
drag on the mean flow.
* As a result of the drag, the simulated jets in both hemispheres
demonstrate a tendency to close in the upper atmosphere. This feature
cannot be reproduced by GCMs with a conventional (low) resolution without
applying an artificial sponge near the model top or an appropriate GW
* In the lower atmosphere, the distributions of wave momentum fluxes
are very patchy, reflecting the highly localized nature of GW sources.
Orographically generated slow waves are filtered in lower layers in the
course of their vertical propagation, and the upper mesosphere is dominated
by harmonics with faster horizontal phase velocities.
Given the lack of observations of GWs in the atmosphere of Mars, our
high-resolution simulations provide the much needed framework for constraining
GW parameterizations, and validating the results obtained with the latter.
Data supporting the figures are available from TK
TK was supported by the Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant Number 24740317, and the Promotion of the Strategic Research
Program for Overseas Assignment of Young Scientists and International
Collaborations titled ”Intensification of International Collaborations for
Planetary Plasma and Atmospheric Dynamics Research based on the Hawaiian
Planetary Telescopes”. The model runs have been performed with the HITACHI
SR16000 System (yayoi) at the Information Technology Center, The University of
Tokyo. This work was partially supported by German Science Foundation (Deutsche
Forschungsgemeinschaft) grant ME2752/3-1 and NASA grant NNX13AO36G.
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1511.00086
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[Corresponding author: ][email protected]
Center for Field Theory and Particle Physics and Department of Physics,
Fudan University, 220 Handan Road, 200433 Shanghai, China
Shadows of black holes surrounded by an optically thin emitting medium have been extensively discussed in the literature. The Hioki-Maeda algorithm is a simple recipe to characterize the shape of these shadows and determine the parameters of the system. Here we extend their idea to the case of a dressed black hole, namely a black hole surrounded by a geometrically thin and optically thick accretion disk. While the boundary of the shadow of black holes surrounded by an optically thin emitting medium corresponds to the apparent photon capture sphere, that of dressed black holes corresponds to the apparent image of the innermost stable circular orbit. Even in this case, we can characterize the shape of the shadow and infer the black hole spin and viewing angle. The shape and the size of the shadow of a dressed black hole are strongly affected by the black hole spin and inclination angle. Despite that, it seems that we cannot extract any additional information from it. Here we study the possibility of testing the Kerr metric. Even with the full knowledge of the boundary of the shadow, those of Kerr and non-Kerr black holes are very similar and it is eventually very difficult to distinguish the two cases.
04.70.-s, 95.30.Sf, 98.62.Js
§ INTRODUCTION
The direct image of a black hole surrounded by an optically thin emitting medium is characterized by the presence of the so-called “shadow”, namely a dark area over a brighter background <cit.>. The shape of the shadow corresponds to the apparent photon capture sphere as seen by a distant observer and it is the result of the strong light bending in the vicinity of the black hole. An accurate observation of the shadow can potentially provide information on the spacetime geometry around the compact object and test general relativity <cit.>. The interest in this topic is today particularly motivated by the possibility of observing the shadow of SgrA$^*$, the supermassive black hole candidate at the center of the Milky Way, with sub-millimeter very long baseline interferometry (VLBI) facilities <cit.>.
If the black hole has an optically thick and geometrically thin accretion disk, its apparent image is substantially different. However, even this “dressed” black hole is characterized by a shadow, whose shape is still determined by the spacetime geometry around the compact object and the viewing angle with respect to the line of sight of the distant observer <cit.>. In this case, the shape of the shadow corresponds to the apparent image of the inner boundary of the accretion disk, which, under certain conditions, should be located at the innermost stable circular orbit (ISCO) <cit.>.
In the Kerr metric, the shape of the shadow is only determined by the values of the black hole spin parameter $a_* = a/M = J/M^2$, where $M$ and $J$ are the black hole mass and spin angular momentum, and of the angle $i$ between the spin and the line of sight of the distant observer. The size of the shadow on the observer's sky is also regulated by the black hole mass and distance. In Ref. <cit.>, Hioki and Maeda proposed a simple algorithm to characterize the shape of the shadow of a Kerr black hole surrounded by an optically thin medium in terms of a distortion parameter $\delta$ and of the shadow radius $R$. If we independently know the mass and the distance of the object, the measurement of $\delta$ and $R$ can provide an estimate of the black hole spin and inclination angle. This algorithm has been later employed for the shadow of non-Kerr black holes as a general approach to characterize their shadow and get a rough estimate of the possibility of testing general relativity with future VLBI observations <cit.>.
With the same spirit, here we introduce two parameters, $\alpha$ and $\beta$, to characterize the shape of the shadow of a dressed black hole. Even in this case, we show that we can eventually measure the black hole spin and the inclination angle. However, it seems we cannot do more. We study the possibility of testing the Kerr metric by considering a non-Kerr background with a spin and a deformation parameter to quantify possible deviations from the Kerr solution. Even introducing a third parameter to describe the shadow shape, we fail to break the degeneracy between the spin and the deformation parameter of the background metric. We also introduce the function $R(\phi)$ to describe the entire shadow shape. Even with $R(\phi)$, we are not able to unambiguously distinguish a Kerr and a non-Kerr black hole. A similar problem was already found in the case of the shadow of a black hole surrounded by an optically thin emitting medium <cit.>, but in that case the shape and the size of the shadow do not change very much if the values of the physical parameters of the system vary. For that of a dressed black hole, both the shape and the size significantly change with the spacetime geometry and the inclination angle. Such a result can be probably understood by noting that the boundary of the shadow of a dressed black hole is the apparent image of the ISCO, whose radius is just one parameter. For a Kerr black hole, it is only determined by the spin of the object. For a non-Kerr black hole, it depends on both the spin and the deformation parameter and therefore there is a degeneracy between these two quantities, in the sense that the same ISCO radius can be obtained from different combinations of the spin and the deformation parameter.
We note that the results of our work have no implications for future observations of SgrA$^*$, as this object has no thin accretion disk. The shadow of a dressed black hole can be potentially observed in the case of stellar-mass black holes in X-ray binaries with X-ray interferometric techniques.
The stellar-mass black hole candidate with the largest angular size on the sky should be that in Cygnus X-1, which can be found in the state with a thin accretion disk and it seems thus a good candidate for this purpose.
However, X-ray interferometric observations will be unlikely possible in the near future.
The content of the paper is as follows. In Section <ref>, we briefly describe our approach to numerically compute the shadow of a dressed black hole. In Section <ref>, we introduce the parameters $\alpha$ and $\beta$ to characterize the shape of the shadow and infer the black hole spin and its inclination angle with respect to the observer's line of sight. In Section <ref>, we discuss how the detection of a shadow may be used to test the Kerr metric and we introduce the function $R(\phi)$ to describe its whole boundary. Summary and conclusions are reported in Section <ref>. Throughout the paper, we employ units in which $G_{\rm N} = c = 1$.
Examples of shadows of dressed black holes. In the left panels, the inclination angle is $i=60^\circ$ and we show the effect of the spin parameter $a_*$. In the right panel, the spin parameter is $a_* = 0.9$ and we change the inclination angle $i$.
§ CALCULATION METHOD
The Kerr solution is a Petrov type-D spacetime and therefore for a suitable choice of coordinates (e.g. in Boyer-Lindquist coordinates) the equations of motion are separable and of first order. This can somehow simplify the calculations of the black hole shadow. However, here we use the general approach and therefore its extension to non-Kerr backgrounds is straightforward. We use the code described in Ref. <cit.>. Since we are only interested in the calculation of the shape of the shadow, not in the intensity map of the direct image, we need to compute the photon trajectories from the image plane of the distant observer to the vicinity of the black hole and check whether the photon hits the thin accretion disk or not. The set of points on the observer's sky whose photons do not hit the disk forms the shadow of the black hole. Such photons can either hit the black hole or cross the equatorial plane between the inner edge of the disk and the black hole and then escape to infinity.
The initial conditions $(t_0, r_0, \theta_0, \phi_0)$ for the photon with Cartesian coordinates $(X,Y)$ on the image plane of the distant observer are <cit.>
t_0 = 0 ,
r_0 = √(X^2 + Y^2 + D^2) ,
θ_0 = arccosY sini + D cosi/√(X^2 + Y^2 + D^2) ,
ϕ_0 = arctanX/D sini - Y cosi .
The initial 3-momentum of the photon, $\bf{k}_0$, is perpendicular to the plane of the image of the observer. The initial conditions for its 4-momentum are thus <cit.>
k^r_0 = - D/√(X^2 + Y^2 + D^2) |k_0| ,
k^θ_0 = cosi - D Y sini + D
cosi/X^2 + Y^2 + D^2/√(X^2 + (D sini - Y cosi)^2) |k_0| ,
k^ϕ_0 = X sini/X^2 + (D sini - Y cosi)^2 |k_0| ,
k^t_0 = √((k^r_0)^2 + r^2_0 (k^θ_0)^2
+ r_0^2 sin^2θ_0 (k^ϕ_0)^2) .
In our calculations, the observer is located at $D = 10^6$ $M$, which is far enough to assume that the background geometry is flat. $k^t_0$ is thus obtained from the condition $g_{\mu\nu}k^\mu k^\nu = 0$ with the metric tensor of a flat spacetime. The photon trajectory is
calculated by solving the second order geodesic equations with the fourth order Runge-Kutta-Nyström method, as described in <cit.>. The trajectory is
numerically integrated backwards in time to check whether the photon hits the black hole, hits the disk, or crosses the equatorial plane between the inner edge of the disk and the black hole and then escapes to infinity. We note that some photons may cross the equatorial plane between the inner edge of the disk and the black hole and then hit either the disk or the black hole. We also note that we employ the usual set-up, in which the disk is on the equatorial plane and the inner edge of the disk is at the ISCO radius.
Fig. <ref> shows some examples of our calculations. In the left panel, the inclination angle is fixed to $i = 60^\circ$ and we change the value of the spin parameter to see its effect on the shape of the shadow. The impact of $a_*$ is definitively different from the case of the shadow of a black hole surrounded by an optically thin emitting medium. In the case of a dressed black hole, the size of the shadow is very sensitive to the black hole spin, as it could have been expected from the fact the ISCO radius ranges from 6 $M$ for a non-rotating Schwarzschild black hole to $M$ for a maximally rotating Kerr black hole and a corotating disk, while the ISCO is at 9 $M$ when the disk of the maximally rotating Kerr black hole is counterrotating. The right panel in Fig. <ref> shows the effect of the inclination angle. Here the black hole has spin parameter $a_* = 0.9$ and the inclination angle is $i = 0^\circ$, $30^\circ$, $60^\circ$, and $80^\circ$. Now the size of the shadow is roughly the same, while the shape changes significantly. If we look at the shadows of black holes surrounded by an optically thin emitting medium (see e.g. the figures in <cit.>), it is easy to conclude that both the shape and the size of the shadow of a geometrically thin and optically thick disk are much more sensitive to the values of the spin and of the inclination angle.
Definition of the parameter $\alpha$ and $\beta$ to characterize the shape of the shadow of a dressed black hole.
Contour maps of $\alpha/M$ (left panel) and of $\alpha/\beta$ (right panel) in the plane spin parameter vs inclination angle for the Kerr metric. See the text for more details.
Left panel: impact of the deformation parameter $\epsilon_3$ on the shape of the shadow of a dressed black hole. The inclination angle is $i = 70^\circ$, the spin parameter is $a_* = 0.5$, and we change the deformation parameter $\epsilon_3$. Right panel: examples of profile of the boundary of the shadow. Here we assume the Kerr metric, the inclination angle is $i=60^\circ$, and we show the profile for a few different spin parameters.
Contour map of $S_1 (a_*, \epsilon_3, i)$ in which the reference model is a Kerr black hole with spin parameter 0.7 and observed from an inclination angle $60^\circ$. Here $i = 60^\circ$ is fixed. See the text for more details.
Left panel: as in Fig. <ref> with $i$ free in the fit. Right panel: contour map of the values of the inclination angle $i$ that minimizes $S_1$ in the left panel. See the text for more details.
Left panel: as in Fig. <ref> for $S_2$. Right panel: as in the left panel in Fig. <ref> for $S_2$. See the text for more details.
§ DETERMINATION OF SPIN AND VIEWING ANGLE
In this section, we want to introduce two parameters to characterize the shadow of a dressed black hole, to be used to infer the values of its spin and viewing angle. Unlike the shadow of a black hole surrounded by an optically thin emitting medium, in general our shadows have not an axis of symmetry and therefore we need to adopt a slightly different approach with respect to that in Ref. <cit.>.
As first step in our algorithm, we find the “center” $C$ of the shadow (see Fig. <ref>). It reminds the center of mass of a body and its coordinates $(X_C,Y_C)$ on the sky are given by
X_C = ∫ρ(X,Y) X dX dY/∫ρ(X,Y) dX dY ,
Y_C = ∫ρ(X,Y) Y dX dY/∫ρ(X,Y) dX dY ,
where $\rho (X,Y) = 1$ inside the shadow and 0 outside. Once we have $C$, we can determine the distance of $C$ from every point of the boundary of the shadow. If $A$ and $B$ are the points on the boundary respectively with the maximum and minimum distance from $C$, we call $\alpha$ the distance $AC$ and $\beta$ the distance $BC$, as shown in Fig. <ref>.
In the case of the Hioki-Maeda algorithm for the shadow of a black hole surrounded by an optically thin emitting medium, we determine two parameters: the shadow radius in units of the apparent size of the gravitational radius on the observer's sky, $R/M$, and the (dimensionless) distortion parameter, $\delta$. With these two quantities we can infer the black hole spin parameter, $a_*$, and the angle between the spin axis and the line of sight of the distant observer, $i$. Here we have the same situation. $\alpha/M$ is the counterpart of $R/M$, while $\alpha/\beta$ plays the role of the Hioki-Maeda distortion parameter $\delta$. In Fig. <ref>, we show the contour maps of $\alpha/M$ (left panel) and $\alpha/\beta$ (right panel). $\alpha/M$ is mainly determined by the black hole spin, while the effect of the inclination angle $i$ is smaller. On the contrary, $\alpha/\beta$ is very sensitive to the exact value of $i$ and it is affected only weakly by $a_*$. If we can determine both $\alpha/M$ and $\alpha/\beta$, we can infer $a_*$ and $i$. We note that in our case of the shadow of a dressed black hole we could estimate the inclination angle $i$ (with some uncertainty) without knowing the apparent size of the gravitational radius on the observer's sky.
§ TESTING THE KERR METRIC
If we characterize the shape of the shadow of a black hole with two parameters, their determination can be used at most to infer two physical parameters of the black hole, like the spin and the inclination angle. In this section we want to figure out if the detection of the shadow of a dressed black hole can test the Kerr metric. To do this, we relax the assumption of the Kerr background and we consider a metric more general than the Kerr solution. The metric is now described by the black hole mass $M$, the spin parameter $a_*$, and, in the simplest case, a deformation parameter that quantifies possible deviations from the Kerr metric. We want to see if it is possible to measure the shadow and infer the three parameters of the system.
As example, we consider the Johannsen-Psaltis metric <cit.>, whose line element reads
ds^2 = - (1 - 2 M r/Σ) (1 + h) dt^2
+ Σ(1 + h)/Δ+ a^2 h sin^2θ dr^2
+ Σdθ^2
- 4 a M r sin^2θ/Σ (1 + h) dt dϕ
+ [ sin^2θ(r^2 + a^2 + 2 a^2 M r sin^2θ/Σ )
+ a^2 (Σ+ 2 M r) sin^4θ/Σ h ] dϕ^2 .
Here $\Sigma = r^2 + a^2 \cos^2\theta$, $\Delta = r^2 - 2 M r + a^2$, and, in the simplest version with only one deformation parameter, $h$ is
h = ϵ_3 M^3 r/Σ^2 .
$\epsilon_3$ is the “deformation parameter” and it is used to quantify possible deviations from the Kerr geometry. The compact object is more prolate (oblate) than a Kerr black hole for $\epsilon_3 > 0$ ($\epsilon_3 < 0$); when $\epsilon_3 = 0$, we exactly recover the Kerr solution. The impact of the deformation parameter on the black hole shadow is shown in the left panel in Fig. <ref>, where $a_*$ and $i$ are fixed and we change $\epsilon_3$.
First, we tried to identify a third parameter to characterize the shape of the shadow of a dressed black holes. We studied a few options (area of the shadow, distance between $C$ and other points on the boundary, etc.). However, we failed, in the sense that we did not find three parameters of the shadow to infer the three physical parameters of the black hole because of the degeneracy between $a_*$ and $\epsilon_3$.
To increase our chances of success, we map the whole shadow boundary. We introduce the function $R(\phi)$ defined as the distance between $C$ and the boundary of the shadow, starting from $\alpha$, i.e. $R(\phi=0) = \alpha$, where $\phi$ is the angle between the the segment $AC$ determining $\alpha$ and the segment between $C$ and the point under consideration. If we do not have an independent measurement of the black hole mass and distance, we can only measure the actual shape of the shadow (not the size) and therefore we can measure $R/\alpha$. Some examples are shown in the right panel in Fig. <ref>, where we have only considered Kerr black holes with inclination angle $i=60^\circ$ and we vary the spin.
With the use of $R$, we can compare the shadow of different black holes. To quantify the similarity between two systems, we use the following estimator
S_1(a_*, ϵ_3, i) = ∑_k (
R(a_*, ϵ_3, i; ϕ_k)/α(a_*, ϵ_3, i)
- R^ref(ϕ_k)/α^ref)^2 ,
where $R(a_*, \epsilon_3, i; \phi_k)$ is the function $R$ at $\phi = \phi_k$ of the black hole under consideration, $\alpha(a_*, \epsilon_3, i)$ is its $\alpha$, while $R^{\rm ref}(\phi_k)$ and $\alpha^{\rm ref}$ are, respectively, the the function $R$ at $\phi = \phi_k$ of some reference black hole and the corresponding $\alpha$.
With this estimator, we are simply considering the least squares method for the normalized radius of the shadow and therefore the “best fits” is obtained when the sum of the squared residuals is minimum.
As example, we consider a reference black hole with $a_*=0.7$, $\epsilon_3 = 0$ (Kerr black hole), and $i=60^\circ$. Fig. <ref> shows $S_1$ with $i=60^\circ$ for every black hole. In the left panel in Fig. <ref>, we show $S_1$ with $i$ free and we have selected the inclination angle that minimizes $S_1$. In the right panel, we show the values of the inclination angles of the left panel. As we can see from Figs. <ref> and <ref>, there is a strong correlation between the spin $a_*$ and the deformation $\epsilon_3$ and it seems it is very difficult to infer the actual values of the physical parameters of the system.
If we have an independent measurement of the black hole mass and distance, we can measure $R$ in units of gravitational radii. In this case, the estimator for the comparison of two shadows is
S_2(a_*, ϵ_3, i) = ∑_k (
R(a_*, ϵ_3, i; ϕ_k)
- R^ref(ϕ_k))^2/M^2 .
Now the least squares method is for the absolute radius of the shadow.
The results for constant $i$ and free $i$ are reported in Fig. <ref>, respectively in left and right panels. Even in the case of the measurement of $R$, it seems we cannot test the Kerr metric because of the degeneracy between the spin and the deformation parameter. A simple explanation is probably that the shadow of a dressed black hole corresponds to the apparent image of the ISCO. The radius of the ISCO is just one parameter. If we have a Kerr black hole, it is only determined by the spin of the object and the measurement of the shadow can be used to infer the spin. In the case of a black hole with a spin and a possible non-vanishing deformation parameter, the same ISCO radius can be obtained with many different combinations of $a_*$ and $\epsilon_3$, namely there is a degeneracy. The measurement of the shape of the shadow cannot thus give an independent estimate of $a_*$ and $\epsilon_3$.
To test the Kerr metric, the shadow constrain should be combined with other observations that are not primarily sensitive to the position of ISCO in order to break the parameter degeneracy. While we have not investigated this point, we can expect that continuum-fitting and iron line measurements are not good to do it because they are strongly affected by the ISCO radius <cit.>. Observations like measurements of QPOs <cit.> or estimate of the jet power <cit.> sounds more promising because based on other properties of the spacetime.
§ CONCLUDING REMARKS
The shadow of a black hole is the dark area appearing in the direct image of the accretion flow. The shadows of black holes in general relativity and in alternative theories of gravity surrounded by an optically thin emitting medium have been extensively discussed in the literature and the interest on the topic is particularly motivated by the possibility of observing the shadow of SgrA$^*$ with VLBI facilities in the next few years. In this paper, we have studied the shadow of a dressed black hole, namely a black hole surrounded by a geometrically thin and optically thick accretion disk. Even these shadows can be potentially observed, but we will probably need to wait for a longer time because the sources are Galactic stellar-mass black holes in X-ray binaries, whose angular size on the sky is about five orders of magnitude smaller. X-ray interferometric techniques may observe these shadows, but there are no scheduled missions at the moment.
The boundary of the shadow of a dressed black hole corresponds to the apparent image of the inner edge of the disk, which, under certain conditions, should be located at the ISCO radius. Following the spirit of the Hioki-Maeda algorithm <cit.>, we have introduced the parameters $\alpha$ and $\beta$ to characterize the shape of the shadow. In the standard set-up with a Kerr black hole, the boundary of the shadow only depends on the black hole spin and inclination angle with respect to the line of sight of the distant observer, like the shadow of a black hole surrounded by an optically thin emitting medium. However, unlike the latter case, both the shape and the size significantly change if we vary $a_*$ and $i$. This may suggest that the shadow of a dressed black hole is more informative than that of a black hole surrounded by an optically thin emitting medium, but this is not what we have eventually found.
If the mass and the distant of the black holes are known, the measurement of $\alpha/M$ and $\alpha/\beta$ can be used to infer the black hole spin parameter $a_*$ and the inclination angle $i$. Actually the two estimates are very weakly correlated, because $\alpha/M$ is mainly sensitive to $a_*$ while $\alpha/\beta$ is essentially determined by $i$. As a result, if the mass and the distant of the black holes are not known and one can only measure $\alpha/\beta$, it is still possible to get an estimate of the inclination angle $i$ with some uncertainty.
As second step, we have checked if an accurate determination of the shadow of a dressed black hole can be used to test the Kerr metric. In the simplest case, the system is now defined by three physical parameters (spin parameter, deformation parameter, viewing angle). While it is an easy job to infer $a_*$ and $i$ in the standard set-up, it seems we cannot test the Kerr metric because of a degeneracy between the spin and the deformation parameter. Even the full knowledge of the boundary of the shadow, which we have here described with the function $R(\phi)$, cannot do it. In other words, the shadow of a dressed black hole is very sensitive to two parameters, the spacetime geometry around the compact object and the inclination angle, but not more.
We thank Cosimo Bambi and Jiachen Jiang for useful discussions and suggestions. This work was supported by the NSFC grant No. 11305038, the Shanghai Municipal Education Commission grant for Innovative Programs No. 14ZZ001, the Thousand Young Talents Program, and Fudan University.
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1Department of Physics, Ehime University, Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan
2Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan
3School of Physics & Astronomy, University of Southampton, Highfield, Southampton, SO17 1BJ, UK
4Research Center for Space and Cosmic Evolution, Ehime University, Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan
5Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan
We report a new sample of obscured active galactic nuclei (AGNs) selected from the serendipitous source
and point-source catalogs. We match X-ray sources with infrared (18 and 90 $\mu$m) sources located at $|b|>10^\circ$
to create a sample consisting of 173 objects. Their optical classifications and absorption column densities
measured by X-ray spectra are compiled and study efficient selection criteria to find obscured AGNs.
We apply the criteria (1) X-ray hardness ratio defined by using the $2-4.5$ keV and $4.5-12$ keV bands $>-0.1$
and (2) EPIC-PN count rate (CR) in the $0.2-12$ keV to infrared flux ratio CR/$F_{90}<0.1$ or CR/$F_{18}<1$,
where $F_{18}$ and $F_{90}$ are infrared fluxes at 18 and 90 $\mu$m in Jy, respectively, to search for
obscured AGNs.
X-ray spectra of 48 candidates, for which no X-ray results have been published,
are analyzed and X-ray evidence for the presence of obscured AGNs such as a convex shape X-ray spectrum
indicative of absorption of $\sim10^{22-24}$ , a very flat continuum, or a strong Fe-K emission line
with an equivalent width of $>700$ eV is found in 26 objects.
Six among them are classified as Compton-thick AGNs, and four are represented by either Compton-thin or Compton-thick
spectral models.
The success rate of finding obscured AGNs combining our analysis and the literature is 92% if the 18 $\mu$m condition is used.
Of the 26 objects, 4 are optically classified as an H2 nucleus and are new “elusive AGNs” in which
star formation activity likely overwhelms AGN emission in the optical and infrared bands.
§ INTRODUCTION
Multiwavelength observations have been finding various populations of active galactic nuclei (AGN).
The population of obscured
AGNs, which constitute a large fraction of AGNs (e.g., Fabian 2004, Gilli et al. 2007), among various classes,
are believed to be important in various aspects including the origin of the Cosmic X-ray
background (XRB; e.g., Gilli et al. 2007), connection between obscuring matter and star formation
activity in the host galaxies (e.g., Wada & Norman 2002), and evolutional paths of AGNs (e.g., Sanders et al., 1988, Hopkins et al. 2006,
Alexander & Hickox 2012).
Modern large area surveys at various wavebands are indeed utilized to find a large number of hidden AGNs and to elucidate
their nature.
Obscured AGNs are found by X-ray emission transmitted through obscuring matter, optical line emission from
extended and ionized regions, infrared emission from dust heated by central AGNs, and so on.
Hard X-ray surveys are one effective way to find obscured AGNs because the photoelectric
cross section decreases as X-ray energy increases, and transmitted X-rays can be observed.
Indeed, more than a dozen of Compton-thick AGNs, which are absorbed by a hydrogen column density
greater than $1.5\times10^{24}$ cm$^{-2}$, show transmitted hard X-rays above 10 keV
(Comastri 2004 and references therein; Burlon et al. 2011).
The all-sky and sensitive hard X-ray surveys conducted by the Swift Burst Alert Telescope (BAT) and INTEGRAL provide an
unprecedented opportunity to search for the heavily absorbed population. While obscured AGNs have been
found in these surveys, the number of Compton-thick sources are not as many as expected from other
studies using, e.g., optical emission-line-selected samples (Gilli et al. 2007) or prediction
from the synthesis models of the XRB (Burlon et al. 2011). This bias is due to the attenuation of X-rays even at
energies above 10 keV for Compton-thick cases
(Wilman & Fabian 1999, Ikeda et al. 2009, Murphy & Yaqoob 2009, Brightman & Nandra 2011a).
Burlon et al. (2011) estimated a correction factor of about four for Compton-thick fraction in their Swift/BAT sample.
Thus, hard X-rays are still biased against heavily absorbed AGNs.
Infrared emission from warm dust heated by the central source is also employed to find obscured AGNs.
Mid-infrared (MIR) emission indeed traces the power of the AGNs regardless of whether they are type 1 or 2,
and the attenuation of MIR in
type 2 AGNs is not large (Horst et al. 2008, Gandhi et al. 2009, Ichikawa et al. 2012, Matsuta et al. 2012).
A weakness of utilizing infrared emission, however, is that emission from dust heated by stars cannot be
separated if spatial resolution is not sufficiently high, and therefore the infrared-selected sample contains
non-negligible fraction of non-AGN galaxies. For example, 13 and 18 out of 126 galaxies
selected at 12 $\mu$m
are optically classified as H2 and H2/AGN composite galaxies, respectively
(Brightman & Nandra 2011b).
The combination of X-ray and infrared selection is a useful way to select a heavily
obscured AGN population and is applied to deep (Fiore et al. 2008, 2009)
or wide surveys (Mateos et al. 2012, Severgnini et al. 2012, Rovilos et al. 2014) to overcome the biases in the selections using
only hard X-ray or infrared emission.
The X-ray to infrared flux ratios, X-ray hardness, infrared colors, and so on are utilized in these selections.
Among the techniques employed, we extend the method used by Severgnini et al. (2012).
They used 25 $\mu$m fluxes ($F_{25})$ measured in the IRAS Point Source Catalog (PSC)
and X-ray data taken from the serendipitous source catalog (2XMM catalog; Watson et al. 2009)
and made a diagnostic plot of X-ray hardness ratio (HR4) and X-ray to infrared flux ratio ($F$(2$-$12 keV)/$\nu_{25}F_{25}$),
where HR4 is defined by using X-ray count rates (CRs) in 2–4.5 keV CR(2–4.5 keV) and in 4.5–12 keV CR(4.5–12 keV) as
\[
{\rm HR4} = \frac{{\rm CR(4.5-12~keV}) - {\rm CR(2.0-4.5~keV)}}{{\rm CR(4.5-12~keV)} + {\rm CR(2.0-4.5~keV)}}.
\]
They defined the region for candidates of Compton-thick AGNs as $F{\rm (2-12 keV})/\nu_{25}F_{25} < 0.02$
and HR4$>-0.2$, and built a sample consisting of 43 candidates. For absorbed sources, the X-ray to infrared ratio becomes small
since X-rays below 12 keV are attenuated by photoelectric absorption. Absorbed sources show flatter X-ray spectra and
therefore larger values of hardness ratios are expected. Thus, their criteria are expected to work to select
heavily absorbed sources.
About 84% of the objects in their sample are confirmed as
Compton-thick AGNs and 20% are newly discovered ones. Thus, the combination of wide field survey data
in the infrared and X-ray bands is promising in the search for heavily obscured AGNs.
In this paper, we combine the infrared all-sky survey data obtained with (Murakami et al. 2007)
and the 2XMM
catalog. We construct diagnostic diagrams to classify activity in galaxies and to
search for obscured AGNs.
We selected 48 candidates for obscured AGNs and analyzed their X-ray spectra.
This paper is organized as follows.
Section 2 describes the selection method of X-ray and infrared sources.
Diagnostic diagrams to classify the selected sources are presented in Section 3.
Results of X-ray spectral analysis are shown in section 4.
Section 5 discusses the results and summaries are given in Section 6.
We adopt $H_0$ = 70 km s$^{-1}$, $\Omega_{\rm M}$ = 0.3, and
$\Omega_\Lambda$ = 0.7 throughout this paper.
Distribution of the separation between the X-ray and infrared positions. Solid line: 18 $\mu$m sources. Dotted-dashed line: 90 $\mu$m sources.
§ THE SAMPLE
§.§ XMM-Newton and AKARI catalogs
We combine two large area survey data in the X-ray and infrared bands.
We used the Serendipitous source catalog Data Release 3 (2XMMi-DR3), which
contains 262902 unique X-ray sources. The median X-ray flux in 0.2–12 keV is $2.5\times10^{-14}$
The typical positional uncertainty is 1.5
(1$\sigma$) (Watson et al. 2009).
The entries listed in this catalog (CRs in 0.2–12 keV and HR4)
are used to create diagnostic diagrams and select candidates for obscured AGNs.
The data from EPIC-PN, which has a larger effective area than those of EPIC-MOS, was used throughout the analysis.
We use sources located at the Galactic latitude $|b|>10^\circ$ in 2902 observations with usable PN data.
Among these observations,
2686 and 216 observations are taken with full window and large window modes, respectively.
There are 2062 and 139 unique fields taken with the full window and large window modes, respectively.
The total number of unique EPIC-PN sources at $|b|>10^\circ$ is 150799.
In the following analysis, sources with EPIC-PN counts in 0.2–12 keV greater than 60 counts (60851 unique sources) are used.
Point Source Catalogs (PSCs) were used as infrared data. surveyed most of the sky
with the two instruments, the Infrared Camera (IRC; Onaka et al. 2007, Ishihara et al. 2010) and the Far-Infrared Surveyor (FIS; Kawada et al. 2007).
The bandpasses of the IRC are centered at 9 and 18 $\mu$m, while the band centers of the FIS
are 65, 90, 140, and 160 $\mu$m. We used 18 and 90 $\mu$m measurements as
mid- and far-infrared data. The 18 $\mu$m is chosen to avoid silicate features, which may affect continuum measurements,
in the 9 $\mu$m
bandpass of the IRC. The 90 $\mu$m band is used among the fir-infrared bands
because this band is most sensitive.
We use only data with the quality flag of FQUAL = 3, which means flux measurements are most
reliable (Yamamura et al. 2010). There are 43865 (18 $\mu$m) and 62326 (90 $\mu$m) sources with FQUAL = 3 located at $|b|>10^\circ$.
Distribution of redshifts for the matched sample using X-ray, 18, and 90 $\mu$m data shown in Table 1.
Distribution of X-ray count rates in 0.2–12 keV. Solid line: all the EPIC-PN sources located at $|b|>10^\circ$ with 0.2–12 keV counts greater than 60.
Dotted-dashed line: 173 objects in our sample after matching with the 18 $\mu$m and 90 $\mu$m sources and excluding Galactic and ultraluminous X-ray sources.
§.§ Cross Correlation of the X-ray and Infrared Catalogs
We compared the positions of X-ray sources in the 2XMM-DR3 and infrared sources in the PSCs and
made a list of X-ray and infrared sources.
We first listed FIS sources within 20 of the X-ray source positions.
If there are multiple sources within the circular region around an X-ray source, the source closest
to the X-ray position is assumed to be a counterpart and is used in the following analysis.
If one source is observed more than twice with , only the data with the largest
number of counts
in 0.2–12 keV is used.
We then matched the IRC and X-ray sources in a similar way using a matching radius of 10.
Finally, the FIS and IRC source lists are combined.
The combined list contains 253 sources.
The histograms of the separation between the X-ray and infrared positions are shown in Figure <ref> for
18 and $90~\mu$m sources. The separations for the $18~\mu$m sources are more concentrated
within a small radius from the X-ray positions compared to the $90~\mu$m sources as expected from
the positional accuracy of the IRC and FIS sources (3$\sigma \approx$6and 18, respectively, for faint sources).
(Upper) Distribution of 18 $\mu$m flux densities.
(Lower) Distribution of 90 $\mu$m flux densities.
Solid line: nrearest neighbor 18 $\mu$m sources to 90 $\mu$m sources within 20 located at $|b|>10^\circ$.
Dotted-dashed line: 173 objects in our sample after matching with the X-ray sources and excluding the Galactic and ultraluminous X-ray sources.
We searched for the most probable counterpart of the X-ray sources using the
NASA/IPAC Extragalactic Database (NED) and SIMBAD Astronomical Database.
Stars, H2 regions, planetary nebulae, young stellar objects, and ultraluminous X-ray sources
are excluded from the sample, resulting in 173 objects.
The sample thus obtained is summarized in Table 1, where the XMM source name,
alternative name, redshift, infrared flux densities,
X-ray CRs in 0.2$-$12 keV, and hardness ratios (HR4)
are shown.
Redshifts are available for 171 of the 173 sources. Their distribution is shown in Figure <ref> as a histogram.
The distribution of X-ray CRs in 0.2–12 keV for all the EPIC-PN sources located at $|b|>10^\circ$
with 0.2–12 keV counts greater than 60 and the 171 objects in our sample is shown in Figure <ref>.
Comparison of these histograms indicates that X-ray brighter objects tend to have a
possible infrared counterpart detected both in the 18 and 90 $\mu$m bands.
None of X-ray sources fainter than 0.004 counts s$^{-1}$ in 0.2–12 keV are matched with infrared sources.
We also compared distributions of infrared flux densities. The solid histograms in Figure <ref> are
distributions of infrared flux densities for 90 $\mu$m sources having 18 $\mu$m source(s) within 20$\arcsec$
and their nearest neighbor 18 $\mu$m sources (3535 objects in total)
located at $|b|>10^\circ$ with FQUAL = 3. The dotted-dashed histograms are those for the 173 objects
matched with EPIC-PN sources. A clear difference before and after matching with X-ray sources is shown as
the double peak structure in the histogram for the 18 $\mu$m sources. The peak at a higher infrared flux density
is likely to be composed of Galactic sources because the peak is more enhanced if we use objects at low Galactic longitude
(e.g., $10^\circ<b<15^\circ$) and because the peak disappeared after excluding Galactic sources in our selection procedure.
The distribution of 90 $\mu$m fluxes for X-ray-matched objects are flatter than that for all the 90 $\mu$m sources with nearby
18 $\mu$m sources. This fact indicates that objects brighter at 90 $\mu$m tend to have a greater probability of being matched with
an X-ray source.
Distribution of observed X-ray luminosities in the 2–10 keV band for objects showing AGN activity for
the matched sample using X-ray, 18, and 90 $\mu$m data shown in Table 1 (shaded histogram),
the 90 $\mu$m sample (hatched histogram), and the 18 $\mu$m (open histogram) sample.
Distribution of infrared luminosities $\nu L_\nu$ at 18 $\mu$m (upper) and 90 $\mu$m (lower)
for the matched sample using X-ray, 18, and 90 $\mu$m data shown in Table 1 (shaded histogram),
the 90 $\mu$m sample (hatched histogram), and the 18 $\mu$m (open histogram) samples.
We summarize the X-ray and infrared luminosities for the X-ray-IR matched sample in Table <ref>.
The distribution of the luminosities are shown in Figures <ref> and <ref> as shaded histograms.
Distances to nearby objects with a redshift parameter smaller than 0.003
are taken from the literature shown in Table <ref>, except for Mrk 59 for which a redshift-independent distance is not available.
Distances to objects at $z>0.003$ and Mrk 59 are calculated from source redshifts and the assumed cosmology.
Observed X-ray luminosities in the 2–10 keV band (source rest frame) are obtained from the literature or our analysis presented in section 4
if X-ray spectra clearly show the presence of an AGN.
Most X-ray fluxes or luminosities are taken from the literature, like for absorption column densities.
References are given in Table <ref> only if a reference different from that for $N_{\rm H}$ is used.
When we use published fluxes in 2–10 keV to calculate luminosities, we made a simple $K$-correction by assuming
a simple power law with a photon index of 1.8. Since most of the objects in our samples are at a low redshift ($z< \sim 0.3$),
the assumption on the spectral shape only slightly affects the correction. If a photon index of 1.4 is assumed instead,
the luminosity would be lower by 10% at $z=0.3$. This amount is much smaller for lower redshift sources.
In some cases, we obtained observed luminosities from the literature and converted them to our assumed cosmology.
Infrared luminosities ($\nu L_{\nu}$ for 18 and/or 90 $\mu$m) are calculated from the AKARI measurement of infrared fluxes.
We applied a $K$-correction by assuming a template spectral energy distribution for Seyfert 2s by Poletta et al. (2007).
Again, because of low redshifts for our sample, the amount of the correction is relatively small.
A correction factor is at most about 10% for a 18 $\mu$m luminosity at $z=0.3$.
We compiled optical classifications from the literature or spectra in the archives.
The sources are classified into
Seyfert, Low-Ionization Nuclear Emission line Region (LINER), H2 nucleus, transition between LINER and H2, BL Lac object,
or normal galaxy.
Seyfert, LINER, H2, and, transition objects are defined based on
the location of the optical emission line ratios ([N2]$\lambda$6584/H$\alpha$, [S2]$\lambda\lambda$6716, 6730/H$\alpha$,
and [O3]$\lambda$5007/H$\beta$)
on the excitation diagrams. Among various definitions of the boundary among
the classes (Baldwin et al. 1981, Veilleux & Osterbrock 1987, Ho et al. 1997, Kewley et al. 2006)
we adopted criteria of Ho et al. (1997) because many objects in our sample are
contained in the Ho et al. sample in which stellar absorption lines were carefully treated
in measuring emission line fluxes.
Some objects are in the boundary region of two activity classes on the
excitation diagrams, or use of different emission lines results in different classifications.
For such ambiguous cases, both activity classes are shown such as Seyfert/LINER.
The types (1, 1.2, 1.5, 1.8, 1.9, and 2) of Seyferts, LINERs, and
transition objects are also shown if available in the literature.
The classifications for some objects are not published in the literature, and we
classified their optical spectra from the archives of the Sloan Digital Sky Survey, the 6dF Galaxy survey, or
the Updated Zwicky Catalog (Falco et al. 1999), if available.
If optical spectra show only absorption lines, they are classified as a normal galaxy.
No classifications are available
for some of the objects in the sample. They are denoted as “unclassified.”
These classifications are shown in Table 1 and summarized as histograms in Figure <ref>.
In the histograms, Seyfert 1, 1.2, and 1.5
are treated as “Seyfert 1", while Seyfert 1.8, 1.9, and 2 are regarded as “Seyfert 2."
Summary of optical classifications for
the matched sample using X-ray, 18, and 90 $\mu$m data shown in Table 1.
Sy1: Seyfert 1, 1.2, and 1.5; Sy2: Seyfert 1.8, 1.9, and 2; L: LINER; T: transition object between LINER and HII nucleus; HII: HII nucleus; Normal: normal galaxy; BL: BL Lac object; Un: Unclassified.
Since one of our aims is to search for obscured AGNs, we compiled
absorption column densities () measured using X-ray spectra from the literature, as
shown in Table 1.
values are shown for Seyferts and LINERs in which X-ray emission is dominated by AGNs.
Some galactic nuclei classified as H2 show evidence for the presence of an AGN.
Their values are also shown.
If there are multiple published results, we put priority on the
results of systematic analysis of a large sample, and results based on better
quality of data. For objects showing a signature of heavy absorption exceeding
$\sim10^{24}$ , we use results based on wide-band spectra covering
hard X-rays above 10 keV whenever possible.
Objects showing a strong Fe-K emission line with an equivalent width (EW) exceeding 700 eV
and/or a very flat spectral slope
are regarded as $>10^{24}$ even if only X-ray spectra below 10 keV
are available. The boundary of the EW (700 eV) was chosen based on
analysis of X-ray spectra for a large sample of AGNs (Guainazzi et al. 2005a; Fukazawa et al. 2011) and
theoretical predictions (Awaki et al. 1991; Leahy & Creighton 1993;
Ghisellini et al. 1994; Ikeda et al. 2009; Murphy & Yaqoob 2009;
Brightman, & Nandra 2011a).
If an value is not explicitly presented in the literature and if X-ray spectral shape does not
show a clear signature of absorption, is regarded as small as noted in Table <ref>.
Hardness ratio (HR4) versus count rate (CR$_{0.2-12}$) / infrared flux ($F_{18}$ or $F_{90}$) ratio diagram.
Different symbols represent optical classifications;
open circles: Seyfert 1,
filled circles: Seyfert 2,
open squares: LINER,
open stars: transition object,
filled stars: H2 nucleus,
asterisk: BL Lac object,
filled triangles: unclassified.
(Left) Diagram using
18 $\mu$m flux as infrared flux. (Right) Diagram using 90 $\mu$m flux as infrared flux.
Hardness ratio (HR4) versus count rate (CR$_{0.2-12}$) / infrared flux ($F_{18}$ or $F_{90}$) diagram same as Figure 2 but
only AGNs with measured X-ray spectra are plotted. Different symbols represent measured from X-ray spectra;
open circles: $<10^{22}$ ,
open squares: $10^{22}$ $<$ $<$ $10^{23}$ ,
filled circles: $10^{23}$ $<$ $<$ $10^{24}$ ,
filled triangles: $>10^{24}$ .
The solid lines are tracks expected for various column densities. EPIC-PN count rate (0.2–12 keV) to infrared flux (18 $\mu$m or 90 $\mu$m)
ratio of 10 is assumed at = 0 cm$^{-2}$. The plus symbols are marked at = (0, 0.5, 1, 5, 10, 50, 100, 150)$\times 10^{22}$ cm$^{-2}$ from the upper-left most to the lowest most point.
See the text for details of the assumed spectral model.
Flux to count rate conversion factor for various values.
The expected count rates in 0.2–12 keV for a source with a flux of $10^{-14}$ erg s$^{-1}$ cm$^{-2}$
in 2–12 keV are shown.
See the text for details of the assumed spectral model.
Count rate (CR$_{0.2-12}$) / infrared flux ($F_{90}$) ratio versus
infrared flux ratio ($F_{18}/F_{90}$) diagram. The symbols are same as in Figure <ref>.
§ DIAGNOSTIC DIAGRAMS
§.§ Hardness and X-ray/Infrared Ratio
We first made a diagnostic diagram using the hardness ratio HR4 and the ratio between the X-ray CR in
0.2–12 keV and the infrared (18 or 90 $\mu$m) flux density.
These diagrams are essentially the same as that used by Severgnini et al. (2012).
We used X-ray CRs instead of X-ray fluxes because X-ray CRs are values directly derived
from observational data without any assumptions on the X-ray spectral shape.
Our diagrams are shown in the left and right panels of Figure <ref> for 18 $\mu$m and 90 $\mu$m, respectively.
Different symbols are used to represent optical classifications of the activity.
Seyfert 1, 1.2, and 1.5
are denoted as “Seyfert 1", while Seyfert 1.8, 1.9, and 2 are shown as “Seyfert 2" in
the diagrams.
Seyfert 1s are located in the upper left part of the diagram, while
Seyfert 2s tend to be located in the lower right.
The larger values of hardness ratios and lower X-ray counts relative to
infrared fluxes of Seyfert 2s are due to the suppression of lower energy X-rays
via photoelectric absorption.
H2 galaxies are located in the lower left side.
The location of H2 galaxies indicates that X-rays are
relatively weak compared to AGNs at a given infrared power.
In order to examine the effect of absorption,
we defined three groups sorted by and plotted them using different symbols in Figure <ref>.
The ordinate and abscissa are same as in Figure <ref>. The open circles, open squares, filled circles, and filled triangles
represent the groups of $<10^{22}$ , $= 10^{22-23}$ , $= 10^{23-24}$ , and $>10^{24}$ ,
respectively. As expected, objects with lower and higher tend to be located at around the upper left and
lower right part of the diagrams, respectively.
We made tracks of expected hardness ratios and
X-ray/infrared ratios for various values of .
A power-law spectrum with a photon index of 1.8 was assumed as
incident emission. One percent of the incident emission is assumed to appear as
scattered emission keeping the spectral shape.
The Galactic absorption of $2\times10^{20}$ is assumed as a representative value.
The expected values were calculated for various intrinsic
column densities from = 0 to 1.5$\times10^{24}$ .
The track thus calculated is shown as the solid line in Figure <ref>, where
the values of CR$_{0.2-12}/F_{18}$ and CR$_{0.2-12}/F_{90}$
are assumed to be 10 when = 0 .
The plus signs are marked at = $(0, 0.5, 1, 5, 10, 50, 100, 150)\times10^{22}$
from the upper left-most to the lowestr point.
The trend shows that X-ray counts are suppressed
and that hardness ratio becomes larger for larger values up to $\sim 5\times10^{23}$ .
If is larger than $\sim 5\times10^{23}$ , the flux from the
scattered emission becomes more significant relative to the
absorbed power-law emission, and the hardness ratio becomes
The above considerations suggest that if objects appeared in the lower right part of the
diagram, they are candidates for obscured AGNs.
Some of the objects in this region have no published X-ray spectra.
We study the X-ray spectra of such candidates for obscured AGNs
selected from that region
(CR$_{0.2-12}/F_{90}$ $<0.1$ or
CR$_{0.2-12}/F_{18}$ $<1$)
and HR4 $>-0.1$ in section 4.
A disadvantage of using CRs instead of fluxes is that they depend on the instrument used in observations.
We calculate the conversion factors from X-ray fluxes to observed CRs. The same spectral shape used to
derive the track shown in Figure <ref> is assumed. The on-axis response for the EPIC-PN and an integration radius
of 60", which contains 95% of the total flux, are used.
The expected CRs for an observed flux in 2–12 keV of $10^{-14}$ erg s$^{-1}$ cm$^{-2}$ are shown as
a function of the absorption column density in Figure <ref>.
§.§ Infrared Color and X-ray/Infrared ratio
Figure <ref> shows a diagram CR$_{0.2-12}/F_{90}$ versus infrared flux ratios ($F_{18}/F_{90}$).
This diagram supplements Figure <ref> in dividing AGNs and H2 nuclei. H2 nuclei
show smaller CR$_{0.2-12}/F_{90}$ and $F_{18}/F_{90}$ ratios compared to AGNs and are located
in the lower left part of the diagram; most H2 nuclei have ratios
CR$_{0.2-12}/F_{90} < 0.05$ and $F_{18}/F_{90} < 0.2$, while only several AGNs
are found in this region. AGNs tend to show warmer infrared colors compared to H2 nuclei,
and the distributions of the color for Seyfert 1 and Seyfert 2 are almost identical.
These results imply that the dust in AGNs are warmer than those in H2 nuclei
because of the presence of a hard heating source,
confirming earlier studies (e.g., Wu et al. 2009).
Comparison between type 1 and type 2 AGNs shows that
Mid-IR emission from warm dust near the
central engine is visible in both types of AGNs. Thus, we confirmed earlier results
based on ground-based observations
(Gandhi et al. 2009), or a combination of and hard X-ray surveys
(Ichikawa et al. 2012, Matsuta et al. 2012).
Distribution of redshifts for the 18 + 90 $\mu$m (shaded histogram),
90 $\mu$m (hatched histogram), and 18 $\mu$m (open histogram) samples.
Distribution of X-ray count rates in the 0.2–12 keV band for
18 $\mu$m + 90 $\mu$m sample (solid histogram) and Severgnini et al.'s sample (dashed histogram).
Distribution of flux densities at 18 $\mu$m for objects satisfying the condition CR$_{0.2-12}$/$F_{18}<1$
in 18 $\mu$m + 90 $\mu$m sample (solid histogram) and Severgnini et al.'s sample (dashed histogram)
§ X-RAY SPECTRA
§.§ The Sample and Data Reduction
As described in section 3.1, objects in the lower right part of the
diagram HR4 versus CR/infrared flux
ratio are candidates for
obscured AGNs. In order to explore the nature of the candidates,
we compiled X-ray results from the literature and analyzed X-ray spectra
of objects for which no published results are available.
We first select objects satisfying (CR$_{0.2-12}/F_{90} < 0.1$ or CR$_{0.2-12}/F_{18} < 1$ )
and HR4 $>-0.1$ from the matched sample using X-ray, 18, and 90 $\mu$m.
This sample consists of 85 objects and is denoted as the 18 + 90 $\mu$m sample hereafter.
In addition to this sample,
we selected objects satisfying
CR$_{0.2-12}/F_{90} < 0.1$ and HR4 $>-0.1$, where
we required the condition FQUAL = 3 only for
90 $\mu$m data to increase the size of the sample. 84 objects are selected by these conditions.
The list of objects in this 90$\mu$m sample is shown in Table <ref>.
We also selected objects with FQUAL = 3 at 18 $\mu$m, FQUAL $\neq$ 3 at 90 $\mu$m, and
HR4 $>-0.1$. This 18 $\mu$m sample consisting of 10 objects is shown in Table <ref>.
The probable counterparts, redshifts, optical classifications, and absorption column densities
determined from X-ray spectra taken from the literature
are shown in the tables.
Redshifts are available for 83, 62, and 9 objects in the 18 + 90 $\mu$m, 90 $\mu$m, and 18 $\mu$m samples, respectively.
The distributions of the redshifts are shown in Figure <ref>.
We excluded stellar and off-nuclear sources from the sample as in the
sample shown in Table 1.
We inspected the X-ray data of objects with no published X-ray results
as candidates for detailed studies.
Then the following cases were excluded;
objects located in bright diffuse emission of a cluster of galaxies or an early-type galaxy,
in the outskirt of the point spread function of a bright source, on or near the gap between CCD chips,
in crowded X-ray source regions such as star-forming regions.
The X-ray and infrared luminosities for the 90 and 18$\mu$m samples are
summarized in Table <ref> and Figures <ref> and <ref>, in which both results taken from
the literature and our own analysis presented in this section are shown.
The $K$-corrections and references for X-ray data and distances are treated
in the same manner for the sample presented in section 2.2.
We retrieved the data for the sample
from the Science Archive and examined their spectra.
The Science Analysis Software (SAS) version 13.0.0 and the calibration files
as of 2013 May were used in the data reduction and analysis.
We first made light curves of a region that does not contain bright sources in 10–12 keV
to examine background stability and time intervals with high background rates were
We extracted source spectra from a circular region centered at the source position
with a radius of 4$^{\prime\prime}$– 60$^{\prime\prime}$.
The extraction radii were determined to achieve good signal to noise ratio and to avoid
nearby sources.
Background spectra were made
from an off-source region in the same CCD chip and subtracted from the source spectra.
After data screening and background subtraction, net source counts in 0.2–12 keV for some objects
were found to be lower than 60 counts because of the reduced exposure time.
Such objects were excluded from the following analysis since their photon statistics
are not sufficient to create spectra of reasonable quality.
The final sample consisting of 48 objects for X-ray spectral analysis and the observation log are summarized
in Table <ref>. The infrared sample from which these objects are taken and Hubble type taken from the HyperLeda database
(Paturel et al. 2003) are also shown in Table <ref>.
The response matrix file and
ancillary response file were made by using the SAS. The spectra were binned
so that each bin contains at least one count.
More channels are binned in the figures shown below for presentation purposes.
A maximum-likelihood method using
the modified version of $C$ statistic (Cash 1979) was employed to fit background-subtracted
Spectral fits were performed with XSPEC version 12.8.0.
The errors represent the 90% confidence level for one parameter of interest. Errors are not shown
for the cases that the value of $C$ statistic is much worse than that for the best-fit model.
The Galactic absorption was applied to all the models examined below.
The Galactic absorption column densities (Kalberla et al. 2005)
were obtained by the FTOOL nh and shown in Table <ref>.
phabs or zphabs in XSPEC were used as a photoelectric absorption model.
We examined the presence of an Fe-K emission line at around 6.4 keV by adding a Gaussian component.
The line center energy was left free if the photon statistics were
sufficient to constrain the energy, otherwise 6.4 keV was assumed.
The line width was fixed at a Gaussian $\sigma$ of 10 eV.
For objects with a known redshift, all the model components except for the Galactic absorption
were assumed to be emitted or absorbed at the source redshift. If a redshift is previously unknown
and an Fe-K emission line is visible in our X-ray spectra, the source redshift was treated as a
free parameter and was determined from the line, whose central energy was assumed to be 6.4 keV in the source rest frame.
The redshifts for the X-ray analysis sample
range from 0.00218 to 0.188 with a median of 0.020, where redshifts determined by an Fe-K line are included.
We assumed a redshift of $z=0$ for all the other cases.
Fe line parameters are not shown if the photon statistics around 6.4 keV are not sufficient and
if no meaningful constraints on the line is obtained.
The results of the spectral fits described below are summarized in Tables <ref> and <ref>.
The adopted model is marked with an asterisk in Table <ref>.
In these tables, spectral parameters for the best-fit models and models with fewer model components are shown for comparison.
The observed fluxes and
luminosities corrected for absorption (for objects with a known redshift)
in the 2–10 keV band were derived for the best-fit spectral model and models giving a similar quality of fits to the best fit.
§.§ Results
We first summarize measurements of absorption column densities taken from the literature, and then
provide detailed results of our own spectral analysis for the 48 objects, for which no X-ray results have been published so far.
The 48 objects analyzed are devided into three groups (1) objects with absorbed spectrum, (2) objects showing very flat continuum and/or
strong Fe-K fluorescent line, and (3) objects showing unabsorbed spectrum, and explained in turn.
§.§.§ X-ray Results Taken from the Literature
We compiled the results of X-ray spectral fits for our samples from the literature.
Since our aim is to find obscured AGNs by combining X-ray and infrared data,
we tabulated absorption column densities ($N_{\rm H}$) derived from X-ray spectra in Tables <ref>, <ref>, and <ref>
for the 18 + 90 $\mu$m, 90 $\mu$m, and 18 $\mu$m samples, respectively.
The $N_{\rm H}$ values are classified into three classes “Unabsorbed”,“Compton-thin”, and “Compton-thick”
for objects with $N_{\rm H}<10^{22}$ cm$^{-2}$,
$10^{22}<N_{\rm H}<1.5\times 10^{24}$ cm$^{-2}$, and
$N_{\rm H}>1.5\times 10^{24}$ cm$^{-2}$, respectively.
If only a lower limit on $N_{\rm H}$ of $1\times10^{24}$ cm$^{-2}$ is presented, such sources are regarded as Compton-thick.
These classifications are shown in the “X-ray Class” column in Tables <ref>, <ref>, and <ref>.
The classifications of X-ray absorption are available for 72, 22, and 9 objects
for the 18 + 90 $\mu$m, 90, and 18 $\mu$m samples, respectively.
§.§.§ Absorbed Spectrum
X-ray spectra of 16 of the 48 objects we analyzed
show convex shape, which is a signature of
absorbed emission, implying the presence of an obscured AGN, at energies above $\sim$ 2 keV.
Their spectra are shown in Figure <ref>.
We fitted their spectra by a power-law model absorbed by neutral matter.
First the photon index was treated as a free parameter, and then a model with a photon index fixed at 1.8
was examined. The result for a free photon index is shown if a meaningful constraint on the index is obtained.
The spectrum of 2MASX J05430955$-$0829274 is well fitted with this absorbed power-law model accompanied by an
emission line at 6.4 keV.
The rest of the objects show additional emission at energies below a few keV.
We tried to model this emission by power law, APEC thermal plasma (Smith et al. 2001), or a combination of both.
For the models with one power-law component, the photon index was left free or fixed at 1.8.
The photon indices of the heavily absorbed power law, which dominates hard emission, and
the additional power law representing the soft part of the spectra were fixed at 1.8 for models containing
two power-law components.
A common photoelectric absorption model, which represent the absorption in the host galaxies,
was applied to both of the power-law components.
The APEC component was assumed to be absorbed only by the Galactic column.
The spectra of the two objects 2MASX J05052442$-$6734358 and NGC 5689 are represented by the two power-law model.
Other objects show excess emission around 0.6–1.0 keV, which implies the presence of Fe-L emission lines from hot plasma,
and the APEC model was used to express this feature.
An APEC + absorbed power-law model describes the spectra of
UGC 959 and IC 5264.
The spectra of the rest of the objects were fitted with a combination of two power-law and APEC components.
IRAS 03156$-$1307 required two temperature APEC components in addition to the two power-law components.
The best-fit column densities for the heavily absorbed power-law component are
$\approx 2\times10^{22} - 1.4\times10^{24}$ ,
which is a range expected for obscured AGNs.
The column densities of the two objects IRAS 01356$-$1307 and NGC 2611 exceed $1\times10^{24}$
and the effect of Compton scattering cannot be neglected.
Therefore, we multiplied the cabs model in XSPEC ($e^{-\sigma_{\rm T} N_{\rm H} }$), where $\sigma_{\rm T}$ is
the Thomson scattering cross section.
Although the energy dependence of the cross section is not taken into account,
this model approximates the shape of the continuum transmitted through Compton-thick matter (Ikeda et al. 2009).
This model affects only the normalization of the heavily absorbed power-law component.
The results of the spectral fits are summarized in Table <ref>,
and the adopted models and data/model ratios are shown in Figure <ref>.
If two or more models provide similar $C$ statistics, we adopt models with the best-fit photon index in the range of 1.5–2.1,
which is typical for X-ray spectra of AGNs.
The results of spectral fits to the Fe-K line are shown in Table <ref>.
The improvement of the $C$ statistic by adding a Gaussian line is also shown in Table <ref>.
An Fe-K emission line is seen in the spectra of 2MASX J05052442$-$6734358, 2MASX J05430955 $-$0829274, and ESO 205$-$IG003.
A hint of Fe-K emission is seen in SDSS J085312.35+162619.4.
The photon statistics are poor around 6.4 keV in IRAS 03136$-$1307 and Fe line parameters are not shown
for this object.
The improvement of the $C$ statistic for other objects is small for one additional parameter (normalization of a Gaussian).
§.§.§ Flat Continuum and Strong Fe-K Emission Line
Ten objects show a flat continuum
at energies above a few keV and/or a strong Fe-K emission line at around 6.4 keV.
The spectral shape is much flatter than that typically observed in AGNs and
implies that the spectrum is a combination of heavily absorbed and less absorbed power laws with
typical photon indices for AGNs (two power-law model) or that the spectrum is reflection-dominated.
Therefore, we examined continuum models for these two cases.
We assumed a common photon index of 1.8 for the two power-law model.
The pexrav model in XSPEC was used to represent
a continuum reflected from cold matter (Magdziarz & Zdziarski 1995). The incident spectrum is assumed to be a power law
with a photon index of 1.8 and an exponential cutoff at 300 keV. The reflection scaling factor (rel_refl)
was set to $-1$ to represent reflected emission alone. The inclination angle of the reflector was assumed to be 60$^\circ$, where
the inclination of $0^\circ$ corresponds to face on.
The spectrum of 2MASX J05391963$-$0726190 is fitted either by a pure reflection model or a two power-law model.
A pure reflection model does not describe the spectra of IRAS 00517+4556,
2XMM J052555.5$-$661038, and
2XMM J184540.6$-$630522.
The continuum of 2XMMi J184540.6 $-$630522 is fitted by a flat power-law model or a combination of reflection and slightly absorbed
power law. A combination of reflection and virtually unabsorbed power law or a two power-law model
represent the spectra of IRAS 00517+4556 and 2XMMi J052555.5$-$661038.
The six objects, NGC 1402, IC 614, 2MASX J11594382$-$2006579, NGC 6926, 2MASX J23404437 $-$1151178, and
NGC 7738, show excess emission around 0.6–1.0 keV suggesting the presence of emission from optically thin plasma.
We used an APEC plasma model to represent this feature.
An APEC + reflection model describes the continua of IC 614, 2MASX J11594382$-$2006579, and NGC 6926.
An APEC + absorbed power-law model also provided a similar quality of fit to 2MASX J11594382$-$2006579.
2MASX J23404437$-$1151178 and NGC 7738 require an additional power-law component, where we assumed a common
for the reflection and power-law components since the quality of the data is not sufficient to constrain values for
these components separately.
The spectrum of NGC 1402 is not very flat, and a combination of APEC
and lightly absorbed power law represents the shape of the continuum.
The six objects, NGC 1402, IC 614, 2XMMi J184540.6$-$630522, NGC 6926, 2MASX J23404437 $-$1151178, and NGC 7738,
show an emission line at around 6.4 keV in the source rest frame with an EW in the range of 1.1–4.6 keV.
An Fe-K line at 6.4 keV is not detected in other objects except for a weak hint of a line in NGC 4713.
An emission-line-like excess at around 7.0 keV is seen
in the spectrum of IRAS 12596$-$1529. If this line is assumed to be from H-like Fe at 6.97 keV,
the $C$ statistic is improved by $\Delta C = 5.5$ for one
additional parameter (normalization of a Gaussian).
2MASX J11594382$-$2006579 shows a line-like emission at $\approx 5.8$ keV.
If this emission is an Fe-K line at 6.4 keV in the source rest frame, the redshift is estimated
to be $0.112^{+0.019}_{-0.018}$, though the improvement of the fit is only $\Delta C = 3.0$
for two additional parameters (normalization of a Gaussian and source redshift).
Limits on the EW of Fe-K line at 6.4 keV were derived for objects with sufficient counts
around 6.4 keV.
The results of the spectral fits are summarized in Tables <ref> and <ref>.
The observed spectra, adopted models,
and data/model ratios are shown in Figure <ref>.
If two or more models give fits of similar quality, we adopt one or two models
as the most appropriate ones satisfying the following conditions:
(1) the best-fit photon index is in the range of 1.5–2.1 and
(2) the constraint on the EW on an Fe-K fluorescent line at 6.4 keV is consistent with the
best-fit $N_{\rm H}$ as observed in obscured AGNs (Guainazzi et al. 2005a, Fukazawa et al. 2011).
EPIC-PN spectra of objects showing absorbed continuum.
(Upper panel) Data (crosses) and adopted best-fit model (solid histogram).
Spectral components are shown as dashed, dotted, dot-dot-dot-dashed lines.
(Lower panel) Data/Model ratio.
EPIC-PN spectra of objects showing flat continuum and/or strong Fe-K emission.
EPIC-PN Spectra of unabsorbed objects.
§.§.§ Unabsorbed Spectrum
The rest of the objects do not show a signature of heavy absorption.
We applied an absorbed power-law model to the spectra.
The photon index $\Gamma$ was first treated as a free parameter. If the photon index was not well constrained,
$\Gamma = 1.8$ was assumed.
Good fits were obtained
for 12 objects (2XMM J004330.4$-$180107, 2MASX J02253645$-$0500123,
A426[BM99]183, AKARI J0531228+120057,
2XMMi J053512.2$-$690009, CXO J054532.6$-$001129,
MCG +01$-$27$-$029, UGC 6046, NGC 3953,
NGC 5132, 2XMM J222942.7 $-$204607, and NGC 7617).
An APEC plasma model was also examined instead of power law, and
similar quality of fits were obtained for
A426[BM99]183, AKARI J0531228+120057,
2XMMi J053512.2$-$690009,
UGC 6046, NGC 3953, and 2XMM J222942.7$-$204607.
The spectra of UGC 587 and CGCG 009$-$061A appear very soft, and an APEC model
with a temperature of 0.3–1 keV provided a good description of the data.
The single-component models do not fit the spectra of
eight objects (NGC 35, ESO 264$-$G032,
NGC 4559A, NGC 4696B,
NGC 4713, IRAS 12596$-$1529,
NGC 5350, and 2MASX J14341353 +0209088). Although the spectra of
NGC 5132 and NGC 7617 are fitted by a power-law model, the resulting photon indices
are very steep (3.3 and 2.8, respectively) and may indicate the presence of a soft component.
We therefore examined a two component-model consisting of APEC plasma and absorbed power law
with a fixed photon index of 1.8.
This model describes the spectra of all but NGC 4696B. The addition of a second APEC component
well fits the spectrum of NGC 4696B.
Two objects (NGC 4713 and IRAS 12596$-$1529) show a hint of an emission line at 6–7 keV.
The improvements of the $C$ statistic are 5.3 and 5.5 for one additional parameter (normalization of a Gaussian)
if the line center energies of 6.4 keV and 6.97 keV were assumed for NGC 4713 and IRAS 12596$-$1529,
Other objects do not show an Fe-K emission line feature. The upper limits on the EW of a Gaussian line at 6.4 keV
were derived for objects with sufficient photon statistics around an Fe-K line and summarized in Table <ref>.
The results of the fits are summarized in Table <ref>.
The observed spectra and the adopted models are shown in Figure <ref>.
§ DISCUSSION
§.§ Selection of Obscured AGNs
We made diagnostic diagrams using X-ray CRs, X-ray hardness, and infrared fluxes, which can be used to
select candidates for obscured AGNs. By accumulating the published results of X-ray spectral analysis,
we found that the regions satisfying
(CR$_{\rm 0.2-12}/F_{18}<1.0$ or
CR$_{\rm 0.2-12}/F_{90}<0.1$)
and HR4$>-0.1$ are the loci for obscured AGNs.
We analyzed the X-ray spectra of 48 objects with X-ray counts greater than 60, for
which no X-ray spectra are published.
Of these, 26 show a signature of absorbed X-ray spectra.
Their classifications of X-ray absorption as Compton-thin or Compton-thick are summarized in Table <ref>.
The 16 objects analyzed in section 4.2.2 show a spectral curvature indicative of
the continuum absorbed by $\approx10^{22-24}$ . Their best-fit values are
in the range of $3\times10^{22} - 1.4\times10^{24}$ , which are typically observed in Seyfert 2 galaxies.
The two objects showing the largest best-fit
($1.05\times10^{24}$ cm$^{-2}$ for IRAS 01356$-$1307 and $1.4\times10^{24}$ cm$^{-2}$ for NGC 2611)
are regarded as Compton-thin in the discussion below
since these values are slightly below the boundary of Compton-thin/thick
column densities ($1.5\times10^{24}$ cm$^{-2}$; Comastri 2004).
Note, however,
that the boundary value of between Compton-thin/thick depends on the assumed abundance of the absorber
(Yaqoob et al. 2010).
The 10 objects analyzed in section 4.2.3 show a flat continuum and/or a strong Fe-K emission line.
Six of these objects are most likely to be Compton-thick AGNs judging from their flat continuum and
strong Fe-K emission line.
The Fe-K emission line is not significant in four
(IRAS 00517+4556, 2MASX J05255807$-$6610523,
2MASX J05391963$-$0726190, and
2MASX J11594382$-$2006579) of the 10 objects, and
their flat continuum could be interpreted as either a reflection-dominated spectrum or
a combination of mildly absorbed ($4\times10^{22}-4\times10^{23}$ ) and less absorbed components.
These objects are tentatively regarded as Compton-thick in the following discussion and denoted as “Thick?" in
the X-ray class column of Table <ref>.
Thus, 26 objects in total are most likely obscured AGNs, for which X-ray signatures of the presence of AGNs
are reported for the first time in this paper.
The 22 objects analyzed in section 4.2.4, on the other hand, show no clear evidence for the presence of obscured AGNs.
By combining the X-ray results taken from the literature and our own analysis,
Table <ref> summarizes the number of objects in the 18 + 90 $\mu$m, 90, and 18 $\mu$m samples,
objects with X-ray measurements of absorption column densities, and unabosrbed/Compton-thin/Compton-thick objects.
Usable X-ray data are available for 84% of the combined sample.
Among the 151 objects with X-ray measurements, 113 (75%) are absorbed by Compton-thin or Compton-thick matter.
If only sources securely detected at 18 $\mu$m are used, 82 of 89 objects (92%) are absorbed.
The MIR band is better for detecting emission from warm dust heated by an AGN compared to
the far-infrared band. Therefore,
we first discuss the success rate to find obscured AGNs selected by the criteria using 18 $\mu$m data.
Among our XMM+AKARI sample, 79 objects satisfy the conditions
CR$_{\rm 0.2-12}/F_{18}<1.0$ and HR4 $>-0.1$.
The absorption column densities of 44 and 29 objects among them are
in $= 1\times10^{22} - 1.5\times10^{24}$ and larger than $1.5\times 10^{24}$ , respectively.
The detection rate of AGNs absorbed by $>10^{22}$ is 73/79 $\approx$92%, which demonstrates
the efficiency of our criteria finding obscured AGNs.
Severgnini et al. (2012) applied a similar selection technique to the sample derived from the IRAS
point source and 2XMM catalogs. They used the
conditions of $F(2-12~{\rm keV})/\nu_{25} F_{25} < 0.02$ and HR4 $>-0.2$ and selected
43 candidates for Compton-thick AGNs.
Of the 43 AGNs, 40 are in common with our sample of obscured AGNs.
Two objects (IRAS 04507+0358 and 3C321) are not selected in our sample since EPIC-PN data are not
available for them. One object (NGC 5194) has an HR4 value of $-0.11$ that is slightly smaller than our
adopted boundary (HR4 $>-0.1$) but satisfies Severgnini et al.'s criterion (HR4 $>-0.2$).
In Severgnini et al.'s sample, 32 of 43 are confirmed to be Compton-thick AGNs.
The classification of Compton thickness for four objects are model-dependent, and seven are Compton-thin.
If the conversion between IRAS 25 $\mu$m and AKARI 18 $\mu$m flux densities of equation (3) in Ichikawa et al. (2012) is assumed,
their criterion translates into CR$_{0.2-12}/F_{18}<$ 0.24 and 0.64 for
assumed of $5\times10^{23}$ and $1\times10^{22}$ , respectively,
where the same spectral shape as in section 3.1 was used.
Thus, our criteria (CR$_{\rm 0.2-12}/F_{18}<1.0$) cover X-ray brighter objects
relative to MIR fluxes, and this might result in the smaller Compton-thick fraction (44/79) derived for our sample.
Another difference between Severgnini et al.'s and our selection criteria is the limited X-ray and infrared flux levels.
Severgnini et al. selected objects with an X-ray flux in 4.5–12 keV larger than $1\times10^{-13}$ erg s$^{-1}$ cm$^{-2}$.
This flux corresponds to $7\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$ in 2–10 keV,
if = $5\times10^{23}$ and the spectral shape used in section 3.1 are assumed.
The fluxes in 2–10 keV of all but one in our sample are larger than this flux. The flux of one object
(2MASX J05391963$-$0726190) is $\sim4\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$ in 2–10 keV, which
is only slightly below the flux limit.
We also compared the distributions of X-ray count rates in 0.2–12 keV.
The solid histogram in Figure <ref> is the distribution for 68 obscured AGN candidates
satisfying the 18 or 90 $\mu$m criteria selected from Table 1.
The distribution of the CRs for the Compton-thick candidates in Severgnini et al. (2012)
is shown as a dashed histogram in the same figure. Our sample contains more X-ray fainter objects
compared to Severgnini et al.'s sample. Therefore, the X-ray flux limit likely explains
the difference in the Compton-thick fraction in part.
The distributions of 18 $\mu$m fluxes for the 70 objects
satisfying the condition CR$_{\rm 0.2-12}/F_{18}<1$ in the 18 + 90 $\mu$m sample
and Severgnini et al's Compton-thick
candidates are also compared in Figure <ref>.
We compiled 18 $\mu$m fluxes measured with AKARI IRC for the latter.
Six sources do not have 18$\mu$m data, and their IRAS 25 $\mu$m fluxes are converted to 18 $\mu$m fluxes
by using the equation (3) in Ichikawa et al. (2012). Although the lower bounds of the distribution are similar,
our sample contains a somewhat larger number of objects at fainter infrared fluxes.
In summary, the combination of the larger numbers of X-ray faint or infrared faint objects compared to Severgnini et al.'s
is likely to be a reason why we were able to find new obscured AGNs not included in Severgnini et al.'s sample.
We also used criteria using far infrared fluxes at 90 $\mu$m, CR$_{\rm 0.2-12}/F_{90}<0.1$ and HR4 $>-0.1$.
The contribution from cold dust heated by stellar processes in the host galaxy is likely to be significant in
the far infrared band unless an AGN overwhelms the emission from the host galaxy.
Therefore, a low CR$_{\rm 0.2-12}/F_{90}$
ratio does not necessarily mean that X-ray emission is weak relative to the infrared
because of the heavy absorption of an AGN.
We thus expect the criteria using far infrared
are less efficient to select obscured AGNs compared to the MIR selection.
On the other hand, the combination of X-ray hardness and a low X-ray/IR ratio
provides us with a chance to find AGNs buried in starburst activity.
There are 83 objects satisfying the conditions CR$_{\rm 0.2-12}/F_{90}<0.1$,
HR4 $>-0.1$, and not securely detected in the 18 $\mu$m band.
The presence of AGNs absorbed by a column density greater than $10^{22}$ cm$^{-2}$ is reported for
31 objects in the literature or in this paper.
The values for 21 and 10 objects are in the range of $1\times10^{22}-1.5\times10^{24}$ and
larger than $1.5\times10^{24}$ , respectively, where three objects classified as “Thick?" are regarded as Compton-thick.
There is no indication of absorption in excess of $10^{22}$ cm$^{-2}$ in 31 objects.
No measurements are available for the rest of the objects (19 objects).
Therefore, the detection rate of obscured AGNs using the far infrared criteria is 31/62=50% for the sample with measurements.
The 22 objects analyzed in section 4.2.4 show no clear evidence for the presence of an obscured AGN.
This result apparently contradicts our selection using hard X-ray spectra measured by the hardness ratio of
HR4 $>-0.1$. A possible reason for this contradiction is the faintness of the sources in the hard X-ray band.
The net counts in the 4.5–12 keV for 15 among the 22 objects are less than 42 according to the 2XMMi
catalog. The uncertainties of the hardness ratios are very large for such faint objects and sources with
an unobscured spectrum could be chosen by our criteria. In one case (UGC 587), almost no photons
are seen above 2 keV in our X-ray spectrum, but the 4.5–12 keV count in the 2XMMi catalog is 110 counts.
This case is likely to be due to a combination of very low real X-ray counts and uncertainties in background
estimation. The 4.5–12 keV counts for rest of the objects are in the range of
120–250 and their HR4 values are around 0.0, which is relatively soft among the objects selected by
our criteria. The inspection of their spectra indicates that the HR4 values are
reliable and consistent with the observed unabsorbed spectra within the errors.
§.§ Obscured AGNs Outside the Selection Criteria
While our selection criteria efficiently select obscured AGNs as discussed in the previous subsection,
there are some obscured AGNs located outside our criteria in Figure <ref>.
We examine the nature of these outliers using the 18 + 90 $\mu$m sample.
None of 29 objects absorbed by $N_{\rm H}>10^{24}$ cm$^{-2}$ are located
in the region CR$_{\rm 0.2-12 keV}$ /$F_{18}<1$.
Of the 26 objects in the range of $N_{\rm H} = 10^{23-24}$ cm$^{-2}$, 9 have
CR$_{\rm 0.2-12 keV}$ /$F_{18}>1$. This ratio scatters from object to object,
and the number of objects with CR$_{\rm 0.2-12 keV}$/$F_{18}>1$ depends on
the choice of the boundary. We set the criteria to efficiently select
more absorbed objects ($>10^{24}$ cm$^{-2}$), which resulted in missing
some moderately absorbed objects ($N_{\rm H} = 10^{23-24}$ cm$^{-2}$) in
our criteria.
Seven out of 29 sources absorbed by $N_{\rm H}>10^{24}$ cm$^{-2}$
and 12 out of 26 absorbed by $N_{\rm H}=10^{23-24}$ cm$^{-2}$ are outside our criteria
using 90 $\mu$m data.
Inspection of Figure <ref> (left and right) clearly shows that scatters
in X-ray to infrared ratios are much larger if 90 $\mu$m data are used.
A possible reason is that there is a wide range of the contribution of
infrared emission from relatively cold dust
heated by sources other than AGNs depending on the nature of host galaxies.
While the scatters naturally worsen the success rate to find obscured AGNs,
there is a higher probability of finding AGN activity hidden behind starbursts.
Among 55 sources with $N_{\rm H}>10^{23}$ cm$^{-2}$,
three objects do not satisfy the condition HR4 $>-0.1$.
In these objects (NGC 3690, HR4=$-0.267$;
Mrk 1, HR4=$-0.101$; NGC 2623, HR4=$-0.169$),
there is a considerable contribution from soft thermal emission to the band pass
used to calculate HR4, where the 2.0–4.5 keV band is used as the soft band.
Since their absorption column densities are above $10^{24}$ cm$^{-2}$,
their X-ray spectra of AGN component below 10 keV are reflection-dominated and the low-energy
cutoff due to photoelectric absorption is not clearly seen. The combination of
the significant contribution of soft thermal emission and the absence of
absorbed direct emission results in the relatively small HR4 values.
If starburst activity coexists with obscured AGNs, soft thermal emission from starburst
contributes to X-rays. In the infrared band, emission from cool dust associated with
star formation activity results in smaller X-ray to 90 $\mu$m ratios. Such cases
tend to be missed if our criteria are applied.
§.§ Optically Elusive AGNs
There are several objects classified as an H2 nucleus showing a relatively large hardness ratio.
In the 18 + 90 $\mu$m sample, the hardness ratios HR4 of nine objects
(IRAS 01173+1405, NGC 695, NGC 3877, IRAS 12550$-$2929, IRAS 12596$-$1529, NGC 5253, IRAS 20551$-$4250,
IRAS 23128$-$5919, and NGC 7738) are greater than $-0.1$. The presence of an AGN is known in
NGC 695, IRAS 12550$-$2929, IRAS 20551$-$4250, and IRAS 23128$-$5919
(Brightman & Nandra 2011a, Severgnini et al. 2012) and the latter three show
significant absorption. A signature of the presence of a Compton-thick AGN is
clearly seen in our spectrum of NGC 7738. All of these AGNs satisfy the
conditions CR$_{0.2-12}/F_{18} < 1$ and CR$_{0.2-12}/F_{90} < 0.1$.
Fifteen objects classified as H2 nuclei in the 90 $\mu$m sample
have HR4 $>-0.1$. Obscured AGNs are found in NGC 1402,
SDSS J085312.35+162619.4, IRAS 13443+0802NE, and IC 5264.
The former object is Compton-thick and the latter three are Compton-thin.
Seven objects show no clear indication of the presence of an obscured AGN
(section 4.2.4; IRAS 10190+1322, Teng et al. 2010; NGC 3314, Hudaverdi et al. 2006)
and four are excluded from our analysis sample (section 4.1).
These AGNs with an optical classification of an H2 nucleus are
a class of “optically elusive" AGNs.
Examples of optically elusive AGNs have been reported from
infrared-selected samples. Classical examples are
the discovery of Compton-thick AGN in infrared bright starburst
galaxies (NGC 4945, Iwasawa et al. 1993; NGC 6240, Iwasawa & Comastri 1998;
Arp 299, Della Ceca et al. 2002).
Maiolino et al. (2003) compiled a sample of non-Seyfert galaxies
selected by infrared luminosity,
infrared color, and the presence of a compact radio core.
AGNs are found in at least 6 of 13 objects in their sample
and AGN fraction becomes higher for more infrared luminous samples.
Most of such elusive AGNs are found to be Compton-thick.
Brightman and Nandra (2011a) found clear evidence for AGNs
in four H2-like objects in the 12 $\mu$m selected sample.
Two of them (Arp 299 and ESO 148$-$IG002) are Compton-thick.
Our results, together with these previous findings, demonstrate
the efficacy of a combination of infrared and X-ray selection to find hidden AGNs in galaxies with significant
star formation activity.
We examined infrared color ($F_{18}/F_{90}$) for the elusive AGNs
in the 18 and 90 $\mu$m sample. The $F_{18}/F_{90}$ ratios
are in the range from 0.038 to 0.067. This color is typical for
H2 nuclei and much colder than usual Seyferts (Figure <ref>).
The cold infrared color also supports that these AGNs are hidden
behind star formation activity.
Reliable intrinsic X-ray luminosities are available for Compton-thin
objects. The logarithm of intrinsic luminosities in 2–10 keV are 41.6 and 42.3
for NGC 695 and IRAS 20551$-$4250, respectively.
These luminosities are relatively low for Seyfert nuclei, and could be easily
overwhelmed by star formation activity in optical and infrared.
§ SUMMARY
We cross-correlated 18 and 90 $\mu$m sources in the PSC
and X-ray sources in the 2XMMi-DR3 catalog, and made a sample of infrared/X-ray-selected galaxies.
As the parent sample of X-ray sources,
we used objects located at $|b|>10^\circ$ and with EPIC-PN counts in 0.2–12 keV greater than 60 counts
(60851 unique sources). Infrared sources at $|b|>10^\circ$ with reliable flux measurements at 18 $\mu$m and/or
90 $\mu$m FQUAL=3) are used. There are 43865 and 62326 18 and 90 $\mu$m sources, respectively,
satisfying these criteria. The matched sample combining 18, 90 $\mu$m, and X-ray sources consist of 173 objects.
Most of them are at a low redshift; the highest redshifts is 0.31 and 90% of objects are at a redshift smaller than 0.05.
The sample was divided into various activity classes and groups of various absorption column densities
derived from X-ray spectra.
Diagnostic diagrams using X-ray hardness (HR4) and X-ray CR to infrared flux density ratios
were made using 173 objects in the matched sample of 18 $\mu$m, 90 $\mu$m, and X-ray sources.
AGNs obscured by a column density greater than $10^{23}$ are located in the lower right part of
the diagrams
HR4 versus CR$_{\rm 0.2-12}$/$F_{18}$ and
HR4 versus CR$_{\rm 0.2-12}$/$F_{90}$. We selected objects
in the region HR4$>-0.1$ and CR$_{\rm 0.2-12}$/$F_{90} <0.1$
without published X-ray results to search for obscured AGNs.
An object not detected in the 90 $\mu$m band and having HR4$>-0.1$ and CR$_{\rm 0.2-12}$/$F_{18} <1$
was also selected as a candidate obscured AGN.
We analyzed X-ray spectra of 48 objects in total after excluding Galactic sources,
sources in complex fields, or sources with very low X-ray counts after data screening.
X-ray spectra of 26 among the 48 objects (54%) show clear evidence for the presence of
absorbed AGNs. 16 objects (33%) show a continuum absorbed by a column density
ranging from $3\times10^{22}$ to $1.4\times 10^{24}$ .
Six objects (13%) show a strong Fe-K emission line and a flat continuum, indicating the presence of a
Compton-thick AGN.
The spectra of four objects are explained by either Compton-thin or Compton-thick AGN.
22 objects (46%) show no clear evidence for the presence of an obscured AGN.
These objects are either very faint in hard X-rays or hardness ratio is modest
Reliable constraints on X-ray absorption are available for
151 among 179 objects, satisfying the conditions
HR4$>-0.1$ and (CR$_{\rm 0.2-12}$/$F_{18} <1$ or CR$_{\rm 0.2-12}$/$F_{90} <0.1$).
113 objects show clear evidence for the presence of absorbed AGNs, resulting in the
success rate of 75%. If only objects satisfying the 18 $\mu$m condition are used,
the detection rate of absorbed AGN becomes 92%.
At least seven objects with an optical classification of an H2 nucleus
show evidence for the presence of obscured AGNs, four of which are reported
for the first time in this paper. These “optically elusive" AGNs have cold infrared
color ($F_{18}/F_{90}$) typical for H2 nuclei. Their optical classifications
and infrared colors are consistent with the idea that the star formation activity
overwhelms their AGN in the optical and infrared wavelengths.
The authors thank an anonymous referee for constructive comments that
improved the clarity of the paper.
This research is based on observations
obtained with XMM-Newton, an ESA science mission
with instruments and contributions directly funded by ESA Member States and NASA,
and , a JAXA project
with the participation of ESA.
This research made use of the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration,
and the HyperLeda database (http://leda.univ-lyon1.fr).
Facilities: XMM-Newton, AKARI
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|
1511.00137
|
Department of Mathematics, University of Michigan ([email protected]).
The Chebyshev points are commonly used for spectral differentiation
in non-periodic domains. The rounding error in the Chebyshev approximation
to the $n$-the derivative increases at a rate greater than $n^{2m}$
for the $m$-th derivative. The mapping technique of Kosloff and Tal-Ezer
(J. Comp. Physics, vol. 104 (1993), p. 457-469) ameliorates
this increase in rounding error. We show that the argument used to
justify the choice of the mapping parameter is substantially incomplete.
We analyze rounding error as well as discretization error and give
a more complete argument for the choice of the mapping parameter.
If the discrete cosine transform is used to compute derivatives, we
show that a different choice of the mapping parameter yields greater
§ INTRODUCTION
The Chebyshev points $x_{j}=\cos(j\pi/n)$, $n=0,1,\ldots,n$, are
commonly used to discretize the interval $[-1,1]$ when the boundary
conditions are not periodic. The $m$-th derivative $f^{(m)}(x)$
may be approximated as $\sum_{k=0}^{m}f(x_{k})w_{k,m}$ where $w_{k,m}$
are differentiation weights. The rounding error in the $m$-th derivative
increases faster than $n^{2m}$ (precise asymptotics will be given
in section 2). In contrast, the rounding error error in Fourier spectral
methods increases at the much milder rate of $n^{m}$ <cit.>
or $n^{m+1}$.
Kosloff and Tal-Ezer <cit.> introduced a mapping
technique to control the growth in rounding errors while preserving
spectral accuracy. The central idea is to replace the function $f(x)$
by the function $F(\xi)=f\left(g(\xi)\right)$ where $g:[-1,1]\rightarrow[-1,1]$,
\begin{equation}
\end{equation}
is a mapping function that depends upon the parameter $\alpha\in[0,1]$.
The grid in $\xi$ is still Chebyshev with $\xi_{j}=\cos(j\pi/n)$,
and is used to define the mapped grid in $x$ as $x_{j}=\xi_{j}$
for $j=0,1,\ldots n$. The derivative is approximated using
\[
\frac{df}{dx}=\frac{1}{g'(\xi)}\frac{dF}{d\xi}.
\]
The derivative $dF/d\xi$ is obtained using spectral differentiation
at Chebyshev points and then scaled by $1/g'(\xi)$ to obtain $df/dx$.
Higher derivatives are obtained by iteration of this technique.
The points $x_{j}$ converge to Chebyshev and equi-spaced points,
respectively, in the limits $\alpha\rightarrow0$ and $\alpha\rightarrow1$.
For $\alpha$ in-between, and usually quite close to $1$, the grid
is nearly equi-spaced and still retains spectral accuracy. Since the
grid points are not clustered quadratically near the endpoints $\pm1$,
the growth of rounding errors is milder <cit.>.
The function $F(\xi)$ will have a singularity in the complex plane,
due to the mapping, even if $f(x)$ is an entire function. Inspection
of (<ref>) shows that there are singularities at
$\xi=\pm1/\alpha$. If $f(x)$ is an entire function, such as $f(x)=\sin Kx$,
the interpolation error in $F(\xi)$ using Chebyshev points and in
$f(x)$ using the mapped grid are both controlled by the singularity
locations $\pm1/\alpha$. Kosloff and Tal-Ezer <cit.>
recommended the choice of $\alpha$ determined by
\begin{equation}
\left(\frac{1-\sqrt{1-\alpha^{2}}}{\alpha}\right)^{n}=u\label{eq:intro-kt-balance-eqn}
\end{equation}
where $u$ is the desired accuracy. Don and Solomonoff <cit.>
showed that taking $u$ to be the machine precision leads to accurate
derivatives. We prefer to take $u$ to be the unit roundoff (for double
precision arithmetic, the unit roundoff is $u=2^{-53}$ and the machine
epsilon is $2^{-52}$ <cit.>) because $u$ is the quantity
that comes up naturally in rounding error analysis. However, the distinction
between unit roundoff and machine epsilon has no real consequence
in this situation. The solution of (<ref>)
is given by $\alpha=2/(t+1/t)$ with $t=u^{-1/n}$.
A plausible argument for (<ref>) is that
it balances the discretization error on the left hand side with the
rounding error on the right hand side. Balancing errors is the right
idea, but it begs the question of why the $n^{2m}$ or $n^{2m+1}$
increase in rounding error is not showing up in (<ref>).
In this article we give a systematic treatment of both rounding and
discretization errors and show that (<ref>)
is still the right equation regardless of the order of the differentiation
$m$. The order of differentiation $m$ introduces prefactors into
both discretization and rounding error, and these cancel off fortuitously
to leave (<ref>) as the correct equation
for the mapping parameter $\alpha$ regardless of $m$.
Computation of derivatives at Chebyshev points incurs more error when
the discrete cosine transform is used <cit.>, in
comparison with carefully computed differentiation matrices <cit.>.
However, the discrete cosine transform is much faster. We show that
(<ref>) can be modified to choose $\alpha$
in a way that yields slightly more accurate derivatives when the discrete
cosine transform is employed.
Sections 2 and 3 present analyses of rounding and discretization errors,
respectively, showing how the pre-factors cancel leading to (<ref>).
When $n$ is small the total error is dominated by discretization
error and when $n$ is large the total error is dominated by rounding
error. In section 3, we show that the value of $n$ at which the total
error transitions from discretization error to rounding error does
not depend upon $m$, the order of differentiation.
In section 4, we specialize arguments to the mapping (<ref>).
We consider the slightly more general balancing equation
\begin{equation}
\left(\frac{1-\sqrt{1-\alpha^{2}}}{\alpha}\right)^{n}=n^{\beta}u\label{eq:intro-balance-general}
\end{equation}
and find that $\beta=0$ is a good choice when accurate differentiation
matrices are used and $\beta=0.5$ is a better choice for the discrete
cosine transform.
§ ROUNDING ERROR ANALYSIS OF FINITE DIFFERENCING
Spectral differentiation at Chebyshev points is a special case of
finite differencing. In this section, we derive rounding error bounds
assuming the method of partial products. The method of partial products
is an efficient way to calculate finite difference weights <cit.>.
The rounding error bounds here include the errors that arise during
the calculation of finite difference weights. Some quantities that
arise will recur in the analysis of discretization error. Comparison
to rounding error bounds which assume that the finite difference weights
are exact shows that computation of finite difference weights introduces
only a modest amount of error. Finally we give asymptotic estimates
of the error in the limit $n\rightarrow\infty$.
For floating point arithmetic, we mostly follow Higham <cit.>,
with a few modifications from <cit.>. The
axiom of floating point arithmetic is $\text{fl}(x.\text{op}.y)=(x.\text{op}.y)(1+\delta)$
with $|\delta|\leq u$, where $u$ is the unit-roundoff ($2^{-53}$
for double precision arithmetic). To handle the accumulation of rounding
error, we denote $(1+\delta_{1})^{\rho_{1}}(1+\delta_{2})^{\rho_{2}}\ldots(1+\delta_{n})^{\rho_{n}}$,
with each $\rho_{i}$ equal to $+1$, $0$, $-1$ and $|\delta_{i}|\leq u$,
by $1+\theta_{n}$. In our convention, each occurrence of $\theta_{n}$
is local, which means that two occurrences of $\theta_{n}$ , even
in the same equation, are not assumed to be equal. The quantity $\theta_{n}$
stands for any quantity that may be realized as the accumulated relative
error of $n$ or fewer multiplications and divisions. It satisfies
$|\theta_{n}|\leq\gamma_{n}$, where $\gamma_{n}=nu/(1-nu)$, as long
as $nu<1$. Whenever $\gamma_{n}$ occurs, it is implicitly assumed
that $nu<1$.
Computed quantities are hatted. Thus if $s=x_{1}+\cdots+x_{n}$, with
each $x_{i}$ a floating point number, the computed quantity is denoted
$\hat{s}$. If the addition is from left to right, we may write
\[
\hat{s}=x_{1}(1+\theta_{n-1})+x_{2}(1+\theta_{n-1})+x_{3}(1+\theta_{n-2})+\cdots+x_{n}(1+\theta_{1}).
\]
Conventions stated above allow us to rewrite this as
\[
\hat{s}=x_{1}(1+\theta_{n-1})+x_{2}(1+\theta_{n-1})+x_{3}(1+\theta_{n-1})+\cdots+x_{n}(1+\theta_{n-1}).
\]
This device will be employed frequently. Notice that it is a mistake
to factor out $(1+\theta_{n-1})$ in the right hand side, because
each $\theta_{n-1}$ is a local variable and two distinct instances
are not necessarily equal. However, we may write $\hat{s}$ as $\sum x_{j}(1+\theta_{n-1})$,
with the assumption that each $\theta_{n-1}$ inside the summation
is different.
§.§ Bounds for rounding error
Assume the $n+1$ grid points to be $x_{0},x_{1},\ldots,x_{n}$. The
weight $w_{k,m}$ in the finite difference formula $f^{(m)}(x)=\sum_{k=0}^{n}w_{k,m}f(x_{k})+\text{error}$
is given by
\begin{equation}
w_{k,m}=\frac{d^{m}\ell_{k}(x)}{dx^{m}}=w_{k}\frac{d^{m}}{dx^{m}}\prod_{j=0,j\neq k}^{n}(x-x_{j}),\label{eq:bnds-wkm-full}
\end{equation}
where $\ell_{k}(x)$ is the Lagrange cardinal function $\prod_{j\neq k}(x-x_{j})/\prod_{j\neq k}(x_{k}-x_{j})$
and $w_{k}$ is the Lagrange weight $1/\prod_{j\neq k}(x_{k}-x_{j})$.
If we assume $x=0$, by shifting the grid if necessary, then
\begin{equation}
w_{k,m}=(-1)^{n-m}m!w_{k}S_{n-m}\left(\left\{ x_{0},\ldots,x_{n}\right\} -\left\{ x_{k}\right\} \right),\label{eq:bnds-wkm-atzero}
\end{equation}
where $S_{n-m}$ is the elementary symmetric function of order $n-m$
<cit.>. The elementary symmetric function $S_{n-m}$
is the sum of $\binom{n}{n-m}$ terms each of which is a product of
a selection of $n-m$ entries out of the $n$ (all grid points excluding
$x_{k}$). $S_{0}$ is defined as $1$.
In the method of partial products <cit.>, the
weight $w_{k,m}$ is computed as follows. The polynomials $\prod_{j=0}^{k}(x-x_{j})$
and $\prod_{j=k}^{n}(x-x_{j})$ are denoted by $L_{k}$ and $R_{k}$,
respectively. Define
\begin{align}
w'_{k,m} & =\text{coeff of \ensuremath{x^{m}}in }L_{k-1}R_{k+1}\nonumber \\
& =(-1)^{n-m}\sum_{m_{1},m_{2}}S_{k-m_{1}}\left(x_{0},\ldots,x_{k-1}\right)S_{n-k-m_{2}}\left(x_{k+1},\ldots,x_{n}\right),\label{eq:bnds-wkm-convolution}
\end{align}
where the sum is taken over nonzero integers $m_{1},m_{2}$ satisfying
$m_{1}+m_{2}=m$, $k-m_{1}\geq0$, and $n-k-m_{2}\geq0$. The finite
difference weight $w_{k,m}$ is obtained as $m!w_{k}w'_{k,m}$, where
$w_{k}$ is the Lagrange weight at $z_{k}$.
The elementary symmetric functions that appear in (<ref>)
are computed by forming the products $L_{k}$ and $R_{k}$, recursively
<cit.>. In effect the recurrence
\begin{equation}
y_{N}S_{N-1}(y_{1},\ldots,y_{N-1})\quad\text{if \ensuremath{m=0}}\\
S_{N-m}(y_{1},\ldots,y_{N-1})+y_{N}S_{N-m-1}\left(y_{1},\ldots,y_{N-1}\right)\quad\text{if \ensuremath{N>m>0}}\\
1\quad\text{if \ensuremath{m=N}}
\end{cases}\label{eq:bnds-SNm-recurrence}
\end{equation}
is used for the computation of symmetric functions.
To prove an upper bound on the rounding error in computing $\sum_{k=0}^{n}w_{k,m}f(x_{k})$,
we begin with the following lemma.
If the recurrence (<ref>) is used to calculate
$S_{N-m}(y_{1},y_{2},\ldots,y_{N})$, the computed quantity may be
represented as
\[
\hat{S}_{N-m}=\sum_{i_{1}<\cdots<i_{N-m}}y_{i_{1}}y_{i_{2}}\ldots y_{i_{N-m}}\left(1+\theta_{f(N,m)}\right)
\]
with $f(N,m)=2(N-1)-m$ for $0\leq m\leq N$.
One may easily verify that $f(1,0)=f(2,0)=f(1,1)=0$ and $f(2,0)=f(2,1)=1$
suffice. If we inductively assume the lemma for $S_{N-m}(y_{1},\ldots,y_{N-1})$
and $S_{N-m-1}(y_{1},\ldots,y_{N-1})$, and apply the floating point
axiom to the recurrence, we get
\[
\]
for $N>2$, along with $f(N,N)=0$ and $f(N,0)=1+f(N-1,0)$. It may
be easily verified that $f(N,m)=2(N-1)-m$ satisfies these relations.
Next we turn to the roundoff analysis of $w'_{k,m}$ computed using
The computed value of $w'_{k,m}$ may be represented as
\[
\hat{w}'_{k,m}=(-1)^{n-m}\sum_{i_{1}<\cdots<i_{n-m}}x_{i_{1}}x_{i_{2}}\ldots x_{i_{n-m}}(1+\theta_{2n+1})
\]
where the summation is over $i_{j}\in\left\{ 0,1,\ldots,n\right\} -\left\{ k\right\} $.
The number of terms in the summation in (<ref>)
is at most $m+1$ and each term is formed using a single multiplication.
Therefore we may represent the computed quantity as
\[
\]
Applying Lemma (<ref>) to $\hat{S}_{k-m}$ (with $N=k$
and $m=m_{1}$)and $\hat{S}_{n-k-m_{2}}$(with $N=n-k$ and $m=m_{2}$),
we get a representation of $\hat{w}'_{k,m}$ that completes the proof.
The following lemma occurs as a part of Higham's rounding error analysis
of the barycentric formula <cit.>.
The computed Lagrange weight $\hat{w}_{k}$ is given by
\[
\hat{w}_{k}=w_{k}(1+\theta_{2n})
\]
where $w_{k}$ is the exact Lagrange weight.
The exact Lagrange weight is given by
\[
w_{k}=\frac{1}{\prod_{j\neq k}(x_{k}-x_{j})}.
\]
The $\theta_{2n}$ in the lemma is a result of $n$ subtractions,
$n-1$ multiplications, and a single division.
The computed weight $w_{k,m}$ may be represented as
\[
\hat{w}_{k,m}=(-1)^{n-m}m!w_{k}\,\sum_{i_{1}<\cdots<i_{n-m}}x_{i_{1}}x_{i_{2}}\ldots x_{i_{n-m}}(1+\theta_{4n+3})
\]
where the summation is over $i_{j}\in\left\{ 0,1,\ldots,n\right\} -\left\{ k\right\} $.
The finite-difference weight $w_{k,m}$ is computed as $m!w_{k}w_{k,m}$.
This lemma is proved using the previous two lemmas and incrementing
the subscript of $\theta$ by $2$ to account for multiplication by
$m!$ and $w_{k}$.
If the derivative is being approximated at $x=\zeta$, the computed
weight $w_{k,m}$ may be represented as
\[
\hat{w}_{k,m}=(-1)^{n-m}m!w_{k}\,\sum_{i_{1}<\cdots<i_{n-m}}\left(x_{i_{1}}-\zeta\right)\left(x_{i_{2}}-\zeta\right)\ldots\left(x_{i_{n-m}}-\zeta\right)(1+\theta_{5n-m+3})
\]
where the summation is over $i_{j}\in\left\{ 0,1,\ldots,n\right\} -\left\{ k\right\} $.
The finite difference weights are computed at $x=\zeta$ by shifting
the grid by $-\zeta$ and then using the algorithm for $x=0$. Thus
compared to the previous lemma, the subscript of $\theta$ is incremented
by $n-m$ to allow for $n-m$ subtractions inside the summation. There
is no need to redo the analysis of $w_{k}$ because $w_{k}$ is unchanged
by the shift and it is assumed that $w_{k}$ is computed prior to
The theorem below introduces $U_{\mathcal{R}}$ which is an upper
bound of the rounding error.
The magnitude of the roundoff error in the computation of the finite
difference approximation
\[
\sum_{k=0}^{n}w_{k,m}f(x_{k})
\]
to $f^{(m)}(\zeta)$ is upper bounded by
\begin{equation}
U_{\mathcal{R}}=\gamma_{6n-m+4}|f|\sum_{k=0}^{n}m!\,|w_{k}|\, S_{n-m}\left(\left\{ |x_{0}-\zeta|,\ldots,|x_{n}-\zeta|\right\} -\left\{ |x_{k}-\zeta\right\} \right),\label{eq:bnds-UR}
\end{equation}
where $|f|$ is equal to $\max_{j}|f(x_{j})|$.
For the computed value of $w_{k,m}$, we may use the previous lemma.
In forming the sum $\sum_{k=0}^{n}w_{k,m}f(x_{k})$, a total of $n+1$
terms are added and each term is formed through a single multiplication.
Therefore the computed value of $\sum_{k}w_{k,m}f(x_{k})$ is
\[
\sum_{k=0}^{m}f(z_{k})(-1)^{n-m}m!w_{k}\,\sum_{i_{1}<\cdots<i_{n-m}}\left(x_{i_{1}}-\zeta\right)\left(x_{i_{2}}-\zeta\right)\ldots\left(x_{i_{n-m}}-\zeta\right)(1+\theta_{6n-m+4}).
\]
Here $(1+\theta_{6n-m+4})$ is obtained from $(1+\theta_{5n-m+3})(1+\theta_{n+1})$.
The upper bound is obtained by subtracting the true value of $\sum_{k}w_{k}f(x_{k})$,
taking absolute values, and using $|\theta_{6n-m+4}|\leq\gamma_{6n-m+4}$.
If the weights $w_{k,m}$ are exact, except for the inevitable roundoff
in floating point representation, the computed value of $\sum_{k=0}^{n}w_{k,m}f(x_{k})$
\[
\sum_{k=0}^{n}w_{k,m}f(x_{k})(1+\theta_{k+2})
\]
assuming right to left summation. Thus the magnitude of the rounding
error is bounded by
\begin{equation}
\end{equation}
For other orders of summation the $\gamma_{k+2}$ may be reordered.
§.§ Asymptotics
To obtain asymptotics for $U_{\mathcal{R}}$ and $U'_{\mathcal{R}}$
in the limit of increasing $n$, we introduce three quantities $\mathcal{W}_{\ell}$,
$\mathcal{E}_{m}^{\ell}$, and $\mathcal{E}_{m}^{\ell,k}$. The first
of these $\mathcal{W}_{\ell}$ is defined as $\prod_{j=0,j\neq\ell}^{n}(x_{\ell}-x_{j})$.
It is the inverse of the Lagrange weight. If $x_{0},x_{1},\ldots,x_{n}$
are the Chebyshev points, it is well-known (see <cit.>
for example) that
\begin{equation}
\mathcal{W}_{\ell}=\begin{cases}
(-1)^{\ell}\frac{2n}{2^{n-1}} & \quad\text{for \ensuremath{\ell=0,n}}\\
(-1)^{\ell}\frac{n}{2^{n-1}} & \quad\text{otherwise.}
\end{cases}\label{eq:bnds-Wl}
\end{equation}
\begin{equation}
\mathcal{E}_{m}^{\ell}=\sum_{i_{1}<\cdots<i_{m}}\frac{1}{\left(x_{\ell}-x_{i_{1}}\right)\left(x_{\ell}-x_{i_{2}}\right)\cdots\left(x_{\ell}-x_{i_{m}}\right)}\label{eq:bnds-El}
\end{equation}
where $i_{j}\in\left\{ 0,1,\ldots,n\right\} -\left\{ \ell\right\} $.
If $\ell\neq k$, define
\begin{equation}
\mathcal{E}_{m}^{\ell,k}=\sum_{i_{1}<\cdots<i_{m}}\frac{1}{\left(x_{\ell}-x_{i_{1}}\right)\left(x_{\ell}-x_{i_{2}}\right)\cdots\left(x_{\ell}-x_{i_{m}}\right)}\label{eq:bnds-Elk}
\end{equation}
where $i_{j}\in\left\{ 0,1,\ldots,n\right\} -\left\{ k,\ell\right\} $.
If $m=0$, both $\mathcal{E}_{m}^{\ell}$ and $\mathcal{E}_{m}^{\ell,k}$
are defined to be $1$.
\[
\mathcal{P}_{r}^{\ell}=\sum_{j=0,j\neq\ell}^{n}\frac{1}{(x_{\ell}-x_{j})^{r}}\quad\text{and}\quad\mathcal{P}_{r}^{\ell,k}=\mathcal{P}_{r}^{\ell}-\frac{1}{(x_{\ell}-x_{k})^{r}}.
\]
For the Chebyshev points, or indeed for any set of points, the rounding
errors are maximized at the edges as evident from inspection of (<ref>).
Later we will see that discretization errors too tend to be the greatest
at the edges. Therefore we set $\ell=0$, and find that
\[
\frac{1}{x_{\ell}-x_{j}}=\frac{1}{1-x_{j}}\sim\frac{2n^{2}}{j^{2}\pi^{2}}.
\]
It follows that
\begin{equation}
\mathcal{P}_{r}^{0}\sim\frac{2^{r}\zeta(2r)}{\pi^{2r}}n^{2r},\label{eq:bnds-Pr0-asym}
\end{equation}
\begin{equation}
\mathcal{P}_{r}^{0,k}\sim\frac{2^{r}}{\pi^{2r}}n^{2r}\left(\zeta(2r)-\frac{1}{k^{2r}}\right),\label{eq:bnds-Prk-asym}
\end{equation}
where $\zeta(\cdot)$ is the zeta function.
The Newton identities relating symmetric functions give
\begin{align}
\mathcal{E}_{1}^{0} & =\mathcal{P}_{1}^{0}\nonumber \\
\mathcal{E}_{2}^{0} & =\mathcal{E}_{1}^{0}\mathcal{P}_{1}^{0}-\mathcal{P}_{2}^{0}\nonumber \\
\mathcal{E}_{3}^{0} & =\mathcal{E}_{2}^{0}\mathcal{P}_{1}^{0}-\mathcal{E}_{1}^{0}\mathcal{P}_{2}^{0}+\mathcal{P}_{3}^{0}.\label{eq:bnds-newton-ids}
\end{align}
Similar identities related $\mathcal{E}_{r}^{0,k}$ and $P_{r}^{0,k}$.
If $x_{0},x_{1},\ldots,x_{n}$ are the Chebyshev points, the upper
bound $U_{\mathcal{R}}$ for the rounding error with $\zeta=x_{0}=1$
has the following asymptotics in the limit of increasing $n$:
\begin{align*}
U_{\mathcal{R}} & \sim\gamma_{6n+3}|f|\left(\frac{n^{2}}{3}+\sum_{k=1}^{n}\frac{4n^{2}}{\pi^{2}}\right)=\gamma_{6n+3}|f|0.9995\ldots n^{2}\\
& \sim\gamma_{6n+2}|f|\left(\frac{n^{4}}{30}+\sum_{k=1}^{n}\frac{4n^{4}}{\pi^{2}k^{2}}\left(\frac{1}{3}-\frac{2}{\pi^{2}k^{2}}\right)\right)=\gamma_{6n+2}|f|0.1665\ldots n^{4}\\
& \sim\gamma_{6n+1}|f|\left(\frac{n^{6}}{630}+\sum_{k=1}^{n}\frac{2n^{6}}{15\pi^{6}k^{6}}\bigl|\pi^{4}k^{4}-20\pi^{2}k^{2}+120\bigr|\right)=\gamma_{6n+1}|f|0.01109\ldots n^{6}\\
& \sim\gamma_{6n}|f|\left(\frac{n^{8}}{22680}+\sum_{k=1}^{n}\frac{2n^{8}}{315\pi^{8}k^{8}}\bigl|\pi^{6}k^{6}-42\pi^{4}k^{4}+840\pi^{2}k^{2}-5040\bigr|\right)=\gamma_{6n}|f|0.00039\ldots n^{8},
\end{align*}
for order of differentiation $m=1,2,3,4$, respectively. As before,
If we go back to (<ref>), which defines $U_{\mathcal{R}}$,
and look at the $k=0$ term with $\zeta=x_{0}=1$, it can be written
\begin{align*}
w_{0}S_{n-m}(1-x_{1},1-x_{2},\ldots,1-x_{n}) & =\frac{S_{n-m}(1-x_{1},1-x_{2},\ldots,1-x_{n})}{\prod_{j=1}^{n}(1-x_{j})}\\
& =\mathcal{E}_{m}^{0}.
\end{align*}
A term with $k>0$ may be written as
\begin{align*}
|w_{k}|S_{n-m}\left(\left\{ 0,1-x_{1},\ldots,1-x_{n}\right\} -\left\{ 1-x_{k}\right\} \right) & =\frac{S_{n-m}\left(\left\{ 1-x_{1},\ldots,1-x_{n}\right\} -\left\{ 1-x_{k}\right\} \right)}{\prod_{j=0,j\neq k}^{j=n}|x_{k}-x_{j}|}\\
& =\frac{|\mathcal{W}_{0}|}{|\mathcal{W}_{k}|}\frac{S_{n-m}\left(\left\{ 1-x_{1},\ldots,1-x_{n}\right\} -\left\{ 1-x_{k}\right\} \right)}{\prod_{j=1}^{n}(1-x_{j})}\\
& =\frac{|\mathcal{W}_{0}|}{|\mathcal{W}_{k}|}\frac{\mathcal{E}_{m-1}^{0,k}}{1-z_{k}}.
\end{align*}
So the summation in (<ref>) becomes
\begin{equation}
\mathcal{E}_{m}^{0}+\sum_{k=1}^{n}\frac{|\mathcal{W}_{0}|}{|\mathcal{W}_{k}|}\frac{\mathcal{E}_{m-1}^{0,k}}{1-z_{k}}.\label{eq:bnds-tmp1}
\end{equation}
The proof is completed using the asymptotics for $\mathcal{P}_{r}^{0}$
and $\mathcal{P}_{r}^{0,k}$ in (<ref>) and (<ref>),
along with Newton identities (<ref>), to obtain
the asymptotics of $\mathcal{E}_{m}^{0}$ and $\mathcal{E}_{m-1}^{0,k}$.
These along with the formula (<ref>) for $\mathcal{W}_{\ell}$
are substituted into (<ref>)to obtain the asymptotics
of that quantity.
The methods that utilize accurate versions of the spectral differentiation
matrix <cit.> are often
employed with $m=1$. Therefore we limit the next theorem to $m=1$.
If $m=1$ and $\zeta=x_{0}=1$, the upper bound $U_{\mathcal{R}}'$
defined by (<ref>) satisfies
\[
U'_{\mathcal{R}}\precsim2u|f|\left(\frac{n^{2}}{3}+\sum_{k=1}^{n}\frac{4n^{2}(k+2)}{\pi^{2}k^{2}}\right)\sim\frac{8}{\pi^{2}}u|f|n^{2}\log n,
\]
where $u$ is the unit-roundoff, and with the assumption $nu<1/2$.
If $\zeta=x_{0}=1$, the formula for $w_{k,1}$ (<ref>)with
$\zeta$ shifted to $0$ becomes
\[
w_{k}S_{n-1}\left(\left\{ 0,1-x_{1},\ldots,1-x_{n}\right\} -\left\{ 1-x_{k}\right\} \right).
\]
If $k=0$, we have
\[
\]
and if $k>1$, we have
\[
\]
To complete the proof, we may obtain asymptotics for $w_{k,1}$ as
in the previous proof and use $\gamma_{k+1}\leq2(k+1)u$ which holds
under the assumption $nu<1/2$.
Comparison of Theorems <ref> and <ref>
gives an indication of the advantage obtained by computing the weights
$w_{k,m}$ accurately followed by careful summation. Since $\gamma_{n}\approx nu$,
the bound for the first derivative in Theorem <ref>
increases at the rate $n^{3}$. In Theorem <ref>,
the rate is $n^{2}\log n$. Thus an $n$ is replaced by $\log n$.
This is very similar to the advantage obtained using compensated summation
and other methods of precise summation <cit.>. The comparison
also shows that a sound method for calculating the weights $w_{k,m}$
introduces only a modest amount of error.
§ DISCRETIZATION ERROR
In this section, we will give a discussion of the discretization error.
We show that the discretization error goes up a factor of $n^{2}$
with every additional derivative just like the rounding error. This
implies that the value of $n$ where the total error transitions from
mostly due to discretization to mostly due to rounding is independent
of the order of the derivative. This implication is illustrated computationally.
The Lagrange interpolant may be augmented with the remainder term
as follows <cit.>:
\[
\]
Here the $f[]$ notation is for divided differences. The finite difference
approximation to $f^{(m)}(x)$ is obtained by differentiating the
Langrange interpolant $m$ times. Therefore the discretization error
for the $m$-th derivative at $x=\zeta$ is equal to
\[
\frac{d^{m}}{d\zeta^{m}}f[x_{0},\ldots,x_{n},\zeta](\zeta-x_{0})(\zeta-x_{1})\ldots(\zeta-x_{n}).
\]
The product rule for differentiation gives
\begin{equation}
\sum_{j=0}^{m}\binom{m}{j}\frac{d^{j}}{d\zeta^{j}}f[z_{0},\ldots,z_{n},\zeta]\:(m-j)!\, S_{n+1-m+j}\left(\zeta-x_{0},\ldots,\zeta-x_{n}\right).\label{eq:bnds-discerr-divdiff'}
\end{equation}
Using standard properties of divided differences, and assuming $f$
to be differentiable as many times as necessary, we may write the
discretization error as
\[
\sum_{j=0}^{m}m!f[x_{0},\ldots,x_{n},\zeta^{(j+1)}]\, S_{n+1-m+j}\left(\zeta-x_{0},\ldots,\zeta-x_{n}\right),
\]
where $\zeta^{(j+1)}$ stands for $\zeta$ repeated $j+1$ times
in the divided difference. Here we have used an identity for differentiating
a divide difference <cit.>. If $x_{0},\ldots,x_{n}$
are the Chebyshev points and the divided differences are assumed to
be relatively uniform throughout the domain, this expression above
shows that the discretization error too is likely to be maximum at
the edges. Therefore we take $\zeta=x_{0}=1$ to get
\[
U_{\mathcal{D}}=\sum_{j=0}^{m}m!f[1^{(j+2)},x_{1},\ldots x_{n}]\, S_{n+1-m+j}(0,1-x_{1},\ldots,1-x_{n}).
\]
We will denote the divided difference $f[1^{(j+2)},x_{1},\ldots x_{n}]$
by $D_{j+2}$. The expression for the discretization error becomes
\begin{equation}
\end{equation}
As $n$ increases, the asymptotics of $U_{\mathcal{D}}$ are given
\begin{align}
U_{\mathcal{D}} & \sim4\left(\frac{D_{2}}{n2^{n}}\right)n^{2}\nonumber \\
& \sim\frac{8}{3}\left(\frac{D_{2}}{n2^{n}}\right)n^{4}+\frac{8nD_{3}}{2^{n}}\nonumber \\
& \sim\frac{4}{5}\left(\frac{D_{2}}{n2^{n}}\right)n^{6}+\frac{8n^{3}D_{3}}{2^{n}}+\frac{24nD_{4}}{2^{n}}\nonumber \\
& \sim\frac{16}{105}\left(\frac{D_{2}}{n2^{n}}\right)n^{8}+\frac{16n^{5}D_{3}}{5.2^{n}}+\frac{32n^{3}D_{4}}{2^{n}}+\frac{96nD_{5}}{2^{n}}\label{eq:disc-UD-asym}
\end{align}
for orders of differentiation $m=1,2,3,4$, respectively. The symmetric
function $S_{n+1-m+j}$ in (<ref>) is equal to $\mathcal{W}_{0}\mathcal{E}_{m-j}^{0}$.
From this point the symmetric functions may be estimated as in the
previous section to derive (<ref>).
We do not attempt to estimate the divided differences $D_{j+2}$.
However they may be estimated using contour integration as shown in
<cit.>. If $f(x)=\sin Kx$, and $K=n\pi/\eta$
with $\eta>\pi$, implying more than $\pi$ points per wavelength,
the divided difference $D_{2}$ decreases exponentially with $n$.
On the other hand, if $K$ has a fixed value such as $K=2\pi$ the
divided difference decreases super-exponentially with $n$. For functions
such as $f(x)=\sin\pi x$, the divided differences $D_{1},$$D_{2}$,
and so on typically vary only by constant factors and the asymptotics
in (<ref>) may be expected to be dominated by the
$D_{2}$ term.
The interpolation error at $x=\zeta$ is given by
\[
\]
Comparison with (<ref>) shows that the interpolation error
is approximately
\begin{equation}
\end{equation}
for $\zeta\in(x_{1},x_{0})$ and where $C$ is $\mathcal{O}(1)$.
Comparison of Theorem <ref> and (<ref>)
suggests that the transition from discretization error to rounding
error should be relatively independent of the order of the derivative.
Every time the order goes up by $1$, both estimates increases by
a factor $n^{2}$, ignoring constants. Therefore the transition should
be at about the same value of $n$ independently of the order of differentiation.
This phenomenon is illustrated in Figure <ref>.
Plots of error vs $n$. The plots on the left are for $f(x)=\sin2\pi x$.
The plots on the right are for $f(x)=\sin Kx$ with $K=n\pi/4$ implying
$4$ points per wavelength. The solid line is the actual error. The
dashed line is the discretization error computed using (<ref>),
with divided differences computed in extended precision. The dotted
lines are the asymptotic rounding error bounds of Theorem <ref>.
The dotted lines with circles replace $\gamma_{6n+4-m}$ by $nu$
and the dotted lines with squares replace that quantity by $u$.
§ CHOICE OF THE MAPPING PARAMETER
The choice of the parameter $\alpha$ is taken to be given by
\begin{equation}
\left(\frac{1-\sqrt{1-\alpha^{2}}}{\alpha}\right)^{n}=n^{\beta}u\label{eq:choice-balance-eqn}
\end{equation}
with $\beta=0$ <cit.> and with
$u$ being the unit roundoff. We will attempt to justify this choice
for all orders of derivative $m$.
Given a function $f(x)$, such as $f(x)=\sin Kx$, the mapped function
is $F(\xi)=f(g(\xi))$ where $g(\cdot)$ is the mapping (<ref>).
We will first argue for (<ref>) as a balance
between the discretization error and the rounding error in interpolation.
The analysis of rounding error that arises in spectral differentiation
is precise. The indeterminacy in the rounding error is limited to
a factor of $n$, as may be seen from Figure <ref>.
However, the discretization errors cannot be estimated as precisely
because the divided differences that arise in (<ref>)
and (<ref>) are not known within factors of
If $f(x)=1$ then $F(\xi)=g(\xi)=\arcsin\alpha\xi/\arcsin\alpha$.
The Chebyshev series of $F(\xi$) may be computed from the Laurent
series of $F((z+1/z)/2)$ centered at $z=0$ (if $z=e^{i\theta}$
then $\xi=\cos\theta$, and $(z^{n}+1/z^{n})/2=\cos n\theta$ is the
Chebyshev polynomial $T_{n}(\xi)$. If $F(\xi)=g(\xi)$, the singularities
are at
\[
\]
Therefore the coefficients of $z^{\pm n}$ in the Laurent series fall
off in magnitude at the rate
\[
\left(\frac{1-\sqrt{1-\alpha^{2}}}{\alpha}\right)^{n}
\]
and so does the coefficient of $T_{n}(\xi)$ in the Chebyshev series
of $g(\xi)$. In fact, one can be more precise. Because the singularities
of $g(\xi)$ at $\xi=\pm1/\alpha$ are of the type $(\xi\pm1/\alpha)^{1/2},$
the coefficients will fall off at the rate
\begin{equation}
\end{equation}
This may be taken as an estimate of the discretization error in $g(\xi)$.
When $f(x)=\sin Kx$, the estimate (<ref>) for
interpolation error will still hold but with additional modulation
factors of the type $n^{\beta}$ with $\beta>0$. These modulating
factors are not precisely known but they certainly exist. For example,
if $K=n\pi/4$, implying $4$ points per wavelength, the number of
terms in the expansion of $\sin Kg(\xi)$ (in powers of $g(\xi)$)
before the exponentially decay of coefficients kicks in, is greater
than $\mathcal{O}(n)$.
As far as the rounding error in interpolation is concerned, this quantity
is bounded by $Cn\log nu$, with $C$ being a small constant <cit.>.
Thus balancing of discretization and rounding errors leads to (<ref>)
but with an indeterminacy in the exact value of $\beta$ and in constants.
The appropriate balance, ignoring constants, is given by (<ref>).
Empirically, $\beta=0$ is found to be a good choice although other
$\beta$ such as $\beta=-1.5$ seem to do just as well. See Figure
Graphs of error vs $n$ for $\sin2\pi x$ and $\sin n\pi x/4$. The
mapping parameter $\alpha$ is determined using (<ref>).
The errors are for the $2$nd derivative.
As long as constants are ignored, (<ref>) remains
the right equation for balancing errors for the first derivative as
well. The derivative $F'(\xi)$ is approximated by spectral differencing
at the Chebyshev points. The discretization error as well as the interpolation
error at the edge $\xi=1$ go up by a factor of $n^{2}$ from Theorem
<ref>, (<ref>), and (<ref>).
The errors are pulled back into the $x$-domain through the same $g^{-1}$
transformation, and the balancing equation remains the same.
For higher derivatives $F^{(m)}(\xi)$ the balancing equation again
remains the same, ignoring constants. With every increase in $m$
by $1$, the discretization and rounding errors both go up by a factor
of $n^{2}$. Both errors are pulled back using the same transformation
$g^{-1}$. If derivatives $f^{(m)}(x)$ are computed by successively
taking the first derivative (as in <cit.>), rather
than using a differencing scheme for the $m$-th derivative directly,
the argument changes only slightly.
The discrete cosine transform is a faster method of approximating
$F'(\xi$). However, it appears to incur greater rounding error <cit.>.
This suggests trying to balance errors in (<ref>)
with $\beta>0$, as the greater error can only be due to rounding.
In Figure (<ref>), $\beta=0.5$ does give smaller
errors for $f(x)=\sin2\pi x$ and the rounding errors vary more smoothly
for $f(x)=\sin n\pi x/4$.
Graphs of error vs $n$ for $\sin2\pi x$ and $\sin n\pi x/4$. The
mapping parameter $\alpha$ is determined using (<ref>).
The errors are for the $3$rd derivative.
§ ACKNOWLEDGEMENTS
I am very grateful to Hans Johnston for many helpful discussions.
This research was partially supported by NSF grant DMS-1115277.
|
1511.00179
|
Search for exotic transitions of muon neutrinos to electron neutrinos with MINOS
Marianna Gabrielyan
(for the MINOS and MINOS+ Collaborations)
The observed neutrino flavor transitions are currently explained by the three flavor neutrino oscillation phenomenon, considered to be the leading order mechanism behind the flavor transitions. Currently existing data from LSND, MiniBooNE and reactor experiments demonstrate anomalies that could potentially be indications of non-standard neutrino phenomena. MINOS can probe transitions from muon neutrinos to electron neutrinos and search for anomalous behavior that cannot be explained by standard model neutrino oscillations. Here we present the search for second order effects in the flavor transitions by analyzing the MINOS $\nu_\mu \to \nu_e$ channel.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
One of the motivations for exploring the sterile neutrino oscillations in the $\nu_\mu \to \nu_e$ appearance channel is the existing tension between the short (SBL) and long (LBL) baseline neutrino experiments. For example, the Liquid Scintillator Neutrino Detector (LSND) and MiniBooNE experiments report anomalous excess of $\overbar{\nu_e}$ appearance in a $\overbar{\nu_\mu}$ beam over a short baseline, which cannot be explained by standard three flavor oscillations <cit.>.
LSND and MiniBooNe allowed regions for the sterile mixing angle and mass splitting. Shows tension with the OPERA and ICARUS long baseline experiments.
One of the explanations is an existence of one or more sterile neutrino states that are not interacting via the weak interaction.
Adding a new neutrino type to 3$\nu$ model introduces another mass splitting and three mixing angles to parameter space.
Figure <ref> shows the allowed regions for the additional mass splitting and $\theta_{\mu e}$ mixing angle.
MINOS+ $\nu_e$ appearance analysis uses (3+1)$\nu$ model to investigate the sterile oscillation hypothesis in the parameter space allowed by LSND and MiniBooNE.
§ MINOS EXPERIMENT
The Main Injector Neutrino Oscillation Search (MINOS) experiment <cit.> is a long-baseline neutrino oscillation experiment utilizing the Neutrinos at the Main Injector (NuMI) neutrino beam. The experiment consists of two functionally identical tracking calorimeters, primarily optimized to detect $\mu^{+}/\mu^{-}$, separated by 735 km distance. The main goals of the experiment are to make precise measurements of the neutrino oscillation parameters to other active neutrino flavors, explore the possibility of sterile oscillations as well as to study exotic scenarios such as non-standard neutrino interactions. Near Detector (ND) is located at Fermilab $\sim1$ km away from NuMI target, is measuring the neutrino flux, cross section, energy spectrum and the flavor composition of the beam. Far Detector (FD) is located at Soudan Underground Laboratory at Northern Minnesota, $\sim735$ km away from NuMI target, measures the same beam parameters. Functionally identical design was chosen to effectively cancel out systematic uncertainties related to the neutrino beam flux and neutrino interaction cross sections.
Simulated NuMI neutrino energy spectrum for MINOS, MINOS+ and NO$\nu$A experiments.
Since the MINOS detectors were not optimized for $\nu_e$ CC events, a new statistical selection
algorithm was implemented to distinguish $\nu_\mu \to \nu_e$ candidates from background interactions - particularly NC events. $\nu_e$ appearance provides sensitivity to $\theta_{13}$, $\theta_{23}$ octant, mass hierarchy and $\delta_{CP}$.
MINOS+ is a continuation of the MINOS experiment, operating the same detectors. It will run concurrently with the NO$\nu$A experiment with an upgraded NuMI beam.
MINOS+ allows access to higher energy region to study sterile neutrino oscillations as well as non-standard neutrino interactions, with beam covering wide range of energies and peaking at 7 GeV as can be seen from Figure <ref>.
§ NUMI BEAMLINE
The $\nu_\mu$ neutrino beam is produced by colliding 120 GeV proton beam from the Main Injector with graphite target. Mesons ($\pi^{\pm}/K^{\pm}$), produced as a result of these collisions, are then focused into the decay pipe by focusing magnetic horns, where they are allowed to decay (Figure <ref>).
Schematic view of NuMI production beamline
This admixture then passes through an absorber, which stops all charged particles from moving forward. As a result wide-band on-axis $\nu_\mu$ beam is produced and directed towards the MINOS detectors. By switching the magnetic horn polarity, one can switch from neutrino to antineutrino beam production mode.
§ STERILE NEUTRINO SENSITIVITY IN $\NU_E$ APPEARANCE
The signature of $\nu_\mu \to \nu_s$ oscillations in MINOS manifests itself as an excess of $\nu_e$-CC events in the Far Detector, depending upon the particular $\Delta m^2_{41} $ and neutrino energy at the same time causing energy-dependent depletion in both NC and $\nu_\mu$-CC energy spectra. This is demonstrated in Figure <ref> for the shown parameters.
Monte Carlo Signal and Background reconstructed energy distributions. Solid lines correspond to 3$\nu$ model predictions, while the dashed lines correspond (3+1)$\nu$ model predictions for the given values of parameters.
Signal and Background Monte Carlo distributions show the excess in the signal and depletion in the background energy spectra in the presence of nonzero $\theta_{14}$ and $\theta_{24}$ mixing angles.
The sensitivity plots were generated by applying Gaussian intervals to the log-likelihood surfaces produced for specified values of
$\Delta m^2_{41}$. At each $\Delta m^2_{41}$ value, a new surface was produced by profiling the combinations of $\theta_{34}$, $\delta_{CP}$, and $\delta_{eff}$.
Figure <ref> shows the 90% confidence level contour for MINOS+ in the SBL parameter space. The shaded blue region is excluded MINOS+.
90% confidence level contour shows the MINOS+ allowed region in the SBL parameter space, assuming both
$\theta_{14}$, $\theta_{24} < 0.25\pi $.
The MINOS+ experiment is capable of decoupling the $\theta_{14}$ and $\theta_{24}$. In Figure <ref> the shaded regions correspond to LSND allowed regions. The blue dashed curves show the $\Delta m^2_{41} = 0.15 eV^2$ LSND region, while the red dashed lines outline the $\Delta m^2_{41} = 0.50 eV^2$ LSND region. The solid lines show the MINOS+ 90% C.L. contours for the same two values of $\Delta m^2_{41}$. MINOS+ excludes the parameter space to the upper-right of the solid contours. For $\Delta m^2_{41} = 0.15 eV^2$ MINOS+ can exclude most of the LSND signal region, and although for $\Delta m^2_{41} = 0.50 eV^2$ MINOS's sensitivity is relatively poor, it still can exclude some portions of the LSND signal parameter space.
Higher values of $\Delta m^2_{41} (1-10^2 eV^2)$ can also be probed, but this will require incorporating oscillations in the Near Detector when extrapolating to Far Detector.
90% C.L. in two-dimensional parameter space for two chosen values of $\Delta m^2_{41}$, assuming $\theta_{14}$, $\theta_{24} < 0.25\pi $. MINOS+ excludes the parameter space to the upper-right of the solid contours. The shaded bands represent the LSND signal region associated with the indicated values of $\Delta m^2_{41}$.
§ SUMMARY
In conclusion, MINOS+ is collecting new high-statistics data with medium energy beam. This will allow new precision measurements of 3-flavor oscillation parameters in an unexplored energy range. It opens pathway to probe sterile neutrino oscillation search in $\nu_e$ appearance channel as well as explore non-standard neutrino interaction scenarios in the 6-12 GeV region.
MINOS+ $\nu_e$ appearance can be used to set new constraints on low-mass sterile mixing parameters.
This work was supported by the U.S. DOE; the United Kingdom STFC; the U.S. NSF; the State and University of Minnesota; Brazil's FAPESP, CNPq and CAPES. We are grateful to the Minnesota Department of Natural Resources and the personnel of the Soudan Laboratory and Fermilab for their contributions to the experiment.
LSND Collaboration: A.Aguilar et al., Evidence for neutrino oscillations from the observation of $\nu_e$ appearance in a $\nu_\mu$ beam, Phys. Rev. D64, 112007 (2001).
MiniBooNE Collaboration: A.A. Aguilar-Arevalo et al., Improved Search for $\nu_\mu \to \nu_e$ Oscillations in the MiniBooNE Experiment, Phys. Rev. Lett. 110, 161801 (2013).
MINOS Collaboration: D.G Michael et al., The magnetized steel and scintillator calorimeters of the MINOS experiment, Nucl. Instrum. Meth. A 596, 190 (2008).
|
1511.00056
|
1Astronomy Program, Department of Physics and Astronomy, Seoul National University,
Seoul 151-742, Korea
2Department of Astronomy and Space Science, Chungnam National University,
Daejeon 305-764, Korea
The first observational evidence for the violation of the maximum turn-around radius on the galaxy group scale is presented.
The NGC 5353/4 group is chosen as an ideal target for our investigation of the bound-violation because of its proximity, low-density environment,
optimal mass scale, and existence of a nearby thin straight filament. Using the observational data on the line-of-sight velocities and three-dimensional
distances of the filament galaxies located
in the bound zone of the NGC 5353/4 group, we construct their radial velocity profile as a function of separation distance from the group
center and then compare it to the analytic formula obtained empirically by <cit.> to find the best-fit value of an adjustable
parameter with the help of the maximum likelihood method.
The turn-around radius of NGC 5353/4 is determined to be the separation distance where the adjusted analytic formula for the radial velocity
profile yields zero. The estimated turn-around radius of NGC 5353/4 turns out to substantially exceed the upper limit predicted by the spherical
model based on the $\Lambda$CDM cosmology. Even when the restrictive condition of spherical symmetry is released, the estimated value
is found to be only marginally consistent with the $\Lambda$CDM expectation.
§ INTRODUCTION
After the epoch of recombination, the growth of an overdense region in the primordial matter density field would be driven by
the competition between its self-gravity and the cosmic expansion. The simplest approach to understanding the gravitational evolution
of an initially overdense region is to regard it as a bound mini-universe with positive curvature and then to solve the Friedmann
equations under the assumption that the region has a uniform density and a spherically symmetric shape <cit.>.
The Friedmann equations have a well known solution for this case, according to which the radius of the spherical region varies with time as a
cycloid. The maximum radius that the spherical overdense region with uniform density will reach during its cycloidal change is called the
"turn-around radius," after which it begins to shrink back since the effect of its self-gravity becomes stronger than that of the cosmic
expansion. In the subsequent nonlinear evolution, the region will go through relaxation, virialization and the eventual
formation of a bound object <cit.>.
The virial radius of a bound object formed at the site of an initial overdense region develops only weak dependence on the background
cosmology since it is determined mainly by the total mass contained in the region <cit.>.
Whereas, the turn-around radius of an initial overdense region depends sensitively on the background cosmology as well as on the
total mass, since at the turn-around moment the pulling effect of its self gravity is cancelled out by the pushing effect
of dark energy on the region <cit.>.
For a background universe that has a larger amount of dark energy with a more negative equation of state, the turn-around radii of
overdense regions would become smaller since their self-gravity becomes comparable earlier to the anti-gravity of dark
energy <cit.>. Hence, if the turn-around radii of the initial overdense regions corresponding to the galaxy groups and clusters could be
estimated with high accuracy, they must provide a powerful independent diagnostics for the density and equation of state of dark
energy <cit.>.
By using an analytic argument, <cit.> recently put a stringent upper bound, $r_{\rm t, max}$, on the turn-around radius that a bound
object with mass $M$ can have in a flat $\Lambda$CDM (cosmological constant $\Lambda$ and Cold Dark Matter) universe:
\begin{equation}
\label{eqn:rt_max}
r_{\rm t, max} = {\cal{A}}\left(\frac{ 3MG}{\Lambda c^{2}}\right)^{1/3}\, ,
\end{equation}
where $\cal{A}$ is a proportionality factor whose value depends on the geometrical shape of an initial overdense region from which the
bound object originates. Only, provided that the shape of an overdense region possesses a perfect spherical symmetry,
the factor $\cal{A}$ equals unity, being independent of $M$. For the realistic case of a non-spherical region whose gravitational
growth tends to proceed quite anisotropically <cit.>, $\cal{A}$ slightly exceeds unity (roughly $1.3$)
taking on the mass dependence <cit.>. From here on, the spherical bound (non-spherical bound) refers to
$r_{\rm t, max}$ with ${\cal A}=1$ (${\cal A}>1$) in Equation (<ref>).
<cit.> suggested that the "zero-velocity surface" around a bound object should be a good approximation of the true turn-around
radius of the initial overdense site at which the object formed. Here the zero-velocity surface around a bound object is defined
as the radial distance from the object center at which the peculiar motions generated by the object's gravity become equal in magnitude to
the Hubble expansion. <cit.> claimed that a violation of the spherical (non-spherical) bound given in Equation (<ref>)
by any bound object observed in the universe would contest the standard $\Lambda$CDM cosmology as a smoldering (smoking)
gun counter-evidence, urging exploration of such bound-violating objects on the galaxy group and cluster scales.
Unfortunately, however, the observational estimate of the turn-around radius of a galaxy group/cluster requires us to
fulfill a tough mission: accurately measuring the peculiar velocities of the neighbor galaxies located in the bound-zone
of the galaxy group/cluster (see Section <ref> for the definition of a bound-zone) .
Here, we develop a novel methodology to estimate the turn-around radius of a galaxy group/cluster without directly measuring
the peculiar velocities. It utilizes the universal formula for the radial velocity profile empirically derived by <cit.>
to estimate the turn-around distances where the total radial velocities of the bound-zone galaxies vanish. We apply this new methodology
to the NGC 5353/4 group, a galaxy group with mass $3\times 10^{13}\, M_{\odot}$, at a distance of $34.7$ Mpc from us, appeared to be
surrounded by the low-density environment <cit.>.
Throughout this paper, to be consistent with <cit.>, we assume the standard Planck model with
$\Omega_{\rm m}=0.315,\ \Omega_{\Lambda}=0.685,\ h=0.673$ <cit.> whenever the background cosmology has to be specified.
§ THEORETICAL PREDICTION
The regions outside of the virial radius, $r_{\rm v}$, of an isolated massive object in the expanding universe can be divided into three
distinct zones. The nearest to the object is the infall zone where the object's gravity defeats the cosmic expansion. If a lower-mass
object is located in this zone, it is expected to undergo infall into the potential well of the massive object to eventually become its satellite
The farthest from the object is the Hubble zone where the cosmic expansion wins over the object's gravity. If a smaller object is located in
this zone, then its effective motion would be just the Hubble expansion since the gravitational influence from the massive object should be
The in-between is the bound zone where the object's gravity is not dominant but strong enough to slow down the Hubble flow.
The three zones (the infall, the bound, and the Hubble zones) correspond roughly to the following ranges of the separation distances, $r$, from
the object center: $r\le 2r_{\rm v}$, $3r_{\rm v}\le r < 10r_{\rm v}$ and $r > 10r_{\rm v}$, respectively <cit.>.
<cit.> has discovered the existence of the following universal profile of the bound-zone peculiar velocities:
\begin{equation}
\label{eqn:vp}
{\bf v}_{p}\cdot\hat{\bf r} = - V_{\rm c}\left(\frac{r}{r_{\rm v}}\right)^{-n_{\rm v}}\, .
\end{equation}
Here ${\bf v}_{p}$ is the peculiar velocity of a test particle at a separation distance of $r$ from a massive object with virial mass
$M_{\rm v}$, $\hat{\bf r}$ is the unit vector in the radial direction from the center of the object to the position of a test particle,
$r_{\rm v}$ is the virial radius of the object related to its virial mass as $M_{\rm v} = 4\pi\Delta_{c}r^{3}_{v}/3$
where $\Delta_{c}$ is $93.7$ times the critical density $\rho_{\rm crit,0}$, and $V_{\rm c}$ is the central velocity of a test
particle at $r_{\rm v}$ given as $V_{\rm c}\approx \left(GM_{\rm v}/r_{\rm v}\right)^{1/2}$. The negative sign in the right-hand side (RHS)
of Equation (<ref>) reflects that the object's gravity is in the direction of $-\hat{\bf r}$.
The universality of the above profile is manifested by the independence of the power-law index $n_{\rm v}$ from the masses and
redshifts of the objects. Analyzing the numerical data around the cluster halos identified from high-resolution $N$-body simulations for a flat
$\Lambda$CDM cosmology, <cit.> have demonstrated that Equation (<ref>) with a constant power-law index of
$n_{\rm v}\approx 0.42$ approximates the peculiar velocity profile of dark matter particles in the bound zones well, no matter what masses
and redshifts the cluster halos have. Their numerical result implied that the expectation value of the peculiar velocity at any point of the
bound zone around a cluster could be theoretically evaluated.
It has also been found by <cit.> that the matter-to-halo bias does not alter the functional form of Equation (<ref>) by
demonstrating that it validly approximates not only the peculiar velocity profile of dark matter particles, but also that of the galactic halos
located in the bound zone. This numerical finding has elevated the practicality of Equation (<ref>) since what is directly
observable is not the positions of dark matter particles but those of the bound-zone galaxies.
Here, we claim that the very existence of the universal peculiar velocity profile in the bound zone of a massive object
(like a galaxy group and cluster) should enable us to estimate its turn-around radius without directly measuring the peculiar velocity field
in the bound-zone.
Note that the bound-zone of a galaxy cluster is similar to the linear regime when the initial proto-cluster evolved until the turn-around moment.
Accordingly, the peculiar velocities of the bound-zone galaxies predicted by Equation (<ref>) can be treated as the linear perturbations to
the zero mean and thus must be a good approximation to the peculiar velocity of the initial proto-cluster region. Now, extrapolating
Equation (<ref>) to the turn-around moment, we determine the value of $r_{\rm t}$ as the separation distance at which the following
equality holds true.
\begin{equation}
\label{eqn:rt}
H_{\rm 0}r_{\rm t} = V_{\rm c}\left(\frac{r_{\rm t}}{r_{\rm v}}\right)^{-n_{\rm v}}\, .
\end{equation}
The left-hand side (LHS) of Equation (<ref>) represents nothing but the Hubble expansion while the RHS is the peculiar velocity of an initial
proto-cluster region in the radial direction at the turn-around radius $r_{\rm t}$ approximated by the predictable peculiar velocity of a bound-zone
galaxy in the radial direction at the same distance $r_{\rm t}$ .
To solve Equation (<ref>) in practice, we consider the total radial velocity profile, $v_{r}(r)$, defined as the combination of
the Hubble expansion with the peculiar velocity in the radial direction:
\begin{equation}
\label{eqn:vr}
v_{r}(r) = H_{\rm 0}r - V_{\rm c}\left(\frac{r}{r_{\rm v}}\right)^{-n_{\rm v}}\, ,
\end{equation}
and then look for the range of $r$ where the function, $v_{r}(r)$, crosses the zero line. It should be emphasized that
finding a solution to $v_{r}(r_{\rm t})=0$ with Equation (<ref>) requires information only on the radial positions of the bound-zone galaxies
but not on their peculiar velocities.
We plot Equation (<ref>) for the case of $M_{\rm v}=3\times 10^{13}\, M_{\odot}$ (the virial mass of NGC 5353/4, see <cit.>)
as the gray region in Figure <ref>. Given the numerical result of $n_{\rm v} = 0.42\pm 0.16$ and $V_{c}=(0.8\pm 0.2)(GM_{\rm v}/r_{\rm v})^{1/2}$
obtained by <cit.>, we let the power-law index $n_{\rm v}$ and the multiplicative constant
$a\equiv V_{\rm c}(GM_{\rm v}/r_{\rm v})^{-1/2}$ vary in the ranges of $2.6\le n_{\rm v}\le 5.8$ and $0.6\le a\le 1.0$, respectively, to
plot Equation (<ref>).
The section of the dotted line ($v_{r}=0$) inside the gray region represents the expected range of the turn-around radius of a galaxy group
with $M_{\rm v}=3\times 10^{13}\, M_{\odot}$ for a flat $\Lambda$CDM cosmology. The blue and green solid lines
indicate the locations of the spherical and non-spherical bounds, respectively.
The comparison of the estimated range of $r_{\rm t}$ with the spherical and non-spherical bounds in Figure <ref> leads us to
answer the question of whether it is possible to find a bound-violating galaxy group with mass $M_{\rm v}=3\times 10^{13}\, M_{\odot}$ in a flat
$\Lambda$CDM universe. As can be seen, the crossing between the blue (green) solid and the black dotted lines occurs inside (outside) the gray region.
This result indicates that although it is not impossible for such a galaxy group to violate the spherical bound in a flat $\Lambda$CDM model,
the violation of the non-spherical bound would rarely occur on that mass scale.
To see whether this critical prediction depends on the mass scale, we vary the values of $M_{\rm v}$ in Equation (<ref>) and find the
turn-around radius $r_{\rm t}$ as a function of $M_{\rm v}$, which is shown in Figure <ref> as gray region. The blue and green solid lines
correspond to the spherical and non-spherical bounds as a function of $M_{\rm v}$, respectively. As can be seen, the blue (green) solid line is
inside (outside) the gray region at all mass scales. Thus, the $\Lambda$CDM prediction against the bound-violation is extended to all mass scales:
it is quite unlikely to find a bound object whose turn-around radius exceeds the non-spherical bound in the standard flat $\Lambda$CDM cosmology,
no matter what mass scale the object has.
§ NGC 5353/4: A BOUND-VIOLATING STRUCTURE
§.§ Radial Velocity Profile of the Filament Galaxies around NGC 5353/4
<cit.> suggested that the optimal target for the investigation of the bound violation should be a "nearby galaxy group located in the
low-density environment."
The proximity condition is necessary to minimize the observational uncertainties. A low-density environment around a target is required to
minimize the difference between the virial and the turn-around masses. Note that $M$ in Equation (<ref>) represents the
turn-around mass enclosed by the turn-around radius $r_{\rm t}$ which is expected to be larger than the virial mass $M_{\rm v}$
<cit.>. The turn-around mass is often approximated by the virial mass since the former is unmeasurable while a variety of methods
has been developed to measure the latter. This approximation, however, works well for the case that a given target is located in the low-density
The reason for the preference of the groups to the clusters lies in the hierarchical nature of structure formation process: the galaxy
groups are more relaxed systems than the galaxy clusters since the former must have formed earlier than the latter.
Thus, the systematic errors produced by the deviation of the true dynamical state from complete relaxation would contaminate
the measurements of the virial masses of the galaxy groups less severely than those of the galaxy clusters.
Here, we require one more condition in addition to the above three for an optimal target: the presence of a thin straight filament
in the bound-zone. According to <cit.>, the total radial velocities and three-dimensional positions of the galaxies could be
readily inferred from the observable line-of-sight velocities and the two-dimensional projected positions, if the bound-zone galaxies
are located along one-dimensional filament. If the galaxies are distributed along one-dimensional filaments, they exhibit coherent
motions along the filaments, which in turn makes it easier to judge whether or not the observed galaxies are located in the bound-zone
of a target (see also S. Kim et al. 2015, in preparation).
In our previous work <cit.>, which reconstructed the radial velocity profile of the Virgo cluster and compared it to Equation
(<ref>), it was confirmed that the presence of a thin straight filament is a key ingredient for the accurate reconstruction of the radial
velocity profile of the bound-zone galaxies.
We find that the NGC 5353/4 group meets all of the above four conditions. It is only $34.7$ Mpc away from us and observed to be located in
the low-density environment <cit.>.
The three-dimensional position of the center of the NGC 5353/4 group (say, ${\bf x}_{c}$) is measured by using available
information on its equatorial coordinates and comoving distance. To estimate the virial mass, $M_{\rm v}$, of NGC 5353/4, <cit.> used
two distinct methods: one was based on the measurements of the velocity dispersions of the NGC 5353/4 satellites and
the other employed the projected mass estimator given by <cit.>. They took the average over the two estimates to find
$M_{\rm v}=2.1\times 10^{13}\, M_{\odot}$. Recently, however, <cit.> has updated $M_{\rm v}$ to a slightly higher value,
$3\times 10^{13}\, M_{\odot}$. The corresponding virial radius of NGC 5353/4 is determined to be $r_{\rm v}=1.1$ Mpc
by using the relation of $r_{\rm v} = \left[3M_{\rm v}/(4\pi \Delta_{c})\right]^{1/3}$. Adopting the conservative definition of
<cit.>, we confine the bound-zone of the NGC 5353/4 group to the region enclosed by a spherical shell whose inner
and outer radii equal $3r_{\rm v}$ and $8r_{\rm v}$, respectively.
S. Kim et al. (2015, in preparation) recently detected around the NGC 5353/4 group a thin straight filament, which is mainly composed of
dwarf galaxies with B-band magnitudes $12.83\le m_{\rm B}\le 19.54$ in the redshift range of $0.007\le z\le 0.011$.
Noting that the filament is elongated along the direction toward the NGC 5353/4 group and analyzing the line-of-sight velocities of the filament
galaxies, S. Kim et al. (2015, in preparation) have concluded that the filament is located in the bound zone of the NGC 5353/4 group.
Since the majority of the filament galaxies has been found to be the dwarf galaxies, the most gravitational influence that the filament galaxies
in the bound zone would feel should come from the NGC 5353/4 group. Information on the comoving distances of $17$ member galaxies of the
filament are found available at the NASA/IPAC Extragalactic Database[https://ned.ipac.caltech.edu].
The three-dimensional positions, ${\bf x}$, of the $17$ filament galaxies are determined and their separation displacement vectors ${\bf r}$
from the group center are calculated as ${\bf r}\equiv {\bf x}-{\bf x}_{c}$. Then, we select only those among the $17$ filament galaxies
whose separation distances, $r\equiv \vert{\bf r}\vert$, satisfy the bound-zone condition of $3\le r/r_{\rm v}\le 8$. From here on, the filament
galaxies, which have their comoving distances measured and belong to the bound-zone of NGC 5353/4, are referred to as the bound-zone
filament galaxies of NGC 5353/4. A total of four bound-zone filament galaxies is selected.
The angle, $\beta$, at which the radial direction, $\hat{\bf r}\equiv {\bf r}/r$, of each bound-zone filament galaxy is inclined to the line-of-sight
direction of the group center, $\hat{\bf x}_{c}\equiv {\bf x}_{c}/x_{c}$, can be determined as $\cos\beta =\hat{\bf r}\cdot\hat{\bf x}_{c}$.
Now, the radial velocity of each bound-zone filament galaxy in unita of km s$^{-1}$ can be evaluated from the measurable inclination angle $\beta$
and the redshift difference $\Delta z$ between the center of the NGC 5353/4 and its bound-zone filament galaxies,
\begin{equation}
\label{eqn:vlos}
v_{r}(r) = \frac{c\Delta z}{\cos\beta}\, ,
\end{equation}
where $c\Delta z$ is basically the line-of-sight velocity of a bound-zone filament galaxy relative to the NGC 5353/4 center.
§.§ Testing the $\Lambda$CDM Cosmology with NGC 5353/4
In Section <ref> we have measured the radial velocities of the bound-zone filament galaxies of NGC 5353/4 from observations.
We are now ready to adjust Equation (<ref>) to this observational result and to eventually estimate the turn-around radius of NGC 5353/4
by equating the RHS of Equation (<ref>) to zero. The adjustable parameter is nothing but the power-law index, $n_{\rm v}$, in the RHS of
Equation (<ref>).
As mentioned in Section <ref>, <cit.> found $n_{\rm v}=0.42\pm 0.16$ from their numerical experiment for the standard flat
$\Lambda$CDM cosmology. Without ruling out the possibility that the true universe deviates from the standard flat $\Lambda$CDM cosmology,
it is not unreasonable for us to suspect that the value of $n_{\rm v}$ might deviate from the estimate of <cit.>.
Moreover, our previous work has found that, although the functional form of Equation (<ref>) itself describes well the
reconstructed radial velocity profile of the Virgo cluster from observational data, the best agreement is reached when the power-law
index $n_{\rm v}$ has a lower (negative) value than $0.42$ <cit.>.
Varying the power-law index of $n_{\rm v}$, we fit the RHS of Equation (<ref>) to the measured radial velocities of the bound-zone
filament galaxies from information on $\beta$ and $c\Delta z$ (see Eq.(<ref>)).
Then, we search for the best-fit value of $n_{\rm v}$ that maximizes the following likelihood function:
\begin{equation}
\label{eqn:like1}
p(n_{\rm v}\vert M_{\rm v}, r_{i}, z_{i}, \beta_{i}) \propto \exp\left(-\frac{\chi^{2}_{\nu}}{2}\right)\, ,
\end{equation}
where $\chi^{2}$ is the reduced chi-square given as
\begin{equation}
\label{eqn:like2}
\chi^{2}_{\nu}(n_{\rm v}\vert M_{\rm v}, r_{i}, z_{i}, \beta_{i}) = \frac{1}{\nu}\sum_{i=1}^{N_{\rm g}}
\left[v^{\rm ob}(z_{i},\beta_{i}) - v^{\rm th}(r_{i}; M_{\rm v}, n_{\rm v})\right]^{2}\, .
\end{equation}
Here $N_{\rm g}$ denotes the number of the bound-zone filament galaxies, $v^{\rm ob}(r_{i})\equiv c\Delta z_{i}/\cos\beta_{i}$
represents the observational radial velocity of the $i$th bound-zone filament galaxy with redshift difference $\Delta z_{i}$ and inclination
angle $\beta_{i}$ estimated by Equation (<ref>), and $v^{\rm th}(r_{i}; n_{\rm v})$ is the theoretical radial velocity at a separation
distance $r_{i}$ predicted by Equation (<ref>). Note that the degree of freedom $\nu$ equals $N_{\rm g}-1$ since Equation (<ref>)
has only one adjustable parameter, $n_{\rm v}$.
Normalizing $p(n_{\rm v})$ to satisfy $\int dn_{\rm v}\,p(n_{\rm v}) = 1$, we plot it as a black solid line in Figure <ref>.
As can be seen, the likelihood function $p(n_{\rm v})$ reaches its sharp peak at $n_{\rm v,p}=-0.13$. The evaluation of the upper and
lower errors (say $\sigma_{\rm u}$ and $\sigma_{\rm l}$) associated with the best-fit value $n_{\rm v,p}$ as
\begin{equation}
\label{eqn:sigu}
\int_{n_{\rm v,p}}^{n_{\rm v,p}+\sigma_{\rm u}}\, dn_{\rm v}^{\prime}\, p(n_{\rm v}^{\prime} \vert M_{\rm v}, r_{i}, z_{i}, \beta_{i})=
\int^{n_{\rm v,p}}_{n_{\rm v,p}-\sigma_{\rm l}}\, dn_{\rm v}^{\prime}\, p(n_{\rm v}^{\prime} \vert M_{\rm v}, r_{i}, z_{i}, \beta_{i}) = 0.34
\end{equation}
yields $\sigma_{\rm u}=0.14$ and $\sigma_{\rm l}=0.16$, respectively.
Now that the best-fit power-law index, $n_{\rm v, p}$, for Equation (<ref>) has been determined, the corresponding turn-around
radius $r_{\rm t}$ can be calculated by equating Equation (<ref>) to zero. Figure <ref> plots how the turn-around radius
varies with the power-law index $n_{\rm v}$ as a red solid line. The location of the best-fit value, $n_{\rm v,p}$, and the amount of the
associated errors, $\sigma_{\rm u}$ and $\sigma_{\rm l}$, obtained by means of the maximum likelihood method are shown as the black
dashed line and gray regions, respectively. The blue and the green solid lines indicate the locations of the spherical and nonspherical
bounds, respectively.
Projecting the section of the red solid line overlapped with the gray region onto the vertical axis in Figure <ref> allows us
to determine the range of the turn-around radius of NGC 5353/4. As can be seen, the best-fit value $r_{\rm t}$ is one $\sigma$ higher
than the upper bound predicted by the $\Lambda$CDM model. Although the signal of the bound-violation is not so statistically significant
less than $2\sigma$, we believe that the NGC 5353/4 group is a strong candidate for the bound-violation on the group scale since its
best-fit turn-around radius turns out to exceed not only the spherical, but also the non-spherical, upper bound.
We also examine how robust the fitting result is against the change of the bound-zone range from
$3\le r/r_{\rm v}\le 8$ into $3\le r/r_{\rm v}\le 7$ and into $3\le r/r_{\rm v}\le 10$.
The likelihood functions, $p(n_{\rm v})$ for these two cases of the bound zone range are shown as red dashed and blue dotted lines,
respectively, in Figure <ref>. As can be seen, the change of the bound-zone range produces a thicker tail of
$p(n_{\rm v})$ in the high $n_{\rm v}$ section and moves the location of the peak, $n_{\rm v,p}$, to a slightly higher value.
Figures <ref> and <ref> plot the same as Figure <ref> but for the cases of
$3\le r/r_{\rm v}\le 7$ and $3\le r/r_{\rm v}\le 10$, respectively, which demonstrate that the change of the bound-zone range enlarges
the errors of $r_{\rm t}$, but does not alleviate the bound-violation by the NGC 5353/4 group much.
Table <ref> lists the numbers of the bound-zone filament galaxies, the corresponding best-fit power-law index and the
resulting ranges of the turn-around radius $r_{\rm t}$ for three different cases of the bound-zone limit.
§ SUMMARY AND DISCUSSION
Our work was inspired and urged by the seminal paper of <cit.> which theoretically proved the existence of the maximum
turn-around radii, $r_{\rm t, max}$, of massive objects and proposed it as an independent test of the $\Lambda$CDM model by
showing its sensitive dependence on the density and equation of state of dark energy <cit.>.
In the current work, we have taken a step toward realizing the ingenious idea of <cit.> by accomplishing the following. First, we
have devised a new methodology to infer the turn-around radii of galaxy groups/clusters without information on the peculiar velocity
field. Based on the numerical finding of <cit.> that the radial velocity profile of dark matter particles in the bound zone of
massive objects displays universality, this new methodology employs the extrapolation of the universal profile to the turn-around moment
at which the radial velocities would vanish.
Second, we have applied this new methodology to the nearby galaxy group, NGC 5353/4, around which a thin straight
filament composed mainly of the dwarf galaxies has been detected (S. Kim et al. 2015, in preparation). Assuming that the radial velocity profile
of the filament galaxies in the bound zone of NGC 5353/4 follows the universal form well, we have shown that
having information on the spatial positions of the filament galaxies suffices to determine the best-fit value of the
characteristic parameter of the profile. Then, we have constrained the ranges of the turn-around radius, $r_{\rm t}$, of NGC 5353/4 by
locating the separation distance at which the radial velocity profile with the best-fit parameter crosses the zero line. The comparison of the
constrained range of $r_{\rm t}$ with the theoretical upper bound, $r_{\rm t, max}$, derived by <cit.> has led us to detect
a $2\sigma$ signal of the violation of the spherical bound and a $1.3\sigma$ signal of the violation of the non-spherical bound.
Weak as the signal may appear at first sight, our result is significant because it reports the first case of possible violation of the
non-spherical bound on the galaxy group scale.
All of the previously reported cases of the bound-violation occurred on the cluster or supercluster scales but none on the group scale
<cit.>. Lack of evidence for the bound-violating cases on the galaxy group scale had been interpreted
as an indication that the signals of the bound-violation detected on the cluster scale might be spurious ones generated by
large uncertainties in the measurements of the virial masses of the clusters as well as the peculiar velocities of their bound-zone
galaxies <cit.>.
Our result provides the first observational counter-evidence against the $\Lambda$CDM prediction for $r_{\rm t, max}$ on the galaxy
group scale, which has two crucial implications. First, the observed signals of the bound-violation on the cluster scale should not be
interpreted just as false ones but deserve thorough reinvestigation by making efforts to estimate the turn-around radii more accurately.
Second, the turn-around radii of those galaxy groups, which were estimated and found to match the $\Lambda$CDM prediction well in previous
studies <cit.> may need to be reestimated without placing too much confidence on the measurements of the peculiar
velocities of their bound-zone galaxies. Since the galaxy groups have weaker self-gravity and a smaller number of bound-zone galaxies than
the galaxy clusters, it is in fact delicately more difficult to infer their turn-around radii by directly measuring the peculiar velocities of the
bound-zone galaxies. Our methodology has allowed us to overcome this difficulty, being capable of producing a robust result that is expected
to be less severely contaminated by observational errors. Yet, as mentioned in <cit.>, given the inherent stochastic nature of the
structure formation process, it requires more counter-evidence on the galaxy group scale to contest the $\Lambda$CDM cosmology
with the $r_{\rm t}$ values estimated by our methodology.
We admit that our result may suffer from the poor-number statistics. Only four bound-zone filament galaxies have been
used in our analysis to determine the best-fit value of the power-law index of the radial velocity profile around NGC 5353/4, simply because it is only
those four galaxies that already have their comoving distances independently measured . It would definitely be desirable to measure the comoving
distances of more bound-zone filament galaxies of NGC 5353/4 and then to reestimate the best-fit value of the power-law index of the radial velocity
profile by using them all, which is, however, beyond the scope of this paper.
We also admit that not all possible errors including hidden systematics have been taken into account to derive our results. For instance, if
the uncertainties in the measurements of the virial mass of NGC 5353/4 and the separation distances to the bound-zone filament
galaxies were known and accounted for, the errors in the final estimate of $r_{\rm t}$ would become larger, which might in turn alleviate
the tension with the $\Lambda$CDM prediction for $r_{\rm t, max}$. Furthermore, although the NGC 5353/4 group appears to be located
in the low-density environment <cit.>, non-negligible difference between the virial and the turn-around masses of NGC 5353/4 may
exist and could produce other uncertainties regarding the estimate of its turn-around radius.
Recently, <cit.> studied how the deviation of gravitational law from the General Relativity would affect the value of the maximum
turn-around radii and derived a general expression for $r_{\rm t, max}$ in modified gravity (MG) models from the first principles. Our
methodology will be useful to put an independent constraint on the MG models by efficiently estimating the turn-around radii of several
nearby galaxy groups and comparing the estimates with the MG predictions given by <cit.>.
However, some numerical work has to precede such a test with our methodology. Strictly speaking, the functional
form of the radial velocity profile (Eq.(<ref>)) empirically drawn by <cit.> from $N$-body simulations is applicable only for a flat
$\Lambda$CDM cosmology and it is not guaranteed that the same functional form can be applied for the cases of MG models.
It will be necessary to investigate if and how not only the value of the power-law index but also the functional form itself of the radial
velocity profile of the bound-zone galaxies differs from Equation (<ref>) in MG models. We plan to work on this project as well as on
testing the robustness of the current result by using more improved data sets. We hope to report the result elsewhere in the near future.
This work was supported by a research grant from the National Research Foundation (NRF) of Korea to the Center for
Galaxy Evolution Research (NO. 2010-0027910). J.L. also acknowledges the financial support of the Basic Science Research
Program through the NRF of Korea funded by the Ministry of Education (NO. 2013004372).
S.C.R. acknowledges the support of the Basic Science Research Program through the NRF funded by the Ministry of Education,
Science, and Technology (NRF-2015R1A2A2A01006828).
S.K. acknowledges support from the National Junior Research Fellowship of NRF (No. 2011-0012618).
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Radial velocity profile of the galaxies ($v_{r}$) located in the bound zone of a galaxy group with virial mass
$M_{\rm v}=3\times 10^{13}\,M_{\odot}$ (gray region) determined by Equation (<ref>). The turn-around radius ($r_{\rm t}$) of the galaxy
group is in the range of the separation distances between the bound-zone galaxies and the group center, $r$, where the gray region
touches the dotted-line ($v_{\rm r}=0$). The blue and green solid lines indicate the locations of the spherical and the non-spherical bounds
predicted by the standard $\Lambda$CDM cosmology <cit.>.
Variation of the turn-around radius ($r_{\rm t}$) with the virial mass of a bound object (gray region) estimated
by solving Equation (<ref>).
The spherical and non-spherical upper bounds, $r_{\rm t, max}$ predicted by the $\Lambda$CDM model are plotted
as blue and green solid lines, respectively.
Posterior probability density distributions of the power-law index $n_{\rm v}$ in the analytic formula of <cit.>
fitted to the observed radial velocities of the bound-zone filament galaxies around NGC 5353/4. The blue dotted, black solid, and
red dashed lines correspond to the cases in which the separation distances $r$ of the bound-zone filament galaxies from NGC 5353/4 are
in the range of $3\le r/r_{\rm v}\le 7$, $3\le r/r_{\rm v}\le 8$, and $3\le r/r_{\rm v}\le 10$, respectively, where $r_{\rm v}$ is the virial
radius of NGC 5353/4.
Variation of the turn-around radius with the power-law index $n_{\rm v}$ of the radial velocity profile
in the bound zone of a galaxy group with mass $M_{\rm v}=3\times 10^{13}\, M_{\odot}$ (red solid line).
The dashed line and the gray region indicate the location of the best-fit $n_{\rm v}$ and the width of the associated $1\sigma$ scatter
around the best-fit $n_{\rm v}$, respectively, which are estimated by fitting the analytic formula of <cit.> to the observed
radial velocities of the bound-zone filament galaxies around NGC 5353/4 where the bound-zone range is given as
$3\le r/r_{\rm v}\le 8$. The blue and green solid lines indicate the locations of the spherical and non-spherical bounds
(i.e., the upper limit of the turn-around radius, $r_{\rm t, max}$, predicted by the standard $\Lambda$CDM model, respectively.
Same as Figure <ref>, but for the case in which the bound-zone range is given as $3\le r/r_{\rm v}\le 7$.
Same as Figure <ref>, but for the case in which the bound-zone range is given as $3\le r/r_{\rm v}\le 10$.
Ratios of the separation distances ($r$) of the bound-zone filament galaxies to the virial radius
($r_{\rm v}$) of NGC 5353/4, the number of the bound-zone filament galaxies ($N_{\rm g}$), and the best-fit value of the
power-law index of the radial velocity profile ($n_{\rm v}$) and the estimated range of the turn-around radius of NGC 5353/4
($r_{\rm t}$).
$r/r_{\rm v}$ $N_{\rm g}$ $n_{\rm v}$ $r_{\rm t}$
$[3\, , 7]$ $3$ $-0.10^{+0.16}_{-0.18}$ $[4.67,\ 9.21]$
$[3\, , 8]$ $4$ $-0.13^{+0.15}_{-0.16}$ $[4.93,\ 9.49]$
$[3\, ,10]$ $5 $ $-0.13^{+0.17}_{-0.20}$ $[4.79,\ 10.79]$
|
1511.00177
|
Sensitivity and Discovery Potential of the PROSPECT Experiment
Karin Gilje
on behalf of the PROSPECT Collaboration
Measurements of the reactor antineutrino flux and spectrum compared to model predictions have revealed an apparent deficit in the interaction rates of reactor antineutrinos and an unexpected spectral deviation.
PROSPECT, the Precision Reactor Oscillation Spectrum measurement, is designed to make a precision measurement of the antineutrino spectrum from a research reactor and search for signs of an eV-scale sterile neutrino.
PROSPECT will be located at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory and make use of a Highly Enriched Uranium reactor for a measurement of the pure U-235 antineutrino spectrum.
An absolute measurement of this spectrum will constrain reactor models and improve our understanding of the reactor antineutrino spectrum.
Additionally, the planned 3-ton lithium-doped liquid scintillator detector is ideally suited to perform a search for sterile neutrinos.
This talk will focus on the sensitivity and discovery potential of PROSPECT and the detector design to achieve these goals.
DPF 2015
The Meeting of the American Physical Society
Division of Particles and Fields
Ann Arbor, Michigan, August 4–8, 2015
§ INTRODUCTION
Describing and understanding the flavor oscillation between neutrinos remains a primary goal in the field of neutrino physics.
Many types of experiments have been conducted using neutrinos from several different sources.
One such source is nuclear reactor cores which produce antineutrinos from the fission processes that occur within them.
The general method to detect antineutrinos coming from a reactor is through the mechanism of Inverse Beta Decay (IBD):
\begin{equation}
\bar{\nu}_e + p \rightarrow e^+ + n.
\end{equation}
Typically, a prompt signal, closely related to the antineutrino energy, is detected from the positron slowing and annihilation and a delay signal is detected through the capture of the free neutron on a nucleus such as lithium, gadolinium or hydrogen.
In order to use the antineutrino emissions of nuclear reactors for oscillation studies or other various applications, accurate predictions of the rate and the spectral shape are required.
There are two common methods used to model reactor antineutrino emissions: ab initio and $\beta$-spectrum conversion.
The ab initio approach calculates the expected neutrino spectrum by using fission daughter product yields and $\beta$-branch information obtained from nuclear databases such as the Evaluated Nuclear Structure Data File (ENSDF).
The $\beta$-spectrum conversion method examines the $\beta$ spectrum measured in dedicated experiments and uses unfolding techniques to infer the associated $\bar{\nu}_e$ spectrum <cit.>.
For commercial cores, a predicted reactor antineutrino spectrum must be known separately for all four primary fission isotopes (${}^{235}\textrm{U}$, ${}^{238}\textrm{U}$, ${}^{239}\textrm{Pu}$, and ${}^{241}\textrm{Pu}$).
For the most recent suite of $\theta_{13}$ experiments, the pure conversion approach of Huber and a hybrid method of Mueller have received the most attention <cit.>.
A recently produced ab initio prediction from Dwyer and Langford has also received significant attention <cit.>.
Two issues have emerged: there is a 10% excess in the number of observed events around 5 MeV in prompt energy and there is an overall 5% deficit in the total number of observed neutrino events, relative to model predictions in recent experiments.
The modeling approaches are necessarily approximate, and rely upon incomplete nuclear data.
Thus, one solution to the flux and spectrum anomalies is that important nuclear processes or $\beta$-decay data are being inappropriately neglected in the reactor emission models.
Given the great complexity of the fission reactor system, direct, high-precision measurements of the reactor antineutrino spectrum provide an efficient path to greatly improving our understanding of these emissions <cit.>.
An alternate explanation for the apparent deficit in the rate of observed neutrinos is the existence of an eV-scale sterile neutrino, $\nu_s$, that does not interact via the weak interaction.
However, active neutrinos can still oscillate to a sterile state leading to the overall effect of a decrease in the rate of observed events.
There are several different anomalies that could be understood with the introduction of a new neutrino beyond the Standard Model at a $\Delta m^2 \approx 1 \textrm{eV}^2$ such as the LSND, MiniBooNE and Gallium experiments <cit.>.
Short baseline reactor experiments specifically measure the neutrino mixing matrix element $|U_{e4}|^2$ through $\bar{\nu}_e$ disappearance measurements which are complementary to $\nu_\mu$ disappearance measurements like those planned for the Fermilab Short Baseline Neutrino (FSBN) program which measures a mixture of $|U_{e4}|^2$ and $|U_{\mu4}|^2$.
§ THE PROSPECT DESIGN
PROSPECT, the Precision Reactor Oscillation and Spectrum experiment, has two goals: make a precise measurement of the ${}^{235}\textrm{U}$ reactor antineutrino spectrum and perform a sterile neutrino search focusing on the parameter space around $\Delta m^2 \approx 1 \textrm{eV}^2$.
PROSPECT will be located near the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory (ORNL).
HFIR is a compact-core research reactor that operates at 85 MW.
The core is a cylinder with a radius of 0.25 m and a height of 0.5 m.
It is one of the last remaining Highly Enriched Uranium (HEU) fuel research reactors in the US in which nearly all fissions come from ${}^{235}\textrm{U}$.
Most power or commercial reactors are a mixture of the four main fuel types listed above and are termed Low Enriched Uranium (LEU) reactors.
This will enable PROSPECT to cleanly extract the ${}^{235}\textrm{U}$ spectrum and help future LEU experiments produce better constrained predictions for other future oscillation measurements <cit.>.
HFIR runs for 41% of the year allowing for in-depth studies of cosmogenic backgrounds during reactor-off periods.
Phase I detector.
Detector placement at HFIR.
(a) The inner structure of the Phase I detector.
The segmented inner volume can be seen with a PMT readout.
The shielding package consists of lead, polyethylene, and water.
(b) The positions of the Phase I (near) and Phase II (far) detectors at the HFIR complex relative to the reactor core (red cylinder).
The PROSPECT experiment has been planned in phases <cit.>.
This has several benefits as it will be able to address experimental situations in a timely manner, to mitigate any risks that arise, to provide systematic controls and increased physics reach, and allow flexibility in response to results from Phase I.
The Phase I detector will be located $\approx 7$ m from the reactor core, as seen in Figure <ref>.
It consists of a 12 x 10 grid of optically separated segments that are 15 cm x 15 cm x 1.19 m.
These segments are filled with ${}^6$Li-loaded liquid scintillator.
The optical separators are reflective in order to efficiently collect light to attain a $4.5\%/\sqrt{\textrm{E}}$ energy efficiency at 1 MeV.
At each end of the segment are 5 inch photomultiplier tubes (PMTs).
In order to extend the baseline coverage of the Phase I detector, a Phase I+ is planned where the entire Phase I detector in its shielding package is moved away from the reactor core giving a total coverage from 7-12 meters.
Logistical and engineering considerations for multiple Phase I detector locations have been confirmed with HFIR engineers.
This will provide greater physics reach through an increased L/E range and the possibility for systematic cross-checks by comparing detector data at the same baseline measured in different detector segments.
In addition to this movable detector, a later addition of a Phase II detector will be situated $\approx$ 18 m from the reactor core, which will provide greatly enhanced coverage over the low $\Delta m_{14}^2$ parameter space.
Since PROSPECT will be conducted in phases, the collaboration can respond to the information gained in Phase I and adjust the design of Phase II in order to perform a more precise measurement.
§ PROSPECT PHYSICS REACH
Three models of the ${}^{235}\textrm{U}$ $\bar{\nu}_e$ energy spectrum are shown relative to a smooth approximation of a spectrum to be measured by PROSPECT.
PROSPECT will measure the reactor antineutrino energy spectrum with a precision sufficient to be able to distinguish between several common models (Figure <ref>).
This precise measurement of the single component spectrum of a ${}^{235}\textrm{U}$ fueled reactor will directly constrain models which, in turn, will aid experiments probing the neutrino mass hierarchy and improve existing and future measurements by reducing uncertainties on the reactor $\bar{\nu}_e$ background.
In addition, there are further applications in the field of nuclear nonproliferation and in the applied nuclear physics community.
(The ratio of the L/E distributions of the oscillated spectrum divided by the unoscillated spectrum using the Kopp best fit point.
The depth of the oscillation shown in the figure is determined by $\sin^2 2\theta_{14}$ and the frequency is determined by $\Delta m^2_{14}$.
The predicted sensitivity reach into the $\Delta m^2_{14}-\sin^2 2\theta_{14}$ parameter space.
Shown are the 3$\sigma$ sensitivity curves for Phase I at 1 and 3 years of data taking and the 3$\sigma$ and $5\sigma$ curves for Phase II at 3 years of data taking in addition to 3 years of Phase I data taking.
The sterile neutrino search will be a relative measurement between the segments of the detector, and therefore will be totally independent of an absolute spectrum model prediction.
All inputs for the sensitivity calculations are based either on direct prototype measurements at HFIR or on data-benchmarked PROSPECT simulations.
Input parameters of primary importance are: 4.5%/$\sqrt{\textrm{E}}$ energy resolution, 15 cm position resolution, 1:1 signal to background ratio at the front position of the Phase I detector, and 41% reactor up-time.
The sensitivity of PROSPECT is shown in Figure <ref> for the different planned phases of the experiment.
An example of the L/E distribution for the Kopp best fit point is shown in <ref>.
The blue and red shaded regions represent the best fit sterile oscillation parameter space, as calculated by Kopp <cit.>, of the short baseline reactor anomaly and the global fit from all $\nu_e$ disappearance experiments, respectively.
The green shaded region is the anomalous region according to the Light Sterile Neutrino whitepaper (RAA) fit <cit.>.
Currently, there is little consensus on the appropriate global fit to compare with.
Although these fits have many similarities, there are a several differences that lead to significant changes in the allowed parameter space.
Each fit includes constraints from the Bugey3 spectrum shape, but RAA also includes the ILL spectrum shape which shifts the anomalous region to the higher $\Delta m_{14}^2$ and higher $\sin^2 2\theta_{14}$.
Both fits use information from the SAGE and GALLEX experiments, but with slightly different cross sections and thus different corrected rates.
Lastly, when calculating the best-fit region for all $\nu_e$ disappearance experiments, Kopp included limits from long baseline reactor experiments, carbon-12 experiments, and solar experiments.
All of these differences contribute to a visible divergence between the 95% confidence level curves of the global fits.
PROSPECT has been designed to cover the Kopp best-fit point with a single year of data in Phase I at $3\sigma$ and cover the majority of the RAA parameter space at $5\sigma$ in the same period.
It will also cover the majority of all suggested parameter space at $5\sigma$ though the combination of Phase I and Phase II.
§ CONCLUSION
PROSPECT will make a precise measurement of the ${}^{235}\textrm{U}$ spectrum which will provide new constraints on reactor antineutrino emission models.
This measurement will prove complimentary to current and future measurements at LEU fueled reactors.
PROSPECT will also perform a search for a sterile neutrino on the $1 \textrm{eV}^2$ scale and, within one year of Phase I data taking, have 3$\sigma$ coverage over the Kopp global best fit.
These studies of $\bar{\nu}_e$ disappearance will be complementary to the current Fermilab short baseline program which will focus on a search for sterile neutrinos through the measurements of $\nu_\mu$ to $\nu_e$ appearance and $\nu_\mu$ disappearance.
This material is based upon work supported by the U.S. Department of Energy Office of Science.
Additional support for this work is provided by Yale University and the Illinois Institute of Technology.
We gratefully acknowledge the support and hospitality of the High Flux Isotope Reactor and the Physics Division at Oak Ridge National Laboratory, managed by UT-Battelle for the U.S. Department of Energy.
K. Schreckenbach, G. Colvin, W. Gelletly and F. Von Feilitzsch,
Phys. Lett. B 160, 325 (1985).
P. Huber,
Phys. Rev. C 84, 024617 (2011)
[Phys. Rev. C 85, 029901 (2012)]
[arXiv:1106.0687 [hep-ph]].
T. A. Mueller et al.,
Phys. Rev. C 83, 054615 (2011)
[arXiv:1101.2663 [hep-ex]].
D. A. Dwyer and T. J. Langford,
Phys. Rev. Lett. 114, no. 1, 012502 (2015)
[arXiv:1407.1281 [nucl-ex]].
A. C. Hayes, J. L. Friar, G. T. Garvey, D. Ibeling, G. Jungman, T. Kawano and R. W. Mills,
Phys. Rev. D 92, no. 3, 033015 (2015)
[arXiv:1506.00583 [nucl-th]].
F. An et al. [JUNO Collaboration],
arXiv:1507.05613 [physics.ins-det].
J. Kopp, P. A. N. Machado, M. Maltoni and T. Schwetz,
JHEP 1305, 050 (2013)
[arXiv:1303.3011 [hep-ph]].
J. Ashenfelter et al. [PROSPECT Collaboration],
arXiv:1309.7647 [physics.ins-det].
K. N. Abazajian et al.,
[arXiv:1204.5379v1 [hep-ph]].
For details on background surveys at the HFIR complex, see J. Ashenfelter et al. [PROSPECT Collaboration],
accepted to Nucl. Instrum. Meth. A (2015) doi:10.1016/j.nima.2015.10.023
arXiv:1506.03547 [physics.ins-det].
For details on a PROSPECT prototype detector, see J. Ashenfelter et al. [PROSPECT Collaboration],
arXiv:1508.06575 [physics.ins-det].
|
1511.00523
|
In this paper, we study the problem of minimizing regret in
discounted-sum games played on weighted game graphs. We give algorithms
for the general problem of computing the minimal regret of the
controller (Eve) as well as several variants depending on which
strategies the environment (Adam) is permitted to use. We also consider
the problem of synthesizing regret-free strategies for Eve in each of
these scenarios.
§ INTRODUCTION
Two-player games played by and on weighted graphs is a well
accepted mathematical formalism for modelling quantitative aspects of a controller ()
interacting with its environment (). The outcome of the interaction between the two
players is an infinite path in the weighted graph and a value is associated to
this infinite path using a measure such as the mean-payoff of the
weights of edges traversed by the infinite path, or the discounted sum of those weights.
In the classical model, the game is considered to be zero sum: the two players
have antagonistic goals–one of the player want to maximize the value associated
to the outcome while the other want to minimize this value. The main solution
concept is then the notion of winning strategy and the main decision problem
asks, given a threshold $c$, whether has a strategy to ensure that, no matter
how plays, that the outcome has a value larger than or equal to $c$.
When the environment is not fully antagonistic, it is reasonable to study other
solution concepts. One interesting concept to explore is the
concept of regret minimization <cit.> which is as
When a strategy of is fixed, we can identify the set of 's strategies
that allow her to secure the best possible outcome against this strategy.
This constitutes 's best response. Then we define the regret of a
strategy $\sigma$ of as the difference between 's best response;
and the payoff she secures thanks to her strategy $\sigma$. So, when
trying to minimize the regret associated to a strategy, we use best responses as
a yardstick. Let us now illustrate this with an example.
[va,right=of A](B)$x$;
[va,below=of A](C)$v$;
[va,right=of C](D)$y$;
(A) edge node[el]$1$ (B)
(A) edge[bend right] node[el,swap]$0$ (C)
(C) edge[bend right] node[el,swap]$0$ (A)
(C) edge node[el,swap]$M$ (D)
(D) edge[loopright] node[el,swap]$M$ (D)
(B) edge[loopright] node[el,swap]$1$ (B)
A game in which waiting is required to minimize regret.
[node distance=0.5cm]
[ve,initial above](A);
[va,left=1cm of A,yshift=-0.5cm](B);
[va,right=1cm of A,yshift=-0.5cm](C);
[ve,below=of B](D);
[ve,below=of C](E);
[va,below=of D](G);
[va,left=2cm of G](F);
[va,below=of E](H);
[ve,below=of F](I);
[ve,below=of G](J);
[ve,below=of H](K);
(A) edge[bend right] node[el,swap]S (B)
(A) edge[bend left] node[el]B (C)
(B) edge[bend right] node[el,swap]H,$-4$ (D)
(B) edge[bend left] node[el]L,$12$ (D)
(C) edge[bend right] node[el,swap]H,$-2$ (E)
(C) edge[bend left] node[el]H,$8$ (E)
(D) edge node[el,swap]S (F)
(D) edge node[el]B (G)
(E) edge node[el]B (H)
(F) edge[bend right] node[el,swap]H,$-4$ (I)
(F) edge[bend left] node[el]L,$12$ (I)
(G) edge[bend right] node[el,swap]H,$-2$ (J)
(G) edge[bend left] node[el]L,$8$ (J)
(H) edge[bend right] node[el,swap]H,$-2$ (K)
(H) edge[bend left] node[el]L,$8$ (K)
A game that models different investment strategies.
HH HL LH LL Worst-case Regret
SS $-7.7616$ $7.6048$ $7.9784$ $23.2848$ $-7.7616$ $3.8808$
SB $-5.8408$ $3.7632$ $9.8392$ $19.4432$ $-5.8408$ ${\bf 3.8416}$
BB $-3.8808$ $5.7232$ $5.9192$ $15.5232$ ${\bf -3.8808}$ $7.7616$
The possible rate configuration for the rate of interests are given as the first four columns, the follows the worst-case performance and the regret associated to each strategy of that are given in rows.
Entries in bold are the values that are maximizing the worst-case (strategy BB) and
minimizing the regret (strategy SB).
[Investment advice]
Consider the discounted sum game of Fig. <ref>. It models the
rentability of different investment plans with a time horizon of two periods. In
the first period, it can be decided to invest in treasure bonds (B) or to
invest in the stock market (S). In the former case, treasure bonds (B) are chosen for two periods. In the latter case, after one period, there is
again a choice for either treasure bonds (B) or stock market (S).
The returns of the different investments depend on the fluctuation of the rate
of interests. When the rate of interests is low (L) then the return for the
stock market investments is equal to $12$ and for the treasure bonds it is equal
to $8$. When the interest rate is high (H) then the returns for the stock
market investments is equal to $-4$ and for the treasure bonds it is equal to
$-2$. To model time and take into account the inflation rate, say equal to $2$
percent, we consider a discount factor $\lambda=0.98$ for the returns. In this
example, we make the hypothesis that the fluctuation of the rate of interests is
not a function of the behavior of the investor. It means that this
fluctuation rate is either one of the following four possibilities: HH,
HL, LH, LL. This corresponds to playing a word
strategy in our terminology. The discounted sum of returns obtained under the
$12$ different scenarios are given in Table <ref>.
Now, assume that you are a broker and you need to advise one of your customers regarding
his next investment. There are several ways to advise your customer.
First, if your customer is strongly risk averse, then you should be able
to convince him that he has to go for the treasure bonds (B). Indeed, this
is the choice that maximizes the worst-case: if the interest rates stay
high for two periods (HH) then the loss will be $-3.8808$ while it will be
higher for any other choices. Second, and maybe more interestingly, if
your customer tolerates some risks, then you may want to keep him happy so that
he will continue to ask for your advice in the future! Then you should propose the
following strategy: first invest in the stock market (S) then in treasure
bonds (B) as this strategy minimizes regret. Indeed, at the end of the two
investment periods, the actual interest rates will be known and so your
customer will evaluate your advices ex-post. So, after the two periods,
the value of the choices made ex ante can be compared to the best strategy
that could have been chosen knowing the evolution of the interest rates. The regret of
SB is at most equal to $3.8416$ in all cases and it is minimal: the regret
of BB can be as high as $7.7616$ if LL is observed, and the regret
of SS can be as high as $3.8808$.
Finally, let us remark that if the investments are done in financial markets that
are subject to different interest rates, then instead of considering the
minimization of regret against word strategies, then we could consider the
regret against all strategies. We also study this case in this paper.
Previous works.
In <cit.>, we studied regret minimization in the context of reactive
synthesis for shortest path objectives. Recently in <cit.>, we studied the
notion of regret minimization when we assume different sets of strategies from
which chooses. We have considered three cases: when the
is allowed to play any strategy, when he is restricted to play a memoryless strategy, and when he plays word strategies. We refer the
interested reader to <cit.> for motivations behind each of these
definitions. In that paper, we studied the regret minimization problem for the following classical
quantitative measures: $\inf$, $\sup$, $\lim \inf$, $\lim \sup$ and the
mean-payoff measure. In this paper, we complete this picture by studying the
regret minimization problem for the discounted-sum measure. Discounted-sum
is a central measure in quantitative games but we did not consider it
in <cit.> because it requires specific techniques which are more involved
than the ones used for the other quantitative measures. For example, while for
mean-payoff objectives, strategies that minimize regret are memoryless when the
can play any strategy, we show in this paper that
pseudo-polynomial memory is necessary (and sufficient)
to minimize regret in discounted-sum games.
The need for memory is illustrated by the following example.
Consider the example in Figure <ref> where $M \gg 1$.
can play the following strategies in this game: let $i \in \nat \cup \{
\infty \}$, and note $\sigma^i$ the strategy that first plays $i$ rounds the
edge $(v_I,v)$ and then switches to $(v_I,x)$. The regret values associated to
those strategies are as follows.
The regret of $\sigma^{\infty}$ is $\frac{1}{1-\lambda}$ and it is witnessed
when never plays the edge $(v,y)$. Indeed, the discounted sum of the
outcome in that case is $0$, while if had chosen to play $(v_I,x)$ at the
first step instead, then she would have gained $\frac{1}{1-\lambda}$. The
regret of $\sigma^i$ is equal to the maximum between $\frac{1}{1-\lambda} -
\lambda^{2i} \frac{1}{1-\lambda}$ and $\lambda^{2i+1} \frac{M}{1-\lambda} -
\lambda^{2i} \frac{1}{1-\lambda}$. The maximum is either witnessed when
never plays $(v,y)$ or plays $(v,y)$ if the edge $(v_I,x)$ has been chosen $i+1$
times (one more time compared to $\sigma^i$).
So the strategy that minimizes regret is the strategy $\sigma^N$ for $N >
\frac{-\log M}{2\log{\lambda}} - \frac{1}{2}$ (so that $\lambda^{2N+1} M <1$), i.e. the strategy needs to count up to $N$.
We describe algorithms to decide the regret
threshold problem for games in three cases: when there is no
restriction on the strategies that can play, when can only play memoryless strategies, and when can only play word strategies. For this last case, our problem is closely
related to open problems in the field of discounted-sum automata, and we also consider variants given as $\epsilon$-gap promise problems.
We also study the complexity of the special case when the threshold is $0$, i.e. when we ask for the existence of regret free strategies. We show that that problem is sometimes easier to solve.
Our results on the complexity of both the regret threshold and the regret-free
problems are summarized in Table <ref>. All our results are for fixed discount factor $\lambda$.
Any strategy Memoryless strategies Word strategies
regret threshold
(Thm. <ref>),
-c ($\epsilon$-gap)
(Thm. <ref>) -h
(Thm. <ref>)
(Thm. <ref>,
Thm. <ref>)
(Thm. <ref>),
-c (Thm. <ref>)
(Thm. <ref>)
-h (Thm. <ref>)
Complexity of deciding the regret threshold and regret-free problems for fixed
Other related works.
A Boolean version of our regret-free strategies has been described
in <cit.>. In that paper, they are called remorse-free strategies.
These correspond to strategies which minimize regret in games with
$\omega$-regular objectives. They do not establish lower bounds on the
complexity of realizability or synthesis of remorse-free strategies and they
only consider word strategies for .
In <cit.>, we established that regret minimization when plays word strategies only
is a generalization of the notion of good-for-games
automata <cit.> and determinization by pruning (of a
refinement) <cit.>.
The notion of regret is closely related to the notion of
competitive ratios used for the analysis of online algorithms <cit.>: the performance
of an online
algorithm facing uncertainty (about the future incoming requests or data)
is compared to the performance of an offline algorithm (where uncertainty is
resolved). According to this quality measure, an online algorithm is better if
its performance is closer to the performance of an optimal offline solution.
Structure of the paper.
In Sect. 2, we introduce the necessary definitions and notations. In Sect. 3,
we study the minimization of regret when the second player plays any strategy.
Finally, in Sect. 4, we study the minimization of regret when the second player
plays a memoryless strategy and in Sect. 5 when he plays a word strategy.
§ PRELIMINARIES
A weighted arena is a tuple $G = (V,\VtcE,E,w,v_I)$ where $(V,E,w)$ is an
edge-weighted graph (with rational weights), $\VtcE \subseteq V$, and $v_I \in
V$ is the initial vertex. For a given $v \in V$ we denote by $\succ{u}$ the set
of successors of $u$ in $G$, that is the set $\{v \in V \st (u,v) \in E\}$.
We assume w.l.o.g. that no vertex is a sink, $\forall v \in V
: |\succ{v}| > 0$, and that every vertex has more than one successor,
$\forall v \in \VtcE : |\succ{v}| > 1$. In the sequel, we depict vertices
in $\VtcE$ with squares and vertices in $V \setminus \VtcE$ with circles. We
denote the maximum absolute value of a weight in a weighted arena by $W$.
A play in a weighted arena is an infinite sequence of vertices $\pi =
v_0v_1 \dots$ where $(v_i,v_{i+1}) \in E$ for all $i$. Given a
play $\pi = v_0v_1 \dots$ and integers $k,l$ we define $\pi[k..l] \defeq
v_k\dots v_l$, $\pi[..k] \defeq \pi[0..k]$, and $\pi[l..] \defeq
v_lv_{l+1}\dots$, all of which we refer to as play prefixes. To improve
readability, we try to adhere to the following convention: use $\pi$ to denote
plays and $\rho$ for play prefixes. The length of a play $\pi$, denoted
$|\pi|$, is $\infty$, and the length of a play prefix $\rho = v_0 \dots
v_n$, $|\rho|$, is $n+1$.Added a note on when $\pi$ and when
$\rho$, also added the notation for length
A strategy for () is a function $\sigma$ that maps play
prefixes ending with a vertex $v$ from $\VtcE$ ($V \setminus \VtcE$) to a
successor of $v$. A strategy has memory $m$ if it can be realized as the output
of a finite state machine with $m$ states (see <cit.> for a formal
definition). A memoryless (or positional) strategy is a strategy with
memory $1$, that is, a function that only depends on the last element of the
given partial play. A play $\pi = v_0v_1 \dots$ is consistent with a
strategy $\sigma$ for () if whenever $v_i \in \VtcE$ ($v_i \in
V\setminus \VtcE$), then $\sigma(\pi[..i]) = v_{i+1}$. We denote by
$\StrAllE(G)$ ($\StrAllA(G)$) the set of all strategies for () and by
$\StrE^{m}(G)$ ($\StrA^{m}(G)$) the set of all strategies for () in
$G$ that require memory of size at most $m$, in particular $\StrPosE(G)$
($\StrPosA(G)$) is the set of all memoryless strategies for () in $G$.
We omit $G$ if the context is clear.
Given strategies $\sigma, \tau$, for and respectively, and $v \in V$,
we denote by $\out{v}{\sigma}{\tau}$ the unique play starting from $v$ that is
consistent with $\sigma$ and $\tau$. If $v$ is omitted, it is assumed to be
A weighted automaton is a tuple $\Gamma=(Q, q_I, A, \Delta, w)$ where $A$
is a finite alphabet, $Q$ is a finite set of states, $q_I$ is the initial state,
$\Delta \subseteq Q \times A \times Q$ is the transition relation, $w : \Delta
\rightarrow \mathbb{Q}$ assigns weights to transitions. A run of $\Gamma$
on a word $a_0 a_1 \dots \in A^\omega$ is a sequence $\rho = q_0 a_0 q_1 a_1
\dots \in (Q\times A)^\omega$ such that $(q_i,a_i,q_{i+1}) \in \Delta$, for all
$i \ge 0$, and has value $\Val(\rho)$ determined by the sequence of
weights of the transitions of the run and the payoff function $\Val$. The value
$\Gamma$ assigns to a word $w$, $\Gamma(w)$, is the supremum of the values of
all runs on the word. We say the automaton is deterministic if $\Delta$ is
Safety games.
A safety game is played on a non-weighted arena by and . The
goal of is to perpetually avoid traversing edges from a set of bad
edges, while attempts to force the play through any unsafe edge. More
formally, a safety game is a tuple $(G,B)$ where $G = (V,\VtcE,E,v_I)$ is a
non-weighted arena and $B \subseteq E$ is the set of bad edges. A play $\pi =
v_0 v_1 \dots$ is winning for if $(v_i,v_{i+1}) \not\in B$, for all
$i \ge 0$, and it is winning for otherwise. A strategy for () is winning for her (him) in the safety game if all plays consistent with
it are winning for her (him). A player wins the safety game if (s)he has a
winning strategy.
Safety games are positionally determined: either has a
positional winning strategy or has a positional strategy.
Determining the winner in a safety game is decidable in linear time.
A play in a weighted arena, or a run in a weighted automaton, induces an
infinite sequence of weights. We define below the discounted-sum payoff
function which maps finite and infinite sequences of rational weights to real
numbers. In the sequel we refer to a weighted arena together with a payoff
function as a game. Formally, given a sequence of weights
$\chi=x_0 x_1 \dots$ of length $n \in \mathbb{N} \cup \{\infty\}$, the
discounted-sum is defined by a rational discount factor $\lambda \in
\( \textstyle
\discfun{\lambda}(\chi) \defeq \sum_{i=0}^n \lambda^i
\)
For convenience, we apply payoff functions directly to plays, runs, and
prefixes. For instance, given a play or play prefix $\pi = v_0 v_1 \dots$ we
write $\discfun{\lambda}(\pi)$ instead $\discfun{\lambda}(w(v_0,v_1) w(v_1,v_2)
\dots)$.
Consider a fixed weighted arena $G$, and a discounted-sum payoff function
$\Val=\discfun{\lambda}$ for some $\lambda \in (0,1)$. Given strategies
$\sigma, \tau$, for and respectively, and $v \in V$, we denote the
value of $\out{v}{\sigma}{\tau}$ by
\(
\StratVal{G}{v}{\sigma}{\tau} \defeq
\PlayVal{\out{v}{\sigma}{\tau}}.
\)
We omit $G$ if it is clear from the context. If $v$ is omitted, it is
assumed to be $v_I$.
Antagonistic & co-operative values.
Two values associated with a weighted arena that we will use
throughout are the antagonistic and co-operative values,
defined for plays from a vertex $v \in V$ as:
\[ \textstyle
\aVal^v(G) \defeq \sup_{\sigma \in \StrAllE} \inf_{\tau \in \StrAllA}
\StratVal{}{v}{\sigma}{\tau}
\qquad
\cVal^v(G) \defeq \sup_{\sigma \in \StrAllE} \sup_{\tau \in \StrAllA}
\StratVal{}{v}{\sigma}{\tau}.
\]
Again, if $G$ is clear from the context it will be omitted, and if $v$ is
omitted it is assumed to be $v_I$. We note that, as memoryless strategies are
sufficient in discounted-sum games <cit.>, $\aVal$ can be computed in time
polynomial (in $\frac{1}{1-\lambda}$, $|V|$, and $\log_2 W$). If $\lambda$ is
given as part of the input, this becomes exponential (in the size of the input).
Regardless of whether $\lambda$ is part of the input, $\cVal$ is computable in
polynomial time, determining if $\aVal$ is bigger (or smaller) than a given
threshold is decidable and in $\NP \cap \coNP$, and the values $\cVal$ and
$\aVal$ are representable using a polynomial number of bits.
A useful observation used by Zwick and Paterson in <cit.>, and which is
implicitly used throughout this work, is the following.
For all $u \in V$, $\cVal^u(G) = \max\{w(u,v) + \lambda\cVal^v(G)
\st (u,v) \in E\}$. For all $u \in \VtcE$, $\aVal^u(G) = \max\{w(u,v) +
\lambda\aVal^v(G) \st (u,v) \in E\}$. For
all $u \in \VtcA$, $\aVal^u(G) = \min\{w(u,v) +
\lambda\aVal^v(G) \st (u,v) \in E\}$.
We say a strategy $\sigma$ for is worst-case optimal (maximizing)
from $v \in V$ if it holds that $\inf_{\tau \in \StrAllA} \StratVal{
}{v}{\sigma}{\tau} = \aVal^v(G)$. Similarly, a strategy $\tau$ for is
worst-case optimal (minimizing) from $v \in V$ if it holds that
$\sup_{\sigma \in \StrAllE} \StratVal{ }{v}{\sigma}{\tau} = \aVal^v(G)$. Also, a
pair of strategies $\sigma,\tau$ for and , respectively, is said to be
co-operative optimal from $v \in V$ if $\StratVal{ }{v}{\sigma}{\tau} =
\cVal^v(G)$.
The following hold:
* there exists $\sigma \in \StrAllE$ which is worst-case
optimal maximizing from all $v \in V$,
* there exists $\tau \in \StrAllA$ which is worst-case
optimal minimizing from all $v \in V$,
* there are $\sigma \in \StrAllE$ and $\tau \in \StrAllA$
which are co-operative optimal from all $v \in V$.
We now recall the definition of a strongly co-operative optimal strategy
$\sigma$ for . Formally, for any play prefix $\rho = v_0 \dots v_n$
consistent with $\sigma$, and such that $v_n \in \VtcE$ if $\sigma(\rho) = v'$,
then $v' \in \copt{v_n}$; where $\copt{u} \defeq \{v \in V \st (u,v) \in E
\text{ and } \cVal^{u}(G) = w(u,v) + \lambda \cVal^{v}(G)\}$. Finally,
we define a new type of strategy for : co-operative worst-case
optimal strategies. A strategy is of this type if, for any play prefix $\rho =
v_0 \dots v_n$ consistent with $\sigma$, and such that $v_n \in \VtcE$, if
$\sigma(\rho) = v'$ then $v' \in \wcopt{v_n}$ and
\[
w(v_n,v') + \lambda \cVal^{v'}(G) = \max\{w(v_n,v'') +
\lambda \cVal^{v''}(G) \st v'' \in \wcopt{v_n}\},
\]
where $\wcopt{u} \defeq \{v \in V \st (u,v) \in E \text{ and } \aVal^u(G) =
w(u,v) + \lambda \aVal^v(G)\}$.
It is not hard to verify that strategies of the above types always exist for .
There exist strongly co-operative optimal strategies and co-operative
worst-case optimal strategies for .
Let $\StrE \subseteq \StrAllE$ and $\StrA \subseteq \StrAllA$ be sets of
strategies for and respectively. Given $\sigma \in \StrE$ we define
the regret of $\sigma$ in $G$ w.r.t. $\StrE$ and $\StrA$ as:
\[ \textstyle
\regret{\sigma}{G}{\StrE,\StrA} \defeq \sup_{\tau \in \StrA}
(\sup_{\sigma' \in \StrE} \StratVal{}{}{\sigma'}{\tau} -
\StratVal{}{}{\sigma}{\tau}).
\]
A strategy $\sigma$ for is then said to be regret-free w.r.t.
$\StrE$ and $\StrA$ if $\regret{ \sigma}{G}{\StrE,\StrA} = 0$. We define the
regret of $G$ w.r.t. $\StrE$ and $\StrA$ as:
\[ \textstyle
\Regret{G}{\StrE,\StrA} \defeq \inf_{\sigma \in \StrE}
\regret{\sigma}{G}{\StrE,\StrA}.
\]
When $\StrE$ or $\StrA$ are omitted from $\regret{{}}{\cdot}{{}}$ and
$\Regret{\cdot}{{}}$ they are assumed to be the set of all strategies for and .
In the unfolded definition of the regret of a game,
\[ \textstyle
\Regret{G}{\StrE,\StrA} \defeq \inf_{\sigma \in \StrE} \sup_{\tau \in
\StrA} (\sup_{\sigma' \in \StrE} \StratVal{}{}{\sigma'}{\tau} -
\StratVal{}{}{\sigma}{\tau}),
\]
let us refer to the witnesses $\sigma$ and $\sigma'$ as the primary
strategy and the alternative strategy respectively. Observe that for any
primary strategy for and any one strategy for , we can assume plays to maximize the payoff ( co-operates) against the alternative strategy
once it deviates (necessarily at an vertex) or to minimize against the
primary strategy—again, once it deviates. Indeed, since the deviation
yields different histories, the two strategies for can be combined without
conflict. More formally,Rewrote the line about deviation and maximizing
Consider any $\sigma \in \StrAllE$, $\tau \in \StrAllA$, and
corresponding play $\out{}{\sigma}{\tau} = v_0 v_1 \dots$. For all $i
\ge 0$ such that $v_i \in \VtcE$, for all $v' \in
\succ{v_i} \setminus \{v_{i+1}\}$ there exist $\sigma' \in \StrAllE$,
$\tau' \in \StrAllA$ for which
* $\out{}{\sigma'}{\tau}[..i+1] = \out{}{\sigma}{\tau}[..i]
\cdot v'$,
* $\PlayVal{\out{}{\sigma'}{\tau'}[i+1..]} = \cVal^{v'}(G)$,
* $\out{}{\sigma}{\tau} = \out{}{\sigma}{\tau'}$.
Consider any $\sigma \in \StrAllE$, $\tau \in \StrAllA$, and
corresponding play $\out{}{\sigma}{\tau} = v_0 v_1 \dots$. For all $i
\ge 0$ such that $v_i \in \VtcE$, for all $v' \in
\succ{v_i} \setminus \{v_{i+1}\}$ there exist $\sigma' \in \StrAllE$,
$\tau' \in \StrAllA$ for which
* $\out{}{\sigma'}{\tau}[..i+1] = \out{}{\sigma}{\tau}[..i]
\cdot v' = \out{}{\sigma}{\tau'}[..i] \cdot v'$,
* $\PlayVal{\out{}{\sigma'}{\tau'}[i+1..]} = \cVal^{v'}(G)$,
* $\PlayVal{\out{}{\sigma}{\tau'}[i+1..]} \le
\aVal^{v_{i+1}}(G)$.
Fixed the last claim $(iii)$ which had to be $\le$ instead of
Both claims follow from the definitions of strategies for and and
from Lemma <ref>.
In the remaining of this work, we will assume that $\lambda$ is not given as
part of the input.Something else we want to assume?
§ REGRET AGAINST ALL STRATEGIES OF
In this section we describe an algorithm to compute the (minimal) regret of a
discounted-sum game when there are no restrictions placed on the strategies of
. The algorithm can be implemented by an alternating machine guaranteed to
halt in polynomial time. We show that the regret value of any game is
achieved by a strategy for which consists of two strategies, the first
choosing edges which lead to the optimal co-operative value, the second choosing
edges which ensure the antagonistic value. The switch from the former to the
latter is done based on the “local regret” of the vertex (this is formalized
in the sequel). The latter allows us to claim -membership of the regret
threshold problem.
following theorem summarizes the bounds we obtain:
Deciding if the regret value is less than a given threshold (strictly or
non-strictly), playing against all strategies of , is in .
Let us start by formalizing the concept of
local regret. Given a play or play prefix $\pi = v_0 \dots$ and integer
$0 \le i < |\pi|$ such that $v_i \in \VtcE$, define $\locreg(\pi,i)$ as follows:
\[
\begin{cases}
\lambda^i \left(\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\pi[i..]} \right) & \text{if } \pi \text{ is a play,}\\
\lambda^i \left(\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\pi[i..j]}\right) -
\lambda^j \aVal^{v_j}(G) & \text{if } \pi \text{ is a prefix
of length } j+1 > i+1,\\
\lambda^i \left(\cVal^{v_i}(G) -
\aVal^{v_i}(G) \right) & \text{if } \pi \text{ is a prefix
of length } i+1,
\end{cases}
\]
\(
\cVal^{v_i}_{\lnot v_{i+1}}(G) = \max \{ w(v_i,v) + \lambda
\cVal^{v}(G) \st (v_i,v) \in E\text{ and } v \neq v_{i+1}\}.
\)
Intuitively, for $\pi$ a play, $\locreg(\pi,i)$ corresponds to the difference
between the value of the best deviation from position $i$ and the value of
$\pi$. For $\pi$ a play prefix, $\locreg(\pi,i)$ assumes that after position $j
= |\pi| - 1$ will play a worst-case optimal strategy.
Deciding 0-regret.
We will now argue that the problem of
determining whether has a regret-free strategy can be
decided in polynomial time. Furthermore, if no such strategy for exists, we
will extract a strategy for which, against any strategy of , ensures
non-zero regret. To do so, we will reduce the problem to that of deciding
whether wins a safety game. The unsafe edges are determined by a function
of the antagonistic and co-operative values of the original game.
Critically, the game is played on the same arena as the original regret game.
Deciding if the regret value is $0$, playing against all strategies of
, is in .
We define a partition of the edges leaving vertices from $\VtcE$ into
good and bad for . A bad edge is one which witnesses non-zero local
We then show that can ensure a regret
value of $0$ if and only if she has a strategy to avoid ever traversing
bad edges. More formally, let us assume a given weighted arena $G = (V,
\VtcE, v_I, E, w)$ and a discount factor $\lambda \in (0,1)$. We define
the set of bad edges $\mathcal{B} \defeq \{ (u,v) \in E \st u \in \VtcE$
and $w(u,v) + \lambda \aVal^v(G) < \cVal^{u}_{\lnot
Note that strategies for either player in the newly defined safety game
are also strategies for them in the original game (and vice versa as
well). We now claim that winning strategies for in the
safety game $\hat{G} = (V,\VtcE,v_I,E,\mathcal{B})$ ensure that,
regardless of the strategy of , its regret will be strictly
positive. The idea behind the claim is that, can force to traverse
a bad edge and from there, play adversarially against the primary
strategy and co-operatively with an alternative strategy.Added the
intuition recommended by JF
If $\tau \in \StrAllA$ is a winning strategy for in
$\hat{G}$, then there exist $\tau' \in \StrAllA$ and $\sigma'
\in \StrAllE$ such that
\(
\forall \sigma \in \StrAllE:
\StratVal{}{}{\sigma'}{\tau'} -
\StratVal{}{}{\sigma}{\tau'} \ge \lambda^{|V|}
\min\{ \cVal^{u}_{\lnot v}(G) - w(u,v) - \lambda
\aVal^v(G) \st (u,v) \in \mathcal{B} \text{ and } u \in \VtcE \} >
\)
The claim follows from the definitions and
Lemma <ref>. Conversely, winning strategies for
in $\hat{G}$ are actually regret-free.
If $\sigma \in \StrAllE$ is a winning strategy for in
$\hat{G}$, then $\regret{\sigma}{G}{} = 0$.
Our argument to prove this claim requires we
first show that a winning strategy for ensures the antagonistic
value of $G$ from $v_I$. For completeness, a proof for this claim is
included in appendix.
The desired result then follows from Lemma <ref> and
from the fact that membership of an edge in $\mathcal{B}$ can be decided
by computing $cVal$ and a threshold query regarding $\aVal$, thus in
polynomial time.
We observe the proof of Theorem <ref>—more
precisely, Claim <ref>—implies that, if there is no
regret-free strategy for in a game, then the regret of the game is at least
$\lambda^{|V|}$ times the smallest local regret labelling the bad edge from
$\mathcal{B}$ which can force. More formally:
If no regret-free strategy for exists in $G$, then
\(
\Regret{G}{} \ge a_G
\)
\(
a_G \defeq \lambda^{|V|} \min\{
\locreg(uv,0)
%\cVal^{u}_{\lnot v}(G) -
%w(u,v) - \lambda \cdot \aVal^v(G)
\st u \in \VtcE \text{ and } (u,v) \in \mathcal{B}\}.
\)
Deciding r-regret.
It will be useful in the sequel to define the regret of a play and the
regret of a play prefix. Given a play $\pi = v_0 v_1 \dots$, we define
the regret of $\pi$ as:
\[ \textstyle
%\regret{}{\pi}{} \defeq \sup \{ \lambda^i (\cVal^{v_i}_{\lnot
% v_{i+1}}(G) - \PlayVal{\pi[i..]}) \st v_i \in V_\exists\} \cup
% \{0\},
\regret{}{\pi}{} \defeq \left( \sup \{ \locreg(\pi,i) \st v_i \in
V_\exists\} \cup \{0\} \right).
\]
Intuitively, the
local regrets give lower bounds for the overall regret of a play. We will also
let the regret of a play prefix $\rho = v_0 \dots v_j$ be equal to
\[ \textstyle
\max \left(\{ \lambda^i
(\cVal^{v_i}_{\lnot v_{i+1}}(G) - \PlayVal{\rho[i..j]}) \st 0 \le
i < j \text{ and } v_i \in V_\exists\} \cup
\{0\}\right).
\]
(root) at (0,5) $v_I$;
(leftcorner) at (-4,0) ;
(rightcorner) at (4,0) ;
(root) edge (leftcorner)
(root) edge (rightcorner);
(bottom) at (0,0) $\out{}{\sigma}{\tau}$;
[->,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(root) edge[decorate] (bottom);
[ve,fill=white] (alt) at (0,3) $v_i$;
(altbottom) at (3.5,0) $\out{}{\sigma'}{\tau}$;
[->,green,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(alt) edge[decorate] (altbottom);
Depiction of a play and a “better alternative play”.
(root) at (0,5) $v_I$;
(leftcorner) at (-4,0) ;
(rightcorner) at (4,0) ;
(root) edge (leftcorner)
(root) edge (rightcorner);
(bottom) at (0,0) $\out{}{\sigma}{\tau}$;
[->,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(root) edge[decorate] (bottom);
[ve,fill=white] (alt) at (0,3) $v_i$;
(altbottom) at (3.5,0) $\out{}{\sigma'}{\tau}$;
[->,green,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(alt) edge[decorate] (altbottom);
[dashed,-,gray] (-3.8,2.5) – (4,2.5);
[gray] at (-4,2.5) $j$;
[ve,fill=white] (alt2) at (0,2) $v_k$;
(altbottom2) at (-2.5,0) $\out{}{\sigma''}{\tau}$;
[->,red,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(alt2) edge[decorate] (altbottom2);
A deviation from $v_k$ cannot be a best alternative to $\out{
}{\sigma}{\tau}$ if $j \ge N(\StratVal{ }{ }{\sigma'}{\tau} -
\StratVal{ }{ }{\sigma}{\tau})$.
Let us give some more intuition regarding the regret of a play. Consider a pair
of strategies $\sigma$ and $\tau$ for and , respectively. Suppose there is
an alternative strategy $\sigma'$ for , such that, against $\tau$, the
obtained payoff is greater than that of $\out{ }{\sigma}{\tau}$. It should be
clear that this implies there is some position $i$ such that, from vertex $v_i
\in \VtcE$ $\sigma'$ and $\tau$ result in a different play from $\out{
}{\sigma}{\tau}$ (see Figure <ref>).
We will sometimes refer to this deviation, the play $\out{
}{\sigma'}{\tau}$, as a better alternative to $\out{ }{\sigma}{\tau}$.
We can now show the regret of a strategy for in fact corresponds to the
supremum of the regret of plays consistent with the strategy.
For any strategy $\sigma$ of Eve,
\(
\regret{\sigma}{G}{} = \sup \{ \regret{}{\pi}{}
\st \pi\text{ is consistent with }\sigma\}.
\)
We note that for any play $\pi$, the sequence $\langle\lambda^i
(\cVal^{v_i}_{\lnot v_{i+1}}(G) - \PlayVal{\pi[i..]})\rangle_{i \ge 0}$
converges to $0$ because $(\cVal^{v_i}_{\lnot v_{i+1}}(G) - \PlayVal{\pi[i..]})$
is bounded by $\frac{2W}{(1 - \lambda)}$. It follows that if we have a non-zero
lower bound for the regret of $\pi$, then there is some index $N$ such that the
witness for the regret occurs before $N$. Moreover, we can place a
polynomial upper bound on $N$.
More precisely:
Let $\pi$ be a play in $G$ and suppose $0 < r \leq \regret{}{\pi}{}$.
\[
N(r) \defeq \left \lfloor (\log r + \log (1-\lambda) -
\log(2W))/\log \lambda \right \rfloor + 1.
\]
Then $\regret{}{\pi}{} = \regret{}{\pi[..{N(r)}]}{} -
\lambda^{N(r)} \PlayVal{\pi[{N(r)}..]}$.
The above result gives us a bound on how far we have to unfold a game after
having witnessed a non-zero lower bound, $r$, for the regret. If we consider the
example from Figure <ref>, this translates into a bound on how
many turns after $v_i$ a deviation can still yield bigger local regret (see
Figure <ref>).
Corollary <ref> then gives us the required lower bound to be
able to use Lemma <ref>.
If $\Regret{G}{} \ge a_G$ then $\Regret{G}{}$ is equal to
\[
\inf_{\sigma \in \StrAllE} \sup\{
\regret{}{\pi[..N({a_G})]}{} - \lambda^{N({a_G})}
\aVal^{v_{N({a_G})}}(G) \st \pi = v_0 v_1 \dots \text{
is consistent with } \sigma\}.
\]
This already implies we can compute the regret value in alternating polynomial
time (or equivalently, deterministic polynomial space <cit.>).
The regret value is computable using only polynomial space.
We first label the arena with the antagonistic and co-operative values
and solve the safety game described for
Theorem <ref>. The latter can be done in
polynomial time.
If the wins the safety game, the regret value is $0$. Otherwise,
we know $a_G > 0$ is a lower bound for the regret value.
We now simulate $G$ using an alternating Turing machine which halts in
at most $N(a_G)$ steps. That is, a polynomial number of steps. The
simulated play prefix is then assigned a regret value as per
Lemma <ref> (recall we have already
pre-computed the antagonistic value of every vertex).
As a side-product of the algorithm described in the above proof we get that
finite memory strategies suffice for to minimize her regret in a
discounted-sum game.
Let $\mu \defeq |\Delta|^{N(a_G)}$, with $N(0) = 0$. It holds that
\[
\Regret{G}{\StrE^{\mu},\StrAllA} = \Regret{G}{\StrAllE,\StrAllA}.
\]
Simple regret-minimizing behaviours.
We will now argue that has a simple strategy which ensures regret of
at most $\Regret{G}{}$. Her strategy will consist in “playing co-operatively”
(, a strategy that attempts to maximize the co-operative payoff) for
some turns (until a high local regret has already been witnessed) and
then switch to a co-operative worst-case optimal strategy (, a strategy
attempting to maximize the co-operative payoff while achieving at least the
antagonistic payoff).
We will now define a family of strategies which switch from co-operative
behaviour to antagonistic, after a specific number of turns have elapsed (in
fact, enough for the discounted local regret to be less than the desired
regret). Denote by $\sigma^{\mathsf{co}}$ a strongly co-operative strategy for
in $G$ and
by $\sigma^{\mathsf{cw}}$
a co-operative worst-case optimal strategy for in $G$. Recall that, by
Lemma <ref>, such strategies for her always exist.
Finally, given a co-operative strategy $\sigma^{\mathsf{co}}$, a
co-operative worst-case optimal strategy $\sigma^{\mathsf{cw}}$, and $t \in
\mathbb{Q}$ let us define an optimistic-then-pessimistic strategy for
$\switch{\sigma^{\mathsf{co}}}{t}{\sigma^{\mathsf{cw}}}$. The strategy is
such that, for any play prefix $\rho = v_0 \dots v_n$ such that $v_n \in \VtcE$
\[
\switch{\sigma^{\mathsf{co}}}{t}{\sigma^{\mathsf{cw}}}(\rho) =
\begin{cases}
\sigma^{\mathsf{co}}(\rho) & \text{if } |\copt{v_n}| = 1
\text{ and }
\locreg({\rho\cdot\sigma^{\mathsf{cw}}(\rho)},{n+1}) > t\\
\sigma^{\mathsf{cw}}(\rho) & \text{otherwise.}
\end{cases}
\]
We claim that, when we set $t = \Regret{G}{}$, an optimistic-then-pessimistic
strategy for ensures minimal regret. That is
Let $\sigma^{\mathsf{co}}$ be a strongly co-operative strategy
for , $\sigma^{\mathsf{cw}}$ be a and a co-operative worst-case
optimal strategy for , and $t = \Regret{G}{}$. The strategy $\sigma
= \switch{\sigma^{\mathsf{co}}}{t}{\sigma^{\mathsf{cw}}}$ has the
property that $\regret{\sigma}{G}{} = \Regret{G}{}$.
This is a refinement of the strategy one can obtain from applying
the algorithm used to prove
Proposition <ref>.[In fact, our proof of
Prop. <ref> relies in requiring finite memory,
to minimize her regret.] The latter tells us that a regret-minimizing
strategy of eventually switches to a worst-case optimal behaviour. For
vertices where, before this switch, another edge was chosen by , we argue
that she must have been playing a co-operative strategy. Otherwise, she could
have switched sooner. A full proof is provided in
Appendix <ref>.
We have
shown the regret value can be computed using an algorithm which requires
polynomial space only. This algorithm is based on a polynomial-length unfolding
of the game and from it we can deduce that the regret value is representable
using a polynomial number of bits. (Indeed, all exponents ocurring in the
formula from Lemma <ref> will be polynomial according
to Lemma <ref>.) Also, we have argued that has a “simple”
strategy $\sigma$ to ensure minimal regret. Such a strategy is defined by two
polynomial-time constructible sub-strategies and the regret value of the game.
Hence, it can be encoded into a polynomial number of bits itself. Furthermore,
$\sigma$ is guaranteed to be playing as its co-operative worst-case optimal
component after $N(\Regret{G}{})$ turns (see, again, Lemma <ref>),
which is a polynomial number of turns.
Given a regret threshold $r$, we claim we can
verify that $\sigma$ ensures regret at most $r$ in polynomial time. This can be
achieved by allowing to play in $G$, and against $\sigma$, with the
objective of reaching an edge with high local regret before $N(\Regret{G}{})$
turns. An possible formalization of this idea follows. Consider the product of
$G$ with a counter ranging from $1$ to $N(\Regret{G}{})$ where we
make all vertices belong to . In this game $H$, we make edges leaving vertices
previously belonging to go to a sink and define a new weight function $w'$
which assigns to these edges their negative non-discounted local regret: going
from $u$ to $v$ when $\sigma$ dictates to go to $v'$ yields $w(u,v') + \lambda
\aVal^{v'}(H \times \sigma) - w(u,v) + \lambda \cVal^v(H)$.
Lemma <ref> allows us to show that $\sigma$ ensures regret at
most $r$ in $G$ if and only if the antagonistic value of a discounted-sum game
played on $H$ with weight function $w'$ is at most $-r$.
It follows that the regret threshold problem is in , as stated in
Theorem <ref>.
We revisit the discounted-sum game from Figure <ref>. Let us
instantiate the values $M = 100$ and $\lambda = \frac{9}{10}$. According
to our previous remarks on this arena, after $i$ visits to $v$ without
choosing $(v,y)$, could achieve $(\frac{9}{10})^{2i}10$ by
going to $x$ or hope for $(\frac{9}{10})^{2i + 1}1000$ by going to $v$
again. Her best regret minimizing strategy corresponds to $\sigma^{22}$
which ensures regret of at most $9.9030 = 10 - (\frac{9}{10})^{44}10$.
It is easy to see that cannot win the safety game $\hat{G}$
constructed from this arena and that the lower bound $a_G$ one can
obtain from $\hat{G}$ is equal to $1.2466 = (\frac{9}{10})^{4}(10 -
(\frac{9}{10})^2 10)$. As expected, when plays
her optimal regret-minimizing (optimistic-then-pessimistic)
strategy any better alternative must deviate before $N(a_G) = 71$
turns. Indeed, we have already argued that the regret $9.9030$ is
witnessed by choosing the edge $(v,y)$ for any strategy of going to $v$ more than $22$ times.
§ REGRET AGAINST POSITIONAL STRATEGIES OF
In this section we consider the problem of computing the (minimal) regret when
Adam is restricted to playing positional strategies.
Deciding if the regret value is less than a given threshold (strictly or
non-strictly), playing against positional strategies of , is
in .
Playing against an , when he is restricted to playing
memoryless strategies gives the opportunity to learn some of 's
strategic choices. However, due to its decaying nature, with the discounted-sum
payoff function must find a balance between exploring too quickly, thereby
presenting lightly discounted alternatives; and learning too slowly, thereby
heavily discounting her eventual payoff.
A similar approach to the one we have adopted in Section <ref>
can be used to obtain an algorithm for this setting. For reasons of space we
defer its presentation to the appendix. The claimed lower bound follows from
Theorem <ref>.
Deciding 0-regret.
As in the previous section, we will reduce the problem of deciding if the game
has regret value $0$ to that of determining the winner of a safety game. It will
be obvious that if no regret-free strategy for exists in the original game,
then we can construct, for any strategy of hers, a positional strategy of which ensures non-zero regret. Hence, we will also obtain a lower bound on the
regret of the game in the case wins the safety game.
Let us fix some notation. For a set of edges $D \subseteq E$, we denote by $G
\restriction D$ the weighted arena $(V,\VtcE,v_I, D,w)$. Also, for a
positional strategy $\tau : (V\setminus\VtcE) \to E$ for in $G$, we denote
by $G \times \tau$ the weighted arena resulting from removing all edges not
consistent with $\tau$. Next, for an edge $(s,t) \in E$ we define
$\learned(st) \defeq \{(u,v) \in E \st \text{if } u=s \text{
then } v = t \text{ or } u \in \VtcE\}$. We extend the latter to play prefixes
$\rho = v_0 \dots v_n$ by (recursively) defining $\learned(\rho) \defeq
\learned(\rho[..n-1]) \cap \learned(v_{n-1} v_n)$. If $\pi$ is a play, then $E
\supseteq \learned(\pi[..i]) \supseteq \learned(\pi[..j])$ for all $0 \le i \le
j$. Hence, since $E$ is finite, the value $\learned(\pi) \defeq \lim_{i \ge 0}
\learned(\pi[..i])$ is well-defined. Remark that $\learned(\pi)$ does not
restrict edges leaving vertices of .
The following properties directly follow from our definitions.
Let $\pi$ be a play or play prefix consistent with a positional strategy for
. It then holds that:
* for every $v \in \VtcA$ there is some edge $(v,\cdot) \in
\learned(\pi)$,
* $\pi$ is consistent with a strategy $\tau \in \StrPosA(G)$
if and only if $\tau \in \StrPosA(G \restriction
\learned(\pi))$,
* every strategy $\tau \in \StrPosA(G \restriction
\learned(\pi))$ is also an element from
To be able to decide whether regret-free strategies for exist, we define a
new safety game. The arena we consider is $\hat{G} \defeq (\hat{V}, \hat{\VtcE},
\hat{v_I}, \hat{E})$ where $\hat{V} \defeq V \times \pow(E)$, $\hat{\VtcE}
\defeq \VtcE \times \pow(E)$, $\hat{v_I} \defeq (v_I,E)$, and $\hat{E}$ contains
the edge $\left( (u,C),(v,D) \right)$ if and only if $(u,v) \in E$ and $D = C
\cap \learned(uv)$.
Deciding if the regret value is $0$, playing against
positional strategies of , is in .
A safety game is constructed as in the proof of
Theorem <ref>. Here, we consider $\tilde{G}$ and
the set of bad edges $\tilde{\mathcal{B}} \defeq \{\left( (u,C),(v,D)
\right) \in \hat{E} \st u \in \VtcE \text{ and } \exists \tau \in
\StrPosA(G\restriction C), w(u,v) + \lambda \cVal^v(G \times
\tau) < \cVal^u_{\lnot v}(G \times \tau)\}$. We then have the safety
game $\tilde{G} = (\hat{V}, \hat{\VtcE}, \hat{v_I}, \hat{E},
\tilde{\mathcal{B}})$. Note that there is an obvious bijective mapping
from plays (and play prefixes) in $\tilde{G}$ to plays (prefixes) in $G$
which are consistent with a positional strategy for . One can then
show the following properties hold:
If $\tau \in \StrAllA(\tilde{G})$ is a winning strategy for in
$\tilde{G}$, then for all $\sigma \in \StrAllE(G)$,
there exist $t_{\tau\sigma} \in \StrPosA(G)$ and $s_{\tau\sigma}
\in \StrAllE(G)$ such that
\(
\StratVal{}{}{s_{\tau\sigma}}{t_{\tau\sigma}} -
\StratVal{}{}{\sigma}{t_{\tau\sigma}} \ge
\lambda^{|V|(|E| + 1)}
\)
\(
\min\{
\cVal^u_{\lnot v}(G \times \tau)
-w(u,v) - \lambda
\cVal^v(G \times \tau)\st
\left( (u,C),(v,D) \right) \in
\tilde{\mathcal{B}}, \tau
\in \StrPosA(G\restriction C)\}.
\)
The claim follows from positional determinacy of safety games and
Lemma <ref> (see
Appendix <ref>).
If $\sigma \in \StrAllE(\tilde{G})$ is a winning strategy for in
$\tilde{G}$, then there is $s_\sigma \in \StrAllE(G)$ such that
$\regret{s_\sigma}{G}{\StrAllE,\StrPosA} = 0$.
It then follows from the determinacy of safety games that wins the
safety game $\tilde{G}$ if and only if she has a regret-free strategy.
We provide full proofs for these claims in appendix.
We observe that simple cycles in $\tilde{G}$ have length at most
$|V|(|E|+1)$. Thus, we can simulate the safety game until we complete a
cycle and check that all traversed edges are good, all in alternating
polynomial time. Indeed, an alternating Turing machine can simulate the
cycle and then (universally) check that for all edges, for all positional
strategies of the , the inequality holds.
If no regret-free strategy for exists in $G$, then
\ge b_G$ where $b_G \defeq \lambda^{|V|(|E|+1)} \min\{
\cVal^u_{\lnot v}(G \times \tau)
-w(u,v) - \lambda
\cVal^v(G \times \tau) \st
\left( (u,C),(v,D) \right) \in \tilde{\mathcal{B}} \text{ and } \tau
\in \StrPosA(G\restriction C)\}$.
Lower bounds.
We claim that both $0$-regret and $r$-regret are -hard.
This can be shown by
adapting the
reduction from $2$-disjoint-paths
given in <cit.>
the regret threshold problem against memoryless adversaries.
For completeness, we provide the reductions here in appendix.
Let $\lambda \in (0,1)$ and $r \in \mathbb{Q}$ be fixed. Deciding if the
regret value is less than $r$ (strictly or non-strictly), playing
against positional strategies of , is
§ PLAYING AGAINST WORD STRATEGIES OF
In this section, we consider the case where is restricted to playing
word strategies. First, we show that the regret threshold problem can be
solved whenever the discounted sum automata associated to the game structure can
be made deterministic. As the determinization problem for discounted sum
automata has been solved in the literature for only sub-classes of discount
factors, and left open in the general case, we complement this result by two
other results. First, we show how to solve an $\epsilon$-gap promise
variant of the regret threshold problem, and second, we give an algorithm to
solve the $0$ regret problem. In the two cases, we obtain completeness results
on the computational complexities of the problems.
The formal definition of the $\epsilon$-gap promise problem is given below. We
first define here the necessary vocabulary. We say that a strategy of is
a word strategy if his strategy can be expressed as a function $\tau :
\mathbb{N} \to [\max\{\outdeg{v} \st v \in V\}]$, where $[n] = \{i \st 1 \le i
\le n\}$. Intuitively, we consider an order on the successors of each vertex. On every turn, the strategy $\tau$ of will tell him to move to the
$i$-th successor of the vertex according to the fixed order. We denote by
$\StrWordA$ the set of all such strategies for . A game in which plays word strategies can be reformulated as a game played on a weighted
automaton $\Gamma=(Q, q_I, A, \Delta, w)$ and strategies of —of the form
$\tau : \mathbb{N} \to A$—determine a sequence of input symbols, i.e. an omega
word, to which has to react by choosing $\Delta$-successor states starting
from $q_I$. In this setting a strategy of which minimizes regret defines a
run by resolving the non-determinism of $\Delta$ in $\Gamma$, and ensures the
difference of value given by the constructed run is minimal w.r.t. to the value
of the best run on the word spelled out by .
Deciding 0-regret.
We will now show that if the regret of an arena (or automaton) is $0$, then we
can construct a memoryless strategy for which ensures no regret is
incurred. More specifically, assuming the regret is $0$, we have the existence
of a family of strategies of which ensure decreasing regret (with limit
$0$). We use this fact to choose a small enough $\epsilon$ and the corresponding
strategy of hers from the aforementioned family to construct a memoryless
strategy for with nice properties which allow us to conclude that its
regret is $0$. Hence, it follows that an automaton has zero regret if and only
a memoryless strategy of ensures regret $0$. As we can guess such a
strategy and easily check if it is indeed regret-free (using the obvious
reduction to non-emptiness of discounted-sum automata or one-player
discounted-sum games), the problem is in .
A matching lower bound follows from a reduction from SAT which was first
described in <cit.>. We sketch it, for completeness, in the appendix.
Deciding if the regret value is $0$, playing against word
strategies of , is -complete.
Deciding r-regret: determinizable cases.
When the weighted automaton $\Gamma$ associated to the game structure can be
made deterministic, we can solve the regret threshold problem with the following
algorithm. In <cit.> we established that, against eloquent adversaries,
computing the regret reduced to computing the value of a quantitative simulation
game as defined in <cit.>. The game is obtained by taking the product of
the original automaton and a deterministic version of it. The new weight
function is the difference of the weights of both components (for each pair of
transitions). In <cit.>, it is shown how to determinize discounted-sum
automata when the discount factor is of the form $\frac{1}{n}$, for $n \in
\nat$. So, for this class of discount factor, we can state the following
Deciding if the regret value is less than a given threshold (strictly or
non-strictly), playing against word strategies of , is in for
$\lambda$ of the form $\frac{1}{n}$.
The $\epsilon$-gap promise problem.
Given a discounted-sum automaton $\calA$, $r \in \mathbb{Q}$, and $\epsilon >
0$, the $\epsilon$-gap promise problem adds to the regret threshold problem the
hypothesis that $\calA$ will either have regret $\leq r$ or $> r + \epsilon$. We
observe that an algorithm which gives:
* a YES answer implies that
\(
\Regret{\mathcal{A}}{\StrE,\StrWordA} \le r+\epsilon,
\)
* whereas a NO answer implies
\(
\Regret{\mathcal{A}}{\StrE,\StrWordA} > r.
\)
will decide the $\epsilon$-gap promise problem.
In <cit.>, it is shown that there are discounted-sum automata which
define functions
that cannot be realized with deterministic-sum automata. Nevertheless, it is also
shown in that paper that given a discounted-sum automaton it is always possible
to construct a deterministic one that is $\epsilon$-close in the following
formal sense. A discounted-sum automaton $\mathcal{A}$ is $\epsilon$-close
to another discounted sum automaton $\mathcal{B}$, if for all words $x$ the absolute
value of the difference between the values assign by $\mathcal{A}$ and
$\mathcal{B}$ to $x$ is at most $\epsilon$. So, it should be clear that we can
apply the algorithm underlying Theorem <ref> to $\Gamma$
and a determinized version $\mathcal{D}_\Gamma$ of it (which is $\epsilon$-close
to $\Gamma$) and solve the $\epsilon$-gap promise problem. We can then prove the
following result.
Deciding the $\epsilon$-gap regret problem
is in .
The complexity of the algorithm follows from the fact that the value of a
(quantitative simulation) game, played on the product of $\Gamma$ and
$\mathcal{D}_\Gamma$ we described above, can be determined by simulating the
game for a polynomial number of turns. Thus, although the automaton constructed
using the techniques of Boker and Henzinger <cit.> is of size exponential,
we can construct it “on-the-fly” for the required number of steps and then
Lower bounds.
We claim the $\epsilon$-gap promise problem is -hard even if both
$\lambda$ and $\epsilon$ are not part of the input. To establish the result, we
give a reduction from QSAT which uses the gadgets depicted in
Figures <ref> and <ref>. For space
reasons we defer the reduction to Appendix <ref>.
Let $\lambda \in (0,1)$ and $\epsilon \in (0,1)$ be fixed. As input,
assume we are given $r \in \mathbb{Q}$ and weighted arena $\calA$
such that $\Regret{\mathcal{A}}{\StrE,\StrWordA} \le r$ or
$\Regret{\mathcal{A}}{\StrE,\StrWordA} > r + \epsilon$. Deciding if the
regret value is less than a given threshold, playing against word
strategies of , is -hard.
It follows that the general problem is also -hard (even if $\epsilon$ is
set to $0$).
Let $\lambda \in (0,1)$. For $r \in \mathbb{Q}$, weighted arena $G$,
determining whether $\Regret{G}{\StrAllE,\StrWordA} \lhd r$, for $\lhd
\in \{<,\le\}$, is -hard.
§ MISSING PROOFS FROM SECTION <REF>
§.§ Proof of Lemma <ref>
Consider any $\sigma, \sigma' \in \StrAllE$ and $\tau \in \StrAllA$ such
that $\out{}{\sigma}{\tau} \neq \out{}{\sigma'}{\tau}$. Let us write
$\out{}{\sigma}{\tau} = v_0 v_1 \dots$ and $\out{}{\sigma'}{\tau} = v'_0
v'_1 \dots$ and denote by $\ell$ the length of the longest common
prefix of $\out{}{\sigma}{\tau}$ and $\out{}{\sigma'}{\tau}$. We claim
\begin{equation}\label{equ:inverse-combine-behavior1}
\lambda^{\ell} \bigl(
\cVal^{v_\ell}_{\lnot
v_{\ell + 1}}(G) -
\PlayVal{\out{}{\sigma}{\tau}} \bigr)
\ge
\lambda^{\ell} \bigl(
\PlayVal{\out{}{\sigma'}{\tau}[\ell..]}-
\PlayVal{\out{}{\sigma}{\tau}[\ell..]}
\bigr).
\end{equation}
Indeed, if we assume it is not the case, we then get that
$\cVal^{v'_{\ell+1}}(G) < \PlayVal{\out{}{\sigma'}{\tau}[\ell + 1..]}$,
which contradicts the definition of $\cVal$. Note that
Lemma <ref> actually tells us that there is
another strategy $\tau'$ for and a second alternative strategy
$\sigma''$ for which give us equality in the above equation. More
formally, from Equation <ref> and
Lemma <ref> we get that for all $\sigma \in
\StrAllE$, if there are $\tau \in \StrAllA$ and $\sigma' \in \StrAllE$
such that $\out{}{\sigma}{\tau} \neq \out{}{\sigma'}{\tau}$ then
\begin{equation}\label{equ:full-combine-behavior1}
\sup_{\tau,\sigma' \text{ s.t. }
\out{}{\sigma}{\tau} \neq \out{}{\sigma'}{\tau}}
\lambda^{\ell} \bigl(
\PlayVal{\out{}{\sigma'}{\tau}[\ell..]}-
\PlayVal{\out{}{\sigma}{\tau}[\ell..]}
\bigr) =
\lambda^{\ell} \bigl(
\cVal^{v_\ell}_{\lnot
v_{\ell + 1}}(G) -
\PlayVal{\out{}{\sigma}{\tau}} \bigr).
\end{equation}
We are now able to prove
the result. That is, for any strategy $\sigma$ for :
\begin{align*}
& \sup\{ \regret{}{\pi}{} \st \pi \text{ is
consistent with } \sigma\} & \\
= & \sup_{\tau \in \StrAllA} \regret{}{\out{}{\sigma}{\tau} =
v_0 v_1 \dots }{}
& \text{def. of } \out{}{\sigma}{\tau}\\
= & \sup_{\tau \in \StrAllA} \max\left\{0,
\sup_{\substack{i \ge 0\\
v_i \in \VtcE}} \lambda^i \left(
\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\out{}{\sigma}{\tau}[i..]} \right) \right\}
& \text{def. of } \regret{}{\out{}{\sigma}{\tau}}{}\\
= & \sup_{\tau \in \StrAllA} \max\left\{0,
\sup_{\sigma' \text{s.t.}\out{}{\sigma}{\tau} \neq
\out{}{\sigma'}{\tau}}
\lambda^\ell \left(
\PlayVal{\out{}{\sigma'}{\tau}[\ell..]} -
\PlayVal{\out{}{\sigma}{\tau}[\ell..]} \right) \right\}
& \text{by Eq.~\eqref{equ:full-combine-behavior1}}\\
= & \sup_{\tau \in \StrAllA} \max\left\{0,
\sup_{\sigma' \text{s.t.}\out{}{\sigma}{\tau} \neq
\out{}{\sigma'}{\tau}}
\left(
\StratVal{}{}{\sigma'}{\tau} -
\StratVal{}{}{\sigma}{\tau} \right) \right\}
& \text{def. of } \PlayVal{\cdot},\ell\\
= & \sup_{\tau \in \StrAllA}
\sup_{\sigma' \in \StrAllE}
\left(
\StratVal{}{}{\sigma'}{\tau} -
\StratVal{}{}{\sigma}{\tau} \right)
& 0 \text{ when } \out{}{\sigma}{\tau} = \out{}{\sigma'}{\tau}
\end{align*}
as required.
§.§ Proof of Lemma <ref>
Observe that $N(r)$ is such that $\frac{2W\lambda^{N(r)}}{1-\lambda}<r$.
Hence, we have that for all $i \ge N(r)$ such that $v_i \in \VtcE$ it
holds that $\lambda^i (\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\pi[i..]}) \leq \frac{2W\lambda^{N(r)}}{1-\lambda} < r$. It
follows that
\begin{align*}
\regret{}{\pi}{} &=
\sup\{
\lambda^i (\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\pi[i..]})\st i \ge 0 \text{ and } v_i \in \VtcE\}\\
&= \max_{\substack{0 \leq i < {N(r)}\\
v_i \in \VtcE}}
\lambda^i \left(\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\pi[i..{N(r)}]}\right) - \lambda^{N(r)}
\PlayVal{\pi[{N(r)}..]}
\end{align*}
as required.
§.§ Proof of Lemma <ref>
First, note that if $\Regret{G}{} > 0$ then there cannot be any
regret-free strategies for in $G$. It then follows from
Corollary <ref> that $\Regret{G}{} \ge a_G$. Next, using
Lemma <ref> and the definition of the regret of a play we
have that $\Regret{G}{}$ is equal to
\[
\inf_{\sigma \in \StrAllE} \sup\{ \regret{}{\pi[..N(a_G)]}{} -
\lambda^{N(a_G)} \PlayVal{\pi[N(a_G)..]} \st \pi \text{ is
consistent with } \sigma\}.
\]
Finally, note that it is in the interest of to maximize the value
$\lambda^{N(a_G)} \PlayVal{\pi[N(a_G)..]}$ in order to minimize
regret. Conversely, tries to minimize the same value. Thus, we
can replace it by the antagonistic value from $\pi[N(a_G)..]$ discounted
accordingly. More formally, we have
\begin{align*}
&\inf_{\sigma \in \StrAllE} \sup\{ \regret{}{\pi[..N(a_G)]}{} -
\lambda^{N(a_G)} \PlayVal{\pi[N(a_G)..]} \st \pi \text{ is
consistent with } \sigma\} \\
=&\inf_{\sigma \in \StrAllE}
\sup_{\tau \in \StrAllA}
\regret{}{\out{}{\sigma}{\tau}[..N(a_G)]}{} -
\lambda^{N(a_G)}
\PlayVal{\out{}{\sigma}{\tau}[N(a_G)..]}\\
=&\inf_{\substack{\sigma \in \StrAllE\\\sigma' \in \StrAllE}}
\sup_{\substack{\tau \in \StrAllA\\\tau' \in \StrAllA}}
\regret{}{\out{}{\sigma}{\tau}[..N(a_G)] = \dots v}{} -
\lambda^{N(a_G)} \StratVal{}{v}{\sigma'}{\tau'}\\
=&\inf_{\sigma \in \StrAllE}
\sup_{\tau \in \StrAllA}
\regret{}{\out{}{\sigma}{\tau}[..N(a_G)] = \dots v}{} +
\inf_{\sigma' \in \StrAllE} \sup_{\tau' \in \StrAllA}
\left(
\StratVal{}{v}{\sigma'}{\tau'}
\right)\\
=&\inf_{\sigma \in \StrAllE}
\sup_{\tau \in \StrAllA}
\regret{}{\out{}{\sigma}{\tau}[..N(a_G)] = \dots v}{} -
\lambda^{N(a_G)}
\left(
\sup_{\sigma' \in \StrAllE} \inf_{\tau' \in \StrAllA}
\StratVal{}{v}{\sigma'}{\tau'}
\right)\\
=&\inf_{\sigma \in \StrAllE}
\sup_{\tau \in \StrAllA}
\regret{}{\out{}{\sigma}{\tau}[..N(a_G)] = \dots v}{} -
\lambda^{N(a_G)}
\aVal^v(G)
\end{align*}
as required.
§.§ Proof of Claim <ref>
As a first step towards proving the result, we first make the
observation that any winning strategy of in $\hat{G}$ also ensures
a value of at least $\aVal(G)$ in the discounted-sum game played on $G$.
More formally,
If $\sigma \in \StrAllE$ is a winning strategy for in
$\hat{G}$, then
\begin{equation}\label{eqn:ensure-aval}
\forall \tau \in \StrAllA, \forall i \ge 0 :
\PlayVal{\out{ }{\sigma}{\tau}[i..] = v_i\dots}
\ge \aVal^{v_i}(G).
\end{equation}
Consider a winning strategy $\sigma \in \StrAllE$ for in
$\hat{G}$. Since safety games are positionally determined (see,
<cit.>) we can assume w.l.o.g. that $\sigma$ is
To convince the reader that $\sigma$ has the property from
Equation (<ref>), we consider the synchronized
product of $G$ and $\sigma$—that is, the synchronized product
of $G$ and the finite Moore machine realizing $\sigma$. As
$\sigma$ is memoryless, then this product, which
we denote in the sequel by $G \times \sigma$, is finite.
Now, towards a contradiction, suppose that
Equation (<ref>) does not hold for $\sigma$.
Further, let us consider an alternative (memoryless) strategy
$\sigma'$ of which ensures $\aVal^v(G)$ from all $v \in V$.
The latter exists by definition of $\aVal(G)$ and memoryless
determinacy of discounted-sum games (see, <cit.>).
Let $H$ denote a copy of $G \times \sigma$ where all edges
induced by $E$ from $G$ are added—not just the ones allowed
by $\sigma$—and $H \arestriction \sigma'$ denote the
sub-graph of $H$ where only edges allowed by $\sigma'$ are left.
Since, by assumption, $\sigma$ does not have the property of
Equation (<ref>) then the edges present in at
least one vertex from $H \arestriction \sigma'$ and $G \times
\sigma$ differ. Note that such a vertex $u$ is necessarily such
that $u \in \VtcE$. Furthermore, from our definition of a
strategy, we know that there is a single outgoing edge from it
in both structures. Let us write $(u,v)$ for the
edge in $G \times \sigma$ and $(u,v')$ for the edge in $H
\arestriction \sigma'$. Recall that $\sigma$ is winning for in $\hat{G}$. Thus, we have that $(u,v) \not\in
\mathcal{B} = \{ (u,v) \in E \st u \in \VtcE$ and $w(u,v) +
\lambda \aVal^v(G) < \cVal^{u}_{\lnot v}(G)\}$. It follows
\begin{align*}
w(u,v) + \lambda \aVal^v(H) & \ge
\max_{x \neq v}\{ w(u,x) + \lambda
\cVal^{x}(H) \} & \\
&\ge \max_{x \neq v}\{ w(u,x) + \lambda
\aVal^{x}(H) \} &
\text{as }
\cVal^{x}(H) \ge \aVal^{x}(H)\\
&=\aVal^u(H) & \text{because } u \in
\end{align*}
Thus, the strategy $\sigma''$ of which takes $(u,v)$
instead of $(u,v')$ and follows $\sigma'$ otherwise—indeed,
this might mean $\sigma''$ is not memoryless—also achieves at
least $\aVal^u(H)$ from $u$ onwards and is therefore
an worst-case optimal antagonistic strategy in $G$ ( it has
the property of Equation (<ref>)). Notice that
this process can be repeated for all vertices in which the two
structures differ. Further, since both are finite, it will
eventually terminate and yield a strategy of which plays
exactly as $\sigma$ and for which Equation (<ref>)
holds, which is absurd.
Once more, consider a winning strategy $\sigma \in \StrAllE$ for in
$\hat{G}$. We will now show that
\[
\forall \tau \in \StrAllA, \forall \sigma' \in \StrAllE
\setminus \{\sigma\}: \StratVal{}{}{\sigma}{\tau}
\ge \StratVal{}{}{\sigma'}{\tau}.
\]
The desired result will then directly follow.
Consider arbitrary strategies $\tau \in \StrAllA$ and $\sigma' \in
\StrAllE \setminus \{\sigma\}$. Suppose that $\out{ }{\sigma}{\tau} \neq
\out{ }{\sigma'}{\tau}$, as our claim trivially holds otherwise. Let
$\iota$ be the maximal index $i \ge 0$ such that, if we write $\out{
}{\sigma}{\tau} = v_0 v_1 \dots$ and $\out{ }{\sigma'}{\tau} = v'_0 v'_1
\dots$, then $v_i = v'_i$. That is, $\iota$ is the maximal index for
which the outcomes of $\sigma$ and $\tau$, and $\sigma'$ and $\tau$
coincide. Note that $v_\iota$ is necessarily an vertex,
$v_\iota \in \VtcE$. We observe that, by definition of $\cVal$,
it holds that
\begin{equation}\label{equ:cval}
\PlayVal{\out{ }{\sigma'}{\tau}[\iota + 1..]} \le
\cVal^{v'_{\iota + 1}}(G).
\end{equation}
Furthermore, we know from the fact that $\sigma$ is winning for in
$\hat{G}$ that the edge $(v_\iota,v_{\iota + 1})$ is such that
\begin{equation}\label{equ:winning}
w(v_\iota,v_{\iota + 1}) + \lambda \aVal^{v_{\iota +
1}}(G) \ge \max_{t \neq v_{\iota + 1}}\{ w(v_\iota,t) +
\lambda \cVal^{t}(G) \}.
\end{equation}
In particular, this implies that $w(v_\iota,v_{\iota + 1}) + \lambda
\aVal^{v_{\iota + 1}}(G) \ge w(v_\iota,v'_{\iota+1}) + \lambda
\cVal^{v'_{\iota + 1}}(G)$. It is then easy to verify that
$w(v_\iota,v_{\iota + 1}) + \lambda \aVal^{v_{\iota + 1}}(G) =
\aVal^{v_\iota}(G)$ using the observation that $v_\iota \in V_\exists$.
From Claim <ref> we also get that
\begin{equation}\label{equ:from-claim}
\PlayVal{\out{ }{\sigma}{\tau}[\iota..]} \ge \aVal^{v_\iota}(G).
\end{equation}
Putting all the above inequalities together, we have
\begin{align*}
\PlayVal{\out{ }{\sigma}{\tau}[\iota..]} & \ge
\aVal^{v_\iota}(G) = w(v_\iota,v_{\iota + 1}) + \lambda
\aVal^{v_{\iota + 1}}(G)
& \text{by Eqn.~\eqref{equ:from-claim}} \\
& \ge w(v_\iota,v'_{\iota+1}) + \lambda \cVal^{v'_{\iota +
1}}(G) & \text{by Eqn.~\eqref{equ:winning}} \\
& \ge \PlayVal{\out{}{\sigma'}{\tau}[\iota..]} & \text{by
\end{align*}
which, in turn, implies $\StratVal{}{}{\sigma}{\tau} \ge
\StratVal{}{}{\sigma'}{\tau}$ since $\out{}{\sigma}{\tau}[..\iota] =
\out{}{\sigma'}{\tau}[..\iota]$.
§.§ Proof of Proposition <ref>
Let us start by showing that the regret of a play $\pi$ is
bounded (from above) by the discounted local regret from any index $i$, where
from the $i$-th turn onwards plays a worst-case optimal strategy. More
Let $\pi = v_0 v_1 \dots$ be a play. Assume there is some $i \in
\mathbb{N}$ such that
* $v_i \in \VtcE$;
* $\regret{}{\pi}{} \le \lambda^i \regret{}{\pi[i..]}{}$; and
* $\aVal^{v_j}(G) = w(v_j,v_{j+1}) +
\lambda \aVal^{v_{j+1}}(G)$, for all $j \ge i$.
It then holds that
$\regret{}{\pi}{} \le \lambda^i \left( \cVal^{v_i}(G) -
\aVal^{v_i}(G) \right)$.
If $\regret{}{\pi}{} = 0$ then the claim holds trivially. Hence, let us
assume $\regret{}{\pi}{} > 0$. It follows from
Lemma <ref> and Assumption $(ii)$ that there
exists $k \ge i$ such that $v_k \in \VtcE$ and
\[
\regret{}{\pi}{} = \lambda^k \left(\cVal^{v_k}_{\lnot
v_{k+1}}(G) - w(v_k,v_{k+1}) - \lambda
\aVal^{v_{k+1}}(G)\right).
\]
Observe that $\cVal^{v_k}(G) \ge \cVal^{v_k}_{\lnot v_{k+1}}(G)$, by
definition, and that from Assumption $(iii)$ we have that
$\aVal^{v_{k}}(G) \le w(v_k,v_{k+1}) + \lambda
\aVal^{v_{k+1}}(G)$. Thus, we get that
$\regret{}{\pi}{} \le \lambda^k \left( \cVal^{v_k}(G) -
\aVal^{v_k}(G) \right)$. Also, note that by definition of $\cVal$ we
have that
\[
\cVal^{v_j}(G) \ge w(v_j,v_{j+1}) + \lambda \cVal^{v_{j+1}}(G)
\]
for all $j \ge 0$. It thus follows from Assumption $(iii)$ and the
previous arguments that $\regret{}{\pi}{} \le \lambda^i \left(
\cVal^{v_i}(G) - \aVal^{v_i}(G) \right)$ as required.
We are now ready to prove the Proposition holds.
The zero case.
If $\Regret{G}{} = 0$, then it follows from
our reduction to safety games that has a co-operative worst-case
optimal strategy which minimizes regret. Indeed, it is straightforward
to show that the strategy for obtained from the safety game does
not only ensure at least the antagonistic value, but it is also
co-operative worst-case optimal. Thus, since
$\switch{\sigma^{\mathsf{co}}}{0}{\sigma^{\mathsf{cw}}}$ is
clearly equivalent to $\sigma^{\mathsf{cw}}$ in this case, the result
Non-zero regret.
Let us assume that $\Regret{G}{} > 0$. It
then follows from Lemma <ref> that has a
finite memory strategy $\sigma$ which ensures regret of at most
$\Regret{G}{}$ (see Corollary <ref>) and which,
furthermore, can be assumed to switch after turn $N(a_G)$ to a
co-operative worst-case optimal strategy $\sigma^{\mathsf{cw}}$ for (since such a strategy ensures at least the antagonistic value of the
vertex from which starts playing it). We will further assume,
w.l.o.g., that
for all play prefixes $\pi = v_0 \dots v_n$ with $n \le N(a_G)$, $v_n
\in \VtcE$ and having $\sigma^{\mathsf{cw}}(\pi) \neq
\sigma^{\mathsf{co}}(\pi) = \sigma(\pi)$, if $\sigma$ switches to
$\sigma^{\mathsf{cw}}$ from $\pi$ onwards—that is, for all prefixes
extending $\pi$—then the regret of the resulting strategy is strictly
greater than $\Regret{G}{}$. Otherwise, one can consider the strategy
resulting from the previously described switch instead of $\sigma$.
We will now argue that for all play prefixes $\pi = v_0 \dots v_n$ with
$n \le N(a_G)$ and $v_n \in \VtcE$, if $\sigma(\pi) \neq
\sigma^{\mathsf{cw}}$ then $\copt{v_n}$ is a singleton and
$\locreg{\pi[..n]\cdot \sigma^{\mathsf{cw}}(\pi[..n])}{n+1} >
\Regret{G}{}$. The desired result will follow since in order for our
assumption of $\regret{}{\sigma}{} = \Regret{G}{}$ to be true must
then choose the unique edge leading to the single element in
Let us consider two cases.
First, if $\locreg{\pi[..n]\cdot
\sigma^{\mathsf{cw}}(\pi[..n])}{n+1} \le \Regret{G}{}$, we can switch to
$\sigma^{\mathsf{cw}}$ fron $\pi[..n]$ onwards. Contradicting our
initial assumption.
Second, if $|\copt{v_n}| > 1$ and $\locreg{\pi[..n]\cdot
\sigma^{\mathsf{cw}}(\pi[..n])}{n+1} > \Regret{G}{}$, then by
Lemma <ref> we get that the regret of
the play (if we switched to $\sigma^{\mathsf{cw}}$) is bounded
above by $\lambda^n \left(\cVal^{v_n}(G) -
\aVal^{v_n}(G)\right)$. Also, since $\copt{v_n}$ is not a
singleton, if does not switch, then she cannot ensure a
local regret of less than $\lambda^n \left(\cVal^{v_n}(G) -
\aVal^{v_n}(G)\right)$—particularly, not even by taking an
edge leading to a vertex in $\copt{v_n}$. This contradicts the
assumption that that switching to $\sigma^{\mathsf{cw}}$ yields
strictly more regret.
§.§ Lower bound
We now establish a lower bound for computing the
minimal regret against any strategy by reducing from the problem of determining
the antagonistic value of a discounted-sum game. More precisely, from a
weighted arena $G$ we construct, in logarithmic space, a weighted arena $G'$
such that the antagonistic value of $G$ is equal to the regret value of $G'$.
This gives us:
Computing the regret of a discounted-sum game is at least as hard as
computing the antagonistic value of a (polynomial-size) game with the
same payoff function.
[inner sep=2mm, ve/.style=rectangle,
draw,va/.style=circle, draw, node distance=1cm]
[ve,initial above](A)$v_I'$;
[ve,dotted,left=of A](B)$v_I$;
[va,right=of A](C);
[va,right=of C, yshift=.5cm](D);
[va,right=of C, yshift=-.5cm](E);
(A) edge node[el]$0$ (B)
(A) edge node[el]$0$ (C)
(C) edge node[el]$K+1$(D)
(C) edge node[el,swap]$-3K-2$(E)
(D) edge[loopright, looseness=6, in=135, out=45] node[el,swap]$0$ (D)
(E) edge[loopright, looseness=6, in=-135, out=-45] node[el]$0$ (E);
Gadget to reduce a game to its regret game.
Suppose $G$ is a weighted arena with initial vertex $v_I$. Consider the
weighted arena $G'$ obtained by adding to $G$ the gadget of
Figure <ref> with $K \defeq \frac{W}{1 - \lambda}$. The
initial vertex of $G'$ is set to be $v'_I$. We will show that
\(
\aVal(G) = K+1-{\Regret{G'}{}}/{\lambda}.
\)
At $v_I'$ has a choice: she can choose to remain in the gadget or
she can move to the original game $G$. If remains in the gadget
her payoff will be $\lambda (-3K-2)$ while could choose to enter
the game and achieve a payoff of $\lambda \cdot \cVal(G)$. In this case
her regret is $\lambda (\cVal(G)+3K+2) \geq \lambda (2K+2)$. Otherwise,
if she chooses to play into $G$ she can achieve at most $\lambda \cdot
\aVal(G)$. The strategy of which maximizes regret against this
choice of is the one which remains in the gadget. The payoff for
is $\lambda(K+1)$ in this case. Hence, the regret of the game in
this scenario is $\lambda(K+1 - \aVal(G)) \leq \lambda(2K + 1)$.
Clearly she will choose to enter the game and $\Regret{G'}{} =
\lambda(K+1-\aVal(G))$.
§ MISSING PROOFS FROM SECTION <REF>
§.§ Proof of Claim <ref>
We will now argue that if $\tau \in \StrAllA(\tilde{G})$ is a winning strategy
for in $\tilde{G}$, then for all $\sigma \in \StrAllE(G)$, there exist
$t_{\tau\sigma} \in \StrPosA(G)$ and $s_{\tau\sigma} \in \StrAllE(G)$ such
\(
\StratVal{}{}{s_{\tau\sigma}}{t_{\tau \sigma}} -
\StratVal{}{}{\sigma}{t_{\tau \sigma}}
\) is at least
\begin{equation}\label{eqn:min-unsafe}
\lambda^{|V|(|E| + 1)}
\min_{\substack{
\left( (u,C),(v,D) \right) \in
\tilde{\mathcal{B}}\\
\tau
\in \StrPosA(G\restriction C)
\cVal^u_{\lnot v}(G \times \tau)
-w(u,v) - \lambda
\cVal^v(G \times \tau)
\}.
\end{equation}
The argument is straightforward and based on the bijection between plays from
$G$, which are consistent with positional strategies of , and plays in
$\tilde{G}$. Recall that safety games are positionally determined. That is,
either has a positional strategy which allows her to perpetually avoid the
unsafe edges against any strategy for , or has a positional strategy
which ensures that—regardless of the behaviour of —the play eventually
traverses some unsafe edge. Thus, since we assume $\tau \in \StrAllA(\tilde{G})$
is winning for in $\tilde{G}$ we can assume that $\tau$ is in fact a
positional strategy for in $\tilde{G}$. Now consider an arbitrary strategy
$\sigma$ for in $G$. We note, once more, that $\tau$ is a strategy for
in $G$, not only in $\tilde{G}$. Furthermore, $\tau$ is a positional
strategy for in $G$. Conversely, $\sigma$ is a valid strategy for in
$\tilde{G}$. These facts follow from the definition of $\learned(\cdot)$ and
construction $\tilde{G}$. Since $\tau$ is winning for in $\tilde{G}$, the
play $\tilde{\out{ }{\sigma}{\tau}}$ traverses an unsafe edge. In fact, since
$\tau$ is positional, the unsafe edge is necessarily traversed in at most
$|V|(|E| + 1)$ steps—that is, at most the length of the longest simple path in
$\tilde{G}$. Let us write $(\tilde{v}_i, \tilde{v}_{i+1}) = \left(
(v_i,C_i),(v_{i+1},C_{i+1}) \right)$ for the traversed unsafe edge at step $i
\le |V|(|E| + 1)$. By definition of $\tilde{\mathcal{B}}$ we have that there
exists $t_{\tau \sigma} \in \StrPosA(G\restriction C_i)$ such that
\[
\cVal^{v_i}_{\lnot v_{i+1}}(G \times t_{\tau \sigma})
-w(v_i,v_{i+1}) - \lambda
\cVal^{v_i}(G \times t_{\tau \sigma}).
\]
We now move from the game $\tilde{G}$ back to the original game $G$. Henceforth,
we consider the play $\out{ }{\sigma}{\tau} = v_0 v_1 \dots$ in $G$ which corresponds to
$\tilde{\out{ }{\sigma}{\tau}} = (v_0,C_0) (v_1,C_1) \dots$ in $\tilde{G}$.
It is easy to see that $\out{ }{\sigma}{\tau}[..i]$ is consistent with
$t_{\tau \sigma}$. Hence, $\out{ }{\sigma}{t_{\tau \sigma}}$ traverses edge
$(v_i, v_{i+1})$ corresponding to bad edge $(\tilde{v}_i, \tilde{v}_{i+1})$ in
$\tilde{G}$. Finally, by determinacy of discounted-sum games and by virtue of $G
\times t_{\tau \sigma}$ being a finite weighted arena, we have that there is a
strategy $s_{\tau \sigma} \in \StrAllE(G \times t_{\tau \sigma})$ such that
$\StratVal{G}{v_i}{s_{\tau \sigma}}{t_{\tau \sigma}} = \cVal^{v_i}(G \times t_{\tau
\sigma})$. It then follows from the definition of $\cVal$ and $G \times s_{\tau
\sigma}$ that
\(
\StratVal{G}{v_I}{s_{\tau\sigma}}{t_{\tau \sigma}} -
\StratVal{G}{v_I}{\sigma}{t_{\tau \sigma}}
\) is at least the value from Equation (<ref>),
just as required.
§.§ Proof of Claim <ref>
Let us show that if $\sigma \in \StrAllE(\tilde{G})$ is a winning strategy for
in $\tilde{G}$, then there is $s_\sigma \in \StrAllE(G)$ such that
$\regret{s_\sigma}{G}{\StrAllE,\StrPosA} = 0$. The intuition behind the argument
is the same as for the proof of Claim <ref>. However, in
this case we first need to describe how to construct the strategy for in
$G$ from a strategy for her in $\tilde{G}$.
*A regret-free strategy from $\tilde{G}$.
Observe that, by construction of $\tilde{G}$, for any vertex $(u,C) \in
\hat{\VtcE}$ and any edge $(u,v) \in E$ there is exactly one corresponding edge
in $\tilde{G}$: $\left( (u,C), (v,C) \right)$. Given a vertex $(u,C)$ from
$\tilde{G}$, denote by $\proj{(u,C)}{1}$ the vertex $u$. Now, given a strategy
$\sigma \in \StrAllE(\tilde{G})$ we define $s_\sigma \in \StrAllE(G)$ as follows
\[
s_\sigma(v_0 v_1 v_2 \dots) = \proj{\sigma( (v_0,C_0) (v_1, C_1 = C_0 \cap
\learned(v_0 v_1)) (v_2, C_1 \cap \learned(v_1 v_2)) \dots)}{1}
\]
where $C_0 = E$. It follows from the fact that we have a bijective mapping from
plays in $\tilde{G}$ to plays in $G$ which are consistent with positional
strategies for , that $s_\sigma$ is a valid strategy for in $G$ when
playing against a positional adversary. Additionally, it is easy to see that
$s_\sigma$ can be realized using finite memory only. The memory required
corresponds to the subsets of $E$. The current memory element is determined by
the applying the operator $\learned(\cdot)$ to the current play prefix.
Now that we have our strategy $s_\sigma$ for in $G$, we proceed by proving
the analogue of Claim <ref> in this setting.
If $\sigma \in \StrAllE(\tilde{G})$ is a winning strategy for in
$\tilde{G}$, then
\begin{equation}\label{eqn:ensure-aval2}
\forall \tau \in \StrPosA(G), \forall i \ge 0 :
\PlayVal{\out{ }{s_\sigma}{\tau}[i..] = v_i\dots}
\ge \cVal^{v_i}(G \times \tau).
\end{equation}
To convince the reader that $s_\sigma$ has the property from
Equation (<ref>), we consider the synchronized
product of $G$ and $s_\sigma$—that is, the synchronized product
of $G$ and the finite Moore machine realizing $s_\sigma$. As
$s_\sigma$ is a finite memory strategy, then this product, which
we denote in the sequel by $G \times s_\sigma$, is finite.
Now, towards a contradiction, suppose that
Equation (<ref>) does not hold for $s_\sigma$. That is,
there is some $\tau \in \StrPosA(G)$ for which the property fails.
Further, let us consider an alternative (memoryless) strategy
$\sigma'$ of which ensures $\cVal^v(G \times \tau)$ from all $v \in V$.
The latter exists by definition of $\cVal(G \times \tau)$ and memoryless
determinacy of discounted-sum games (see, <cit.>).
Let $H$ denote a copy of $G \times s_\sigma$ where all edges
induced by $E$ from $G$ are added—not just the ones allowed
by $s_\sigma$—and $H \arestriction \sigma'$ denote the
sub-graph of $H$ where only edges allowed by $\sigma'$ are left.
Intuitively, both $G \times s_\sigma$ and $H \arestriction \sigma'$
are sub-structures of $\tilde{G}$ with a weight function $\tilde{w}$
lifted from $w$ to the blown-up vertex set $\tilde{V}$. This is due to
the way in which we constructed $s_\sigma$.
Since, by assumption, $s_\sigma$ does not have the property of
Equation (<ref>) then the edges present in at
least one vertex from $H \arestriction \sigma'$ and $G \times
\sigma$ differ. Note that such a vertex $(u,C)$ is necessarily such
that $u \in \VtcE$—and $C$ is a “memory element” from the machine
realizing $s_\sigma$ corresponding to a subset of $E$ obtained via
$\learned(\cdot)$. Furthermore, from our definition of a
strategy, we know that there is a single outgoing edge from it
in both structures. Let us write $(u,v)$—instead of $\left(
(u,C),(v,D) \right)$—for the
edge in $G \times s_\sigma$ and $(u,v')$ for the edge in $H
\arestriction \sigma'$. Recall that $s_\sigma$ is winning for in $\tilde{G}$. Thus, we have that $(u,v) \not\in
\tilde{\mathcal{B}} = \{\left( (u,C),(v,D)
\right) \in \hat{E} \st u \in \VtcE \text{ and } \exists \tau' \in
\StrPosA(G\restriction C), w(u,v) + \lambda \cVal^v(G \times
\tau') < \cVal^u_{\lnot v}(G \times \tau')\}$. It follows
\[
w(u,v) + \lambda \cVal^v(H \times \tau) \ge \cVal^{v'}(H \times
\tau).
\]
Thus, the strategy $\sigma''$ of which takes $(u,v)$
instead of $(u,v')$ and follows $\sigma'$ otherwise—indeed,
this might mean $\sigma''$ is no longer memoryless—also achieves at
least $\cVal^u(H \times \tau)$ from $u$ onwards. Notice that
this process can be repeated for all vertices in which the two
structures differ. Further, since both are finite, it will
eventually terminate and yield a strategy of which plays
exactly as $s_\sigma$ and for which, since $\tau$ was chosen
arbitrarily, Equation (<ref>) holds. Contradiction.
It follows immediately that $\regret{s_\sigma}{G}{\StrAllE,\StrPosA} = 0$.
Indeed, if we suppose that this is not the case, then there exists a strategy
$\sigma' \in \StrAllE(G)$ such that
\[
\exists \tau \in \StrPosA(G) : \StratVal{}{}{s_\sigma}{\tau} <
\StratVal{}{}{\sigma'}{\tau}.
\]
The above directly contradicts Claim <ref>.
§.§ Proof of Theorem <ref>
In this section we present sufficient modifications to our definitions from
Section <ref> in order for the techniques used therein to be
adapted for this case. Particularly, our notion of regret of a play and the
safety game used to decide the existence of regret-free strategies need to take
into account the fact that witnessing edges taken by affects previously
observed local regrets. That is, we formalize the intuition that alternative
plays must also be consistent with the behaviour of that we have witnessed
in the current play.
We are now ready to define the regret of a play in a game against a positional
adversary. Given a play $\pi = v_0 v_1 \dots$, we let
\[
\regret{}{\pi}{} \defeq \sup\{\lambda^i(\cVal^{v_i}_{\lnot v_{i+1}}(G
\restriction \learned(\pi)) - \PlayVal{\pi[i..]} \st v_i \in \VtcE \}
\cup \{0\}.
\]
Consider now a play prefix $\rho = v_0 \dots v_j$. We let the regret of $\rho$ be
\[
\max\{\lambda^i(\cVal^{v_i}_{\lnot v_{i+1}}(G
\restriction \learned(\rho[i..j])) - \PlayVal{\rho[i..j]} \st 0 \le i < j \text{
and } v_i \in \VtcE \}
\cup \{0\}.
\]
We will now re-prove Lemma <ref> in the current
For any strategy $\sigma$ of Eve,
\[
\regret{\sigma}{G}{\StrAllE,\StrPosA} = \sup \{ \regret{}{\pi}{}
\st \pi\text{ is consistent with } \sigma \text{ and some } \tau
\in \StrPosA\}.
\]
Consider any $\sigma, \sigma' \in \StrAllE$ and $\tau \in \StrPosA$ such
that $\out{}{\sigma}{\tau} \neq \out{}{\sigma'}{\tau}$. Let us write
$\out{}{\sigma}{\tau} = v_0 v_1 \dots$ and $\out{}{\sigma'}{\tau} = v'_0
v'_1 \dots$ and denote by $\ell$ the length of the longest common
prefix of $\out{}{\sigma}{\tau}$ and $\out{}{\sigma'}{\tau}$. We claim
\begin{equation}\label{equ:inverse-combine-behavior2}
\lambda^{\ell} \bigl(
\cVal^{v_\ell}_{\lnot
v_{\ell + 1}}(G \restriction
\learned(\out{}{\sigma}{\tau})) -
\PlayVal{\out{}{\sigma}{\tau}}[\ell..] \bigr)
\ge
\lambda^{\ell} \bigl(
\PlayVal{\out{}{\sigma'}{\tau}[\ell..]}-
\PlayVal{\out{}{\sigma}{\tau}[\ell..]}
\bigr).
\end{equation}
Indeed, if we assume it is not the case, we then get that
\[
\cVal^{v'_{\ell+1}}(G \restriction
\learned(\out{}{\sigma}{\tau})) <
\PlayVal{\out{}{\sigma'}{\tau}[\ell + 1..]}.
\]
However, recall that $G \times \tau$ is a sub-arena of
$G \restriction \learned(\out{}{\sigma}{\tau})$. Thus, the co-operative
value can obtain in the former, say by playing $\sigma'$, must be
at most that which she can obtain in the latter. Contradiction.
Note that there is another positional strategy $\tau'$ for and a
second alternative strategy $\sigma''$ for which do give us
equality for Equation (<ref>). For this
purpose, we choose $\tau'$ so that $\tau' \in \StrPosA(G \restriction
\learned(\out{}{\sigma}{\tau}))$—so that $\out{ }{\sigma}{\tau}$ is
also consistent with $\tau'$, thus $\learned(\out{ }{\sigma}{\tau}) =
\learned(\out{ }{\sigma}{\tau'})$ (see
Lemma <ref>)—and also such that
\[
\cVal^{v'_{\ell+1}}(G \times \tau') = \cVal^{v'_{\ell+1}}(G
\restriction \learned(\out{ }{\sigma}{\tau})).
\]
We choose $\sigma''$ so that it follows $\sigma$ for $\ell$ turns, goes
to $v'$, and then plays co-operatively with $\tau'$ from $v'$. More
formally, let $\sigma''$ be a strategy for such that
$\out{}{\sigma}{\tau}[..\ell] = \out{}{\sigma''}{\tau}[..\ell]$ and
therefore, by choice of $\tau'$, such that
$\out{}{\sigma}{\tau'}[..\ell] = \out{}{\sigma''}{\tau'}[..\ell]$
and so that
\[
\PlayVal{\out{ }{\sigma''}{\tau'}[\ell..]} =
\cVal^{v'_{\ell+1}}(G \times \tau').
\]
It follows from Equation (<ref>) and the
above arguments that for all $\sigma \in \StrAllE$, if there are $\tau
\in \StrPosA$ and $\sigma' \in \StrAllE$ such that $\out{}{\sigma}{\tau}
\neq \out{}{\sigma'}{\tau}$ then
\begin{equation}\label{equ:full-combine-behavior2}
\sup_{\tau,\sigma' \st
\out{}{\sigma}{\tau} \neq \out{}{\sigma'}{\tau}}
\lambda^{\ell} \bigl(
\PlayVal{\out{}{\sigma'}{\tau}[\ell..]}-
\PlayVal{\out{}{\sigma}{\tau}[\ell..]}
\bigr) =
\lambda^{\ell} \bigl(
\cVal^{v_\ell}_{\lnot
v_{\ell + 1}}(G \restriction \learned(\out{
}{\sigma}{\tau})) -
\PlayVal{\out{}{\sigma}{\tau}} \bigr).
\end{equation}
We are now able to prove
the result. That is, for any strategy $\sigma$ for :
\begin{align*}
& \sup\{ \regret{}{\pi}{} \st \pi \text{ is
consistent with } \sigma \text{ and some } \tau \in \StrPosA\} & \\
= & \sup_{\tau \in \StrPosA} \regret{}{\out{}{\sigma}{\tau} =
v_0 v_1 \dots }{}
& \text{def. of } \out{}{\sigma}{\tau}\\
= & \sup_{\tau \in \StrPosA} \max\left\{0,
\sup_{\substack{i \ge 0\\
v_i \in \VtcE}} \lambda^i \left(
\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction \learned(\out{
}{\sigma}{\tau})) -
\PlayVal{\out{}{\sigma}{\tau}[i..]} \right) \right\}
& \text{def. of } \regret{}{\out{}{\sigma}{\tau}}{}\\
= & \sup_{\tau \in \StrPosA} \max\left\{0,
\sup_{\sigma' \st \out{}{\sigma}{\tau} \neq
\out{}{\sigma'}{\tau}}
\lambda^\ell \left(
\PlayVal{\out{}{\sigma'}{\tau}[\ell..]} -
\PlayVal{\out{}{\sigma}{\tau}[\ell..]} \right) \right\}
& \text{by Eq.~\eqref{equ:full-combine-behavior2}}\\
= & \sup_{\tau \in \StrPosA} \max\left\{0,
\sup_{\sigma' \st \out{}{\sigma}{\tau} \neq
\out{}{\sigma'}{\tau}}
\left(
\StratVal{}{}{\sigma'}{\tau} -
\StratVal{}{}{\sigma}{\tau} \right) \right\}
& \text{def. of } \PlayVal{\cdot},\ell\\
= & \sup_{\tau \in \StrPosA}
\sup_{\sigma' \in \StrAllE}
\left(
\StratVal{}{}{\sigma'}{\tau} -
\StratVal{}{}{\sigma}{\tau} \right)
& 0 \text{ when } \out{}{\sigma}{\tau} = \out{}{\sigma'}{\tau}
\end{align*}
as required.
We will now state and prove a restricted version of
Lemma <ref>. Intuitively,
for a play $\pi$, we will not be able to consider a deviation with respect to a
prefix of $\pi$. Rather, we are forced to take the co-operative value with
respect to the set $\learned(\pi)$—that is, the edges consistent with any
positional strategy might be playing—even after the bound on where the
best deviation occurs.
Let $\pi$ be a play in $G$ and suppose $0 < r \leq \regret{}{\pi}{}$.
\[
N(r) \defeq \left \lfloor (\log r + \log (1-\lambda) -
\log(2W))/\log \lambda \right \rfloor + 1.
\]
Then $\regret{}{\pi}{}$ is equal to
\[
\max_{\substack{0 \le i < N(r)\\ v_i \in
\VtcE}}\{\lambda^i(\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction
\learned(\pi)) - \PlayVal{\pi[i..N(r)]}\} - \lambda^{N(r)}
\PlayVal{\pi[{N(r)}..]}.
\]
Observe that $N(r)$ is such that $\frac{2W\lambda^{N(r)}}{1-\lambda}<r$.
Hence, we have that for all $i \ge N(r)$ such that $v_i \in \VtcE$ it
holds that $\lambda^i (\cVal^{v_i}_{\lnot v_{i+1}}(G) -
\PlayVal{\pi[i..]}) \leq \frac{2W\lambda^{N(r)}}{1-\lambda} < r$.
Clearly, since $\cVal^{v_i}_{\lnot v_{i+1}}(H) \le \cVal^{v_i}_{\lnot
v_{i+1}}(G)$ holds for any sub-arena $H$ of $G$, we have that
\[
\lambda^i (\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction
\learned(\pi)) - \PlayVal{\pi[i..]})
\leq \frac{2W\lambda^{N(r)}}{1-\lambda} < r.
\]
It thus follows that
\begin{align*}
\regret{}{\pi}{} &=
\sup\{
\lambda^i (\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction \learned(\pi)) -
\PlayVal{\pi[i..]})\st i \ge 0 \text{ and } v_i \in \VtcE\}\\
&= \max_{\substack{0 \leq i < {N(r)}\\
v_i \in \VtcE}}
\lambda^i \left(\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction
\learned(\pi)) -
\PlayVal{\pi[i..{N(r)}]}\right) - \lambda^{N(r)}
\PlayVal{\pi[{N(r)}..]}
\end{align*}
as required.
(root) at (0,5) $v_I$;
(leftcorner) at (-4,0) ;
(rightcorner) at (4,0) ;
(root) edge (leftcorner)
(root) edge (rightcorner);
[va] (adam-choice) at (0,1) $v_j$;
(bottom2) at (-1.5,0) $\pi$;
(bottom1) at (1.5,0) $\pi'$;
[->,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(root) edge[decorate] (adam-choice)
(adam-choice) edge[decorate,green] (bottom2)
(adam-choice) edge[decorate,red] (bottom1)
[ve,fill=white] (alt1) at (0,3) $v_{i'}$;
[label=right:$\cVal^{v_{i'}}_{\lnot v_{i'+1}}(G \restriction
\learned(\pi))$] (alttarget1) at (2,2.6) ;
(alt1) edge[green] (alttarget1)
[ve,fill=white] (alt2) at (0,4) $v_{i}$;
[label=left:$\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction
\learned(\pi'))$] (alttarget2) at (-1.3,3.5) ;
(alt2) edge[red] (alttarget2)
[dashed,-,gray] (-3.4,2.3) – (4,2.3);
[gray] at (-4,2.3) $N(b_G)$;
Let $\rho$ denote the play prefix $v_0 \dots v_j$. The alternative play from
$v_{i'}$ is better than the one from
$v_i$ w.r.t $\rho$. However, for play
$\pi'$ extending $\rho$, the alternative play from $v_i$ becomes better
than the one from $v_{i'}$
if $\lambda^{i' -i}\cVal^{v_{i'}}_{\lnot v_{i' + 1}}(G \restriction
\learned(\pi'))$ is smaller than $\cVal^{v_i}_{\lnot v_i + 1}(G
\restriction \learned(\pi')) - \PlayVal{\rho[i..i']}$.
(root) at (0,5) $v_I$;
(leftcorner) at (-4,0) ;
(rightcorner) at (4,0) ;
(root) edge (leftcorner)
(root) edge (rightcorner);
[va] (adam-choice) at (0,1) $v_j$;
(bottom2) at (-1.5,0) $\pi$;
(bottom1) at (1.5,0) $\pi'$;
[->,decoration=zigzag,segment length=4,amplitude=.9,
post=lineto,post length=2pt]
(root) edge[decorate] (adam-choice)
(adam-choice) edge[decorate,green] (bottom2)
(adam-choice) edge[decorate,red] (bottom1)
[ve,fill=white] (alt1) at (0,3) $v_{i'}$;
[label=right:$\cVal^{v_{i'}}_{\lnot v_{i'+1}}(G \restriction
\learned(\rho))$] (alttarget1) at (2,2.6) ;
(alt1) edge[green] (alttarget1)
[ve,fill=white] (alt2) at (0,4) $v_{i}$;
[label=left:$\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction
\learned(\pi'))$] (alttarget2) at (-1.3,3.5) ;
(alt2) edge[red] (alttarget2)
[dashed,-,gray] (-3.4,2.3) – (4,2.3);
[gray] at (-4,2.3) $N(b_G)$;
[dashed,-,gray] (-3.4,1.8) – (4,1.8);
[gray] at (-4,1.8) $\nu(b_G)$;
A play $\pi'$ extending $\rho$ in a way such that $\StrPosA(G
\restriction \learned(\pi')) \cap \mrs(\rho) = \emptyset$ cannot have
more regret than a play $\pi$ extending $\rho$ for which $\StrPosA(G
\restriction \learned(\pi)) \cap \mrs(\rho) \neq \emptyset$—for $\rho$
longer than $\nu(b_G)$.
The main difference between the problem at hand and the one we solved in
Section <ref> is that, when playing against a positional adversary,
information revealed to in the present can affect the best alternatives to
her current behaviour. Some definitions are in order. Let $\rho = v_0 \dots v_j$
be a play prefix.
The maximal-regret points of $\rho$, denoted by $\mrp(\rho)$, is the set
\[
\{0 \le i < j \st v_i \in \VtcE \text{ and }
\lambda^i\left( \cVal^{v_i}_{\lnot v_{i+1}}(G\restriction
\learned(\rho[..j])) - \PlayVal{\rho[i..j]}\right) = \regret{}{\rho}{} \};
\]
and the maximal-regret strategies of $\rho$, written $\mrs(\rho)$, is
equal to
\[
\left\{ \tau \in \StrPosA(G\restriction \learned(\rho[..j]))
\st
\bigvee_{i \in
\mrp(\rho)} \cVal^{v_i}_{\lnot v_{i+1}}(G\restriction \learned(\rho[..j]))
= \cVal^{v_i}_{\lnot v_{i+1}}(G \times \tau)\right\}.
\]
The above definitions are meant to capture the intuition that, upon witnessing a
new choice of , we can reduce the size of the set of possible positional
strategies he could be using. Consider a play prefix $\rho$. The maximal-regret
points of $\rho$ correspond to the positions at which best alternatives to
$\rho$ occur. The maximal-regret strategies of $\rho$ is the set of
positional strategies of , $\rho$ consistent
with them, such that at least one of the best alternatives to $\rho$ is
consistent with them. Recall from Lemma <ref> $(ii)$ that
a play prefix $\rho$ is consistent with a positional strategy $\tau \in
\StrPosA(G)$ if and only if $\tau \in \StrPosA(G \restriction \learned(\rho))$.
We can, therefore, think of the set of edges $\learned(\rho)$ as representing
the set of all positional strategies for in $G$ that $\rho$ is consistent
with, $\{\tau \in \StrPosA(G) \st \rho \text{ is consistent with } \tau\}$.
Let us write $\StrPosA(G,\rho)$ for the set we just described. Let $\beta$ be
the value of one of the best alternatives to $\rho$. If $\beta' < \beta$ is
the value of one of the best alternatives to $\rho'$, then we know the
best alternatives to $\rho$ are not consistent with any strategy from
$\StrPosA(G,\rho')$. Then, according to our definition of maximal-regret
strategies, this also means that $\mrs(\rho) \cap \StrPosA(G,\rho') =
\emptyset$. The converse is also true.
As an example, consider the situation depicted in
Figure <ref>. If, from $v_j$, the play $\pi'$ is obtained
and we have that $\StrPosA(G \restriction \learned(\pi') \cap \mrs(\rho)$ is
empty, then the deviation from $v_{i'}$ might no longer be a best alternative.
Indeed, there is no positional strategy of which allows the deviation from
$v_{i'}$ to obtain the value we assumed (from just looking at the prefix $\rho$)
and which is also consistent with $\pi'$. In order to deal with this, we need
some more definitions.
Assume that $\Regret{G}{\StrAllE,\StrPosA} \ge b_G$. For a play prefix $\rho =
v_0 \dots v_n$ with $n \ge N(b_G)$, let us define the value $\delta_\rho$
($\delta$ for drop) as
\[
\min_{\substack{0 \le i \le j < N(b_G)\\
\tau,\tau' \in \StrPosA(G \restriction \learned(\rho))}}
\left| \lambda^i \left(
\cVal^{v_i}_{\lnot v_{i+1}}(G \times \tau) -
\PlayVal{\rho[i..j]} \right) - \lambda^{j} \cVal^{v_j}_{\lnot
v_{j+1}}(G \times \tau'))
\right|.
\]
Intuitively $\delta_\rho$ is the minimal drop of the regret achievable by a
better alternative (given the information we can extract from $\rho$).
*The smallest possible drop.
Let us derive a universal lower bound on $\delta_\rho$ for all $\rho$ of length
at least $N(b_G)$. In order to do so we will recall “the shape” of the co-operative
value of $G$. Recall the $\cVal$ in a discounted-sum game can be obtained by
supposing controls all vertices and computing $\aVal$ instead. It then
follows from positional determinacy of discounted-sum games that the $\cVal$ is
achieved by a lasso in the arena $G$. More formally, we know that there
is a play $\pi$ in $G$ of the form
\[
\pi = v_0 \dots v_{k-1} (v_k \dots v_\ell)^\omega
\]
where $0 \le k < \ell \le |V|$, and such that $\PlayVal{\pi} = \cVal^{v_0}(G)$.
Let us write $\lambda = \frac{\alpha}{\beta}$ with $\alpha,\beta \in
\mathbb{Z}$. One can then verify that
For all sub-arenas $H$ of $G$, for all vertices $v \in V$, there exists
$N \in \mathbb{Z}$ such that $\cVal^v(H) = \frac{N}{D}$ where
\(
D \defeq \beta^{|V|} (\beta^{|V|} - \alpha^{|V|}).
\)
It then follows from the definition of $\delta_\rho$ that:
For all play prefixes $\rho = v_0 \dots v_n$
such that $n \ge N(b_G)$ we have that
\[
\delta_\rho > \frac{1}{\beta^{N(b_G)} D}.
\]
*Formalizing our claims.
We can now prove a replacement for Lemma <ref> holds in this context.
Let $\pi$ be a play in $G$ and assume $\Regret{G}{\StrAllE,\StrPosA} > 0$.
$\nu(b_G)$ denote the value
\[
% |V|(|E| + 1) & \text{if } b_G = 0\\
N(b_G) + \left\lfloor \frac{\log(1 - \lambda) - \log W - (N(b_G)
+ |V|)\log \beta - \log(\beta^{|V|} -
\alpha^{|V|})}{\log \lambda} \right\rfloor + 1.% &
% \text{otherwise.}
\]
Then for all $\sigma \in \StrAllE$,
\[
\sup_{\tau \in \StrPosA} \regret{}{\out{}{\sigma}{\tau}}{} =
\sup_{\tau \in \StrPosA}
\regret{}{\out{}{\sigma}{\tau}[..{\nu(b_G)}]}{} - \lambda^{\nu(b_G)}
\PlayVal{\out{}{\sigma}{\tau}[{\nu(b_G)}..]}.
\]
Let us consider throughout this argument an arbitrary $\sigma \in
\StrAllE$.
From Lemma <ref> and the fact that
$\nu(b_G)$ is such that $N(b_G)$, we know that $\sup_{\tau \in \StrPosA}
\regret{}{\out{}{\sigma}{\tau} = v_0 \dots}{}$ equals
\[
\sup_{\tau \in \StrPosA}
\max_{\substack{0 \le i < \nu(b_G)\\ v_i \in
\VtcE}}\{\lambda^i(\cVal^{v_i}_{\lnot v_{i+1}}(G \restriction
\learned(\out{}{\sigma}{\tau})) -
\PlayVal{\out{}{\sigma}{\tau}[i..\nu(b_G)]}\} - \lambda^{\nu(b_G)}
\PlayVal{\out{}{\sigma}{\tau}[{\nu(b_G)}..]}.
\]
Now, also note that $\nu(b_G)$ was chosen so that
\[
\frac{W\lambda^{\nu(b_G)}}{1-\lambda} < \frac{1}{\beta^{N(b_G)
+ |V|} D}.
\]
Hence, for all $\tau' \in \StrPosA$ if we write
$\out{ }{\sigma}{\tau'} = v'_0 \dots$, then
for all $j \ge \nu(b_G)$ such that $v'_j \in \VtcE$ it holds that
\[
-\frac{1}{\beta^{N(b_G) + |V|} D} <
\lambda^i
\PlayVal{\out{ }{\sigma}{\tau'}[i..]}) <
\frac{1}{\beta^{N(b_G) + |V|} D}.
\]
It then follows from Lemma <ref> and the definition
of $\delta_{\out{ }{\sigma}{\tau'}[..\nu(b_G)]}$ that, if there exists
$\ell \ge \nu(b_G)$ such that for all $0 \le k \le \nu(b_G)$ with $v'_k
\in \VtcE$
\[
\cVal^{v'_k}_{\lnot v'_{k+1}}(G \restriction \learned(\pi[..\ell])) <
\cVal^{v'_k}_{\lnot v'_{k+1}}(G \restriction
\learned(\pi[..\nu(b_G)]))
\]
then $\regret{}{\out{ }{\sigma}{\tau'}}{} < \regret{}{\out{
}{\sigma}{\tau''}}{}$ for all $\tau'' \in \mrs(\pi'[..\nu(b_G)])$. This
is due to the fact that
that $\out{ }{\sigma}{\tau''}[..\nu(b_G)] = \out{
}{\sigma}{\tau'}[..\nu(b_G)]$ and
\[
\cVal^{v'_k}_{\lnot v'_{k+1}}(G \times \tau'')
= \cVal^{v'_k}_{\lnot v'_{k+1}}(G \restriction
\learned(\out{ }{\sigma}{\tau''}[..\nu(b_G)])).
\]
The above implies
that for all $\sigma \in \StrAllA$ the value $\sup_{\tau
\in \StrPosA} \regret{}{\out{}{\sigma}{\tau} = v_0 \dots}{}$ equals
\[
\max \{
\cVal^{v_i}_{\lnot v_{i+1}}(G\restriction
\learned(\out{}{\sigma}{\tau}[..\nu(b_G)])) -
\lambda^{\nu(b_G)}
\PlayVal{\out{
}{\sigma}{\tau}[..\nu(b_G)]} \st
0 \le i \le N(b_G) \text{ and }
v_i \in \VtcE
\}
\]
and therefore (by definition of regret of a prefix) we have that
\[
\sup_{\tau \in \StrPosA} \regret{}{\out{}{\sigma}{\tau}}{} =
\sup_{\tau \in \StrPosA}
\regret{}{\out{}{\sigma}{\tau}[..{\nu(b_G)}]}{} - \lambda^{\nu(b_G)}
\PlayVal{\out{}{\sigma}{\tau}[{\nu(b_G)}..]}.
\]
as required.
*Putting everything together.
Let us go back to our example to illustrate how to use $\nu(b_G)$ and the drop
of a prefix. Consider now the situation from
Figure <ref>. Recall we have assumed $\pi'$ is a play
extending $\rho$ with $\StrPosA(G \restriction \learned(\pi')) \cap \mrs(\rho)
=\emptyset$. It follows that all
best alternatives to $\pi'$ achieve a payoff strictly smaller than
$\cVal^{v'}_{\lnot v_{i' + 1}}(G \restriction \learned(\rho))$. Thus, the regret
of $\pi'$ can only be bigger than the regret of a play $\pi$ with $\StrPosA(G
\restriction \learned(\pi)) \cap
\mrs(\rho) \neq \emptyset$ if the minimal
index $k > j$ such that $\StrPosA(G \restriction \learned(\pi'[..j])) \cap
\mrs(\rho) = \emptyset$— the turn at which revealed he was not
playing a strategy from $\mrs(\rho)$—is small enough. In other words, the drop
in the value of the best alternative has to be compensated by a similar drop in
the value obtained by , and the discount factor makes this impossible after
some number of turns.
If $\Regret{G}{\StrAllE,\StrPosA} \ge b_G$ then
$\Regret{G}{\StrAllE,\StrPosA}$ is equal to
\[
\inf_{\sigma \in \StrAllE}
\sup\{ \regret{}{\pi[..\nu(b_G)]}{} - \lambda^{\nu(b_G)}
\aVal^{\hat{u}}(\hat{H}) \st \pi = v_0 v_1 \dots \text{ cons.
with } \sigma \text{ and some }\tau \in \StrPosA\}
\]
* $\hat{u} \defeq (v_{\nu(b_G)},\learned(\pi[..\nu(b_G)]))$ and
* $\hat{H} \defeq \hat{G} \restriction \{\left( (C,u),(D,v)
\right) \st \StrPosA(G\restriction D) \cap
\mrs(\pi[..\nu(b_G)]) \neq \emptyset \}$.
First, note that if $\Regret{G}{\StrAllE,\StrPosA} > 0$ then there
cannot be any regret-free strategies for in $G$ when playing
against a positional adversary. It then follows from
Corollary <ref> that
$\Regret{G}{\StrAllE,\StrPosA} \ge b_G$.
Now using Lemma <ref> together with the definition of the
regret of a play we get that $\Regret{G}{\StrAllE,\StrPosA}$ is equal
\[
\inf_{\sigma \in \StrAllE} \sup\{ \regret{}{\pi[..\nu(b_G)]}{} -
\lambda^{\nu(b_G)} \PlayVal{\pi[\nu(b_G)..]} \st \pi \text{
cons. } \sigma \text{ and some } \tau \in \StrPosA \}.
\]
Finally, note that it is in the interest of to maximize the value
$\lambda^{\nu(b_G)} \PlayVal{\pi[\nu(b_G)..]}$ in order to minimize
regret. Conversely, tries to minimize the same value with a
strategy from $\mrs(\pi[..\nu(b_G)])$: critically, the strategy is such
that the prefix $\pi[..\nu(b_G)]$ is consistent with it.
Thus, we
can replace it by the antagonistic value from $\pi[\nu(b_G)..]$ discounted
accordingly. In this setting we also want to force to play a
positional strategy which is consistent with deviations before $N(b_G)$
which achieve the assumed regret of the prefix $\pi[..\nu(b_G)]$.
More formally, we have
\begin{align*}
&\inf_{\sigma \in \StrAllE}
\sup_{\tau \in \StrPosA}
\regret{}{\out{}{\sigma}{\tau}[..\nu(b_G)]}{} -
\lambda^{\nu(b_G)}
\PlayVal{\out{}{\sigma}{\tau}[\nu(b_G)..]}\\
=&\inf_{\substack{\sigma \in \StrAllE\\\sigma' \in \StrAllE}}
\sup_{\substack{\tau \in \StrPosA\\
\tau' \in \mrs(\out{}{\sigma}{\tau}[..\nu(b_G)])
\regret{}{\out{}{\sigma}{\tau}[..\nu(b_G)]}{} -
\lambda^{\nu(b_G)} \StratVal{}{}{\sigma'}{\tau'}\\
=&\inf_{\sigma \in \StrAllE}
\sup_{\tau \in \StrAllA}
\regret{}{\out{}{\sigma}{\tau}[..\nu(b_G)]}{} +
\inf_{\sigma' \in \StrAllE} \sup_{
\substack{
\tau' \in \mrs(\out{}{\sigma}{\tau}[..\nu(b_G)])
\left(
\StratVal{}{}{\sigma'}{\tau'}
\right).
\end{align*}
It should be clear that the RHS term of the sum is equivalent to
\[
\]
as required.
The above result allows us to claim an algorithm (when $\lambda$ is
not fixed) to compute the regret of a game. As in
Section <ref>, we simulate the game using an alternating
machine which halts in at most a pseudo-polynomial number of steps which depends
on $\nu(b_G)$ and, in turn, on $b_G$. After that, we must compute the
antagonistic value of $\hat{G}$. As a first step, however, we compute the safety
game $\tilde{G}$ and determine its winner.
Computing the regret value of a game, playing against a positional
adversary, can be done in time $\mathcal{O}(\max\{|V|(|E| + 1),
\nu(b_G)\})$ with an
alternating Turing machine.
The memory requirements for are as follows:
Let $\eta \defeq |\Delta|^{d}$ where $d = \max\{|V|(|E| + 1),
\nu(b_G)\}$. It then holds that $\Regret{G}{\StrE^{\eta},\StrPosA} =
\Regret{G}{\StrAllE,\StrPosA}$.
§.§ Lower bounds
In the main body of the paper, namely in Section <ref>, we
have claimed that the regret threshold problem is -hard when $\lambda$ is
fixed. The proof of this claim is provided in
Appendix <ref>. In the next section we shall prove the
following result which applies for when $\lambda$ is not fixed.
For a discount factor $\lambda \in (0,1)$, regret threshold $r \in
\mathbb{Q}$, and weighted arena $G$, determining whether
$\Regret{G}{\StrAllE,\StrPosA} \lhd r$, for $\lhd \in \{<,\le\}$, is
§.§.§ Proof of Lemma <ref>
[clause/.style=isosceles triangle, shape border
rotate=90,inner sep=3pt, draw,dotted,node distance=0.5cm]
[ve, right=of A, yshift=-0.5cm](B)$x_0$;
[ve, right=of A, yshift=0.5cm](B')$\overline{x_0}$;
[va, right=of B, yshift=0.5cm](C);
[ve, right=of C, yshift=-0.5cm](D)$x_1$;
[ve, right=of C, yshift=0.5cm](D')$\overline{x_1}$;
[right=of D, yshift=0.5cm](E)$\dots$;
[ve,right=of E, yshift=-0.5cm](F)$x_m$;
[ve,right=of E, yshift=0.5cm](F')$\overline{x_m}$;
[ve,right=of F, yshift=0.5cm](G)$\Phi$;
[below=0.5cm of B, clause,xshift=-0.8cm](X)$C_i$;
[below=0.5cm of B](Y)$\dots$;
[below=0.5cm of B, clause,xshift= 0.8cm](Z)$C_j$;
(A) edge (B)
(B) edge (C)
(A) edge (B')
(B') edge (C)
(C) edge (D)
(C) edge (D')
(D) edge (E)
(D') edge (E)
(E) edge (F)
(E) edge (F')
(F) edge (G)
(F') edge (G)
(G) edge[loopright] node[el,swap]$A$ (G)
(B) edge[bend left] (X.apex)
(B) edge[bend left] (Z.apex)
(X.apex) edge[bend left] (B)
(Z.apex) edge[bend left] (B)
Depiction of the reduction from QBF.
[ve](A) at (2,6) $x_i$;
(rA) at (4,7) ;
(lA) at (0,7) ;
[va](B) at (2,4) ;
[ve](C) at (4,3) ;
[va](D) at (4,1) ;
[ve](E) at (6,0) $\overline{x_j}$;
[ve](F) at (2,0) ;
[va](G) at (0,1) ;
[ve](H) at (0,3) ;
[ve](I) at (-2,0) $x_k$;
[ve](P) at (2,2) ;
(A) edge[dotted] node[el] (rA)
(lA) edge[dotted] node[el] (A)
(A) edge[bend left] node[el] (B)
(B) edge[bend left] node[el] (A)
(B) edge node[el]$C$ (C)
(C) edge node[el,swap] (P)
(C) edge node[el]$C$ (D)
(D) edge[bend left] node[el] (E)
(E) edge[bend left] node[el] (D)
(D) edge node[el]$C$ (F)
(F) edge node[el,swap] (P)
(F) edge node[el]$C$ (G)
(G) edge node[el]$C$ (H)
(H) edge node[el]$C$ (B)
(G) edge[bend left] node[el] (I)
(I) edge[bend left] node[el,swap] (G)
(H) edge node[el,swap] (P)
(P) edge[out=45,in=135,loop,looseness=8] node[el,swap]$B$ (P);
Clause gadget for the QBF reduction for clause $x_i \lor \lnot x_j
\lor x_k$.
The QSAT Problem asks whether a given fully quantified boolean
formula (QBF) is satisfiable. The problem is known to be
-complete <cit.>. It is known the result holds even if the
formula is assumed to be in conjunctive normal form with three literals
per clause (also known as $3$-CNF). Therefore, w.l.o.g., we consider an
instance of the QSAT Problem to be given in the following form:
\[
\exists x_0 \forall x_1 \exists x_2 \dots \Phi(x_0, x_1,
\dots, x_m)
\]
where $\Phi$ is in $3$-CNF. Let $n$ be the number of clauses from
In the sequel we describe how to construct, in polynomial time, a
weighted arena in which ensures regret of at most $r$ if and only
if the QBF is true.
We first describe the value-choosing part of the game (see
Figure <ref>). $\VtcE$ contains vertices for every
existentially quantified variable from the QBF and $\VtcA$ contains
vertices for every universally quantified variable. At each of this
vertices, there are two outgoing edges with weight $0$ corresponding to
a choice of truth value for the variable. For the variable $x_i$ vertex,
the true edge leads to a vertex from which can choose to
move to any of the clause gadgets corresponding to clauses where the
literal $x_i$ occurs (see dotted incoming edge in
Figure <ref>) or to advance to $x_{i+1}$. The
false edge construction is similar. From the vertices encoding
the choice of truth value for $x_m$ can either visit the clause
gadgets for it or move to a “final” vertex $\Phi \in \VtcE$. This
final vertex has a self-loop with weight $A$.
Our reduction works for values of $\lambda$, $r$, $A$, $B$, and $C$ such
that the following constraints are met:
* $A < B < C$,
* $\lambda^2\left(\frac{C}{1 - \lambda}\right) -
\lambda^{2nm - 2}\left(C +
\lambda^2 \frac{B}{1-\lambda}\right) < r$,
* $\lambda^{2nm-2}\left(C + \lambda^2
\frac{B}{1-\lambda}\right) >
\lambda^2\left(\frac{C \frac{1-\lambda^4}{1-\lambda}}{1 -
\lambda^8}\right)$,
* $\lambda^2\left(C + \lambda^2 \frac{B}{1-\lambda}\right) -
\lambda^{2nm}\left(\frac{A}{1-\lambda}\right) < r$, and
* $\lambda^{2nm-2}\left(\frac{C}{1-\lambda}\right) -
\lambda^{2nm}\left(\frac{A}{1-\lambda}\right) \ge r$.
(See below for a sample concrete assignment.)
*Value-choosing strategies.
To conclude the proof, we describe the strategy of which ensures
the desired property if the QBF is satisfiable and a strategy of which ensures the property is falsified otherwise.
Assume the QBF is true. It follows that there is a strategy of
the existential player in the QBF game such that for any strategy of the
universal player the QBF will be true after they both choose values for
the variables. now follows this strategy while visiting all clause
gadgets corresponding to occurrences of chosen literals. At every gadget
clause she visits she chooses to enter the gadget. If now decides to take the weight $C$ edge, can go to the center-most
vertex and obtain a payoff of at least
\[
\lambda^{2nm -2}\left(C + \lambda^2 \frac{B}{1-\lambda}\right),
\]
with equality holding if helps her at the very last clause visit
of the very last variable gadget. In this case,
the claim holds by $(i)$. We therefore focus in
the case where chooses to take back to the vertex from which
she entered the gadget. She can now go to the next clause gadget and
repeat. Thus, when the play reaches vertex $\Phi$, must have
visited every clause gadget and has chosen to disallow a weight
$C$ edge in every gadget. Now can ensure a payoff value of
$\lambda^{2nm}(\frac{A}{1-\lambda})$ by
going to $\Phi$. As she has witnessed that in every clause gadget there
is at least one vertex in which is not helping her, alternative
strategies might have ensured a payoff of at most $\lambda^{2}
(C + \lambda^2 \frac{B}{1-\lambda})$, by playing to the center of some
clause gadget, or
\[
\lambda^2\left(\frac{C
\frac{1-\lambda^4}{1-\lambda}}{1-\lambda^8}\right)
\]
by playing in and out of some adjacent clause gadgets. By $(iii)$, we
know it suffices to show that the former is still not enough to make the
regret of at least $r$. Thus, from $(iv)$, we get that her regret
is less than $r$.
Conversely, if the universal player had a winning strategy (or, in other
words, the QBF was not satisfiable) then the strategy of consists
in following this strategy in choosing values for the variables and
taking out of clause gadgets if she ever enters one. If the play
arrives at $\Phi$ we have that there is at least one clause gadget that
was not visited by the play. We note there is an alternative strategy of
which, by choosing a different valuation of some variable, reaches
this clause gadget and with the help of achieves value of at least
$\lambda^{2nm-2}(\frac{C}{1-\lambda})$. Hence, by $(v)$,
this strategy of ensures regret of at least $r$. If avoids
reaching $\Phi$ then she can ensure a value of at most $0$, which means
an even greater regret for her.
*Example assignment.
For completeness, we give one assignment of the positive rationals
$\lambda$, $r$, $A$, $B$, and $C$ which satisfies the inequalities.
It will be obvious the chosen values can be encoded
into a polynomial number of bits w.r.t. $n$ and $m$.
We can assume, w.l.o.g., that $2 \le 2m \le n$. Intuitively, we want
values such that $(i)$ $A < B < C$ and such that the discount factor
$\lambda$ is close enough to $1$ so that going to the center of a clause
gadget at the end of the value-choosing rounds, is preferable for compared to doing some strange path between adjacent clauses—this is captured
by item $(iii)$. A $\lambda$ which is close to $1$ also gives us item
$(v)$ from $(i)$. In order to ensure wins if she does visit the
center of a clause gadget, we also would like to have $C - A < r
\lambda^{-2} (1-\lambda)$, which would imply items $(ii)$ and $(iv)$
from the inequality list. It is not hard to see that the following
assignment satisfies all the inequalities:
* $\lambda \defeq 1 - \frac{1}{2^{n^3}}$,
* $A \defeq 2$,
* $B \defeq 3$,
* $C \defeq 4$, and
* $r \defeq 3(2^{n^6} - 1)$.
§.§.§ Proof of
Theorem <ref>
[va] (A) at (0,1.5) $v$;
[ve] (B) at (2,1.5) ;
[va] (C) at (4,1.5) $t_1$;
[va] (D) at (2,0) ;
[va] (E) at (4,0) $s_2$;
(A) edge (B)
(B) edge (C)
(B) edge (D)
(D) edge[loopleft] (D)
(D) edge (E)
Regret gadget for $2$-disjoint-paths reduction.
The $2$-disjoint-paths Problem on directed
graphs is known to be -complete <cit.>.
We sketch how to translate a given instance of the
$2$-disjoint-paths Problem into a weighted arena in which can ensure regret value of $0$ if, and only if, the answer
to the $2$-disjoint-paths Problem is negative.
Consider a directed graph $G$ and distinct vertex pairs $(s_1,t_1)$ and
$(s_2,t_2)$. W.l.o.g. we assume that for all $i \in \{1,2\}$:
* $s_i \neq t_i$,
* $t_i$ is reachable from $s_i$, and
* $t_i$ is a sink (has no outgoing edges).
in $G$. We now describe the changes we apply to $G$ in order to get the
underlying graph structure of the weighted arena and then comment on the
weight function. Let all vertices from $G$ be vertices and $s_1$
be the initial vertex. We replace all edges $(v, t_1)$ incident on $t_1$
by a copy of the gadget shown in Figure <ref>. Next, we
add self-loops on $t_1$ and $t_2$ with weights $A$ and $B$,
respectively. Finally, the weights of all remaining edges are $0$.
Our reduction works for any value of $A$ and $B$ such that
* $\lambda^{|V|} \frac{A}{1-\lambda} > r$, and
* $\lambda^{|V|} \frac{B}{1-\lambda} - \lambda
\frac{A}{1-\lambda} > r$.
For instance, consider $\alpha \defeq \frac{r+1}{\lambda^{|V|}}$. It is
easy to verify that setting $A \defeq (1-\lambda)\alpha$ and $B\defeq
(1-\lambda)\alpha^2$ satisfies the inequalities. Furthermore, $A$ and
$B$ are rational numbers which can be represented using a polynomial
number of bits w.r.t. $|V|$ and the size of the representation of both
$\lambda$ and $r$.
We claim that, in this new weighted arena, can ensure a regret
value of $0$ if in $G$ the vertex pairs $(s_1,t_1)$ and $(s_2,t_2)$
cannot be joined by vertex-disjoint paths. If, on the contrary, there
are vertex-disjoint paths joining the pairs of vertices, then can
ensure a regret value strictly greater than $r$. Indeed, we claim that
the strategy that minimizes the regret of is the strategy that, in
states where has a choice, tells her to go to
First, let us prove that this strategy has regret $0$
if, and only if, there are no two paths disjoint paths in the graph between
the pairs of states $(s_1,t_1)$, $(s_2,t_2)$. Assume there are no
disjoint paths, then if chooses to always avoid $t_1$ then the
regret is $0$. If $t_1$ is reached, then the choice of ensures a
value of at least $\lambda^{|V|} \frac{A}{1-\lambda}$.
The only alternative strategy of is to have chosen to go to
$s_2$. As there are no disjoint paths, we know that either the path
constructed from $s_2$ by never reaches $t_2$, and then the value
of the path is $0$ and the regret is $0$ for or the path
constructed from $s_2$ reaches $t_1$ again, and so the regret is also
equal to $0$ since the discount factor ensures the value of this play is
lower than the one realized by the current strategy of .
Now assume that there are disjoint paths, if would
have chosen to put the game in $s_2$ (instead of choosing $t_1$) then
has a strategy which allows to reach $t_2$ and get a payoff
of at least $\lambda^{|V|} \frac{B}{1-\lambda}$ while she achieves at
most $\lambda \frac{A}{1 - \lambda}$. From $(i)$ we have that the regret
in this case is greater than $r$.
To conclude the proof, let us show that any other strategy of has a
regret greater than $0$. Indeed, if decides to go to $s_2$ (instead
of choosing to go to $t_1$) then can choose to loop on $s_2$ and
the payoff in this case is $0$. The regret of is non-zero in this
case since she could have achieved at least $\lambda^{|V|}
\frac{A}{1-\lambda}$ by going to $t_1$. It follows from $(ii)$ that this
ensures a regret value greater than $r$.
§ MISSING PROOFS FROM SECTION <REF>
§.§ Proof of Theorem <ref>
We reduce the problem to determining the winner of a reachability game
on an exponentially larger arena. Although the arena is exponentially
larger, all paths are only polynomial in length, so the winner can be
determined in alternating polynomial time, or equivalently, polynomial
The idea of the construction is as follows. Given a discounted-sum
automaton $\mathcal{A}$, we determinize its transitions via a subset
construction, to obtain a deterministic, multi-valued discounted-sum
automaton $D_{\mathcal{A}}$. Then we decide if Eve is able to simulate,
within the regret bound, the $D_{\mathcal{A}}$ on $\mathcal{A}$ for all
finite words up to a length (polynomially) dependent on
$\epsilon$. If we simulate the automaton for a sufficient number of
steps, then any significant gap between the automata will be
unrecoverable regardless of future inputs, and we can give a
satisfactory answer for the $\epsilon$-gap regret problem.
More formally, given a discounted-sum automaton $\mathcal{A} = (Q,q_0,A,
\delta, w)$, a regret value $r$ and a precision $\epsilon>0$, we
construct a reachability game $G_\mathcal{A}^\epsilon(r)$ as follows.
\[
N \defeq \left\lfloor
\log_{\lambda}\left(\frac{\epsilon(1-\lambda)}{4W}\right)\right\rfloor +
\]
where $W$ is the maximum absolute value weight occurring in
$\mathcal{A}$, so that $ \frac{\lambda^N \cdot W}{1-\lambda} <
\frac{\epsilon}{4}$. Let $P = \{\discfun{\lambda}(\pi) \st \pi\in
Q^*\text{ is a finite run of $\mathcal{A}$ with }|\pi|\leq N\}$ denote
the (finite) set of possible discounted payoffs of words of length at
most $N$. Let $\mathcal{F}$ be the set of functions $f:Q \to
\mathbb{R}\cup\{\bot\}$, and for $f \in \mathcal{F}$, let $\supp{f} =
\{q \in Q \st f(q) \neq \bot\}$. Intuitively, each $f \in \mathcal{F}$
represents a weighted subset of $Q$ ($\supp{f}$ being the corresponding
unweighted subset), where $f(q)$ for $q \in \supp{f}$ corresponds to the
maximal weight over all (consistent) paths ending in $q$ (scaled by a
power of $\lambda$). Given $f \in \mathcal{F}$ and $\alpha \in A$
the $\alpha$-successor of $f$ is the function $f_\alpha$ defined as:
\[
f_\alpha(q') \defeq
\begin{cases}
\displaystyle
\max_{\substack{q \in \supp{f}\\(q,\alpha,q') \in
\delta }} \{\lambda^{-1}\cdot f(q) +
w(q,\alpha,q') \} & \text{if this set is not empty}\\
\bot & \text{otherwise.}
\end{cases}
\]
We define $\mathcal{F}_0 = \{f_0\}$ where $f_0(q_0) = 0$ and $f_0(q) =
\bot$ for all $q \neq q_0$; and for all $n\geq 0$, we define
$\mathcal{F}_{n+1} \defeq \{f_\alpha \st f \in \mathcal{F}\text{ and
}\alpha \in A\}$. For convenience, let $F = {\bigcup}_{i=0}^N
\mathcal{F}_i$ (considered as a disjoint union).
The game $G_{\mathcal{A}}^\epsilon(r) = (V,\VtcE,E,v_0,T)$ is defined as
* $V = (Q \times F \times P) \cup (Q \times F \times P \times
* $\VtcE = (Q \times F \times P \times A)$;
* $\big((q,f,c),(q,f,c,\alpha)\big) \in E$ for all $q \in
Q$, $f \in F\setminus \mathcal{F}_N$, $c \in P$, and
$\alpha \in A$;
* $\big((q,f,c,\alpha),(q',f',c')\big) \in E$ for all $q,q'
\in Q$, $f \in F\setminus \mathcal{F}_N$, $c \in P$, and
$\alpha \in A$ such that $(q,\alpha,q') \in
\delta$, $f' = f_\alpha$, and $c'=c+\lambda \cdot
* $v_0 = (q_0,f_0,0)$; and
* $(q,f,c) \in T$ if, and only if, $f \in \mathcal{F}_N$ and
$\max_{s \in \supp{f}}\lambda^{N-1}\cdot f(s) \leq
We claim that determining the winner of $G_\mathcal{A}^\epsilon(r)$ yields a
correct response for the $\epsilon$-gap promise problem.
Let $G_{\mathcal{A}}^\epsilon(r)$ be
defined as above. Then:
* If Eve wins $G_{\mathcal{A}}^\epsilon(r)$ then
$\Regret{\mathcal{A}}{\StrE,\StrWordA} \le r+\epsilon$, and
* if Adam wins $G_{\mathcal{A}}^\epsilon(r)$ then
$\Regret{\mathcal{A}}{\StrE,\StrWordA} > r$.
It is easy to see that a play of $G_{\mathcal{A}}^\epsilon(r)$
results in Adam choosing a word $w \in A^*$ of length $N$,
and Eve selecting a run, $\pi$, of $w$ on $\mathcal{A}$ by
resolving non-determinism at each symbol. Further, if the play
terminates at $(q,f,c)$ then $c=\discfun{\lambda}(\pi)$ and, as
$f$ contains the maximal weights of all paths (scaled by a power
of $\lambda$), $\mathcal{A}(w) = \lambda^{N-1}(\max_{s \in
\supp{f}} f(s))$. Since $|w|=N$ we have, for any
infinite word $w' \in A^\omega$ and for any run,
$\pi'$, of $\mathcal{A}$ on $w'$ from $q$, $\pi'$:
\begin{eqnarray*}
|\mathcal{A}(w\cdot w')-\mathcal{A}(w)| &\leq&
\frac{\lambda^N \cdot W}{1-\lambda} <
\frac{\epsilon}{4}, \text{ and}\\
|\discfun{\lambda}(\pi\cdot \pi') -
\discfun{\lambda}(\pi)|&\leq& \frac{\lambda^N \cdot
W}{1-\lambda} < \frac{\epsilon}{4}.
\end{eqnarray*}
It follows that:
\begin{equation}\label{eqn:approx}
(\mathcal{A}(w)-\discfun{\lambda}(\pi)) -
\frac{\epsilon}{2} <
\mathcal{A}(w\cdot w') - \discfun{\lambda}(\pi\cdot\pi') <
(\mathcal{A}(w)-\discfun{\lambda}(\pi)) + \frac{\epsilon}{2}.
\end{equation}
Now suppose Eve wins $G_\mathcal{A}^\epsilon(r)$. Then, for
every word $w$ with $|w|=N$, Eve has a strategy $\sigma$ that
construct a run, $\pi$, on $\mathcal{A}$ such that
$\mathcal{A}(w) \leq \discfun{\lambda}(\pi) + r +
\frac{\epsilon}{2}$. We extend this strategy to infinite words
by playing arbitrarily after the first $N$ symbols. It follows
from Equation <ref> that for every infinite word
$\hat{w}$, the resulting run, $\hat{\pi}$,
\[
\mathcal{A}(\hat{w}) - \discfun{\lambda}(\hat{\pi}) <
(\mathcal{A}(w)-\discfun{\lambda}(\pi)) + \frac{\epsilon}{2}
\leq r + \epsilon.
\]
Since $\regret{\sigma}{\StrE,\StrWordA}{\mathcal{A}} =
\sup_{\hat{w} \in A^\omega}(\mathcal{A}(\hat{w}) -
\discfun{\lambda}(\pi))$, we have
$\Regret{\mathcal{A}}{\StrE,\StrWordA} \le r+\epsilon$.
Conversely, suppose Adam wins $G_\mathcal{A}^\epsilon(r)$. Then
for any strategy of Eve, Adam can construct a word $w$, with
$|w|=N$ such that the run, $\pi$, of $\mathcal{A}$ on $w$
determined by Eve's strategy satisfies $\mathcal{A}(w) >
\discfun{\lambda}(\pi) + r + \frac{\epsilon}{2}$. Again, from
Equation <ref> it follows that for any infinite word
$\hat{w}$ with $w$ as its prefix and any
consistent run $\pi'$,
\[
\mathcal{A}(\hat{w}) - \discfun{\lambda}(\hat{\pi}) >
(\mathcal{A}(w)-\discfun{\lambda}(\pi)) -
\frac{\epsilon}{2} > r.
\]
As this is valid for any strategy of Eve, we have
$\Regret{\mathcal{A}}{\StrE,\StrWordA}>r$ as required.
Now every path in $G_\mathcal{A}^\epsilon(r)$ has length at most $N$,
and as the set of successors of a given state can be computed on-the-fly
in polynomial time, the winner can be determined in alternating
polynomial time. Hence a solution to the $\epsilon$-gap promise problem is
constructible in polynomial space.
§.§ Proof of Theorem <ref>
[state] (sink0) at (0,2) $\bot_0$;
[state] (linter) at (2,2) ;
[state,initial above] (vi) at (4,2) ;
[state] (rinter) at (6,2) ;
[state] (sink2) at (8,2) $\bot_Z$;
(lbinter) at (2,0) ;
(rbinter) at (6,0) ;
(sink0) edge[loop] node[el,swap] $A,0$ (sink0)
(sink2) edge[loop] node[el,swap] $A,Z$ (sink2)
(vi) edge node[el,swap] $A,0$ (linter)
(linter) edge node[el,swap] $bail, 0$ (sink0)
(vi) edge node[el] $A,0$ (rinter)
(rinter) edge node[el] $bail, 0$ (sink2)
(linter) edge node[el,swap] $A\setminus \{bail\},0$ (lbinter)
(rinter) edge node[el] $A\setminus \{bail\},0$ (rbinter)
Initial gadget used in reduction from QBF.
[every initial by arrow/.style=dotted]
at (-2.8,7.2) …;
[state] (sink0) at (-2,0) $\bot_0$;
[state] (sinkZ) at (0,0) $\bot_Z$;
[state] (xnleft) at (0,2) $\overline{x_n}$;
[state] (axnleft) at (0,4) ;
[state] (bx1left) at (0,6) ;
[state] (x1left) at (0,8) $x_1$;
[state,initial above] (ax1left) at (0,10) ;
at (2.5,7.2) …;
[state] (sinkX) at (5,0) $\bot_X$;
[state] (sinkY) at (8,0) $\bot_Y$;
[state] (xnright) at (5,2) $x_k$;
[state] (axnright) at (5,4) ;
[state] (bx1right) at (5,6) ;
[state] (notx1right) at (4,8) $\overline{x_j}$;
[state] (x1right) at (6,8) $x_j$;
[state,initial above] (ax1right) at (5,10) ;
(ax1left) edge node[el]$A,0$ (x1left)
(x1left) edge node[el]$\lnot b,0$ (bx1left)
(x1left) edge[bend right] node[el,swap]$A \setminus \lnot b, 0$ (sink0)
(bx1left) edge[dotted] (axnleft)
(axnleft) edge node[el]$A,0$ (xnleft)
(xnleft) edge node[el]$b,0$ (sinkZ)
(xnleft) edge[bend right,pos=0.2] node[el,swap]$A \setminus b, 0$ (sink0)
(ax1right) edge node[el,swap]$A,0$ (notx1right)
(ax1right) edge node[el]$A,0$ (x1right)
(notx1right) edge node[el,swap,pos=0.2]$\lnot b,0$ (bx1right)
(notx1right) edge[bend left] node[el,swap]$A \setminus \lnot b, 0$ (sinkY)
(x1right) edge[pos=0.2] node[el]$b,0$ (bx1right)
(x1right) edge[bend left] node[el]$A \setminus b, 0$ (sinkY)
(bx1right) edge[dotted] (axnright)
(axnright) edge node[el]$A,0$ (xnright)
(xnright) edge[dotted] node[el,align=right]$\lnot b,0$
$b,0$ (sinkX)
(xnright) edge[bend left,pos=0.2] node[el]$A \setminus \{b, \lnot b\}, 0$ (sinkY)
Left and right sub-arenas of the reduction from QBF. Clause $i$ shown
on the left; existential and universal gadgets for variables $x_j$ and $x_k$,
respectively, on the right.
Given an instance of the QSAT Problem – a fully quantified
boolean formula (QBF) – we construct, in polynomial time, a weighted
arena such that the answer to the regret threshold problem is positive
if, and only if, the QBF is true. The main idea behind our reduction is
to build an arena with two disconnected sub-graphs joined by an initial
gadget in which we force to go into a specific sub-arena.
In order for her to ensure the regret is not too high she
must now make sure all alternative plays in the other part of the arena
do not achieve too high values. In the sub-arena where finds
herself, we will simulate the choice of values for the boolean variables
from the QBF while in the other sub-arena these choices will affect which
alternative paths can achieve high discounted-sum values based on the
clauses of the QBF. We describe the reduction for $\le$. It will be clear
how to extend the result to $<$.
The QSAT Problem asks whether a given fully quantified boolean
formula (QBF) is satisfiable. The problem is known to be
-complete <cit.>. It is known the result holds even if the
formula is assumed to be in conjunctive normal form with three literals
per clause (also known as $3$-CNF). Therefore, w.l.o.g., we consider an
instance of the QSAT Problem to be given in the following form:
\[
\exists x_0 \forall x_1 \exists x_2 \dots \Phi(x_0, x_1,
\dots, x_n)
\]
where $\Phi$ is in $3$-CNF.
We now give the details of the construction. Our reduction works for
values of positive rationals $r$, $X$, $Y$, and $Z$ such that
* $\lambda^2 \frac{Z}{1 - \lambda} > r + \epsilon$,
* $\lambda^{2n} \frac{Z}{1-\lambda} - \lambda^{2n}
\frac{X}{1-\lambda} > r + \epsilon$,
* $\lambda^{2n} \frac{Z}{1-\lambda} - \lambda^{2n}
\frac{Y}{1-\lambda} \le r$,
* $\lambda^3 \frac{Y}{1-\lambda} - \lambda^{2n}
\frac{X}{1-\lambda} \le r$.
The alphabet of the new weighted arena is $A = \{bail, b, \lnot b\}$.
*Example assignment. In order to convince the reader
that values which satisfy the above inequalities indeed exist
for all possible valuations of $n$ and $\epsilon$ we give such
a valuation. Let $f : \mathbb{Q} \to \mathbb{Q}$ be
defined as $f(x) \defeq \frac{(1-\lambda)x}{\lambda^{2n}}$. Note that,
w.l.o.g., we can assume that $n \ge 2$. Consider the valuation
* $r \defeq \lambda^{3-2n} (1+\epsilon)$,
* $Z \defeq f(r + \epsilon + 2)$,
* $X \defeq f(1)$,
* $Y \defeq f(2+\epsilon)$.
Clearly, inequalities $(i)$–$(iii)$ hold. Regarding $(iv)$, it will be
useful to consider the equivalent inequality
\[
\lambda^{3 - 2n}Y - X \le \frac{r(1-\lambda)}{\lambda^{2n}}.
\]
We observe that the LHS is smaller than $\lambda^{3-2n}(Y - X)$.
Furthermore the difference $Y - X$ is equivalent to $\frac{(1 +
\epsilon)(1-\lambda)}{\lambda^{2n}}$. Finally, by choice of $r$ we have
that the RHS is equivalent to
\[
\lambda^{3-2n}\left(\frac{(1+\epsilon)(1-\lambda)}{\lambda^{2n}} \right).
\]
Hence, $(iv)$ holds as well. Note that the chosen values can be encoded
into a polynomial number of bits w.r.t. $\lambda$ and $n$ as well as the
size of the representation of $\epsilon$.
*Initial gadget. The weighted arena we construct starts as is
shown in Figure <ref>. Here, has a to make a
choice: she can go left or right. If she goes left, then can play
$bail$ and force her into $\bot_0$ giving her a value of $0$ while an
alternative play goes into $\bot_Z$ achieving a value of $\lambda^2
\frac{Z}{1-\lambda}$. By $(i)$ we get that the regret of this strategy
is greater than $r + \epsilon$. Thus, we can assume that will
always play to the right.
*Choosing values. For each existentially quantified variable
$x_i$ we will create a “diamond gadget” to allow to choose a
different state depending on the value she wants to assign to $x_i$.
From the corresponding states, will have to play $b$ or $\lnot b$,
respectively, otherwise he allows her to get to $\bot_Y$. For
universally quantified variables we have a $2$-transition path which
allows to choose $b$ or $\lnot b$ (in the second step). The right
path shown in Figure <ref> depicts this construction.
From $(iii)$ it follows that if cheats at any point during this
simulation of value choosing phase of the QSAT game, then the play
reaches $\bot_Y$ and the regret is at most $r$. Hence, we can
assume that does not cheat and the play eventually reaches
$\bot_X$. Observe that the choice of values in this gadget is made as
follows: at turn $2i$ after having entered the gadget, the value of
$x_i$ is decided.
*Clause gadgets. For every clause from $\Phi$ we create a path
in the new weighted arena such that every literal $\ell_i$ in the clause
is synchronized with the turn at which the value of $x_i$ is decided in
the value-choosing gadget. That is to say, there are $2i - 1$ states
that must be visited before arriving at the state corresponding to
$\ell_i$. At state $\ell_i$, if the value of $x_i$ corresponding to
literal $\ell_i$ is chosen, the play deterministically goes to $\bot_0$.
Otherwise, traversal of the clause-path continues.
It should be clear that if the QBF is true, then has a
value-choosing strategy such that at least one literal from every clause
holds. That means that every alternative play in the left sub-arena of
our construction has been forced into $\bot_0$ while has ensured a
discounted-sum value of $\lambda^{2n} \frac{X}{1-\lambda}$ by reaching
$\bot_X$. From $(iv)$ it follows that has ensured a regret of at
most $r$. Conversely, if has a value-choosing strategy
in the QSAT problem so the QBF is show to be false, then he can use his
strategy in the constructed arena so that some alternative path in the
left sub-arena eventually reaches $\bot_Z$. In this case, from $(ii)$ we
get that the regret value is greater than $r + \epsilon$, as expected.
§.§ Proof of Theorem <ref>
Consider a fixed weighted automaton $\calA = (Q,q_I,A,\Delta,w)$
and a discount factor $\lambda \in (0,1)$. Further, we suppose
the regret of $\calA$ is $0$.
Let us start by defining a
set of values which, intuitively, represent lower bounds on the regret
can get by resolving the non-determinism of $\calA$ on the fly.
First, let us introduce some additional notation. Define $\calA^q \defeq
(Q,q,A,\Delta,w)$, the automaton $\calA$ with new initial state $q$.
For states $q,q' \in Q$, let $\mu(q,q') \defeq \sup\left(\{
\calA^{q'}(x) - \calA^q(x) \st x \in A^\omega \} \cup
\{0\}\right)$. We are now ready to describe our set of values:
\[
M \defeq \{ |w(p,\sigma,q') - w(p,\sigma,q) + \lambda \cdot
\mu(q,q')| \st p \in Q \text{ and } q,q' \text{ are }
\sigma \text{-successors of } p \}.
\]
Note that since $\calA$ is assumed to be total (, every state-action
pair has at least one successor) then $M$ cannot be empty. Observe
that, by definition, $M$ only contains non-negative values.
Since $\calA$ has regret $0$, then we know that for all
$d \in (0,1)$, there is a strategy $\sigma_d$ of such that
$\regret{\sigma_d}{\calA}{\StrAllE, \StrWordA} = 0$. If $M \neq \{0\}$,
we let $\epsilon < \lambda^{|Q|} \cdot \left(\min M \setminus
\{0\}\right)$. Denote by $\tilde{Q}$ the set of states reachable from
$q_I$ by reading some finite word $x$ of length at most $|Q|$,$x \in
A^{\le |Q|}$, according to $\sigma_\epsilon$. If $M = \{0\}$, let
$\tilde{Q} = Q$. We now define a memoryless strategy $\sigma$ of as
follows: if $M = \{0\}$ then $\sigma$ is arbitrary, otherwise
$\sigma(p,a) = q$ implies $q \in \tilde{Q}$. To conclude, we then show
that $\sigma$ ensures regret $0$.
[state,initial above](I) at (2,6) ;
[state](A) at (0,4) ;
(B) at (2,4) …;
[state](C) at (4,4) ;
[state](D) at (0,2) ;
(E) at (2,2) …;
[state](F) at (4,2) ;
[state](G) at (2,0) $\bot_1$;
(I) edge node[el,swap]$1$ (A)
(I) edge node[el]$n$ (C)
(A) edge node[el,swap]# (D)
(C) edge node[el]# (F)
(D) edge node[el,swap]$1$ (G)
(F) edge node[el]$n$ (G)
(G) edge[loopbelow] node[el,swap]$A,1$ (G)
Clause choosing gadget for the SAT reduction. There are as many paths
from top to bottom ($\bot_1$) as there are clauses ($n$).
[state,initial above](I) at (3,6) ;
[state](A) at (1,4) $x_1$;
[state](At) at (0,2) $1_{true}$;
[state](Af) at (2,2) $1_{false}$;
[state](C) at (5,4) $x_2$;
[state](Ct) at (4,2) $2_{true}$;
[state](Cf) at (6,2) $2_{false}$;
[state](G) at (3,-1) $\bot_1$;
(I) edge node[el,swap]$1,2,3$ (A)
(I) edge node[el]$1,2,3$ (C)
(A) edge node[el,swap]# (At)
(A) edge node[el]# (Af)
(C) edge node[el,swap]# (Ct)
(C) edge node[el]# (Cf)
(At) edge node[el,swap]$1$ (G)
(Af) edge node[pos=0.4,el,swap]$2,3$ (G)
(Ct) edge node[pos=0.4,el]$1,2$ (G)
(Cf) edge node[el]$3$ (G)
(G) edge[loopbelow] node[el,swap]$A,1$ (G)
Value choosing gadget for the SAT reduction. Depicted is the
configuration for $(x_1 \lor x_2) \land (\lnot x_1 \lor x_2)
\land(\lnot x_1 \lor \lnot x_2)$.
We give a reduction from the SAT problem, i.e. satisfiability
of a CNF formula. The construction presented is based on a proof
in <cit.>. The idea is simple: given boolean formula $\Phi$ in CNF
we construct a weighted automaton $\Gamma_\Phi$ such that can
ensure regret value of $0$ with a positional strategy in $\Gamma_\Phi$
if and only if $\Phi$ is satisfiable. Note that this restriction of to positional strategies is no loss of generality. Indeed,
we have shown that if the regret of a game against an eloquent adversary
is $0$, then she has a positional strategy with regret $0$.
Let us now fix a boolean formula $\Phi$ in CNF with $n$ clauses and $m$
boolean variables $x_1,\ldots,x_m$. The weighted automaton
$\Gamma_\Phi = (Q, q_I, A, \Delta, w)$ has alphabet $A = \{bail,\#\}
\cup \{i \st 1 \le i \le n\}$. $\Gamma_\Phi$ includes an initial gadget
such as the one depicted in Figure <ref>. Recall that
this gadget forces to play into the right sub-arena. As the left
sub-arena of $\Gamma_\Phi$ we attach the gadget depicted in
Figure <ref>. All transitions shown have weight
$1$ and all missing transitions in order for $\Gamma_\Phi$ to be
complete lead to a state $\bot_0$ with a self-loop on every symbol from
$A$ with weight $0$. Intuitively, as must go to the right sub-arena
then all alternative plays in the left sub-arena correspond to either
choosing a clause $i$ and spelling $i \# i$ to reach $\bot_1$ or
reaching $\bot_0$ by playing any other sequence of symbols. The right
sub-arena of the automaton is as shown in
Figure <ref>, where all transitions shown have
weight $1$ and all missing transitions go to $\bot_0$ again. Here, from
$q_0$ we have transitions to state $x_j$ with symbol $i$ if the $i$-th
clause contains variable $x_j$. For every state $x_j$ we have
transitions to $j_{true}$ and $j_{false}$ with symbol $\#$. The idea is
to allow to choose the truth value of $x_j$. Finally, every state
$j_{true}$ (or $j_{false}$) has a transition to $\bot_1$ with symbol $i$ if
the literal $x_j$ (resp. $\lnot x_j$) appears in the $i$-th
The argument to show that can ensure regret of $0$ if and only if
$\Phi$ is satisfiable is straightforward. Assume the formula is indeed
satisfiable. Assume, also, that chooses $1 \le i \le n$
and spells $i \# i$. Since $\Phi$ is satisfiable there is a choice of
values for $x_1,\ldots,x_m$ such that for each clause
there must be at least one literal in the $i$-th clause which makes the
clause true. transitions, in the right sub-arena from $q_0$ to the
corresponding value and when plays $\#$ she chooses the correct
truth value for the variable. Thus, the play reaches $\bot_1$ and, as
$W = 1$ in the left and right sub-arenas of $\Gamma_\Phi$,
it follows that her regret is $0$. Indeed, her payoff will be
$\lambda^2/(1-\lambda)$—recall the first two turns are spent in the
initial gadget, where all transitions leading to both sub-arenas are
$0$-weighted—which is the maximal payoff obtainable in either
If does not
play as assumed then we know all plays in $\Gamma_\Phi$ reach $\bot_0$
and again her regret is $0$. Note that this strategy can be realized
with a positional strategy by assigning to each $x_j$ the choice of
truth value and choosing from $q_0$ any valid transition for all $1 \le
i \le n$.
Conversely, if $\Phi$ is not satisfiable then for every valuation of
variables $x_1,\ldots, x_m$ there is at least one clause which is not
true. Given any positional strategy of in $\Gamma_\Phi$ we can
extract the corresponding valuation of the boolean variables. Now chooses $1 \le i\le n$ such that the $i$-th clause is not satisfied by
the assignment. The play will therefore end in $\bot_0$ while an
alternative play in the left sub-arena will reach $\bot_1$. Hence the
regret of in the game is non-zero.
|
1511.00520
|
We prove the strong Suslin reciprocity law conjectured by Alexander Goncharov and describe it's corollaries for the theory of scissor congruence of polyhedra in hyperbolic space. The proof is based on the study of Goncharov conjectural description of certain rational motivic cohomology groups of a field. Our main result is a homotopy invariance theorem for these groups.
I would like to thank S. Bloch, A. Suslin and A. Goncharov for multiple invaluable discussions and suggestions.
§ INTRODUCTION
Let $F$ be an arbitrary field. For each pair of integer indices $p,q$ one can define motivic cohomology of $F$ with rational coefficients $H_{\mathcal{M}}^{p,q}(F,\mathbb{Q}).$ These groups are functorial with respect to inclusions and finite extensions of fields. For every discrete valuation of $F$ the corresponding residue maps could be defined. One of the most fundamental properties, which motivic cohomology satisfy, is the homotopy invariance property, which can be formulated via the following exact sequence:
0 \longrightarrow H_{\mathcal{M}}^{p,q}(F,\mathbb{Q}) \longrightarrow H_{\mathcal{M}}^{p,q}(F(t),\mathbb{Q}) \stackrel{\oplus \partial_P}{\longrightarrow} \bigoplus_{\substack{P \neq \infty} } H_{\mathcal{M}}^{p-1,q-1}(F_P,\mathbb{Q}) \longrightarrow 0.
Here the summation goes over all points of projective line $\mathbb{P}^1_F$, $F_P$— stands for the residue field in point $P$, and $\partial_P$ — is the residue map on motivic cohomology.
Besides that, motivic cohomology satisfies the following Suslin reciprocity law: for every smooth curve $\mathcal{X}$ over $\mathbb{C}$ the full residue map
Res \colon H_{\mathcal{M}}^{p,q}(\mathbb{C}(\mathcal{X})
,\mathbb{Q}) \longrightarrow H_{\mathcal{M}}^{p-1,q-1}(\mathbb{C},\mathbb{Q}),
which is defined as a sum of local residue maps $\partial_P$ over all points of $\mathcal{X},$ is identically zero.
There exist several general definitions of motivic cohomology groups (<cit.>), but only in few particular cases explicit presentations of these groups via generators and relations are known. In all these cases such presentations lead to connections between motivic cohomology groups of a field and more classical objects, like symbols in algebraic number theory or scissor congruence invariants of hyperbolic polytopes. In view of these connections general properties of motivic cohomology, like Suslin reciprocity law, manifest themselves in nontrivial theorems of algebraic number theory (for instance, Gauss reciprocity law), algebraic geometry (Weil reciprocity law) or hyperbolic geometry (computation of the cokernel of Dehn invariant).
Explicit description of motivic cohomology, mentioned above, is known in two cases. Firstly, one can show that for $p=q=n>0$ motivic cohomology groups of a field $F$ coincide with Milnor $K-$theory groups of a field $F$ and can be expressed via generators and relations in the following way:
H_{\mathcal{M}}^{n,n}(F,\mathbb{Q}) \cong\dfrac{\stackrel{n}{\bigwedge} F^*}{S},
where $F^*$ is the multiplicative group of a field $F$ and by $S$ we mean the subgroup, generated by so-called Steinberg elements $t_1 \wedge t_2 \wedge \ldots \wedge t_n,$ for which $t_1+t_2=1.$
To get an idea on where the connections with number theory and algebraic geometry described above come from, notice that the Hilbert symbol of two rational numbers, corresponding to some prime, is a function on $H_{\mathcal{M}}^{2,2}(\mathbb{Q}, \mathbb{Z}) \cong K^M_2(\mathbb{Q}),$ and Weil symbol of two functions on a smooth curve $\mathcal{X}$ over $\mathbb{C}$ in some point is a functional on $H_{\mathcal{M}}^{2,2}(\mathbb{C}(\mathcal{X}),\mathbb{Q}).$
Next, let us describe the second case, where there is known an explicit definition of motivic cohomology groups of a field. In <cit.> Suslin showed that for $p=1, q=2$ the motivic cohomology group coincides with so-called Bloch group. To define the latter, consider the subgroup $R_2(F)$ of the free over all points of projective line $\mathbb{Q}[\mathbb{P}^{1}(F)], $ generated by elements
\{0\}_2, \{ \infty \}_2 \qquad \mbox{and} \qquad \sum_{i=1}^{5}(-1)^{i}\{r(x_1, \ldots , \hat{x}_i, \ldots ,x_5)\}_2,
where $x_1,\ldots,x_5$ — are points of $\mathbb{P}^{1}(F),$ $r(a,b,c,d)=\dfrac{(a-b)(c-d)}{(c-b)(a-d)}$ is the four points cross-ratio and symbol $\{x\}_2$ is used to denote the generator of $\mathbb{Q}[\mathbb{P}^{1}(F)], $ corresponding to the point $x \in \mathbb{P}^{1}(F).$ The group $B_2(F)$ is defined as a following factor:
It is easy to show that the map $\delta_2 \colon \mathbb{Q}[\mathbb{P}^{1}(F)] \longrightarrow F^* \wedge F^*,$ taking the generator $\{x\}_2 \in \mathbb{Q}[\mathbb{P}^{1}(F)]$ to the Steinberg element $x \wedge (1-x),$ vanishes on $R_2(F).$ The Bloch group of $F$, which, according to Suslin's theorem, coincides with the motivic cohomology group $H_{\mathcal{M}}^{1,2}(F,\mathbb{Q}),$ is the kernel of the map
$$\delta_2 \colon B_2(F) \longrightarrow F^* \wedge F^*.$$
Alexander Goncharov has suggested hypothetical description of motivic cohomology groups of arbitrary weight, see <cit.>. In the sequel we will define the group $H_{\mathcal{G}}^{n-1,n}(F,\mathbb{Q})$, which, according to Goncharov conjectures, should coincide with $H_{\mathcal{M}}^{n-1,n}(F,\mathbb{Q}).$ We will show that these groups satisfy the homotopy invariance property and Suslin reciprocity law. As a first application we prove the strong Suslin reciprocity law, conjectured by Goncharov in <cit.>. As the other application of our results we prove that each three meromorphic functions on a smooth curve define a scissor congruence class of hyperbolic polytope with interesting properties.
§ MAIN RESULTS AND MOTIVATION
We will denote by $H_{\mathcal{G}}^{n-1,n}(F,\mathbb{Q})$ the kernel of the map
B_2(F) \otimes_a \stackrel{n-2}{\bigwedge} F^*
\stackrel{\delta_n}{\longrightarrow} \stackrel{n}{\bigwedge} F^*.
Here $B_2(F) \otimes_a \stackrel{n-2}{\bigwedge} F^*$ is the factor of the group $B_2(F) \otimes \stackrel{n-2}{\bigwedge} F^*$ by a subgroup, generated by elements $\{x\}_2 \otimes x \wedge x_1 \wedge \ldots \wedge x_{n-3}$ for some $x, x_1, \ldots , x_{n-3} \in F,$ and map $\delta_n$ is defined by the formula
\delta_n(\{x\}_2 \otimes x_1 \wedge \ldots \wedge x_{n-2})= x \wedge (1-x) \wedge x_1 \wedge \ldots \wedge x_{n-2}.
Note that the cokernel of $\delta_n$ coincides with the Milnor $K-$theory group $H_{\mathcal{M}}^{n,n}(F,\mathbb{Q}).$
Below we will explain the motivation behind this definition. In <cit.> Beilinson and Deligne suggested that for arbitrary field $F$ there should exist a Lie coalgebra $\mathcal{L}_\mathcal{M}(F),$ graded in degrees $1, 2, \ldots, $ such that
H^{p}_{[q]}(\mathcal{L}_\mathcal{M}(F))=H_M^{p,q}(F, \mathbb{Q}).
Here by symbol $H^{p}_{[q]}$ we mean the $p-$cohomology in grading $j.$ If such Lie coalgebra exists, than its $1$-component should coincide with the group $F^*$, and $2$-component — with the group $B_2(F)$, defined in the previous section. The map $\delta_2$ is the coproduct. Alexander Goncharov suggested similar description of coalgebra $\mathcal{L}_\mathcal{M}(F)$ in degree $3$. According to it, $\mathcal{L}_\mathcal{M}(F)_{3}$ coinsides with the group $B_3(F),$ which is a factor of the free group on all points of the projective line over $F$ over some subgroup $R_3$, describing functional equations for trilogarithm:
The coproduct is given by the map $\gamma$:
\gamma \colon B_3(F) \longrightarrow B_2(F) \otimes F^*,
defined on the generators $\delta(\{x\}_3)$ of $ B_3(F)$ by formula $\delta(\{x\}_3)=\{x\}_2 \otimes x.$
Consider the last three terms of the cochain complex for Lie coalgebra $\mathcal{L}_\mathcal{M}(F)$ in degree $n$:
B_3(F) \otimes \stackrel{n-3}{\bigwedge} F^* \stackrel{\gamma_n}{\longrightarrow} B_2(F) \otimes \stackrel{n-2}{\bigwedge} F^*
\stackrel{\delta_n}{\longrightarrow} \stackrel{n}{\bigwedge} F^*.
Obviously, its first cohomology group coincides with $H_{\mathcal{G}}^{n-1,n}(F,\mathbb{Q}),$ defined above.
According to the Goncharov conjectures it should coincide with $H_{\mathcal{M}}^{n-1,n}(F,\mathbb{Q}).$
Consider a field $L$ with discrete valuation $\nu$ and residue field $\overline{L}.$
Let us define the residue map
\partial_\nu \colon B_2(L) \otimes_a \stackrel{n-2}{\bigwedge} L^* \longrightarrow B_2(\overline{L}) \otimes_a \stackrel{n-3}{\bigwedge} \overline{L}^*
on element $X=\{x\}_2 \otimes x_1 \wedge \ldots \wedge x_{n-2}$
according to the following rules:
if $\nu(x) \neq 0$, we put $\partial(X)=0;$ if all the valuations of $x_i$ vanish, we also put $\partial(X)=0;$ finally, if $\nu(x_1)=1, \nu(x_2)=0,..., \nu(x_{n-2})=0,$ let
\partial(X)=\{\overline{x}\}_2 \otimes \overline{x_2} \wedge \ldots \wedge \overline{x_{n-2}}.
On the other elements of $B_2(L) \otimes_a \stackrel{n-2}{\bigwedge} L^*$ residue is extended by skew-linearity. Similarly we define the map
\partial_\nu \colon \stackrel{n}{\bigwedge} L^* \longrightarrow \stackrel{n-1}{\bigwedge} \overline{L}^*.
Finally, we get a chain map $\partial_{\nu}$:
\begin{CD}
B_2(L) \otimes_a \stackrel{n-2}{\bigwedge} L^*@>\delta>>\stackrel{n}{\bigwedge} L^*\\
@VV\partial_{\nu} V @VV\partial_{\nu} V \\
B_2(\overline{L}) \otimes_a \stackrel{n-3}{\bigwedge} \overline{L}^*@>\delta>> \stackrel{n-1}{\bigwedge} \overline{L}^* \\
\end{CD}. $$
It is not hard to see that $\partial_\nu$ induces the classical residue map on Milnor $K-$theory.
The case, when $L$ is a function field of a smooth projective curve over $F$, and $\nu$ is a discrete valuation, associated with a point $P$ of the curve will be especially important for us. In this case the residue field will be denoted by $F_P,$ and the residue map — by $\partial_P.$
The following homotopy invariance theorem is our main result:
For an arbitrary field $F$ the following sequence is exact:
0 \longrightarrow H_{\mathcal{G}}^{n-1,n}(F,\mathbb{Q}) \longrightarrow H_{\mathcal{G}}^{n-1,n}(F(t),\mathbb{Q}) \stackrel{\oplus \partial_P}{\longrightarrow} \bigoplus_{\substack{P \neq \infty} } H_{\mathcal{G}}^{n-2,n-1}(F_P,\mathbb{Q}) \longrightarrow 0.
In this theorem the injection is induced by the inclusion of field $F$ into field $F(t),$ and
the surjection is a direct sum of the residue maps in all points of $\mathbb{P}^1_F$ except the $\infty$.
The proof of this theorem occupies the rest of the paper. Now we will explain two its corollaries. We start with the strong Suslin reciprocity law.
For a smooth projective curve $\mathcal{X}$ over $\mathbb{C}$ the full residue chain map
\begin{CD}
B_2(\mathbb{C}(\mathcal{X}) \otimes_a \mathbb{C}(\mathcal{X}) ^*@>\delta_3>>\stackrel{3}{\bigwedge} \mathbb{C}(\mathcal{X}) ^*\\
@VV Res V @VV Res V \\
B_2(\mathbb{C}) @>\delta_3>>\mathbb{C}^* \wedge \mathbb{C}^* \\
\end{CD}, $$
defined as a sum of the maps $\partial_P$ over all points of the curve $\mathcal{X},$ is chain homotopy equivalent to zero. So, there exists a contracting map
h \colon \stackrel{3}{\bigwedge} \mathbb{C}(\mathcal{X}) ^* \longrightarrow B_2(\mathbb{C}),
such that $h \circ \delta =Res$ and $\delta \circ h =Res.$
For Milnor $K-$theory groups the following exact sequence was proved by Milnor:
0 \longrightarrow H_{\mathcal{M}}^{n,n}(F,\mathbb{Q}) \longrightarrow H_{\mathcal{M}}^{n,n}(F(t),\mathbb{Q}) \stackrel{\oplus \partial_P}{\longrightarrow} \bigoplus_{\substack{P \neq \infty} } H_{\mathcal{M}}^{n-1,n-1}(F_P,\mathbb{Q}) \longrightarrow 0.
This exact sequence shows the homotopy invariance of Milnor $K-$theory groups.
In <cit.> Suslin used this sequence to construct the norm map on Milnor $K-$theory of $F$ and used it to prove that for a smooth projective curve $\mathcal{X}$ over $\mathbb{C}$ the full residue map
Res \colon H_{\mathcal{M}}^{n,n}(\mathbb{C}(\mathcal{X})
,\mathbb{Q}) \longrightarrow H_{\mathcal{M}}^{n-1,n-1}(\mathbb{C},\mathbb{Q}),
Repeating Suslin's argument word by word starting with exact sequence in theorem <ref> one can show that for a smooth projective curve $\mathcal{X}$ over $\mathbb{C}$ the full residue map
Res \colon H_{\mathcal{G}}^{n-1,n}(\mathbb{C}(\mathcal{X})
,\mathbb{Q}) \longrightarrow H_{\mathcal{G}}^{n-2,n-1}(\mathbb{C},\mathbb{Q}),
These to results together are equivalent to the fact that the chain map $Res$ is homotopy equivalent to zero, since we are working with vector spaces over $\mathbb{Q}.$ From this the existence of the map $h$ with desired properties follows as well.
To formulate the next corollary, we need to recall basics of the scissor congruence theory for polytopes in the hyperbolic space $\mathbb{H}^3$. The detailed explanation could be found in <cit.>.
Let $\mathcal{P}(\overline{\mathbb{H}}^3)$ be the abelian group, generated by symbols $[T]$, corresponding to hyperbolic polytopes $T,$ possibly, with vertices on the boundary sphere of the hyperbolic space. These symbols, called the scissor congruence classes of polytopes, satisfy two types of relations. Firstly, for polytopes $T_1$ and $T_2$ with non-overlaping interiors, the class of their union $[T_1 \sqcup T_2] $ equals to the sum of $[T_1]$ and $[T_2].$ Secondly, the class of a polytope is preserved by the hyperbolic isometry.
To each complex number $z \in \mathbb{C}$ one associates a class $[z] \in \mathcal{P}(\overline{\mathbb{H}}^3),$ corresponding to a tetrahedron with vertices on the boundary of $\mathbb{H}^3$ with coordinates $\infty, 0,1, z$ in the Poincare model.
For each element of $[T] \in \mathcal{P}(\overline{\mathbb{H}}^3)$ its hyperbolic volume $Vol(T) \in \mathbb{R}$ and Dehn invariant $D(T) \in \mathbb{R} \otimes \mathbb{R}/2 \pi \mathbb{Z}$ are correctly defined. The Dehn invariant of a polytope with edges length $l_i$ and corresponding dihedral angles $\alpha_i$ is defined as a sum
D(T)=\sum l_i \otimes \alpha_i.
Informally, the second corollary of theorem <ref> claims that to each triple of invertible meromorphic functions
on a smooth Riemann surface one can associate a scissor congruence class of a hyperbolic polytope. This correspondence is antisymmetric and poly-linear. Moreover, there exists a closed formula for Dehn invariant of the obtained polytope and the volume of this polytope equals to the integral over the Riemann surface of some explicitly defined differential form. Desire to prove that such a construction exists was the primary motivation of the author's interest to the topic.
Let us consider the projection $p \colon \mathbb{C}^* \wedge \mathbb{C}^* \longrightarrow \mathbb{R} \otimes \mathbb{R}/2 \pi \mathbb{Z},$ defined by the formula
$p(z \wedge w) = z \wedge w - \overline{z} \wedge \overline{w}.$ A simple check shows that $D([z])=p(z \wedge (1-z)).$
For a smooth projective curve $\mathcal{X}$ over $\mathbb{C}$ there exists a linear map
H \colon \mathbb{C}(\mathcal{X})^* \wedge \mathbb{C}(\mathcal{X})^* \wedge \mathbb{C}(\mathcal{X})^* \longrightarrow \mathcal{P}(\overline{\mathbb{H}})
satisfying the following properties:
1. [Dehn invariant] $D \circ H = \sum \limits_{P \in \mathcal{X}} p \circ \partial_P,$
2. $H (f \wedge (1-f) \wedge g) = \sum \limits_{P \in \mathcal{X}} ord_P(g)\cdot [f(P)],$
3. [Volume] $Vol \circ H (f_1 \wedge f_2 \wedge f_3)=\frac{1}{2 \pi i} \int \limits_{\mathcal{X}(\mathbb{C})} r_2(f_1, f_2 , f_3),$
r_2(f_1, f_2 , f_3)=Alt_3\left ( \frac{1}{6} \log{|f_1|} \, d\log{|f_2|} \wedge d \log{|f_3|}-
\frac{1}{2} \log{|f_1|} \, d arg(f_2) \wedge d arg(f_3) \right ).
By corollary <ref> there exists a map
h \colon \stackrel{3}{\bigwedge} \mathbb{C}(\mathcal{X}) ^* \longrightarrow B_2(\mathbb{C}),
such that $h \circ \delta =Res$ and $\delta \circ h =Res.$
Let $p\colon B_2(\mathbb{C}) \longrightarrow \mathcal{P}(\overline{\mathbb{H}})
$ be defined on generators by the formula $p(\{z\}_2)=[z].$ It satisfies the identity
$$ \sum_{i=1}^{5}(-1)^{i}p\left (\{r(x_1, \ldots , \hat{x}_i, \ldots ,x_5)\}_2\right )=0,
because the union of some pair of polytopes $p\left (\{r(x_1, \ldots , \hat{x}_i, \ldots ,x_5)\}_2\right )$ coincides with the union of the three remaining. Let's define the map $H$ as a composition $p \circ h.$ By the properties of map $h, $ formulated in the corollary <ref>, $H$ is skew-linear and satisfies properties 1 and 2. The property 3 follows from the fact that the volume of the polytope $[z]$ is proportional to the $Bloch-Wigner$ dilogarithm of $z$ and theorem 6.10 in <cit.>.
§ PREPARATION FOR THE PROOF OF THEOREM <REF>.
First we will prove an important technical statement about the group $B(F(t)).$ Unfortunately, we have not yet found an independent prove of these results, so we have to use a result of A. Suslin from <cit.>. Let's fix some notation.
Throughout the paper, we call a polynomial "irreducible" if it is either a constant or a monic irreducible polynomial. We start with defining the notion of a degree extending that for a polynomial. Classically, degree is a partial ordering $\succ$ on the set of all irreducible polynomials over $F$. Fix any linear ordering on this set, extending the partial ordering, given by the degree. We will call an element
\[
a_1 \wedge a_2 \wedge \ldots \wedge a_n \in \stackrel{n}{\bigwedge} F(t)^*
\]
monomial, if all nonconstant $a_i$ are monic irreducible polynomials with $a_1 \succ a_2 \succ \ldots \succ a_n$. It's degree is the degree of $a_1$. Every element of $ \stackrel{n}{\bigwedge} F(t)^*$ can be presented in the unique way as an integer linear combination of monomial elements. Degree of an element in $ \stackrel{n}{\bigwedge} F(t)^*$ is the maximal degree of it's monomials.
We say that an element of $B(F(t))$ has degree $d$ if its image in $ \stackrel{2}{\bigwedge} F(t)^*$ has degree $d.$ Subgroup of elements of degree less or equal than $d$ is denoted by $B_{d}(F(t))$ or simply $B_d.$
For an irreducible polynomial $f$ of degree $d \geq 1$ and three nonzero polynomials $a, b$ and $r$ of degree less than $d$ such that $f | (ab-r)$ we denote a rational function $\frac{ab}{r}$ by $x_f(a,b).$ We will use the same symbol $x_f(a,b)$ in case when $a$ and $b$ are arbitrary nonzero elements of the field $F[t]/(f)$ meaning by that $\dfrac{\overline{a} \:\overline{b}}{r}$, where $\overline{a}$ and $ \overline{b}$ are representatives of the classes $a$ and $b$ of $F[t]/(f)$ having minimal degree. These elements have several properties, which are obvious from the definition:
1. $x_f(a,b)=x_f(b,a),$
2. $x_f(a,b)=1$ if $deg(a)+deg(b) < deg(f),$
For $d\geq 1\:\:\: B_d(F(t))$ is generated over $B(F)$ by elements of the form
$\{x_f(a,b) \}$ with $deg(f) \leq d$ and
$\left \{ \dfrac{g}{h} \right \}$ for monic irreducible polynomials $g$ and $h$ with $deg(g) =deg (h) \leq d$.
Consider the following commutative diagram with exact raws:
\begin{CD}
0 @>>> B(F) @>>> B(F(t)) @>>> B(F(t))/ B(F) @>>> 0 \\
@. @VVV @VVV @VVV @. \\
0 @>>> F^* \wedge F^* @>>> F(t)^* \wedge F(t)^* @>>> F(t)^* \wedge F(t)^*/ F^* \wedge F^* @>>> 0 \\
\end{CD}
From <cit.> it follows that $Ker(B(F) \rightarrow F^* \wedge F^*)$ is isomorphic to the $Ker(B(F(t)) \rightarrow F(t)^* \wedge F(t)^*).$ By a theorem of Matsumuto,
$$Coker(B(F) \rightarrow F^* \wedge F^*) \cong K^M_2(F)$$
$$Coker(B(F(t)) \rightarrow F(t)^* \wedge F(t)^*) \cong K^M_2(F(t)).$$
Next, it is a result of Milnor that the following sequence is exact:
\[
0 \longrightarrow K_2^M(F) \longhookrightarrow K_2^M(F(t)) \stackrel{\oplus \partial_P}{ \longtwoheadrightarrow} \bigoplus_{P \neq \infty }F(P)^* \longrightarrow 0,
\]
Applying the Snake Lemma and combining all the quoted results we deduce that the following sequence is exact:
\[
0 \longrightarrow B_2(F(t))/B_2(F) \stackrel{\delta}{\longhookrightarrow} F(t)^* \wedge F(t)^*/ F^* \wedge F^*\stackrel{\oplus \partial_P}{\longtwoheadrightarrow}
\bigoplus_{P \neq \infty } F(P)^* \longrightarrow 0.
\]
From this it follows that the lemma will hold if we prove that $Ker(\oplus \partial_P)$ is generated by elements of the form $x_f(a,b) \wedge (1- x_f(a,b))$ and $ \frac{g}{h} \wedge (1-\frac{g}{h}).$ We use induction on $d.$
Suppose that the statement is proved for $d-1$ (for $d=1$ the statement will be proved unconditionally).
Let $X$ be an arbitrary element in $Ker(\oplus \partial_P).$ It can be presented as a sum of monomials. We will show that after subtracting some number of elements $\delta_2 \{x_f(a,b) \}$ with $deg(f) = d$ and $\delta_2 \left \{ \frac{g}{h} \right \}$ with $deg(g)= deg(h)= d,$ we get an element of degree less or equal than $d-1,$ to which the induction hypothesis might be applied.
First, for all monomials in $\delta_2 (X)$ of the form $A \wedge B$ with $deg(A)=deg(B)=d$ we subtract
\[
-\delta_2 \left \{ \frac{A}{B} \right \}=A \wedge B + B \wedge (B-A) -A \wedge (B-A).
\]
After that the only monomials of degree $d$ will be of the form $f_P \wedge C,$ where $f_P$ is the uniformizer in a point $P$ and $C$ will be irreducible polynomial of degree less than $d.$ For each point $P$ the sum of the monomials, containing $f_P$ will equal to
$f_P \wedge \dfrac{C_1 C_2 \ldots C_n}{D_1 D_2 \ldots D_m},$ with $deg (C_i), deg (D_i) \leq (d-1)$ and such that
$\dfrac{C_1 C_2 \ldots C_n}{D_1 D_2 \ldots D_m}$ is congruent to $1$ modulo $f_P$. It remains to note that $x_{f_P}(a,b) \wedge (1- x_{f_P}(a,b)) = f_P \wedge a + f_P \wedge b - f_P \wedge r$ modulo monomials of degree less than $d,$ so the element $f_P \wedge \dfrac{C_1 C_2 \ldots C_n}{D_1 D_2 \ldots D_m}$ is equal to the sum of elements $\delta_2(\{x_{f_P}\})$ modulo elements of degree less than $d$. From this lemma follows.
Next, we deduce a corollary of the lemma. Fix an irreducible monic polynomial $f$ of degree $d \geq 1.$ By $F(t)^*_f$ we denote the subgroup of $F(t)^*$, generated by element $f$ and polynomials of degree less than $d$. By $B_f(F(t))$ we denote the subgroup of $B(F(t))$, consisting of elements with image in $F(t)^*_f \wedge F(t)^*_f.$ Note that $B_{d-1}(F(t)) \subset B_f(F(t)).$
$B_f(F(t))$ is generated over $B_{d-1}(F(t))$ by elements of the form $x_f(a,b).$
The statement follows from the proof of lemma <ref>.
Next, we formulate three main computations that we will prove in the last section of the paper.
For a monic irreducible polynomial $P$ of degree $d \geq 1$ and arbitrary $G_1,G_2,G_3 \in F[t]$ nonzero modulo $P$ the following linear combination of elements in $B_{d}(F(t))$ lies in $B_{d-1}(F(t))$
\[
\{ x_P(G_1,G_2) \} - \{ x_P(G_1 \cdot G_3, G_2 ) \} - \{ x_P(G_1,G_3) \} + \{ x_P(G_1 \cdot G_2, G_3) \} \in B_{d-1}(F(t)) .
\]
For a monic irreducible $P$ of degree $d \geq 1,$ and arbitrary polynomials $F_1, F_2, G_1, G_2 \in F[t]$ of degree less than $d$ nonzero modulo $P$ the element below lies in $B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*$
\[
\{ x_P(F_1, F_2)\} \otimes x_P(G_1, G_2) + \{ x_P(G_1, G_2)\} \otimes x_P(F_1, F_2) \in B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*.
\]
For a monic irreducible polynomial $P$ and arbitrary rational function $X \in F(t)$, such that it is integer with respect to a valuation in $P$ denote the residue of $X$ in $F(t)/(P)$ by $\overline{X}.$
Let $P$ be a monic irreducible polynomial of degree $d \geq 1.$ Let $R_1$, $R_2$ be arbitrary nonzero elements of $F(P).$ Define recursively $R_{i+1} = \frac{1-R_i}{R_{i+1}}.$ Note that $R_{i+5}=R_{i}.$ Then
\[
\sum_{i=1}^{i=5} \left\{ \overline{R_i}
\right\} \otimes P +
\sum_{i=1}^{i=5} \left\{ \dfrac{\overline{R_{i-1}} \: \overline{R_{i+1}}}{\overline{1-R_i}} \right\} \otimes \overline{R_i}
\in B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*.
\]
§ LOCAL PART OF THE PROOF OF THEOREM <REF>.
In this section we begin to prove theorem <ref>.
Consider group $\bigwedge F(t)^*.$ Let's denote by $L_d$ its subgroup, generated by monomials of degree less or equal than $d$. Obviously, $\bigwedge F(t)^* = \bigcup L_d$ and $\bigwedge F^* =L_0$.
We additionally put $L_{-1}=0.$
By $L_d^1$ we denote a subgroup of $L_d$, generated by monomials with not more than one term of degree exactly $d.$ Since we are working with vector spaces over $\mathbb{Q}$,
$$\bigwedge F(t)^* = \bigcup_{d=0}^{\infty} \left( L_d/L_d^1 \oplus L_d^1 /L_{d-1} \right ). $$
It is not hard to find free basis for vector spaces $L_d/L_d^1$ and $ L_d^1 /L_{d-1} .$ The first one is generated by monomials $f_1 \wedge f_2 \wedge ... \wedge f_k$ with irreducible $f_i$ such that $f_1 \succ f_2 \succ \ldots \succ f_k$ and $deg(f_1)=deg(f_2)=d.$
The second one — by monomials $f \wedge u_1 \wedge \ldots \wedge u_k$ with irreducible $f$ and $u_i$ such that $deg(f)=d$, $deg(u_i) < d$ and $u_1\succ u_2 \succ \ldots \succ u_k.$ Consider the following map:
\[
\partial_d^1 : L_d \longrightarrow \bigoplus_{deg(P)=d } \bigwedge F(P)^*.
\]
Obviously, it vanishes on $L_{d-1}.$ Denote it's kernel by $X_{d}$ and the kernel of its restriction to $L_d^1$ by $X^1_d.$ Group $L^1_d/L_{d-1}$ and $X^1_d/L_{d-1}$ are direct sums of there subgroups, generated by monomials with the only term of degree $d$ coinciding with uniformizer $f_P$ for some point $P$ of degree $d$. We denote the summands $L_P$ and $X_P$ respectively. So we have the following decompositions:
\[
L^1_d/L_{d-1}= \bigoplus_{deg(P)=d} L_P/L_{d-1}, \quad X^1_d/L_{d-1}= \bigoplus_{deg(P)=d} X_P/L_{d-1} .
\]
Next, we introduce similar filtration on the group $B(F(t)) \otimes_{a} \bigwedge F(t)^*.$ We put
\[
M_d := B_d \otimes_a L_d, \quad M_P:= B_P \otimes_a L_P. \quad M_d^1:= \sum_{deg(P)=d} M_P.
\]
Looking at the image of $\delta$ one can deduce the following direct sum decomposition:
\[
M_d^1/M_{d-1}:= \bigoplus_{deg(P)=d} M_P/M_{d-1}.
\]
Obviously, the following map vanishes on $M_{d-1}$:
\[
\partial_d^2 : M_d \longrightarrow \bigoplus_{deg(P)=d } B(F(P)) \otimes_a \bigwedge F(P)^*.
\]
Denote its kernel by $Y_d$, kernel of its restriction to $M_{d}^1$ by $Y_d^1$ and on $M_P$ by $Y_P.$ Now we have introduced all the notation to formulate the proposition, which is the main goal of this section:
The map $\delta$ is an isomorphism between $Y_d^1/M_{d-1}$ and $X_d^1/L_{d-1}.$
Because of the direct sum decompositions
\[
X^1_d/L_{d-1}= \bigoplus_{deg(P)=d} X_P/L_{d-1} , \:\:\:\:Y^1_d/M_{d-1}= \bigoplus_{deg(P)=d} Y_P/M_{d-1},
\]
it is enough to prove that the map $\delta$ induces isomorphism between $X_P/L_{d-1}$ and $Y_P/M_{d-1}. $
These groups can be put in the following diagram:
\begin{CD}
0 @>>> Y_P/M_{d-1} @>>> M_P/M_{d-1} @>>> B(F(P)) \otimes_{a} \bigwedge F(P)^* @>>> 0\\
@. @VV \delta V @VV \delta V @VV \delta V @. \\
0 @>>> X_P/L_{d-1} @>>> L_P/L_{d-1} @>>> \bigwedge F(P)^* @>>>0\\
\end{CD}
To show that $X_P/L_{d-1}$ and $Y_P/M_{d-1} $ are isomorphic we first find generators and relations for $X_P/L_{d-1} $, and then the set of generators of $Y_P/M_{d-1} $, whose image coincides with the generators of $X_P/L_{d-1}$. We finish the proof of the proposition, showing that generators of $Y_P/M_{d-1} $ satisfy the same relations, as generators of $X_P/L_{d-1}$.
It is not hard to see that elements
\[
x_P(a,b) \wedge f \wedge u_1 \wedge \ldots \wedge u_n,
\]
such that $a,b \in F(P)$ and $deg(u_i) <d,$ lie in $X_P/L_{d-1}$. Indeed,
$\partial_P(x_P(a,b) \wedge f \wedge u_1 \wedge \ldots \wedge u_k)=-\overline{x_P(a,b)} \wedge \overline{u_1} \wedge \ldots \wedge \overline{u_k},$
which vanishes in the group $ \bigwedge k(P)^* $, since $x_P(a,b)=\dfrac{ \overline{a}\: \overline{b}}{ \overline{ab}} \cong_f 1.$ It is also not hard to find some relations that they satisfy. To begin with, these elements are symmetric in first two variables, multilinear and antisymmetric in the last $n.$ Next, if $deg(a)+deg(b)<d$, then they vanish. The last two relations are a bit more interesting. From the cocycle equation $ x_P(a,b)x_P(ab,c)=x_P(a,c)x_P(ac,b)$ it follows that
\[
x_P(a,b)\wedge f \wedge u_1 \wedge \ldots \wedge u_n +x_P(ab,c)\wedge f \wedge u_1 \wedge \ldots \wedge u_n-
\]
\[
x_P(a,c)\wedge f \wedge u_1 \wedge \ldots \wedge u_n -x_P(ac,b)\wedge f \wedge u_1 \wedge \ldots \wedge u_n=0.
\]
The most involved one recovers the symmetry between first variable and the others:
\[
x_P(a,b)\wedge f \wedge \overline{c} \wedge u_2 \ldots \wedge u_n + x_P(a,b)\wedge f \wedge \overline{d} \wedge u_2 \ldots \wedge u_n-
\]
\[
-x_P(a,b)\wedge f \wedge \overline{cd} \wedge u_2 \ldots \wedge u_n + x_P(c,d)\wedge f \wedge \overline{a} \wedge u_2 \ldots \wedge u_n-
\]
\[
x_P(c,d)\wedge f \wedge \overline{b} \wedge u_2 \ldots \wedge u_n + x_P(c,d)\wedge f \wedge \overline{ab} \wedge u_2 \ldots \wedge u_n=0.
\]
We want to prove that these elements indeed generate $X_P/L_{d-1}$ and that all relations between them follow from those described above. It will be a corollary of the following two lemmas.
Suppose that $d>0.$ We remind that $ F_{d-1}(t)^*$ is the subgroup of $ F(t)^*$, generated by irreducible polynomials of degree less than $d$. We introduce specialization map $s_P \colon F_{d-1}(t)^* \longrightarrow F(P)^*$ in the usual way:
$$s_P(x)=\partial_P(f \wedge x).$$
The kernel of the map $F_{d-1}(t)^*\stackrel{s_P}{\longrightarrow} F(P)^*$
is generated by symbols $y(a, b) $ for $a, b \in F(P)^*,$
satisfying the following three relations:
\[
y(a,b)=0$ if $deg(\overline{a})+deg(\overline{b}) < d,
\]
\[
y(a, b)=y(b, a),
\]
\[
y(a, b)+y(ab, c) -y(a, c) -y(ac, b)=0.
\]
Image of $y(a,b)$ in $ F_{d-1}(t)^* $ is $\overline{a} + \overline{b} - \overline{ab} .$
Let $A$ be an arbitrary uniquely divisible abelian group. Consider the last three terms of the Bar complex for $A$ with coefficients in $\mathbb{Q}$:
\mathbb{Q}[A] \otimes \mathbb{Q}[A] \otimes \mathbb{Q}[A] \stackrel{d_3}{\longrightarrow} \mathbb{Q}[A] \otimes \mathbb{Q}[A] \stackrel{d_2}{\longrightarrow} \mathbb{Q}[A]/\mathbb{Q}.
It is known that
\dfrac{Ker(d_2)}{Im(d_3)} \cong H_2(A, \mathbb{Q}) \cong \bigwedge^2 A, \quad \dfrac{ \mathbb{Q}[A]}{<Im(d_3),\mathbb{Q}>} \cong H_1(A, \mathbb{Q}) \cong A.
From this it follows that the sequence below is exact:
0\longrightarrow \bigwedge^2 A \longhookrightarrow \dfrac{\mathbb{Q}[A] \otimes \mathbb{Q}[A]}{Im(d_3)} \longrightarrow \dfrac{\mathbb{Q}[A]}{\mathbb{Q}[1]}\longtwoheadrightarrow A \longrightarrow 0.
In this sequence the first map sends $a \wedge b$ to $a \otimes b -b \otimes a.$
Now, let $A=F(P)^*.$ Then, $ F_{d-1}(t)^* \cong \mathbb{Q}[A]/I,$
where $I$ is a subgroup, generated by elements $[a]+[b]-[ab]$ with $deg(a)+deg(b) <d.$ From the exact sequence above it follows that
$$Ker(s_P) \cong \dfrac{\mathbb{Q}[F(P)^*] \otimes \mathbb{Q}[F(P)^*]}{<Im(d_3),Im(\bigwedge^2 F(P)^*),I>}. $$
This is nothing but the statement of the lemma.
Next we need an additional linear algebra lemma:
Let $A, B$ and $C$ be three vector spaces over $\mathbb{Q}$, such that
\[
0 \longrightarrow A\stackrel {i}{\longhookrightarrow} B \stackrel {p}{\longrightarrow} C \longtwoheadrightarrow 0.
\]
Then, the following sequence is exact:
\[
A \otimes A \otimes \bigwedge B \stackrel {q}{\longrightarrow} A \otimes \bigwedge B \stackrel {i \otimes id}{\longrightarrow} \bigwedge B \stackrel {\bigwedge p}{\longtwoheadrightarrow} \bigwedge C \longrightarrow 0.
\]
Here the map $q$ is defined on generators by the formula
\[
q(a_1 \otimes a_2 \otimes b)= a_1 \otimes i(a_2) \wedge b + a_2 \otimes i(a_1) \wedge b.
\]
Let $X$ be a complex $ A\stackrel {i}{\longrightarrow} B.$ By Kunneth's theorem, complex
$Tot(\bigotimes \limits_{i=1}^{n} X)$ will have only top homology group, isomorphic to $\bigotimes \limits_{i=1}^{n} C.$
Symmetric group $S_n$ acts on $Tot(\bigotimes \limits_{i=1}^{n} X)$ by permuting summands and multiplying on sign.
Since $S_n$ is finite, functor of taking coinvariants of $S_n$ is exact. The desired exact sequence is nothing but the truncated complex
\[
Tot(\bigotimes_{i=1}^{n} X)^{S_n} \longrightarrow \stackrel {n}{\bigwedge} C.
\]
Using this lemma we can find generators and relations for $X_P/L_{d-1}.$
The group $X_P/L_{d-1}$ is a free group on generators
$T_f (a,b,u_1,...,u_n)$ for $a, b \in L$ and monic irreducible polynomials $u_i$ of degrees less than $d,$ factorized over the following four relations:
${\bf (R1)} \quad T_f (a,b,u_1,\ldots,u_n)=0 ,$ if $deg(\overline{a})+deg(\overline{b})<d,$
${\bf (R2)} \quad T_f (a,b,u_1,\ldots,u_n)$ is symmetric in the first two variables and antisymmetric in the last $n.$
${\bf(R3)} \quad T_f (a,b,u_1,\ldots,u_n) +T_f (ab,c,u_1,\ldots,u_n)-$
$-T_f (a,c,u_1,\ldots,u_n)-T_f (ac,b,u_1,\ldots,u_n)=0$
${\bf (R4)} \quad T_f (a, b,c, u_2,\ldots,u_n) + T_f (a , b, d,u_2,\ldots,u_n) - $
$-T_f (a, b ,\overline{cd},u_2,\ldots,u_n) +T_f (c, d ,a,u_2,\ldots,u_n) +$
$+ T_f (c, d ,b,u_2,\ldots,u_n)) - T_f (c , d ,\overline{ab},u_2,\ldots,u_n)=0.$
The image of $T_f (a,b,u_1,\ldots,u_n)$ in $X_P/L_{d-1}$ equals to
$x_f(a,b) \wedge f \wedge u_1 \wedge \ldots \wedge u_n. $
The group $X_P/L_{d-1}$ is isomorphic to the kernel of the map
$$f \wedge \bigwedge k_{d-1}(t) \longrightarrow \bigwedge F(P)^*,$$
so the presentation by generators and relations follows from lemmas <ref> and <ref>.
For simplicity we will identify symbols $T_f $ with their image in the group $X_P/L_{d-1} $. Our next step is to find elements in the group $Y_P/M_{d-1} $, whose image under $\delta$ equals to $T_f (a,b,u_1,\ldots,u_n)$ and prove that they generate $Y_P/M_{d-1}. $ But this is not hard: it is enough to take elements
$$\{x_f(a,b) \} \wedge u_1 \wedge \ldots \wedge u_n.$$
We finish this section with the proof of <ref>:
Let's denote elements $\{x_f(a,b) \} \wedge u_1 \wedge \ldots \wedge u_n$ by $\widehat{T}_f (a,b,u_1,\ldots,u_n).$
Suppose that $\overline{a}\overline{b}=f q+\overline{ab}.$ Then,
\[
\delta \left( \widehat{T}_f (a,b,u_1, \ldots ,u_n) \right )=\delta \left( \widehat{T}_f (\{x_f(a,b) \} \wedge u_1 \wedge \ldots \wedge u_n) \right )=
\]
\[
=x_f(a,b) \wedge f \wedge u_1 \wedge \ldots \wedge u_n+x_f(a,b) \wedge \dfrac{q}{\overline{ab}} \wedge u_1 \wedge \ldots \wedge u_n.
\]
Since the second summand vanishes in $X_P/L_{d-1},$ we have shown that $\delta(\widehat{T}_f (a,b,u_1,\ldots,u_n))=T_f (a,b,u_1,\ldots,u_n).$
It remains to prove that elements $\widehat{T}_f$ generate $Y_P/M_{d-1} .$ We denote by $K$ the group, generated by these elements. Obviously, $K \subset Y_P/M_{d-1} .$ To prove the opposite inclusion we construct a map $r \colon B(F(P)) \otimes_a \bigwedge F(P)^* \longrightarrow M_P/\left ( M_{d-1} + K\right ),$ inverse to $\partial_P.$
It is defined by the following formula:
$$r(\{w\} \otimes v_1 \wedge \ldots \wedge v_n)=\{\overline{w}\} \otimes \overline{v_1} \wedge \ldots \wedge \overline{v_n} \wedge{f_P}.$$
At first, we need to show that it is defined correctly, that it satisfies the five-term relation in the first variable and is linear in other variables. Linearity is easier: since
\[
\{a\}\otimes b \otimes (1-b) = \{b\}\otimes a \otimes (1-a),
\]
\[
\{\overline{w}\} \otimes \overline{v_1} \wedge f + \{\overline{w}\} \otimes \overline{v_2} \wedge f -\{\overline{w}\} \otimes \overline{v_1 v_2} \wedge f = \{\overline{w}\} \otimes \dfrac{\overline{v_1} \: \overline{v_2} }{\overline{v_1 v_2} } \wedge f=
\]
\[
=\{\overline{w}\} \otimes x_f(v_1,v_2) \wedge (1-x_f(v_1,v_2) )=\{x_f(v_1,v_2) \} \otimes \overline{w} \wedge (1-\overline{w}) \in K.
\]
The five-term relation follows from lemma <ref>.
Since the composition $\partial_P \circ r$ trivially equals to the identity, it only remains to prove that $r \circ \partial_P$ equals to the identity, which can be shown for each generator of $M_P$ separately. By corollary <ref>, we need to show it only for two types of elements:
$\left \{ \dfrac{g_1}{g_2} \right \} \wedge f \wedge u_1 \wedge \ldots \wedge u_n$ with $deg(g_1) = deg(g_2), deg(u_i) < d$ and
$\left \{ x_g(a,b) \right \} \wedge f \wedge u_1 \wedge \ldots \wedge u_n$ with $deg(g), deg(u_i) < d.$
For both we apply the five-term relation in the same way. We check the claim only for the first type of elements. If $\dfrac{g_1}{g_2} \equiv h,$
\[
\left \{ \dfrac{g_1}{g_2} \right \} \wedge f-r(\left \{ h \right \} )=\left \{ \dfrac{g_1}{g_2} \right \} \wedge f- \left \{ h \right \} \wedge f=
\]
\[
=\left \{ \dfrac{g_1}{g_2 \: h} \right \} \wedge f-\left \{ \dfrac{g_2 - g_1}{g_2 ( 1-h)} \right \} \wedge f+ \left \{ \dfrac{(g_2 - g_1)h}{g_1 ( 1-h)} \right \} \wedge f.
\]
It is easy to see that all three terms are in $K.$ After similar check for the second type of the elements above we conclude that $\widehat{T}_f$ generate $Y_P/M_{d-1} .$ This finishes the proof of the proposition.
§ GLOBAL PART OF THE PROOF OF THEOREM <REF>.
The goal of this section is the following proposition:
The map $\delta$ is an isomorphism between $M_d/M_{d}^1$ and $L_d/L_{d}^1.$
To show the claim we introduce additional increasing filtrations $M_d^k$ and $L_{d}^k.$
The group $L_{d}^k$ is generated by monomials $f_1 \wedge f_2 \wedge \ldots \wedge f_s \wedge u_1 \wedge \ldots \wedge u_t,$ for $s \leq k$, monic irreducible $f_i$ of degree $d$ and $u_j$ of degree less than $d$. Monomials with $f_1 \succ \ldots \succ f_s \succ\ldots \succ u_1\succ \ldots \succ u_t$ form a basis. $M_{d}^k$ is a subgroup of $M_d$, generated by monomials with image in $L_{d}^k.$ The proposition will follow from the next two lemmas.
$M_d^2/M_d^1$ is isomorphic to $L_d^2/L_d^1$.
By lemma <ref> the group $M_d^2/M_d^1$ is generated by elements of three types:
\[
\{x_f(a,b)\} \otimes g \wedge u_1 \wedge \ldots \wedge u_l,
\]
\[
\{u\} \otimes f \wedge g \wedge u_1 \wedge \ldots \wedge u_l,
\]
\[
\left \{\frac{g}{f} \right \} \otimes u_1 \wedge \ldots \wedge u_l.
\]
Here we suppose that $f, g$ are monic irreducible of degree $d$, $u_j$ and $u$ have degree less than $d$ and $f \succ g\succ u\succ u_1 \succ \ldots \succ u_l$. To prove the lemma we need to show that $M_d^2/M_d^1$ is generated by elements of the third type only. Indeed,
\[
\delta \left ( \left \{ \dfrac{g}{f} \right \} \otimes u_1 \wedge \ldots \wedge u_l \right )=\dfrac{g}{f} \wedge \left(1- \dfrac{g}{f} \right) \wedge u_1 \wedge \ldots \wedge u_l)=f\wedge g \wedge u_1 \wedge \ldots \wedge u_l,
\]
so these elements are mapped to the basis of $L_d^2/L_d^1$.
For elements of the second type the result follows from the identity
\[
\{u\} \otimes f\wedge g=-\{u\} \otimes \dfrac{f}{g} \wedge \dfrac{g-f}{g}= \left \{ \dfrac{f}{g} \right \} \otimes (1-u)\wedge u.
\]
So, it remains to prove the claim for elements of the first type. For this, let's denote by $p$ residue of $x_f(a,b)$ modulo $g.$ By the five-term relation
\[
\{x_f(a,b)\} \otimes g = p \otimes g
- \left \{ \dfrac{ab}{rp} \right \}\otimes g + \left \{ \dfrac{fq}{r(1-p)} \right \} \otimes g - \left \{ \dfrac{fqr}{ab(1-r)} \right \}\otimes g.
\]
We claim that each of these summands lies in the subgroup, generated by $ \left \{ \dfrac{f}{g} \right\} \otimes L_{d-1}$ and $M_d^1.$ For the first summand this is obvious. The proof for the other three summands is similar, we will show it for $\left \{ \dfrac{ab}{rp} \right \} \otimes g $. Let $rp-ab=gs,$ where degree of $s$ is less than $d.$ We have
\[
\left \{ \dfrac{ab}{rp} \right \}\otimes g= \left \{ \dfrac{rp-ab}{rp} \right \}\otimes g= \left \{ \dfrac{gs}{rp} \right \}\otimes g=- \left \{ \dfrac{gs}{rp} \right \}\otimes \dfrac{s}{rp}.
\]
By lemma <ref>, $ \left \{ \dfrac{gs}{rp} \right \}- \left \{ \dfrac{f}{g} \right \} \in M_d^1,$ from which the claim follows.
$M_d^k/M_d^{k-1}$ is isomorphic to $L_d^k/L_d^{k-1}$ for $k>2.$
Using the arguments from the proof of the previous lemma, we see that $M_d^k/M_d^{k-1}$ is generated by elements of the form
\[
\left \{ \dfrac{g}{f} \right \} \otimes h \wedge f_1 \wedge \ldots \wedge f_s \wedge u_1 \wedge \ldots \wedge u_l.
\]
We may suppose that $h\succ f_1\succ f_2 \succ \ldots \succ f_s \succ u_1 \succ \ldots \succ u_l.$ To prove the claim it remains to show that we can interchange $f, g$ and $h$ modulo $M_d^{k-1}$ in arbitrary way. Interchanging $f,g$ is not a problem.
We claim that $\left \{ \dfrac{f}{g} \right \} \otimes h - \left \{ \dfrac{f}{h} \right \} \otimes g \in M_d^{k-1}$. Indeed, let $p$ be a polynomial of degree less than $d$, such that $h|(f-pg),\:pg+hq=f.$ Then
\[
\left \{ \dfrac{f}{g} \right \} \otimes h = p \otimes h
- \left \{ \dfrac{f}{gp} \right \}\otimes h + \left \{ \dfrac{g-f}{g(1-p)} \right \}\otimes h - \left \{ \dfrac{(g-f)p}{f(1-p)} \right \}\otimes h=
\]
\[
=\left \{ \dfrac{f}{gp} \right \}\otimes g - \left \{ \dfrac{g-f}{g(1-p)} \right \} \otimes g + \left \{ \dfrac{(g-f)p}{f(1-p)} \right \}\otimes f=- \left \{ \dfrac{f}{g} \right \} \otimes g + \left \{ \dfrac{(g-f)p}{f(1-p)} \right \}\otimes \dfrac{f}{g}=
\]
\[
= \left \{ \dfrac{(g-f)p}{f(1-p)} \right \}\otimes g= \left \{ \dfrac{f}{h} \right \} \otimes g.
\]
Putting these two lemmas together we get the proposition.
The third proposition we need for the homotopy invariance theorem follows from the proof of the Milnor's theorem.
Let $z$ be an element of $M_{d+1}$, such that $\delta(z) \in L_d.$ Then, there exists an element $z_1 \in M_d$ such that $\delta(z) = \delta(z_1).$
We will show that
L_d/(L_{d-1}+M_d) \cong \bigoplus_{deg(P)=d } K^M(F(P)).
From this the result will follow, since $\delta(z)$ vanishes in $K^M(F(x))$, so do all its residues.
To prove the isomorphism, we construct the inverse map to the residue map $\oplus \partial_P.$ For each $P$ this map is defined by the formula $s_P(u_1\wedge...\wedge u_k)=f_P \wedge \overline{u_1}\wedge...\wedge \overline{u_k}.$ Simple check shows that it is linear and satisfies Steinberg relation. To show that it is, indeed, inverse, note that by proposition <ref> $L_d/(L_{d-1}+M_d) \cong L_d^1/(L_{d-1}+M_d)$ and that the composition is, obviously, trivial on $L_d^1.$
§ THE END OF THE PROOF OF THEOREM <REF>.
In this section we finish the proof of the theorem <ref>.
Consider the following diagram:
\begin{CD}
\left ( B(F(t)) \otimes_{alt} \bigwedge F(t)^* \right )/ \left ( B(F) \otimes_{alt} \bigwedge F^* \right )@> \delta> >\bigwedge F(t)^*/ \bigwedge F^* \\
@VV\oplus \partial_P^2 V @VV\oplus \partial_P^1 V \\
\bigoplus_{P \neq \infty } B(F(P))\otimes_{alt} \bigwedge F(P)^* @>\delta >> \bigoplus_{P \neq \infty } \bigwedge F(P)^* \\
\end{CD}
To prove that $\oplus \partial_P^2$ is quasi-isomorphism we need to show that $\oplus \partial_P^2$ is an isomorphism between the kernels of the horizontal maps.
First, we show that it is injective. For this, take $x$ such that $\oplus \partial_P^2(x)=0$ and $\delta(x)=0.$ Let's take the minimal $d$ such that $x \in M_d$. By proposition <ref>, $x \in M_d^1.$ So, $x\in Y_d^1.$ After that, applying proposition <ref> we get that $x \in M_{d-1}$, which contradicts minimality of $d.$
To show surjectivity, we can take arbitrary point $P$ and element $y \in B_2(F(P))\otimes_{alt} \bigwedge F(P)^*$ such that $\delta(y)=0.$ Consider the diagram
\begin{CD}
M_P/ M_{d-1}@> \delta> >L_P/L_{d-1} \\
@VV \partial_P^2 V @VV \partial_P^1 V \\
B(F(P))\otimes_{alt} \bigwedge F(P)^* @>\delta >> \bigwedge F(P)^* \\
\end{CD}
Obviously, $\partial_P^2 $ and $\partial_P^1 $ are surjective, so from proposition <ref> its raws are quasi-isomorphic. So, there exist $z \in M_P$ with $\delta(z)\in L_{d-1}$ and $\partial_P^2(z)=y.$ Application of the proposition <ref> finishes the proof.
§ PROOFS OF THE LEMMAS
For a monic irreducible polynomial $P$ of degree $d \geq 1$ and arbitrary $G_1,G_2,G_3 \in F[t]$ nonzero modulo $P$ the following linear combination of elements in $B_{d}(F(t))$ lies in $B_{d-1}(F(t))$
$$\{ x_P(G_1,G_2) \} - \{ x_P(G_1 \cdot G_3, G_2 ) \} - \{ x_P(G_1,G_3) \} + \{ x_P(G_1 \cdot G_2, G_3) \} \in B_{d-1}(F(t)) .$$
We can suppose that $G_1,G_2, G_3 \in F[t]$ and $deg(G_1), deg(G_2), deg(G_3) \leq d-1$.
Let's apply division algorithm and find $Q_{1,2}, R_{1,2},Q_{1,3}, R_{1,3}, Q_{12,3}, R_{12,3}, Q_{13,2} , R_{13,2} \in F[t]$ with $deg(R_{1,2}), deg(R_{1,3}), deg(R_{12,3}), deg(R_{13,2}) \leq d-1$ such that
$G_1 \cdot G_2= Q_{1,2} \cdot P+R_{1,2}, \:\:\:G_1 \cdot G_3= Q_{1,3}\cdot P+R_{1,3},$
$R_{1,2} \cdot G_3= Q_{12,3}\cdot P+R_{12,3}, \:\:\:R_{1,3} \cdot G_2= G\cdot P+R_{13,2}.$
Obviously, $deg(Q_{1,2})= deg(G_1 \cdot G_2- R_{1,2}) - deg(P) \leq d-1.$ Similarly, $deg(Q_{1,3}), deg(Q_{12,3}),deg(Q_{13,2} ) \leq d-1.$
Computing $G_1 \cdot G_2 \cdot G_3 $ in two ways we get that
$ (Q_{1,2}P+R_{1,2})\cdot G_3=(Q_{1,2} G_3+Q_{12,3})\cdot P+ R_{12,3}$ and $(Q_{1,3} P+R_{1,3}) \cdot G_2=(Q_{1,3} G_2+Q_{13,2} )\cdot P + R_{13,2},$
so $R_{12,3}=R_{13,2}.$
Next, we compute in two ways $R_{1,3} \cdot G_1\cdot G_2$ and get that
$(Q_{1,2} P+R_{1,2})R_{1,3} = G_1\cdot (Q_{13,2} P+R_{13,2}),$ so $(Q_{1,2} R_{1,3}-G_1 Q_{13,2} )\cdot P = G_1R_{13,2}-R_{1,2} R_{1,3}.$ So, $deg(Q_{1,2} R_{1,3}-G_1 G) \leq d-1,$
We need three more computations: $Q_{1,2} R_{13,2}-R_{1,2} Q_{13,2} =G_2\cdot (Q_{1,2} R_{1,3} -G_1 Q_{13,2} ), \:\:\:R_{1,2} Q_{1,3}-Q_{12,3} G_1= Q_{1,2} R_{1,3} -G_1Q_{13,2} , \:\:\:Q_{1,3} R_{12,3} -R_{1,3} Q_{12,3}= G_3\cdot (R_{1,2} Q_{1,3}-Q_{12,3} G_1).$
We apply five-term relationship twice:
$$ \left \{\dfrac{G_1G_2}{R_{1,2}} \right \} - \left \{\dfrac{R_{1,3}G_2}{R_{13,2}} \right \}= - \left \{\dfrac{R_{1,3}R_{1,2}}{G_1R_{13,2}} \right \} + \left \{\dfrac{R_{1,3} \cdot (G_1 G_2 - R_{1,2})}{G_1(R_{1,3} G_2-R_{13,2})} \right \} -\left \{\dfrac{(R_{1,2}-G_1G_2)R_{13,2}}{R_{1,2} \cdot (R_{13,2}-R_{1,3}G_2)} \right \}, $$
$$\left \{\dfrac{R_{1,2} G_3}{R_{12,3}} \right \} - \left \{\dfrac{G_1 G_3}{R_{1,3}} \right \}= \left \{\dfrac{R_{1,3} R_{1,2}}{G_1R_{12,3}} \right \} -\left \{\dfrac{R_{1,2} \cdot(G_1 G_3 - R_{1,3})}{G_1\cdot (R_{1,2}G_3-R_{12,3})} \right \} + \left \{\dfrac{(R_{1,3}-G_1G_3)R_{12,3}}{R_{1,3}\cdot (R_{12,3}-R_{1,2}G_3)} \right \}.$$
Adding this two expressions, we get that
$ \{ x_P(G_1,G_2) \} - \{ x_P(G_1 \cdot G_3, G_2 ) \} - \{ x_P(G_1,G_3) \} + \{ x_P(G_1 \cdot G_2, G_3) \}= $
$=\left \{\dfrac{Q_{1,2}R_{1,3}}{Q_{13,2} G_1} \right \} - \left \{\dfrac{Q_{1,2} R_{13,2}}{R_{1,2} Q_{13,2} } \right \} + \left \{\dfrac{Q_{1,3} R_{12,3}}{R_{1,3} Q_{12,3}} \right \} - \left \{\dfrac{Q_{1,3}R_{1,2}}{G_1 Q_{12,3}} \right \}.$
All four terms are elements of $\left [ B(F(t)) \right ]_{d-1}.$ Let's check it, for example, for $\left \{\dfrac{Q_{1,2}R_{1,3}}{Q_{13,2} G_1} \right \}.$
Indeed, from the computations above we know that $deg(Q_{1,2}), deg(R_{1,3}),deg(Q_{13,2} ),deg(G_1),deg(Q_{1,2}R_{1,3}- Q_{13,2} G_1) \leq d-1. $
For a monic irreducible $P$ of degree $d \geq 1,$ and arbitrary polynomials $F_1, F_2, G_1, G_2 \in F[t]$ of degree less than $d$ nonzero modulo $P$ the element below lies in $B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*$
$$ \{ x_P(F_1, F_2)\} \otimes x_P(G_1, G_2) + \{ x_P(G_1, G_2)\} \otimes x_P(F_1, F_2) \in B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*.$$
Throughout the prove we work with elements in $B_{P}(F(t)) \otimes_{a} F(t)_{P}^*$ modulo $B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*.$ Let's introduce notation:
$$S[ F_1, F_2 ; G_1 , G_2 ] = \left\{ \dfrac{\overline{F_1} \cdot \overline{F_2}}{\overline{F_1 F_2}} \right\} \otimes \dfrac{\overline{G_1} \cdot \overline{G_2}}{\overline{G_1 G_2}} + \left\{ \dfrac{\overline{G_1} \cdot \overline{G_2}}{\overline{G_1 G_2}} \right\} \otimes \dfrac{\overline{F_1} \cdot \overline{F_2}}{\overline{F_1 F_2}} .$$
Obviously, this expression is symmetric in the first two and the last two variables. We divide the prove into several steps.
STEP 1
From the five-term relation it follows that for arbitrary $a, b \neq 0,1$
$\{a\} \otimes b + \{b\} \otimes a = \left ( \left \{ \dfrac{1-a}{1-b} \right \} + \left \{ \dfrac{(1-a)b}{(1-b)a} \right \} \right ) \otimes \dfrac{a}{b} .$
From this it follows that the expression $\{a\} \otimes b + \{b\} \otimes a $ vanishes (modulo $B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*$ ) if $a$, $b$ lie in $F(t)_{d-1}^*$ and $1-a, 1-b, a-b$ are product of $P$ with elements of $F(t)_{d-1}^*$.
STEP 2 If $ F_1= G_1$ or $F_1 F_2 \equiv G_1 G_2$, then $S[ F_1, F_2 ; G_1 , G_2 ]=0.$
Taking $a=\dfrac{\overline{F_1} \cdot \overline{F_2}}{\overline{F_1 F_2}}, b= \dfrac{\overline{G_1} \cdot \overline{G_2}}{\overline{G_1 G_2}}$ and applying Step 1 finishes the proof.
STEP 3 $ S[ F_1, F_2 ; G_1 , G_2 ]= S[ F_1, F_2 ; G_1 \cdot F_1 , G_2 ].$
By lemma <ref>
$$\left\{ \dfrac{\overline{F_1} \cdot \overline{G_1}}{\overline{F_1 G_1}} \right\}
+\left\{ \dfrac{\overline{F_1G_1} \cdot \overline{G_2}}{\overline{F_1 G_1 G_2}} \right\} =
\left\{ \dfrac{\overline{G_1} \cdot \overline{G_2}}{\overline{G_1 G_2}} \right\}
+\left\{ \dfrac{\overline{G_1 G_2} \cdot \overline{F_1}}{\overline{F_1 G_1 G_2}} \right\}=0.$$
$$\left\{ \dfrac{\overline{F_1} \cdot \overline{F_2}}{\overline{F_1 F_2}} \right\} \otimes \left ( \dfrac{\overline{F_1} \cdot \overline{G_1}}{\overline{F_1 G_1}}
\cdot \dfrac{\overline{F_1G_1} \cdot \overline{G_2}}{\overline{F_1 G_1 G_2}} \right ) =
\left\{ \dfrac{\overline{F_1} \cdot \overline{F_2}}{\overline{F_1 F_2}} \right\} \otimes \left ( \dfrac{\overline{G_1} \cdot \overline{G_2}}{\overline{G_1 G_2}}
\cdot \dfrac{\overline{G_1 G_2} \cdot \overline{F_1}}{\overline{F_1 G_1 G_2}} \right ).
Summing this two expressions we get that
$$S[ F_1, G_1 ; F_1 , F_2 ]+ S[ F_1 G_1, G_2 ; F_1 , F_2 ] = S[ G_1, G_2 ; F_1 , F_2 ]+S[ G_1 G_2, F_1 ; F_1 , F_2 ].$$
By Step 2 we know that $S[ F_1, G_1 ; F_1 , F_2 ]$ and $S[ G_1 G_2, F_1 ; F_1 , F_2 ]$ vanish. From this the result follows.
STEP 4 $S[ F_1,F_2 ; G_1 , G_2 ]= - S\left [ F_1, G_2 ; G_1 , \dfrac{ F_2}{G_1} \right ].$
Applying the five-term relation with $a=\dfrac{F_1 \cdot F_2}{\overline{F_1 \cdot F_2}}$ and $b=\dfrac{F_1 \cdot G_2}{\overline{F_1 \cdot G_2}}$ and tensoring with $\dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}}$ one gets that
$$\left \{ \dfrac{F_1 \cdot F_2}{\overline{F_1 \cdot F_2}} \right \} \otimes \dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}} -
\left \{\dfrac{F_1 \cdot G_2}{\overline{F_1 \cdot G_2}} \right \} \otimes \dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}}=$$
$$=-\left \{ \dfrac{F_2 \cdot \overline{F_1 \cdot G_2}}{\overline{\overline{F_1 \cdot F_2} \cdot G_2}} \right \} \otimes \dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}}.$$
The same expression tensored with $\dfrac{G_1 \cdot G_2}{\overline{G_1 G_2}}$ gives
$$\left \{ \dfrac{F_1 \cdot F_2}{\overline{F_1 \cdot F_2}} \right \} \otimes \dfrac{G_1 \cdot G_2}{\overline{G_1 G_2}} -
\left \{\dfrac{F_1 \cdot G_2}{\overline{F_1 \cdot G_2}} \right \} \otimes \dfrac{G_1 \cdot G_2}{\overline{G_1 G_2}}=
-\left \{ \dfrac{F_2 \cdot \overline{F_1 \cdot G_2}}{\overline{\overline{F_1 \cdot F_2} \cdot G_2}} \right \} \otimes \dfrac{G_1 \cdot G_2}{\overline{G_1 G_2}}.$$
Applying the five-term relation with $a=\dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}}$ and $b=\dfrac{G_1 \cdot G_2}{\overline{G_1 \cdot G_2}}$ and tensoring with $\dfrac{F_1 \cdot F_2}{\overline{F_1 F_2}}$ one gets that
$$\left \{ \dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}} \right \} \otimes \dfrac{F_1 \cdot F_2}{\overline{F_1 F_2}} -
\left \{\dfrac{G_1 \cdot G_2}{\overline{G_1 \cdot G_2}} \right \} \otimes \dfrac{F_1 \cdot F_2}{\overline{F_1 F_2}}=-\left \{ \dfrac{\overline{G_1 G_2} \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2} \cdot G_2} \right \} \otimes \dfrac{F_1 \cdot F_2}{\overline{F_1 F_2}}.$$
The same expression tensored with $\dfrac{F_1 \cdot G_2}{\overline{F_1 G_2}}$ gives
$$\left \{ \dfrac{G_1 \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2}} \right \} \otimes \dfrac{F_1 \cdot G_2}{\overline{F_1 G_2}} -
\left \{\dfrac{G_1 \cdot G_2}{\overline{G_1 \cdot G_2}} \right \} \otimes \dfrac{F_1 \cdot G_2}{\overline{F_1 G_2}}=-\left \{ \dfrac{\overline{G_1 G_2} \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2} \cdot G_2} \right \} \otimes \dfrac{F_1 \cdot G_2}{\overline{F_1 G_2}}.$$
The sum of the right parts of the four expressions above vanishes by Step 1 with
$a=\dfrac{F_2 \cdot \overline{F_1 \cdot G_2}}{\overline{\overline{F_1 \cdot F_2} \cdot G_2}}$ and $b=\dfrac{\overline{G_1 G_2} \cdot \left ( \overline{\dfrac{F_1 F_2}{G_1}} \right )}{\overline{F_1 F_2} \cdot G_2}.$ From this we get that
$$S \left [ F_1, F_2 ; G_1 , \dfrac{F_1 F_2}{G_1} \right ] - S \left [ F_1, G_2 ; G_1 , \dfrac{F_1 F_2}{G_1} \right ] - S[ F_1, F_2 ; G_1 , G_2 ] +S[ F_1 , G_2; G_1 , G_2 ]=0.$$
By Step 2 we know that $S\left[ F_1, F_2 ; G_1 , \dfrac{F_1 F_2}{G_1} \right ]$ and $S[ F_1 , G_2; G_1 , G_2 ]$ vanish.
By Step 3 $S\left [ F_1, G_2 ; G_1 , \dfrac{F_1 F_2}{G_1}\right ] =S\left [ F_1, G_2 ; G_1 , \dfrac{F_2}{G_1} \right]. $ From this the result follows.
STEP 5 $2 \cdot S[ F_1, F_2 ; G_1 , G_2 ] =0.$
$ S[ F_1, F_2 ; G_1 , G_2 ]= \Xi[ F_1, F_2 G_1 ; G_1 , G_2 ]=- \Xi[ F_1, G_2 ; G_1 , F_2],$ so
$ S[ F_1, F_2 ; G_1 , G_2 ]=- S[ F_1, G_2 ; G_1 , F_2 ]=S[ F_1, G_1 ; G_2 , F_2 ]=- S[ F_1, F_2 ; G_1, G_2 ].$
For arbitrary $R \in F(P)$ the following holds:
if $d=2k+1$, there exist $A, B \in F[t]$ such that $R=\frac{A}{B} \in F[t]$ and $deg(A), deg(B) \leq F.$
if $d=2k$, there exist $A, B \in F[t]$ such that $R=\frac{A}{B} \in F[t]$ and $deg(A) \leq k-1,$ $deg(B) \leq k$
For $R=0$ the statement holds for trivial reasons. Assume that $R$ is invertible.
We prove the statement for $d=2k,$ the second case is similar.
$k(P)$ is a vector space over $k$ of degree $d$ with bases $1, t, t^2, ... , t^{d-1}.$ Denote by $F(P)_s$ a linear subspace with bases $1, t, t^2, ... , t^{s}$ of dimension $s+1.$
Multiplication by $R$ is an invertible linear operator on F(P), we denote in by $M_P$.
With this notation the lemma can be stated as follows:
$$M_P \left ( F(P)_{k-1} \right ) \cap F(P)_{k} \neq \emptyset.$$
Since $M_P$ is invertible, $dim\left (M_P \left ( F(P)_{k-1} \right) \right) =
dim \left ( F(P)_{k-1} \right)=k,$
$$dim\left (M_P \left ( F(P)_{k-1} \right) \right) + dim \left ( F(P)_{k} \right) =2k+1 > dim(F(P)).$$
From this the statement follows.
For a monic irreducible polynomial $P$ and arbitrary rational function $X \in F(t)$, such that it is integer with respect to a valuation in $P$ denote the residue of $X$ in $F(t)/(P)$ by $\overline{X}.$
Let $P$ be a monic irreducible polynomial of degree $d \geq 1.$ Let $R_1$, $R_2$ be arbitrary nonzero elements of $F(P).$ Define recursively $R_{i+1} = \dfrac{1-R_i}{R_{i+1}}.$ Note that $R_{i+5}=R_{i}.$ Then
$$\sum_{i=1}^{i=5} \left\{ \overline{R_i}
\right\} \otimes P +
\sum_{i=1}^{i=5} \left\{ \frac{\overline{R_{i-1}} \: \overline{R_{i+1}}}{\overline{1-R_i}} \right\} \otimes \overline{R_i}
\in B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*.
We start with a remark. Denote by $H$ a projection from the group $F(t)^* \wedge F(t)^*$ to $B(F(t))/B_{d-1}(F(t)),$ sending elements of $F(t)_P^* \wedge F(t)_P^*$ to $B_P(F(t))/B_{d-1}(F(t))$ and $F(t)_{d-1}^* \wedge F(t)_{d-1}^*$ to $0.$ Existence of such a map follows from lemma <ref>. The main its property, important for us, will be the following: for $X \in F(t)_{d-1}$ such that $\nu_P(1-X) > 0$ and $\dfrac{1-X}{P} \in F(t)_{d-1}, \:\:\: H(X)=\{ X \}.$ Particularly,
$$H(F_1,P)+H(F_2,P)-H(\overline{F_1F_2},P)=H\left( \dfrac{F_1 \cdot F_2}{\overline{F_1F_2}} ,P \right )= \{x_P(F_1,F_2)\}.$$
Using this definition, one can restate lemma <ref> as follows:
$$H\left( \dfrac{F_1 \cdot F_2}{\overline{F_1F_2}} ,P \right ) \otimes \dfrac{G_1 \cdot G_2}{\overline{G_1G_2}}=-
H\left( \dfrac{G_1 \cdot G_2}{\overline{G_1G_2}} ,P \right ) \otimes \dfrac{F_1 \cdot F_2}{\overline{F_1F_2}}.$$
Suppose $deg(P)=d=2k+1,$ the even case is similar. Using Polynomial Thue's lemma we can find $X_1=\frac{A_1}{B_1}$ and $X_2=\frac{A_2}{B_2}$ with
$deg(A_1), deg(A_2), deg(B_1), deg(B_2) \leq k$ such that $X_1 \equiv R_1$ and $X_2 \equiv R_2.$ Define recursively $X_{i+1} = \frac{1-X_i}{X_{i+1}}.$ This gives
$X_1=\frac{A_1}{B_1}; 1-X_1=\frac{B_1-A_1}{B_1}, $
$X_2=\frac{A_2}{B_2} ; 1-X_2=\frac{B_2-A_2}{B_2},$
$X_3=\frac{B_1(B_2-A_2)}{A_1B_2}; 1-X_3=\frac{A_1B_2-B_1B_2+B_1A_2}{A_1B_2},$
$X_4=\frac{A_1B_2-B_1B_2+B_1A_2}{A_1A_2}; 1-X_4=\frac{(B_1-A_1)(B_2-A_2)}{A_1A_2},$
$X_5=\frac{B_2(B_1-A_1)}{A_2B_1}; 1-X_5=\frac{A_2B_1-B_2B_1+B_2A_1}{A_2B_1}.$
One can see that for arbitrary $i \:\:\:X_i, 1-X_i \in F(t)_{d-1}.$ For simplicity we write $R_i$ instead of $\overline{R_{i}}.$ Let $X_i=P Q_i + R_i.$ Since each $X_i$ is a ratio of two polynomials of degree less than $d$ and $R_i$ has degree less than $d$, $Q_i$ is a ratio of two polynomials of degree less than $d$. Below all the computations are performed modulo $B_{d-1}(F(t)) \otimes_{a} F(t)_{d-1}^*.$ From the five-term relation it follows that
$$\left\{ X_i \right\} \otimes P-\left\{ R_i \right\} \otimes P= - \left\{ \dfrac{R_i}{X_i} \right\} \otimes P + \left\{ \dfrac{1-R_i}{1-X_i}\right\} \otimes P - \left\{ \dfrac{(1-R_i)X_i}{R_i(1-X_i)} \right\} \otimes P=$$
$$= - H \left ( \dfrac{R_i}{X_i} , \dfrac{Q_i P}{X_i} \right ) \otimes P + H \left ( \dfrac{1-R_i}{1-X_i} , \dfrac{Q_i P}{1-X_i} \right ) \otimes P - H \left ( \dfrac{(1-R_i)X_i}{R_i(1-X_i)} , \dfrac{Q_iP}{R_i(1-X_i)} \right ) \otimes P =$$
$$= H \left ( \dfrac{R_i}{X_i} , \frac{Q_i P}{X_i} \right ) \otimes \dfrac{Q_i }{X_i} - H \left ( \dfrac{1-R_i}{1-X_i} , \dfrac{Q_i P}{1-X_i} \right ) \otimes \dfrac{Q_i}{1-X_i} + H \left ( \dfrac{(1-R_i)X_i}{R_i(1-X_i)} , \dfrac{Q_iP}{R_i(1-X_i)} \right ) \otimes \dfrac{Q_i}{R_i(1-X_i)} =
$$= H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes \dfrac{Q_i }{X_i} - H \left ( \dfrac{1-R_i}{1-X_i} , P \right ) \otimes \dfrac{Q_i}{1-X_i} + H \left ( \dfrac{(1-R_i)X_i}{R_i(1-X_i)} , P \right ) \otimes \dfrac{Q_i}{R_i(1-X_i)} =
$$= H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes Q_i - H \left ( \dfrac{1-R_i}{1-X_i} , P \right ) \otimes Q_i+ H \left ( \dfrac{(1-R_i)X_i}{R_i(1-X_i)} , P \right ) \otimes Q_i - H \left ( \frac{R_i}{X_i} , P \right ) \otimes X_i+$$
$$ + H \left ( \frac{1-R_i}{1-X_i} , P \right ) \otimes (1-X_i) - H \left ( \frac{(1-R_i)X_i}{R_i(1-X_i)} , P \right ) \otimes R_i(1-X_i) =(*).$$
Since $H$ is bilinear,
$$H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes Q_i - H \left ( \dfrac{1-R_i}{1-X_i} , P \right ) \otimes Q_i+ H \left ( \dfrac{(1-R_i)X_i}{R_i(1-X_i)} , P \right ) \otimes Q_i =0.$$
Since $ H \left ( \dfrac{R_i}{X_i} , \dfrac{P Q_i}{X_i} \right ) \otimes \dfrac{R_i}{X_i} =0$
we get that
$H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes X_i = H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes R_i. $
So, (*)=
$$=- H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes R_i + H \left ( \dfrac{1-R_i}{1-X_i} , P \right ) \otimes (1-X_i) - H \left ( \dfrac{(1-R_i)X_i}{R_i(1-X_i)} , P \right ) \otimes R_i(1-X_i) =$$
$$= H \left ( 1-X_i , P \right ) \otimes R_i - H \left ( 1-R_i , P \right ) \otimes R_i + H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes (1-X_i).$$
Since $X_{i+1} = \dfrac{1-X_i}{X_{i-1}},$ by the five-term relation $\sum_{i=1}^{i=5} \left\{ X_i \right\} =0.$
Summing over all $i$ we get
$- \sum_{i=1}^{i=5} \left\{ R_i\right\} \otimes P =$
$$= \sum_{i=1}^{i=5} H \left ( 1-X_i , P \right ) \otimes R_i - \sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i
+ \sum_{i=1}^{i=5} H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes (1-X_i)=$$
$$= \sum_{i=1}^{i=5} H \left ( X_{i-1}X_{i+1} , P \right ) \otimes R_i - \sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i
+ \sum_{i=1}^{i=5} H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes X_{i-1}X_{i+1}= $$
$$= \sum_{i=1}^{i=5} H \left ( X_{i-1} , P \right ) \otimes R_i +\sum_{i=1}^{i=5} H \left ( X_{i+1} , P \right ) \otimes R_i - \sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i+$$
$+ \sum_{i=1}^{i=5} H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes X_{i-1} + \sum_{i=1}^{i=5} H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes X_{i+1}=$
$ =\sum_{i=1}^{i=5} H \left ( X_{i-1} , P \right ) \otimes R_i +\sum_{i=1}^{i=5} H \left ( X_{i} , P \right ) \otimes R_{i-1} - $
$$\sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i
+ \sum_{i=1}^{i=5} H \left ( \dfrac{R_i}{X_i} , P \right ) \otimes X_{i-1} + \sum_{i=1}^{i=5} H \left ( \dfrac{R_{i-1}}{X_{i-1}} , P \right ) \otimes X_{i}=$$
$$=-\sum_{i=1}^{i=5} H \left ( \dfrac{R_{i-1}}{X_{i-1}} , P \right ) \otimes \dfrac{R_{i}}{X_{i}} -
\sum_{i=1}^{i=5} H \left ( \dfrac{R_{i}}{X_{i}} , P \right ) \otimes \dfrac{R_{i-1}}{X_{i-1}} $$
$$- \sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i + \sum_{i=1}^{i=5} H \left ( R_{i-1} , P \right ) \otimes R_i
+\sum_{i=1}^{i=5} H \left (R_{i} , P \right ) \otimes R_{i-1}.
By Lemma, for any $i$
$$H \left ( \frac{R_{i-1}}{X_{i-1}} , P \right ) \otimes \frac{R_{i}}{X_{i}} +
H \left ( \frac{R_{i}}{X_{i}} , P \right ) \otimes \frac{R_{i-1}}{X_{i-1}} =0.$$
From this we finally get
$ -\sum_{i=1}^{i=5} \left\{ R_i\right\} \otimes P =$
$$- \sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i + \sum_{i=1}^{i=5} H \left ( R_{i-1} , P \right ) \otimes R_i
+\sum_{i=1}^{i=5} H \left ( R_{i} , P \right ) \otimes R_{i-1}=
$$- \sum_{i=1}^{i=5} H \left ( 1-R_i , P \right ) \otimes R_i + \sum_{i=1}^{i=5} H \left ( R_{i-1} , P \right ) \otimes R_i
+\sum_{i=1}^{i=5} H \left ( R_{i+1} , P \right ) \otimes R_{i}=
$$=\sum_{i=1}^{i=5} H \left ( \dfrac{R_{i-1} R_{i+1}}{1-R_i}, P \right) \otimes R_i=
\sum_{i=1}^{i=5} \left \{ \dfrac{R_{i-1} R_{i+1}}{1-R_i} \right \} \otimes R_i.$$
B Bloch, S. Algebraic cycles and higher K-theory. Adv. in Math. 61 (1986), no. 3, 267Đ304.
BD Beilinson A.A., Deligne P, Interpretation motivique de la conjecture de Zagier. Symp. in Pure Math., v. 55, part 2, (1994).
DS Dupont J., Sah C.-H, Scissors congruences II. J. Pure Appl. Algebra, v. 25, (1982),
G1 Goncharov, A. B. Geometry of configurations, polylogarithms and motivic cohomology. Adv. Math., 114 (1995), pp. 197Đ318.
G2 Goncharov, A. B. Polylogarithms, regulators and Arakelov motivic complexes. ArXiv:math/0207036 [math.NT].
MVW Mazza, C., Voevodsky, V. and Weibel, C. Lecture Notes on motivic cohomology. Clay Mathematics Monographs, 2006.
S1 Suslin, A. A. . $K_3$ of a field, and the Bloch group. Trudy Mat. Inst. Steklov (in Russian) (1990), 180Đ199.
|
1511.00588
|
We study the spherically symmetric collapsing star in terms of
dynamical instability. We take the framework of extended
teleparallel gravity with non-diagonal tetrad, power-law form of
model presenting torsion and matter distribution as non-dissipative
anisotropic fluid. The vanishing shear scalar condition is adopted
to search the insights of collapsing star. We apply first order
linear perturbation scheme to metric, matter and $f(T)$ functions.
The dynamical equations are formulated under this perturbation
scheme to develop collapsing equation for finding dynamical
instability limits in two regimes such as Newtonian and
post-Newtonian. We obtain constraint free solution of perturbed time
dependent part with the help of vanishing shear scalar. The
adiabatic index exhibits the instability ranges through second
dynamical equation which depend on physical quantities such as
density, pressure components, perturbed parts of symmetry of star,
etc. We also develop some constraints on positivity of these
quantities and obtain instability ranges to satisfy the dynamical
instability condition.
Keywords: $f(T)$ gravity; Instability; Shear-free; Newtonian
and post-Newtonian regimes.
PACS: 04.50.Kd; 04.25.Nx; 04.40.Dg.
§ INTRODUCTION
The gravitational collapse of self-gravitating objects has become
widely discussed phenomena in general relativity (GR) as well as in
modified theories of gravity. This contains the evolutionary
development and constancy of these objects during collapse process
and rests importantly at the center of structure formation. This
process occurs when a stable matter becomes unbalanced and
ultimately undergoes a collapse which results different structures
like stars, stellar groups and planets. In this way,
self-gravitating objects go across various dynamical states which
may be analyzed through dynamical equations. The dynamical
instability was firstly investigated by Chandrasekhar <cit.>
with the help of adiabatic index $\Gamma$ of a spherical star with
isotropic pressure. This index depicts the consequences of various
structural quantities of a fluid on the instability ranges. For
instability ranges in Newtonian and post-Newtonian regimes, Herrera
et al. <cit.> explored dissipative, non-adiabatic spherically
symmetric collapsing star.
The adiabatic index develops the instability ranges in GR as well as
in modified theories of gravity which induces that these ranges
depending on dark source terms in addition to usual terms. Under
different conditions for cylindrically and spherically symmetric
collapsing matters in $f(R)$ gravity, the instability ranges have
been explored through adiabatic index<cit.>. Taking
expansion-free condition, Skripkin <cit.> developed a model for
non-dissipative spherically symmetric fluid distribution with
isotropy and constant energy density and remarked that a Minkowskian
cavity is observed at a center of fluid. Under this condition, the
instability for spherically and cylindrically symmetric anisotropic
fluids in Newtonian, post-Newtonian regimes is explored in GR
<cit.> as well as $f(R)$ gravity <cit.>. In Brans-Dicke
gravity, Sharif and Manzoor <cit.> explored the instability
ranges of spherically symmetric collapsing star.
The physical aspects such as isotropy, radiation, anisotropy, shear,
dissipation, expansion are main sources of cause for the
gravitational evolution. Among these factors, the shear leads to the
formation of naked singularities. That is, it contribute to the
formation of an apparent horizon which results in a black hole of
the evolving cloud. Thus, the shear tensor occupies a direction of
well-motivation to study structure formation and its consequences on
the dynamical instability range of a self-gravitating body.
In context of extended teleparallel gravity (ETG) (or $f(T)$
gravity) which is the generalization of teleparallel gravity, the
gravitational collapse is discussed with and without expansion
scalar by Sharif and Rani <cit.>. They found that the
physical properties invade a vast impact of dynamical instability in
studying the self-gravitating objects with expansion. Without
expansion, they obtain the instability ranges for Newtonian (N) and
post-Newtonian (pN) regimes. In this paper, we assume shear-free
condition instead expansion-free and explore the instability ranges
of a collapsing star in ETG.
The scheme of the paper is given by: In section 2, we give
the basics of ETG and provide the construction of field equations in
two ways, simple and covariant form. Section 3 contains the
basic equations for the static spherically symmetry. Also, junction
conditions are given for dynamical instability of a spherically
symmetric collapsing star in the context of ETG gravity. In the next
section, we represent perturbation scheme and ETG model and apply to
all matter, metric and $f(T)$ functions. In section 5, we
formulate dynamical collapsing equation and found the instability
ranges in N and pN regimes. The last section summarizes the results
and elaborate the comparison.
§ EXTENDED TELEPARALLEL GRAVITY
In this section, we provide the basics such as tetrad field and the
Weitzenböck connection of ETG. We give field equations in simple
form as well as its covariant construction.
§.§ Tetrad Field
The geometry of ETG is unambiguously descried thorough an
orthonormal set having three spacelike and one timelike fields
called tetrad field. The trivial tetrad field has the form
$e_a=\delta_{a}^\mu\partial_\mu,~e^b=\delta^b_\mu dx^\mu$, where
$\delta^a_\mu$ named as the Kronecker delta. This is the simplest
field and less important due to zero torsion. The non-trivial tetrad
field allows non-zero torsion and grants the construction of
teleparallel as well as ETG theory. It is given by
\begin{equation}\label{1.1.6}
h_a={h_a}^\mu\partial_\mu,\quad h^b={h^b}_\nu dx^\nu
\end{equation}
satisfying the following properties
\begin{equation}\label{1.1.7}
\end{equation}
The metric tensor is demonstrated as a by product of this field whih
is as follows
\begin{equation}\label{1.1.4}
\end{equation}
§.§ The Weitzenböck Connection
The basic phenomenon of teleparallel gravity and ETG is the parallel
transport of tetrad field in Weitzenböck spacetime which is
carried out by the significant component Weitzenböck connection.
By applying covariant derivative w.r.t spacetime of tetrad field, we
\begin{equation}\label{1.2.8}
\Delta_{\nu}{h^a}_\mu=\partial_{\nu}{h^a}_\mu-{\widetilde{\Gamma}^\alpha}_{~\mu\nu}{h^a}_\alpha\equiv0,
\end{equation}
is the Weitzenböck connection. We obtain torsion tensor by the
antisymmetric part of this connection as follows
\begin{equation}\label{1.2.10}
\end{equation}
which is antisymmetric in its lower indices, i.e.,
${T^\alpha}_{\mu\nu}=-{T^\alpha}_{\nu\mu}$. This absolute
parallelism passed away rapidly the curvature of the Weitzenböck
connection identically. The following relation is also satisfied by
the Weitzenböck connection
\begin{equation}\label{1.2.12}
\end{equation}
here ${{\Gamma}^\alpha}_{\mu\nu},~{K^\alpha}_{\mu\nu}$ appear as the
usual Levi-Civita connection (torsionless) and the contorsion
tensor, respectively, which can be defined as follows
\begin{eqnarray}\label{1.2.13}
\end{eqnarray}
§.§ Field Equations
To generalize the action of teleparallel gravity, we just substitute
a general function of torsion scalar by itself as follows
\begin{equation}\label{1.3.8}
\mathcal{S}=\frac{1}{2\kappa^2}\int h(f(T)+\mathcal{L}_m)d^{4}x.
\end{equation}
where, $h=\textmd{det}({h^a}_\lambda),~\mathcal{L}_m$ is the matter
Lagrangian and $f$ is the function of torsu=ion scalar. The torsion
scalar is defined as
\begin{equation}\label{1.3.5}
\end{equation}
\begin{equation}\label{1.3.6}
\delta^{\nu}_{\alpha}{T^{\theta\mu}}_{\theta}].
\end{equation}
called the superpotential tensor. Applying the variation of action
(<ref>) w.r.t tetrad field, we will get field equations as
\begin{equation}\label{1.3.9}
\end{equation}
where $f_{_T}$ is the first order derivative and $f_{_{TT}}$
represent second order derivative of $f$ with respect to $T$.
The field equations (<ref>) turn out to be extremely different
from Einstein's equations on account of tetrad components and
partial derivatives. Since tetrad are not completely eradicated
which causes difficulty to compare teleparallel gravity (and ETG)
with GR. To obtain equivalent description of field equations
(<ref>) with the other modified theories, we will apply
covariant formalism <cit.>. We replace all partial derivatives
in Eqs.(<ref>), (<ref>), (<ref>) by covariant
derivatives using the condition on metric tensor,
$\nabla_{\sigma}g_{\mu\nu}=0$, i.e., the compatibility of the metric
tensor, we get
\begin{eqnarray*}
\delta^{\nu}_{\alpha}\eta^{ab}h^{\sigma}_{a}\nabla_{\sigma}h^{\mu}_{b}
\end{eqnarray*}
where we have applied the following relations
\begin{eqnarray}\label{111a}
\end{eqnarray}
In this case, the Weitzenböck connection becomes zero, while the
Riemann tensor turns out to be
\begin{eqnarray}\nonumber
\end{eqnarray}
The corresponding Ricci tensor becomes
\begin{eqnarray*}
\end{eqnarray*}
Using relations (<ref>) along with
${S^\nu}_{\alpha\nu}=-2{T^\nu}_{\alpha\nu}=2{K^\nu}_{\alpha\nu}$, we
\begin{eqnarray}\label{116a}
\end{eqnarray}
The covariant derivative of torsion tensor in the last equation
shows the only difference between Ricci and torsion scalars.
After some calculations, we will get
\begin{equation}\label{1117a}
\end{equation}
where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ appears as the
Einstein tensor. By using this in (<ref>), we attain the
required field equations in $f(T)$ gravity
\begin{equation}\label{118a}
\end{equation}
here $D_{\mu\nu}={S_{\nu\mu}}^{\alpha}\nabla_{\alpha}T$. It can be
observed that (<ref>) has an equivalent structure such as
$f(R)$ gravity and reduces to GR for $f(T)=T$. Here, the trace of
the above equation is
\begin{equation}\label{119a}
\end{equation}
with $D={D^\nu}_{\nu}$ and $\mathcal{T}={\mathcal{T}^{\nu}}_{\nu}$.
The $f(T)$ field equations can also defined as
\begin{equation}\label{120a}
\end{equation}
Here $\mathcal{T}_{\mu\nu}^{m}$ represents the matter fluid and
torsion contribution is
\begin{equation}\label{122a}
\mathcal{T}_{\mu\nu}^{T}=\frac{1}{\kappa^2}[-D_{\mu\nu}f_{TT}-
\frac{1}{4}g_{\mu\nu}(\mathcal{T}-Df_{TT}+Rf_{T})].
\end{equation}
§ BASIC EQUATIONS
The general spherically symmetric metric in the interior region is
\begin{equation}\label{1c5}
ds^2_-=X^2dt^{2}-Y^2dr^{2}-R^2(d\theta^{2}+\sin^2\theta d\phi^{2}),
\end{equation}
where $X,~Y$ and $R$ are functions of $t$ and $r$. The line element
for exterior spacetime (the Schwarzschild metric) is <cit.>
\begin{equation}\label{21c5}
\end{equation}
where $\upsilon$ is the retarded time and $M$ represents the total
mass of the bounded surface. Also, the anisotropic energy-momentum
tensor can be defined as follows
\begin{equation}\label{zc3}
\mathcal{T}^{\mu}_{~\nu}=(\rho+p_{\perp})u^{\mu}u_{\nu}-p_{\perp}\delta^{\mu}_{\nu}
\end{equation}
in the interior region while
$\rho=\rho(t,r),~p_r=p_r(t,r),~p_{\perp}=p_{\perp}(t,r)$. The four
velocity $u_\mu=\frac{1}{X}\delta^0_\mu$ and unit four vector
directed towards radial component $v_\mu=\frac{1}{Y}\delta^1_\mu$
satisfy the relations $u_\mu u^\mu=1,~v_\mu v^\mu=-1,~u_\mu
v^\mu=0$. We take the non-diagonal tetrad for the interior spacetime
X & 0 & 0 & 0 \\
0 & Y\sin\theta \cos\phi & R\cos\theta\cos\phi & -R\sin\theta\sin\phi \\
0 & Y\sin\theta \sin\phi & R\cos\theta\sin\phi & R\sin\theta\cos\phi \\
0 & Y\cos\theta & -R\sin\theta & 0 \\
\end{array}\right).$
The description of gravitational collapsing star takes kinematics of
the dynamical equations of spherically symmetric models. This needs
acceleration, expansion, rotation and distortion or shear. Due to
shearfree condition, there have been many interesting results such
as this condition depicts the physical aspects of compact bodies in
the relativistic astrophysics phenomena. The shear scalar and tensor
are defined by
\begin{equation}\label{26c5++}
\sigma^2=\frac{1}{2}\sigma^{\alpha\beta}\sigma_{\alpha\beta},
\end{equation}
\begin{equation}\label{26c5+}
\sigma_{\alpha\beta}=V_{(\alpha;\beta)}+a_{(a}V_{\beta)}-\frac{1}{3}\Theta(g_{\alpha\beta}+V_\alpha
V_\beta), \quad \Theta=V^{\alpha}_{~;\alpha}, \quad
\end{equation}
For interior spacetime, this tensor yields the following scalar
\begin{equation}\label{26c5}
\sigma=\frac{1}{X}\left(\frac{\dot{Y}}{Y}-\frac{\dot{R}}{R}\right).
\end{equation}
Using the tetrad along with Eq.(<ref>) in (<ref>), the
field equations are
\begin{eqnarray}\nonumber
\frac{\dot{R}}{R}-\left(\frac{X}{Y}\right)^2
\left[\frac{2R''}{R}+\left(\frac{R'}{R}\right)^2-\frac{2Y'R'}{YR}
-\left(\frac{Y}{R}\right)^2\right] \\\label{13c5}
\frac{X^2}{\kappa^2}\left\{\frac{Tf_{_T}-f}{2}-\frac{1}{Y^2}\left(\frac{R'}{R}-
\frac{Y}{R}\right)f_T'\right\}\right],
\\\label{14c5}
=\frac{\dot{R}f_{_T}'}{R f_{_T}}, \\\label{142c5}
\frac{Y}{R}\right)\frac{f_{_T}'}{f_{_T}},\\\nonumber
-\frac{\dot{R}}{R}\right) \frac{\dot{R}}{R}\right]
\\\label{15c5}&&=\frac{\kappa^2}{f_{_T}}
\left[p_rY^{2}-\frac{Y^2}{\kappa^2}\left(\frac{Tf_{_T}-f}{2}
\left(\frac{\dot{Y}}{Y}+\frac{\dot{R}}{R}\right)+\frac{\dot{Y}\dot{R}}{YR}\right]
\frac{1}{2}\right.\right.\\\label{16c5}
\frac{1}{Y^2}\left(\frac{X'}{X}+\frac{R'}{R}-\frac{Y}{R}\right)\right)f_{_T}'\right\}\right].
\end{eqnarray}
The torsion scalar takes the form
\begin{eqnarray}\nonumber
\frac{1}{R^2}\right.\\\label{17c5}&-&\left.\frac{2}{YR}\left(\frac{X'}{X}+\frac{R'}{R}\right)\right].
\end{eqnarray}
Also, from Eqs.(<ref>) and (<ref>), we obtain a
relationship as follows
\begin{eqnarray}\label{xyc5}
\frac{\dot{R}}{R}=\frac{\dot{T}}{T'}\left(\frac{R'}{R}+\frac{Y}{R}\right).
\end{eqnarray}
To analyze the properties of collapsing star, the non-trivial
contracted identities yield the dynamical equations which are very
useful. These are given by
\begin{eqnarray}\label{18c5}
\left(\overset{m}{\mathcal{T}^{\mu\nu}}+\overset{T}{\mathcal{T}^{\mu\nu}}\right)_{;\nu}u_{\mu}=0,\quad
\left(\overset{m}{\mathcal{T}^{\mu\nu}}+\overset{T}{\mathcal{T}^{\mu\nu}}\right)_{;\nu}
\end{eqnarray}
Using these equation, the dynamical equations become
\begin{eqnarray}\nonumber
\frac{\dot{R}}{XR}\right]+\frac{X}{\kappa^2}\left[\left(\frac{Tf_{_T}-f}{2X^2}-
\frac{\dot{R}T'f_{_T}'}{X^2Y^2R\dot{T}}\right)_{,0}\right.\\\nonumber&&
+ \left.\frac{\dot{X}}{X^3}(Tf_{_T}-f)+\left(\frac{\dot{R}f_{_T}'}
\frac{1}{X^2R}\left\{\frac{2\dot{R}}{XY^2}\left(\frac{\dot{X}T'}{\dot{T}}-2X'\right)
\left(\frac{2\dot{Y}}{Y}+\frac{\dot{R}}{R}\right)\frac{\dot{T}}{T'}+\frac{\dot{R}}{Y^2}
\left(\left(\frac{{\dot{R}}}{R}+\frac{\dot{Y}}{Y}\right)
\frac{T'}{\dot{T}}-\frac{Y'}{Y}-\frac{2R'}{R}\right)\right\}f_{_T}'\right]=0,
\\\nonumber
\left[\left(\frac{\dot{R}f_{_T}'}{X^2Y^2R}\right)_{,0}-\frac{1}{Y^2}\right.\\\nonumber
\frac{\dot{R}\dot{T}f_{_T}'}{X^2Y^2RT'}\right)_{,1}+
\frac{1}{XY^2R}\right.\\\nonumber&&\left.\times\left\{-\frac{X'\dot{R}T'}{Y^2\dot{T}}+
\frac{\dot{R}}{X}\left(\frac{3\dot{Y}}{Y}+\frac{\dot{X}}{X}\right)
\left(1-\frac{1}{Y^2}\right)-\frac{\dot{R}}{XY}\right.\right.
\\\label{20c5}&&\left.\left.\times\left(Y'+\frac{Y'}{Y^2}+
\frac{X'}{XY}\right)\frac{\dot{T}}{T'}\right\}f_T'\right]=0.
\end{eqnarray}
§.§ Junction Conditions
In order to join smoothly the interior and exterior spacetimes over
a hypersurface $\Sigma^{(e)}$, we apply junction conditions. The
collapse problems are dealt by Darmois junction conditions in
appropriate manner. These conditions require the continuity of
intrinsic and extrinsic curvatures over the hypersurface, i.e.,
$(ds^2)_{\Sigma}=(ds^2_{-})_{\Sigma}=(ds^2_{+})_{\Sigma}$ and
respectively. The Misner-Sharp mass function is given by
\begin{equation*}
\end{equation*}
where a spherical object of radius $R$ contributes total energy and
contributes to study Darmois junction conditions. For
Eq.(<ref>), it takes the form
\begin{equation}\label{22c5}
\end{equation}
In order to match exterior region with interior, it requires that
$r=r_{\Sigma^{(e)}}=constant$ on the boundary surface $\Sigma^{(e)}$
<cit.> which results
\begin{eqnarray}\label{23c5}
\end{eqnarray}
\begin{eqnarray}\nonumber
\frac{\dot{R}}{R}\right) \frac{\dot{R}}{R}\right]\\\label{24c5}
\end{eqnarray}
Substituting the field equations (<ref>) and (<ref>) in
the above equation, we get
\begin{equation}\label{25c5}
\end{equation}
where $f(T_c)$ represents a constant value for constant torsion
scalar $T_c$.
§ LINEAR PERTURBATION STRATEGY AND POWER-LAW $F(T)$ MODEL
In $f(R)$ gravity, the gravitational collapse is widely discussed
taking power-law form of model which is simply generalizes GR. We
take particular $f(T)$ model in power-law form analogy to $f(R)$
model <cit.> like $f(R)=R+\gamma R^2$ to analyze the evolution
of collapsing star. The power-law $f(T)$ model has contributed as a
most viable model due to its simple form and we may directly compare
our results with GR. We assume the ETG model as follows
\begin{equation}\label{29c5}
f(T)=T+\delta T^2,
\end{equation}
where $\delta$ is an arbitrary constant. For this model, we obtain
accelerated expansion universe in phantom phase, possibility of
realistic wormhole solutions and instability conditions for a
collapsing star. We assume the linear perturbation strategy to
construct the dynamical equations in order to explore instability
ranges for the underlying scenario. For this purpose, we assume the
system initially in static equilibrium. That is, metric and matter
parts are at zero order perturbation are only radial dependent only
which also become time dependent for the first order perturbations.
These perturbations are described as follows <cit.>-<cit.>,
\begin{eqnarray}\label{41cc5}
X(t,r)&=&X_0(r)+\epsilon \Pi(t)x(r),\\\label{42cc5}
Y(t,r)&=&Y_0(r)+\epsilon \Pi(t)y(r),\\\label{43cc5}
R(t,r)&=&R_0(r)+\epsilon \Pi(t)c(r),\\\label{44cc5}
\rho(t,r)&=&\rho_0(r)+\epsilon {\hat{\rho}}(t,r),\\\label{45cc5}
p_r(t,r)&=&p_{r0}(r)+\epsilon {\hat{p}_r}(t,r), \\\label{46cc5}
m(t,r)&=&m_0(r)+\epsilon \hat{m}(t,r), \\\label{49'cc5}
T(t,r)&=&T_0(r)+\epsilon \Pi(t)e(r),
\end{eqnarray}
where the quantities with zero subscript denotes static parts of
corresponding functions and $0<\epsilon\ll1$. The perturbed
vanishing shear scalar and $f(T)$ model takes the form
\begin{eqnarray}\label{a+}
f(T)&=&T_0(1+\delta T_0)+\epsilon \Pi e(1+2\delta
T_0),\\\label{51'cc5} f_{_T}(T)&=&1+2\delta T_0+2\epsilon \delta \Pi
e,\\\label{a} \frac{y}{Y_0}&=&\frac{c}{R_0}.
\end{eqnarray}
Taking into account shear-free condition ($\sigma=0$),
Eq.(<ref>) yields $\frac{\dot{Y}}{Y}=\frac{\dot{R}}{R}$. The
solution of this equation turns out as $Y=\alpha R$ where $\alpha$
is an arbitrary function of $r$ taken as 1 without loss of
generality and using the freedom to rescale the radial coordinate,
we take $R_0=r$ which is also the Schwarzschild coordinate. The
condition under which an initially shear-free flow remains
shear-free all along the evolution, has been studied by Herrera et
al. <cit.>. One of the consequences of such a study is that the
pressure anisotropy may affect the propagation in time, of the
shear-free condition. The shear-free condition is unstable, in
particular, against the presence of pressure anisotropy. An
expansion scalar and a scalar function insured the departures from
the shear–free condition for the geodesic case. These scalars are
defined in purely physical variables such as in terms of the Weyl
tensor, anisotropy of pressure and the shear viscosity. It is
remarked that one can consider such a case that pressure anisotropy
and density inhomogeneity are present in a way that the scalar
function appearing in orthogonal splitting of Reimann tensor
vanishes, implying non-homogeneous anisotropic stable shear-free
flow. Since we are dealing with fluid evolving under shearfree
condition, so we shall make use of this condition while evaluating
the components of field equations and also in conservation
Now we evaluate zero order as well as first order configurations.
The zero order perturbation of the field equations
(<ref>)-(<ref>) is given by
\begin{eqnarray}\label{50cc5}
\\\label{51cc5} &&\frac{1}{1+2\delta T_0}\left[\kappa^2
&&\frac{1}{1+2\delta T_0}\left[\kappa^2
\left(\frac{X_0'}{X_0}-\frac{Y_0'}{Y_0}\right)\right].
\end{eqnarray}
The first dynamical equation (<ref>) fulfills the zero order
perturbation identically while second dynamical equation
(<ref>) becomes
\begin{equation}\label{j3c5}
\frac{2}{r}(p_{r0}-p_{\perp0})+p_{r0}'+\frac{X_0'}{X_0}(\rho_0+p_{r0})
\end{equation}
In static background, the matching condition, mass function and
torsion scalar contribute for static equilibrium as
\begin{eqnarray}\label{63'ccc5}
p_{r0}\overset{\Delta^{(e)}}{=}\frac{\delta T_0^2}{2\kappa^2}, \quad
\end{eqnarray}
Applying the perturbed quantities to the field equations, we obtain
\begin{eqnarray}\nonumber
\left(\frac{y}{Y_0}\right)'
\right]\\\nonumber &=&\frac{2\Pi
\hat{\rho}}{1+2\delta T_0}+\frac{\Pi\delta}{(1+2\delta
T_0)}\left[T_0e-\frac{2\kappa^2 \rho_0 y}{1+2\delta
T_0}\right.\\\label{53cc5} &-&\left.\frac{\delta T_0^2y}{1+2\delta
T_0}-\frac{cT_0'}{eY_0^2r}\left(2e'-\frac{4\delta T_0'e}{1+2\delta
T_0'c}{r(1+2\delta T_0)},
\\\label{54ccc5}
\frac{e(1-Y_0)\delta}{r(1+2\delta T_0)},
\\\nonumber
\frac{2\Pi}{r}\left[\left(\frac{x}{X_0}\right)'+
\left(r\frac{X_0'}{X_0}+1\right)\left(\frac{c}{r}\right)'\right]\\\nonumber
&=&\frac{\kappa^2 Y_0^2\hat{p_r}}{1+2\delta T_0}+\frac{2\kappa^2\Pi
Y_0^2}{1+2\delta T_0}\left(\frac{c}{r}-\frac{\delta e}{1+2\delta
T_0}\right)p_{r0}-\frac{\delta \Pi T_0Y_0^2}{1+2\delta T_0}
\left(e+\frac{cT_0}{r}\right. \\\label{55cc5} &-&\left.\frac{\delta
Y_0T_0e}{1+2\delta T_0}\right), \\\nonumber
\left[\left(\frac{x}{X_0}\right)''
\left(\frac{x}{X_0}\right)'\right.\\\nonumber &-&\left.
\left(\frac{Y_0'}{Y_0}-\frac{1}{r}\right)
\left(\frac{c}{r}\right)'\right]=\frac{\kappa^2\hat{p}_{\perp}}{1+2\delta
T_0} \\\nonumber&-&\frac{2\Pi e\delta\kappa^2}{(1+2\delta
T_0)^2}p_{\perp0}+\frac{2\Pi c}{rY_0^2}\left[\frac{X_0''}{X_0}
\left(\frac{X_0'}{X_0}-\frac{Y_0'}{Y_0}\right)\right]\\\nonumber
&+&\frac{\delta \Pi}{1+2\delta T_0}\left[\frac{e\delta
T_0^2}{1+2\delta T_0}-eT_0 +\frac{X_0'e'}{X_0Y
T_0)}\right. \\\label{12''c5}
-\frac{2c\delta T_0'^2}{rY_0^2(1+2\delta T_0)}\right].
\end{eqnarray}
Applying Eqs.(<ref>)-(<ref>) to the Bianchi identities,
it follows that
\begin{eqnarray}\label{j4c5}
\dot{\hat{\rho}}+\left[(3\rho_0+p_{r0}+2p_{\perp0})\frac{c}{r}+J_0\right]\dot{\Pi}=0,
\end{eqnarray}
\begin{eqnarray}\nonumber
\right.\\\nonumber&-&\left.\frac{4c\delta
\end{eqnarray}
Integrating Eq.(<ref>) with respect to time, we have
\begin{eqnarray}\label{62cc5}
\hat{\rho}&=&-\left[(3\rho_0+p_{r0}+2p_{\perp0})
\frac{c}{r}+J_0\right]\Pi.
\end{eqnarray}
The perturbed second Bianchi identity is given by
\begin{eqnarray}\nonumber
-(\hat{\rho}+\hat{p}_r)\frac{X_0'r}{X_0\Pi r}+J_1\\\label{j9c5}
c}-\frac{c}{\Pi r}\hat{p}_r'=0,
\end{eqnarray}
\begin{eqnarray}\nonumber
\frac{3cY_0'T_0^2}{rY_0}\right.\\\nonumber
\\\nonumber&+&
\left.+\frac{2cX_0'T_0'}{erX_0Y_0^2}\left(1-\frac{m_0}{r}\right)'\left(1-\frac{m_0}{r}\right)
\left(2e'-\frac{xT_0'}{X_0}-\frac{4cT_0'}{r}\right)\right].
\end{eqnarray}
The junction condition, torsion scalar and mass function are
\begin{eqnarray}\label{622c5}
\hat{p_r}&\overset{\Sigma^{(e)}}{=}&\frac{\delta}{\kappa^2}\left(\Pi
T_0e-\frac{2T_0'\dot{\Pi}c}{rX_0Y _0}\right),\\\nonumber
\frac{x}{X_0}+\frac{y}{Y_0}+\frac{c}{r}\right)\right\}\right.
\\\label{6222c5}&-&\left.
\frac{cY_0}{r^2}+\frac{xX_0'}{X_0^2}-\frac{x'}{X_0}-\left(\frac{c}{r}\right)'
\bar{m}&=&-\frac{\Pi}{Y_0^2}\left[r\left(c'-\frac{y}{Y_0}\right)+\frac{c}{2}(1-Y
_0^2) \right].
\end{eqnarray}
Substituting zero order and first order matching conditions in
Eq.(<ref>), we have
\begin{equation}\label{j5c5}
\ddot{\Pi}\overset{\Sigma^{(e)}}{=}0,
\end{equation}
yields the general solution of this equation is
\begin{equation}\label{ac5}
\Pi(t)=h_1t+h_2,
\end{equation}
where $h_1$ and $h_2$ are arbitrary constants. It is remarked that
we do not need to impose any extra condition on this solution due to
vanishing shear scalar condition for collapse to occur. In
<cit.>-<cit.>, we need to apply some constraint
for the static solution in order to discuss instability analysis.
§ COLLAPSE EQUATION AND DYNAMICAL INSTABILITY
In this section, we construct the collapse equation in order to work
for dynamical instability in different regimes with shear-free
condition. The Harison-Wheeler equation of state is used in this
regard which is given by <cit.>
\begin{equation}\label{j7c5}
{\hat{p}_{ir}}=\hat{\rho} \frac{p_{i0}}{\rho_0+p_{i0}}\Gamma.
\end{equation}
We use this index in order to examine instability ranges in the
context of ETG with $\sigma=0$. The adiabatic index $\Gamma$ finds
the rigidity of the fluid and evaluates the change of pressure to
corresponding density. Substituting the value of $\hat{\rho}$, it
follows that
\begin{eqnarray}\label{j8c5}
{\hat{p}_r}&=&-\Pi \left[\frac{2c}{r}\frac{\rho_0+p_{\perp0}}{\rho_0
{\hat{p}_\perp}&=&-\Pi \left[\frac{c}{r}\frac{\rho_0+p_{r0}}{\rho_0
\end{eqnarray}
Finally, the collapse equation used to analyze instability ranges of
collapsing star in N and pN regimes, we insert all the corresponding
values in Eq.(<ref>), we have
\begin{eqnarray}\nonumber
&&\frac{\delta T_0
\\\nonumber&&\left.+\frac{c}{r}(\rho_0+p_{r0})\right]
\frac{p_{\perp0}}{\rho_0+p_{\perp0}}\right)\right.
\\\label{j12c5}&&\times\left. J_0\right]+\frac{\Gamma
\end{eqnarray}
We consider the N and pN approximations to obtain the instability
ranges in the following.
§.§ Newtonian limit
The N approximations are given by
\begin{equation}\nonumber
X_0=1=Y_0, \quad \rho_0\gg p_{r0},\quad \rho_0\gg p_{\perp0},\quad
p_{r0}\gg p_{\perp0}.
\end{equation}
Inserting these conditions as well as Eq.(<ref>) in
(<ref>), the collapse equation yields
\begin{eqnarray}\label{j133c5}
\frac{\delta T_0 T'_0}{\kappa^2}-\frac{xr
\rho_0}{c}+J_{1(N)}-2p_{r0}\left(\frac{r}{c}\left(\frac{c}{r}\right)'-\frac{1}{r}\right)+
\Gamma\left[\frac{3cp_{r0}}{r}+\frac{3r}{c}\left(\frac{cp_{r0}}{r}\right)_{,1}\right]=0,
\end{eqnarray}
\begin{eqnarray}\nonumber
\end{eqnarray}
To check the instability range for N approximation, we obtain
\begin{eqnarray}\label{j13c5}
\Gamma<\frac{1}{I_0}\left[-\frac{\delta T_0 T'_0}{\kappa^2}+\frac{xr
\rho_0}{c}-J_{1(N)}+2p_{r0}\left(\frac{r}{c}\left(\frac{c}{r}\right)'-\frac{1}{r}\right)\right],
\end{eqnarray}
This equation expresses the dependence of adiabatic index on torsion
terms along with physical properties such as anisotropic pressure
and energy density. As long as the above inequality maintains, the
collapsing system stays unstable. To be hold for dynamical
instability condition the terms in collapse equation should be
positive in the inequality. It requires that
\begin{eqnarray}\nonumber
-\frac{\delta T_0 T'_0}{\kappa^2}+\frac{xr
\rho_0}{c}-J_{1(N)}+2p_{r0}\left(\frac{r}{c}\left(\frac{c}{r}\right)'-\frac{1}{r}\right)>0,
\end{eqnarray}
where $I_0>0$ holds. In GR <cit.>, the adiabatic index
represents a numerical value corresponding to dynamical instability
of an isotropic sphere. This is given by
* It is found that the dynamical stability is achieved for
$\Gamma>\frac{4}{3}$ when the weight of outer layers is weaker in
contrast to the pressure in star.
* The case $\Gamma=\frac{4}{3}$ corresponds to the hydrostatic equilibrium condition.
* When the weight of outer layers increases very fast as compared to pressure inside the star gives the
collapse and $\Gamma<\frac{4}{3}$ constitutes the dynamical
The expressions in Eq.(<ref>) constitute the following
\begin{eqnarray}\nonumber
(i) \quad\left[-\frac{\delta T_0 T'_0}{\kappa^2}+\frac{xr
\rho_0}{c}-J_{1(N)}+2p_{r0}\left(\frac{r}{c}\left(\frac{c}{r}\right)'-\frac{1}{r}\right)\right]=I_0,\\\nonumber
(ii) \quad\left[-\frac{\delta T_0 T'_0}{\kappa^2}+\frac{xr
\rho_0}{c}-J_{1(N)}+2p_{r0}\left(\frac{r}{c}\left(\frac{c}{r}\right)'-\frac{1}{r}\right)\right]<I_0,\\\nonumber
(iii) \quad\left[-\frac{\delta T_0 T'_0}{\kappa^2}+\frac{xr
\rho_0}{c}-J_{1(N)}+2p_{r0}\left(\frac{r}{c}\left(\frac{c}{r}\right)'-\frac{1}{r}\right)\right]>I_0.
\end{eqnarray}
In the first possibility together with Eq.(<ref>), we obtain
the instability range as $0<\Gamma<1$. The case $(ii)$ yields that
faction in Eq.(<ref>) always less than 1 but depending on the
values of dynamical terms. The third case $(iii)$ represents the
different numerical values for different values of dynamical terms.
Thus, it shows that the collapsing star remains unstable for
$\Gamma>1$. Also, Eq.(<ref>) depicts that the adiabatic index
contains the physical quantities like $f(R)$ gravity
§.§ Post-Newtonian limit
We consider the following pN approximations in terms of metric
component expressions given by
\begin{eqnarray}\label{j14c5}
X_0=1-\frac{m_0}{r},\quad Y_0=1+\frac{m_0}{r},
\end{eqnarray}
upto order $\frac{m_0}{r}$. Introducing these approximations in
Eq.(<ref>), we obtain
\begin{eqnarray}\nonumber
&&\frac{\delta T_0
\left(\frac{c}{r}\right)'-\frac{1}{r}\right]+
\frac{(\Gamma+1)m_0}{rc}\left[\frac{c}{r}(3\rho_0+p_{r0}+2p_{\perp0})+J_{0(pN)}\right]
\\\nonumber&&+\Gamma\left[\frac{c}{r}\left(p_{r0}-\frac{\rho_0+p_{r0}}{\rho_0
\\\nonumber&&\left.
\frac{p_{\perp0}}{\rho_0+p_{\perp0}}\right)
J_{0(pN)}\right]+\frac{\Gamma r}{c} \left[\frac{c}{r}p_{r0}
\end{eqnarray}
where $J_{1(pN)}$ and $J_{0(pN)}$ are those terms which belong to pN
regime in $J_1$ and $J_0$ expressions are given below.
\begin{eqnarray}\nonumber
\\\nonumber
\left(1-\frac{2m_0}{r}\right)\right\}_{,1}+T_0^2\left(\frac{c}{r}\right)'
\right.\\\nonumber &+&\left.\left(1+\frac{m_0}{r}\right)'
\left(1-\frac{m_0}{r}\right)\left(2T_0e-
\frac{3cT_0^2}{r}+\left(\frac{c}{r}\right)'T_0^2\right)-\frac{2cT_0^2}{r^2}
\\\nonumber&+&
\left.\frac{2cx'T_0'^2}{er}\left(1-\frac{m_0}{r}\right)+
\frac{2cT_0'}{er}\left(2e'-xT_0'\left(1+\frac{m_0}{r}\right)-\frac{4cT_0'}{r}\right)\right].
\end{eqnarray}
Similar to the N approximation strategy, the instability range is
given by
\begin{eqnarray}\label{j15c5}
\Gamma<\frac{\mathcal{A}}{\mathcal{B}},
\end{eqnarray}
\begin{eqnarray*}
\mathcal{A}&=&-\frac{\delta T_0
\left(\frac{c}{r}\right)'-\frac{1}{r}\right]-
\frac{m_0}{rc}\left[\frac{c}{r}(3\rho_0+p_{r0}+2p_{\perp0})+J_{0(pN)}\right],\\\nonumber
\mathcal{B}&=&\frac{m_0}{rc}\left[\frac{c}{r}(3\rho_0+p_{r0}+2p_{\perp0})+J_{0(pN)}\right]
\left(\frac{p_{r0}}{\rho_0+p_{r0}}-
\frac{p_{\perp0}}{\rho_0+p_{\perp0}}\right) J_{0(pN)}\right]
\\\nonumber&+&\frac{r}{c} \left[\frac{c}{r}p_{r0}
\end{eqnarray*}
We obtain instability ranges of collapsing star with vanishing shear
as long as this inequality holds. We analyze that the adiabatic
index evinces the dependence of instability ranges on relativistic
and torsion terms under zero order configuration.
For the dynamical instability condition, it is required that the
right hand side (expressions $\mathcal{A}$ and $\mathcal{B}$) in
inequality (<ref>) remains positive. Similarly to the N
regime, we have three cases such as
\mathcal{A}>\mathcal{B}$. These cases constitute the instability
ranges as $0<\Gamma<1$ for the first and second case while
$\Gamma>1$ for the last case.
§ CONCLUDING REMARKS
The collapse in a star occurs due to state of disequilibrium between
inwardly acting gravitational pull and outwardly drawn pressure in
it. In modified theories of gravity, the ranges of dynamical
instability depends on dark source terms in addition to usual terms
(that is terms of GR) determined by the adiabatic index. We have
analyzed the dynamical instability of a collapsing star taking
vanishing shear scalar in ETG gravity. We have taken interior metric
as general spherically symmetric metric while Schwarzschild metric
is considered in exterior region to $\Sigma^{(e)}$ in anisotropic
matter distribution. The contracted Bianchi identities are used to
find two dynamical equations corresponding to a star experiencing
collapse process. A well-known power-law ETG model is considered to
analyze these ranges. We have used perturbation strategy on all
functions such that metric components, energy density, pressure
components, mass, torsion, shear scalar to examine the evolution of
collapse with respect to time. We have applied zero order and first
order perturbed configurations on the field and dynamical equations.
The collapse equations has been constructed through second dynamical
equation. The results are given as follows.
The adiabatic index ($\Gamma$) plays a vital role to examine the
instability ranges for a collapsing star with shear-free condition.
We can analyze the instability regions through this index by
applying N and pN approximations on collapsing equation. In both
cases of approximations, we have observed that this index depends
upon on various quantities such as energy density, anisotropy of
pressure and on some other constraints. This index shows that how we
can modify the instability range of collapsing star under
relativistic as well as torsion terms. Hence, the physical
properties play a crucial role in analyzing the self-gravitating
objects in dynamical instability. Similar to the case of $f(R)$
gravity, the results of present paper contain physical quantities.
It is pointed out that the instability ranges are found as
$0<\Gamma<1$ and $\Gamma>1$ for both approximation i.e., system
yields unstable behavior and meets $\Gamma< \frac{4}{3}$ in the N
and pN perfect fluid limit <cit.>.
We would like to point out here that some authors have discussed the
instability analysis by imposing some constraints for finding the
solutions <cit.>-<cit.>. However, we have found
the solutions without imposing any extra constraints.
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C 74(2014)2760.
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432(2013)264; ibid. 434(2013)2529; Eur. Phys. J. C
73(2013)2633; Astropart. Phy. 56(2014)19.
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44(2012)1143; Sharif, M. and Azam, M: Gen. Relativ. Gravit.
44(2012)1181; Mon. Not. R. Astron. Soc.
S22 Sharif, M. and Kausar, H.R.: J. Cosmol. Astropart. Phys. 07
(2011)022; Sharif, M. and Yousaf, Z.: Phys. Rev. D
rm Sharif, M. and Manzoor, R.: Astrophys. Space Sci. 354(2014)2122.
sr Sharif, M. and Rani, S.: Mon. Not. Roy. Astron. Soc.
sr1 Sharif, M. and Rani, S.: Int. J. Theor. Phys.
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Phys. 2014(2014)691497.
S4 Jawad, A. and Rani, S.: Eur. Phys. J. C
S31 Li, B., Sotiriou, T.P. and Barrow, J.D.: Phys. Rev. D
83(2011)064035; ibid. 83(2011)104017; Sotiriou,
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Liu, D. and Rebouças, M.J.: Phys. Rev. D
18c5 Chan, R., Herrera, L. and Santos, N.O.: Mon. Not. R. Astron. Soc.
2c5 Herrera, L. and Santos, N.O.: Astrophys. J.
3c5 Chan, R.: Mon. Not. Roy. Astron. Soc.
5c5 Sharif, M. and Azam, M.: J. Cosmol. Astropart. Phys.
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ac5 Harrison, B.K., Thorne, K.S., Wakano, M. and
Wheeler, J.A.: Gravitation Theory and Gravitational Collapse,
(University of Chicago Press, 1965); Chan R., Kichenassamy K., Le
Denmat G. and Santos N.O.: Mon. Not. R. Astron. Soc.
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1511.00275
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Department of Physics and Institute of Theoretical Physics and
Astrophysics,Xiamen University, Xiamen 361005, China.
We study the heat conduct behavior of a lattice model with asymmetry
harmonic inter-particle interactions in this paper. Normal heat conductivity
independent of the system size is observed when the lattice chain is long
enough. Because only the harmonic interactions are involved, the result
confirms without ambiguous interpretation that the asymmetry plays the key
role in resulting in the normal heat conduct of one dimensional momentum
conserving lattices. Both equilibrium and non-equilibrium simulations are
performed to support the conclusion.
05.60.Cd, 44.10.+i, 66.70.-f, 63.20.-e
The heat transport properties of low-dimensional systems have evoked
intensive studies for decades [1-13], aiming at to verify whether the
Fourier's law of heat conduction
\begin{equation}
J=-\kappa \nabla T \label{Four}
\end{equation}
valid in low-dimensional materials. Here $J$ is the heat current, $\nabla T$
is the temperature gradient along the sample, $\kappa $ is the thermal
conductivity. At present, for momentum-conserving 1D fluids and lattices, it
is generally believed that the thermal conductivity should diverge as $%
\kappa \sim L^{\alpha }$ with the increase of the system size $L$ for
momentum conserving low-dimensional systems[14-19]. Meanwhile, some
counterexamples with size-independent thermal conductivities have been also
found, such as the rotator model [20,21], a 1D lattice in effective magnetic
fields [22], the variant ding-a-ling model [23].
Recently we find that momentum-conserved lattice models with asymmetric
interparticle interactions [24] can also result the normal heat conduction.
This result is confirmed by investigating the time-dependent behavior of
current autocorrelation functions in one-dimensional lattice systems with
the asymmetric and symmetric interaction potentials [25]. The current
autocorrelation is defined as
\begin{equation*}
C(t)=\langle J(t)J(0)\rangle ,
\end{equation*}
where $J(t)$ represents the current fluctuation at time $t$ and Here $%
\langle \cdot \rangle $ denotes the equilibrium thermodynamic average.
Following the linear response theory [27], the thermal conductivity may be
calculated by the Green-Kubo formula
\begin{equation}
\kappa =\lim_{\tau \rightarrow \infty }\lim_{N\rightarrow \infty }\frac{1}{\
2k_{B}T}\int_{0}^{\tau }C(t)dt,
\end{equation}
once the correlation function $C(t)$ is obtained, where $\tau $ is the time
of evolution, $L$ is the linear dimension of the system along which the
current flows, $k_{B}$ is the Boltzmann constant, $T$ is the temperature of
the system. Differing from the direct nonequilibrium calculation based on
equition (1), the thermal conductivity here is calculated with current
fluctuations in the euilibrium system. After studying different types of
interaction potentials, it is found [25] that with proper degree of
asymmetry, the current autocorrelation may show rapid decay which led to the
convergence of the Green-Kubo formula.
It is well-known that the asymmetry interaction may induce the thermal
expansion while the symmetry one may not, and real materials usually show
thermal expansion effect[26]. Thus, our finding has particular importance
for real materials. It implies that low-dimensional materials may also have
the size-independent thermal conductivity in the thermal limit as the bulk
materials, and the Fourier's law of heat conduction is generally valid also
for low-dimensional materials. However, as mentioned above, at present it is
generally accepted that the heat current autocorrelation decays in power-law
and the thermal conductivity diverges with the system size in
one-dimensional momentum conserving systems. Meanwhile, the models we have
studied [25,26] have a combination potential of nonlinearity and
asymmetricity. Therefore, whether the convergent thermal conductivity is
resulted by the nonlinearity or the asymmtry feature needs to be clearified.
plot of asymmetric harmonic interaction potential with $r=0.0$, $%
r=0.3$, $r=0.5$ and $r=0.7$.
(a) Temperature profiles for the asymmetric harmonic interaction
potential with fixed $r=0.5$. The temperatures of the two heat baths coupled
to the system are $T_{L}=3$ and $T_{R}=2$ respectively, (b) The heat
conductivity $\protect\kappa $ vs the number of particles $N$ for $r=0.3$, $%
r=0.5$ and $r=0.7$, (c) The mass density function respectively for the
asymmetric harmonic interaction potential model with fixed $r=0.3$, $r=0.5$
and $r=0.7$ and the Fermi-Pasta-Ulam-$\protect\beta $ (FPU-$\protect\beta $)
model. The system size is $N=2000$. Other parameters are $T_{L}=3$ and $%
This paper studies an one-dimensional momentum conserving lattice with
simple asymmetric interparticle interactions: The compress and stretch are
govern by different harmonic potentials. In more detail, we study the
lattice described by the Hamiltonian
\begin{equation}
\end{equation}
where $p_{i}$ and $x_{i}$ represent the momentum and the deviation from its
equilibrium position of the $i$th particle respectively. The potential is as
\begin{equation}
\begin{array}{ll}
{\frac{1}{2}}(1+r)x^{2} & \text{if $x<0$ ,} \\
{\frac{1}{2}}(1-r)x^{2} & \text{otherwise .}%
\end{array}%
\right.
\end{equation}
where $r$ control the degree of the asymmetry. The potentials with several $r
$ are plotted in Fig.1. This potential and its higher order derivatives are
continuous at $x=0$ except the second derivative.
To perform the nonequilibrium simulations, the Nose-Hoover heat baths [28]
with temperatures $T_{L}$ and $T_{R}$ are coupled to the left and right
particles respectively. The fixed boundary conditions are applied in the
simulation. The equations of the motion of particles in heat baths is given
\begin{equation}
\dot{x}_{1,N}=\frac{p_{1,N}}{\mu },\quad \dot{p}_{1,N}=-\frac{\partial H}{%
\partial x_{1,N}}-\varsigma _{\pm }p_{1,N},\quad \dot{\varsigma}_{\pm }=%
\frac{p_{i}^{2}}{T_{\pm }}-1,
\end{equation}
and the motions of $N-2$ other particles are described by
\begin{equation}
\dot{x}_{i}=\frac{p_{i}}{\mu },\quad \dot{p}_{i}=-\frac{\partial H}{\partial
\end{equation}
We integrate the equations of motion by using the leap-frog integrating
algorithm. The local temperature and local heat current at the $i$th site
are calculated by $T_{i}\equiv \langle p_{i}^{2}\rangle $ and $J_{i}\equiv
\langle \dot{x}_{i}\frac{\partial H}{\partial x_{i}}\rangle $
respectively[8]. The simulations are performed with a sufficient long time,
usually $t>10^{7}$, to ensure the system reaching a stationary state. In
such a state, the local current is equal to the global flux, $J_{i}=J$. To
avoid the finite-size effect, the system size $N$ is extended until the the
temperature profiles would fit with each other by rescaling the $x$ variable
with factor $1/N$, in which case $dT/dx\symbol{126}N/(T_{L}-T_{R})$ and the
thermal conductivity can thus be calculated following $\kappa =\langle
J\rangle N/(T_{L}-T_{R})$.
The autocorrelation function of the heat flux,$C_{j}(t)$, for the
asymmetric harmonic interaction potential with $r=0.5$.
Figure.2(a) shows the temperature profiles with $r=0.5$ for several system
sizes. One can see that for $N>10^{4}$ the temperature profiles are
well-rescaled together. Figure.2 (b) shows the thermal conductivity as a
function of the system size. It can be seen that $\kappa $ converges to be a
size-independent constant gradually. A remarkable difference to the case of
models with symmetry potential, such as the results shown in refs.
[4-7,10,24] where the finite-size effect disappears usually before $N>10^{3}$
, is that the convergence threshold of the system size is quite long in this
model. This feature appears also in our previous work [24,25] where a
different asymmetry interaction potential is applied. We guss it may be one
of the reason why previous researchers have not observed convergent thermal
conductivity even they also investigated certain asymmetry-potential lattice
The convergence threshold is related to the degree of the asymmetry. In
Fig.2, we also show the results with $r=0.3$ and $r=0.7$ respectively. It is
clear that the threshold of convergent thermal conductivity increases with
the decrease of the asymmetry degree. However, further increase $r$ may
result simulation difficult. We have to apply a very small intergral step to
guaratee the intergral precision.
As in reference [24], we calculate the mass density $\rho (x)$ along the
lattice chain at non-equilibrium stationary states. The results are shown in
Fig. 2(c). It can be seen that mass gradients do set up alone the chain in
the asymmetry cases, while in the sysmmetry case of $r=0$ there is no such a
gradient. The density is inverse proportional to the temperature. This is a
result of positive $r$, in which case the compress is difficult than the
stretch. If one apply a negative $r$ to the potential, in which case the
compress is easy than the stretch, he shall found that the mass desity is
proportional to the temperature. This is a qualitative different property
between symmetry and asymmertry lattice systems, and may provide clue to
understand why qualitative difference in heat transport is resulted. We guss
that the mass gradient can induce additional scattering of the current and,
together with other scattering mechanism, result the normal heat condduct
The decay behavior of the current autocorrelation can further confirm that
the asymmetry interparticle interactions may result the normal heat conduct
of lattice systems. To calculate the current autocorrelation function, we
first evolute the system for a sufficient long time to relax the system to
its equilibrium state. Then the currelation function $C(t)=\langle
J(t)J(0)\rangle $ is calculated by applying the emporal current fluctuations
$J(t)=\sum\limits_{i=1}^{N}J_{i}(t)$. The decay behavior of $C(t)$
determines whether the heat flux violates the Fourier law of heat
conduction. If it decays as $C(t)\sim t^{-\gamma }$ with $\gamma <1$, the $%
\kappa $ will divergent following the Green-Kubo formula. If it decays
faster than $\gamma =1$, particularly with exponential decay of $C(t)\sim
e^{-\delta t}$, the Green-Kubo formula converges and the thermal
conductivity is size-indepedent in the termaldynamical limit. The Fourier
law is thus obeyed. Figure 3 shows the current autocorrelation functions
corresponding to the parameter sets applied in Fig. 2. To perform the
simulation, periodic boundary conditions are applied with several simulation
sizes. The temperature is set to be $T=2.5$ which is corresponding to the
average temperature applied in the nonequilibrium simulations. One may seen
that the curves with different simulation sizes overlap with each other,
indicating that the finite-size effect is avoided. It is clear, either with
the log-log plot or semi-log plot, the decay of the autocorrelation function
is quite fast, even approaches the exponential decay manner, indicating a
convergent thermal conductivity.
In conclusion, the lattice model even with asymmetric harmonic
inter-particle interactions shows normal thermal conduction behavior. Our
nonequilibrium simulations obtain a size-independent thermal conductivity
when the simulation size is sufficient long. Our equilibrium simulations
show that the current autocorrelation decay faster than the power-law decay
of $C(t)\sim t^{-1}$, implying a convergent thermal conductivity according
to the Green-Kubo formula. Because this model involves only the asymmety
harmonic interactions, our results thus confirm that it is the asymmetry of
interaction potentials resulting the normal thermal conduct behavior of
one-dimensional momentum conserving lattices.
This model has another obvious adventage. With a scale transformation $(%
\widetilde{x},\widetilde{t})\rightarrow (\alpha x,t)$, the Hamiltonian
changs as $\widetilde{H}\rightarrow \alpha ^{2}H$. Therefore, the dynamics
of the system keep unchange with the scale transformation. In more detail,
the systems with Hamiltonian $\widetilde{H}$ at temperature $\widetilde{T}$
and with Hamiltonian $H$ at temperature $T$ are identical, where $\widetilde{%
T}=\alpha ^{2}T$. Therefore, the conclusion of normal thermal conductivity
can be directly extended to any temperature.
We have observed that the mass density is set up alone the lattice chain in
the case of asymmetry potentials. This phenomenon may be important in
understanding the microscopic mechanism of the normal thermal conductivity
in nonequilibrium systems. In equilibrium systems, there is no such a
stationary gradient of mass density. How a rapid decay of the current
autocorrelation is arisen is an open problem and asks further studies.
We thank Shunda Chen for profitable discussions. This work is supported by
the NNSF (Grants No. 10925525, No. 10975115) and SRFDP (Grant No.
20100121110021) of China.
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1511.00371
|
Differentiable stratified groupoids]Differentiable stratified groupoids and a de Rham theorem for inertia spaces
Department of Mathematics, University of Colorado at Boulder, Campus
Box 395, Boulder, CO 80309-0395, USA
Department of Mathematics, University of Colorado at Boulder, Campus
Box 395, Boulder, CO 80309-0395, USA and Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, 04103 Leipzig, Germany
Department of Mathematics and Computer Science,
Rhodes College, 2000 N. Parkway, Memphis, TN 38112, USA
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid,
generalizations of Lie groupoids in which the spaces of objects and arrows have the
structures of differentiable spaces, respectively differentiable stratified spaces,
compatible with the groupoid structure. After studying basic properties of these groupoids
including Morita equivalence, we prove a de Rham theorem for
locally contractible differentiable stratified groupoids.
We then focus on the study of the inertia groupoid associated to a proper Lie groupoid.
We show that the loop and the inertia space of a proper Lie groupoid
can be endowed with a natural Whitney B stratification, which we call the
orbit Cartan type stratification. Endowed with this stratification,
the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable
stratified groupoid.
§ INTRODUCTION
The theory of Lie group actions on smooth manifolds is fundamental for several areas in mathematics and
has a long tradition. But examples of natural Lie group actions do not only comprise actions on smooth manifolds but
also on singular spaces such as orbifolds <cit.> or manifolds with boundary or corners
<cit.>. Lie group actions on singular spaces arise also in singular symplectic
reduction <cit.> and the transverse cotangent bundle <cit.>.
In these cases, the corresponding translation groupoid, while not a
Lie groupoid, has significantly more structure than merely that of a topological groupoid. In particular,
the spaces of objects and arrows inherit differentiable stratified space structures compatible with the
groupoid structure maps. One of the main goals of this paper is to define and study the category of such groupoids
and their singular structures.
In this paper, we propose in Definition <ref> the notion of
a differentiable stratified groupoid, a class
of groupoids within the category of differentiable stratified spaces.
That definition is designed so that, under mild hypotheses,
the restriction of a differentiable stratified groupoid to a stratum of the orbit space is a Lie groupoid;
cf. Proposition <ref>.
One major motivating example is that of a compact Lie group acting
differentiably on a differentiable stratified space in such a way that the strata are permuted by the action.
We will see in Proposition <ref> that the corresponding translation
groupoid will always satisfy our definition, from which it follows that our definition includes many of
the examples described above. Indeed, most examples are locally translation differentiable stratified
groupoids, a particular subclass of differentiable stratified groupoid which can be locally described as
a translation groupoid in a compatible way; see Definitions <ref> and
We study the properties of these groupoids and their object spaces, including the appropriate notion
of Morita equivalence in these categories, and prove a de Rham theorem that relates the singular
cohomology of the orbit space of a locally translation differentiable stratified groupoid
fulfilling a local contractibility hypotheses to the cohomology of basic differential forms on the object
space; see Theorem <ref>. To prove this theorem, we first localize
to groupoid charts and then prove a Poincaré lemma for basic forms in the context of locally translation
differentiable stratified groupoids.
A particularly important example which we consider here is that of the inertia groupoid
of a proper Lie groupoid $\sfG$.
The inertia groupoid is presented as the translation groupoid of $\sfG$ acting on the so-called
loop space $\inertianull{\sfG}\subset \sfG$ which consist of all arrows having the same source and target.
When $\sfG$ is an orbifold groupoid, the loop space $\inertianull{\sfG}$ is a smooth manifold so that
the inertia groupoid presents an orbifold, the inertia orbifold. The inertia orbifold plays
an import role in the geometry and index theory of orbifolds; see <cit.>. However,
when $\sfG$ is not an orbifold groupoid, the loop space $\inertianull{\sfG}$ becomes singular,
and very little is known about its singularity structure in general. One of the goals of this paper is to deepen the
understanding of this groupoid by exhibiting an explicit stratification.
The inertia groupoid represents the inertia stack $\inertia{\mathfrak{X}}$ of
the differentiable stack $\mathfrak{X}$ represented by $\sfG$, which plays an important role in
the string topology of $\mathfrak{X}$; see <cit.>. The inertia stack can be interpreted
as the collection of hidden loops, elements of the free loop stack
$\operatorname{L}\mathfrak{X}$ of $\mathfrak{X}$ that vanish on the course moduli space
$|\mathfrak{X}|$; see <cit.> for more details.
Note that when the adjoint action of $G$ on itself is considered, the loop space corresponds to the set
of pairs of commuting elements in $G$, which has been considered in
<cit.>. Similarly, one may consider the space of
(conjugacy classes of) commuting $m$-tuples of elements of $G$
as well as the spaces of conjugacy classes of homomorphisms $\pi\to G$ where $\pi$ is a
finitely generated discrete group <cit.>.
When $\sfG$ is the translation groupoid $G \ltimes M$ associated to the smooth action of
a compact Lie group $G$ on a manifold $M$, the orbit space of the $G$-action on the corrresponding loop space
- here this orbit space is called the inertia space - was first considered in <cit.>.
In the previous paper <cit.>, we introduced an explicit Whitney B stratification of the loop space
and its associated orbit space. The stratification given there does in general not yield a well-defined global
stratification of the loop space of a proper Lie groupoid, though. Specifically, the local stratifications from that paper
may not coincide on intersections of charts.
Here, we will give a modification of the stratification in <cit.> that yields a well-defined
stratification of the loop and inertia spaces of an arbitrary proper Lie groupoid. We call this stratification the
orbit Cartan type stratification. For the translation groupoid of a compact group action
the orbit Cartan type stratification presented here is in general coarser than the one considered
in our previous paper <cit.>.
Because the construction is local, we carry it out on a
single groupoid chart, which, up to Morita equivalence, is given by the translation groupoid
associated to a finite-dimensional $G$-representation $V$ where $G$ is a proper Lie groupoid, see Theorem
<ref>. For this case, the main ideas of our construction rely on providing
a stratification of slices, see Section <ref> for details. To achieve this,
we first decompose the Cartan subgroups of the stabilizers into equivalence classes determined by
their fixed sets in $V$, which motivates the name of the stratification. We then use these
decompositions together with the fixed sets of the stabilizers to stratify the slice and take
saturations to stratify the loop and orbit spaces. In Section <ref> we
demonstrate that our orbit Cartan type decomposition indeed satisfies all the properties of
a stratification and turns the loop and the inertia space into differentiable stratified spaces.
Moreover, in Section <ref>, we show that the orbit Cartan type stratification
is Whitney B regular. That this stratification is given explicitly allows us finally to verify in Theorem <ref> that
the local contractibility hypotheses of Definition <ref> is fulfilled for
the inertia space, hence the de Rham Theorem <ref> holds in this case.
Let us mention that the latter result provides the main tool for the proof of Brylinski's
claim <cit.>, which is not fully proven in his paper,
that the complex of basic differential forms on the loop space of a
transformation groupoid $G\ltimes M$ is acyclic. Since this
complex of basic differential forms naturally coincides with the Hochschild homology of the
convolution algebra on $G\ltimes M$, the $E^2$-term of the spectral sequence from Hochschild to cyclic
homology is given by the cohomology of the sheaf which the sheaf complex of basic differential forms
resolves. The results of our paper will be crucial to extend Brylinski's observations
to the proper Lie groupoid case. Work on this is in progress.
C.F. would like to thank the Harish-Chandra Research Institute (Allahabad, India) and the
DST Center for Interdisciplinary Mathematical Sciences of Banares Hindu University (Varanasi, India) for hospitality
during work on this manuscript.
C.S. was supported by a Rhodes College Faculty Development Grant, the E.C. Ellett Professorship in Mathematics, and
the Meyers fund. In addition, C.S. would like to thank the University of Colorado at Boulder for hospitality during
work on this manuscript.
M.J.P. would like to thank the Max-Planck Institute for the Mathematics of the Sciences in Leipzig, Germany, for
hospitality and support. NSF support under contract DMS 1105670 is also kindly acknowledged.
§ FUNDAMENTALS
In this section, we give the definitions and basic properties of the groupoids under consideration.
See Appendices <ref> and <ref> for a review of the notions
of differentiable and differentiable stratified spaces used in this paper.
§.§ Topological groupoids
Recall that by a groupoid $\sfG$ one understands a small category
with object set $\sfG_0$ and arrow set $\sfG_1$ such that all arrows are invertible. We use
$s, t\co\sfG_1\to\sfG_0$ to denote the source and target maps, respectively, write $u\co \sfG_0\to\sfG_1$
for the unit map, $i\co\sfG_1\to\sfG_1$ for the inverse map, and
finally denote by $m\co\sfG_1\sttimes\sfG_1\to\sfG_1$ the multiplication or composition map.
The maps $s$, $t$, $u$, $i$, and $m$ are collectively referred to as the structure maps of the groupoid.
If $\sfG$ and $\sfH$ are groupoids, a morphism $f\co\sfG\to\sfH$ is a functor,
i.e. a pair of functions $f_0\co\sfG_0\to\sfH_0$ and $f_1\co\sfG_1\to\sfH_1$ that commute
with each of the structure maps.
The orbit through a point $\obj{x}\in \sfG_0$ is defined as the set of all $\obj{y} \in \sfG_0$
for which there exists a $\arr{g}\in \sfG_1$ such that $s(\arr{g})=\obj{x}$ and
$t(\arr{g})=\obj{y}$. It is denoted by $\sfG \obj{x}$.
Obviously, the object space is partitioned into orbits. We denote the set of orbits of a groupoid $\sfG$
by $|\sfG|$, and the canonical projection from the object to the orbit space by $\pi\co \sfG_0 \to |\sfG|$.
A groupoid $\sfG$ is called a topological groupoid if $\sfG_0$ and $\sfG_1$ are
topological spaces and each of the structure maps are continuous. This implies in particularly that
the unit map is a homeomorphism onto its image. If the topological groupoid $\sfG$ is
Hausdorff, which means that $\sfG_1$ is Hausdorff, the image $u(\sfG_0)$ is closed in $\sfG_1$,
cf. <cit.>. The orbit space of a topological
groupoid $\sfG$ is always assumed to carry the quotient topology with respect to the canonical projection
$\pi\co \sfG_0 \to |\sfG |$. If $\sfG$ and $\sfH$ are topological groupoids and $f\co\sfG\to\sfH$ a
morphism of groupoids, then $f$ is a morphism of topological groupoids, if in addition $f_0$ and $f_1$ are
continuous functions.
If the source map of a topological groupoid $\sfG$ is an open
map one says that $\sfG$ is an open topological groupoid. Note that as $i$ is a homeomorphism
and $t = s\circ i$, the target map of an open topological groupoid is open as well. A topological groupoid $\sfG$
for which the source map $s$ is a local homeomorphism is called an étale groupoid.
$\sfG$ is called a quasi-proper groupoid if $s\times t \co \sfG_1\times\sfG_1 \to \sfG_0$ is a proper
map, and a proper groupoid if it is Hausdorff and quasi-proper. Finally, we say that a topological
groupoid $\sfG$ is compact, respectively locally compact, if $\sfG_1$ as a topological space has the
respective property. Note that $\sfG_0$ is then as well compact, respectively locally compact, by
the proof of <cit.>.
In this paper we follow the definitions of quasi-compact, compact, and locally compact spaces by
Bourbaki <cit.>.
That means, a topological space $X$ is called quasi-compact if every open cover of $X$
admits a finite subcover, compact if $X$ is both Hausdorff and quasi-compact, and finally
locally compact if it is Hausdorff and every point $x\in X$ has a compact neighborhood.
If each point of a topological space $X$ possesses a Hausdorff (respectively quasi-compact) neighborhood,
we say that $X$ is locally Hausdorff (respectively locally quasi-compact).
A continuous map
$f\co X \to Y $ between not necessarily Hausdorff spaces $X$ and $Y$ is called proper
if $f\times \id_Z \co X \times Z \to Y \times Z$ is a closed map for every topological space $Z$,
cf. <cit.> or <cit.>.
By <cit.>, properness of $f$ is equivalent to the property that $f$ is
closed and $f^{-1}(y)$ is quasi-compact for each $y\in Y$.
Let $\sfG$ be a topological groupoid.
Then the following holds true:
* If $\sfG$ is proper and the object space $\sfG_0$ is locally compact, then $\sfG$ is a
locally compact groupoid.
* The quotient map $\pi\co \sfG_0 \to |\sfG|$ is an open map of topological spaces
if $\sfG$ is an open groupoid.
* The orbit space of $\sfG$ is Hausdorff if $\sfG$ is open,
$\sfG_0$ is locally compact, and
$(s,t)(\sfG_1)$ is closed in $\sfG_0 \times \sfG_0$. In particular, this is the case if
$\sfG$ is a proper open groupoid.
* The orbit space of $\sfG$ is locally compact if
$\sfG$ is a locally compact open groupoid
and $(s,t)(\sfG_1)$ is locally closed in $\sfG_0 \times \sfG_0$.
Let $\arr{g} \in \sfG_1$, $K$ be a compact neighborhood of $s(\arr{g})$, and $L$ be a compact neighborhood of $t(\arr{g})$.
Then $(s,t)^{-1} (K\times L)$ is a compact neighborhood of $\arr{g}$. Since $\sfG_1$ is Hausdorff, the
first claim is proved. Point (<ref>) is an immediate consequence of <cit.>.
The claims (<ref>) and (<ref>) follow from
§.§ Differentiable groupoids
Let $\sfG$ be an open topological groupoid. We say that $\sfG$ is a differentiable groupoid
if $\sfG_0$ and $\sfG_1$ are differentiable spaces and the structure maps
$s$, $t$, $i$, $u$, and $m$ are morphisms of differentiable spaces. We say
that $\sfG$ is a reduced differentiable groupoid if $\sfG_0$ and $\sfG_1$ are reduced differentiable spaces.
A morphism of topological groupoids $f\co\sfG\to\sfH$ between differentiable groupoids $\sfG$ and $\sfH$ is called
a morphism of differentiable groupoids if the functions $f_0$ and $f_1$ are both morphisms of
differentiable spaces.
The requirement that the structure maps $s\co\sfG_1\to\sfG_0$ and $t\co\sfG_1\to\sfG_0$
are morphisms of the differentiable spaces $\sfG_0$ and $\sfG_1$ implies by
<cit.> that the fibred product
$\sfG_1\sttimes\sfG_1$ inherits the structure of a differentiable space. It is with respect to this structure
that we require that $m\co\sfG_1\sttimes\sfG_1\to\sfG_1$ is a morphism
of differentiable spaces. As the inverse map $i\co\sfG_1\to\sfG_1$ is clearly invertible
with $i^{-1}=i$, it follows that $i$ is an isomorphism of differentiable spaces. Similarly,
$u\co\sfG_0\to\sfG_1$ is an embedding of differentiable spaces.
If $\sfG$ is a differentiable groupoid, the underlying topological spaces
$\sfG_0$ and $\sfG_1$ need not be Hausdorff. However,
because both carry the structure of a differentiable space, $\sfG_0$ and $\sfG_1$ are locally Hausdorff and
locally quasi-compact, see <cit.>.
Let $G$ be a Lie group and let $X$ be a Hausdorff differentiable space with a differentiable $G$-action,
cf. <cit.>.
Then $G\ltimes X$ is a differentiable groupoid with space of objects $X$ and space of arrows
$G \times X$, the latter being a differentiable space by <cit.>.
The source map $s\co G \times X \to X$ is the projection, hence smooth and open, and
the target map $t\co G \times X \to X$ is given by
$(g, x) \mapsto gx$, which is differentiable by the definition of a differentiable action.
The unit map $u\co x \mapsto (e, x)$ is easily seen to be a closed embedding since $\{ e \}$ is closed
in $G$. The domain of the product map $m$ is a differentiable subspace of $G\times X \times G \times X$
by <cit.> and the proof of <cit.>.
Moreover, the map $G\times G \times X \to (G\times X) \sttimes (G \times X)$,
$(g,h,x) \mapsto \big((g, hx), (h, x)\big)$
is an isomorphism of differentiable spaces onto the domain of $m$. Since $m$ pulled back by this isomorphism
is the smooth map $G \times G \times X \to G \times X$, $(g, h, x) \mapsto (gh, x)$,
multiplication on $G\ltimes X$ is smooth. Similarly one shows that the inverse map
$i \co G \times X \to G \times X$, $(g, x) \mapsto (g^{-1}, gx)$ is smooth.
By definition, the quotient space $X/G$ of the $G$-space $X$ coincides with the orbit space
of the groupoid $|G\ltimes X|$. Hence, if $X$ is Hausdorff, the orbit space becomes
a Hausdorff space by Prop. <ref> and
even is a differentiable space by <cit.> when equipped with the sheaf
of $G$-invariant functions on $X$ as the structure sheaf.
* In general, the orbit space $|\sfG |$ of a differentiable
groupoid $\sfG$ need not admit a differentiable structure with respect to which the quotient
map $\sfG_0 \to |\sfG |$ is a smooth map. As a well-known example, consider the
action of $\Z$ on the circle $S^1$ by an irrational rotation. Then the translation groupoid
$\Z\ltimes S^1$ is a (non-quasi-proper) differentiable groupoid whose orbit space $|\Z\ltimes S^1|$
is not locally Hausdorff and hence can not carry the structure of a differentiable space.
Let $\sfG$ be a reduced differentiable groupoid and $Y$ a differentiable subspace of $\sfG_0$.
A bisection of $\sfG$ over $Y$ then is a smooth map $\sigma\co Y\to \sfG_1$ such that
$s\circ\sigma = \id_Y$, and such that $t\circ\sigma$ is an isomorphism of $Y$ onto the differentiable
subspace $t\circ\sigma(Y)$ of $\sfG_0$, cf. <cit.>.
Note that we will consider bisections over sets $Y$ which may not be open in $\sfG_0$.
In the case that $\sfG$ is proper, the question of whether $|\sfG |$ admits a
differentiable structure remains open in general.
However, from Example <ref> (<ref>), it is clear that the
quotient map of the action groupoid of a differentiable action on a Hausdorff differentiable space
is a morphism of differentiable spaces. Many of the examples we consider will be of this form, at least
locally, which motivates the following.
We say that a reduced differentiable groupoid $\sfG$ is locally translation if
the following conditions are satisfied for every $\obj{x} \in \sfG_0$.
The isotropy group $\sfG_\obj{x}$ becomes a Lie group with the induced topological and differentiable structures.
There is an open Hausdorff neighborhood $U_\obj{x}$ of $\obj{x}$ in $\sfG_0$ and a
relatively closed connected reduced differentiable subspace $Y_\obj{x} \subset U_\obj{x}$ containing $\obj{x}$
together with a smooth $\sfG_\obj{x}$-action on $Y_\obj{x}$ such that $\obj{x}$ is a fixed point of the
$\sfG_\obj{x}$-action and such that the restriction $\sfG_{|U_\obj{x}}$ is isomorphic as a differentiable groupoid to
the product of the translation groupoid $G_\obj{x}\ltimes Y_\obj{x}$ and the pair groupoid
$O_\obj{x} \times O_\obj{x}$, where $O_\obj{x}$ is an open neighborhood of $\obj{x}$ in its orbit.
For each $\obj{z} \in Y_\obj{x}$, we may choose $U_\obj{z}$ and $Y_\obj{z}$ with $U_\obj{z} \subset U_\obj{x}$ and $Y_\obj{z} \subset Y_\obj{x}$.
For each arrow $\arr{g} \in \sfG$ with $s(\arr{g}) =\obj{x}$
there exist open neigborhoods $U_\obj{x}$ of $\obj{x}$ and $U_\obj{y}$ of
$\obj{y} := t(\arr{g})$ as in (LT<ref>) and a
bisection $\sigma$ over $U_\obj{x}$ such that
$\sigma(\obj{x}) = \arr{g}$ and such that the morphism of differentiable
groupoids $f \co \sfG_{|U_\obj{x}} \to \sfG_{|U_\obj{y}} $ with components
$f_0 := t\circ\sigma\co U_\obj{x} \to U_\obj{y}$ and
$f_1 \co \sfG_{|U_\obj{x},1} \to \sfG_{|U_\obj{y},1}$, $\arr{h} \mapsto \sigma (t(\arr{h})) \arr{h}\big(\sigma (s(\arr{h}))\big)^{-1} $
is an isomorphism.
A neighborhood $U_\obj{x}$ as in (LT<ref>) is called a
trivializing neighborhood of $\obj{x}$, a differentiable subspace
$Y_\obj{x}$ as in (LT<ref>) a $\sfG$-slice or
groupoid-slice of $\obj{x}$.
By the following observation one can assume, possibly after shrinking, that a $\sfG$-slice $Y_\obj{x}$ of
$\obj{x}$ possesses a $\sfG_\obj{x}$-equivariant singular chart
$\iota \co Y_\obj{x} \hookrightarrow T_\obj{x}Y_\obj{x} \cong \R^{\operatorname{rk}\obj{x}}$ with $\iota
(\obj{x})=0$; see Appendix <ref> for the definition of a singular chart.
Let $G$ be a compact Lie group, $Y$ a differentiable space carrying a differentiable $G$-action, and $x\in Y$ a
fixed point. Then the Zariski tangent space $T_xY$ inherits a natural $G$-action from the $G$-action on $Y$.
Moreover, there exists an open $G$-invariant neighborhood $W$ of $x$ in $Y$ and a $G$-equivariant singular chart
$\iota: W \hookrightarrow T_xY$ mapping $x$ to the origin.
The proof is literally identical to the proof of <cit.> when replacing “holomorphic” with “smooth”.
Let $\sfG$ be a locally translation differentiable groupoid. Then the orbit space
$|\sfG|$ inherits the structure of a differentiable space with respect to which
the quotient map $\pi\co \sfG_0 \to |\sfG|$ is a smooth map. Specifically, the structure
sheaf of $|\sfG|$ is given by the differentiable functions on $\sfG_0$ that are constant
on orbits.
We define the structure sheaf of $|\sfG|$ to be the sheaf of continuous functions
on $|\sfG|$ which pull back under the projection $\pi$ to $\sfG$-invariant smooth functions
on $\sfG_0$.
For each orbit $\sfG \obj{x}\in |\sfG|$, we may use condition (LT<ref>)
in Definition <ref> to identify a neighborhood of
$\sfG \obj{x}$ with $|G_\obj{x}\ltimes Y_\obj{x}|$, a Hausdorff differentiable space as explained in
Example <ref> (<ref>).
That these local identifications are well defined and isomorphisms of Hausdorff differentiable
spaces are consequences of (LT<ref>) and (LT<ref>).
§.§ Differentiable stratified groupoids
Recall that by a stratified submersion (respectively stratified immersion), one understands
a morphism $f\co X\to Y$ of reduced differentiable stratified spaces such that the restriction of $f$
to a connected component of a stratum of the maximal decomposition of $X$ is a submersion
(respectively immersion), cf. <cit.>. A stratified surjective submersion
is a stratified submersion that maps connected components of strata onto connected components
of strata, and a stratified embedding is a stratied immersion that is injective
on connected components of strata.
A differentiable stratified groupoid is a reduced differentiable groupoid
$\sfG$ such that the following properties hold true.
$\sfG_0$ and $\sfG_1$ are differentiable stratified spaces with respective
stratifications $\mathcal{S}^0$ and $\mathcal{S}^1$.
The structure maps $s$, $t$, $i$, $u$, and $m$ are stratified mappings.
The maps $s$ and $t$ are stratified surjective submersions, and $u$ is a stratified embedding.
For every $\obj{x}\in \sfG_0$ and arrow $\arr{g} \in s^{-1} (\obj{x})$ the germ $ [s^{-1}(\mathcal{S}_\obj{x}^0)]_\arr{g}$
is a subgerm of $\mathcal{S}_\arr{g}^1$.
Let $\obj{x}\in\sfG_0$, $\arr{g}\in\sfG_1$ with $t(\arr{g}) = \obj{x}$, and $U$ be an open
connected neighborhood of $\obj{x}$ within the stratum of $\sfG_0$ containing $\obj{x}$.
Assume that $\sigma\co U\to \sfG_1$ is
a bisection of $\sfG$. Then the map
$L_\sigma\co t^{-1}(U)\to\sfG_1$ defined by $\arr{h}\mapsto \sigma(t(\arr{h}))\arr{h}$
satisfies $L_\sigma(\mathcal{S}_\arr{g}^1) = \mathcal{S}_{L_\sigma(\arr{g})}^1$.
If $\sfG$ and $\sfH$ are differentiable stratified groupoids, a morphism of differentiable stratified groupoids
is a morphism of differentiable groupoids $f\co\sfG\to\sfH$ such that $f_0$ and $f_1$ are in addition
stratified mappings.
A differentiable stratified groupoid $\sfG$ is called a Lie groupoid
if in addition the following axiom holds true.
The stratifications $\mathcal{S}^0$ and $\mathcal{S}^1$ are induced
by $\sfG_0$ and $\sfG_1$, respectively, which in other words means
that both $\sfG_0$ and $\sfG_1$ are smooth manifolds and their
stratifications have only one stratum.
We say that $\sfG$ is a differentiable Whitney stratified groupoid if
the stratifications of $\sfG_0$ and $\sfG_1$, as well as the induced
stratification of $\sfG_1 \sttimes\sfG_1$, are Whitney (B)-regular. Similarly,
$\sfG$ is topologically locally trivial if
the stratifications of $\sfG_0$, $\sfG_1$, and the induced stratification of
$\sfG_1 \sttimes\sfG_1$ are topologically locally trivial
(cf. <cit.>).
In the definition, $[s^{-1}(\mathcal{S}_\obj{x}^0)]_\arr{g}$ of course means the germ of
$s^{-1}(S)$ at $\arr{g}$, where $S$ is a set defining the germ $\mathcal{S}_\obj{x}^0$ at
$\obj{x}$. Condition (DSG<ref>) is a kind of equivariance
condition for the stratification. It entails that every set germ of a source
fiber is contained in a stratum of the arrow space, so that the stratification
of the arrow space is not unnecessarily fine. This will be used to show that
$\sfG$-orbits are locally contained in strata; see
Lemma <ref> below. Likewise, (DSG<ref>)
requires that bisections defined on open subsets of strata act on $\sfG_1$ in
a way compatibe with its stratification. Note that a bisection
$\sigma\co S \to \sfG_1$ as in (DSG<ref>) has image in the
stratum of $\sfG_1$ through $\arr{g} = \sigma (\obj{x})$ by (DSG<ref>).
The existence of such bisections $\sigma$ is demonstrated by
Lemma <ref> and Corollary <ref>
* Though $\sfG_0$ and $\sfG_1$ are locally Hausdorff spaces, we do not require that they are Hausdorff.
See Appendix <ref> for stratifications
of locally Hausdorff spaces.
* One readily checks that Definition <ref> reduces to the standard definition of a
Lie groupoid if (DSG<ref>) is fulfilled. Observe that in the case of a Lie groupoid,
conditions (DSG<ref>) and (DSG<ref>) become trivial.
In order for the requirement that $m\co \sfG_1 \sttimes \sfG_1 \to \sfG_1$ is a stratified
mapping to make sense, it must be that $\sfG_1 \sttimes \sfG_1$ is a stratified space.
We will always let $\sfG_1 \sttimes \sfG_1$ carry the stratification induced by the
stratification of $\sfG_1$, the existance of which is guaranteed by
Lemma <ref>.
Our definition of a differentiable Whitney stratified groupoid is stronger than the one of a
stratified Lie groupoid given in <cit.> in that we require conditions
(DSG<ref>) and (DSG<ref>).
The following examples illustrate the kinds of behavior we preclude by
requiring these conditions.
Consider the translation groupoid $\sfG = \sphere^1 \ltimes \sphere^1$ where the action is
by left-translation.
We decompose $\sfG_0 = \sphere^1$ into pieces $\{ 1\}$ and $\sphere^1\smallsetminus\{1\}$,
and $\sfG_1 = \sphere^1 \times \sphere^1$ into pieces $\{(1,1)\}$,
$\{ (a,1)\mid a\neq 1\}$, $\{(a, a^{-1})\mid a\neq 1\}$, and
$\{ (a,b)\mid b\neq 1,\; a\neq b^{-1}\}$. Then it is immediate to check that
(DSG<ref>), (DSG<ref>), and
(DSG<ref>) are satisfied.
However, condition (DSG<ref>) fails, as
the germ of $s^{-1}(1) = \{ (a,1)\mid a\in \sphere^1 \}$ is not contained in
the stratum $\{(1,1)\}$.
Note that $\sfG_0$ consists of a single connected orbit that is given an
“artificial" stratification that is too fine.
Of course, $\sfG$ is a Lie groupoid when given the trivial stratifications of $\sfG_0$
and $\sfG_1$.
Let $G$ be a compact Lie group of positive dimension and let $\sfG = G \ltimes \{p\}$
be the translation groupoid associated to the trivial action of $G$ on a one-point space.
Define a stratification of $\sfG_1 = G \times\{p\}$ by the decomposition into
$\{ (1,p)\}$ and $\{(g,p)\mid g\neq 1\}$.
Obviously, the source and target maps are stratified submersions, the unit map
is a stratified embedding, and the inverse map is a stratified mapping. Though
the multiplication map $\sfG_1 \sttimes \sfG_1 = \sfG_1\times\sfG_1 \to \sfG_1$
given by $((g,p),(h,p))\mapsto(gh,p)$ is not a stratified mapping with respect
to the induced stratification, the space $\sfG_1 \sttimes \sfG_1$ does admit a
stratification with respect to which the multiplication map $m$ is stratified.
Specifically, we may decompose $\sfG_1 \sttimes \sfG_1$ into the pieces
\[
\{((g^{-1},p),(g,p))\mid g \in G \}
\quad\mbox{and}\quad
\{((g,p),(h,p))\mid g\neq h^{-1}\},
\]
and then $m$ becomes stratified. In this case, while $\sfG_0$ consists of a
single stratum, the stratification of $\sfG_1$ is “too fine" so
that (DSG<ref>) again fails;
the preimage $s^{-1}(p) = \sfG_1$ is not contained in the stratum $\{ (1,p)\}$
through $(1,p)$.
Let $\sfG$ be the pair groupoid on the topological disjoint union $\R\sqcup\{p\}$.
Give $\sfG_0$ and $\sfG_1$ the stratifications
by connected components and natural differentiable structures. One checks that
(DSG<ref>), (DSG<ref>), (DSG<ref>), and
(DSG<ref>) are satisfied. However, let
$\sigma \co \{ p \} \to \sfG_1$ be the
bisection with the single value $\sigma(p) = (0,p)$.
For any arrow of the form $(p,y)$ one has
$\sigma(t(p,y))(p,y) = (0,y)$. Hence the mapping $\arr{h} \mapsto \sigma(t(\arr{h}))\arr{h}$
maps the stratum $\{ (p,y)\mid y\in\R\}$ of $\sfG_1$ into the stratum $\R^2$
as the $y$-axis. Thus (DSG<ref>) fails.
We next collect some useful consequences of Definition <ref>.
We hereby assume for the remainder of this section that $\sfG$ denotes a
differentiable stratified groupoid.
Let $\arr{g}\in\sfG_1$ with $s(\arr{g})=\obj{x}$ and $t(\arr{g}) = \obj{y}$. Then one has
\begin{align}
\label{Ite1} \mathcal{S}_\arr{g}^1 & = [s^{-1}(\mathcal{S}_\obj{x}^0)]_\arr{g}, \text{ i.e.~$\mathcal{S}^1$ is the
pullback of $\mathcal{S}^0$ via $s$, \cite[(2.3)]{MatherStratMap}},\\
\label{Ite2} \mathcal{S}_\obj{x}^0 & =[s(\mathcal{S}_\arr{g}^1)]_\obj{x}, \\
\label{Ite3} \mathcal{S}_\arr{g}^1 & =[t^{-1}(\mathcal{S}_\obj{y}^0)]_\arr{g},\\
\label{Ite4} \mathcal{S}_\obj{y}^0 &= [t(\mathcal{S}_\arr{g}^1)]_\obj{y}, \text{ and}, \\
\label{Ite5} \mathcal{S}_\obj{y}^0 &= [t(s^{-1}(\mathcal{S}_\obj{x}^0))]_\obj{y} .
\end{align}
Let $R_\arr{g}$ be the connected component of the stratum of $\sfG_1$ containing $\arr{g}$,
and let $S_\obj{x}$ and $S_\obj{y}$ be the connected components of the strata of $\sfG_0$
containing $\obj{x}$ and $y$, respectively. Then $s_{|R_\arr{g}}$ and $t_{|R_\arr{g}}$ are, respectively, surjective submersions onto $S_\obj{x}$ and $S_\obj{y}$.
By (DSG<ref>), there exists a relatively open neighborhood
$U_\obj{x}$ of $\obj{x}$ in $S_\obj{x}$ and an open neighborhood $V_\arr{g}$ of $\arr{g}$ in $\sfG_1$ such
that $V_\arr{g}\cap s^{-1}(U_\obj{x}) \subset R_\arr{g}$. As $s_{|R_\arr{g}}$ is a smooth map onto $S_\obj{x}$,
$s^{-1}(U_\obj{x})$ is a relatively open neighborhood of $\arr{g}$ in $R_\arr{g}$,
proving (<ref>). Since $s_{|R_\arr{g}}$ is a submersion, hence an open map,
$s(V_\arr{g}\cap s^{-1}(U_\obj{x}))$ is an open neighborhood of $\obj{x}$ in $S_\obj{x}$, which gives
(<ref>). Then (<ref>) and (<ref>) follow from the fact that
$t = s\circ i$ and that $i$ is a stratified mapping with $i^2 = \id_{\sfG_1}$.
Finally, (<ref>) is a consequence of (<ref>) and (<ref>).
Let $\sfG$ be a differentiable stratified groupoid and let
$\obj{x} \in \sfG_0$ be a point. Then the connected component of $\obj{x}$
in the orbit $\sfG \obj{x}$ through $\obj{x}$ is contained in the stratum of $\sfG_0$
containing $\obj{x}$.
Let $S_\obj{x}$ be the connected component of the stratum of $\sfG_0$ containing $\obj{x}$.
Suppose for contradiction that the germ of $\sfG \obj{x}$ at $\obj{x}$ is not a subgerm of
$\mathcal{S}_\obj{x}^1$. Then one may construct a sequence $( \obj{x}_n )_{n\in \N}$ of
elements of $\sfG \obj{x}$ such that $\lim_{n\to\infty} \obj{x}_n = \obj{x}$, yet each $\obj{x}_n$ is
not contained in $S_\obj{x}$. Since the set $K = \{ \obj{x}_n\}_{n\in \N}\cup\{ \obj{x}\}$ is compact and
$\sfG$ is proper,
the preimage $(s,t)^{-1}(K)$ is a quasi-compact subset of $\sfG_1\times\sfG_1$.
Hence projecting onto the first factor yields a quasi-compact subset
$C := \operatorname{pr}_1((s,t)^{-1}(S))$ of $\sfG_1$. Note that $C$ is the
set of elements of $\sfG_1$ with source in $K$.
For each $n$, choose an arrow $\arr{g}_n\in\sfG_1$ with source $\obj{x}$ and target $\obj{x}_n$. As
each $\arr{g}_n$ is in the quasi-compact set $C$, there is a subsequence
$(\arr{g}_{n_k})_{k\in \N}$ with limit $\arr{g} \in\sfG_1$. However, (DSG<ref>) entails the existence of an open neighborhood $U_\obj{x}$ of $\obj{x}$ in $S_\obj{x}$ and a neighborhood $V_\arr{g}$ of $\arr{g}$ in $\sfG_1$ such that
$V_\arr{g} \cap s^{-1}(U_\obj{x})$ is contained in the connected component $R_\arr{g}$ of the stratum of $\sfG_1$ containing $\arr{g}$. Infinitely many of the $\arr{g}_{n_k}$ must be contained in $V_\arr{g}$ and each $\arr{g}_{n_k}$
is an element of $s^{-1}(U_\obj{x})$ by construction, so infinitely many of the $\arr{g}_{n_k}$ are contained
in $R_\arr{g}$. But since $t$ is a stratified mapping, infinitely many of the $\obj{x}_{n_k}$
are contained in $S_\obj{x}$, which is a contradiction. It follows that the germ of $\sfG \obj{x}$ at $\obj{x}$ is a subgerm of
$\mathcal{S}_\obj{x}$, hence that the connected component of $\sfG \obj{x}$ containing $\obj{x}$ is a
subset of $S_\obj{x}$.
The following property is important in realizing the consequences of
(DSG<ref>) and is proven as in the case of a Lie groupoid;
cf. <cit.>.
Let $\obj{x}, \obj{y}$ be two points of the differentiable stratified groupoid $\sfG$ and $\arr{g}$ an arrow
with $s(\arr{g}) = \obj{x}$ and $t(\arr{g}) = \obj{y}$. Denote by $S_\obj{x}$ and $S_\obj{y}$ the connected
components of the strata of $\sfG_0$ containing $\obj{x}$ and $ \obj{y}$, respectively.
If $\dim S_\obj{x} \leq \dim S_\obj{y}$, then there exists a relatively open neighborhood
$U_\obj{x}$ of $\obj{x}$ in $S_\obj{x}$ and a bisection $\sigma$ of $\sfG$ on $U_\obj{x}$ such that
$\sigma$ is a stratified mapping and $\sigma(\obj{x}) = \arr{g}$.
Of course, since $\sigma$ is only defined on a subset of the stratum through $\obj{x}$,
$\sigma$ being a stratified mapping means that its image is contained in the stratum
containing $\arr{g}$. The
hypothesis that $\dim S_\obj{x} \leq \dim S_\obj{y}$ will be seen to be unnecessary below.
Let $R_\arr{g}$ be the connected component of the stratum of $\sfG$ containing $\arr{g}$. Then
$s_{|R_\arr{g}}$ and $t_{|R_\arr{g}}$ are surjective submersions onto $S_\obj{x}$ and $S_\obj{y}$, respectively.
Moreover, there are relatively open neighborhoods of $\arr{g}$ in $s^{-1}(\obj{x})$ and $t^{-1}(\obj{y})$
contained in $R_\arr{g}$ by (DSG<ref>) and
Lemma <ref> (<ref>). Then there are subspaces
$E \subset F$ of the tangent space $T_\arr{g} R_\arr{g}$ such that
$T_\arr{g} R_\arr{g} = T_\arr{g} (s^{-1}(\obj{x})) \times E = T_\arr{g}(t^{-1}(\obj{y})) \times F$.
Choose a local section $\sigma\co U_\obj{x} \to R_\arr{g}$ such that $\sigma(\obj{x}) = \arr{g}$ and
such that the image of $T_\obj{x}\sigma$ is $E$. Then $T_\obj{x} (t\circ\sigma)$ is
injective so that we can shrink $U_\obj{x}$ in a way that $t\circ\sigma$ is a
diffeomorphism onto its image.
As an important consequence, we may now conclude that the strata of $\sfG_0$
that meet the orbit $\sfG \obj{x}$ of a point $\obj{x} \in \sfG_0$ must all have the same
dimension. The proof follows <cit.>.
Let $\sfG$ be a differentiable stratified groupoid, and
let $S$ be a connected component of a stratum of $\sfG_0$. Then the following
holds true.
Each stratum of $\sfG_1$ contained in $s^{-1}(S)$ has the same dimension.
The rank of $t$ on $s^{-1}(S)$ is constant.
Each connected component $S^\prime$ of a stratum of $\sfG_0$ such that
$s^{-1}(S)\cap t^{-1}(S^\prime) \neq \emptyset$ has the same dimension as $S$.
Let $S_1$ and $S_2$ be (not necessarily distinct) connected components of strata of
$\sfG_0$ such that $s^{-1}(S_1)\cap t^{-1}(S_2) \neq \emptyset$. We assume
$\dim S_1 \leq \dim S_2$. Otherwise, we may switch roles and apply the inverse
map, so this hypothesis introduces no loss of generality.
As $s$ and $t$ are stratified surjective submersions, $s^{-1}(S_1)$
and $t^{-1}(S_2)$ are both unions of connected components of strata of $\sfG_1$. Hence
their intersection is a union of connected components of strata. Let $\arr{g} \in \sfG_1$ with
$s(\arr{g}) = \obj{x} \in S_1$ and $t(\arr{g}) = \obj{y} \in S_2$. Then there exists by
Lemma <ref> a bisection $\sigma$ of $\sfG$ on a relatively open
neighborhood $U_\obj{x}$ of $\obj{x}$ in $S_1$ such that $\sigma(\obj{x}) = \arr{g}$. Let $R_{u(\obj{x})}$ and $R_\arr{g}$
denote the connected components of the strata of $\sfG_1$ containing $u(\obj{x})$ and $\arr{g}$,
respectively. By (DSG<ref>), there is a relatively open
neighborhood $V_{u(\obj{x})}$ of $u(\obj{x})$ in $R_{u(\obj{x})}$ such that $L_\sigma(V_{u(\obj{x})})$ is a
relatively open neighborhood of $\arr{g}$ in $R_\arr{g}$. The requirement that $t\circ\sigma$ is
injective implies that $L_\sigma$ is injective, hence that $R_{u(\obj{x})}$ and $R_\arr{g}$ have the
same dimension. Since $S_2$ and $\arr{g}$ were arbitrary, (<ref>) follows.
Moreover, from the definition of $L_\sigma$, we have
$t_{|L_\sigma(V_{u(\obj{x})})} \circ L_\sigma = (t\circ\sigma)\circ t_{|V_{u(\obj{x})}}$.
Then, as $L_\sigma(V_{u(\obj{x})})$ is an open neighborhood of $\arr{g}$ in $R_\arr{g}$ and $t\circ\sigma$ is
a diffeomorphism onto its image, the ranks of $t_{|R_{u(\obj{x})}}$ at $u(\obj{x})$ and $t_{|R_\arr{g}}$ at $\arr{g}$
coincide, yielding (<ref>).
Since $t$ is a stratified surjective submersion, (<ref>) is immediate.
In particular, the hypothesis $\dim S_\obj{x} \leq \dim S_\obj{y}$ in Lemma <ref>
is now seen to be superfluous. Indeed, given the other hypotheses, we always have
$\dim S_\obj{x} = \dim S_\obj{y}$.
Note that if $S_1$ and $S_2$ are connected components of strata of $\sfG_0$,
even connected components of the same stratum, it need not be the case that the strata of
$s^{-1}(S_1)$ and $s^{-1}(S_2)$ have the same dimension. As an example, let $G$ and $H$
be compact Lie groups, and let $\sfG$ be the disjoint union of $G\ltimes \{p\}$ and
$H \ltimes \{ q\}$. Then $\sfG_0$ is the discrete set $\{p,q\}$. The maximal stratification
of $\sfG_0$ contains a single stratum with two one-point connected components, yet the space of
arrows of $\sfG_{|\{p\}}$ and $\sfG_{|\{q\}}$ have dimensions $\dim G$ and $\dim H$, respectively,
which clearly need not coincide.
Assume that $\sfG_0$ and $\sfG_1$ are topologically locally trivial, and let
$S \subset \sfG_0$ be a connected component of a stratum of $\sfG_0$.
Let $\mathcal{P}$ be a collection of connected components of strata of
$\sfG_0$ such that $S \in\mathcal{P}$, and such that for each
$S^\prime \in\mathcal{P}$ the relation
$s^{-1}(S)\cap t^{-1}(S^\prime)\neq\emptyset$ is satisfied. Letting
$P = \bigcup_{S^\prime \in\mathcal{P}} S^\prime$, the restriction $\sfG_{|P}$ then
is a Lie groupoid.
By Corollary <ref>, each element of $\mathcal{P}$ has the same dimension
as $S$, hence each connected component of a stratum of $\sfG_1$ contained in
$s^{-1}(P)$ has the same dimension as well. The hypothesis that $\sfG_0$ is a
topologically locally trivial implies that connected components of strata of the
same dimension are separated from one another's closures so that $P$ is a manifold. The same holds for
$\sfG_1$ so that $s^{-1}(P)\cap t^{-1}(P)$ is a union of strata of the same dimension
and therefore a manifold. Hence $P$ and $s^{-1}(P)\cap t^{-1}(P)$ are manifolds,
which can both be given their trivial stratifications. The claim follows.
Of course, the hypothesis that $\sfG_0$ and $\sfG_1$ are topologically locally trivial is
only required so that the strata of $\sfG_0$ and $\sfG_1$ of the same dimension are not
contained in one another's closures. Any other hypothesis that ensures this is as well
We also note the following, which will be useful in the sequel.
Assume that $\sfG$ is proper and let $\obj{x}\in\sfG_0$. Then each
connected component of the orbit $\sfG \obj{x}$ is a smooth submanifold of
$\sfG_0$. If $\sfG_0$ and $\sfG_1$ are in addition topologically locally
trivial, then $\sfG \obj{x}$ is a smooth submanifold of $\sfG_0$.
Let $S_\obj{x}$ be the connected component of the stratum containing $\obj{x}$. Then the
connected component of $\sfG \obj{x}$ containing $\obj{x}$ is contained in $S_\obj{x}$ by Lemma <ref>.
The restricted groupoid $\sfG_{|S_\obj{x}}$ is a Lie groupoid by Proposition <ref>. Since $\sfG \obj{x}\cap S_\obj{x} = (\sfG_{|S_\obj{x}}) \obj{x}$
this implies that the connected components of $\sfG \obj{x}$ contained in
$S_\obj{x}$ are smooth submanifolds of $S_\obj{x}$, hence of $\sfG_0$.
If $\sfG_0$ and $\sfG_1$ are topologically locally trivial, then the restriction of
$\sfG$ to the saturation of $S_\obj{x}$ is as well a Lie groupoid, and the same argument
Finally, we include the following example to demonstrate that the hypothesis that
$\sfG_0$ and $\sfG_1$ are topologically locally trivial (or a similar requirement)
is necessary in Proposition <ref> and Corollary <ref>,
cf. <cit.>.
Let $\sfG_0 = S_1 \cup S_2 \subset \R^2$ where $S_1 = \{0\}\times (-1,1)$ and
$S_2 = \{ (x,y)\in \R^2\mid x > 0, y = \sin 1/x \}$, and let $\sfG$ be the pair
groupoid on $\sfG_0$. Then $S_1$ and $S_2$, both of dimension $1$, are the pieces
of a decomposition of $\sfG_0$ with $S_1 \subset \overline{S_2}$. The decomposition
of $\sfG_1$ is into pieces of the form $S_i \times S_j$, each of dimension $2$.
Then $\sfG$ is a differentiable stratified groupoid. However, the single orbit
$\sfG_0$ is not locally connected and hence not a manifold, and $\sfG$ is not
a Lie groupoid. Clearly, however, the restriction of $\sfG$ to either stratum
of $\sfG_0$ is a Lie groupoid.
§.§ Morita equivalence
Let $\sfG$ be a topological groupoid, $Y$ a topological space, and
$f \co Y \to \sfG_0$ a continuous function. Following <cit.>,
we denote by $\sfG[Y]$ the groupoid with object space
$\sfG[Y]_0 := Y$ and arrow space
\[
\sfG[Y]_1 :=
(Y\times Y)\fgtimes{(f,f)}{(t,s)} \sfG_1
\{ (\obj{y}, \obj{z}, \arr{g}) \in Y \times Y \times \sfG_1 \mid
t(\arr{g}) = f(\obj{y}), \: s(\arr{g}) = f(\obj{z}) \},
\]
and structure maps given as follows.
The source of $(\obj{y},\obj{z}, \arr{g}) \in \sfG[Y]_1$ is $\obj{z}$,
its target is $\obj{y}$. Multiplication maps
$ (\obj{y}, \obj{w}, \arr{g}), (\obj{w}, \obj{z}, \arr{h}) \in \sfG[Y]_1$ with $s(\arr{g})=t(\arr{h})$ to
(\obj{y}, \obj{w}, \arr{g}) \cdot (\obj{w}, \obj{z}, \arr{h}) := (\obj{y}, \obj{z}, \arr{g}\arr{h}).
The unit map is $\sfG[Y]_0 \to \sfG[Y]_1$, $y \mapsto (\obj{y}, \obj{y}, u(f(\obj{x})) )$, and the inverse map
$\sfG[Y]_1 \to \sfG[Y]_1$, $(\obj{y}, \obj{z}, \arr{g}) \mapsto (\obj{z}, \obj{y}, \arr{g}^{-1})$. Then $\sfG[Y]$ is a subgroupoid
of $Y \times Y \times \sfG$ where $Y \times Y$ denotes the pair groupoid and
where we identify $\sfG[Y]_0 = Y$ with
$Y \fgtimes{f}{\operatorname{id}} \sfG_0$.
By <cit.>, $\sfG[Y]$ is a locally closed
subgroupoid of $Y \times Y \times \sfG$, if $\sfG_0$ is locally Hausdorff.
Moreover, $\sfG[Y]$ is locally compact, if $\sfG$ and $T$ are locally compact.
Finally, by <cit.>, $\sfG[Y]$ is proper,
if $\sfG$ is proper.
Now suppose that $\sfG$ is a differentiable groupoid, $Y$ a differentiable space,
and $f$ a differentiable map. Then it is straightforward to see that $\sfG[Y]$ is a
differentiable subgroupoid of $Y \times Y \times \sfG$. In particular,
both $\sfG[Y]_0$ and $\sfG[Y]_1$ are defined as fibered products.
Similarly, if $\sfG$ and $\sfH$
are differentiable stratified groupoids, $Y$ is a differentiable stratified space,
and $f$ a differentiable stratified surjective submersion, then
$\sfG[Y]$ is a differentiable stratified groupoid as well, where $\sfG[Y]_0$
and $\sfG[Y]_1$ are given the induced stratifications.
Two open topological groupoids $\sfG$ and $\sfH$ are called
Morita equivalent as topological groupoids, if there exists a topological space $Y$ together
with open surjective continuous functions $f \co Y \to \sfG_0$ and
$g \co Y \to \sfH_0$ such that $\sfG[Y]$ and $\sfH[Y]$ are isomorphic as
topological groupoids. If $\sfG$ and $\sfH$ are differentiable groupoids,
$Y$ is a differentiable space, and $f$ and $g$ are
differentiable maps, then $\sfG$ and $\sfH$ are called Morita equivalent as differentiable groupoids,
if $\sfG[Y]$ and $\sfH[Y]$ are isomorphic as differentiable groupoids.
Similarly, if $\sfG$ and $\sfH$ are differentiable stratified groupoids, $Y$ is a
differentiable stratified space, and $f$ and $g$ are differentiable stratified
surjective submersions, then $\sfG$ and $\sfH$ are called Morita equivalent as differentiable
stratified groupoids, if $\sfG[Y]$ and $\sfH[Y]$ are isomorphic as differentiable
stratified groupoids.
One verifies immediately that Morita equivalence is transitive.
Specifically, if the maps $\sfG_0\overset{f_1}{\longleftarrow}Y\overset{g_1}{\longrightarrow}\sfH_0$
realize a Morita equivalence between (topological, differentiable, or differentiable
stratified) groupoids $\sfG$ and $\sfH$, and
$\sfH_0\overset{f_2}{\longleftarrow}Y^\prime\overset{g_2}{\longrightarrow}\mathsf{K}_0$ is
a Morita equivalence between $\sfH$ and $\mathsf{K}$, then
$\sfG_0 \xleftarrow{f_1\circ pr_1} Y\fgtimes{g_1}{f_2}Y^\prime \xrightarrow{g_2\circ pr_2} \mathsf{K}_0$
induces a Morita equivalence between $\sfG$ and $\mathsf{K}$.
If $\sfG$ and $\sfH$ are Morita equivalent open topological groupoids,
then the orbit spaces $|\sfG|$ and $|\sfH|$ are homeomorphic. Moreover,
if $\sfG$ and $\sfH$ are Morita equivalent differentiable groupoids
such that $|\sfG|$ and $|\sfH|$ admit the structure of differentiable spaces
and the quotient maps are differentiable, then
$|\sfG|$ and $|\sfH|$ are isomorphic as differentiable spaces.
Because the orbit spaces $|\sfG[Y]|$ and $|\sfH[Y]|$ are clearly homeomorphic,
it is sufficient to show that $|\sfG|$ is homeomorphic to $|\sfG[Y]|$.
To see this, define the map $\sfG[Y]_0 \to \sfG_0$ by $(\obj{y}, \obj{x}) \mapsto \obj{x}$.
Given an arrow $\arr{g}\in\sfG_1$ from $\obj{x}$ to $\obj{x}^\prime$, there exists, by
the surjectivity of $f$, a $\obj{y}^\prime$ such that $f(\obj{y}^\prime) = \obj{x}^\prime$
and hence an arrow $(\obj{y}, \obj{y}^\prime, \arr{g})$ from $(\obj{y}, \obj{x})$ to $(\obj{y}^\prime, \obj{x}^\prime)$.
Conversely, if $(\obj{y}, \obj{y}^\prime, \arr{g})$ is an arrow from $(\obj{y}, \obj{x})$ to $(\obj{y}^\prime, \obj{x}^\prime)$,
then $\arr{g}$ is by definition an arrow from $\obj{x}$ to $\obj{x}^\prime$. Hence
$(\obj{y},\obj{x}) \mapsto \obj{x}$ maps orbits to orbits. In the differentiable case, this map is
obviously differentiable, so that if the quotient map $\sfG_0 \to |\sfG|$ is
differentiable, then its composition with $(\obj{y},\obj{x}) \mapsto \obj{x}$ is also differentiable.
For Lie groupoids, the notion of Morita equivalence is often defined in terms
of morphisms called weak equivalences. We introduce a similar notion
as follows.
A morphism $f\co\sfG\to\sfH$ of differentiable stratified groupoids is called
a weak equivalence if it is essentially surjective and fully faithful, i.e.
if the following two conditions are satisfied.
The map
$t\circ\operatorname{pr}_1\co \sfH_1\fgtimes{s}{f_0}\sfG_0 \to \sfH_0$
is a stratified surjective submersion.
(FF) The arrow space $\sfG_1$ is a fibered product via the diagram
\[
\xymatrix{
\sfG_1
\ar[d]_{(s,t)} \ar[r]^{f_1} &\sfH_1 \ar[d]^{(s,t)}
\\
\sfG_0\times\sfG_0
\ar[r]^{(f_0,f_0)}
\sfH_0\times\sfH_0.
\]
There is an analogous notion of a weak equivalence between differentiable groupoids,
where axiom (ES) is replaced with the requirement that
$t\circ\operatorname{pr}_1$ is an open surjective map that admits local sections.
Note that if the object and arrow spaces of $\sfG$ and $\sfH$ are topologically
locally trivial, then the restrictions of $\sfG$ and $\sfH$ to
connected components of strata are Lie groupoids by Proposition <ref>.
One immediately checks that the restriction of a weak equivalence to strata yields a
weak equivalence of Lie groupoids. We have the following.
Let $\sfG$ and $\sfH$ be differentiable stratified groupoids. The following are
$\sfG$ and $\sfH$ are Morita equivalent as differentiable stratified groupoids.
There is a differentiable stratified groupoid $\mathsf{K}$ together with
weak equivalences $h\co\mathsf{K}\to\sfG$ and $k\co\mathsf{K}\to\sfH$.
Assume that (<ref>) holds true. Then there exists a differentiable
stratified space $Y$ together with open surjective
differentiable stratified submersions $f\co Y\to \sfG_0$ and $g\co Y\to \sfH_0$
such that $\sfG[Y]$ and $\sfH[Y]$ are isomorphic. Define the
groupoid $\mathsf{K}$ by setting $\mathsf{K}_0 = Y$ and
$\mathsf{K}_1 = \sfG_1 \fgtimes{(f\circ s_{\sfG},f\circ t_{\sfG})}
{(g\circ s_{\sfH},g\circ t_{\sfH})} \sfH_1$, and put
$s_{\mathsf{K}} := f\circ s_{\sfG} = g\circ s_{\sfH}$ and
$t_{\mathsf{K}} := f\circ t_{\sfG} = g\circ t_{\sfH}$.
The unit, inverse, and multiplication maps are defined component-wise. Then it is straightforward to
see that $\mathsf{K}$ is a differentiable stratified groupoid, where $\mathsf{K}_1$
is given the induced stratification. We define a morphism
$h \co \mathsf{K} \to \sfG$ by setting $h_0 = f$ and $h_1 = \operatorname{pr}_1$.
Then $h_0 = f$ is in fact surjective by hypothesis so that
$t_{\sfG}\circ\operatorname{pr}_1\co \sfG_1\fgtimes{s}{h_0}\mathsf{K}_0 \to \sfG_0$
is a composition of stratified surjective submersions, and $\mathsf{K}_1$
is a fibered product by construction, so $h$ is a weak equivalence. The weak
equivalence $k\co\mathsf{K}\to\sfH$ is defined identically.
Now assume that we have a weak equivalence $h\co\mathsf{K}\to\sfG$. Let
$Y = \sfG_1 \fgtimes{s_{\sfG}}{h_0} \mathsf{K}_0$ with the induced stratification,
and let $f = \operatorname{pr}_2 \co Y \to \mathsf{K}_0$ and
$g = s_{\sfG}\circ\operatorname{pr}_1 \co Y \to \sfG_0$. Then $f$ is obviously
an open surjective differentiable stratified submersion, while the same for $g$
follows from condition (ES) in Definition <ref> (along with applying
the inverse map). By definition, $\mathsf{K}[Y]$ and $\sfG[Y]$ both have $Y$ as object
space. The arrow spaces are given by
\[
\mathsf{K}[Y]_1=
(\sfG_1 \fgtimes{s_{\sfG}}{h_0} \mathsf{K}_0 \times
\sfG_1 \fgtimes{s_{\sfG}}{h_0} \mathsf{K}_0)
\fgtimes{(\operatorname{pr}_2,\operatorname{pr}_2)}
\]
\[
\sfG[Y]_1 =
(\sfG_1 \fgtimes{s_{\sfG}}{h_0} \mathsf{K}_0 \times
\sfG_1 \fgtimes{s_{\sfG}}{h_0} \mathsf{K}_0)
\fgtimes{(s_{\sfG}\circ\operatorname{pr}_1,s_{\sfG}\circ\operatorname{pr}_1)}
\]
We define an isomorphism $\mathsf{K}[Y] \to \sfG[Y]$ as the identity on objects
and by applying $h_1$ to the last factor on arrows. This is obviously a differentiable
stratified mapping. The fact that it is an isomorphism follows from condition (FF)
in Definition <ref>.
We will need the following, whose proof is that of
<cit.> with minor modifications.
Let $\sfG$ be a differentiable stratified groupoid. Suppose $Y$ is a locally closed
differentiable stratified subspace of $\sfG_0$ such that $\sfG_{|Y}$ is a differentiable
stratified subgroupoid. Then the inclusion map $\iota\co\sfG_{|Y} \to \sfG_{|\Sat(Y)}$
is a weak equivalence, where
\[
\Sat(Y) := \{ \obj{y} \in \sfG_0 \mid \obj{y} = t(\arr{g})
\text{ for some $\arr{g}\in\sfG_1$ with $s(\arr{g}) \in Y$}\}
\]
denotes that saturation of $Y$.
Note that $(\sfG_{|\Sat(Y)})_1 \fgtimes{s}{\iota_0} Y = \{ \arr{g}\in\sfG_1\mid s(\arr{g})\in Y\}$,
so that $t$ clearly restricts to this set as a map that is surjective onto $Y$. Moreover,
$t$ restricts to a stratified surjective submersion $(\sfG_{|Y})_1\to Y$ by hypothesis so that
(ES) is satisfied. The condition (FF) clearly follows from the definition of $\Sat(Y)$.
With this, we make the following.
Let $\sfG$ be a differentiable stratified groupoid.
We say that $\sfG$ is a
locally translation differentiable stratified groupoid if
for every $\obj{x}\in \sfG_0$ the conditions
(LT<ref>) to (LT<ref>)
from Definition <ref> are satisfied
and if in addition the following holds true:
The slice $Y_\obj{x}$ can be chosen in such a way that
$Y_\obj{x}$ is a differentiable stratified subspace of $U_\obj{x}$,
and such that $Y_\obj{x}$ has the form $Z_\obj{x} \times R_\obj{x}$,
where $Z_\obj{x}$ is a $\sfG_\obj{x}$-invariant subspace of $Y_\obj{x}$
and $R_\obj{x}$ is the stratum through $\obj{x}$. Moreover, the isomorphism
between $\sfG_{|U_\obj{x}}$ and
$(O_\obj{x} \times O_\obj{x}) \times (\sfG_\obj{x} \ltimes Y_\obj{x})$
then becomes an isomorphism of differentiable stratified groupoids,
where $O_\obj{x}$ carries the trivial stratification.
We will see that important examples of differentiable stratified groupoids are
locally translation differentiable stratified groupoids. Under additional hypotheses,
we have the following.
Let $\sfG$ be a locally translation differentiable stratified groupoid, and suppose
that for each $x \in \sfG_0$, the set $Y_\obj{x}$ as in
(LT<ref>) of
Definition <ref> can be chosen so that on each stratum
of $Y_\obj{x}$ the $\sfG_\obj{x}$-orbit type is constant.
Then the assignment
\[
|\sfG| \ni \sfG \obj{x} \mapsto \mathcal{S}_{\sfG x} = \pi(\mathcal{S}^0_\obj{x})
\]
defines a stratification of $|\sfG|$ with respect to which the orbit map $\pi\co\sfG_0\to|\sfG|$
is a stratified surjective submersion.
Note that $|\sfG|$ is a differentiable space by Proposition <ref>.
Choose $U_\obj{x}$, $Y_\obj{x}$, etc. as in Definitions <ref> and
<ref>, and note that we may shrink $U_\obj{x}$ if necessary to assume that
the stratification of $Y_\obj{x}$ consists of a finite number of strata. By Lemma
<ref> and Proposition <ref>, the inclusion of
$Y_\obj{x}$ into its saturation in $\sfG_0$ induces an isomorphism of the differentiable spaces
$Y_\obj{x}/\sfG_\obj{x}$ and $|\sfG_{\Sat(Y_\obj{x})}|$. As $\Sat(Y_\obj{x})$ contains
$U_\obj{x}$ by definition, and as the orbit map is open by Proposition
<ref>(<ref>), it follows that $Y_\obj{x}/\sfG_\obj{x}$ is
isomorphic as a differentiable space to an open neighborhood of $\sfG \obj{x}$ in $|\sfG|$.
Moreover, by (LT<ref>), the embedding of $Y_\obj{x}$ into its saturation
in $\sfG_0$ preserves strata, so it is sufficient to show that the stratification of $Y_\obj{x}$
induces a stratification of the orbit space $Y_\obj{x}/\sfG_\obj{x}$.
Now, each stratum $S$ of $Y_\obj{x}$ is a smooth manifold with $\sfG_\obj{x}$-action so that
$S/\sfG_\obj{x}$ is stratified by $\sfG_\obj{x}$-orbit types. Since each $S$ has a single orbit type by
hypothesis, the stratification of $S/\sfG_\obj{x}$ is trivial, i.e. $S/\sfG_\obj{x}$ is a smooth submanifold
of $Y_\obj{x}/\sfG_\obj{x}$. As $Y_\obj{x}$ is assumed to have finitely many strata, the resulting
stratification of $Y_\obj{x}/\sfG_\obj{x}$ is clearly finite. As $\sfG_\obj{x}$ is compact, the orbit map
$Y_\obj{x} \to Y_\obj{x}/\sfG_\obj{x}$ is closed so that $\overline{(S/\sfG_\obj{x})} = \overline{S}/\sfG_\obj{x}$.
Therefore, if $S/\sfG_\obj{x} \cap \overline{S^\prime/\sfG_\obj{x}}\neq\emptyset$ for strata $S$ and $S^\prime$
of $Y_\obj{x}$, then $S\cap \overline{S^\prime}\neq\emptyset$, implying $S\subset\overline{S^\prime}$
and hence $S/\sfG_\obj{x} \subset \overline{S^\prime/\sfG_\obj{x}}$. The pieces $S/\sfG_\obj{x}$ therefore satisfy
the condition of frontier and define a stratification of $Y_\obj{x}/\sfG_\obj{x}$.
Note that the hypotheses of Proposition <ref> need not be satisfied even in the
case of a proper Lie groupoid. For instance, when $G$ is a compact Lie group, a smooth $G$-manifold
equipped with the trivial stratification satisfies these hypotheses if and only if $G$ acts with
a single orbit type.
§ EXAMPLES OF DIFFERENTIABLE STRATIFIED GROUPOIDS
Many examples of differentiable stratified groupoids arise naturally from
differentiable actions of Lie groupoids on stratified differentiable spaces.
Recall that if $\sfG$ is a topological groupoid and $X$ is a topological space,
an action of $\sfG$ on $X$ is given by a continuous anchor map
$\alpha\co X \to \sfG_0$ together with a continuous map
$\cdot\co\sfG_1 \fgtimes{s}{\alpha} X \to X$ such that for all
$\obj{x} \in X$ and $\arr{g},\arr{h}\in\sfG_1$ with $s(\arr{g}) = \alpha(\obj{x})$ and
$t(\arr{g}) = s(\arr{h})$ the relations
\[
\alpha(\arr{g}\cdot \obj{x}) = t(\arr{g}), \quad
\arr{h}\cdot(\arr{g}\cdot x) = (\arr{h}\arr{g})\cdot \obj{x}, \quad \text{and} \quad
u(\alpha(\obj{x}))\cdot \obj{x} = \obj{x}
\]
hold true. As in the case of group actions, the $\sfG$-orbit of $\obj{x} \in X$ is defined to be
$\{ \arr{g}\cdot \obj{x} \mid s(\arr{g})= \obj{x} \}$. The translation groupoid $\sfG\ltimes X$
associated to the action has object space $(\sfG\ltimes X)_0 = X$ and arrow space
$(\sfG\ltimes X)_1 = \sfG_1\fgtimes{s}{\alpha} X$. The structure maps are given by
\begin{equation}
\label{eq:StrucMapsTranslation}
\begin{split}
& s_{\sfG\ltimes X} (\arr{g}, \obj{x}) = \obj{x},
\quad
t_{\sfG\ltimes X} (\arr{g}, \obj{x}) = \arr{g}\cdot \obj{x},
\quad
u_{\sfG\ltimes X} (\obj{x}) = (u\circ\alpha(\obj{x}), \obj{x}),
\\
& i_{\sfG\ltimes X} (\arr{g}, \obj{x}) = (\arr{g}^{-1}, \arr{g}\cdot \obj{x}),
\quad
m_{\sfG\ltimes X} \big( (\arr{h},\arr{g}\cdot x) ,(\arr{g}, \obj{x})\big)
= (\arr{h}\arr{g}, \obj{x}).
\end{split}
\end{equation}
If $\sfG$ is a differentiable groupoid and $X$ a differentiable space, we say that
the action is differentiable if $\alpha$ and $\cdot$ are morphisms of differentiable
space. In this case, the arrow space $(\sfG\ltimes X)_1 = \sfG_1\fgtimes{s}{\alpha} X$ inherits a
differentiable structure by <cit.>. Then, as each of
the structure maps of $\sfG\ltimes X$ is defined in terms of the structure maps of $\sfG$
and the differentiable maps $\alpha$ and $\cdot$, $\sfG\ltimes X$ is
a differentiable groupoid. For actions of differentiable stratified groupoids, one even has the following.
Let $X$ be a differentiable stratified space that is topologically locally trivial. Further let
$\sfG$ be a differentiable stratified groupoid with $\sfG_0$ and $\sfG_1$
topologically locally trivial that acts differentiably on $X$ in such a way that the $\sfG$-orbit
of each $x \in X$ is a subset of the stratum containing $x$. Then the translation groupoid
$\sfG\ltimes X$ is a differentiable stratified groupoid.
Let $\mathcal{Z}$ denote the maximal decomposition of $X$, and endow
$(\sfG\ltimes X)_1 = \sfG_1\fgtimes{s}{\alpha} X$ with the stratification induced by
those of $\sfG_1$ and $X$. Then by Lemma <ref>, $(\sfG\ltimes X)_1$ is
a differentiable stratified space. By definition of the induced stratification, the
germ of the stratum of $(\sfG\ltimes X)_1$ at the point $(\arr{g}, x)$ is given by the
fibered product over $s$ and $\alpha$ of neighborhoods of strata in $\sfG_1$ and $X$.
Since $i$, $u$, $m$, and $\alpha$ are stratified mappings, the structure maps
$s_{\sfG\ltimes X}$, $t_{\sfG\ltimes X}$, $u_{\sfG\ltimes X}$, $i_{\sfG\ltimes X}$, and $m_{\sfG\ltimes X}$ defined
by Equation (<ref>) are as well stratified
mappings, so that (DSG<ref>) and (DSG<ref>) are satisfied.
To see that $s_{\sfG\ltimes X}$ is a stratified surjective submersion, let $(\arr{g}, x) \in (\sfG\ltimes X)_1$.
Note that as $s$ is a stratified surjective submersion, the restriction of $s$ to the connected
component $R_\arr{g}$ of the stratum of $\sfG_1$ containing $\arr{g}$ is a surjective submersion
onto the connected component $S_{\alpha(x)}$ of the stratum of $\sfG_0$ containing $\alpha(x)$.
As $\alpha$ is a stratified mapping, it maps the connected component $P_x$ of the stratum of $X$
containing $x$ into $S_{\alpha(x)}$. Hence the restriction of $s_{\sfG\ltimes X}$ to
$R_\arr{g} \fgtimes{s}{\alpha} P_x$ is a surjective submersion onto $P_x$.
It follows that $s_{\sfG\ltimes X}$ is a stratified surjective submersion. The
proof for $t_{\sfG\ltimes X}$ is identical. In a similar fashion one shows that $u_{\sfG\ltimes X}$ is a stratified
embedding, because $\alpha$ is a stratified map and $u$ a stratified embedding, so (DSG<ref>)
is satisfied. Property (DSG<ref>) follows from the definition of the induced
stratification, as the connected component of the stratum of $(\sfG\ltimes X)_1$ containing the arrow
$(\arr{g}, x)$ equals the fibered product $R_\arr{g} \fgtimes{s}{\alpha} P_x$ with $R_\arr{g}$
and $P_x$ as above. Therefore, the germ at $(\arr{g}, x)$ of the set of points
$(\arr{h},y)\in (\sfG\ltimes X)_1$ such that $y \in P_x$ is contained in the germ
$[R_\arr{g} \fgtimes{s}{\alpha} P_x]_{(\arr{g}, x)}$ of the stratum through $(\arr{g}, x)$.
To verify (DSG<ref>), consider again $P_x \subset X$, the connected component of
the stratum through $x$. Since by assumption on $X$ the $\sfG$-orbit of each point
in $P_x$ is a subset of the stratum containing $P_x$, the saturation $P:= \Sat P_x$ has to be a union
of connected components of strata, which are separated from one another's closures by
topological local triviality. Assume that $\sigma \co P \to (\sfG\ltimes X )_1$
is a bisection as in (DSG<ref>). Since $\alpha$ is a stratified mapping, $\alpha(P_x)$
is contained in a connected component $S$ of a stratum of $\sfG_0$. The saturation
$\Sat S$ of $S$ by $\sfG$ now is given by $t(s^{-1}(S))$ and hence is a union of connected
components of strata of $\sfG_0$ such that for each such connected component $S^\prime$,
we have $s^{-1}(S)\cap t^{-1}(S^\prime)\neq\emptyset$ by construction. Then by Proposition
<ref>, the restriction $\sfG_{|\Sat S}$ is a Lie groupoid. Moreover,
it follows from (DSG<ref>) that
$\sigma(P ) \subset (\sfG_{| \Sat S}\ltimes P)_1$. Therefore, the map $L_\sigma$ described
in (DSG<ref>) is a left translation of the Lie groupoid
$\sfG_{|\Sat S}\ltimes P$, see <cit.>, implying in particular that
it is a diffeomorphism of the stratum $s_{\sfG\ltimes X}^{-1}(P)$ onto itself. Hence
(DSG<ref>) holds, completing the proof.
A particularly important special case
appears when a Lie group $G$ acts differentiably on a differentiable stratified space $X$ in
such a way that the action restricts to a smooth action on each stratum.
The resulting translation groupoid $G\ltimes X$ then is a differentiable stratified groupoid
by Proposition <ref>.
Many significant and naturally occuring examples of differentiable stratified groupoids are constructed
that way. In the following, we will provide a few.
[Singular symplectic reduction]
Suppose $(M, \omega)$ is a symplectic manifold equipped with a Hamiltonian $G$-action
with moment map $J\co M\to\mathfrak{g}^\ast$ such that $0$ is a singular value for $J$.
By <cit.>, the corresponding symplectic quotient
$J^{-1}(0)/G$ is stratified by orbit types. It is then easy to see that $J^{-1}(0)$
inherits the structure of a smooth stratified space on which $G$ acts in a way such that the
hypotheses of Proposition <ref> are satisfied. Therefore, the
singular symplectic quotient can be realized as the orbit space of the differentiable stratified groupoid
$G\ltimes J^{-1}(0)$. Note that since $J^{-1}(0)$ is equivariantly embedded as a differentiable subspace
in the smooth $G$-manifold $M$ the contractibility hypotheses (LC) in Definition <ref>)
is satisfied. Hence, Theorem <ref> below reduces to the de Rham theorem of
<cit.> in this case.
[Lie Groupoid actions on manifolds with corners]
Manifolds with corners and manifolds with boundary are in a natural way locally trivial differentiable stratified spaces, and even more
$\mathcal{C}^\infty$-cone spaces, see <cit.> and <cit.>.
Compact Lie group actions on manifolds with corners have been considered,
e.g. in <cit.>. By Proposition <ref>,
the corresponding translation groupoids are differential stratified groupoids under
mild hypotheses.
[Semialgebraic actions]
Recall that a semialgebraic set is a locally closed subset of $\R^n$ locally given by the solution of a
finite collection of polynomial equations and inequalities, see <cit.>.
Semialgebraic sets are differentiable stratified spaces in a natural way, since they admit a minimal
Whitney, and hence topologically locally trivial, stratification into semialgebraic manifolds. If a compact
Lie group acts on a semialgebraic set and preserves this stratification, the resulting
translation groupoid is again a stratified differentiable groupoid by
Proposition <ref>. Lie group actions on semialgebraic sets
have been considered in <cit.>.
[Transverse cotangent bundle]
Let $G$ be a compact Lie group and $M$ a $G$-manifold. The transverse cotangent bundle
$T_G^\ast M$ is the subspace of the cotangent bundle $T^\ast M$ consisting of elements that
are conormal to the $G$-orbits in $M$. The transverse cotangent bundle appears in the study
of transversally elliptic operators $G$-invariant pseudodifferential operators on $M$,
see <cit.>.
The action of $G$ on $M$ induces an action of $G$ on
$T_G^* M$. It is not difficult to show that the stratification of $M$ by orbit
types induces a stratification of $T_G^\ast M$ that is compatible with the smooth
structure $T_G^\ast M$ inherits as a subset of $T^\ast M$. Hence the corresponding
translation groupoid is a differentiable stratified groupoid.
If $\sfG$ is a proper Lie groupoid, the transverse cotangent bundle $T_\sfG^\ast \sfG_0 \subset T^\ast \sfG_0$
can be defined in a similar fashion as the subspace of all $\alpha \in T^\ast \sfG_0$ such that
$\langle \alpha , v \rangle =0 $ for all $v \in T_\obj{x} \calO_\obj{x}$, where $\obj{x} \in \sfG_0$ is the
footpoint of $\alpha$ and $\calO_\obj{x}$ the orbit through $\obj{x}$. By the slice theorem
for groupoids as stated in <cit.> recalled in Section <ref>
below, it follows that locally
around a point $\obj{x} \in \sfG_0$ the transverse cotangent bundle $T_\sfG^\ast \sfG_0 $ is isomorphic to the
exterior tensor product of the transverse cotangent bundle $T^\ast_{\sfG_\obj{x}} Y_{\obj{x}}$ of a slice $Y_\obj{x}$
through $\obj{x}$ with the cotangent bundle $T^\ast O$ of an open connected neighborhood $O$ of $\obj{x}$
in the orbit through $\obj{x}$. By the preceeding considerations it follows that the
transverse cotangent bundle of a proper Lie groupoid is locally smoothly contractible as well.
[Singular riemannian foliations]
Let $M$ be a smooth, connected manifold and let $\mathcal{F}$ be a singular riemannian foliation
of $M$, see <cit.>. That is, $\mathcal{F}$ is a partition of $M$ into
connected, immersed submanifolds called leaves such that the module of smooth vector
fields on $M$ that are tangent to the leaves is transitive on each leaf,
and there is a riemannian metric on $M$ with respect to which every
geodesic that is normal to a leaf is normal to every
leaf it intersects. A singular riemannian foliation is an example of a singular Stefan–Sussmann
foliation, see <cit.>. By
<cit.>, $M$ is stratified by unions of leaves of the same dimension; see
also <cit.> and <cit.>.
Suppose $(M, \mathcal{F})$ is almost regular, meaning that the union of leaves of
maximal dimension $k$ is an open, dense subset of $M$. Suppose further that the foliation
$\mathcal{F}$ can be defined by a Lie algebroid of dimension $k$. In <cit.>,
the holonomy groupoid $\sfG$ of $(M,\mathcal{F})$ is constructed as a Lie groupoid with
object space $\sfG_0 = M$ and such that the orbits of $\sfG$ correspond to the leaves $\mathcal{F}$;
see also <cit.>. Giving $M$ the stratification by leaves of the same dimension
described above and $\sfG_1$ the stratification given by the pullback of this stratification via $s$,
it is easy to see that $\sfG$ has, along with its structure as a Lie groupoid, an alternate structure
as a differential stratified groupoid. These two structures coincide if and only if the foliation
$\mathcal{F}$ is regular, i.e. all leaves have the same dimension.
Note that the holonomy groupoid of a general singular Stefan–Sussmann foliation was constructed as a
topological groupoid in <cit.> and coincides with the holonomy groupoid
of <cit.> when the latter is defined. Other hypotheses under which
this holonomy groupoid naturally has the structure of a differentiable stratified groupoid
are not yet clear.
§ THE ALGEBROID OF A DIFFERENTIABLE STRATIFIED GROUPOID
Given a reduced differentiable Whitney stratified groupoid $\sfG$ with $\calS^i$,
$i=0,1$ the decomposition of $\sfG_i$ into its strata we obtain the so-called
stratified tangent bundles
\[
\stratan \sfG_0 := \bigcup_{S \in \calS^0} TS
\text{ and }
\stratan\sfG_1 :=\bigcup_{R \in \calS^1} TR =\bigcup_{S \in \calS^0}TS^1,
\text{ where $S^1 := s^{-1} (S)$ for $S\in \calS^0$}.
\]
Since, by assumption, the spaces $\sfG_0$ and $\sfG_1$ satisfy Whitney's
condition B, hence A, the stratified tangent bundles
$\stratan\sfG_0 $ and $\stratan\sfG_1 $ inherit the
structures of differentiable stratified spaces by <cit.>.
Moreover, we have tangent maps
$Ts \co \stratan \sfG_1 \to \stratan \sfG_0 $ and
$Tt \co \stratan \sfG_1 \to \stratan \sfG_0 $.
Now we can define what we understand by the algebroid of $\sfG$.
Given a differentiable stratified groupoid $\sfG$,
the differentiable stratified algebroid of a differentiable
Whitney stratified groupoid $\sfG$ is defined as the space
\begin{equation}
\label{eq:AlgebroidDef}
\sfA := \bigcup_{S \in \calS^0} \sfA_S ,
\end{equation}
where $\sfA_S := u^*_{|S}\ker T_{|S_1}s$ denotes the Lie algebroid
of the Lie groupoid $\sfG_{|S}$. In other words, $\sfA$ can be identified
with $u^*\ker Ts$, the restriction of $\ker Ts$ to $\sfG_0$. We define the
anchor map of the algebroid $\sfA$ as the map
\[
\varrho\co \sfA \to \stratan\sfG_0 , \quad v \mapsto Tt (v).
\]
Let $\sfG$ be a proper reduced differentiable Whitney stratified groupoid.
Then the differentiable stratified algebroid $\sfA$ of $\sfG$ is a reduced
differentiable stratified space, where the
differentiable structure is that inherited from $\stratan\sfG_1$ and the
stratification is that induced by the decomposition in
Equation (<ref>).
The anchor map $\varrho\co \sfA \to \stratan\sfG_0$ is a differentiable
stratified submersion.
Since $\sfG_1$ is locally compact, $u(\sfG_0)$ is locally closed in $\sfG_1$ by
<cit.>. For each stratum $S$ of $\sfG_0$
the source map $s$ restricts to $S_1$ as a submersion which implies
that $\ker T_{|S_1}s$ is a subbundle, hence closed subset, of $TS_1$.
Hence, if $U$ is an open subset of $\sfG_1$ intersecting finitely many strata,
$\sfA \cap\stratan U$ is a finite union of locally closed sets, therefore
locally closed itself. Then $\sfA$ inherits the structure of a reduced
differentiable space from that of $\stratan \sfG_1$, see
Since each $\sfA_S$ is a subbundle of the restriction of $TS_1$
to $S_1 \cap u(\sfG_0)$ one concludes that each $\sfA_S$ is a smooth
submanifold of $\sfA$.
Note that the projection $\pi\co\stratan\sfG_1\to \sfG_1$ is clearly open
as its restriction to an element of $\calS^1$ is a bundle map.
The fact that $\calS^1$ is locally finite implies then
that $(\sfA_S)_{S\in \calS^0}$ is a locally finite decomposition of $\sfA$.
To verify the condition of frontier, suppose
$\sfA_S \cap \overline{\sfA_{S^\prime}}\neq\emptyset$ for $S, S^\prime\in\calS^0$,
and let $S_1 = s^{-1}(S)$ and $S_1^\prime = s^{-1}(S^\prime)$.
By <cit.>, $\pi\co\stratan\sfG_1\to \sfG_1$ is a
topological projection, hence $S_1 \subset\overline{S_1^\prime}$
and $T S_1 \subset \overline{T S_1^\prime}$ follow.
Choose a tangent vector $v \in \ker T_{|S_1}s$ and assume for simplicity that
$v$ is a unit vector. Let $\obj{x} \in S$ be the footpoint, i.e. $\obj{x}$
is the unique object such that $\pi(v) = u(\obj{x})$.
Choosing a singular chart for $\sfG_1$ at $u(\obj{x})$,
we may reduce to the case where $S_1$ and $S_1^\prime$ are closed subsets of
$\R^n$. Let $\widetilde{s}$ denote a smooth function from $\R^n$ into a
singular chart of $\sfG_0$ at $\obj{x}$ that extends $s$. Note that
$\widetilde{s}$ may not be a submersion. Let $p(t)\co[-1,1]\to S_1$ be a
smooth path in $S_1$ with $p(0) = u(\obj{x})$ and tangent
$p^\prime(0) = v$, and put $\alpha_i = p(1/i)$ for $i\in\N$.
As $\alpha_1 \in S_1 \subset \overline{S_1^\prime}$, we may choose a sequence
$(\beta_{1,j})_{j\in \N} \subset S_1^\prime$ such that
$\lim_{j\to\infty} \beta_{1,j} = \alpha_1$. We then set
$\obj{x}_j=s(\beta_{1,j}) \in S^\prime$ for each $j$. By continuity of $u$ and $s$,
we have $\lim_{j\to\infty}\obj{x}_j = \obj{x}$ and $\lim_{j\to\infty}u(\obj{x}_j) = u(\obj{x})$.
As $s_{|S_1^\prime}$ is a submersion, each $s^{-1}(\obj{x}_j)$ is a closed submanifold of $\R^n$.
Then the intersection of each $s^{-1}(\obj{x}_j)$ with a closed ball in $\R^n$ is compact.
Hence we may define, for each $i > 1$ and each $j \geq 1$, the element $\beta_{i,j}$ of $S_1^\prime$
to be the point on the connected component of $s^{-1}(\obj{x}_j)$ containing $u(\obj{x}_j)$
intersected with a closed annulus of external radius $1/j$ about $u(\obj{x}_j)$
that minimizes the distance to $\alpha_i$. In particular, for each $j$,
$\lim_{i\to\infty}\beta_{i,j} = u(\obj{x}_j)$, and for $i$ sufficiently large,
$\lim_{j\to\infty}\beta_{i,j} = \alpha_i$; if necessary, we restrict to a
subsequence in the $j$ direction so that this intersection is not empty.
With this, we set
\[
v_j := \lim_{i\to\infty} \frac{\beta_{i,j} - u(\obj{x}_j)}{\|\beta_{i,j} - u(\obj{x}_j)\|},
\]
and then $v_j \in T_{\obj{x}_j} S_1^\prime$ for each $j$. Moreover, as $\beta_{i,j}\in s^{-1}(\obj{x}_j)$,
it follows that $v_j \in \ker T_{|S_1^\prime}s$. However,
\[
\lim_{j\to\infty} v_j = \lim_{i\to\infty}\lim_{j\to\infty}
\frac{\beta_{i,j} - u(\obj{x}_j)}{\|\beta_{i,j} - u(\obj{x}_j)\|}
= \lim_{i\to\infty}
\frac{\alpha_i - u(\obj{x}_j)}{\|\alpha_i - u(\obj{x}_j)\|} = v
\]
so that $v \in \overline{\ker T_{|S_1^\prime}s}$. It follows that the decomposition
of $\sfA$ into the $\sfA_S$ satisfies the condition of frontier.
Finally, the anchor map is a differentiable stratified morphism by
Eq. (<ref>) in Lemma <ref> and because
$t$ is a stratified map. Since the restriction of $\varrho$ to each stratum
is a submersion, the anchor map is a stratified submersion.
§ A DE RHAM THEOREM
In this section, we prove a de Rham theorem for proper reduced
differentiable stratified groupoids which are locally translation and satisfy
a certain local contractibility condition. As we will see below, this
hypothesis is satisfied in a number of the examples we have considered.
Throughout this section let $\sfG$ be a proper differentiable
Whitney stratified groupoid. Let $\Omega^\bullet$ be the sheaf complex
of abstract forms on $\sfG_0$ as constructed in Section <ref>.
Recall from <cit.>
that a differential form on the object space of a proper Lie groupoid is called
basic if contraction with any smooth section of the Lie algebroid
vanishes and if the form is invariant under the conormal action of the Lie groupoid.
Let $U \subset |\sfG|$ be an open subset of the orbit space of $\sfG$, and $U_0$ its
preimage under the canonical projection $\pi \co \sfG_0 \to |\sfG|$. One calls an abstract $k$-form
$\omega \in \Omega^k (U_0)$ $\sfG$-horizontal or simply horizontal,
if for each smooth section $\xi \co U_0 \to \sfA$ of the differentiable stratified algebroid of $\sfG$ the
stratawise contracted form $(\varrho \circ \xi) \lrcorner \omega$ vanishes.
Next let $S\subset \sfG_0$ be a (relatively closed and open) component of a stratum of $\sfG_0$ such that the
projection $\pi (S) $ to the orbit space is connected.
For $\obj{x}\in\sfG_0$ consider the Zariski tangent space
$\zartan_x\sfG_0$; see Appendix <ref>.
Note that $\zartan_x\sfG_0$ does in general not coincide with $\stratan_x\sfG_0$, but that the latter is always
contained in the former.
Under the assumption that $\sfG$ is locally translation, the space
$\zartan_{|S} \sfG_0 := \bigcup_{\obj{x}\in S} \zartan_x\sfG_0$ naturally carries the structure of a smooth
vector bundle over $S$.
Given $\obj{x}\in S$ choose a trivializing neighborhood $U_\obj{x}$ and a groupoid
slice $Y_\obj{x}$ with a $\sfG_\obj{x}$-action such that there exists an isomorphism
of differentiable groupoids
$\sfG_{|U_\obj{x}} \to ( O_\obj{x} \times O_\obj{x}) \times (\sfG_\obj{x} \ltimes Y_\obj{x})$,
where $O_\obj{x}$ is an open contractible neighborhood of $\obj{x}$ in its orbit.
Such $U_\obj{x}$, $Y_\obj{x}$, and $O_\obj{x}$ exist according to (LT<ref>).
After possibly shrinking these data, we can assume by (LT<ref>) that the slice
$Y_\obj{x}$ is stratified and is isomorphic as a differentiable stratified space
to the product space $Z_\obj{x} \times R_\obj{x}$, where $Z_\obj{x} \subset Y_\obj{x}$
is a $\sfG_\obj{x}$-invariant subspace, and $R_\obj{x} \subset Y_\obj{x}$
is the stratum through $\obj{x}$.
Hence $\bigcup_{\obj{y}\in O_\obj{x} \times R_\obj{x}} \zartan_{\obj{y}}\sfG_0$ is a vector
bundle isomorphic to $T (O_\obj{x} \times R_\obj{x}) \times \zartan_\obj{x}Z_\obj{x}$.
But the set $O_\obj{x} \times R_\obj{x}$ contains $\obj{x}$ and is relatively open
in the stratum $S$. This means that, locally, $\zartan_{|S} \sfG_0$ is a vector bundle.
By construction, the transition maps arise from the local isomorphisms $\sfG_{U_\obj{x}} \to ( O_\obj{x} \times O_\obj{x})
\times (\sfG_\obj{x} \ltimes Y_\obj{x})$, and are vector bundle isomorphisms, hence the claim follows.
We now have the means to define basic forms on $\sfG$. Note that here we apply the
pull-back morphisms constructed in Appendix <ref>.
Let $U \subset |\sfG|$ be an open subset of the orbit space, and $U_0 := \pi^{-1} (U)$.
One calls an abstract $k$-form $\omega \in \Omega^k (U_0)$ $\sfG$-basic or simply basic
if it is horizontal and if for every $\obj{x} \in U_0$ and every smooth bisection
$\sigma : U_\obj{x} \to \sfG$ defined on an open neighborhood
$U_\obj{x} \subset U_0$ of $\obj{x}$ the equality
$(t\circ\sigma)^* \omega = \omega_{| U_\obj{x}}$ holds true.
By definition, $\Omega^0_\textup{basic} (U)$ coincides with the space of smooth functions
on $U_0$ which are invariant under the $\sfG$-action, hence
$\Omega^0_\textup{basic} (U)$ can be naturally identified with $\calC^\infty (U)$.
Moreover, the spaces $\Omega^k_\textup{basic} (U)$, where $U$ runs through the
open sets of $|\sfG|$, form the space of sections of a sheaf on $|\sfG|$ which will
be denoted by $\Omega^k_\textup{basic}$.
By construction the sheaf $\Omega^k_\textup{basic}$ is a $\calC^\infty_{|\sfG|}$-module.
This entails the first part of the following result.
The sheaves of basic $k$-forms $\Omega^k_\textup{basic}$ are fine. Moreover, the
exterior differential on $\Omega^\bullet$ descends to a differential $d$ on
$\Omega^\bullet_\textup{basic}$ turning $\big( \Omega^\bullet_\textup{basic},d\big)$
into a sheaf of commutative differential graded algebras over the orbit space $|\sfG|$.
It remains to show that $d$ descends to $\Omega^\bullet_\textup{basic}$. But that follows
from the fact that $d$ commutes with the pull-back morphism $(t\circ\sigma)^*$
for every bisection $\sigma : U_\obj{x} \to \sfG$ defined on an open neighborhood
$U_\obj{x}$ of $\obj{x} \in \sfG_0$.
The next result will be needed for a proof of a Poincaré Lemma for basic forms.
Let $V$ be an open subset of the object space of a locally translation
proper differentiable Whitney stratified groupoid, and let $\omega \in \Omega^k (V)$
be an abstract $k$-form which is $\sfG_{|V}$-basic.
Then there exists a unique form $\widehat{\omega}\in \Omega^k \big( \Sat (V) \big)$
which is $\sfG$-basic and whose restriction to $V$ coincides with $\omega$. We call
$\widehat{\omega}$ the basic extension of $\omega$.
Let $\obj{x} \in \Sat (V)$, and choose a bisection $\sigma : U_\obj{x} \to \sfG_1$ defined on an open neighborhood of $\obj{x}$ such that
$(t \circ \sigma) (U_\obj{x}) \subset V$. Then put $\omega_{U_\obj{x}} := (t\circ \sigma)^* (\omega )$.
If $\eta : U_\obj{x} \to \sfG_1$ is another bisection with $U_{\obj{z}} := (t \circ \eta) (U_\obj{x}) \subset V$,
where $\obj{z} := (t \circ \eta) (\obj{x})$, then the bisection $\mu\co U_{\obj{z}} \to \sfG_1$ defined by
\mu (\obj{y}) = \sigma \big( (t \circ \eta)^{-1} (\obj{y}) \big) \cdot \eta^{-1} \big( (t \circ \eta) (\obj{y}) \big)
\quad\text{for $\obj{y} \in U_{\obj{z}}$}
starts and ends in $V$. Hence, by assumption, $(t\circ \mu)^* \omega = \omega_{|U_{\obj{z}}}$.
But $t \circ \mu = (t \circ \sigma) \circ (t \circ \eta)^{-1}$, which implies
\[
(t \circ \eta)^* \omega = (t \circ \eta)^* (t\circ \mu)^* \omega =
(t \circ \sigma)^* \omega.
\]
This shows that $\omega_{U_\obj{x}}$ does not depend on the particular choice of the bisection
$\sigma : U_\obj{x} \to t^{-1} (V) $. An analogous argument proves that
for points $\obj{x}, \obj{y} \in \Sat (V)$ the forms $\omega_{U_\obj{x}}$ and $\omega_{U_\obj{y}}$ coincide over
$U_\obj{x} \cap U_\obj{y}$. Let $\widehat{\omega} \in \Omega^k \big(\Sat (V)\big)$ be the abstract $k$-form such that
$\widehat{\omega}_{|U_\obj{x}} = \omega_{U_\obj{x}}$ for all $ \obj{x} \in \Sat (V)$. Obviously,
$\widehat{\omega}$ is horizontal and basic by construction.
Uniqueness of $\widehat{\omega}$ is clear since the form has to be basic.
Since the kernel of the sheaf morphism $d: \Omega^0_\textup{basic} \to \Omega^1_\textup{basic}$
can be naturally identified with $\R_{|\sfG|}$, the sheaf of locally constant real-valued functions on $|\sfG|$,
we obtain a sheaf complex
\[
0 \longrightarrow \R_{|\sfG|} \longrightarrow \Omega^\bullet_\textup{basic}
\]
which is exact at $\R_{|\sfG|}$ and at $\Omega^0_\textup{basic} $.
Now observe that the orbit space $|\sfG|$ is paracompact, locally contractible and locally path connected
by <cit.>. Since the sheaves $\Omega^k_\textup{basic} $ are fine, the following is
a consequence of <cit.>.
If the sheaf complex $\big( \Omega^\bullet_\textup{basic} , d \big)$ of basic forms on a
proper differentiable Whitney stratified groupoid $\sfG$ is exact, the
basic cohomology $H^\bullet_\textup{basic} (\sfG) :=
H^\bullet \big( \Omega^\bullet_\textup{basic} (|\sfG|) \big)$ of $\sfG$
coincides naturally with the real singular cohomology of $|\sfG|$.
In the remainder of this section we will show that the sheaf complex of basic
forms on a groupoid is exact if a certain local contractibility condition is satisfied.
Before we come to stating the local contractibility condition recall that by
Proposition <ref> every $\sfG$-slice can be, possibly after
shrinking, equivariantly embedded around the fixed point into the Zariski tangent space.
Let $\sfG$ be a proper locally translation differentiable stratified groupoid in the
sense of Definition <ref>. We then say that $\sfG$ fulfills the
local contractibility hypothesis if the following condition holds true.
For each $\obj{x}\in\sfG_0$ there exists a groupoid slice $Y_\obj{x}$ as in
(LT<ref>), a linear $\sfG_\obj{x}$-action on some $\R^n$
together with a singular $\sfG_\obj{x}$-equivariant chart
$\iota: Z_\obj{x} \to\widetilde{V_\obj{x}}\subset \R^n$, $y \mapsto \widetilde{y}$,
and a smooth homotopy
$h \co \widetilde{V_\obj{x}}\times[0,1]\to \widetilde{V_\obj{x}}$ having the
following properties:
* The chart $\iota$ maps $\obj{x}$ to $0$ and the stratum $R_\obj{x}$ through $\obj{x}$ to the subspace of
$\R^n$ fixed by $\sfG_\obj{x}$. Moreover,
$\widetilde{V_\obj{x}}$ is an open neighborhood of $0$ in $\R^n$,
and $\widetilde{Y}_\obj{x} := \iota( Y_\obj{x})$ is relatively closed in
* One has $\im h_0 = \{ 0 \}$ and $h_1 = \id_{\widetilde{V}_\obj{x}}$.
* The homotopy $h$ is a homotopy along $\widetilde{Y}_\obj{x}$ which means that
$h (\widetilde{\obj{y}},t) \in \widetilde{Y}_\obj{x}$ for all $\obj{y}\in Z_\obj{x}$
and $t\in [0,1]$.
* The homotopy $h$ is $G_\obj{x}$-equivariant.
* The homotopy $h$ preserves the stratification which means for each $\obj{y}\in Y_\obj{x}$
and $ t \in (0,1]$ the points $\iota^{-1} h(\widetilde{\obj{y}},t)$
and $\obj{y}$ are in the same stratum.
Let $G$ be a compact Lie group and $M$ is a $G$-manifold. The transverse cotangent bundle described in
Example <ref> satisfies the local contractibility hypothesis. To verify this, observe
first that $T_G^\ast M$ is equivariantly embedded in the $G$-manifold $T^\ast M$, and that
$\R_{\geq 0}$ acts on $T^\ast M$ by fiberwise homotheties. Since the transverse
cotangent bundle is invariant under these homotheties, it contracts
smoothly to the $0$-section which is diffeomorphic to $M$. But $M$ is a smooth manifold, so is locally
smoothly contractible for trivial reasons. Hence $T_G^\ast M$ is locally smoothly contractible as well.
We will illustrate in Section <ref> that the inertia groupoid of a
proper Lie groupoid fulfills the local contractibility hypothesis.
Now we will prove that the sheaf complex of basic forms on a locally
translation proper Lie groupoid fulfilling the contractibility hypothesis
is exact, or in other words satisfies Poincaré's Lemma.
We first consider the case where the groupoid $\sfG$ is of the form
$(O\times O) \times (G \ltimes Y) \rightrightarrows O \times Y$, where $O$ is an open contractible set in some
$\R^m$, $O\times O$ denotes the corresponding pair groupoid, $G$ is a Lie group acting linearly on some $\R^n$,
and $Y \subset \R^n$ is an affine $G$-invariant differentiable stratified space on which $G$ acts by
strata preserving maps. In addition we assume that $0\in Y$ and that there exists a smooth homotopy
$h : \widetilde{V} \times [0,1] \to \widetilde{V}$ defined on an open $G$-invariant subset $\widetilde{V} \subset \R^n$
such that the five conditions of the local contractibility hypothesis
are satisfied with $\iota : Y \hookrightarrow \R^n$ being the identical embedding.
Denote by $I \subset \calC^\infty (O \times \widetilde{V})$ the vanishing ideal
of $O \times Y$. Observe that by <cit.> and the smooth contractibility of $Y$ the
subcomplex $I^\bullet \subset \Omega^\bullet (O\times \widetilde{V} )$ defined by
\[
I^k :=
\begin{cases}
I, & \text{for $k=0$}, \\
I \Omega^k (O\times \widetilde{V}) + dI \wedge \Omega^{k-1} (O\times \widetilde{V}), & \text{for $k=1,\ldots ,n+m$}
\end{cases}
\]
is contractible. More precisely, an (algebraic) contraction is given by
\begin{equation}
\label{eq:AlgContraction}
K\omega =
\begin{cases}
0, & \text{for $\omega \in \Omega^0 (O\times \widetilde{V}) = \calC^\infty (O\times \widetilde{V})$},\\
\int_0^1 H_t^* (\xi_t \lrcorner \omega ) \, dt, &
\text{for $\omega \in \Omega^k (O\times \widetilde{V})$, $k\in \N^*$},
\end{cases}
\end{equation}
where the homotopy $h$ has been extended to a homotopy on $O \times \widetilde{V}$ by putting
$H_t (v, x) = (v,h_t(x))$ for $v\in O$, $x\in \widetilde{V}$, $t \in [0,1]$,
and $\xi_t : O \times \widetilde{V} \to T \widetilde{V}$ is the vector field defined by
$\xi_t := \partial_t H_t$. Cartan's magic formula implies that
\begin{equation}
\label{eq:AlgHomotopyrelation}
\omega - H^*_0 \omega = dK\omega + K d\omega, \quad \text{ for $\omega \in \Omega^k (O\times \widetilde{V})$,
$k\in \N$}.
\end{equation}
But this entails that the restriction of $K$ to $I^\bullet$ is an algebraic contraction indeed, since
every form $\omega \in I^k$ satisfies the relation $H^*_0 \omega =0$.
Now consider the subcomplex $\Omega^\bullet_\textup{r-basic} (O\times \widetilde{V})$ of relative basic forms or more precisely of
basic forms relative $(O\times Y)$. It consists of all
$\omega \in \Omega^k (O\times \widetilde{V})$ which are invariant under the $G$-action and which have the property
that for each stratum $S$ of $O\times Y$ the form $\iota_S^* \omega $ is a basic form for the restricted Lie groupoid $\sfG_{|S}$.
In particular, this implies that $H_0^* \omega = \iota_{O\times \{ 0 \}}^* \omega $ vanishes for each relative basic form $\omega$.
Since $H_t$ commutes with the $G$-action and maps fibers $O\times \{ y\}$ to
$O \times \{ h_t(y)\}$, the algebraic contraction $K$ maps basic forms to basic forms.
One concludes that the complex $\Omega^\bullet_\textup{r-basic} (O\times \widetilde{V})$ is exact, and that
the subcomplex $I^\bullet_\textup{r-basic} := I^\bullet \cap \Omega^\bullet_\textup{r-basic} (O\times \widetilde{V})$
is contractible. Hence the quotient complex $\Omega^\bullet_\textup{r-basic} (O\times \widetilde{V}) / I^\bullet_\textup{r-basic}$
is exact. But one has
\[
\Omega^\bullet_\textup{basic} (Y/G) = \Omega^\bullet_\textup{r-basic} (O\times \widetilde{V}) / I^\bullet_\textup{r-basic}.
\]
This can be seen by averaging a representative of an element $[\omega] \in \Omega^k_\textup{basic} (Y/G)$ over the orbits of the $G$-action
using a bi-invariant Haar measure on $G$. The resulting new representative $\omega$ then is $G$-invariant.
Since $[\omega]$ is basic, the pull-back of such a representative $\omega$ to each stratum $S$ of $O \times Y$ has to be basic as well,
hence $\omega \in \Omega^\bullet_\textup{r-basic} (O\times \widetilde{V})$. So we have shown Poincare's Lemma in the special
case where the groupoid $\sfG$ is of the form $(O\times O) \times (G \ltimes Y) \rightrightarrows O \times Y$ with $Y$ and $O$ as stated above.
Next let us consider the general case of a proper locally translation differentiable Whitney stratified groupoid
$\sfG$ which fulfills the local
contractibility hypothesis. Let $\obj{x} \in \sfG_0$ be a point in the object space. Since $\sfG$ is locally translation, we can choose
a trivializing neighborhood $U_\obj{x} \subset \sfG_0$ of $\obj{x}$, an open contractible neighborhood $O_\obj{x}$ in the orbit through $\obj{x}$ and
a groupoid slice $Y_\obj{x} \subset U_\obj{x}$ with a $\sfG_\obj{x}$-action such that $\sfG_{|U_\obj{x}}$ as a differentiable stratified
groupoid is isomorphic to the groupoid $(O_\obj{x} \times O_\obj{x}) \times (\sfG_\obj{x} \ltimes Y_\obj{x})$.
We will show that $\Omega^\bullet_\textup{basic} (\pi(U_\obj{x}))$ is exact. To this end let $\omega \in \Omega^k (\Sat (U_\obj{x}))$ be
a closed basic $k$-form. The restriction of $\omega$ to $U_\obj{x}$ is a closed and $\sfG_{|U_\obj{x}}$-basic form. By the preceeding considerations
there exists a $\sfG_{|U_\obj{x}}$-basic $(k-1)$-form $\eta \in \Omega^{k-1} (U_\obj{x})$ such that $d \eta = \omega_{|U_\obj{x}}$.
By Proposition <ref>, $\eta$ has a unique extension to a basic form
$\widehat{\eta} \in \Omega^{k-1}_\textup{basic} (\pi(U_\obj{x}))$. By construction of $\widehat{\eta}$ we have $d \eta = \omega$, since
$d$ commutes which each of the isomorphisms $t\circ \sigma$, where $\sigma : U \to \sfG$ is a bisection. This proves exactness of
$\Omega^\bullet_\textup{basic} (\pi(U_\obj{x}))$, and entails the following result.
Let $\sfG$ be a proper locally translation differentiable Whitney stratified groupoid satisfying the local
contractibility hypothesis.
The complex of sheaves $(\Omega_{\textup{basic}}^\bullet, d)$ on $|\sfG|$ then is a fine resolution of the sheaf of locally constant
real-valued functions on $|\sfG|$. In particular this implies that the cohomology of the complex $(\Omega_{\textup{basic}}^\bullet, d)$ of basic
differential forms on $\sfG$ is naturally isomorphic to the singular cohomology of $|\sfG|$ with coefficients in $\R$.
If $\sfG$ is a proper Lie groupoid, then the complex of sheaves
$(\Omega_{\textup{basic}}^\bullet, d)$ coincides with the sheaf of basic differential
forms defined in <cit.> so that Theorem <ref>
reduces to <cit.>.
If $\sfG$ is a locally translation groupoid, then $(s,t)(\sfG_1)$ is necessarily
locally closed in $\sfG_0\times\sfG_0$. Specifically, using the local model of $\sfG$ given by
condition (LT<ref>), the condition that $(s,t)(\sfG_1)$ is locally closed is
equivalent to the requirement that the $G_x$-action on $Y_x$ is a proper group action, which
is automatically satisfied as $G_x$ is compact by <cit.>.
Therefore, $|\sfG|$ is locally compact by Proposition
<ref>(<ref>). If we assume further that
$|\sfG|$ carries a stratification according to Proposition <ref> and that that
stratification fulfills Whitney's condition B, then $|\sfG|$ is a differentiable stratified space with
control data in the sense of Mather by <cit.>.
Hence $|\sfG|$ admits in this case a triangulation subordinate to its stratification,
cf. <cit.>. Therefore, given an open covering of $|\sfG|$, there exists a
subordinate good covering. See <cit.> for more details.
As in the case of Lie groupoids, cf. <cit.>,
one concludes that the singular cohomology of $|\sfG|$ with real coefficients coincides with the Čech
cohomology under these hypotheses, and that it is finite-dimensional if $|\sfG|$ is compact.
§ THE INERTIA GROUPOID OF A PROPER LIE GROUPOID AS A DIFFERENTIABLE STRATIFIED GROUPOID
The goal of this section is to construct an explicit Whitney stratification of the loop
space of a proper Lie groupoid $\sfG$ with respect to which the inertia groupoid $\inertia{\sfG}$ becomes
a reduced differentiable stratified groupoid which is locally translation and satisfies the
local contractibility hypotheses. As the general strategy we hereby use the
slice theorem for proper Lie groupoids to describe the groupoid $\sfG$ locally
in terms of translation groupoids by compact Lie groups.
This will allow us to describe $\inertia{\sfG}$ locally in terms of the inertia groupoid associated
to such a translation groupoid. Using isotropy types and an equivalence relation on the group defined
in terms of this action, we will construct stratifications for inertia groupoids of such translation
groupoids that patch together to a well-defined stratification of $\inertianull{\sfG}$.
§.§ The inertia groupoid of a proper Lie groupoid
Let $\sfG$ be a proper Lie groupoid. Define the loop space of $\sfG$ to be
\[
\inertianull{\sfG} := \{ \arr{h} \in \sfG_1 \mid s(\arr{h}) = t(\arr{h}) \}.
\]
Since the loop space is closed in $\sfG_1$ it inherits the structure of a reduced differentiable space from
the ambient manifold $\sfG_1$. The map $s = t\co \inertianull{\sfG}\to \sfG_0$ serves as an anchor map
for the action of $\sfG$ on $\inertianull{\sfG}$ by conjugation.
More precisely, the action of $\arr{g}\in \sfG_1$ on $\arr{h}\in \inertianull{\sfG}$ with
$s(\arr{g}) = s(\arr{h})$ is given by
\begin{equation}
\label{eq:InertiaActionDef}
\arr{g}\cdot\arr{h} = \arr{g}\arr{h}\arr{g}^{-1}.
\end{equation}
The inertia groupoid of a proper Lie groupoid $\sfG$ is the action
groupoid $\inertia{\sfG} := \sfG\ltimes\inertianull{\sfG}$. The space of its objects
is the loop space $\inertianull{\sfG}$, its space of
arrows is $\sfG_1 \fgtimes{s}{s} \inertianull{\sfG}$.
The inertia space of $\sfG$ then is the orbit space $|\inertia{\sfG}|$.
Note that, while $\inertianull{\sfG}$ is a differentiable subspace of the
smooth manifold $\sfG_1$, the action of $\sfG$ on $\inertianull{\sfG}$ does not necessarily extend to an
action on $\sfG_1$ or an open neighborhood of $\inertianull{\sfG}$ in $\sfG_1$.
§.§ The stratification of the loop space
§.§.§ The compact Lie group action case
Assume that the compact Lie group $G$ acts by diffeomorphisms on the
smooth manifold $M$. The loop space $\inertianull{(G \ltimes M)}$ coincides in this case
with the union $\bigcup_{g\in G} \{ g \} \times M^g$, where $M^g$ denotes the fixed point
space of $g\in G$.
To describe our stratification of $\inertianull{(G \ltimes V)}$ recall first
<cit.> that a closed subgroup $\cartan$ of the Lie group
$G$ is called a Cartan subgroup if it is closed,
topologically cyclic, and of finite index in its normalizer.
As in <cit.>, we say
$\cartan$ is associated to an element $h \in G$ if $h \in \cartan$, and
$\cartan/\cartan^\circ$ is generated by $h\cartan^\circ$.
Now let $(h, x) \in\inertianull{(G \ltimes V)} \subset G \times V$, and choose a slice
$Y_x$ at $x$ for the $G$-action on $M$. Then, after possibly shrinking $Y_x$
and the choice of an appropriate $G$-invariant riemannian metric on $M$, $Y_x$ is the
image under the exponential map of an open ball $B_x \subset N_x$ around the origin of the
normal space $N_x := T_xM/T_x(Gx)$ to the tangent space of the orbit through $x$.
Let $H = G_{(h,x)}$, and note that $H = G_x \cap \centralizer_G(h) = \centralizer_{G_x}\!(h)$ is the centralizer of $h$ in $G_x$.
Let $\cartan_{(h,x)}$ be a Cartan subgroup of $H$ associated to $h$.
Note that if $G_x$ is connected, the relation $h \in (\centralizer_{G_x}\!(h))^\circ = H^\circ$
holds true by <cit.>, so that $\cartan_{(h, x)}$ is a maximal torus of $H^\circ$ containing $h$.
Define an equivalence relation $\simeq$ on $\cartan_{(h,x)}$ by $s \simeq t$ if
$N_x^s = N_x^t$, and let $\cartan_{(h,x)}^{\ast}$ denote the connected
component of the $\simeq$ class containing $h$. Note that by construction,
$s \simeq t$ if and only if the germs of the sets $Y_x^s$ and $Y_x^t$ at $x$ coincide.
Next choose a slice $V_{(h,x)}$ at $(h, x)$ for the $G_x$-action on $G_x \times Y_x$ which
is given by $g(k,y) = (gkg^{-1},gy)$, i.e. by the diagonal action with conjugation on the
$G_x$-factor. Then assign to $(h,x)$ the germ
\begin{equation}
\label{eq:MLocalStratDef}
\mathcal{S}_{(h,x)}
\left[G\left( V_{(h,x)}^H \cap
\left(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\right)\right)\right]_{(h,x)}.
\end{equation}
It will be demonstrated below that this yields a stratification of the loop space $\inertianull{(G \ltimes M)}$;
see Theorem <ref>.
We refer to this as the orbit Cartan type stratification of the loop space of the
Lie groupoid $G \ltimes M$.
The germ $\mathcal{S}_{(h,x)}$ is obviously $G_x$-invariant, and hence, if intersected with
$\inertianull{(G_x\ltimes Y_x)}$ (i.e. take $G_x$-orbits rather than $G$-orbits)
defines a germ in the quotient $|\inertia{(G_x\ltimes Y_x)}|$.
This stratification depends only on the $G$-orbit of $(h,x)$,
and hence defines a germ in the quotient $|\inertia{(G\ltimes M)}|$ as well.
To see this, note that if $g \in G$, then $gY_x$ is a slice at $gx$ for the $G$-action
on $M$, and conjugation by $g$ maps $G_x$ onto $G_{gx}$. Choosing $V_{g(h,x)}$ and
$\cartan_{g(h,x)}$ to be the images of $V_{(h,x)}$ and $\cartan_{(h,x)}$ under the
induced isomorphism $G_x\ltimes Y_x \to G_{gx} \ltimes gY_x$,
the germ $\mathcal{S}_{g(h,x)}$ coincides with $\mathcal{S}_{(h,x)}$.
Then the orbit Cartan type stratification of the inertia space
$|\inertia{(G\ltimes M)}|$ is given by
\begin{equation}
\label{eq:MLocalStratQuotDef}
\mathcal{R}_{(h,x)}
G\backslash\left[G\left( V_{(h,x)}^H \cap
\left(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\right)\right)\right]_{(h,x)}.
\end{equation}
§.§.§ The proper Lie groupoid case
We now turn to the case of a proper Lie groupoid $\sfG$, and endow $\sfG$ with a transversally invariant
riemannian metric. Recall <cit.>
that for each point $\obj{x} \in \sfG_0$, there is an open neighborhood $U$ of $\obj{x}$
in $\sfG_0$ diffeomorphic to $O \times B_{\obj{x}}$ such that $\sfG_{|U}$ is isomorphic to the product of the pair groupoid over $O$ and $\sfG_{\obj{x}} \ltimes B_{\obj{x}}$.
Hereby, $O$ is an open ball around $\obj{x}$ in the orbit
of $\obj{x}$ and $B_{\obj{x}}$ is a $\sfG_{\obj{x}}$-invariant open ball around the origin
in the normal space $N_{\obj{x}} = T_\obj{x} \sfG_0 / T_{\obj{x}}\calO_{\obj{x}}$ to the tangent
space of the orbit through $\obj{x}$.
According to <cit.>, one can
achieve that the corresponding diffeomorphism $O \times B_{\obj{x}} \to \sfG_0$ is given,
over the factor $B_{\obj{x}}$, by the exponential map
with respect to the chosen transversally invariant riemannian metric.
We let $Y_{\obj{x}}\subset \sfG_0$ denote the image of $\{ \obj{x} \}\times B_{\obj{x}}$ under this diffeomorphism
and call it a slice for $\sfG$ at $\obj{x}$.
By <cit.>, $\sfG_{|Y_{\obj{x}}}$ is isomorphic to $\sfG_{\obj{x}} \ltimes B_{\obj{x}}$. Since the latter is isomorphic to $\sfG_{\obj{x}} \ltimes Y_{\obj{x}}$,
we obtain an isomorphism between $\sfG_{|Y_{\obj{x}}}$ and $\sfG_{\obj{x}} \ltimes Y_{\obj{x}}$
which is induced by the exponential map and the canonical action of $\sfG_{\obj{x}}$ on
$N_{\obj{x}}$. This isomorphism gives rise to an embedding
$\sfG_{\obj{x}} \ltimes Y_{\obj{x}} \hookrightarrow \sfG$ of differentiable stratified groupoids.
Note that if $\sfG = G\ltimes M$ is a translation groupoid, then a slice as defined here corresponds to a slice for the $G$-action on $M$, so using the same notation for both will cause no confusion.
To define a stratification of $\inertianull{\sfG}$, we will employ the stratification constructed above
of the loop space $\inertianull{(\sfG_{\obj{x}}\ltimes Y_{\obj{x}})}_0$.
To this end choose $\arr{g} \in \inertianull{\sfG}$ with $s(\arr{g}) = t(\arr{g}) = \obj{x} \in \sfG_0$.
We then define the germ $\mathcal{S}_{\arr{g}}^{\sfG}$ of the stratification of $\inertianull{\sfG}$ as follows.
Let $h \in \sfG_{\obj{x}}$ denote the element such that $(h,\obj{x})$ corresponds to the arrow $\arr{g}$ under the
isomorphism between $\sfG_{|Y_{\obj{x}}}$ and $\sfG_{\obj{x}} \ltimes Y_{\obj{x}}$.
Let $\mathcal{S}_{(h,\obj{x})}$ be the germ of the orbit Cartan type stratification of
$\inertianull{(\sfG_{\obj{x}} \ltimes Y_{\obj{x}})}$, and
let $\Sat (\mathcal{S}_{(h,\obj{x})})$ denote its saturation within $\sfG$, i.e. the germ of the saturation of
a defining set for $\mathcal{S}_{(h,\obj{x})}$.
\begin{equation}
\label{eq:MLocalStratGpoidDef}
\mathcal{S}_{\arr{g}}^{\sfG}:= \Sat (\mathcal{S}_{(h,\obj{x})})
\end{equation}
then defines the orbit Cartan type stratification of the loop space $\inertianull{\sfG}$.
Similarly, we define
\begin{equation}
\label{eq:MLocalStratGpoidDefQuot}
\mathcal{R}_{\inertia{\pi}(\arr{g})}^{\sfG}:= \inertia{\pi}(\Sat (\mathcal{S}_{(h,\obj{x})})),
\end{equation}
where $\inertia{\pi}\co\inertianull{\sfG}\to |\inertia{\sfG}|$ is the orbit map of the inertia
groupoid. That means that the germ $\mathcal{R}_{\inertia{\pi}(\arr{g})}^{\sfG}$ in the orbit space
$|\inertia{\sfG}|$ is defined to be the projection of $\mathcal{S}_{\arr{g}}^{\sfG}$ to the orbit space.
With these definitions, we have the following.
Let $\sfG$ be a proper Lie groupoid. Then Equation (<ref>) defines
a Whitney stratification of the loop space $\inertianull{\sfG}$ with respect to which the
inertia groupoid $\inertia{\sfG}$ is a locally translation differentiable stratified groupoid.
Moreover, the inertia space $|\inertia{\sfG}|$ inherits a differentiable structure, and
Equation (<ref>) defines a stratification with respect to which
the inertia space is a differentiable stratified space and the orbit map
$\inertianull{\sfG}\to|\inertia{\sfG}|$ a differentiable stratified surjective submersion.
In order to prove Theorem <ref>, the primary focus will be to demonstrate
the following.
Let $G$ be a compact Lie group and $M$ a smooth $G$-manifold. Then Equation (<ref>)
defines a $G$-invariant stratification of the loop space $\inertianull{(G\ltimes M)}$
with respect to which $\inertianull{(G\ltimes M)}$ is a differentiable stratified space such
that the $G$-orbits are subsets of strata.
Before turning to the proof of Theorem <ref>, assume it holds. Assume
further that the stratification of $\inertianull{\sfG}$ is well-defined, i.e. that it does not depend on the choice
of a point in an orbit nor a slice at that point, and that the stratification fulfills Whitney's condition B.
Recall that Whitney stratified spaces are topologically locally trivial,
see <cit.>, and that strata contain orbits because they are defined as saturations.
Hence $\inertia{G} = \sfG\ltimes\inertianull{\sfG}$ is a differentiable stratified groupoid by
Proposition <ref>. That $\sfG$ is a locally translation differentiable groupoid
follows from <cit.>. Similarly, by the definition of the
stratification of $\inertianull{\sfG}$ in terms of stratifications of the inertia spaces of slices
$\sfG_{\obj{x}}\ltimes Y_{\obj{x}}$, $\obj{x}\in \sfG_0$, the inertia groupoid $\inertia{\sfG}$ is even a locally
translation differentiable stratified groupoid. Moreover, since the elements of a stratum of a slice
$Y_{\obj{x}}$ all have the same $\sfG_{\obj{x}}$-orbit type, the inertia space is a differentiable
stratified space by Proposition <ref>, and the orbit map is a stratified surjective
Hence, once we demonstrate that the stratification of $\inertianull{\sfG}$ is
well-defined, prove Theorem <ref> and verify Whitney's condition B,
Theorem <ref> will follow. We first show here that the stratification
is well-defined, assuming well-definition of the stratification for a translation groupoid
$G\ltimes M$.
In the following section, we prove Theorem <ref>.
Afterwards we verify Whitney's condition B to hold true for the orbit Cartan type stratification of
the inertia groupoid and the inertia space of a proper Lie groupoid.
Let $\sfG$ be a proper Lie groupoid,
and let $\obj{x}, \obj{y} \in \sfG_0$ be points in the same orbit.
Let $\arr{g}\in\sfG_1$ such that $s(\arr{g}) = t(\arr{g}) = \obj{x}$, and let $\arr{h}\in\sfG_1$ such that
$s(\arr{h}) = \obj{x}$ and $t(\arr{h}) = \obj{y}$. Put $\arr{g}^\prime:= \arr{h}\arr{g}\arr{h}^{-1}$.
Then $\mathcal{S}_{\arr{g}} = \mathcal{S}_{\arr{g}^\prime}$. In particular,
$\mathcal{S}_{\arr{g}}$ does not depend on the choice of a slice $Y_{\obj{x}}$.
Choose slices $Y_{\obj{x}}$ and $Y_{\obj{y}}$ for $\sfG$ at $\obj{x}$ and $\obj{y}$, respectively.
Then there are identifications $\sfG_{|Y_{\obj{x}}} \cong \sfG_{\obj{x}}\ltimes Y_{\obj{x}}$ and
$\sfG_{|Y_{\obj{y}}} \cong \sfG_{\obj{y}}\ltimes Y_{\obj{y}}$ by <cit.>.
Under these identifications let $\arr{g} = (h,\obj{x})$ and $\arr{g}^\prime = (k,\obj{y})$
for some $h\in \sfG_{\obj{x}}$ and $k\in \sfG_{\obj{y}}$.
Choose a local bisection $\sigma \co U \to \sfG_1$ defined on an open neighborhood of $\obj{x}$ such that
$\sigma (\obj{x}) = \arr{h}$ and such that $t\circ\sigma_{|Y_{\obj{x}}}$ induces a diffeomorphism from
$Y_{\obj{x}}$ to $Y_{\obj{y}}$. The
existence of such a bisection, after possibly shrinking $Y_{\obj{x}}$ and $Y_{\obj{y}}$, is guaranteed by
Then we obtain an isomorphism $\Psi\co \sfG_{\obj{x}}\ltimes Y_{\obj{x}} \to \sfG_{y}\ltimes Y_{y}$
which is given by the composition of the diffeomeorphism
\[
\sfG_{|Y_{\obj{x}}} \to \sfG_{|Y_{\obj{y}}}, \quad \arr{k}
\mapsto
\big(\sigma(t(\arr{k}))\big)\arr{k} \big(\sigma(s(\arr{k}))\big)^{-1}
\]
with the identifications $\sfG_{|Y_{\obj{x}}} \cong \sfG_{\obj{x}}\ltimes Y_{\obj{x}}$ and
$\sfG_{|Y_{\obj{y}}} \cong \sfG_{\obj{y}}\ltimes Y_{\obj{y}}$. Note that $\Psi$
obviously restricts to a diffeomorphism from $\inertianull{(\sfG_{\obj{x}}\ltimes Y_{\obj{x}})}$ onto
$\inertianull{(\sfG_{\obj{y}}\ltimes Y_{\obj{y}})}$. Moreover, by construction,
$\Psi (h,\obj{x})$ is the image of
\[
\big(\sigma (\obj{x})\big)\arr{g} \big(\sigma (\obj{x})\big)^{-1} = \arr{h}\arr{g}\arr{h}^{-1} = \arr{g}^\prime
\]
under the identification $\sfG_{|Y_{\obj{y}}} \cong \sfG_{\obj{y}}\ltimes Y_{\obj{y}}$, hence
$\Psi (h,\obj{x}) = (k,\obj{y})$.
Now we choose a slice $V_{(h,\obj{x})}$ at $(h,\obj{x})$ for the $\sfG_{\obj{x}}$-action on $\sfG_{\obj{x}}\ltimes Y_{\obj{x}}$ and
a Cartan subgroup $\cartan_{(h,\obj{x})}$ of $\centralizer_{\sfG_{\obj{x}}}\!(h)$ associated to $h$. Then
$\Psi(V_{(h,\obj{x})})$ is a slice at $(k,\obj{y})$ for the $\sfG_{\obj{y}}$-action on $\sfG_{\obj{y}}\times Y_{\obj{y}}$, and
$\Psi(\cartan_{(h,\obj{x})} \times\{\obj{x}\})$ a Cartan subgroup of $\centralizer_{\sfG_{\obj{y}}}\!(k)$ associated
to $k$.
Moreover, we claim that $\Psi(\cartan_{(h,\obj{x})}^{\ast} \times\{\obj{x}\}) = \cartan_{(k,\obj{y})}^{\ast} \times\{\obj{y}\}$.
To see this, note that $\Psi_0\co Y_{\obj{x}} \to Y_{\obj{y}}$ is a diffeomorphism that is
$\cartan_{(h,\obj{x})}$-equivariant with respect to the isomorphism $\tau\co \cartan_{(h,\obj{x})} \to \cartan_{(k,\obj{y})}$
given by $\tau(k) = \pi_1\circ\Psi(k,\obj{x})$, where $\pi_1$ denotes the projection
$\pi_1\co\sfG_{\obj{y}}\times Y_{\obj{y}}\to\sfG_{\obj{y}}$.
Then for $\obj{z} \in Y_{\obj{y}}$ and $s \in \cartan_{(h,\obj{x})}$,
$\tau(s)\obj{z} = \tau(s)(\Psi_0\circ \Psi_0^{-1}(\obj{z})) = \Psi_0(s (\Psi_0^{-1}(\obj{z})))$.
Hence $\tau(s)\obj{z} = \obj{z}$ if and only if $s (\Psi_0^{-1}(\obj{z}))= \Psi_0^{-1}(\obj{z})$, from which it follows
that $\Psi_0$ maps the set of points of $Y_{\obj{x}}$ fixed by $s\in\cartan_{(h,\obj{x})}$ onto the fixed set of
$\tau(s)$ in $Y_{\obj{y}}$.
The isomorphism $\Psi$ then maps $\mathcal{S}_{(h,\obj{x})}$
onto $\mathcal{S}_{(k,\obj{y})}$. This proves the claim. If $\obj{x} = \obj{y}$, this argument shows that
the stratification does not depend on the choice of the slice $Y_{\obj{x}}$.
§.§ Proof of Theorem <ref>
Because the stratification given by Equation (<ref>) is a variation on the
stratification given by <cit.>, we refer the reader there for
arguments that are identical or only slightly modified. Observe though that the
stratification given here is coarser than the one from <cit.>,
and that the virtue of the new definition is that it is invariant under Morita equivalences,
hence can be glued together via chart changes.
We assume $G\times M$ is equipped with a riemannian metric given by the product of a $G$-invariant
metric on $M$ and a bi-invariant metric on $G$. For points $y\in M$ and
$(h,x)\in \inertianull{(G\ltimes M)}$, we will denote by $Y_y$ always a slice at $y$ for the $G$-action
on $M$ and by $V_{(h,x)}$ a slice at $(h,x)$ for the $G_x$-action on $G_x\times Y_x$.
We use $H$ to denote the isotropy group $G_{(h,x)} = \centralizer_{G_x}\! (h)$ of $(h,x)$ and the symbol
$N_{(h,x)}$ to denote the normal space
$T_{(h,x)} (G_x\times Y_x) / T_{(h,x)} (G_x(h,x))\cong T_{(h,x)} (G \times M) / T_{(h,x)} (G (h,x))$.
It will be helpful to observe the following, which is a slight strengthening of a special
case of <cit.>.
Let $G$ be a compact Lie group, let $h$ and $k$ be elements of a single connected component
of $G$, and let $\cartan_h$ and $\cartan_k$ be Cartan subgroups of $G$ associated to $h$ and $k$,
respectively. Then there is an element $g \in G^\circ$ such that
g\cartan_k g^{-1} = \cartan_h \quad\text{and}\quad g(k\cartan_k^\circ) g^{-1} = h\cartan_h^\circ .
By <cit.>, we know a priori that $\cartan_h$
and $\cartan_k$ are conjugate and hence isomorphic.
Let $\cartan_k \cong \T^\ell \times \Z/r\Z$ for some $\ell$ and $r$. Then the topological
generators of $\T^\ell \times \Z/r\Z$, i.e. the elements that generate a dense subset of
$\T^\ell \times \Z/r\Z$, are given by $(s, \alpha)$ where $s$ is a topological generator of
$\T^\ell$ and $\alpha$ is a generator of $\Z/r\Z$; see
Since $k\cartan_k^\circ$ generates $\cartan_k/\cartan_k^\circ \cong \Z/r\Z$,
we may therefore choose a topological generator $t$ of $\cartan_k$ such that
$t \in k\cartan_k^\circ$.
Note that $t$, $h$, and $k$ are all in the same connected component of $G$.
By the same argument a topological generator of $\cartan_h$ can be chosen to be an
element of $h\cartan_h^\circ$. Hence there is an element
$g \in G^\circ$ such that $g t g^{-1} \in h\cartan_h^\circ$. But then $gtg^{-1}$ generates a
subgroup of $\cartan_h$ that is dense in $g\cartan_k g^{-1}$. Therefore $g\cartan_k g^{-1}$ is
contained in $\cartan_h$. But as $g\cartan_k g^{-1}$ is isomorphic to $\cartan_k$, hence
to $\cartan_h$, we have $g\cartan_k g^{-1} = \cartan_h$.
The following is a simple consequence of the definition of a Cartan subgroup.
Let $(h,x) \in \inertianull{(G \ltimes M)}$ and $H = G_{(h,x)}$.
A Cartan subgroup $\cartan_{(h,x)}$ of $G_x$ associated to $h$
is also a Cartan subgroup of $H$ associated to $h$.
Recall that the equivalence relation $\simeq$ on a Cartan subgroup $\cartan$ of the isotropy group
$G_x$ is defined by setting $s\simeq t$ if and only if $N_x^s = N_x^t$, where $N_x$ denotes the
normal space $T_xM/T_x (Gx)$. We denote by $[s]$ the $\simeq$ class of $s \in \cartan$. Note that
by definition of a slice, $s\simeq t$ holds true if and only $Y_x^s = Y_x^t$ for one, hence all
(sufficiently small) slices $Y_x$ at $x$.
We recall the following properties of the relation $\simeq$, whose proofs are analogous
to <cit.> and hence are omitted.
Let $Y$ be a slice for the $G$-action on $M$ through a point $x\in M$ and let $\cartan \subset G_x $
be a Cartan subgroup. Then the following holds true.
The group $\cartan$ is partitioned into a finite number of $\simeq$ classes, each with a finite
number of connected components. Each $\simeq$ class $[t]$ is an open subset of the closed subgroup
$t^\bullet$ of $\cartan$ defined by
\[
\bigcap\limits_{t \in H_i} H_i
\bigcap\limits_{y \in Y^t} \cartan_y,
\]
where $\{H_0, \ldots, H_r\}$ is the finite set of isotropy groups for the $\cartan$-action on $Y$,
$\cartan_y$ is the isotropy group of $y$ in $\cartan$,
and $\overline{[t]}$ consists of a union of connected components of $t^\bullet$.
If $s, t \in \cartan$ with $[s] \cap \overline{[t]} \neq \emptyset$, then for each connected component
$[s]^\circ$ of $[s]$ and $[t]^\circ$ of $[t]$ such that
$[s]^\circ \cap \overline{[t]^\circ} \neq \emptyset$, we have
$[s]^\circ \subset \overline{[t]^\circ}$.
If $s, t \in \cartan$ such that $s \not\simeq t$ and $[s]$ is diffeomorphic to $[t]$, then
$[s] \cap \overline{[t]} = \emptyset$.
Using the observation that equivariant diffeomeorphisms map $\simeq$ classes in a
Cartan subgroup $\cartan$ onto $\simeq$ classes in the image Cartan subgroup $\cartan^\prime$,
proven as in Proposition <ref> above, it is
straightforward to verify the following; see also <cit.>.
The normalizer $\normalizer_H(\cartan)$ of $\cartan$ in $H=G_{(h,x)}$ acts on the finite set of
$\simeq$ classes in $\cartan$ in such a way that for each $n \in \normalizer_H(\cartan)$ and
$t \in \cartan$, the submanifold $n[t] n^{-1}$ is diffeomorphic to $[t]$. Moreover,
either $n[t] n^{-1} = [t]$ or $\overline{n[t] n^{-1}} \cap [t] = \emptyset$.
The set germ $\mathcal{S}_{(h,x)}$ is contained in the germ at
$(h,x)$ of points in $\inertianull{(G \ltimes M)}$ having the same isotropy type as $(h,x)$
with respect to the $G$-action on $G \times M$. For, if
$(k, y) \in V_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)$,
then $G_x = G_y$, so the isotropy group of $(k,y)$ with respect to the $G$-action on $G \times M$
coincides with the isotropy group of $(k,y)$ with respect to the $G_x$-action on $G_x \times Y_x$.
But this isotropy group is equal to $H$ as $(k,y) \in V_{(h,x)}^H$.
Similarly, using the same argument as for <cit.>, one shows
that the germ $\mathcal{S}_{(h,x)}$ does not depend on the choice of a Cartan subgroup
$\cartan_{(h,x)}$ of $H$ associated to $h$.
In order to prove that the stratification does not depend on the choice of a slice $V_{(h,x)}$,
we will need the following lemma, which essentially demonstrates that two slices
at the same point are related by a local bisection that acts on $\cartan_{(h,x)}$ by conjugation
in a way that fixes the connected component of $h$.
Let $V_{(h,x)}$ and $W_{(h,x)}$ be slices at $(h,x)$ for the $G_x$-action on
$G_x\times Y_x$ and let $H = \centralizer_{G_x}\!(h)$.
Possibly after shrinking the slices, there is a smooth
function $\sigma\co W_{(h,x)} \to G_x$, $(k,y) \to \sigma_{(k,y)}$
such that the map $\tau\co W_{(h,x)}\to G_x\times Y_x$
given by $\tau(k,y) = \sigma_{(k,y)}(k,y)$ is an
$H$-diffeomorphism of $W_{(h,x)}$ onto $V_{(h,x)}$ which
satisfies $\tau(h,x) = (h,x)$, meaning $\sigma_{(h,x)} \in H$.
Moreover, one can choose $\sigma$ in such a way that
$\sigma_{(h,x)}$ is any arbitrary element of $H$.
For any $\sigma$ as in (<ref>), one has
$\sigma(W_{(h,x)}^H)\subset \sigma_{(h,x)} \normalizer_{G_x}(H)^\circ$.
Fix a $\sigma\co W_{(h,x)}\to G_x$ as above and suppose
$\sigma_{(h,x)} \in \normalizer_{G_x}(H)^\circ$. Let $\cartan_{(h,x)}$ be a
Cartan subgroup of $H$ associated to $h$ and
\[
{K} = \normalizer_H(\cartan_{(h,x)})\cap \normalizer_H(h\cartan_{(h,x)}^\circ)\cap H^\circ.
\]
Then ${K}$ is a closed subgroup of $H^\circ$. There is a continuous function
$(W_{(h,x)})^{H} \to {K}\backslash H^\circ$, which we denote
$(k,y)\mapsto{K}g_{(k,y)}$, such that for each
$(k,y) \in W_{(h,x)}^H$ the relations
\[ \:
g_{(k,y)}\sigma_{(k,y)}\cartan_{(h,x)}\sigma_{(k,y)}^{-1}g_{(k,y)}^{-1} = \cartan_{(h,x)} \: \text{ and } \:
g_{(k,y)} \sigma_{(k,y)}\big(h\cartan_{(h,x)}^\circ\big)\sigma_{(k,y)}^{-1} g_{(k,y)}^{-1} = h\cartan_{(h,x)}^\circ
\]
are fulfilled.
In other words, $g_{(k,y)}\sigma_{(k,y)} \in \normalizer_H(\cartan_{(h,x)})$, and the action of
on $\cartan_{(h,x)}$ by conjugation fixes the connected component containing $h$.
Statement (<ref>) is a special case of <cit.> where it is
shown to be true for proper Lie groupoids. The claim that $\sigma_{(h,x)}$ can be an arbitrary
element of $H$ can be verified by choosing an arbitrary $\sigma$ and an $\ell \in H$ and then
redefining $\sigma$ as the map $(k,y) \mapsto \ell\sigma_{(h,x)}^{-1}\sigma_{(k,y)}$.
To prove (<ref>), fix an arbitrary $\sigma$ as in (<ref>).
If $(k,y) \in W_{(h,x)}^H$, then
$G_{\sigma_{(k,y)}(k,y)} = \sigma_{(k,y)} G_{(k,y)} \sigma_{(k,y)}^{-1} = \sigma_{(k,y)} H \sigma_{(k,y)}^{-1}$.
As $\sigma_{(k,y)}(k,y) \in V_{(h,x)}$ implies $G_{\sigma_{(k,y)}(k,y)} \leq H$, one obtains
$G_{\sigma_{(k,y)}(k,y)} = H$. Therefore, $\sigma$ maps $W_{(h,x)}^H$
into $\normalizer_{G_x}(H)$. By the connectedness of $W_{(h,x)}^H$, the funtion
$\sigma$ therefore maps $(W_{(h,x)})^H$ into $\sigma_{(h,x)} \normalizer_{G_x}(H)^\circ$.
We now turn to (<ref>). For each $(k,y) \in W_{(h,x)}^H$ the action of $\sigma_{(k,y)}$
by conjugation on $H$ fixes the connected components of $H$. This follows from
$\sigma_{(k,y)} \in \normalizer_{G_x}(H)^\circ$ and the fact that the induced map from
$\normalizer_{G_x}(H)$ into the symmetric group of the connected components of $H$ is continuous
and hence maps $\normalizer_{G_x}(H)^\circ$ into the identity.
Then $\sigma_{(k,y)}\cartan_{(h,x)}\sigma_{(k,y)}^{-1}$
is a Cartan subgroup of $H$ associated to $\sigma_{(k,y)}h\sigma_{(k,y)}^{-1} \in hH^\circ$.
By Lemma <ref>, there is a $g_{(k,y)} \in H^\circ$ such that
$g_{(k,y)}\sigma_{(k,y)}\cartan_{(h,x)}\sigma_{(k,y)}^{-1}g_{(k,y)}^{-1} = \cartan_{(h,x)}$ and such that
$g_{(k,y)} \big(\sigma_{(k,y)}h\cartan_{(h,x)}^\circ\sigma_{(k,y)}^{-1}\big) g_{(k,y)}^{-1} = h\cartan_{(h,x)}^\circ$.
Then $g_{(k,y)}\sigma_{(k,y)}\in \normalizer_H(\cartan_{(h,x)})$ is clear.
Of course, $g_{(k,y)}$ is not unique; however, if $g_{(k,y)}$ and $g_{(k,y)}^\prime$ are two such choices, then
a routine computation demonstrates that
$g_{(k,y)}^\prime g_{(k,y)} \in \normalizer_H(\cartan_{(h,x)})$, and
$g_{(k,y)}^\prime g_{(k,y)} \in \normalizer_H(h\cartan_{(h,x)}^\circ)$. Similarly, as both $g_{(k,y)}$
and $g_{(k,y)}^\prime$ are elements of $H^\circ$ by construction, $g_{(k,y)} g_{(k,y)}^\prime\in H^\circ$.
That is, $g_{(k,y)}^\prime g_{(k,y)}^{-1} \in {K}$.
Conversely, if $n \in {K}$ and $g_{(k,y)}$ is given as above, then we have $ng_{(k,y)} \in H^\circ$
\[
ng_{(k,y)}\sigma_{(k,y)}\cartan_{(h,x)}\sigma_{(k,y)}^{-1}g_{(k,y)}^{-1}n^{-1} = n\cartan_{(h,x)}n^{-1} = \cartan_{(h,x)},
\]
\[
ng_{(k,y)} \sigma_{(k,y)}\big( h\cartan_{(h,x)}^\circ\big)\sigma_{(k,y)}^{-1} g_{(k,y)}^{-1}n^{-1}
= n\big(h\cartan_{(h,x)}^\circ \big) n^{-1}
= h\cartan_{(h,x)}^\circ,
\]
so that $ng_{(k,y)}$ satisfies the desired properties as well. Therefore, while $g_{(k,y)}$
is not unique, the right coset ${K}g_{(k,y)}$ is determined uniquely. Note that
${K}$ is indeed a closed subgroup of $H$, as
$\normalizer_H(h\cartan_{(h,x)}^\circ)\cap \normalizer_H(\cartan_{(h,x)})$
is a union of connected components of $\normalizer_H(\cartan_{(h,x)})$; this is clear by
considering the homomorphism of $\normalizer_H(\cartan_{(h,x)})$ into the symmetric group
on the connected components of $\cartan_{(h,x)}/\cartan_{(h,x)}^\circ$.
We claim that the assignment $(k,y) \mapsto {K}g_{(k,y)}$ is a continuous function
$W_{(h,x)}^H \to {K}\backslash H^\circ$.
Let $ (k_i,y_i)_{i\in \N} \subset W_{(h,x)}^H$ be a convergent sequence with $\lim_{i\to\infty} (k_i,y_i) = (k,y)$.
Put $g_i := g_{(k_i,y_i)}$ for each $i\in \N$. Without loss of generality,
we may assume that $\lim_{i\to\infty} g_i = g \in H^\circ$. Choose a topological generator
$t$ of $\cartan_{(h,x)}$. As $\cartan_{(h,x)}$ is a Cartan subgroup associated to $h$, we may
assume that $t$ is in the same connected component $h\cartan_{(h,x)}^\circ$ of $\cartan_{(h,x)}$
as $h$; see the proof of Lemma <ref> above.
Then we have $g_i\sigma_{(k_i,y_i)} t \sigma_{(k_i,y_i)}^{-1} g_i^{-1} \in h\cartan_{(h,x)}^\circ$
for each $i$. Since $\sigma$ is a continuous function, we may take the limit to conclude
$g \sigma_{(k,y)} t \sigma_{(k,y)}^{-1} g^{-1} \in h\cartan_{(h,x)}^\circ$. However, as $t$ generates a subgroup
that is dense in $\cartan_{(h,x)}$, we have that $g \sigma_{(k,y)} t \sigma_{(k,y)}^{-1} g^{-1}$ generates
a subgroup that is dense in a subgroup of $\cartan_{(h,x)}$ that is isomorphic to $\cartan_{(h,x)}$,
and then must be equal to $\cartan_{(h,x)}$. This implies
\[
= \cartan_{(h,x)},
\]
\[
g_{(k,y)}\big(\sigma_{(k,y)}h\cartan_{(h,x)}\sigma_{(k,y)}^{-1}\big)^\circ g_{(k,y)}^{-1}
= h\cartan_{(h,x)}^\circ.
\]
Hence, we can take $g_{(k,y)} = g$, determining the unique coset of ${K}\backslash H^\circ$,
and hence the map $W_{(h,x)}^H \to {K}\backslash H^\circ$ given by
$(k,y) \mapsto {K}g_{(k,y)}$ is continuous.
With this, we have the following.
The germ $\mathcal{S}_{(h,x)}$ is independent of the particular choice of the slice $V_{(h,x)}$ at $(h,x)$.
In fact, given slices $V_{(h,x)}$ and $W_{(h,x)}$ at $(h,x)$, the germs of
$V_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)$ and
$W_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)$ coincide at $(h,x)$.
Suppose $V_{(h,x)}$ and $W_{(h,x)}$ are two choices of slices at $(h,x)$ for the
$G_x$-action on $G_x \times Y_x$. Let $\cartan_{(h,x)}$ be a Cartan subgroup of $H$
associated to $h$. As the germ of the stratification does not depend on the choice of Cartan
subgroup, we may assume that the stratum containing $(h,x)$ is defined with respect to each
of the two slices using this Cartan subgroup.
By Proposition <ref> (<ref>) there exists, after shrinking slices if necessary,
a function $\sigma\co W_{(h,x)}\to G_x$ such that $(k,y)\mapsto \sigma_{(k,y)}y$ is a
$H$-diffeomorphism of $W_{(h,x)}$ onto $V_{(h,x)}$. We may assume that
$\sigma_{(h,x)} = 1$. Choose a Cartan subgroup $\cartan_{(h,x)}$ of $H$
associated to $h$ and then, by Proposition <ref> (<ref>),
a continuous function $W_{(h,x)}^H \to {K}\backslash H^\circ$ denoted
$(k,y)\mapsto{K}g_{(k,y)}$ such that for each $(k,y)\in W_{(h,x)}^H$,
we have $g_{(k,y)}\sigma_{(k,y)}\cartan_{(h,x)}\sigma_{(k,y)}^{-1}g_{(k,y)}^{-1} = \cartan_{(h,x)}$, and
$g_{(k,y)} \sigma_{(k,y)}\big(h\cartan_{(h,x)}^\circ\big)\sigma_{(k,y)}^{-1} g_{(k,y)}^{-1} = h\cartan_{(h,x)}^\circ$.
In particular, as $\sigma_{(h,x)} = 1$, it follows that ${K}g_{(h,x)} = {K}$.
Choose a local section for the fiber bundle $H^\circ \to {K}\backslash H^\circ$
near the point ${K}$ such that ${K} \mapsto 1$. Shrinking slices if necessary so that
each $g_{(k,y)}$ is contained in the domain of this section for $(k,y) \in W_{(h,x)}^H$
(which is possible as $g_{(h,x)} = {K}$),
we have a continuous choice of a specific representative $g_{(k,y)}$ of each coset
${K}g_{(k,y)}$. In particular, as $\sigma_{(h,x)} = 1$ (and as the section was chosen such that
${K}\mapsto 1$, we have $g_{(h,x)} = 1$).
For each $(k,y) \in W_{(h,x)}$, we have that $g_{(k,y)}\sigma_{(k,y)} \in \normalizer_{G_x}(\cartan_{(h,x)})$,
and hence conjugation by $g_{(k,y)}\sigma_{(k,y)}$ yields an element of $\AUT(\cartan_{(h,x)})$.
That is, we have a function $W_{(h,x)}^H \to \AUT(\cartan_{(h,x)})$ which is
continuous by construction. However, the automorphism group of $\cartan_{(h,x)}$ is discrete,
see <cit.>, so that as $W_{(h,x)}^H$ is
connected, the map into $\AUT(\cartan_{(h,x)})$ must be constant. Moreover, as
$g_{(h,x)}\sigma_{(h,x)} = 1$, we see that each $g_{(k,y)}\sigma_{(k,y)}$ maps to the trivial
element of $\AUT(\cartan_{(h,x)})$.
We let $U = G_x V_{(h,x)} \cap G_x W_{(h,x)}$ and claim that
\[
U \cap \Big(W_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)\Big)
= U \cap \Big(V_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)\Big)
\]
$(k,y)\in W_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)$.
Using the above construction, we have $\sigma_{(k,y)}(k,y) \in V_{(h,x)}$.
As $\sigma_{(k,y)} \in \normalizer_{G_x}(H)$, we have $\sigma_{(k,y)}(k,y) \in V_{(h,x)}^H$.
Similarly, $\sigma_{(k,y)} \in G_x$ and $y \in Y_x^{G_x}$, so that
$\sigma_{(k,y)}y = y \in Y_x^{G_x}$. Then as $H$ fixes $\sigma_{(k,y)}(k,y)$ and
$g_{(k,y)} \in H$, it follows that
\[
\in
\cartan_{(h,x)}.
\]
But then as $g_{(k,y)}\sigma_{(k,y)}$ acts trivially on $\cartan_{(h,x)}$, we have that
$g_{(k,y)}\sigma_{(k,y)}k\sigma_{(k,y)}^{-1}g_{(k,y)}^{-1} = k$, so that the point
$(k,y) \in V_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\big)$
to begin with. Switching roles completes the proof.
With this, the proof of the following proposition uses identical techniques to those of
<cit.>. In particular, it is a matter of choosing
specific slices in terms of orthogonal complements using the riemannian metric.
Each $\mathcal{S}_{(h,x)}$ is the germ of a smooth $G$-submanifold of $G \times M$ that
intersects $G_x \times Y_x$ as a smooth submanifold. Each $\mathcal{R}_{(h,x)}$ is the
germ of a smooth submanifold of the differentiable space $G\backslash(G \times M)$
that intersects $G_x\backslash(G_x \times Y_x)$ as a smooth submanifold.
In order for the germs $\mathcal{S}_{(h,x)}$ to define a stratification,
one must verify that for each $(h,x) \in \inertianull{(G\ltimes V)}$ there is
a neighborhood $U$ in $\inertianull{(G\ltimes V)}$ and a decomposition $\mathcal{Z}$
of $U$ such that for all $(k, y) \in \inertianull{(G\ltimes V)}$, the germ
$\mathcal{S}_{(k,y)}$ coincides with the germ of the piece of $\mathcal{Z}$
containing $(k, y)$. For the remainder of this section, we fix $(h,x)$,
a slice $Y_x$ at $x$ for the $G_x$-action on $V$, and a slice $V_{(h,x)}$
at $(h,x)$ for the $G_x$-action on $G_x \times Y_x$. Set
$U := GV_{(h,x)} \cap \inertianull{(G\ltimes V)}$. Note that $U$ is indeed an
open neighborhood of $(h,x)$ in $\inertianull{(G\ltimes V)}$, as $G_x V_{(h,x)}$
is an open $G_x$-invariant neighborhood of $(h,x)$ in $\inertianull{(G_x\ltimes Y_x)}$
and so the $G$-saturation is as well by <cit.>.
We now define the decomposition $\mathcal{Z}$ of $U$.
Given $(\tilde{k}, \tilde{y}) \in U$ there is a $\tilde{g} \in G$ such that
$\tilde{g}(\tilde{k}, \tilde{y}) \in V_{(h,x)}$. Put $(k, y) = \tilde{g}(\tilde{k}, \tilde{y})$
and $K = G_{(k,y)} \leq H$, and let $\cartan_{(k,y)}$ be a Cartan subgroup in $K$
associated to $k$. Define $\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y})$
to be the $G$-saturation of the set of points
$(l, z) \in (V_{(h,x)})_K \cap (\cartan_{(k,y)} \times (Y_x)_{G_y})$
such that $\cartan_{(k,y)}$ is also a Cartan subgroup of $K$ associated to $l$ and such that
the $\simeq$ class of $l$ at $z$ in $\cartan_{(k,y)}$ is diffeomorphic to $\cartan_{(k,y)}^{\ast}$.
Define the piece $\mathcal{Z}$ containing $(\tilde{k},\tilde{y})$ to be the connected
component of $\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y})$
containing $(\tilde{k}, \tilde{y})$.
By <cit.>,
the slice representations of
points in the same orbit type are isomorphic.
Hence, if $z \in (Y_x)_{G_y}$, then the slice representations $N_y$ and $N_z$ for the action
of $G_x$ on $Y_x$ at $y$ and $z$, respectively, are isomorphic as $G_y$-representations.
This isomorphism induces an isomorphism of $\cartan_{(k,y)}$-representations, which
induces a diffeomorphism of $\simeq$
classes at $y$ onto $\simeq$ classes at $z$. Then the $\simeq$ class of $l$ at $y$
is diffeomorphic to the $\simeq$ class of $l$ at $z$, and hence to the $\simeq$
class of $k$ at $y$. Therefore, the set
$\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y})$ can be written as
\[
\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y}) =
G \Big( (V_{(h,x)})_K \cap \big(\cartan_{(k,y)}^{\ast\ast} \times (Y_x)_{G_y} \big) \Big),
\]
where $\cartan_{(k,y)}^{\ast\ast}$ denotes the union of $\simeq$ classes in $\cartan_{(k,y)}$
that are diffeomorphic to $\cartan_{(k,y)}^{\ast}$. Choosing representatives
$k_0,\ldots, k_r$ from the collection of such $\simeq$ classes with $k_0 = k$ and noting
that this collection is finite by Proposition <ref>, we can express
\begin{equation}
\label{eq:decUrep}
\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y}) =
G \Big(\bigcup\limits_{i=0}^r (V_{(h,x)})_K \cap \big(\cartan_{(k_i,y)}^{\ast}
\times (Y_x)_{G_y} \big) \Big).
\end{equation}
In particular, we will see below that for every
$(k,y) \in V_{(h,x)}$ with isotropy group $K$ and every Cartan subgroup $\cartan_{(k,y)}$
of $K$ associated to $k$,
the connected component of $\cartan_{(k,y)}$ containing $k$ is the only connected
component that intersects the projection of $V_{(h,x)}$ onto $G_x$.
Note that $\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y})$
is clearly a subset of $\inertianull{(G\ltimes V)}$ as $\cartan_{(k,y)} \leq G_y$.
Using arguments identical to those in <cit.>, one can
demonstrate that $\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y})$ depends only
on the orbit $G(k,y)$, and does not depend on the choice of the Cartan subgroup $\cartan_{(k,y)}$.
Hence, we we may denote $\mathcal{U}_{\tilde{g}}^{\cartan_{(k,y)}}(\tilde{k}, \tilde{y})$ simply
as $\mathcal{U}(k, y)$ and let $\mathcal{U}_{(\tilde{k}, \tilde{y})}$ denote the
connected component of $\mathcal{U}(k, y)$ containing $(\tilde{k}, \tilde{y})$.
The partition $\mathcal{Z}$ of $U$ then can be written as
\begin{equation}
\label{eq:LocalStratPieces}
\mathcal{Z} = \big\{ \mathcal{U}_{(\tilde{k} , \tilde{y})} \subset U \mid
(\tilde{k} , \tilde{y}) \in U \big\} .
\end{equation}
By <cit.>, using the linearity of the action on slices,
one can demonstrate that the exponential map associated to the product metric on $G_x \times Y_x$
maps the subset
\begin{equation}
\label{eq:modspace}
(N_{(h,x)})_K \cap \big( T_h (k_i^\bullet) \oplus (T_x Y_x)_{G_y} \big) \cap B_{(h,x)}
\end{equation}
onto $(V_{(h,x)})_K \cap \left(\overline{\cartan_{(k_i,y)}^{\ast}} \times (Y_x)_{G_y} \right)$.
Recall that $B_{(h,x)}$ is an $H$-invariant ball around the origin in the normal space $N_{(h,x)}$
and $k_i^\bullet$ is defined in Proposition <ref>.
By construction, (<ref>) is a semialgebraic subset of $N_{(h,x)}$ and is invariant
under the action of $t\in (0,1]$. Similarly, because there are only
finitely many $\simeq$ classes in $\cartan_{(k,y)}$, there are
$l_1,\ldots ,l_N \in \cartan_{(k,y)}$
such that each group $l_j^\bullet$, $j=1,\ldots,N$,
has dimension less than $\dim k^\bullet$, and
\[
\cartan_{(k,y)}^{\ast} = \overline{\cartan_{(k,y)}^{\ast}}
\smallsetminus \bigcup_{j =1}^N l_j^\bullet \: .
\]
Then the exponential function maps the semialgebraic set
\begin{equation}
\label{eq:modspace2}
(N_{(h,x)})_K \cap \Big( \big( T_h (k^\bullet ) \smallsetminus \bigcup_{j =1}^N
T_h (l_j^\bullet) \big) \oplus (T_x Y_x)_{G_y} \Big) \cap B_{(h,x)}
\end{equation}
onto $(V_{(h,x)})_K \cap \big(\cartan_{(k,y)}^{\ast} \times (Y_x)_{G_y} \big)$.
Restricting the inverse of the exponential map from $V_{(h,x)}$ to
$V_{(h,x)}\cap\inertianull{(G_x \ltimes Y_x)}$ yields an $H$-equivariant embedding
$\iota$ of a neighborhood of $(h,x)$ in $V_{(h,x)}\cap\inertianull{(G_x \ltimes Y_x)}$
into the normal space $N_{(h,x)}$, where the stratum of $(h,x)$ is mapped onto the subspace
\[
N_{(h,x)}^H \cap \Big( T_h h^\bullet \times (T_x Y_x)^{G_x} \Big).
\]
Using the description of the image of each stratum
given in Equation (<ref>), one sees immediately that the homotopy
defined as multiplication in $N_{(h,x)}$ by scalars $t \in [0,1]$ contracts
the image of $\iota$ onto the origin preserving the image of $\iota$. The linearity
of the $H$-action on $N_{(h,x)}$ ensures that this homotopy is $H$-equivariant,
and scalars $t\in(0,1]$ preserve the images of strata as demonstrated above. We
therefore observe the following.
The inertia groupoid $\inertia{\sfG}$ of a proper Lie groupoid $\sfG$ satisfies
the local contractibility hypothesis of Definition <ref>.
Following <cit.>, it is easy to see the following.
The germs of the $\mathcal{U}\big( G(k,y) \big)$ coincide with the stratification.
That is, for $(\tilde{k}, \tilde{y}) \in U = GV_{(h,x)}$, the germs
$[\mathcal{U}\big( G(k,y) \big)]_{(\tilde{k},\tilde{y})}$,
and $\mathcal{S}_{(\tilde{k},\tilde{y})}$ coincide.
Since the $\mathcal{S}_{(h,x)}$ are germs of smooth
$G$-submanifolds of $G \times M$, and the piece associated to a
point $(\tilde{k},\tilde{y}) \in U$ has the same set germ as
$\mathcal{S}_{(l, z)}$ at $(l, z) \in
\mathcal{U}_{(\tilde{k},\tilde{y})}$, it follows that the
pieces of $\mathcal{Z}$ are smooth submanifolds of $G \times M$
invariant under the $G$-action. The proof of the following
is a minor variation of that of <cit.>.
The partition $\mathcal{Z}$
of $U = GV_{(h,x)}$ given by Equation (<ref>) is finite.
We now verify that $\mathcal{Z}$ is a decomposition indeed,
cf. <cit.>. The proof is similar to that of
The pieces of $\mathcal{Z}$
satisfy the condition of frontier.
Suppose there are points $(h,x)$ and $(k,y)$ with
$\mathcal{U}\big( G(h,x) \big) \cap \overline{\mathcal{U}\big( G(k,y) \big)}\neq\emptyset$.
As the pieces of $\mathcal{Z}$ are defined to be connected components, it is sufficient to
show that $\mathcal{U}\big( G(h,x) \big) \cap \overline{\mathcal{U}\big( G(k,y) \big)}$,
which is obviously closed in $\mathcal{U}\big( G(h,x) \big)$, is also
open in $\mathcal{U}\big( G(h,x) \big)$. Note that we can assume with no loss of generality
that one of the points in question is $(h,x)$, the point used to define $U$, as we may
restrict consideration to a neighborhood of that point. Moreover, as the piece
$\mathcal{U}\big(G(h,x)\big)$ may be defined in terms of any point it contains,
we may assume that
$G(h,x) \subset \mathcal{U}\big( G(h,x) \big) \cap \overline{\mathcal{U}\big( G(k,y) \big)}$.
Similarly, we assume by choosing another representative of the orbit if necessary that
$(k,y) \subset V_{(h,x)}$. By Proposition <ref>, an open
neighborhood of $(h, x)$ in $\mathcal{U}\big( G(h,x) \big)$ is given by
$G\big(V_{(h, x)}^H \cap (\cartan_{(h, x)}^{\ast}\times Y_x^{G_x})\big)$
for a sufficiently small slice $V_{(h, x)}$ at $(h, x)$. We will show that
$G\big(V_{(h, x)}^H \cap (\cartan_{(h, x)}^{\ast}\times Y_x^{G_x})\big)$
is contained in $\overline{\mathcal{U}\big( G(k,y) \big)}$.
Let $K := G_{(k,y)} \leq H$, and then $K^\circ \leq H^\circ$. Then any maximal
torus in $K^\circ$ is contained in a maximal torus in $H^\circ$.
Moreover, as $V_{(h,x)}^K$ is connected and closed under multiplication
by scalars $t \in (0,1]$, taking the limit of $t(k,y)$ as $t\to 0$,
we see that $h \in K$. Similarly, $h$ and $k$ are in the same
connected component of $K$. It follows that we may choose a Cartan subgroup
of $H$ associated to $h$ by taking the group generated by a maximal torus
in $H^\circ$ that contains a maximal torus in $K^\circ$ and $h$. That is,
we may assume that $h \in \cartan_{(k,y)} \leq \cartan_{(h,x)}$ and
$\cartan_{(k,y)} = \cartan_{(h,x)} \cap K$.
Then we have $h \in \overline{\cartan_{(k',y)}^{\ast}}$ for some
$k'$ whose $\simeq$ class at $y$ is diffeomorphic to $\cartan_{(k,y)}^{\ast}$.
Similarly, as $G_y \leq G_x$, it follows that
$Y_x^{G_x} \subset \overline{(Y_x)_{G_y}}$. From these observations, we have
\[
(h,x) \in
V_{(h,x)}^H \cap \Big(\overline{\cartan_{(k',y)}^{\ast}} \times Y_x^{G_x}\Big)
\subset
\overline{(V_{(h,x)})_K} \cap \big(\overline{\cartan_{(k',y)}^{\ast}}
\times \overline{(Y_x)_{G_y}} \big).
\]
Now, let $l \in \cartan_{(h,x)}^{\ast}$ so that $Y_x^l = Y_x^h$. In particular, as $h \in K \leq G_y$
and $y \in Y_x$, it follows that
$l \in G_y$. Similarly, as $l \in \cartan_{(h,x)}$, as $k \in \cartan_{(k,y)} \leq \cartan_{(h,x)}$,
and as $\cartan_{(h,x)}$ is abelian, we have $l(k,y) = (k,y)$ so that $l \in K$. In particular,
$l \in \cartan_{(h,x)} \cap K = \cartan_{(k,y)}$. This demonstrates
$\cartan_{(h,x)}^{\ast} \subset \cartan_{(k,y)}$. As $Y_y \subset Y_x$,
we have that the relation $\simeq$ at $x$ implies $\simeq$ at $y$, so that the $\simeq$ classes at $y$
are the intersection with $\cartan_{(k,y)}$ of a (finite) union of $\simeq$ classes at $y$.
That is, using Proposition <ref> (<ref>),
$\cartan_{(h,x)}^{\ast} \subset\overline{\cartan_{(k',y)}^{\ast}}$. Then
\[
V_{(h,x)}^H \cap \Big(\cartan_{(h,x)}^{\ast} \times Y_x^{G_x}\Big)
\subset
\overline{(V_{(h,x)})_K} \cap \big(\overline{\cartan_{(k',y)}^{\ast}}
\times \overline{(Y_x)_{G_y}} \big)
\subset \overline{\mathcal{U}\big( G(k,y) \big)}.
\]
Considering the $G$-saturations of both sides of this
inclusion, it follows that an open neighborhood of $(h, x)$ in
$\mathcal{U}\big( G(k,x) \big)$ is contained in
$\mathcal{U}\big( G(h,x) \big) \cap \overline{\mathcal{U}\big( G(k,y) \big)}$.
This completes the proof.
§.§ Whitney Condition B
Here, we complete the proof of Theorem <ref> by demonstrating
that the the stratifications of $\inertianull{\sfG}$ and $|\inertia{\sfG}|$ are
Whitney B-regular.
The proof follows <cit.> and <cit.>.
Roughly, the proof involves giving a parameterization of a neighborhood of a point in
$\inertia{\sfG}$ and its tangent space sufficient to describe the secants of points
in neighboring strata. Note that in our argument we use that the pieces satisfy
the condition of frontier, which was shown above.
Let $\sfG$ be a proper Lie groupoid. The orbit Cartan type stratifications of the loop space
$\inertia{\sfG}$ and the inertia space $|\inertia{\sfG}|$ both satisfy Whitney's condition B.
Because the claim is local, we may assume that the groupoid $\sfG$ is given by
the product of $O\times O \rightrightarrows O$ and $G\ltimes Y$ where
$G$ is the isotropy group $\sfG_x$ of some point $x \in \sfG_0$, $O$ is an open
neighborhood of $x$ in its orbit, and $Y$ is a slice through $x$.
Let $(h,x) \in \inertia{\sfG}$, $H = \centralizer_{G}\!(h)$, and $V_{(h,x)}$
a slice at
$(h,x)$ for the $G$-action on $G\times Y$ of the form $\exp (B_{(h,x)})$, where $B_{(h,x)}$ is
a ball around the origin in the normal space $N_{(h,x)}$. Now let us denote by $\mathcal{Z}$ the decomposition
of $\inertianull{\sfG}$ obtained by taking the saturations of the sets defined through Eq. (<ref>),
which amounts to taking their products with $O$. Let $R$ be the piece of $\mathcal{Z}$
containing $(h,x)$, i.e. the set of points of the form $((o,o),(l,z))$ where $o \in O$ and
$(l,z) \in V_{(h,x)}^H \cap \big(\cartan_{(h,x)}^{\ast}\times Y^{G}\big)$.
We show that for any stratum $S \in \mathcal{Z}$ with $(h,x)\in \overline{S}$, Whitney's condition B
is satisfied at $(h,x)$ for the pair of strata $(R,S)$. To describe the stratum $S$ in some more detail,
consider an orbit $G(k,y)$ for $(k, y) \in S$. As in the proof of Proposition <ref>,
we may choose the representative $(k,y)$ of the orbit $G(k,y)$ such that $(k,y) \in V_{(h,x)}$,
$h \in \cartan_{(k,y)} \leq \cartan_{(h,x)}$, and $h \in \overline{\cartan_{(k',y)}^{\ast}}$
for some $k'$. In particular, we then have
$K \leq H$ for the isotropy group $K := \centralizer_{G_y}\!(k)$ of $(k,y)$ and $G_y \leq G$. As
shown above, $S$ coincides with the connected component of $\mathcal{U}\big( G(k,y) \big)$ containing
Suppose now that $((u_i,u_i),(h_i, x_i))_{i \in \N}$ is a sequence in $R$ and $((o_i,o_i),(k_i, y_i))_{i \in \N}$
a sequence in $S$,
and that both sequences converge to $((x,x),(h,x))$. Assume in addition that in a smooth chart around
$((x,x),(h,x))$ the secant lines
\[ \ell_i = \overline{((u_i,u_i),(h_i,x_i)),((o_i,o_i),(k_i,y_i))} \]
converge to a straight line $\ell$, and the tangent spaces
$T_{((o_i,o_i),(k_i,y_i))}S$ converge to a subspace $\tau$. Then we must show that $\ell \subset \tau$.
Note that the hypotheses imply that
$((x,x),(h,x)) \in \mathcal{U}\big( G(h,x) \big)\cap \overline{\mathcal{U}\big( G(k,y) \big)}$. By
the proof of Proposition <ref> and
the choices of $(k,y)$ and $\cartan_{(k,y)} \subset K$ we
obtain the relation
\begin{equation}
\label{eq:closurerel1}
V_{(h,x)}^H \cap \Big(\cartan_{(h,x)}^{\ast} \times Y^{G}\Big)
\subset
\overline{(V_{(h,x)})_K} \cap \big(\overline{\cartan_{(k',y)}^{\ast}}
\times \overline{(Y)_{G_y}} \big).
\end{equation}
Denote by $\mathfrak{g}_x$ the Lie algebra of $G$, by $\mathfrak{h}$ the Lie algebra of $H$,
and let $\mathfrak{m}$ denote the orthogonal complement of $\mathfrak{h}$ in $\mathfrak{g}_x$
with respect to the initially chosen bi-invariant metric on $G$. Then there is a neighborhood
$U \subset \sfG_0 \cong O \times_H V_{(h,x)}$ of $(h,x)$ such that
\[
\Psi \co U \longrightarrow O \times \mathfrak{m} \times N_{(h,x)} , \:
[o, \exp_{|\mathfrak{m}} \xi, \exp_{(h,x)}(v)]\longmapsto (o, \xi, v)
\]
is a smooth chart at $((x,x),(h,x))$, where $\exp_{|\mathfrak{m}}$ denotes the restriction of the exponential
map of the Lie group $G$ to $\mathfrak{m}$, and $\exp_{(h,x)}$ the exponential function restricted to
the open ball $B_{(h,x)} \subset N_{(h,x)}$. After possibly shrinking $U$
there is an open neighborhood $Q$ of $H$ in $G$ such that
\[
\Psi\left(O\times Q\left( V_{(h,x)}^H \cap
(\cartan_{(h,x)}^{\ast}\times Y^{G})\right) \right)
\subset
O\times \mathfrak{m} \times \left( N_{(h,x)}^H \cap T_{(h,x)}
(\cartan_{(h,x)}^{\ast}\times Y^{G})\right).
\]
We may assume that the sequences $((u_i,u_i),(h_i, x_i))_{i \in \N}$ and $((o_i,o_i),(k_i, y_i))_{i \in \N}$
are contained in $U$.
Since $((o_i,o_i),(k_i, y_i)) \in \mathcal{U}\big( G(k,y) \big)$, one knows that
\[
\Psi((o_i,o_i),(k_i,y_i))
\in O \times \mathfrak{m} \times H \left(
(V_{(h,x)})_K \cap \left( \cartan_{(k,y)}^{\ast\ast} \times Y^{G}\right)\right).
\]
Recall that $\cartan_{(k,y)}^{\ast\ast}$ consists of a finite collection of pairwise
disjoint $\simeq$ classes in $\cartan_{(k,y)}$. Moreover, by Lemma
<ref>, each such $\simeq$ class is disjoint from the closures
of the other classes. By passing to a subsequence, we may assume without loss of generality
that each $k_i$ is in one fixed class, i.e.
\[
((o_i,o_i),(k_i, y_i)) \in O\times G \left((V_{(h,x)})_K \cap
\left( \cartan_{(k^\prime,y)}^{\ast} \times Y^{G}\right)\right)
\]
for all $i$ and some fixed $k^\prime \in \cartan_{(k,y)}$.
Note that $\lim o_i = \lim u_i = x$. Moreover, each piece
of $\mathcal{Z}$ is a product of a piece in $GV_{(h,x)} \subset G\times Y$
with the diagonal in $O\times O$, so we may project onto $G\times Y$ and ignore
the $O$-factor.
Choose $\tilde{l}_i \in G$ such that $(\tilde{k}_i,
\tilde{y}_i):= \tilde{l}_i (k_i, y_i) \in (V_{(h,x)})_K$ for
all $i\in \N$. Put $(\tilde{h}_i,\tilde{x}_i):= l_i (h_i,x_i)$.
After possibly passing to a subsequence, $(\tilde{l}_i)_{i\in
\N}$ converges to some $\tilde{l} \in H$, the secant lines
$\tilde{\ell}_i =
\overline{(\tilde{h}_i,\tilde{x}_i),(\tilde{k}_i,\tilde{y}_i)}$
converge to a straight line $\tilde{\ell}$, and the tangent
spaces $T_{(\tilde{k}_i,\tilde{y}_i)}S$ converge to a subspace
$\tilde{\tau}$. By definition, and since $\tilde{l}_i
T_{(k_i,y_i)} S = T_{(\tilde{k}_i,\tilde{y}_i)}S$ for all $i$,
one obtains $\tilde{\ell} = \tilde{l} \ell$, and $\tilde{\tau}
= \tilde{l} \tau$. Hence, the first claim is shown, if
$\tilde{\ell} \subset \tilde{\tau}$. Without loss of
generality we may therefore assume that for all $i \in \N$
\begin{equation}
\label{eq:reductionky}
(k_i, y_i) \in (V_{(h,x)})_K \cap \big( \cartan_{(k^\prime,y)}^{\ast}
\times Y^{G} \big) ,
\end{equation}
and then show $\ell \subset \tau$ for the sequences $(k_i,
y_i)_{i\in \N}$ and $(h_i,x_i)_{i\in \N}$.
Eq. (<ref>) now means in particular that
\[
\Psi (k_i,y_i)
\in \{ 0 \} \times \Big(
(N_{(h,x)})_{K} \cap
\exp^{-1}_{(h,x)}\big(\cartan_{(k^\prime,y)}^{\ast} \times Y^{G}\big)\Big).
\]
Since $\overline{\cartan_{(k^\prime,y)}^{\ast}}$ is an
open and closed subset of a closed subgroup of $G$ and also contains $h$, the set
\[
V:= N_{(h,x)} \cap T_{(h,x)}
\Big( \big(\overline{\cartan_{(k^\prime,y)}^{\ast}}\big) \times Y^{G}\Big)
\]
is a subspace of $N_{(h,x)}$. Let $W$ be the orthogonal
complement of the invariant space $V^H$ in $V$ with respect to
the $H$-invariant scalar product induced from $V_{(h,x)}$. Then
the image under the chart $\Psi$ of every element of $G\big(
V_{(h,x)}^H \cap (\cartan_{(h,x)}^{\ast}\times Y^{G})\big)\cap U$ and every
$(k_i, y_i)$ is contained in
\[
\mathfrak{m} \times (W_K \cup \{ 0 \}) \times V^H .
\]
With respect to this decomposition, $(h,x)$ has coordinates
$(0, 0, 0)$, each element of $G\left( V_{(h,x)}^H \cap(\cartan_{(h,x)}^{\ast}\times Y^{G})\right)$
has coordinates contained in $\mathfrak{m} \times 0 \times V^H$, and each
sequence element $(k_i, y_i)$ has coordinates contained in
$\{ 0 \} \times W_{K} \times V^H$. In particular, let
\[
\Psi(k_i,y_i)
(0 , w_i, v_i)
\]
for every $i$. Since $W_{K}$ is invariant under multiplication by non-vanishing scalars, we have
\[
\begin{split}
(\xi, w, v)
:=\, &
\lim\limits_{i\to\infty} \frac{ \Psi(k_i,y_i) - \Psi(h_i,x_i) }
{\| \Psi(k_i,y_i) - \Psi(h_i,x_i) \|}
\, \in \mathfrak{m} \times \overline{W_{K}} \times V^H \ .
% &=
% \mathfrak{m} \times (W_{(K)} \cup \{ 0 \}) \times V^H.
\end{split}
\]
By compactness of the unit sphere in $W$, the sequence
$\frac{w_i}{\| w_i \|}$ converges to some $\hat{w} \in SW$
after possibly passing to a subsequence. Then $w = \| w
\|\hat{w}$. Since $W_{K}$ is invariant by non-vanishing
scalars, we have
\[
\mathfrak{m} \times \operatorname{span}\: \hat{w} \times V^H \subset \tau,
\]
\[
\ell = \operatorname{span} \: (\xi, \hat{w}, v) \subset \tau,
\]
proving the first claim.
Now let us show that the orbit Cartan type stratification of
$|\inertia{\sfG}|$ satisfies Whitney's condition B as well. To this
end let us first choose a Hilbert basis of $H$-invariant
polynomials $p_1,\ldots , p_\kappa \co
\big(N_{(h,x)}^H\big)^\perp \rightarrow \R$ of the orthogonal
complement of the invariant space $N_{(h,x)}^H$ in $N_{(h,x)}$.
Next let $p_{\kappa + 1},\ldots , p_N \co N_{(h,x)}^H \rightarrow
\R$ with $N=\kappa +\dim N_{(h,x)}^H$ be a linear coordinate
system of the invariant space. We can even choose these $p_i$
in such a way that $p_{\kappa + 1},\ldots , p_{\kappa + \dim
V^H}$ is a linear coordinate system of $V^H$. By construction,
$p_1,\ldots , p_N$ then is a Hilbert basis of the normal space
$N_{(h,x)}$. Denote by $p\co N_{(h,x)} \rightarrow \R^N$ the
corresponding Hilbert map. Recall that $p$ induces a chart of
$|\inertia{\sfG}|$ over $G \backslash U$ by
\[
\widehat{\Psi}\co G \backslash U \rightarrow \R^N, \: G \exp_{(h,x)} (v) \mapsto p(v) .
\]
Note that by $H$-invariance of $p$ and since for every orbit in
$U$ there is a representative in $V_{(h,x)}$, the chart
$\widehat{\Psi}$ is well-defined indeed. A decomposition of
$\widehat{U}:= \widehat{\Psi} ( G \backslash U )$ inducing the
orbit Cartan type stratification on $G \backslash U$ is given
\[
\widehat{\mathcal{Z}}:=
\big\{ \widehat{\Psi} (G\backslash(S\cap (G\times Y)) \mid S \in \mathcal{Z} \big\} .
\]
Let $\widehat{S}\in \widehat{\mathcal{Z}}$ denote the stratum
containing the orbit $G(h,x)$, and $\widehat{S}\in \mathcal{Z}$
a stratum $\neq \widehat{R}$ such that $G(h,x)$ lies in the
closure of $\widehat{S}$. Now consider sequences of orbits
$\big( G(h_i,x_i) \big)_{i\in \N}$ in $\widehat{R}$ and
$\big( G(k_i,y_i) \big)_{i\in \N}$ in $\widehat{S}$ such that both
sequences converge to $G(h,x)$. Moreover, assume that the
sequence of secants $ \overline{\widehat{\Psi} (G(h_i,x_i)) , \widehat{\Psi} (G(k_i,y_i))}$
converges to a line $\widehat{\ell}$, and that the sequence of tangent spaces
$T_{\widehat{\Psi} (G(k_i,y_i))} \widehat{S}$ converges to some
subspace $\widehat{\tau} \subset \R^N$. Using notation from
before, we can choose representatives $(h_i,x_i)$ and
$(k_i,y_i)$ having coordinates in
$ \mathfrak{m} \times (W_K \cup \{ 0 \}) \times V^H \subset N_{(h,x)}$ such that
\begin{equation}
\label{eq:coordinates}
\begin{split}
\Psi (h_i,x_i) & \, = (0,0, v_i^\prime ) \in \{ 0 \} \times \{ 0 \} \times V^H
\: \text{ and } \\
\Psi (k_i,y_i) & \, = (0,w_i,v_i) \in \{ 0 \} \times W_K \times V^H .
\end{split}
\end{equation}
Next observe that by the Tarski–Seidenberg Theorem, the stratum
$\widehat{S}$ is semialgebraic as the image of the
semialgebraic set $(W_K \times V^H) \cap B_{(h,x)}$ under the
Hilbert map $p$. By the same argument, $p(W_K)$ is
semialgebraic, too, and an analytic manifold, since $p(W_K)
\cong \normalizer_H(K)\backslash W_K \cong H \backslash W_{(K)}$.
Moreover, the equality
\[
\widehat{S} = ( p(W_K) \times V^H) \cap p(B_{(h,x)} )
\]
holds true, where we have canonically identified $V^H$ with its
image under the Hilbert map $p$. By Eq. (<ref>),
this entails that
\begin{equation}
\label{eq:tangentlimits}
\widehat{\tau} =
\lim_{i\rightarrow \infty} T_{\widehat{\Psi} (G(k_i,y_i))} \widehat{S} =
\lim_{i\rightarrow \infty} T_{p(w_i)}p(W_K) \times V^H .
\end{equation}
Since $p(W_K)$ is semialgebraic and an analytic manifold,
<cit.> by Łojasiewicz entails that
$p(W_K)$ satisfies Whitney's condition B over the origin. This
means after possibly passing to subsequences, that $\ell_{W_K}
\subset \tau_{W_K}$, where $\ell_{W_K}$ is the limit line of
the secants $\overline{p(w_i),0}$, and $\tau_{W_K}$ the limit
of the tangent spaces $T_{p(w_i)}p(W_K)$ for $i\rightarrow
\infty$. By Eqs. (<ref>) and
(<ref>) this entails that
\[
\widehat{\ell} \subset \ell_{W_K} \times V^H \subset \tau_{W_K} \times V^H = \widehat{\tau} .
\]
This finishes the proof.
Note that $\inertianull{\sfG}$ is clearly topologically locally trivial based
on its description in slices. Therefore,
by Proposition <ref>, the inertia groupoid
$\inertia{\sfG}$ is a differentiable stratified groupoid.
Now, recall that the loop space $\inertianull{\sfG}$ is a
differentiable subspace of the smooth manifold $\sfG_1$, and the space of arrows
$\inertia{\sfG}_1 = \sfG_1 \fgtimes{s}{t} \inertianull{\sfG}$ is a differentiable
subspace of the smooth manifold $\sfG_1 \fgtimes{s}{t} \sfG_1$. Similarly,
if $\obj{x}, \obj{y} \in \sfG_0$ are in the same orbit, then the slices
$Y_\obj{x}$ and $Y_\obj{y}$ for $\sfG$ can be chosen such that $\sfG_{|Y_\obj{x}}$
and $\sfG_{|Y_\obj{y}}$ are isomorphic by <cit.>.
This defines a diffeomorphism between the arrow spaces of $\sfG_{|Y_\obj{x}}$ and
$\sfG_{|Y_\obj{y}}$, both smooth manifolds, whose restriction defines
an isomorphism between $\inertia{\sfG_{|Y_\obj{x}}}$ and $\inertia{\sfG_{|Y_\obj{y}}}$.
It follows that the inertia groupoid satisfies conditions
(LT<ref>) to (LT<ref>)
in Definitions <ref> and <ref>,
hence is locally translation.
Moreover, by Proposition <ref>, the inertia
groupoid satisfies the local contractibility hypothesis of
Definition <ref>.
It is straightforward to verify that a weak equivalence $f\co\sfG\to\sfH$ of proper Lie groupoids
induces a weak equivalence $\inertia{\sfG}\to\inertia{\sfH}$ given by the restriction
of $f_1$ to the loop spaces. In particular, because the stratification of $\inertianull{\sfG}$
is defined in terms of slices for $\sfG$, and the representation of the isotropy group on a
slice is Morita invariant, the stratification
of $\inertianull{\sfG}$ is obviously the pullback of the stratification of
$\inertianull{\sfH}$ via $f_1$. Moreover, as the stratification of the inertia space
$|\inertia{\sfG}|$ can be defined locally in terms of the actions of isotropy
groups on slices, it is as well Morita invariant. This means that the isomorphism between
$|\inertia{\sfG}|$ and $|\inertia{\sfH}|$ from Proposition <ref>
is an isomorphism of differentiable stratified spaces.
We summarize these observations in the following.
Let $\sfG$ be a proper Lie groupoid. Then the inertia groupoid $\inertia{\sfG}$ is a proper differentiable stratified groupoid fulfilling Whitney's condition B.
Moreover the inertia groupoid $\inertia{\sfG}$ is locally translation
and satisfying the local contractibility hypotheses.
Finally, the inertia space $|\inertia{\sfG}|$ inherits from $\inertianull{\sfG}$
via the canoncial projection $\pi:\inertianull{\sfG} \to |\inertia{\sfG}|$
a stratification also fulfilling Whitney's condition B.
§ DIFFERENTIABLE STRATIFIED SPACES
In this appendix we describe the category
of differentiable stratified spaces used throughout this paper. Our notion
of differentiable spaces is that of <cit.> to which we refer the reader
for more details. For the definition of stratified
spaces we follow Mather <cit.> and <cit.>, except
that we relax the assumption that the spaces under consideration are Hausdorff and
only require that they are locally Hausdorff. Hence, a stratified space with smooth
structure as defined in <cit.> or a differentiable stratified
space as defined in <cit.> corresponds to a Hausdorff
differentiable stratified space as defined here.
Note that in addition to <cit.> various other concepts of
structure sheaves respectively structure algebras of smooth functions over
stratified spaces have been introduced in the literature. See for example
the work by Kreck <cit.> on stratifolds,
by Lusala–Śniatycki <cit.> on
stratified subcartesian spaces, by Watts
<cit.> on differential spaces, and finally by
Somberg–Vân Lê–Vanzura <cit.>
on smooth structures on locally conic stratified spaces.
§.§ Differentiable spaces
Let $(X, \mathcal{O})$ be a locally $\R$-ringed space which we always assume to
be commutative.
One says that $(X, \mathcal{O})$ is an affine differentiable space, if
there is a closed ideal $\mathfrak{a} \subset\mathcal{C}^\infty(\R^n)$ such that
$(X, \mathcal{O})$ is isomorphic as a ringed space to the real spectrum of
$\mathcal{C}^\infty(\R^n)/\mathfrak{a}$ equipped with its structure sheaf, which
associates to each open set its localization over that set. Here, we consider the
unique topology with respect to which $\mathcal{C}^\infty(\R^n)$ is a Frechét algebra.
A locally $\R$-ringed space $(X, \mathcal{O})$ is a differentiable space if,
for each $x \in X$, there is an open neighborhood $U$ of $X$ such that the restriction
$(U, \mathcal{O}|_U)$ is an affine differentiable space. A differentiable space
is reduced if for each open subset $U$ of $X$ the map
$\mathcal{O}(U) \to \mathcal{C}(U)$ defined by the evaluation map is injective.
A morphism of differentiable spaces
$(f,\varphi) \co (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ consists of a continuous map
$f\co X \to Y$ and a morphism
$\varphi \co\mathcal{O}_Y\to f_\ast \mathcal{O}_X$ of sheaves
of $\R$-algebras such that for each $x\in X$ the induced morphism
on the stalks $\varphi_x \co\mathcal{O}_{Y,f(x)}\to\mathcal{O}_{X,x}$ is local,
i.e. maps the the maximal ideal $\mathfrak{m}_y \subset \mathcal{O}_{Y,f(x)}$
to the maximal ideal $\mathfrak{m}_x \subset \mathcal{O}_{X,x}$.
Note that if $ (X, \calC^\infty_X) $ and $(Y, \mathcal{C}^\infty_Y)$ are reduced,
a morphism of differentiable spaces
$(f,\varphi)\co (X, \calC^\infty_X) \to (Y, \calC^\infty_Y) $
is fully determined by the map $f \co X \to Y$. The sheaf morphism
$\varphi$ is given in this case over each open $V \subset Y$ by the pullback
map $f^*: \calC^\infty_Y (V) \to \calC^\infty_X(f^{-1} (V))$, $g \mapsto g\circ f_{|V}$.
We therefore sometimes call a morphism between reduced differentiable spaces
a smooth map, and just denote it by the underlying map $f$.
By <cit.>, a differentiable space $(X,\calO)$ is reduced if
and only if each $x \in X$ is contained in an open neighborhood $V$ isomorphic as a
differentiable space to a locally closed subset of the affine space $\R^n$
with structure sheaf given by restrictions of smooth functions from $\R^n$.
We refer to such a $V$ as an affine neighborhood of $x$,
and call an embedding $\iota : V \hookrightarrow \R^n$ such that
$(\iota,\iota^*) : (V,\calO_{|V}) \rightarrow (\iota (V), \calC^\infty_{|\iota (V)})$ is an
isomorphism of locally ringed spaces a singular chart (of rank $n$) for $X$.
We often denote the structure sheaf of a reduced differentiable space $X$ by $\calC^\infty_X$
or shortly by $\calC^\infty$, if no confusion can arise.
By a smooth submanifold of a differentiable space $X$, we mean a differentiable subspace
whose differentiable structure is that of a smooth manifold in the usual sense.
§.§ Stratified spaces
Let $X$ be a paracompact separable locally Hausdorff topological space.
A decomposition $\mathcal{Z}$ of $X$ is a locally finite partition of
$X$ into locally closed subspaces such that each $S\in\mathcal{Z}$ is a
countable union of smooth (not necessarily Hausdorff) manifolds
such that the condition of frontier is satisfied:
(CF) If $R\cap\overline{S}\neq\emptyset$ for $R, S\in\mathcal{Z}$,
then $R\subset\overline{S}$.
If $R\subset\overline{S}$, one writes $R \leq S$ and says that $R$ is incident to $S$.
The incidence relation is an order relation on $\mathcal{Z}$. The elements of a decomposition
$\mathcal{Z}$ are called its pieces.
In the following we provide a generalization of the definition of a stratification by
Mather <cit.> to the case of a locally
Hausdorff space; cf. also <cit.>.
Let $X$ be a locally Hausdorff topological space. A stratification of $X$ is an
assignment to each $x \in X$ of a germ $\mathcal{S}_x$ of subsets of $X$ at $x$ such that
for each $x \in X$ there is a Hausdorff neighborhood $U$ of $x$ in $X$ and a decomposition
$\mathcal{Z}$ of $U$ with the property that for each $y \in U$ the germ $\mathcal{S}_y$
is equal to the germ at $y$ of the piece of $\mathcal{Z}$ containing $y$. The set $X$
along with the stratification $\mathcal{S}$ is called a stratified space.
If, moreover, $X$ is a differentiable space, and for each $x \in X$, the germ
$\mathcal{S}_x$ is that of a smooth submanifold of $X$, then we say $X$ is a
differentiable stratified space.
A continuous function $f\co (X, \mathcal{S})\to (Y,\mathcal{R})$ is a
morphism of stratified spaces if, for each $x \in X$ with $f(x) = y$,
there are open Hausdorff neighborhoods $U$ of $x$ and $V$ of $y$ with $U \subset f^{-1}(V)$
and decompositions of $U$ and $V$ inducing their respective stratifications
such that for every $z \in U$ contained in the piece $S$ of $U$, there is an open neighborhood
$O$ of $z$ in $U$ such that $f_{|S\cap O}$ maps into the piece of $V$ containing $f(z)$.
If $(X, \mathcal{O}_X, \mathcal{S})$ and $(Y, \mathcal{O}_Y, \mathcal{R})$ are
differentiable stratified spaces, a function $f\co X \to Y$ is a differentiable
stratified morphism if it is simultaneously a morphism of differentiable spaces
and a morphism of stratified spaces.
Note that the definition of a differentiable
stratified space coincides, for a Hausdorff space $X$, with that of a stratified space with
$\mathcal{C}^\infty$ structure defined in <cit.>;
see also <cit.>.
If $(X, \mathcal{Z})$ is a decomposed space, then the decomposition induces a stratification
by assigning to $x \in X$ the germ at $x$ of the piece containing $x$; two decompositions of
$X$ are equivalent if they induce the same stratification.
The depth of $x \in X$ with respect to the decomposition $\mathcal{Z}$ is the maximum
$k$ such that $x \in S_0 < S_1 < \cdots < S_k$ for $S_i \in \mathcal{Z}$.
Now, let $(X, \mathcal{S})$ be a stratified space and let $x \in X$. The proof of
<cit.> (see also <cit.>) is local,
hence can be executed on a Hausdorff neighborhood of $x$. It therefore extends to our
case and demonstrates that the depth of $x$ coincides for any decomposition of a Hausdorff
neighborhood of $x$ inducing $\mathcal{S}$. Hence, we may define the depth of $x$
with respect to $\mathcal{S}$ to be the depth with respect to any such decomposition.
In the same way, the proofs of <cit.> and
<cit.>) extend to the situation of locally Hausdorff stratified space.
So $X$ admits a decomposition $\mathcal{Z}$ that induces $\mathcal{S}$ and is maximal in the
sense that for every Hausdorff open subset $U$ of $X$, the restriction of $\mathcal{Z}$ to $U$
is coarser than any decomposition of $U$ that induces $\mathcal{S}$. We will often refer to
$\mathcal{Z}$ simply as the maximal decomposition of $X$.
Its pieces are called the strata of $X$.
Note that if $X$ is a (Hausdorff) differentiable stratified space, the strata of $\mathcal{Z}$
are obviously (Hausdorff) smooth manifolds.
We recall the following from <cit.>.
A stratified space $(X, \mathcal{S})$ is topologically locally trivial if for every
$x \in X$ in the stratum $S$ of $X$, there is a neighborhood $U$, a stratified space
$(F, \mathcal{S}^F)$, a point $o \in F$, and an isomorphism of stratified spaces
$h \co U \to (S \cap U) \times F$ such that $h^{-1}(y,o) = y$ for all $y \in S\cap U$,
and such that $\mathcal{S}_o^F$ is the germ of the set $\{ o \}$.
§.§ Fibered products
By <cit.>, the fibred product of differentiable spaces
has a unique differentiable space structure with respect to which the projection maps
are morphism of differentiable spaces. We now demonstrate that the same holds true for
differentiable stratified spaces. Let $X$, $Y$, and $Z$ be differentiable stratified
spaces with respective stratifications
$\mathcal{S}^X$, $\mathcal{S}^Y$, and $\mathcal{S}^Z$. Suppose
$f\co X\to Z$ and $g\co Y \to Z$ are differentiable stratified mappings.
If $f$ is in addition a stratified submersion, then we define a stratification
of the fibred product $X \fgtimes{f}{g} Y$ as follows. Let
$(x,y) \in X \fgtimes{f}{g} Y$, let $P \subset X$ be a subset whose germ
$[P]_x = \mathcal{S}_x^X$, and let $R \subset Y$ such that
$[R]_y = \mathcal{S}_y^Y$. Then we assign to $(x,y) \in X \fgtimes{f}{g} Y$
the germ $\mathcal{S}_{(x,y)} := [P \fgtimes{f}{g} R]_{(x,y)}$. We refer to
$\mathcal{S}$ as the induced stratification of
$X \fgtimes{f}{g} Y$ by the stratifications $\mathcal{S}^X$ and $\mathcal{S}^Y$.
Suppose $X$, $Y$, and $Z$ are differentiable stratified spaces
and $f\co X\to Z$ and $g\co Y \to Z$ are differentiable stratified mappings.
If $f$ is in addition a stratified submersion, then the induced stratification
$\mathcal{S}$ is a stratification of $X \fgtimes{f}{g} Y$.
Of course, $X \fgtimes{f}{g} Y$ and $Y \fgtimes{g}{f} X$ are isomorphic, so the same holds
true if we assume that $g$ is a stratified submersion.
For simplicity, we work with the maximal decompositions $\mathcal{Z}^X$, $\mathcal{Z}^Y$,
and $\mathcal{Z}^Z$ of $X$, $Y$, and $Z$, respectively,
which is clearly sufficient as the definition of the fibred product is local. Then the fact
that $\mathcal{Z}^X$ and $\mathcal{Z}^Y$ are partitions of $X$ and $Y$ immediately implies
that $\mathcal{Z}:=\{ P \fgtimes{f}{g} R \mid P \in\mathcal{Z}^X, R \in\mathcal{Z}^Y \}$ is
a partition of $X \fgtimes{f}{g} Y$. That each $P \in\mathcal{Z}^X$ and $R\in\mathcal{Z}^Y$
is locally closed implies that $P \times R$ is locally closed in $X\times Y$ and hence
$P \fgtimes{f}{g} R = (P\times R)\cap (X \fgtimes{f}{g} Y)$ is locally closed
in $X \fgtimes{f}{g} Y$. Given $(x,y)\in X \fgtimes{f}{g} Y$, let $U_x$ and $U_y$
be open neighborhoods of $x$ in $X$ and $y$ in $Y$, respectively, that each intersect
finitely many elements $\mathcal{Z}^X$ and $\mathcal{Z}^Y$,
and then $(U_x\times U_y)\cap (X \fgtimes{f}{g} Y)$ is an open neighborhood of $(x,y)$
in $X \fgtimes{f}{g} Y$ that evidently meets finitely many elements of $\mathcal{Z}$.
Therefore, $\mathcal{Z}$ is a locally finite partition of $X \fgtimes{f}{g} Y$ into
locally closed sets.
Now, let $P\in\mathcal{Z}^X$ and $R\in\mathcal{Z}^Y$, and choose connected components
$P_0$ of $P$ and $R_0$ of $R$. Then as $f$ and $g$ are stratified mappings, there is a
piece $S\in\mathcal{Z}^Z$ with $f(P_0), g(R_0) \subset S$; see <cit.>.
Moreover, as $f$ is a stratified submersion, $f_{|P_0}$ is by definition a submersion.
Then by <cit.> and the fact that $f_{|P_0}$
is a submersion implies that $f_{|P_0}$ is transversal to $g_{|R_0}$, we have that
$P_0 \fgtimes{f}{g} R_0$ is a smooth submanifold of $P_0 \times R_0$
and hence of the differentiable space $X \fgtimes{f}{g} Y$. Hence each connected component
of $P \fgtimes{f}{g} R$ is a smooth manifold.
Finally, suppose
$(P \fgtimes{f}{g} R)\cap\overline{(P^\prime \fgtimes{f}{g} R^\prime)}\neq\emptyset$
for $P, P^\prime\in\mathcal{Z}^X$ and $R, R^\prime\in\mathcal{Z}^Y$.
Choose $(x,y) \in (P \fgtimes{f}{g} R)\cap \overline{(P^\prime \fgtimes{f}{g} R^\prime)}$,
and then for any open neighborhoods $U_x$ and $U_y$ of $x$ and $y$ in $X$ and $Y$, respectively,
$U_x\times Y_y$ intersects $P^\prime \fgtimes{f}{g} R^\prime$. It follows that
$P\cap\overline{P^\prime}, R\cap\overline{R^\prime}\neq \emptyset$ so that
$P\subset\overline{P^\prime}$ and $R\subset\overline{R^\prime}$, hence
$P \fgtimes{f}{g} R \subset\overline{(P^\prime \fgtimes{f}{g} R^\prime)}$.
That is, $\mathcal{Z}$ satisfies the condition of frontier and hence is a
decomposition of $X \fgtimes{f}{g} Y$.
§.§ Tangent space
Assume that $(X,\calC^\infty)$ is a differentiable space. Then, given a point $x\in X$,
the maximal ideal $\mfrak_x \subset \calC^\infty_x$ in the stalk at $x$ is finitely generated,
hence the quotient space
$\mfrak_x/\mfrak_x^2$ is a finite dimensional real vector space. One calls this space
the Zariski cotangent space $\zartan_x^* X$ of $(X,\calC^\infty)$ at $x$,
and its dual $(\mfrak_x/\mfrak_x^2)^*$ the Zariski tangent space $\zartan_x X$.
There is another notion of a tangent bundle for a differentiable stratified space
$(X,\calC^\infty)$, namely the stratified tangent space $(T^\textup{st}X,\calC^\infty)$.
If $(X,\calC^\infty)$ fulfills Whitney's condition A, then $(T^\textup{st}X,\calC^\infty)$ is
a differentiable stratified space as well.
See <cit.> for more details on the stratified tangent bundle
§.§ Differential forms
Let $(X,\calC^\infty)$ denote a reduced differentiable stratified space. Let $U$ be an affine open subset
of $X$ and $\iota: U \hookrightarrow \R^n$ be a singular chart of $X$. Denote by $\calI_\iota$
the sheaf of smooth functions vanishing on $\overline{\iota(U)}$. Then define
the sheaf $\Omega^k_\iota$ for $k=0$ as $ \iota^{-1} (\calC^\infty / \calI_\iota) \cong \calC^\infty_{|U}$
and for $k \in \N^*$ as the following inverse image sheaf
\[
\Omega^k_\iota := \iota^{-1}\big( \Omega^k_{\R^n} / (\calI_\iota \Omega^k_{\R^n} +
d \calI_\iota \wedge \Omega^{k-1}_{\R^n}) \big).
\]
Observe that by construction the exterior differential factors
through the $ \Omega^k_\iota$, hence we obtain a differential graded algebra $\big( \Omega^\bullet_\iota , d \big)$.
If $\kappa : V \hookrightarrow \R^m$ is another singular chart of $X$,
there exists a unique sheaf isomorphism
$\eta_{\iota,\kappa}: \Omega^\bullet_{|\kappa(U\cap V)} \to \Omega^\bullet_{|\iota(U\cap V)}$ extending the
isomorphism of sheaves
$\eta_{\iota,\kappa} : (\calC^\infty / \calI_\kappa)_{|\kappa(U\cap V)} \to
(\calC^\infty / \calI_\iota)_{|\iota(U\cap V)}$.
One concludes that the cocycle condition
\begin{equation}
\label{eq:CocycleCondition}
\eta_{\kappa , \iota} =
\eta_{\kappa , \lambda} \circ \eta_{\lambda,\iota}
\end{equation}
is fulfilled if $\lambda : V \hookrightarrow \R^l$ denotes a third singular chart of $X$.
Hence the sheaves $ \Omega^k_\iota$ glue to a globally defined sheaf $\Omega_X^k$ of
so-called abstract $k$-forms on $X$ in such a way that the
gluing maps preserve $d$. So we obtain a sheaf complex
$\big( \Omega^\bullet_X,d\big)$ of differential graded algebras.
The complex of global sections
$\big( \Omega^\bullet (X),d\big)$ will be called the Grauert–Grothendieck complex of $X$.
For $X \subset \C^n$ a complex space, the construction of the complex $\Omega^\bullet (X)$
within the analytic category goes back to Grauert <cit.> and Grothendieck <cit.>.
Let us now describe how one can represent elements of $\Omega^k (X)$.
To this end assume to be given an open covering $\mathcal{U}$ of $X$ by coordinate domains and a family
$(\kappa_U)_{U\in\mathcal{U}}$ of singular charts
$\kappa_U : U \hookrightarrow \widetilde{U} \subset \R^{n_U}$ such that $\widetilde{U}$ is open and
contains $\kappa_U (U) $ as a relatively closed subset.
An element of $\Omega^k (X)$ can then be represented as a family
$([\omega_U])_{U \in \mathcal{U}}$, where $\omega_U \in \Omega^k(\widetilde{U})$
and where one has for any two $U,V \in \mathcal{U}$ over the overlap $U\cap V$
\[
\eta_{\kappa_V,\kappa_U} \big( [\omega_U] \big) = [\omega_V] .
\]
Hereby, $ [\omega_U]$ denotes the equivalence class of $\omega_U$ in $\Omega^k_{\kappa_U}$.
This representation allows for the following useful construction. Assume that $S$ is a stratum of $X$,
and let $\iota_S : S \hookrightarrow X$ denote the canonical embedding. Given an element
$\omega = ([\omega_U])_{U \in \mathcal{U}} \in \Omega^k (X) $ one observes that for any two
$U, V \in \mathcal{U}$ the pulled back forms $\iota^*_{S\cap U} \kappa_U^* (\omega_U)$
and $\iota^*_{S\cap V} \kappa_V^* (\omega_V)$ coincide on the overlap $U\cap V$, hence glue together to a
global form on $S$ which we denote by $\iota^*_S \omega \in \Omega^k (S)$.
By construction, each of the sheaves $\Omega^k_X$ carries the structure of a $\calC^\infty$-module in a natural way.
This observation entails the following result
The Grauert–Grothendieck complex $\big( \Omega^\bullet (X),d\big)$ of a differentiable stratified space $(X,\calC^\infty)$
is a complex of fine sheaves.
Finally in this section, we will define the pull-back morphism $f^* : \Omega^k_Y \to \Omega^k_X$ associated to a smooth
map $f:X\to Y$ between reduced differentiable stratified spaces $(X,\calC^\infty_X)$ and $(Y,\calC^\infty_Y)$.
By the preceeding proposition and the construction of the Grauert–Grothendieck complex it suffices to consider the
case where $X\subset \R^n$ and $Y\subset \R^m$ are affine.
Choose open neighorhoods $U \subset \R^n$ of $X$ and $V \subset \R^m$ of $Y$ such that $X$ is closed in $U$ and
$Y$ in $V$. Choose a smooth function $F:U \to V$ such that $F_{|X}=f$. For $\omega \in \Omega^k (V)$ representing an
abstract $k$-form on $Y$ we put
\[
f^* ([\omega] ) := [F^*\omega] \in \Omega^k(X).
\]
Since $F^*$ maps the vanishing ideal $I_Y \subset \calC^\infty (V)$ to the vanishing ideal $I_X \subset \calC^\infty (U)$
and since $F^*$ commutes with $d$, $F^*$ maps the $I_Y \Omega^k(V) + dI_Y \wedge \Omega^k(V)$ to
$I_X \Omega^k(U) + dI_X \wedge \Omega^k(U)$. Moreover, if $\widetilde{F}:U \to V$ is another smooth function such that
$\widetilde{F}_{|X}=f$, then $F^* g- \widetilde{F}^*g \in I_X $ and
$F^* dg- \widetilde{F}^*dg \in dI_X $ for all $g \in \calC^\infty (V)$, which entails that
$F^* \omega- \widetilde{F}^*\omega \in I_X \Omega^k(U) + dI_X \wedge \Omega^k(U)$. This proves that
$[F^*\omega]$ neither depends on the particular choice of the representative of $[\omega]$ nor on the particular smooth $F$
extending $f$ to an open neighborhood of $X$. Hence $f^* : \Omega^k(Y) \to \Omega^k (X)$ is well-defined.
Obviously, $d$ commutes with $f^*$, since it commutes with $F^*$.
|
1511.00300
|
firstpage–lastpage 2014
Most massive galaxies are thought to contain a supermassive black hole in their centre surrounded by a tenuous gas environment, leading to no significant emission. In these quiescent galaxies, tidal disruption events represent a powerful detection method for the central black hole. Following the disruption, the stellar debris evolves into an elongated gas stream, which partly falls back towards the disruption site and accretes onto the black hole producing a luminous flare. Using an analytical treatment, we investigate the interaction between the debris stream and the gas environment of quiescent galaxies. Although we find dynamical effects to be negligible, we demonstrate that Kelvin–Helmholtz instability can lead to the dissolution of the stream into the ambient medium before it reaches the black hole, likely dimming the associated flare. This result is robust against the presence of a typical stellar magnetic field and fast cooling within the stream. Furthermore, we find this effect to be enhanced for disruptions involving more massive black holes and/or giant stars. Consequently, although disruptions of evolved stars have been proposed as a useful probe of black holes with masses $\gtrsim 10^8 \msun$, we argue that the associated flares are likely less luminous than expected.
black hole physics – hydrodynamics – galaxies: nuclei.
§ INTRODUCTION
Tidal disruption events (TDEs) occur when a star is scattered into a plunging orbit that brings it so close to a supermassive black hole (SMBH) that it is torn apart by strong tidal forces <cit.>. During the disruption, the stellar elements are forced into different trajectories, which causes the debris to subsequently evolve into an elongated gas stream. Half of the debris within this stream is bound to the black hole while the other half is unbound. After a revolution around the black hole, the bound debris returns to the disruption site and forms an accretion disc <cit.>, from which a powerful flare can be emitted . This flare represents a unique probe to detect SMBHs in the centres of otherwise quiescent galaxies. Through this signal, it is also in principle possible to put constraints on the black hole properties as well as to investigate the physics of accretion and relativistic jets around these objects.
The debris evolution within the stream from disruption to its return to pericentre has been the focus of several studies, both numerical and analytical. While the debris follows close to ballistic orbits, the transverse structure of the stream is set by the equilibrium between the different forces acting in this direction. During most of its evolution, internal pressure is balanced by self-gravity, which causes the stream to maintain a narrow profile <cit.>. However, a recent simulation shows that internal pressure inside the stream may be unable to prevent the fragmentation of the debris into self-gravitating clumps, which can form a few years after disruption <cit.>.
Although it is not associated to substantial emission, a gas component is present around SMBHs in the centre of quiescent galaxies. It is commonly assumed to originate from stellar winds released by massive stars surrounding the black hole <cit.>. The impact of this gaseous environment on the stream evolution has so far been largely ignored, owing to a large density contrast between the two components. In a recent study, <cit.> find that it can affect the trajectories of the unbound debris, resulting in its deceleration on parsec scales. Other authors looked into the influence on the bound part of the stream but in specific contexts, such as a possible origin for the G2 cloud <cit.> and the interaction with a fossil accretion disc <cit.>.
In this paper, we investigate the influence of the ambient gas on the bound debris in a general way. Although dynamical effects are negligible, we demonstrate that hydrodynamical instabilities can lead to the dissolution of a significant part of this debris into the gaseous environment before it returns to pericentre. In this situation, we argue that the associated TDE would be significantly dimmer than expected. This effect is enhanced when the disruption involves a giant star and/or a more massive black hole. As a result, TDEs involving black holes of mass $\gtrsim 10^8 \msun$ could be difficult to detect. While main sequence stars are swallowed whole by such black holes leading to no substantial emission <cit.>, disruptions of giant stars could be just as dim owing to the dissolution of the debris into the ambient medium.
This paper is organized as follows. Sections <ref> and <ref> present the models used for the SMBH gaseous environment and the debris stream respectively. Section <ref> investigates the interaction between these two components through both ram pressure and hydrodynamical instabilities. In Section <ref>, we determine the impact on the detectability of TDEs. Our concluding remarks are found in Section <ref>.
§ GASEOUS ENVIRONMENT MODEL
In quiescent galaxies, black holes are surrounded by accretion flows, whose gas is mostly supplied by stellar winds from massive stars. The density distribution within this flow is given by the interplay between their hydrodynamics and the efficiency of the supply mechanism.
The Milky Way is the best example of a quiescent galaxy. It harbours , a central black hole of mass $4.3 \times 10^6 \msun$, surrounded by a gas environment well studied both theoretically and observationally. Analytical models of stellar winds sources find a density profile in the inner region of the flow decreasing as $R^{-1}$ <cit.>, a result consistent with numerical simulations <cit.>.
Based on this example, we adopt a simple gas density profile for the inner region of quiescent galaxies, given by[In our galaxy, a density profile scaling as $R^{-1/2}$ may be more consistent with observations of the inner accretion flow <cit.>. In other quiescent galaxies, this slope can be derived from observations of TDEs featuring outflows, where it is found to be steeper, decreasing as $R^{-5/2}$ <cit.> or $R^{-3/2}$ <cit.>. However, this could be caused by the propagation of the outflow into a previously evacuated funnel.]
(R) = ρ_0 ( R/R_0 )^-1,
For the Milky Way, the normalization is inferred from Chandra X-ray observations at the Bondi radius, which find a density $\rhozmw=2.2 \times 10^{-22} \gcm3$ at $\rzmw=0.04 \pc$.
For galaxies hosting SMBHs of different masses, this profile is scaled using the black hole radius of influence
R_inf=G /σ^2 ≃3 (/4.3 ×10^6 )^7/15,
where $\mh$ is the black hole mass, $\sigma$ is the velocity dispersion of stars in the bulge and the second equality uses the $\mh - \sigma $ relation $\mh= 2 \times 10^8 (\sigma / 200 \kms)^{15/4} \msun$ <cit.>[The $\mh-\sigma$ relation can be steeper than this. However, our results are essentially unchanged when using a steeper $\mh \propto \sigma^{5.3}$ relation <cit.>.]. The normalization radius is then obtained from
R_0= (/4.3 ×10^6 )^7/15 .
The normalization density is computed by assuming spherical accretion at a velocity $v \propto v_{\rm ff} \propto \mh^{1/2} R^{-1/2}$, where $v_{\rm ff}$ is the free-fall velocity. It leads to an accretion rate $\mdot \propto R^2_0 \rho_0 v(R_0) \propto \rho_0\mh^{6/5} $ using equation (<ref>). The gas is supplied to the accretion flow by stellar winds from stars within the black hole sphere of influence. As the mass of stars is similar to that of the black hole within this distance, $\mdot \propto \mh$. This yields
ρ_0= η( /4.3 ×10^6 )^-1/5,
where $\eta$ is a parameter, equal to 1 for the Milky Way. In the following, it is varied up to 1000 to investigate galaxies with denser gas environments. This simple scaling of the gas density profile has also been used by <cit.>. It leads to a similar dependence on $\mh$ as found from a more detailed treatment <cit.>.
width=0.47, file=fig1.pdf
Sketch of a portion of debris stream with an element shown in orange. The element has a cylindrical geometry, with length $\le$ and width $\he$. Its density $\rhoe$ is obtained from equation (<ref>) knowing its mass. At a distance $\re$ from the black hole, it moves through a gaseous environment of density $\rho_{\rm g,e} \equiv \rhog(\re)$ with a velocity $\vect{v_{\rm e}}$, inclined with respect to its longitudinal axis by an angle $\thetae$.
§ TIDAL STREAM MODEL
The disruption of a star of mass $\mstar$ and radius $\rstar$ occurs when it reaches the tidal radius $\rt=\rstar (\mh/\mstar)^{1/3}$. The resulting debris evolves into an elongated stream owing to an orbital energy spread $\de=G\mh \rstar /R^2_{\rm t}$, acquired during the disruption. In this work, we only focus on the bound debris, with orbital energies $\epsilon$ from $-\de$ to 0 and periods $T$ between $\tmin=2 \pi G \mh (2 \de)^{-3/2}$ and $+\infty$.
To model the stream of bound debris, we divide it into cylindrical elements, an example of which is sketched in Fig. <ref>. In the following, the variables associated to a particular element are indicated by the subscript “$\rm e$” to differentiate them from those associated to the debris.
An element of period $\te$ contains debris whose periods satisfy $\te-\dte<T<\te+\dte$. Equivalently, it has an average orbital energy $\ene = -(1/2)(2 \pi G \mh /\te)^{2/3}$ and contains debris with orbital energies in the range $\ene- \dene<\epsilon<\ene+\dene$. To ensure that each element is composed of debris with similar periods, we set $\dte = 10^{-2} \, \tmin \ll \te$.
Following the disruption, each component of the stream is assumed to follow Keplerian orbits with the same pericentre $\rt$ but different orbital energies $\epsilon$. The position $\vect{x_{\rm e}}$ and velocity $\vect{v_{\rm e}}$ of an element are identified with those of the debris with orbital energy $\ene$.
Owing to its cylindrical geometry, the density of an element is obtained by
= /πh^2_e ≤,
where $\me$, $\he$ and $\le$ denote the mass, width and length of the element respectively. We explain how these quantities are computed in the remaining of this section.
Knowing the separation $\vect{\delta x_{\rm e}}$ of its two extremities, the length of an element is obtained by $\le=|\vect{\delta x_{\rm e}}|$. Its velocity $\vect{v_{\rm e}}$ is inclined with respect to its longitudinal direction by an angle $\thetae$ obtained by $\cos \thetae = \vect{v_{\rm e}} \cdot \vect{\delta x_{\rm e}}/ (|\vect{v_{\rm e}}| |\vect{\delta x_{\rm e}}|)$.
The mass $\me$ of an element is obtained from
=∫_- ^+ M ≃2 M/ϵ|_ .
where $\diff M / \diff \epsilon$ is the debris orbital energy distribution. The latter is computed using the analytical model developed by <cit.>, which assumes that the debris energy is given by its depth within the black hole potential when the star is disrupted. This yields
M/ϵ = / ∫_Δr^ 2 π(r) r r,
where $\rhostar$ is the density inside the star and $\Delta r = (\epsilon/\de) \rstar$. This allows to compute the fallback rate of the debris to pericentre, given by
Ṁ_fb = M/ϵ ϵ/T = (2 πG )^2/3/3 M/ϵ T^-5/3,
where the relation $T=2 \pi G \mh (-2 \epsilon)^{-3/2}$ is used in the second equality.
Based on the work by <cit.>, different density profiles are considered corresponding to the evolution of a 1.4 $\msun$ star. They are obtained from a detailed simulation of the star using the stellar evolution code MESA <cit.>. The evolution of the stellar radius is shown in Fig. <ref>, with the main phases of evolution indicated by filled areas and the five stellar density profiles considered later in the paper shown with coloured points. In the main sequence phase (green area), one profile is considered (MS). Two profiles are chosen in the red giant phase (yellow area): when the star is ascending the red giant branch (RG1) and when it reached the tip of this branch (RG2). For the horizontal branch (orange area) and the asymptotic giant branch (red area) phases, two profiles are selected (HB and AGB).
The width $\he$ is obtained by assuming hydrostatic equilibrium in the stream transverse direction. While pressure tends to expand the stream, the tidal force from the black hole and the stream self-gravity oppose this expansion. Note that the tidal force acts inwards since the stream transverse direction is close to that orthogonal to the direction of the black hole. Hydrostatic equilibrium thus reduces to
where $\ape = \nabla \pe / \rhoe \simeq \pe / (\rhoe \he)$ is the pressure acceleration, $\ate \simeq G \mh \he / \re^3 $ is the tidal acceleration and $\age\simeq G \me/ (\he\le)$ is the self-gravity acceleration within the stream, $\re=|\vect{x_{\rm e}}|$ being the distance from the black hole and $\pe$ the pressure in the stream. For the pressure, we assume an adiabatic evolution with $\pe=K \rhoe^{\gamma}$ where $\gamma=5/3$. Although the adiabatic constant $K$ should a priori be different for different elements, we adopt a single value averaged over the volume of the star. This is legitimate as the value of $K$ within the star varies only by a factor of a few around this average. The width $\he$ is obtained by solving equation (<ref>), making use of equation (<ref>). For illustration, in the two limiting cases $\age \gg \ate$ and $\age \ll \ate$, it scales as $\he \propto (\me/\le)^{-1/4}$ and $\he \propto (\re^3/ \mh)^{3/10} (\me/\le)^{-1/5}$ respectively.
width=0.47, file=fig2.pdf
Evolution of the radius of a 1.4 $\msun$ star. The main evolutionary phases are indicated by filled regions: main sequence (green), red giant (yellow), horizontal branch (orange) and asymptotic giant branch (red). The coloured points correspond to the five stellar density profiles considered.
§ TIDAL STREAM - AMBIENT MEDIUM INTERACTIONS
§.§ Hydrodynamical instabilities
As the stream moves through the ambient medium, it is subject to the Kelvin–Helmholtz (K-H) instability. In this section, we evaluate the effect of this instability on each stream element.
Taking a conservative approach, we only consider the second half of each element orbit, i.e. after apocentre passage. This approach is motivated by the fact that an element reaches its lowest density in this part of the orbit and is therefore more easily affected by its interaction with the ambient medium. In this portion of the orbit, an element falls almost radially from apocentre to pericentre. In this configuration, the K-H instability develops on a given stream element for wavenumbers $\ke$ which obey the inequality <cit.>
æ< ρ_g,e/^2-ρ^2_g,e v^2_rel,e
where $\ae$ is the inwards acceleration of the element in the transverse direction, $\rho_{\rm g,e}\equiv \rhog(\re)$ is the density of gas at the position of the element and $v_{\rm rel,e}$ is the relative velocity between the element and the background gas. Although modes with large $\ke$ have fast growth rates, they are also the least disruptive as the associated instability saturates at an amplitude $\sim 1/\ke$. We therefore consider a wavenumber $\ke=1/\he$ which has the slowest growth rate but is the most disruptive since it develops on the whole element width. The transverse acceleration $\ae$ has two inwards components. One is the self-gravity acceleration $\age\simeq G \me / (\he\le)$ and the other is the tidal acceleration $\ate \simeq G \mh \he / R^3_{\rm e}$. With $\ae=\ate+\age$, condition (<ref>) reduces to
+<ρ_g,e v^2_e/≡,
which uses $\rhoe \gg \rho_{\rm g,e}$. The relative velocity is computed by $v_{\rm rel,e} = \ve \cos \thetae \simeq \ve$ where $\ve = |\vect{v_{\rm e}}|$ is the velocity of the element. This uses the approximation $\thetae \ll 1$, which is satisfied along an element orbit, as soon as it leaves its apocentre. In addition, this value of $v_{\rm rel,e}$ assumes that the background gas is at rest. The possibility of a lower relative velocity caused by radially falling back ground gas has been explored and leads to no significant difference. The right-hand side of equation (<ref>) is called $\are$ as it is equivalent to a ram pressure acceleration.
If condition (<ref>) is satisfied, the K-H instability then grows on a timescale
for a given element. Otherwise, the instability does not develop and $\tkhe = +\infty$. The K-H instability has time to fully grow before the element reaches pericentre if
≡∫_/2^ dt/ >1,
where $\te/2$ and $\te$ are the times corresponding to the element apocentre and pericentre passages respectively. Condition (<ref>) can be understood by omitting the temporal dependence of $\tkhe$. In this case, it reduces to $\tkhe < \te/2$ which clearly implies that the K-H instability has time to fully grow during the portion of orbit considered.
width=0.47, file=fig3.pdf
Evolution of $\age$ (dotted lines), $\ate$ (dashed dotted lines), $\age+\ate$ (solid line) and $\are$ (dashed lines) for three elements of a stream produced by the tidal disruption of the star in the red giant phase (profile RG1) by a black hole of mass $\mh=10^8 \msun$ in a galaxy with $\eta = 5$. The elements have different periods $\te=40 \yr$ (blue lines), $\te=160 \yr$ (red lines) and $\te=850 \yr$ (yellow lines). For each element, the filled areas indicate the regions where $\age+\ate<\are$, that is where condition (<ref>) is satisfied. The grey areas indicate the range of periods of elements verifying condition (<ref>), for which the K-H instability has time to fully develop before they return to pericentre.
As an example, the evolution of $\age$ (dotted lines), $\ate$ (dashed dotted lines), the left-hand side of equation (<ref>) $\age+\ate$ (solid line) and its right-hand side $\are$ (dashed lines) is shown in Fig. <ref> for three different elements of a stream produced by the disruption of the star in the red giant phase (profile RG1) by a black hole of mass $\mh=10^8 \msun$ in a galaxy with $\eta=5$. These elements have periods $\te=40 \yr$ (blue lines), $\te=160 \yr$ (red lines) and $\te=850 \yr$ (yellow lines). For all elements, tidal acceleration dominates self-gravity acceleration ($\ate>\age$) in the final part of their orbit, when $\re<\he(\mh/\me)^{1/3}$. The zones where condition (<ref>) is true, are indicated by filled regions for each element. They only exist for the most bound (blue lines) and least bound (yellow lines) of the elements considered. For these two elements, condition (<ref>) is also satisfied and the K-H instability therefore has time to fully develop before they return to pericentre. For the intermediate element (red lines), condition (<ref>) is never verified. This implies $\fkhe=0$ and condition (<ref>) is therefore not satisfied either. The grey areas indicate the range of periods of all the stream elements that satisfy condition (<ref>). On these elements, we expect the K-H instability to fully grow over the course of their orbit.
Fig. <ref> indicates the range of periods of elements that satisfy condition (<ref>), but does not show the precise evolution of $\fkhe$ with $\te$. Actually, the transition between $\fkhe=0$ and $\fkhe>1$ is very sharp. For elements that never satisfy condition (<ref>), $\fkhe=0$. However, as soon as condition (<ref>) is met at some point along an element orbit, $\fkhe\gtrsim 1$, which implies that condition (<ref>) is already marginally satisfied. This is because, in the final part of an element orbit where $\ate \gg \age$, the inequality $\are / \ate \gtrsim 2$ implies $\tkhe \lesssim (G \mh / R^3_{\rm e})^{-1/2}$, where the right-hand side is the infall time from $\re$ to pericentre. Omitting the time dependence of $\tkhe$, this translates to $\fkhe \gtrsim 1$.
The reason why only the most and least bound part of the stream are affected by the K-H instability can be understood by examining condition (<ref>) more in detail in the final part of each element orbit, where $\ate \gg \age$. Using $\ve \simeq (G \mh/\re)^{1/2}$, it reduces to
/≤<ρ_g,e R^2_e,
that is a condition on the stream linear density. Note that this condition is also independent on the element width $\he$. Our results are therefore largely independent on the assumption of hydrostatic equilibrium made to compute this width in Section <ref>. Furthermore, this means that physical mechanisms modifying the stream width, such as fast cooling of the debris, are unlikely to affect our results. One can clearly see that condition (<ref>), and therefore condition (<ref>), is easily satisfied for the most bound part of the stream, which is less massive since it originates from the tenuous outer layer of the star. Although the least bound part of the stream contains more mass, it is stretched owing to different trajectories of neighbouring debris regions and condition (<ref>) is also satisfied.
At this point, one can predict how the impact of the K-H instability depends on the other parameters, namely the black hole mass $\mh$, the evolutionary stage of the star and $\eta$, which relates to the ambient medium density via equation (<ref>). Tidal disruptions by more massive black holes lead to more extended streams. Furthermore, the right-hand side of condition (<ref>) evaluated at $\rt$ scales as $\rho_{\rm g,e} R^2_{\rm e} \simeq \rhog(\rt) R^2_{\rm t} \propto \mh^{9/15}$, which increases with the black hole mass. We therefore anticipate condition (<ref>) to be more easily satisfied when $\mh$ is larger. The stream is therefore likely to be more sensitive to the K-H instability. This trend is also expected for disruptions of evolved stars as they also lead to more extended streams whose debris originates from a more tenuous outer layer. Finally, we anticipate the same tendency when $\eta$ is increased, that is for environments with higher gas density, since $\rho_{\rm g,e} R^2_{\rm e} \propto \eta$. These predictions will be verified explicitly in Section <ref>.
width=0.47, file=fig4.pdf
Evolution of the debris mass fallback rate after a disruption with $\mh=10^8 \msun$ and $\eta=5$ for the five stellar density profiles considered: MS (grey line), RG1 (blue line), RG2 (green line), HB (yellow line) and AGB (red line). The filled areas correspond to the return times of debris satisfying condition (<ref>).
width=0.47, file=fig5.pdf
$M_{\rm h}-\eta$ plane depicting the effect of the K-H instability on different disruption events. Each line corresponds to one of the stellar density profiles considered. The zone in the direction of the arrow corresponds to events affected by the K-H instability, for which $\fkhp>1$. The zone in the opposite direction corresponds to events for which $\fkhp<1$, unaffected by the K-H instability. The purple diamond shows the parameters corresponding to Fig. <ref>, $\mh=10^8 \msun$ and $\eta=5$.
§.§ Ram pressure
As a stream element sweeps up the ambient medium located on its trajectory, it loses momentum and decelerates. This deceleration affects significantly the trajectory of the element once it has swept a mass of ambient gas larger than its own mass. This is equivalent to
≡1/ ∫_0^ A_e t > 1,
where $A_{\rm e}=\he \le \sin \thetae$ is the element area sweeping gas from the ambient medium. As for the K-H instability, we find this condition to be satisfied both for the most and least bound part of the stream. However, $\frame<\fkhe$ in all cases explored, which means that the debris is affected by the K-H instability before their trajectories change due to ram pressure.
§.§ Effect on flare luminosities
We now evaluate the impact of the K-H instability on the flare luminosities produced by the disruption of the star in different evolutionary stages and examine the dependence on the black hole mass $\mh$ and ambient gas density, through the parameter $\eta$.
Fig. <ref> shows the fallback rate, computed using equation (<ref>), of the debris produced by the disruption of the star by a black hole of mass $\mh=10^8\msun$ in a galaxy with $\eta=5$ for the five stellar density profiles considered. The filled areas indicate the times at which elements satisfying condition (<ref>) return to pericentre. For these elements, the K-H instability has time to fully grow over the course of their orbit. For profiles MS and RG1, these zones exist only for the most and least bound debris, as in the example of Fig. <ref>. The debris whose return times correspond to the peak fallback rate are always outside this zone. Instead, for profiles RG2, HB and AGB, all the elements lie in the filled zone, even those returning to pericentre when the fallback rate peaks. It means that the K-H instability has time to fully grow in the whole stream. This confirms our expectation that streams produced by the disruption of evolved stars are more sensitive to the K-H instability.
So far, we have examined for which elements condition (<ref>) is satisfied, that is for which debris the K-H instability fully develop before it reaches pericentre. As these instabilities involve the whole width of the stream, we infer that this debris subsequently dissolves into the ambient medium and does not return to pericentre.
Only the elements reaching pericentre intact can participate to the luminosity emitted from the event. Therefore, if all the stream dissolves into the ambient medium due to the K-H instability, the appearance of the event is likely to be affected, emitting a significantly lower luminosity. We take a conservative approach and state that an event is affected by this instability if even the element corresponding to the peak of the mass fallback rate dissolves into the background gas. According to our criterion, this requires condition (<ref>) to be satisfied for this element, that is $\fkhp > 1$. Fig. <ref> shows the regions of the $\mh-\eta$ plane corresponding to events affected by the K-H instability. Each line is associated to one of the stellar density profiles considered. The zone in the direction of the arrow corresponds to affected events while the zone in the opposite direction corresponds to unaffected events. The example discussed above ($\mh = 10^8 \msun$ and $\eta = 5$), where events corresponding to profiles RG2, HB and AGB are affected, is indicated by a purple diamond. As predicted above, events involving more massive black holes or occurring in galactic nuclei with denser gaseous environment are more sensitive to the K-H instability.
width=0.47, file=fig6_left.pdf
width=0.47, file=fig6_right.pdf
Probability for a disruption event to occur in a given evolutionary stage as a function of $\mh$. The different coloured areas correspond to different phases in the evolution of the star: main sequence (green), red giant (yellow), horizontal branch (orange) and asymptotic giant branch (red). Only the right panel includes the effect of the K-H instability, with the grey area corresponding to affected events for a galaxy with $\eta=5$. The zone swept by the boundary of this area is shown by a blue hatched region for values of $\eta$ varying from 1 to 1000 from right to left.
§ IMPACT ON THE DETECTABILITY OF TDES
In the previous section, we argued that the K-H instability can lead to the dissolution of a significant part of the stream before it comes back to pericentre, which could significantly reduce the luminosity emitted from the associated event. Furthermore, we showed that events involving more massive black holes and/or evolved stars are more sensitive to this effect. In this section, we examine the consequence on the detectability of TDEs produced by the disruption of a $1.4 \msun$ star in different evolutionary stages and by black hole of different masses.
For an event to lead to a substantial flare, the star must be disrupted outside the black hole's Schwarzschild radius $\rs$. Otherwise, it is swallowed whole without significant emission. We investigate the effect of the K-H instability on the detectability of events satisfying this condition. To this aim, we define the probability of such events to occur when the star is in a given evolutionary stage by
f^stage_flaring = N_stage / N_lifetime,
where $N_{\rm stage}$ and $N_{\rm lifetime}$ are the number of events occurring during the evolutionary stage and the whole stellar lifetime respectively. The possibility of an event to be affected by the K-H instability is only included in $N_{\rm stage}$. These numbers are obtained by
N_stage=∫_t_start^t_end Ṅ dt,
N_lifetime=∫_0^t_lt Ṅ dt,
where $t_{\rm start}$ and $t_{\rm end}$ are the starting and ending times of the stage respectively, while $t_{\rm lt}$ is the lifetime of the star. $\dot{N}$ is the disruption rate, which we assumed to scale as $\dot{N} \propto \rt^{1/4}$ following <cit.>. $\chiswa$ and $\chikh$ are binary functions given by
= 0 $\rt \leq \rs$ 1 ,
= 0 $\fkhp \geq 1$ 1 ,
which are respectively zero if the star is swallowed whole and if the stream is affected by the K-H instability according to the criterion defined in Section <ref>.
This probability is shown in Fig. <ref> as a function of the black hole mass for different evolutionary stages. The left panel does not take into account the K-H instability, artificially fixing $\chikh=1$ in equation (<ref>). It reproduces figure 14 (right panel) of <cit.>. For $\mh\gtrsim 10^8\msun$, the evolutionary stage of most disrupted stars switches from main sequence stars to giant stars. This is because $\rt<\rs$ for main sequence stars above this mass. Instead, the right panel of Fig. <ref> includes the effect of the K-H instability. The grey zone indicates affected events in a galaxy with $\eta=5$. For $\mh\gtrsim 10^8\msun$, giant stars as previously become more likely to be tidally disrupted than main sequence stars. However, as giant stars are more sensitive to the K-H instability, all the events are affected by the K-H instability for $\mh\gtrsim 10^9\msun$, which could significantly hamper their detection. The blue hatched region indicates the zone swept by the boundary of the grey area for values of $\eta$ varying from 1 to 1000 from right to left. For $\eta \gtrsim 10$, even the events involving main sequence stars are affected by the K-H instability.
§ DISCUSSION AND CONCLUSION
The interaction between the debris stream produced by TDEs and the background gas of quiescent galaxies has often been neglected, on the basis of their large difference in density. In this paper, we have investigated this interaction for the bound part of the stream, involved in the flaring activity of these events. Through an analytical argument, we have demonstrated that the K-H instability can affect the debris, especially for disruptions involving an evolved star and/or a massive black hole. In this case, a substantial fraction of the tidal stream can dissolve into the background gas before it reaches pericentre, likely leading to a flare dimmer than previously expected.
In order to model the stream, we have used the analytical model of <cit.> for the specific energy distribution within the stream, which assumes that the star is unperturbed until it reaches pericentre. Actually, numerical simulations have shown that the stellar structure is perturbed at pericentre <cit.>. However, this effect can be easily accounted for within the same analytical model, by applying a homologous expansion of the unperturbed model by a factor $\sim 2$ <cit.>, which makes the energy distribution very close to the one obtained through simulations. This leads to a stream slightly more resistant to the K-H instability but does not affect our main conclusions.
Another assumption of the model is a total disruption of the star by the black hole. However, simulations have shown that a surviving core can remain after the disruption <cit.>, which keeps following the initial stellar orbit. This likely causes the marginally bound part of the stream to contain less mass than expected from <cit.>. The debris returning to pericentre at late times would therefore be even more sensitive to the K-H instability.
We also note that we have neglected the effects of magnetic fields in the stream. Such effects might prevent the dissolution of the stream by the K-H instability <cit.>. We can address this issue analytically by adding a term $\ame \simeq B^2_{\parallel} /(\rhoe \he)$ to the left-hand side of condition (<ref>), where $B_{\parallel}$ is the component of the magnetic field parallel to the stream <cit.>. This term is significant only if $\ame \gtrsim \are$, which translates to $B_{\parallel} \gtrsim \rho^{1/2}_{\rm g,e} \ve \simeq 3 \, \mathrm{G} \, (\eta/1)^{1/2} (\rstar/12 \rsun)^{-1} (\mh/10^8 \msun)^{3/10}$, evaluating the right-hand side at $\rt$ and for a stellar radius corresponding to profile RG1. Such values of $B_{\parallel}$ correspond to typical surface magnetic fields for main sequence stars. They probably exceed typical surface magnetic fields in red giants, estimated from magnetic flux conservation in the expansion phase. For example, a expansion by a factor of 10 implies a magnetic field reduced by a factor of 100. In the stream, the critical value for $B_{\parallel}$ is unlikely to be reached for several reasons. Firstly, they require that the star is exactly stretched in the direction of its magnetic field, which is unlikely since the magnetic field orientation is random. Secondly, the magnetic field in the inner region of a star is likely tangled and not ordered in the same direction. In this configuration, magnetic reconnection may also occur in the stretching process, lowering the total magnetic field. In addition, although flux conservation imposes that the magnetic field in the direction of the stream is conserved since the stream stays thin, magnetic diffusion could lead to a decrease of this component as the stream orbits around the black hole. A caveat in these arguments is the ill-known value of the magnetic field strength inside giant stars. Nevertheless, we consider it unlikely that magnetic fields would prevent the K-H instability from developing. However, a definite answer would require to follow the evolution of the stellar magnetic field during the disruption and the fallback of the debris.
Finally, our calculations are made in an ambient medium at rest although an inward velocity of the gas environment could diminish the effect of the K-H instability. We have tested the dependence of our results on this assumption by introducing an radial velocity of the gas, which results in a lower relative velocity in equations (<ref>) and (<ref>). We find that our main conclusions remain unchanged for an infall velocity up to the Keplerian velocity, thus confirming the solidity of our analysis.
The main implication of this study is that any TDEs involving black holes with masses $\gtrsim 10^8 \msun$ might be difficult to detect, a conclusion largely independent of our scaling for the background gas density with black hole mass. This was already known for main sequence stars, which are swallowed whole for this range of masses <cit.>. Here, we show that this is also the case for giant stars, which have their debris stream dissolved into the background gas through the K-H instability.
§ ACKNOWLEDGMENTS
C.B. is grateful to Giovanni Dipierro for hosting him during the completion of this work.
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1511.00435
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This paper proposes a new methodology to maximize the feasible set of power injections
and cross-border power transfers in meshed multi-area power systems.
The approach used polyhedral computation schemes and is an extension to the classic procedure for cross-border transfer capacity assessment in the European power network, including the computation of bilateral cross-border transfer capacities as well as multilateral flow-based approaches.
The focus is the characterization of inter-area exchange limits
required for secure power system operation in the presence of physical transmission constraints, while maximizing the utilization factors of the transmission lines. The numerical examples include a case study of the transmission system.
Power System Capacity Allocation, Net Transfer Capacities, Multi-Area Power Systems, Congestion Management, Polyhedral Computations, Optimization
§ INTRODUCTION
This paper outlines a methodology to maximize the feasible set of the cross-border transfer capacity calculation. We presents an extension to the classic procedure for cross-border capacity assessment in the European power network. The results are compared to the well-established bilateral computation of cross-border transfer capacities and can be used as an preprocessing step of existing multilateral flow-based approaches.
For the determination of cross-border transfer capacities, the key challenge is the effect of multiple parallel flows through neighboring areas. In the European network, for example, a simultaneous power exchange from Belgium to Italy and from the Netherlands to Austria will induce major flows through the German transmission grid. The transfer capacity must be sufficiently conservative to guarantee the secure system operation during all flow conditions, both regarding the physical constraints of cross-border transmission lines as well as internal transmission constraints of the full networks in the area represented by the nodes.
Our contribution is twofold. First, the capacity bounds consider not only neighbors but all areas between which power exchange is possible, similar to the flow-based methods. Second, the capacity bounds are not a set of box constraints as induced by the upper and lower bilateral transfer capacity limits, but rather a polyhedral capacity set that incorporates the effect of parallel load flows. In the mentioned example, Belgium could likely export much more power to Italy, if there is no export occurring from the Netherlands to Italy, freeing capacities in the German transmission system. Subsequently, the often applied bilateral computation is one specific subset of the proposed polyhedral capacity set.
The paper focuses on the characterization of inter-area exchange limits caused by the physical transmission constraints required for secure power system operation while maximizing the utilization factors of the transmission lines. We investigate neither the allocation of these capacities to the bidding zones nor their pricing.
The structure of the paper is as follows: Section <ref> briefly reviews the current European concepts for managing cross-border congestions. Section <ref> and Section <ref> outline the modeling of transfer capacities and the extended capacity selection problem, respectively. Section <ref> presents the simulation results and discusses the policy implications. Finally, Section <ref> is devoted to conclusions and perspectives.
§ EUROPEAN MULTI-AREA CONGESTION MANAGEMENT
In liberalized electricity markets, the transmission system determines the limitation for wholesale and ancillary service trading <cit.>. Consequently, the way in which cross-border transfer capacities are calculated has a substantial impact on the market opportunities. In Europe, the Transmission System Operators (TSOs) bear responsibility for the operation and security of the power system, which includes the determination of the transfer capacity available to market participants' electricity trading. Cross-border capacities are either agreed bilaterally between neighboring countries (contract-based) or determined multilaterally for several areas (flow-based) <cit.>. Both approaches can be used in regional markets implementations. For example, in Central Western Europe (CWE), i.e. the Netherlands, Belgium, France, Luxembourg, and Germany, a co-ordinated contract-based market coupling was launched by , which was superseded by a flow-based day-ahead market coupling . We briefly outline the basics of both state-of-the-art approaches, as our methodology to determine the feasible sets for the transfer capacities can be compared to existing bilaterally agreed values as well as enable flow-based transfer capacity calculations.
§.§ The Available Transfer Capacity Approach
This mechanism is based on bilateral agreements between neighboring TSOs. Based on historical data, i.e. reference days, well-representing seasonal patterns as well as justified security margins, each TSO determines a Total Transfer Capacity (TTC) for each direction on each border of its control area. Thus, the TTC is the upper limit for which the maximum physical flow on a critical network element does not exceed its safety margins, i.e. criterion. Based on the TTC, each TSO deducts a safety margin, referred to as the Transmission Reliability Margin (TRM), as well as used capacities of Long-Term Contracts (LTCs), as a holder of a LTC must always declare by the previous day, whether or not and to what extent the holder intends to use the respective long-term reserved transfer capacities. The Net Transfer Capacity (NTC) available to wholesale trading results from the TTC minus the TRM minus the LTC. Finally, the Available Transmission Capacity (ATC) is the part of NTC that remains available after each phase of the allocation procedure for further commercial activity, i.e. ATC is NTC minus Already Allocated Capacity (AAC). This whole process is operated bilaterally; therefore, if the calculated values deviate between neighboring TSOs, generally the lower ones are selected. By that, borders are considered separately which does not allow for a holistic consideration of the power flows in the power system.
§.§ The Flow-Based Approach
Instead of fixed capacities, the flow-based methodology is based on a reduced network constituting of nodes and lines in order to take into account that electricity can flow via different paths in an highly meshed power system. Each TSO provides input data, which is combined at a regional level. To retrieve a reduced network model, instead of considering each and every line, so-called critical branches are introduced. They consist of tie-lines as well as internal lines that significantly impact a cross-border exchange. This allows to determine which combinations of cross-zonal exchanges may lead to an overload of a critical network element. Based on the physical limit and potential security margins of the line, the physical capacity for each critical branch, i.e. the total maximal flow, is determined. A Flow Reliability Margin (FRM) is to cope with the uncertainty inherent to the process of determining the remaining capacity, and a reference flow to consider the already known long-term nominations. What will eventually be offered to the wholesale market is the so-called Remaining Available Margin (RAM). By that, it is possible to consider the effect of a meshed system.
§ MODELING OF TRANSFER CAPACITIES
This section introduces the mathematical modeling needed for the characterization of transfer capacities.
The modeling is the same, whether it is applied to characterize TTC, NTC, ATC or the flow-based approach.
The first part defines the full power system model and constraint formulation.
The second part defines the aggregation scheme used to obtain a reduced network model.
§.§ Power system modeling and constraints
The power system is modeled using a DC power flow approximation <cit.>.
The network has $\nbus$ buses connected by $\nline$ lines.
Each bus $i$ has a known demand power $\Powd{i}$ and a generator power $\Powg{i}$
that is to be selected during the power flow optimization.
The difference between generation and demand at each bus is the net power injection
\begin{equation}
\Pownet{i} = \Powg{i} - \Powd{i}, \qquad i = 1,2, ..., \nbus \quad ,
\end{equation}
and forms the power system state
$x \in \R{\nbus}$,
\begin{equation}
x = [\Pownet{1}, \Pownet{2}, ..., \Pownet{\nbus} ]^T \quad .
\end{equation}
Furthermore, the vector of active power flows in the $\nline$ lines is defined as $\Powlineall \in \R{\nline}$,
\begin{equation}
\Powlineall = [\Powline{1}, \Powline{2}, ..., \Powline{\nline} ]^T \quad .
\end{equation}
In the DC power flow model, all relations between bus voltage angles, net power injections and power flows in the lines become linear, in particular <cit.>
\begin{equation}
\Powlineall = \Bgl x \quad .
\label{eq:Bgl}
\end{equation}
The net power injections have to satisfy the power balance equality constraint,
\begin{equation}
\Abal^T x = [1, 1, ..., 1] x = \sum_{i=1}^{\nbus} \Pownet{i} = 0\quad.
\end{equation}
There are two sets of inequalities constraining the set of possible net power injections.
The generator powers at each node are bounded by the generation limit
\begin{equation}
0 \leq \Powg{i} \leq \overline{\Powg{i}} \quad ,
\label{eq:pgcon}
\end{equation}
and the power flows in each line are bounded by the thermal limit
\begin{equation}
|\Powline{i}| \leq \overline{\Powline{i}} \quad .
\label{eq:plcon}
\end{equation}
The inequalities define the generator constraint polyhedron $\PG$ and the line constraint polyhedron $\PL$,
\begin{align}
\PG & = \{x \in \R{\nbus} \ : \quad \Agen x \leq \bgen, \ \Abal x = 0 \} \label{eq:PGpoly}\\
\PL & = \{x \in \R{\nbus} \ : \quad \Aline x \leq \bline, \ \Abal x = 0 \},
\end{align}
where the parameters $\{\Agen, \bgen\}$ are computed from pgcon and
the parameters $\{\Aline, \bline\}$ are computed from Bgl and plcon.
A system state $x$ that satisfies all inequalities lies in both polyhedra,
\begin{equation}
x \in (\PG \cap \PL)\quad ,
\end{equation}
and is referred to as feasible, otherwise it is referred to as infeasible.
§.§ Network aggregation
The background of this paper is the aggregation of a detailed power system model as defined in psmodel with the system state $x \in \R{\nbus}$
by assigning the original $\nbus$ buses to one of the $\nbust$ regions, with $\nbust < \nbus$.
The outcome is a reduced power system model with the system state $y \in \R{\nbust}$ denoting the total net power injections of the regions.
The transformation with the bus aggregation matrix $\Tbus$
\begin{equation}
y = \Tbus x
\label{eq:yTx}
\end{equation}
is essentially a summation of the net power injections of all buses associated with a region.
If the original bus $i$ is assigned to region $j$, then the element of $\Tbus$ in column $i$ and row $j$ is 1, and zero otherwise.
The linear map yTx also defines a mapping $\MapPT{\cdot}{\cdot}$ of polyhedral sets from the original to the reduced state space.
For instance,
\begin{align}
\PGt & = \MapPT{\PG}{\Tbus} \nonumber \\
& = \{y\in\R{\nbust} : \exists x \in \PG: y = \Tbus x \}
\end{align}
denotes all reduced system states with a corresponding original state that satisfies the generator constraints.
Since $\nbust < \nbus$, $\PGt$ is a projection on the dimensions defined by the rows of the bus aggregation matrix $\Tbus$.
Linear mappings of polyhedra can be computed exactly or with approximations using existing software implementations <cit.>, <cit.>.
The aggregation transformation is assumed to be known and fixed.
It originates from the organization of the power system into control areas or market zones.
§.§ Feasibility conditions
To characterize feasible power injections in the reduced and original system, two conditions are defined.
Given a generation constraint polyhedron $\PG$, a line constraints polyhedron $\PL$ and a bus aggregation matrix $\Tbus$,
a state $y$ of the reduced system is feasible in the original system if
\begin{equation}
% \exists x \in\R{\nbus} : y = \Tbus x, \quad x \in (\PG\cap\PL) \quad .
\exists x \in\PG : y = \Tbus x, \quad x \in \PL \quad .
\label{eq:feascon}
\end{equation}
For a reduced system state $y$ that satisfies feascon, it is by definition always possible to find a corresponding power injection $x$ that satisfies all generator and line constraints.
However, the identification of the corresponding power injections is not unique.
In fact, some of the power injections that satisfy the generator constraints may violate internal line constraints and have to be avoided through a regional dispatch.
Consequently, $y$ is referred to as strongly feasible in the original system if it satisfies feascon and additionally
\begin{equation}
% \nexists x \in\R{\nbus} : y = \Tbus x, \quad x \in \PG, x \notin\PL \quad.
\nexists x \in\PG : y = \Tbus x, x \notin\PL \quad.
\label{eq:strongfeascon}
\end{equation}
In this case, a regional dispatch is not required since the line constraints are automatically satisfied with the generator constraints.
§ CHARACTERIZATION OF FEASIBLE NETWORK INJECTIONS
This section characterizes the original power system constraints in the reduced power system model.
It enables operational decisions with the reduced model that automatically ensure the absence of constraint violations in the original system.
The required information for decisions is only the reduced system state $y$, the original system model and state are not needed.
Three polyhedral constraint sets of the reduced system state $y$ are presented, that can be calculated in a preprocessing step from the original system model.
The first two sets, parameterizing the NTC approach and the flow-based approach, ensure the feasibility condition feascon.
The third set ensures the strong feasibility condition strongfeascon.
All sets are optimal in the sense that they maximize the transmission line utilization possible with the approach.
§.§ Mapping of NTC constraints
NTC bounds constrain the aggregated active power flow $\Powlinetotal$ through selected transmission lines of the network,
\begin{equation}
\Powlinetotal= \Tline \Powlineall \leq \bntc\quad .
\label{eq:Pntctotal}
\end{equation}
If the original bus $\Powline{i}$ contributes to the $j$'th NTC bound of the NTC vector $\bntc$, then the element of the line aggregation matrix $\Tline$ in column $i$ and row $j$ is 1,
and zero otherwise.
The polyhedron of power injections satisfying the NTC and generation constraints is defined as
\begin{align}
\nonumber
\PNTC = \{x\in\R{\nbus}: \Antc x \leq \bntc , \Agen x & \leq \bgen, \\
\Abal x & = 0 \ \ \} \quad,
\end{align}
with $\Antc = \Tline \Bgl $ combining Bgl and Pntctotal and the other parameter as in PGpoly.
The mapped NTC polyhedron
\begin{equation}
\PNTCt = \MapPT{\PNTC}{\Tbus}
\end{equation}
characterizes the NTC constraints in the reduced system space.
The states in the mapped NTC polyhedron can satisfy the feasibility condition feascon if the NTC bounds $\bntc$ is not chosen too large.
The maximum NTC bounds that still satisfy the feasibility condition are obtained by solving
\begin{equation}
\max_{\bntc}\quad w^T \bntc \quad \text{s.t.} \quad \PNTC \subset (\PG \cap \PL)
\label{eq:ntcopt}
\end{equation}
with the vector $w$ denoting a weighting of the NTC bounds.
Since the polyhedral set constraint in ntcopt complicates the problem,
it is useful to simplify the problem by searching along a nominal NTC directions $\nntc$,
\begin{align}
k_i^* = \min_{k_i, x}\quad & k_i \label{eq:linntc1}\\
\text{s.t.} \quad \Antc x & \leq \nntc\cdot k_i ,\\
\Agen x & \leq \bgen, \\
\label{eq:linntcend} \Alinei^T x & \geq \blinei,
\end{align}
where $\Alinei^T$ denotes the $i$'th row of the line constraint matrix $\Aline$.
The solution of linntc1-linntcend determines the smallest NTC scaling along direction $\nntc$ that violates the $i$'th line constraint.
Repeating the linear program for all line constraints yields the optimal NTC scaling
\begin{equation}
k^* = \min_{i }\quad k_i^* \quad .
\end{equation}
with $\bntc = k^*\nntc$.
The procedure can be repeated for randomly sampled NTC directions to maximize a common objective like the sum of all NTCs or an economic objective.
If an NTC direction and scaling is provided as parameter, for instance from bilateral agreements, the linear programs can still be solved without an objective as a pure feasibility problem to verify that the NTC bounds prevent all constraint violations.
During operation, reduced system states in $\PNTCt$ with $\bntc$ selected using the linear program, satisfy a sufficient condition for the feasibility condition feascon.
The evaluation of a candidate system state $y$ is very fast, requiring only the computation of $\Powlinetotal$ through a matrix multiplication of $y$ and
the verification of the box constraints Pntctotal.
§.§ Mapping of full flow constraints
The necessary and sufficient for the feasibility condition feascon define a set containing exactly all reduced states $y$ that have a corresponding feasible original state $x$.
The computation of this set is a key preprocessing step of the flow-based dispatch since it allows the maximum utilization of the available transmission capacities.
Mathematically, the set is a linear map of the feasible constraint polyhedron to the reduced system state,
\begin{equation}
\PLt= \MapPT{\PG \cap \PL}{\Tbus} \quad .
\end{equation}
The complexity of the projection mainly depends on the dimension of the reduced space and the number of line constraints.
Inner approximations of the projected set can be efficiently obtained using sampling approaches similar to the NTC maximization, since the constraint sets are bounded intersections of convex sets.
During operation, the evaluation of a candidate system state $y$ is still very fast, requiring only the evaluation of the polyhedral set constraints of $\PLt$ through a matrix multiplication and vector comparison.
§.§ Set difference of line violations
The reduced system states $y$ satisfying the strong feasibility conditions form a subset of $\PLt$.
In addition to feascon, they have to satisfy the constraint strongfeascon, which defines a set
\begin{align}
\PFat & = \{ y \in \R{\nbust}:
\nexists x \in\PG : y = \Tbus x, x \notin\PL
\} \quad ,\\
& = \{ y \in \R{\nbust}:
\exists x \in\PG : y = \Tbus x, x \notin\PL
\}{^{\mathrm{c}}} \quad, \\
& = \left(\PFatc\right){^{\mathrm{c}}}
\end{align}
written in the second line using the complement.
The set $\PFat^c$ is a collection of overlapping polyhedra, with
\begin{align}
\PFatc & = \cup_i \ \PFatic\\
& = \cup_i \
\{ y \in \R{\nbust}:
\exists x \in\PG : &&y = \Tbus x, \nonumber \\
& && \Alinei^Tx \geq\blinei
\} \quad . \quad \label{eq:PFatviolate}
\end{align}
The line constraint violation in PFatviolate covers all cases where
$x \notin \PL$.
The final step is the computation of the set $\PFt$ that satisfies both conditions feascon-strongfeascon, resulting in
\begin{align}
\PFt & = \PGt \cap \left( \PFatc\right){^{\mathrm{c}}}\\
& = \PGt \backslash \PFatc \quad.
\label{eq:PbP}
\end{align}
The set difference operation in PbP of a polyhedron $ \PGt$ with a family of overlapping polyhedra $ \PFatc $ is described in <cit.>.
It successively constructs a family of non-overlapping polytopes covering the non-convex set $\PFt$ and is available as software implementation <cit.>.
The outlined approach provides a set of constraints that are necessary and sufficient for the strong feasibility condition.
The resulting set $\PFt$ enables the selection of reduced system states $y$ that are guaranteed to satisfy the line constraints of the original model, no matter what generator settings are selected.
This is particularly useful if during the power system dispatch the actual
distribution of generator capacity in each region is not known, for example due to fluctuating availability of renewable energy sources or restricted communication.
On the downside, the approach is more complex than the other two approaches.
For the preprocessing, a polyhedral set difference computation is required.
The evaluation of a candidate system state $y$, requires the evaluation of potentially many polyhedra defining the non-convex region.
Several approaches to reduce the evaluation complexity exist, but require addition preprocessing <cit.>, <cit.>, <cit.>.
Finally, the strong feasibility condition can quickly become too strict, since some infeasible power system states map to the same reduced system state, resulting in $\PFt$ being the empty set.
A solution can be a more detailed modeling, including the violated transmission lines in the reduced system model.
Alternatively, only the regular feasibility condition can be required as in ntcmap and flowmap, thereby shifting the responsibility for the local line constraints to the regional grid participants.
§ NUMERICAL EXAMPLES
This section applies the proposed methodology to two example systems.
A simple system highlights the different feasibility sets.
Then, a large model of the ENTSO-E network is used to demonstrate the applicability of the method to practical system models.
§.§ Illustrative example
The simple example system consists of six buses that are aggregated into three different areas.
The topology, line constraints and power capacity is given in sixbus.
All lines are 500 km long, using an inductance of $0.09 \text{ p.u.}/\text{km}$ for a base voltage of 380 kV and a base power of 900 MVA.
The southern part of the system is a net power importer from the center or the north.
The limiting factor is caused by the constrained transmission inside the central region.
Topology of the six bus example system, showing bus numbers and line constraints.
The nodes 2-5 are aggregated into the center region.
The demand power at node 6 is 6 GW, at all other nodes 3 GW.
The maximum generation power at nodes 1 and 4 is 10 GW, at all other nodes 3 GW.
1-5 have a demand of 3 GW, node 6 has a demand of 6 GW.
After the aggregation step, only three net power injections remain and $y\in\R{3}$.
Furthermore, one of these free variables becomes dependent on the others due to the power balance constraint.
Therefore, the full feasible set can be illustrated in a 2D plot without any further projections being required.
The resulting polytopes can also be plotted in $\R{3}$ and will lie on a hyperplane in the $[1, 1, 1]^T$ direction.
7bus_ntc shows that the full projection of the line constraints yields a much larger feasible set than the largest set obtainable with pure NTC constraints between the three regions.
The reason is that basically a larger power export in the north is possible whenever the center reduces its power production. This effect requires the flow-based approach that takes into account the overall state of the network, not only isolated lines.
Six bus system aggregated into three regions. Possible net power injections in the center and southern region.
Generator constraints $\PGt$ (red),
line constraints $\PLt$ (yellow) and
NTC constraints $\PNTCt$ (blue).
7bus_strong illustrates the strong feasibility approach.
The goal is to determine the subset of $\PLt$ that not only has some feasible corresponding state in the original system, but for which all the corresponding states that can be realized by the generators do not violate any line constraints.
This requirement is quite strict an required to relax the line constraints of the six bus system.
To this end, the capacity of the two transmission lines in the center are increased to 2700 MW and the two souther line capacities to 3300 MW.
The result shows the non-convexity of the strongly feasible set, caused by the set difference operation.
Six bus system with increased transmission capacity aggregated into three regions. Possible net power injections in the center and southern region.
Generator constraints $\PGt$ (red),
line constraints $\PLt$ (yellow),
NTC constraints $\PNTCt$ (blue) and
strong feasibility constraints $\PFt$ (green).
§.§ European system
The method is now applied to a large model of the ENTSO-E transmission grid, documented in <cit.> with an implementation available in <cit.>.
It consists of 9241 buses, 16049 lines and is aggregated into 23 zones as illustrated in entsoemap.
The resulting system can now be analysed regarding the constraints of the individual zones or larger subgroups of the system.
Two results are shown in this section.
polish_tope illustrates the constraint coupling between the German and the Polish zone.
Note that the figure shows only a projection of the feasibility set onto these two dimensions.
A specific flow pattern between the other control zones is required to realize the maximum possible export of 10 GW from the polish zone.
Using the classical NTC or ATC characterization as outlined in Section II
and <cit.>, simplifies the constraint set but makes it also more conservative.
The illustration applies the Polish NTC values from 2011 <cit.> to the different neighboring lines with a total of 3500 MW of cross-border transfer capacity.
The violation observed along the German dimension is no issue, since additional NTC constraints of other critical system boundaries are commonly used to ensure full feasibility.
Finally, entsoe_tope shows the application of the proposed method
to determine the full constraint set if the ENTSO-E network is operated in four distinct areas.
The grouping into the four regions is also shown in entsoemap.
For the constraint characterisation, all of the 16049 original line constraint are included.
The projection step uses a sampling step to determine the characteristic directions in the reduced network space.
The accuracy is ensured by comparing inner and outer approximations of the constraint set.
The projection result is a polytope in the four dimensions corresponding to the four network regions.
Due to the intersection with a hyperplane representing the power balance constraint, it is possible to display the constraint set in a 3D picture.
For the application, also groupings into more than 4 regions can be considered and add little to the evaluation complexity.
Map of the ENTSO-E system with 9241 buses, aggregated in to 23 zones (circles) according to <cit.>,
and into four large regions to illustrate cross border power exchanges:
South-western (red), northern (black), central (green) and south-eastern region (blue).
Possible net power injections of the Polish zone and the German zone, projected from the full ENTSO-E system model.
Feasible set defined by generator constraints $\PGt$ (red),
the full line constraints $\PLt$ (yellow) and the Polish NTC constraints $\PNTCt$ (blue) according to <cit.>.
Note that relying purely on classical NTC definitions significantly underestimates the admissible region of net power injections.
Possible net power injections of the ENTSO-E system aggregated into four regions, projection of the line constraints $\PLt$.
The feasible set of the south-western, northern and central region is shown.
The south-eastern region is not shown and compensates the total power imbalance of the shown regions to zero.
The polyhedral representations of admissible net power injections allows a fast feasibility assessment of the aggregated system model.
§ CLOSING REMARKS
§.§ Conclusion
This paper demonstrated how the constraints of a detailed large-scale network model can be characterized in a reduced aggregated power system model.
The proposed method enables the operator to take decisions in the reduced model, that automatically ensure the absence of constraint violations in the detailed network model.
We outlined different polyhedral constraint sets that can be calculated in a preprocessing step of the market-driven transfer capacity allocation.
The method was evaluated using a six bus test system as well as a large-scale network model of .
The results show a large potential of the flow-based capacity allocation if the full constraint set can be used, compared to a classical NTC approach.
§.§ Outlook
The next key step is the application of this method to a market model with a realistic bidding process.
The European Union's "Third Energy Package" stipulates a competitive and integrated European electricity market with extensive cross-border trade facilitation.
This not only emphasizes the need for transparent and traceable methods to determine cross-border capacities, it also requires system operators and respective regulators to ensure the effective and optimal determination of those. For a large-scale implementation, the practical applicability of the proposed method needs to be further investigated.
In this paper, the reduced network model is assumed to be given from organizational or political constraints.
The choice of the appropriate aggregation scheme and reduced model has a large impact on the constraint sets as well as computational efforts during the decision making
and is therefore another topic of investigation.
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1511.00485
|
Symplectic homology and the Eilenberg-Steenrod axioms]Symplectic homology and the Eilenberg-Steenrod axioms
Universität Augsburg, Universitätsstrasse 14, D-86159 Augsburg, Germany
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 4 place Jussieu, F-75005, Paris, France
We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and explore some of its consequences.
§ INTRODUCTION
To begin with, a story. At the Workshop on Conservative Dynamics and Symplectic Geometry held at IMPA, Rio de Janeiro in August 2009, the participants had seen in the course of a single day at least four kinds of Floer homologies for non-compact objects, among which wrapped Floer homology, symplectic homology, Rabinowitz-Floer homology, and linearized contact homology. The second author was seated in the audience next to Albert Fathi, who at some point suddenly turned to him and exclaimed: “There are too many such homologies!". Hopefully this paper can serve as a structuring answer: although there are indeed several versions of symplectic homology (non-equivariant, $S^1$-equivariant, Lagrangian, each coming in several flavors determined by suitable action truncations), we show that they all obey the same axiomatic pattern, very much similar to that of the Eilenberg-Steenrod axioms for singular homology. In order to exhibit such a structured behaviour we need to extend the definition of symplectic homology to pairs of cobordisms endowed with an exact filling.
We find it useful to explain immediately our definition, although there is a price to pay regarding the length of this Introduction.
We need to first recall the main version of symplectic homology that is currently in use, which can be interpreted as dealing with cobordisms with empty negative end. This construction associates to a Liouville domain, meaning an exact symplectic manifold $(W^{2n},\omega,\lambda)$, $\omega=d\lambda$ such that $\alpha=\lambda|_{\p W}$ is a positive contact form (see <ref>), a symplectic homology group $SH_*(W)$ which is an invariant of the symplectic completion $(\wh W,\wh\omega)=(W,\omega)\, \cup \, \big([1,\infty)\times\p W, d(r\alpha)\big)$. The generators of the underlying chain complex can be thought of as being either the critical points of a Morse function on $W$ which is increasing towards the boundary, or the positively parameterized closed orbits of the Reeb vector field $R_\alpha$ on $\p W$ defined by $d\alpha(R_\alpha,\cdot)=0$, $\alpha(R_\alpha)=1$. Since the generators of the underlying complex are closed Hamiltonian orbits, we also refer to symplectic homology as being a theory of closed strings (compare with the discussion of Lagrangian symplectic homology, or wrapped Floer homology, further below).
We interpret a Liouville domain $(W,\omega,\lambda)$ as an exact symplectic filling of its contact boundary $(M,\xi=\ker\alpha)$, or as an exact cobordism from the empty set to $M$, which we call the positive boundary of $W$, also denoted $M=\p^+W$.
The implementation of this setup is the following. One considers on $\wh W$ (smooth time-dependent $1$-periodic approximations of) Hamiltonians $H_\tau$ which are identically zero on $W$ and equal to the linear function $\tau r-\tau$, $r\in[1,\infty)$ on the symplectization part $[1,\infty)\times M$, where $\tau>0$ is different from the period of a closed Reeb orbit on $M$.
One then sets
SH_*(W)=\lim\limits^{\longrightarrow}_{\tau\to\infty} FH_*(H_\tau)
where $FH_*(H_\tau)$ stands for Hamiltonian Floer homology of $H_\tau$ which is generated by closed Hamiltonian orbits of period $1$,
and the direct limit is considered with respect to continuation maps induced by increasing homotopies of Hamiltonians. The dynamical interpretation of these homology groups reflects the fact that the Hamiltonian vector field of a function $h(r)$ defined on the symplectization part $[1,\infty)\times M$ is equal to $X_h(r,x)=h'(r)R_\alpha(x)$. A schematic picture for the Hamiltonians underlying symplectic homology of such cobordisms with empty negative end is given in Figure <ref>, in which the arrows indicate the location of the two kinds of generators for the underlying complex, constant orbits in the interior of the cobordism and nonconstant orbits located in the “bending" region near the positive boundary. The vertical thick dotted arrow in Figure <ref> indicates that we consider a limit over $\tau\to\infty$.
Symplectic homology of a domain.
Key to our construction is the notion of Liouville cobordism with filling. The definition of a Liouville cobordism $W^{2n}$ is similar to that of a Liouville domain, with the notable difference that we allow the volume form $\alpha\wedge(d\alpha)^{n-1}$ determined by $\alpha$ on $\p W$ to define the opposite of the boundary orientation on some of the components of $\p W$, the collection of which is called the negative boundary of $W$ and is denoted $\p^-W$, while the positive boundary of $W$ is $\p^+ W=\p W\setminus \p^- W$. In addition, we assume that one is given a Liouville domain $F$ whose positive boundary is isomorphic to the contact negative boundary of $W$, so that the concatenation $F\circ W$ is a Liouville domain with positive boundary $\p^+W$.
Given a Liouville cobordism $W$ with filling $F$, the output of the closed theory is a symplectic homology group $SH_*(W)$. Although we drop the filling $F$ from the notation for the sake of readability, this homology group does depend on $F$. The dependence is well understood in terms of the geometric augmentation of the contact homology algebra of $\p^-W$ induced by the filling, see <cit.>. Symplectic homology $SH_*(W)$ is an invariant of the Liouville homotopy class of $W$ with filling, and the generators of the underlying chain complex can be thought of as being of one of the following three types: negatively parameterized closed Reeb orbits on $\p^-W$, constants in $W$, and positively parameterized closed Reeb orbits on $\p^+W$.
To implement this setup one considers (smooth time-dependent $1$-periodic approximations of) Hamiltonians $H_{\mu,\tau}$ described as follows: they are equal to the linear function $\tau r-\tau$ on the symplectization part $[1,\infty)\times\p^+W$, they are identically equal to $0$ on $W$, they are equal to the linear function $-\mu r+\mu$ on some finite but large part of the negative symplectization $(\delta,1]\times\p^-W\subset F$ with $\delta>0$, and finally they are constant on the remaining part of $F$. Here $\tau>0$ is required not to be equal to the period of a closed Reeb orbit on $\p^+W$,
and $\mu>0$ is required not to be equal to the period of a closed Reeb orbit on $\p^-W$.
Finally, one sets
SH_*(W)=\lim\limits^{\longrightarrow}_{b\to\infty} \lim\limits^{\longleftarrow}_{a\to-\infty} \lim\limits^{\longrightarrow}_{\mu,\tau\to\infty} FH_*^{(a,b)}(H_{\mu,\tau}),
where $FH_*^{(a,b)}$ denotes Floer homology truncated in the finite action window $(a,b)$.
Though the definition may seem frightening when compared to the one for Liouville domains, it is actually motivated analogously by the dynamical interpretation of the groups that we wish to construct. Let us consider the corresponding shape of Hamiltonians depicted in Figure <ref>. (The vertical thick dotted arrows in Figure <ref> indicate that we consider limits over $\mu\to\infty$ and $\tau\to\infty$.) A Hamiltonian $H_{\mu,\tau}$ has $1$-periodic orbits either in the regions where it is constant, or in the small “bending" regions near $\{\delta\}\times\p^-W$ and $\p^\pm W$ where it acquires some derivative with respect to the symplectization coordinate $r$. The role of the finite action window $(a,b)$ in the definition is to take into account only the orbits located in the areas indicated by arrows in Figure <ref>: as $\mu$ and $\tau$ increase, the orbits located deep inside the filling $F$ have very negative action and naturally fall outside the action window. The order of the limits on the extremities of the action window, first an inverse limit on $a\to-\infty$ and then a direct limit on $b\to\infty$, is important. It has two motivations: (i) the inverse limit functor is not exact except when applied to an inverse system consisting of finite dimensional vector spaces. Should one wish to exchange the order of the limits on $a$ and $b$, such a finite dimensionality property would typically not hold on the inverse system indexed by $a\to-\infty$, and this would have implications on the various exact sequences that we construct in the paper. (ii) With this definition, symplectic homology of a cobordism is a ring with unit (see <ref>). Should one wish to reverse the order of the limits on $a$ and $b$, this would not be true anymore.
Symplectic homology of a cobordism.
It turns out that the full structure of symplectic homology involves in a crucial way a definition that is yet more involved, namely that of symplectic homology groups of a pair of filled Liouville cobordisms. To give the definition of such a pair it is important to single out the operation of composition of cobordisms which we already implicitly used above. Given cobordisms $W$ and $W'$ such that $\p^+W=\p^-W'$ as contact manifolds, one forms the Liouville cobordism $W\circ W' = W \ _{\p^+ W}\!\cup_{\p^-W'} \ W'$ by gluing the two cobordisms along the corresponding boundary. The resulting Liouville structure is well-defined up to homotopy. A pair of Liouville cobordisms $(W,V)$ then consists of a Liouville cobordism $(W,\omega,\lambda)$ together with a codimension $0$ submanifold with boundary $V\subset W$ such that $(V,\omega|_V,\lambda|_V)$ is a Liouville cobordism and $(W\setminus V,\omega|,\lambda|)$ is the disjoint union of two Liouville cobordisms $W^{bottom}$ and $W^{top}$ such that $W=W^{bottom}\circ V\circ W^{top}$. We allow any of the cobordisms $W^{bottom}$, $W^{top}$, or $V$ to be empty. If $V=\varnothing$ we think of the pair $(W,\varnothing)$ as being the cobordism $W$ itself. A convenient abuse of notation is to allow $V=\p^+W$ or $V=\p^-W$, in which case we think of $V$ as being a trivial collar cobordism on $\p^\pm W$. This setup does not allow for $V=\p W$ in case the latter has both negative and positive components,
but one can extend it in this direction without much
difficulty at the price of somewhat burdening the notation, see
Remark <ref> and Section <ref>.
A pair of Liouville cobordisms with filling is a pair $(W,V)$ as above, together with an exact filling $F$ of $\p^-W$. In this case the cobordism $V$ inherits a natural filling $F\circ W^{bottom}$. See Figure <ref>.
Cobordism pair $(W,V)$ with filling $F$.
Given a cobordism pair $(W,V)$ with filling $F$ we define a symplectic homology group $SH_*(W,V)$ by a procedure similar to the above, involving suitable direct and inverse limits and based on Hamiltonians that have the more complicated shape depicted in Figure <ref>. The Hamiltonians depend now on three parameters $\mu,\nu,\tau>0$ and the vertical thick dotted arrows in Figure <ref> indicate that we consider limits over $\mu,\nu,\tau\to\infty$. One sets
SH_*(W,V)=\lim\limits^{\longrightarrow}_{b\to\infty} \lim\limits^{\longleftarrow}_{a\to-\infty} \lim\limits^{\longrightarrow}_{\mu,\tau\to\infty} \lim\limits^{\longleftarrow}_{\nu\to\infty} FH_*^{(a,b)}(H_{\mu,\nu,\tau}).
This is as complicated as it gets. The definition is again motivated by the dynamical interpretation of the groups that we wish to construct. For a given finite action window and for suitable choices of the parameters the orbits that are taken into account in $FH_*(H_{\mu,\nu,\tau})$ are located in the regions indicated by arrows in Figure <ref>. They correspond (from left to right in the picture) to negatively parameterized closed Reeb orbits on $\p^-W$, to constants in $W^{bottom}$, to negatively parameterized closed Reeb orbits on $\p^- V$, to positively parameterized closed Reeb orbits on $\p^+V$, to constants in $W^{top}$, and finally to positively parameterized closed Reeb orbits on $\p^+W$ (see <ref>).
We wish to emphasise at this point the fact that the above groups of periodic orbits cannot be singled out solely from action considerations. Filtering by the action and considering suitable subcomplexes or quotient complexes is the easiest way to extract useful information from some large chain complex, but this is not enough for our purposes here. Indeed, getting hold of enough tools in order to single out the desired groups of orbits was one of the major difficulties that we encountered. We gathered these tools in <ref>, and there are no less than four of them: a robust maximum principle due to Abouzaid and Seidel <cit.> (Lemma <ref>), an asymptotic behaviour lemma which appeared for the first time in <cit.> (Lemma <ref>), a new stretch-of-the-neck argument tailored to the situation at hand (Lemma <ref>), and a new mechanism to exclude certain Floer trajectories asymptotic to constant orbits (Lemma <ref>).
The simultaneous use of the first three of these tools is illustrated by the proof of the Excision Theorem <ref>. The fourth tool is not used in the paper, though it does provide alternative proofs for some confinement results.
Symplectic homology of a cobordism pair.
Important particular cases of such relative symplectic homology groups are the symplectic homology groups of a filled Liouville cobordism relative to (a part of) its boundary. Recalling that we think of a contact type hypersurface in $W$ as a trivial collar cobordism, we obtain groups $SH_*(W,\p^\pm W)$. It turns out that these can be equivalently defined using Hamiltonians of a much simpler shape, as shown in Figure <ref> below. It is then straightforward to define also symplectic homology groups $SH_*(W,\p W)$, which play a role in the formulation of Poincaré duality, see <ref>.
We refer to <ref> for the details of the definitions.
Symplectic homology of a cobordism relative to its boundary.
Our previous conventions for Liouville pairs do not allow to interpret $SH_*(W,\p W)$ as symplectic homology of the pair $(W,[0,1]\times\p W)$ in case $\p W$ has both negative and positive components. To remedy for this one needs to further extend the setup to pairs of multilevel Liouville cobordisms with filling, see <ref>.
The mnemotechnic rule for all these constructions is the following:
To compute $SH_*(W,V)$ one must use a family of Hamiltonians that vanish on $W\setminus V$, that go to $-\infty$ near $\p V$ and that go to $+\infty$ near $\p W\setminus \p V$.
Some of these shapes of Hamiltonians already appeared, if only implicitly, in Viterbo's foundational paper <cit.>, as well as in <cit.>. We make their use systematic.
These constructions have Lagrangian analogues, which we will refer to as
the open theory. The main notion is that of an
exact Lagrangian cobordism with filling, meaning an exact
Lagrangian submanifold $L\subset W$ of a Liouville cobordism $W$,
which intersects $\p W$ transversally, and such that $\p^-L=L\cap
\p^-W$ is the Legendrian boundary of an exact Lagrangian submanifold
$L_F\subset F$ inside the filling $F$ of $W$. We call $L_F$ an
exact Lagrangian filling. There is also an obvious notion of
exact Lagrangian pair with filling. The open theory associates
to such a pair $(L,K)$ a Lagrangian symplectic homology group
$SH_*(L,K)$, which is an invariant of the Hamiltonian isotopy class
preserving boundaries of the pair $(L,K)$ inside the Liouville pair
$(W,V)$. (In the case of a single Lagrangian $L$ with empty negative
boundary this is known under the name of wrapped Floer homology
of $L$.) Formally the implementation of the Lagrangian setup is the
same, using exactly the same shapes of Hamiltonians for a Lagrangian
Floer homology group. The generators of the relevant chain complexes
are then Hamiltonian chords which correspond either to Reeb chords
with endpoints on the relevant Legendrian boundaries, or to constants
in the interior of the relevant Lagrangian cobordisms. One can also
mix the closed and open theories together as in <cit.>,
see <ref>, and there are also $S^1$-equivariant closed
theories, see <ref>. In order to streamline the
discussion, we shall restrict in this Introduction to the
non-equivariant closed theory described above.
Remark (grading).
For simplicity we shall restrict in this paper to
Liouville domains $W$ whose first Chern class vanishes. In this case
the filtered Floer homology groups are $\Z$-graded by the
Conley-Zehnder index,
where the grading depends on the choice of a trivialisation of the canonical
bundle of $W$ for each free homotopy classes of loops. If $c_1(W)$ is
non-zero the groups are only graded modulo twice the minimal Chern
As announced in the title, one way to state our results is in terms of the Eilenberg-Steenrod axioms for a homology theory. We define a category which we call the Liouville category with fillings whose objects are pairs of Liouville cobordisms with filling, and whose morphisms are exact embeddings of pairs of Liouville cobordisms with filling. Such an exact embedding of a pair $(W,V)$ with filling $F$ into a pair $(W',V')$ with filling $F'$ is an exact codimension $0$ embedding $f:W\hookrightarrow W'$, meaning that $f^*\lambda'-\lambda$ is an exact $1$-form, together with an extension $\bar f:F\circ W\hookrightarrow F'\circ W'$ which is also an exact codimension $0$ embedding, and such that $f(V)\subset V'$. A cobordism triple $(W,V,U)$ (with filling) is a topological triple such that $(W,V)$ and $(V,U)$ are cobordism pairs (with filling).
Symplectic homology with coefficients in a field $\mathfrak{k}$ defines a contravariant functor from the Liouville category with fillings to the category of graded $\mathfrak{k}$-vector spaces. It associates to a pair $(W,V)$ with filling the symplectic homology groups $SH_*(W,V)$, and to an exact embedding $f:(W,V)\hookrightarrow (W',V')$ between pairs with fillings a linear map
f_!:SH_*(W',V')\to SH_*(W,V)
called Viterbo transfer map, or shriek map. This functor satisfies the following properties:
(i) (homotopy) If $f$ and $g$ are homotopic through exact embeddings, then
(ii) (exact triangle of a pair) Given a pair $(W,V)$ for which we denote the inclusions $V\stackrel{i}\longrightarrow W\stackrel{j}\longrightarrow (W,V)$, there is a functorial exact triangle in which the map $\p$ has degree $-1$
\begin{equation*}
\xymatrix
SH_*(W,V) \ar[rr]^{j_!} & &
SH_*(W) \ar[dl]^{i_!} \\ & SH_*(V) \ar[ul]^-\p_-{[-1]}
\end{equation*}
Here we identify as usual a cobordism $W$ with the pair $(W,\varnothing)$.
(iii) (excision) For any cobordism triple $(W,V,U)$, the transfer map induced by the inclusion $(W\setminus \mathrm{int}(U), V\setminus \mathrm{int}(U))\stackrel{i}\longrightarrow (W,V)$ is an isomorphism:
i_!:SH_*(W,V)\stackrel\simeq\longrightarrow SH_*(W\setminus \mathrm{int}(U),V\setminus \mathrm{int}(U)).
These are symplectic analogues of the first Eilenberg-Steenrod axioms
for a homology theory <cit.>. The one fact that may be puzzling
about our terminology is that we insist on calling this a homology
theory, though it defines a contravariant functor. Our arguments
are the following. The first one is geometric: With
$\Z/2$-coefficients we have an isomorphism $SH_*(T^*M)\simeq H_*(\cL
M)$ between the symplectic homology of the cotangent bundle of a
closed manifold $M$ and the homology of $\cL M$, the space of free
loop on $M$. Moreover, the product structure on $SH_*(T^*M)$ is
isomorphic to the Chas-Sullivan product structure on $H_*(\cL M)$, and
the latter naturally lives on homology since it extends the
intersection product on $H_*(M)\cong H_{n+*}(T^*M,T^*M\setminus
M)$. The second one is algebraic and uses the $S^1$-equivariant
version of symplectic homology (see <ref>): We wish that
$S^1$-equivariant homology with coefficients in any ring $R$ be
naturally a $R[u]$-module, with $u$ a formal variable of degree $-2$,
and that multiplication by $u$ be nilpotent. In contrast,
$S^1$-equivariant cohomology should naturally be a $R[u]$-module, with
$u$ of degree $+2$, and
multiplication by $u$ should typically not be nilpotent. This
is exactly the kind of structure that we have on the $S^1$-equivariant
version of our symplectic homology groups. The third one is an
algebraic argument that refers to the $0$-level part of the
$S^1$-equivariant version of a filled Liouville cobordism: Given such
a cobordism $W^{2n}$, this $0$-level part is denoted
$SH_k^{S^1,=0}(W)$ and can be expressed either as the degree $n+k$
part of $H_*(W,\p W)\otimes R[u^{-1}]$, with $R$ the ground ring and
$u$ of degree $-2$, or as the degree $n-k$ part of $H^*(W)\otimes
R[u]$. Since $H^*(W)\otimes R[u]$ is nontrivial in arbitrarily
negative degrees, it is only the first expression that allows
the interpretation of $SH_*^{S^1,=0}(W)$ as the singular (co)homology
group of a topological space via the Borel construction. This natural
emphasis on homology determines our interpretation of the induced maps
as shriek or transfer maps.
Our bottom line is that the theory is homological in nature, but contravariant because the induced maps are shriek maps.
Note that in the case of a pair $(W,V)$ the simplest expression for $SH_k^{S^1,=0}(W,V)$ is obtained by identifying it with the degree $n-k$ part of the cohomology group $H^*(W,V)\otimes R[u^{-1}]$. To turn this into homology one needs to use excision followed by Poincaré duality, and the expression gets more cumbersome. A similar phenomenon happens for the non-equivariant version $SH_*^{=0}(W,V)$. In order to simplify the notation we always identify the $0$-level part of symplectic homology with singular cohomology throughout the paper.
Remark (coefficients).
The symplectic homology
groups are defined with coefficients in an arbitrary ring $R$, and
statement (i) in Theorem <ref> is valid with arbitrary
coefficients too. Field coefficients are necessary only for
statements (ii) and (iii). As a general fact, the statements in this
paper which involve exact triangles are only valid with field
coefficients, and the proof of excision does require such an exact
triangle, see <ref>. The reason is that we define our
symplectic homology groups as a first-inverse-then-direct-limit over
symplectic homology groups truncated in a finite action window. The
various exact triangles involving symplectic homology are obtained by
passing to the limit in the corresponding exact triangles for such
finite action windows, at which point arises naturally the question of
the exactness of the direct limit functor and of the inverse limit
functor. While the direct limit functor is exact, the inverse limit
functor is not. Nevertheless, the inverse limit functor is exact when
applied to inverse systems consisting of finite dimensional vector
spaces, which is the case for symplectic homology groups truncated in
a finite action window. In order to extend the exact triangle of a
pair (and also the other exact triangles which we establish in this
paper) to arbitrary coefficients one would need to modify the
definition of our groups by passing to the limit at chain level and
use a version of the Mittag-Leffler condition, a path that we shall
not pursue here. See also the discussion of factorisation homology
below, the discussion in <ref>, and Remark <ref>.
More generally, one can define symplectic homology with coefficients in a local system with fibre $\mathfrak{k}$, see <cit.>, and most of the results of this paper adapt in a quite straightforward way to that setup. One notable exception are the duality results in <ref>, in which the treatment of local coefficients would be more delicate.
In view of the above discussion, we henceforth adopt the following convention:
Convention (coefficients). In this paper we use constant coefficients in a field $\mathfrak{k}$.
Let us now discuss briefly the two other Eilenberg-Steenrod axioms, namely the direct sum axiom and the dimension axiom, and explain why they do not, and indeed cannot, have a symplectic counterpart.
(I) The direct sum axiom expresses the fact that the homology of an arbitrary disjoint union of topological spaces is naturally isomorphic to the direct sum of their homologies, whereas in contrast a cohomology theory would involve a direct product. The distinction between direct sums and direct products is not relevant in the setup of Liouville domains, which are by definition compact and therefore consist of at most finitely many connected components. Passing to arbitrary disjoint unions would mean to go from Liouville domains to Liouville manifolds as in <cit.>, and the contravariant nature of the functor would imply that it behaves as a direct product. This is one of the reasons why <cit.> refers to the same object as “symplectic cohomology". However, in view of the extension of the definition to cobordisms this appears to be only a superficial distinction. The deeper fact is that, whichever way one turns it around, symplectic homology of a cobordism with nonempty negative boundary is an object of a mixed homological-cohomological nature because its definition involves both a direct limit (on $b\to\infty$) and an inverse limit (on $a\to-\infty$). We actually present in <ref> an example showing that algebraic duality fails already in the case of symplectic homology of a trivial cobordism.
(II) The dimension axiom of Eilenberg and Steenrod expresses the fact that the value of the functor on any pair homotopy equivalent to a pair of CW-complexes is determined by its value on a point. This fact relies on the homotopy axiom and illustrates the strength of the latter: since any ball is homotopy equivalent to a point, the homotopy axiom allows one to go up in dimension for computations. As a matter of fact the dimension of a space plays no role in the definition of a homology theory in the sense of Eilenberg and Steenrod, although it is indeed visible homologically via the fact that the homology of a pair consisting of an $n$-ball and of its boundary is concentrated in degree $n$. In contrast, symplectic homology is a dimension dependent theory. Moreover, it cannot be determined by its value on a single object. No change in dimension is possible, and no dimension axiom can exist. For example, symplectic homology vanishes on the $2n$-dimensional ball since the latter is subcritical, but the theory is nontrivial. The symplectic analogue of the class of CW-complexes is that of Weinstein manifolds, and the whole richness of symplectic homology is encoded in the way it behaves under critical handle attachments, see <cit.>. One could say that it is determined by its value on the elementary cobordisms corresponding to a single critical handle attachment, but that would be an essentially useless statement, since it would involve all possible contact manifolds and all their possible exact fillings. The complexity of symplectic homology reflects that of Reeb dynamics and is such that there is no analogue of the dimension axiom.
We show in <ref> how to interpret Poincaré duality by defining an appropriate version of symplectic cohomology, and we establish in <ref> a Mayer-Vietoris exact triangle.
It is interesting to note at this point a formal similarity with the recent development of factorisation homology, see the paper <cit.> by Ayala and Francis as well as the references therein. Roughly speaking, a factorisation homology theory is a graded vector space valued monoidal functor defined on some category of open topological manifolds of fixed dimension $n$, with morphism spaces given by topological embeddings, and which obeys a dimension axiom involving the notion of an $E_n$-algebra. (Such a category includes in particular that of closed manifolds of dimension $n-1$, which are identified with open trivial cobordisms of one dimension higher, a procedure very much similar to our viewpoint on contact hypersurfaces as trivial cobordisms.) If one forgets the monoidal property then one essentially recovers the restriction of an Eilenberg-Steenrod homology theory to a category of manifolds of fixed dimension. Conjecturally the symplectic analogue of a factorisation homology theory should involve some differential graded algebra (DGA) enhancement of symplectic homology in the spirit of <cit.>, and the axioms satisfied by factorisation homology should provide a reasonable approximation to the structural properties satisfied by such a DGA enhancement.
One other lesson that the authors have learned from Ayala and Francis <cit.> is that the Eilenberg-Steenrod axioms can, and probably should, be formulated at chain level. More precisely, the target of a homological functor is naturally the category of chain complexes up to homotopy rather than that of graded $R$-modules. This kind of formulation in the case of symplectic homology seems to lie at close hand using the methods of our paper, but we shall not deal with it.
A fruitful line of thought, pioneered by Viterbo in the case of Liouville domains <cit.>, is to compare the symplectic homology groups of a pair $(W,V)$ with the singular cohomology groups, the philosophy being that the difference between the two measures the amount of homologically interesting dynamics on the relevant contact boundary. The singular cohomology $H^{n-*}(W,V)$ is visible through the Floer complex generated by the constant orbits in $W\setminus V$ of any of the Hamiltonians $H_{\mu,\nu,\tau}$, see Figure <ref>, with the degree shift being dictated by our normalisation convention for the Conley-Zehnder index, and this Floer complex coincides with the Morse complex since we work on symplectically aspherical manifolds and the Hamiltonian is essentially constant in the relevant region <cit.>. Note also that these constant orbits are singled out among the various types of orbits involved in the computation of $SH_*(W,V)$ by the fact that their action is close to zero, whereas all the other orbits have negative or positive action bounded away from zero. Accordingly, we denote the resulting homology group by $SH_*^{=0}(W,V)$, with the understanding that we have an isomorphism
SH_*^{=0}(W,V)\simeq H^{n-*}(W,V).
In the case of a Liouville domain (Figure <ref>) we see that these constant orbits form a subcomplex since all the other orbits have positive action. As such, for a Liouville domain there is a natural map $H^{n-*}(W)\to SH_*(W)$. In the case of a cobordism or of a pair of cobordisms such a map does not exist anymore since the orbits on level zero do not form a subcomplex anymore. The correct way to heal this apparent ailment is to consider symplectic homology groups truncated in action with respect to the zero level, which we denote
SH_*^{>0}(W,V),\qquad SH_*^{\ge 0}(W,V),\qquad SH_*^{\le 0}(W,V),\qquad SH_*^{<0}(W,V).
Their meaning is the following. Each of them respectively takes into account, among the orbits involved in the definition of $SH_*(W,V)$, the ones which have strictly positive action (on $\p^+V$ and $\p^+W$), non-negative action (on $\p^+V$, $\p^+W$, and $W\setminus V$), non-positive action (on $\p^-V$, $\p^-W$, and $W\setminus V$), negative action (on $\p^-V$ and $\p^-W$). We refer to <ref> and <ref> for the definitions.
We have maps $SH_*^{<0}(W,V)\to SH_*^{\le 0}(W,V)\to SH_*(W,V)$
induced by inclusions of subcomplexes, and also maps $SH_*(W,V)\to
SH_*^{\ge 0}(W,V)\to SH_*^{>0}(W,V)$ induced by projections onto
quotient complexes. The group $SH_*^{=0}(W,V)$ can be thought of as a
homological cone since it completes the map $SH_*^{<0}(W,V)\to
SH_*^{\le 0}(W,V)$ to an exact triangle. The various maps which
connect these groups are conveniently depicted as forming an
octahedron as in diagram (<ref>). The continuous arrows preserve the degree, whereas the dotted arrows decrease the degree by $1$. Among the eight triangles forming the surface of the octahedron, the four triangles whose sides consist of one dotted arrow and two continuous arrows are exact triangles (see Proposition <ref>), and the four triangles whose sides consist either of three continuous arrows or of one continuous arrow and two dotted arrows are commutative. The structure of this octahedron is exactly the same as the one involved in the octahedron axiom for a triangulated category <cit.>, and for a good reason: this tautological octahedron can be deduced from the octahedron axiom of a triangulated category starting from (the chain level version of) a commuting triangle which involves $SH_*^{<0}$, $SH_*^{\le 0}$, and $SH_*$, and in which the composition of the natural maps $SH_*^{<0}\to SH_*^{\le 0}\to SH_*$ is the natural map $SH_*^{<0}\to SH_*$. Turning this around, this action-filtered octahedron can serve as an interpretation of the octahedron axiom for a triangulated category fit for readers with a preference for variational methods over homological methods.
\begin{equation} \label{eq:SHoctahedron}
\xymatrix{
& & SH_* \ar[ddl] \ar[ddr] & \\
& & & \\
& SH_*^{\ge 0} \ar@{.>}[dl]^{[-1]} \ar '[r] [rr] & & SH_*^{>0} \ar@{.>}[dl]_{[-1]} \ar@{.>}@/^/[dddll]^{[-1]} \\
SH_*^{<0} \ar[rr] \ar[uuurr] & & SH_*^{\le 0} \ar[ddl] \ar[uuu] & \\
& & & \\
& SH_*^{=0} \simeq H^{n-*} \ar@{.>}[uul]^{[-1]} \ar '[uu] [uuu] & &
\end{equation}
Our uniform and emotional notation for these groups is
SH_*^\heartsuit(W,V),\qquad \heartsuit\in\{\varnothing, >0,\ge 0, =0, \le 0, <0\},
with the meaning that $SH_*^\varnothing=SH_*$.
A functor from the Liouville category with fillings to the category of graded $\mathfrak{k}$-vector spaces satisfying the conclusions of Theorem <ref> is called a Liouville homology theory.
For $\heartsuit\in\{\varnothing, >0,\ge 0, =0, \le 0, <0\}$ the action filtered symplectic homology group $SH_*^\heartsuit$ with coefficients in a field $\mathfrak{k}$ defines a Liouville homology theory.
The octahedron (<ref>) defines a diagram of natural transformations which is compatible with the functorial exact sequence of a pair.
In particular, each of the symplectic homology groups $SH_*^\heartsuit$ defines a Liouville homotopy invariant of the pair $(W,V)$. Note that such an invariance statement can only hold provided we truncate the action with respect to the zero value, which is the level of constant orbits. Indeed, answering a question of Polterovich and Shelukhin, we can define symplectic homology groups $SH_*^{(a,b)}(W,V)$ truncated in an arbitrary action interval $(a,b)\subset \R$, see <ref>, and the exact triangle of a pair still holds for $SH_*^{(a,b)}$. However, the homotopy axiom would generally break down and the resulting homology groups would not be Liouville homotopy invariant, except if the interval is either small and centered at $0$, or semi-infinite with the finite end close enough to zero, which are the cases that we consider. Failure of Liouville homotopy invariance for most truncations by the action can be easily detected by rescaling the symplectic form. We believe this action filtration carries interesting information for cobordisms in the form of spectral invariants, or more generally persistence modules <cit.>.
What do we gain from this extension of the theory of symplectic homology from Liouville domains to Liouville cobordisms, and from having singled out the action filtered symplectic homology groups $SH_*^\heartsuit$ ? Firstly, a broad unifying perspective. Secondly, new computational results. We refer to <ref>, <ref>, and <ref> for a full discussion, and give here a brief overview.
(a) Our point of view encompasses symplectic homology, wrapped Floer homology, Rabinowitz-Floer homology, $S^1$-equivariant symplectic homology, linearized contact homology, non-equivariant linearized contact homology. Indeed:
In view of <cit.> Rabinowitz-Floer homology of a separating contact hypersurface $\Sigma$ in a Liouville domain $W$ is $SH_*(\Sigma)$, understood to be computed with respect to the natural filling $\mathrm{int}(\Sigma)$.
We show in <ref> that Viterbo's $S^1$-equivariant symplectic homology $SH_*^{S^1}$ and its flavors $SH_*^{S^1,\heartsuit}$ define Liouville homology theories, and the same is true for negative and periodic cyclic homology. The Gysin exact sequences are diagrams of natural transformations which are compatible with the exact triangles of pairs and with the octahedron (<ref>).
In view of <cit.> linearized contact homology is encompassed by $SH_*^{S^1,>0}$ and non-equivariant linearized contact homology is encompassed by $SH_*^{>0}$. Moreover, our enrichment of symplectic homology to (pairs of) cobordisms indicates several natural extensions of linearized contact homology theories which blend homology with cohomology and whose definition involves the “banana", i.e. the genus zero curve with two positive punctures, see also <cit.> and Remark <ref>. Indeed, such an enrichment should exist at the level of contact homology too, i.e. non-linearized.
(b) Most of the key exact sequences established in recent years for symplectic invariants involving pseudo-holomorphic curves appear to us as instances of the exact triangle of a pair. Examples are the critical handle attaching exact sequence <cit.>, the new subcritical handle attaching exact sequence of <ref>, see also <cit.>, the exact sequence relating Rabinowitz-Floer homology and symplectic homology <cit.>, the Legendrian duality exact sequence <cit.>. We discuss these in detail in <ref>. Our point of view embeds all these isolated results into a much larger framework and establishes compatibilities between exact triangles, e.g. with Gysin exact triangles, see <ref>.
(c) Since our setup covers Rabinowitz-Floer homology, it clarifies in particular the functorial behaviour of the latter. Unlike for symplectic homology, a cobordism does not give rise to a transfer map but rather to a correspondence
SH_*(\p^- W)\longleftarrow SH_*(W)\longrightarrow SH_*(\p^+W).
This allows us in particular to prove invariance of Rabinowitz-Floer homology under subcritical handle attachment and understand its behaviour under critical handle attachment as a formal consequence of <cit.>. See <ref>.
(d) We describe in <ref> which of the symplectic
homology groups carry product structures, with respect to which
transfer maps are ring homomorphisms as in the classical case of
symplectic homology of a Liouville domain. As a consequence we
construct a degree $-n$ product on Rabinowitz-Floer homology which
induces a degree $1-n$ coproduct on positive symplectic homology.
(e) We give a uniform treatment of vanishing and finite dimensionality results in <ref>.
(f) We establish in <ref> Mayer-Vietoris exact triangles for all flavors $SH_*^\heartsuit$. To the best of our knowledge such exact triangles have not appeared previously in the literature.
A word about our method of proof. We already mentioned the confinement lemmas of <ref>. There are two other important ingredients in our construction: continuation maps and mapping cones. We now describe their roles. It turns out that the key map of the theory is the transfer map
i_!:SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)
induced by the inclusion $i:V\hookrightarrow W$ for a pair of Liouville cobordisms $(W,V)$ with filling, see <ref>. It is instrumental for our constructions to interpret this transfer map as a continuation map determined by a suitable increasing homotopy of Hamiltonians. (Compare with the original definition <cit.> of the transfer map for Liouville domains, where its continuation nature is only implicit and truncation by the action plays the main role.) The next step is to interpret the homological mapping cone of the transfer map as being isomorphic to the group $SH_*^\heartsuit(W,V)$ shifted in degree down by $1$ (Proposition <ref>). This is achieved via a systematic use of homological algebra for mapping cones, see <ref>, in which a higher homotopy invariance property of the Floer chain complex plays a key role (Lemma <ref>). While it is possible to show directly starting from the definitions that the groups $SH_*(W,V)$, $SH_*(W)$, and $SH_*(V)$ fit into an exact triangle, we did not succeed in proving this directly for the truncated versions $SH_*^\heartsuit$. The situation was unlocked and the arguments were streamlined upon adopting the continuation map and mapping cone point of view.
We implicitly described the structure of the paper in the body of the Introduction, so we shall not repeat it here. The titles of the sections should now be self-explanatory. We end the Introduction by mentioning two further directions that seem to unfold naturally from the story presented in this paper. The first one is an extension of symplectic homology, which is a linear theory in the sense that its output is valued in graded $R$-modules possibly endowed with a ring structure, to a nonlinear theory at the level of DGAs. This is accomplished for $SH_*^{>0}$ of Liouville domains in <cit.>, but the other flavors may admit similar extensions too. The second one is a further categorical extension of the theory to the level of the wrapped Fukaya category, in the spirit of <cit.> where this is again accomplished for Liouville domains. We expect in particular a meaningful theory of wrapped Fukaya categories for cobordisms, with interesting computational applications.
Acknowledgements. Thanks to Grégory Ginot for inspiring discussions on factorisation homology, to Mihai Damian for having kindly provided the environment that allowed us to overcome one last obstacle in the proof, to Stéphane Guillermou and Pierre Schapira for help with correcting a sign error in <ref>,
and to the referee for valuable suggestions.
The authors acknowledge the hospitality of the Institute for Advanced
Study, Princeton, NJ in 2012 and of the Simons Center for Geometry and
Physics, Stonybrook, NY in 2014, when part of this work was carried out.
K.C. was supported by DFG grant CI 45/6-1.
A.O. was supported by the European Research Council via the Starting Grant StG-259118-STEIN and by the Agence Nationale de la Recherche, France via the project ANR-15-CE40-0007-MICROLOCAL. A.O. acknowledges support from the School of Mathematics at the Institute for Advanced Study in Princeton, NJ during the Spring Semester 2017, funded by the Charles Simonyi Endowment.
§ SYMPLECTIC (CO)HOMOLOGY FOR FILLED LIOUVILLE COBORDISMS
Symplectic homology for Liouville domains was introduced by
Floer–Hofer <cit.> and Viterbo <cit.>.
In this section we extend their definition to filled Liouville
cobordisms. Since symplectic homology is a well established theory, we
will omit many details of the construction and concentrate on the new
aspects. For background we refer to the excellent
account <cit.>.
§.§ Liouville cobordisms
A Liouville cobordism $(W,\lambda)$ consists of a compact
manifold with boundary $W$ and a $1$-form $\lambda$ such that
$d\lambda$ is symplectic and $\lambda$ restricts to a contact form on
$\p W$. We refer to $\lambda$ as the Liouville form. If the
dimension of $W$ is $2n$ the last condition means that
$\lambda\wedge(d\lambda)^{n-1}$ defines a volume form on $\p W$. We
denote by $\p^+W\subset \p W$ the union of the components for which
the orientation induced by $\lambda\wedge(d\lambda)^{n-1}$ coincides
with the boundary orientation of $W$ and call it the convex
boundary of $W$. We call $\p^-W=\p W\setminus \p^+W$ the
concave boundary of $W$.
The convex/concave boundaries of $W$ are contact manifolds $(\p^\pm
W,\alpha^\pm:=\lambda|_{\p^\pm W})$.[Unless otherwise stated
our contact manifolds will be always cooriented and equipped with
chosen contact forms.]
We refer to <cit.> for an
exhaustive discussion of Liouville cobordisms and their homotopies.
A Liouville domain is a Liouville cobordism such that $\p W=\p^+W$.
Given a Riemannian manifold $(N,g)$, its unit
codisk bundle $D^*_rN:=\{(q,p)\in T^*N\;\bigl|\; \|p\|_g\le r\}$
is a Liouville domain with the canonical Liouville form
$\lambda=p\,dq$, whereas $T^*_{r,R}N:=D^*_RN\setminus
\mathrm{int}\, D^*_rN$ for $r<R$ is a Liouville cobordism with
concave boundary given by $S^*_rN:=\p D^*_rN$.
Define the Liouville vector field $Z\in\cX(W)$ by
$\iota_Zd\lambda=\lambda$ and denote by $\alpha^\pm$ the restriction
of $\lambda$ to $\p^\pm W$. It is a consequence of the definitions
that $Z$ is transverse to $\p W$ and points outwards along $\p^+W$,
and inwards along $\p^- W$. The flow $\phi_Z^t$ of the vector field
$Z$ defines Liouville trivialisations of collar neighborhoods
$\NN^\pm$ of $\p^\pm W$
\Psi^+:\big((1-\eps,1]\times \p^+ W,r\alpha^+\big) \to (\NN^+,\lambda),
\Psi^-:\big([1,1+\eps)\times \p^- W,r\alpha^-\big) \to (\NN^-,\lambda),
via the map
(r,x)\mapsto\varphi_Z^{\ln r}(x).
Given a contact manifold $(M,\alpha)$, its
symplectization is given by $(0,\infty)\times M$ with the
Liouville form $r\alpha$. We call $(0,1]\times M$ and
$[1,\infty)\times M$ (both equipped with the form $r\alpha$) the
negative, respectively positive part of the symplectization.
Given a Liouville cobordism $(W,\lambda)$, we define its
completion by
\wh W = ((0,1]\times \p^- W) \sqcup _{\Psi^-} W \ _{\Psi^+}\!\!\sqcup \ [1,\infty)\times \p^+W,
with the obvious Liouville form still denoted by $\lambda$.
Given a contact manifold $(M,\alpha)$ we define a
(Liouville) filling to be a Liouville domain $(F,\lambda)$ together
with a diffeomorphism $\varphi:\p F\to M$ such that $\varphi^*\alpha=
\lambda|_{\p F}$.
We view a Liouville cobordism $(W,\omega,\lambda)$ as a morphism from the
concave boundary to the convex boundary, $W:(\p^- W,\alpha^-)\to (\p
We view a Liouville domain $W$ as a cobordism from $\varnothing$ to its convex boundary.
Given two Liouville cobordisms $W$ and $W'$ together with an
identification $\varphi:(\p^-W,\alpha^-)\stackrel \cong\to
(\p^+W',\alpha'^+)$, we define their composition by
W\circ W' = W \sqcup_{\varphi:\p^-W\stackrel\cong\to \p^+W'} W'.
The gluing is understood to be compatible with the trivialisations
$\Psi^-$ and $\Psi'^+$, so that the Liouville forms glue smoothly.
§.§ Filtered Foer homology
A contact manifold $(M,\alpha)$ carries a canonical Reeb
vector field $R_\alpha\in\cX(M)$ defined by the conditions
$i_{R_\alpha}d\alpha=0$ and $\alpha(R_\alpha)=1$. We refer to the
closed integral curves of $R_\alpha$ as closed Reeb orbits, or
just Reeb orbits. We
denote by $\mathrm{Spec}(M,\alpha)$ the set of periods of closed Reeb
orbits. This is the critical value set of the action functional given
by integrating the contact form on closed loops, and a version of
Sard's theorem shows that $\mathrm{Spec}(M,\alpha)$ is a closed
nowhere dense subset of $[0,\infty)$. If $M$ is compact the set
$\mathrm{Spec}(M,\alpha)$ is bounded away from $0$ since the Reeb
vector field is nonvanishing.
Consider the symplectization $((0,\infty)\times M,r\alpha)$ and let
$h:(0,\infty)\times M\to \R$ be a function that depends only on the
radial coordinate, i.e. $h(r,x)=h(r)$. Its Hamiltonian vector field,
defined by $d(r\alpha)(X_h,\cdot)=-dh$, is given by
The $1$-periodic orbits of $X_h$ on the level $\{r\}\times M$ are
therefore in one-to-one correspondence with the closed Reeb orbits
with period $h'(r)$. Here we understand that a Reeb orbit of negative
period is parameterized by $-R_\alpha$, whereas a $0$-periodic Reeb
orbit is by convention a constant.
Let $(W,\lambda)$ be a Liouville domain and $\wh W$ its completion.
We define the class
\cH(\wh W)
of admissible Hamiltonians on $\wh W$
to consist of functions $H:S^1\times\wh W\to \R$ such that in the
complement of some compact set $K\supset W$ we have $H(r,x)=ar+c$ with
$a,c\in\R$ and $a\notin \pm\mathrm{Spec}(\p W,\alpha)\cup\{0\}$. In
particular, $H$ has no $1$-periodic orbits outside the compact set
An almost complex structure $J$ on the symplectization
$((0,\infty)\times M,r\alpha)$ is called cylindrical if it
preserves $\xi=\ker\alpha$, if $J|_\xi$ is independent of $r$ and compatible
with $d(r\alpha)|_\xi$, and if $J(r\p_r)=R_\alpha$. Such almost
complex structures are compatible with $d(r\alpha)$ and are
invariant with respect to dilations $(r,x)\mapsto (cr,x)$, $c>0$. In
the definition of Floer homology for admissible Hamiltonians on
$\wh W$ we shall use almost complex structures which are
cylindrical outside some compact set that contains $W$, which we
call admissible almost complex structures on $\wh W$.
Consider an admissible Hamiltonian $H$ and an admissible almost
complex structure $J$ on the completion $\wh W$ of a Liouville domain
$W$. To define the filtered Floer homology we use the same notation
and sign conventions as in <cit.>, which
match those of <cit.>:
d\lambda(\cdot,J\cdot)=g_J \qquad\text{(Riemannian metric)},
d\lambda(X_H,\cdot)=-dH,\qquad X_H=J\nabla H \qquad\text{(Hamiltonian vector field)},
\cL\wh W:=C^\infty(S^1,\wh W), \qquad S^1=\R/\Z \qquad \text{(loop space)},
A_H:\cL\wh W\to \R,\qquad A_H(x):=\int_{S^1}x^*\lambda -
\int_{S^1}H(t,x(t))\,dt \qquad\text{(action)},
\nabla A_H(x)=-J(x)(\dot x-X_H(t,x)) \qquad\text{($L^2$-gradient)},
u:\R\to\cL W,\qquad \p_su=\nabla A_H(u(s,\cdot)) \qquad\text{(gradient
\begin{equation}\label{eq:Floer}
\Longleftrightarrow \p_s u + J(u)(\p_t u-X_H(t,u))=0
\qquad\text{(Floer equation)},
\end{equation}
\cP(H):={\rm Crit}(A_H) = \{\text{$1$-periodic orbits of the Hamiltonian vector field $X_H$}\} ,
\hspace{.5cm} \mathcal M(x_-,x_+;H,J)=\{u:\R\times S^1\to W \mid
\p_su= \nabla A_H(u(s,\cdot)),\ u(\pm\infty,\cdot)=x_\pm\}/\R
\mbox{(moduli space of Floer trajectories connecting $x_\pm\in\cP(H)$)},
\dim \cM(x_-,x_+;H,J)=CZ(x_+)-CZ(x_-)-1,
A_H(x_+)-A_H(x_-) = \int_{\R\times S^1}|\p_su|^2ds\,dt = \int_{\R\times S^1}u^*(d\lambda-dH\wedge dt).
Here the formula expressing the dimension of the moduli space in terms
of Conley-Zehnder indices is to be understood with respect to a
symplectic trivialisation of $u^*TW$.
Let $\mathfrak{k}$ be a field and $a<b$ with $a,b\notin{\rm
Spec}(\p W,\alpha)$. We define the filtered Floer
chain groups with coefficients in $\mathfrak{k}$ by
FC_*^{<b}(H) = \bigoplus_{\scriptsize \begin{array}{c} x\in \cP(H)
\\ A_H(x)<b \end{array}} \mathfrak{k}\cdot x,\qquad
FC_*^{(a,b)}(H) = FC_*^{<b}(H)/FC_*^{<a}(H),
with the differential $\p:FC_*^{(a,b)}(H)\to FC_{*-1}^{(a,b)}(H)$
given by
\p x_+=\sum_{\tiny CZ(x_-)=CZ(x_+)-1} \#\mathcal M(x_-,x_+;H,J)\cdot x_-.
Here $\#$ denotes the signed count of points with respect to suitable
We think of the cylinder $\R\times S^1$ as the twice punctured Riemann
sphere, with the positive puncture at $+\infty$ as incoming, and the
negative puncture at $-\infty$ as outgoing. This terminology makes
reference to the corresponding asymptote being an input, respectively
an output for the Floer differential. Note that the differential
decreases both the action $A_H$ and the Conley-Zenhder index.
The filtered Floer homology is now defined as
FH_*^{(a,b)}(H) = \ker\p/\im\p.
Note that for $a<b<c$ the short exact sequence
0 \to FC_*^{(a,b)}(H) \to FC_*^{(a,c)}(H) \to FC_*^{(b,c)}(H) \to 0
induces a tautological exact triangle
\begin{equation}\label{eq:taut1}
FH_*^{(a,b)}(H) \to FH_*^{(a,c)}(H) \to FH_*^{(b,c)}(H)
\to FH_*^{(a,b)}(H)[-1].
\end{equation}
Remark. We will suppress the field $\mathfrak{k}$ from the notation. As noted in the Introduction, the definition can also be given with coefficients in a commutative ring, and more generally with coefficients in a local system as
in <cit.>.
§.§ Restrictions on Floer trajectories
We shall frequently make use of the following three lemmas to exclude
certain types of Floer trajectories.
The first one is an immediate consequence of Lemma 7.2
in <cit.>, see also <cit.>.
Since our setup differs slightly from the one there,
we include the proof for completeness.
Let $H$ be an admissible Hamiltonian on a completed Liouville domain
$(\wh W,\om,\lambda)$. Let $V\subset\wh W$ be a compact subset with
smooth boundary $\p V$ such that $\lambda|_{\p V}$ is a positive contact form,
$J$ is cylindrical near $\p V$, and $H=h(r)$ in cylindrical
coordinates $(r,x)$ near $\p V=\{r=1\}$.
If both asymptotes of a Floer cylinder $u:\R\times
S^1\to\wh W$ are contained in $V$, then $u$ is entirely contained in $V$.
The result continues to hold if $H_s$ depends on the coordinate
$s\in\R$ on the cylinder $\R\times S^1$ such that $\p_sH_s\leq0$ and
the action $A_{h_s}(r)=rh_s'(r)-h_s(r)$ satisfies $\p_s A_{h_s}(r)\leq
0$ for $r$ near $1$.
Assume first that $H$ is $s$-independent.
Arguing by contradiction, suppose that $u$ leaves the set $V$.
After replacing $V$ by the set $\{r\leq r_0\}$ for a constant $r_0>1$
close to $1$, we may assume that $u$ leaves $V$ and is transverse to $\p V$.
In cylindrical coordinates near $\p V$ we have $X_H=h'(r)R$ and
$\lambda=r\alpha$, where $R$ is the Reeb vector field of
$\alpha=\lambda|_{\p V}$, so the functions $H=h(r)$ and
$\lambda(X_H)=rh'(r)$ are both constant along $\p V$.
Note that their difference equals the action $A_h(r)$.
Now $S:=u^{-1}(\wh W\setminus{\rm Int}\,V)$ is a compact surface with
boundary. We denote by $j$ and $\beta$ the restrictions of the complex
structure and the $1$-form $dt$ from the cylinder $\R\times S^1$ to $S$,
so that on $S$ the Floer equation for $u$ can be written as
$\bigl(du-X_H(u)\otimes\beta\bigr)^{0,1}=0$. We estimate the energy of $u|_S$:
\begin{align*}
E(u|_S) &= \frac12\int_S|du-X_H\otimes\beta|^2{\rm vol}_S \cr
&= \int_S(u^*d\lambda-u^*dH\wedge\beta) \cr
&= \int_S d\bigl(u^*\lambda-(u^*H)\beta\bigr) + \int_S (u^*H)d\beta \cr
&\leq \int_{\p S} \bigl(u^*\lambda-(u^*H)\beta\bigr) \cr
&= \int_{\p S} \lambda\bigl(du-X_H(u)\otimes\beta\bigr) \cr
&= \int_{\p S} \lambda\Bigl(J\circ\bigl(du-X_H(u)\otimes\beta\bigr)\circ(-j)\Bigr) \cr
&= \int_{\p S} dr\circ du\circ(-j) \cr
& \leq 0.
\end{align*}
Here the inequality in the 4-th line follows from Stokes' theorem and
$d\beta\equiv 0$. The equality in the 5-th line holds because the
$r$-component of $u|_{\p S}$ equals $r_0$ and thus
\int_{\p S} u^*\bigl(\lambda(X_H)-H\bigr)\beta
= \int_{\p S}A_h(r_0)\beta
= \int_{S}A_h(r_0)d\beta = 0.
The equality in the 6-th line follows from the Floer equation,
and the equality in the 7-th line from $\lambda\circ J=dr$ and $dr(X_H)=0$
along $\p V$. The last inequality follows from the fact that for each tangent
vector $\xi$ to $\p S$ defining its boundary orientation, $j\xi$
points into $S$, so $du(j\xi)$ points out of $V$ and $dr\circ
du(j\xi)\geq 0$. Since $E(u|_S)$ is nonnegative, it follows that
$E(u|_S)=0$, and therefore $du-X_H(u)\otimes\beta\equiv 0$. So each
connected component of $u|_S$ is contained in an $X_H$-orbit, and
since $X_H$ is tangent to $\p V$, $u(S)$ is entirely contained in $\p
V$. This contradicts the hypothesis that $u$ leaves $V$ and the lemma
is proved for $s$-independent $H$.
If $H_s$ is $s$-dependent we get an additional term
$\int_S (u^*\p_sH_s)ds\wedge dt\leq 0$ in the third line, so the 4-th
inequality continues to hold. The equality in the 5-th line becomes an
inequality $\leq$ due to the nonpositive additional term in
\int_{\p S}A_{h_s}(r_0)\beta
= \int_{S}A_{h_s}(r_0)d\beta + \int_S\p_sA_{h_s}(r_0)ds\wedge dt \leq 0.
This proves the lemma for $s$-dependent $H_s$.
The proof shows that Lemma <ref> continues to hold if the cylinder
$\R\times S^1$ is replaced by a general Riemann surface $S$ with a
$1$-form $\beta$ satisfying $H\,d\beta\leq 0$ and $A_h(r)d\beta\leq 0$
for all $r$ near $1$. In this case we can allow $H$ to depend on $s$
in holomorphic coordinates $s+it$ on a region $U\subset S$ in which
$\beta=c\,dt$ for a constant $c\geq 0$, with the requirements
$\p_sH_s\leq0$ and $\p_s A_{h_s}(r)\leq 0$ as before.
This generalization underlies the definition of product
structures in Section <ref>.
The second lemma summarises an argument that has appeared first
in <cit.>. Since the conventions
in <cit.> differ from ours, we include the short
proof for completeness.
Let $(\R_+\times M,r\alpha)$ be the symplectization of a contact
manifold $(M,\alpha)$. Let $H=h(r)$ be a Hamiltonian depending only on
the radial coordinate $r\in\R_+$, and let $J$ be a cylindrical
almost complex structure. Let $u=(a,f):{\color{black}\R_\pm}\times S^1\to
\R_+\times M$ be a solution of the Floer equation (<ref>)
with $\lim_{s\to\pm\infty} u(s,\cdot)=(r_\pm,\gamma_\pm(\cdot))$ for
suitably parameterized Reeb orbits $\gamma_\pm$.
(i) Assume $h''(r_-)>0$. Then either there exists
$(s_0,t_0)\in\R\times S^1$ such that $a(s_0,t_0)>r_-$, or
$u$ is constant equal to $(r_-,\gamma_-)$.
(ii) Assume $h''(r_+)<0$. Then either there exists
$(s_0,t_0)\in\R\times S^1$ such that $a(s_0,t_0)>r_+$, or
$u$ is constant equal to $(r_+,\gamma_+)$.
In coordinates $(s,t)\in\R_\pm\times S^1$, the Floer equation for
$u=(a,f)$ with Hamiltonian $H=h(r)$ writes out as
\begin{equation}\label{eq:Floer-symp}
\p_sa-\alpha(\p_tf)+h'(a)=0,\quad
\p_ta+\alpha(\p_sf)=0, \quad
\pi_\xi\p_sf+J(f)\pi_\xi\p_tf=0,
\end{equation}
where $\pi_\xi:TM\to\xi=\ker\alpha$ is the projection along the Reeb
vector field $R$. In case (i), suppose $h''(r_-)>0$ and $a(s,t)\leq r_-$ for
all $(s,t)\in\R_-\times S^1$. After replacing $\R_-\times S^1$ by a
smaller half-cylinder we may assume that $h''(a(s,t))\geq 0$ for all
$(s,t)\in\R_-\times S^1$. Then the average $\ol a(s):=\int_0^1a(s,t)dt$ satisfies
\begin{align*}
\ol a'(s) &= \int_0^1\p_sa(s,t)dt \cr
&= \int_0^1\alpha(\p_sf)(s,t)dt - \int_0^1h'\bigl(a(s,t)\bigr)dt
\cr
&\geq \int_0^1f^*\alpha(s) - \int_0^1h'(r_-)dt \cr
&\geq \int_{\gamma_-}\alpha - h'(r_-) = h'(r_-) - h'(r_-) = 0.
\end{align*}
Here the second equality follows from the first equation
in (<ref>), the first inequality from $a(s,t)\leq r_-$
and $h''(a(s,t))\geq 0$, and the second inequality from Stokes' theorem
and $f^*d\alpha\geq 0$. For the third equality observe that
$x_-(t)=\bigl(r_-,\gamma_-(t)\bigr)$ is a $1$-periodic orbit of
$X_H=h'(r)R$ iff $\dot\gamma_-=h'(r_-)R$, so that $\int_{\gamma_-}\alpha=h'(r_-)$.
Now $\ol a'(s)\geq 0$ and $\ol a(-\infty)=r_-$ imply that $\ol
a(s)\geq r_-$ for all $s$, which is compatible with $a(s,t)\leq r_-$
only if $a(s,t)=r_-$ for all $(s,t)$. Then all of the preceding
inequalities are equalities, in particular $f^*d\alpha\equiv 0$, and
therefore $u(s,t)=\bigl(r_-,\gamma_t(t)$ for all $(s,t)$. This proves
case (i). Case (ii) follows from case (i) by replacing $h$ by $-h$ and
$u(s,t)$ by $u(-s,-t)$.
Lemma <ref> can be rephrased by saying that nonconstant Floer trajectories
must rise above their output asymptote if the Hamiltonian is convex at
the asymptote, and they must rise above their input asymptote if the
Hamiltonian is concave at the asymptote. Combined with
Lemma <ref>, it forbids Floer trajectories of the kind
shown in Figure <ref>.
Such Floer trajectories are forbidden by Lemma <ref> in combination with Lemma <ref>.
The third lemma follows from a neck stretching argument using the
compactness theorem in symplectic field theory (SFT). We refer to
Figure <ref> for a sketch of a situation in which a
certain kind of Floer trajectory is forbidden by this technique.
Let $H$ be an admissible Hamiltonian on a completed Liouville domain
$(\wh W,\lambda)$. Let $V\subset\wh W$ be a compact subset with
smooth boundary $\p V$ such that $H\equiv c$ near $\p V$
and $\lambda|_{\p V}$ is a positive contact form. Let $J_R$ be the
compatible almost complex structure on $\wh W$ obtained from $J$ by
inserting a cylinder of length $2R$ around $\p V$. Then for
sufficiently large $R$ there exists no $J_R$-Floer cylinder
$u:\R\times S^1\to\wh W$ with asymptotic orbits $x_\pm$ at
$\pm\infty$ such that
* $x_-\subset {\rm int}V$ and $x_+\subset\wh W\setminus V$ with $A_H(x_+)<-c$, or
* $x_+\subset V$ and $x_-\subset\wh W\setminus V$ with $A_H(x_-)>-c$.
Such Floer trajectories are forbidden if $-c>A_H(x_+)$.
Let us first describe more precisely the neck stretching along $M=\p
V$. Pick a tubular neighborhood $[-\eps,\eps]\times M$ of $M$ in $\wh
W$ on which $H\equiv c$ and $\lambda=e^\rho\alpha$, where
$\alpha=\lambda|_M$ and $\rho$ denotes the coordinate on $\R$. Let $J$
be a compatible almost complex structure on $\wh W$ whose restriction
$J_0$ to $[-\eps,\eps]\times M$ is independent of $\rho$ and maps
$\xi=\ker\alpha$ to $\xi$ and $\p_\rho$ to $R_\alpha$. Let $\phi_R$ be
any diffeomorphism $[-R,R]\to[-\eps,\eps]$ with derivative $1$ near
the boundary. Then we define $J_R$ on $\wh W$ by
$(\phi_R\times\id)_*J_0$ on $[-\eps,\eps]\times M$, and by $J$ outside
$[-\eps,\eps]\times M$.
Consider a $J_R$-Floer cylinder $u:\R\times S^1\to\wh W$ with
asymptotic orbits $x_\pm$. Its Floer energy is given by
A_H(x_+)-A_H(x_-) = \int_{\R\times S^1}|\p_su|^2ds\,dt = \int_{\R\times S^1}u^*(d\lambda-dH\wedge dt).
Set $\Sigma=u^{-1}([-\eps,\eps]\times M)$ and write the restriction of $u$ to $\Sigma$ as
u|_\Sigma=(\phi_R\circ a,f),\qquad (a,f):\Sigma\to[-R,R]\times M.
Let $\psi:[-R,R]\to[e^{-\eps},e^\eps]$ be any nondecreasing function which equals $e^{\phi_R}$
on the boundary. Using non-negativity of the integrand in the Floer energy, vanishing of $dH$ on $[-\eps,\eps]\times M$, and Stokes' theorem, we obtain
\begin{align*}
&\geq \int_{\Sigma}u^*(d\lambda-dH\wedge dt)
= \int_{\Sigma}u^*d\lambda\cr
&= \int_{\Sigma}(a,f)^*d(e^{\phi_R}\alpha)
= \int_{\Sigma}(a,f)^*d(\psi\alpha)\cr
&= \int_{\Sigma}\Bigl(\psi'(a)da\wedge f^*\alpha+\psi(a)f^*d\alpha\Bigr).
\end{align*}
Since $(a,f)$ is $J_0$-holomorphic, $da\wedge f^*\alpha$ and
$f^*d\alpha$ are nonnegative $2$-forms on $\Sigma$. Since
$\psi'(a)\geq 0$ and $\psi(a)\geq e^{-\eps}$,
and $\psi$ was arbitrary with the given boundary conditions, this
yields a uniform bound (independent of $R$) on the Hofer energy of
$(a,f)$ (see <cit.>).
Now suppose that there exists a sequence $R_k\to\infty$ and
$J_{R_k}$-Floer cylinders $u_k:\R\times S^1\to\wh W$ with asymptotic
orbits $x_\pm$ lying on different sides of $M$. By the SFT
compactness theorem <cit.>, $u_k$ converges in the limit to a
broken cylinder consisting of components in the completions of $V$ and
$\wh W\setminus V$ satisfying the Floer equation and $J_0$-holomorphic
components in $\R\times M$, glued along closed Reeb orbits in
$M$. Since $x_\pm$ lie on different sides of $M$, the punctures
asymptotic to $x_\pm$ lie on different components. Hence for
large $k$ there exists a separating embedded loop
$\delta_k\subset\R\times S^1$ such that $u_k\circ\delta_k$ is
$C^1$-close to a (positively parameterized) closed Reeb orbit $\gamma$
on $M$ (which we view as a loop in $\wh W$ lying on $\p V$). Here
$\delta_k$ is parameterized as a positive boundary of the
component of $\R\times S^1$ that is mapped to $\wh V$. Now we
distinguish two cases.
Case (i): $x_-\subset V$ and $x_+\subset\wh W\setminus V$. Then
$\delta_k$ winds around the cylinder in the positive $S^1$-direction,
and since the Hamiltonian action increases along Floer cylinders we
A_H(x_+)\geq A_H(\gamma)\geq A_H(x_-).
Since $\int_\gamma\lambda=\int_\gamma\alpha\geq 0$, we obtain
$A_H(\gamma)=\int_\gamma\lambda-\int_0^1c\,dt\geq -c$
and hence $A_H(x_+)\geq -c$.
Case (ii): $x_+\subset V$ and $x_-\subset\wh W\setminus V$. Then
$\delta_k$ winds around the cylinder in the negative $S^1$-direction,
and since the Hamiltonian action increases along Floer cylinders we
A_H(x_+)\geq A_H(-\gamma)\geq A_H(x_-).
Since $\int_\gamma\lambda=\int_\gamma\alpha\geq 0$, we obtain
$A_H(-\gamma)=-\int_\gamma\lambda-\int_0^1c\,dt\leq -c$
and hence $A_H(x_-)\leq -c$.
Our fourth lemma prohibits certain trajectories asymptotic to constant
Hamiltonian orbits.
We consider the setup consisting of a completed Liouville domain $\wh W$, a cobordism $V\subset W$ such that $(W,V)$ is a Liouville pair, i.e. $W=W^{bottom}\circ V\circ W^{top}$, and a Hamiltonian $H:\wh W\to\R$ which is constant on $V$, which depends only on the radial coordinate $r$ in an open neighborhood of $\p V$, and which is either strictly convex or strictly concave as a function of $r$ outside $V$ in each component of the given neighborhood of $\p V$. Denote by $c$ the value of $H$ on $V$.
Let $f:V\to\R$ be a Morse function which depends only on the radial coordinate $r$ in some neighborhood of $\p V$ and such that $\p^\pm V$ are regular level sets.
We require the gradient of $f$ to point inside/outside $V$ along $\p^-V$ if $H$ is concave/convex near $\p^-V$, and
to point inside/outside $V$ along $\p^+V$ if $H$ is concave/convex near $\p^+V$.
Given $\epsilon>0$ we denote by $V^\epsilon = ([1-\epsilon,1]\times \p^- V)\cup V \cup ([1,1+\epsilon]\times\p^+ V)$ an $\epsilon$-thickening of $V$ inside $\wh W$. For $\epsilon>0$ small enough let
H_{f,\epsilon}:S^1\times\wh W\to\R
be a smooth Hamiltonian which is equal to $c+\epsilon^2 f$ on $V$, which is equal to $H$ outside $V^\epsilon$, and which smoothly interpolates between $H$ and $c+\epsilon^2 f$ on $[1-\epsilon,1]\times\p^- V$ and $[1,1+\epsilon]\times\p^+ V$ as a function of $r$ which is either concave or convex, according to $H$ being concave or convex on each of these regions.
We consider admissible almost complex structures on $\wh W$ which are time-independent on $V$, cylindrical near $\p V$, and such that the gradient flow of $f$ is Morse-Smale.
Let $f:V\to\R$ be a Morse function and $H_{f,\epsilon}$ a Hamiltonian as above. For $\epsilon>0$ small enough the following hold:
(1) If the gradient of $f$ points inside $V$ along $\p^- V$, then there is no Floer trajectory for $H_{f,\epsilon}$ which is asymptotic at the positive end to a constant orbit given by a critical point of $f$ and which is asymptotic at the negative end to an orbit in $W^{bottom}$.
(2) If the gradient of $f$ points outside $V$ along $\p^- V$, then there is no Floer trajectory for $H_{f,\epsilon}$ which is asymptotic at the negative end to a constant orbit given by a critical point of $f$ and which is asymptotic at the positive end to an orbit in $W^{bottom}$.
To prove (1) we argue by contradiction and assume without loss of generality that there is a sequence of positive real numbers $\epsilon_n\to 0$ and a sequence of Floer trajectories $u_n:\R\times S^1\to \wh W$ solving $\p_s u_n + J_t(u_n)(\p_t u_n -X_{H_{f,\epsilon_n}}(u_n))=0$ such that $\lim_{s\to\infty}Ęu_n(s,t)=p_+$, $\lim_{s\to-\infty}Ęu_n(s,t)=x_-(t)$, with $p_+$ a critical point of $f$, $x_-:S^1\to\wh W$ a $1$-periodic orbit of $H$ inside $W^{bottom}$, and $J=(J_t)$ an admissible almost complex structure which is time-independent on $V$ and such that the flow of the gradient of $f$ for the corresponding Riemannian metric is Morse-Smale.
We interpret $V$ as a Morse-Bott critical manifold with boundary for the action functional $A_H$, and we view $H_{f,\epsilon_n}$, $n\ge 1$ as determining a sequence of Morse perturbations of $A_H$ along $V$. The Morse-Bott compactness theorem proved in a more restricted Hamiltonian setting in <cit.>, and in a
general SFT setting in <cit.>, applies to our situation. Indeed, the fact that the Morse-Bott manifold $V$ has boundary plays no role and the proof of <cit.> carries over mutatis mutandis.
It follows that, up to extracting a subsequence, the sequence $u_n$ converges in the terminology of <cit.> to a
broken Floer trajectory $\mathbf{[u]}$ with gradient fragments. The critical manifold $V$ may be disconnected, but all its components are located on the same action level $A_H=-c$. Since Floer trajectories for $H$ strictly increase the action from the asymptote at the negative puncture to the asymptote at the positive puncture, we infer that each level of the limit $\mathbf{[u]}$ contains at most one gradient trajectory of $f$. Moreover, $\mathbf{[u]}$ has a representative $\mathbf{\bar u}=(\mathbf{u}_1,\dots,\mathbf{u}_\ell)$ described as follows: there exists $1\le i\le \ell$ such that
* $\mathbf{u}_1,\dots,\mathbf{u}_{i-1}$ are Floer trajectories for $H$, with $\mathbf{u}_1(-\infty)=x_-$, $\mathbf{u}_j(+\infty)=\mathbf{u}_{j+1}(-\infty)$ for $1\le j\le i-2$.
* $\mathbf{u}_i$ is a Floer trajectory with one gradient fragment, i.e. $\mathbf{u}_i=(u_i,\gamma_i)$ with $u_i$ a Floer trajectory for $H$ and $\gamma_i:[0,+\infty)\to V$ a negative gradient trajectory for $f$, i.e. solving $\dot\gamma_i=-\nabla f (\gamma_i)$, subject to the following conditions: $\mathbf{u}_{i-1}(+\infty)=\mathbf{u}_i(-\infty)$ if $i>1$ and $\mathbf{u}_i(-\infty)=x_-$ if $i=1$; $u_i(+\infty)=\gamma_i(0)\in V$; and $\gamma_i(+\infty)=p_+$ if $i=\ell$.
* $\mathbf{u}_{i+1},\dots,\mathbf{u}_\ell$ are negative gradient trajectories $\mathbf{u}_j=\gamma_j:\R\to V$ for $f$, i.e. solving $\dot\gamma_j=-\nabla f(\gamma_j)$, $j=i+1,\dots,\ell$, subject to the conditions $\gamma_j(-\infty)=\gamma_{j-1}(+\infty)$ for $j=i+1,\dots,\ell$, and $\gamma_\ell(+\infty)=p_+$.
We now focus on the level $\mathbf{u}_i=(u_i,\gamma_i)$. Three situations can arise:
Case 1: $\gamma(0)\in V\setminus \p V$. Then the Floer trajectory $u_i$ solves the Cauchy-Riemann equation $\p_s u + J(u) \p_t u=0$ on some half-cylinder $[s_0,+\infty)\times S^1$ for $s_0\gg 0$. We identify biholomorphically $[s_0,+\infty)\times S^1$ with a punctured disc $\dot D$ and, by assumption, $u:\dot D\to V$ admits a continuous extension at the puncture. Thus $0\in D$ is a removable singularity and we can view $u_i:\R\times S^1\to \wh W$ as being defined on a Riemann sphere with a single negative puncture, on which it solves a Floer equation. The asymptote at the negative puncture is located in $W^{bottom}$ by assumption, and the image of $u_i$ intersects $\p^- V$. Then Lemma <ref> gives a contradiction.
Case 2: $\gamma(0)\in \p^+ V$. Pick $\delta>0$ such that $[1-\delta,1]\times\p^+V$ does not contain critical points of $f$. Since $\mathbf{[u]}$ is the limit of the sequence $u_n$, there exists $n_0\ge 1$ such that the image of $u_n$ intersects the set $(1-\delta,1]\times\p^+ V$. By assumption both asymptotes of $u_n$ are located in $W^{bottom}\cup V \setminus ([1-\delta,1]\times\p^+ V)$, and Lemma <ref> again gives a contradiction.
Case 3: $\gamma(0)\in \p^- V$. The map $\gamma_i:[0,\infty)\to V$ solves $\dot\gamma_i=-\nabla f(\gamma_i)$ and enters $V$ in positive time, but at the same time $-\nabla f$ points outwards along $\p V$, which is a contradiction.
The proof of (2) is entirely analogous: cases 1 and 2 are treated exactly in the same way, while case 3 is proved similarly to (1) using that negative gradient trajectories of a Morse function on $V$ whose gradient points outwards along $\p V$ must exit $V$ in negative time.
The conclusions of Lemma <ref> most likely do not hold if one exchanges “positive" and “negative" in either of the statements (1) or (2). Although we do not have an explicit example involving Floer trajectories, i.e. twice punctured spheres, we can easily give an example involving pairs of pants. Consider to this effect a Liouville domain $W$ and the trivial cobordism $V=[\frac 1 2,1]\times\p W$ over the boundary. As discussed in <ref>, the symplectic homology group $SH_*^{\le 0}(V)=SH_*^{\le 0}(\p W)$ is a unital graded commutative ring, and the unit maps to $1\in H^{n-*}(\p W)$ under the
projection $SH_*^{\le 0}(V)\to SH_*^{=0}(V)\simeq H^{n-*}(\p W)$. Assume now that the map $SH_*^{<0}(V)\to SH_*^{\le 0}(V)$ is nontrivial – which holds for example in the case of unit cotangent bundles of closed manifolds – and consider a class $\alpha\neq 0$ in its image. Since $1\cdot \alpha=\alpha\neq 0$ we infer the existence of at least one solution to a Floer equation defined on a pair of pants with two positive punctures and one negative puncture, asymptotic at one of the positive punctures to a constant orbit inside $V$, and asymptotic at the two other punctures to orbits located in $W^{bottom}=W\setminus V$.
§.§ Symplectic homology of a filled Liouville cobordism
Let $(W,\lambda)$ be a Liouville cobordism and $(F,\lambda)$ a
Liouville filling of $(\p^-W,\alpha^-=\lambda_{\p^-W})$. We compose $F$ and $W$
to the Liouville domain
W_F := F\circ W
and denote its completion by $\wh W_F$. We define the class
\cH(W;F)
of admissible Hamiltonians on $\wh W_F$ with respect to the
filling $F$ to consist of functions $H:S^1\times \wh W_F\to \R$ such
that $H\in\cH(\wh W_F)$ and $H= 0$ on $W$. When there is no danger of
confusion we shall use the notation
\cH(W)
for the set $\cH(W;F)$ and refer to its elements as admissible Hamiltonians on $W$.
For the purposes of this section it would have been enough to
define admissible Hamiltonians by the condition $H\le 0$ on
$W$. This would have allowed for cofinal families consisting of
Hamiltonians with nondegenerate $1$-periodic orbits. The definition
that we have adopted requires to use small perturbations in order to
define Floer homology and is slightly cumbersome in that
respect. However, it will prove very convenient when we come to the
definition of symplectic homology groups for pairs.
Next we consider continuation maps. Let $H_-\ge H_+$ be admissible
Hamiltonians and $H_s$, $s\in\R$ be a decreasing homotopy through
admissible Hamiltonians such that $H_s=H_\pm$ near $\pm\infty$. Let
$J_s$ be a homotopy of admissible almost complex structures. Solutions
of the Floer equation $\p_s u+J_s(u)(\p_t u-X_{H_s}(u))=0$ satisfy a
maximum principle in the region where all the Hamiltonians $H_s$ are
linear and all the almost complex structures are cylindrical, and
their count defines continuation maps $FH_*(H_+)\to FH_*(H_-)$. Since
the homotopy is decreasing, the action increases along solutions of
the preceding $s$-dependent Floer equation, so it decreases under
the continuation map. We infer from this the existence of filtered
continuation maps $FH_*^{(-\infty,b)}(H_+)\to
FH_*^{(-\infty,b)}(H_-)$, $b\in\R$, and more generally the existence
of filtered continuation maps
FH_*^{(a,b)}(H_+)\to FH_*^{(a,b)}(H_-), \qquad a<b.
For an admissible Hamiltonian $H$ we also have natural morphisms
determined by inclusions of and quotients by appropriate subcomplexes
FH_*^{(a,b)}(H)\to FH_*^{(a',b')}(H),\qquad a\le a',\ b\le b'.
These morphisms commute with the continuation morphisms, and we obtain
more general versions of the latter
FH_*^{(a,b)}(H_+)\to FH_*^{(a',b')}(H_-), \qquad a\le a',\ b\le b'.
Given real numbers $-\infty<a<b<\infty$, we define the
filtered symplectic homology groups of $W$ (with respect
to the filling $F$) to be
\begin{equation}\label{eq:SH*abW}
SH_*^{(a,b)}(W)=\lim^{\longrightarrow}_{H\in\cH(W;F)} FH_*^{(a,b)}(H).
\end{equation}
The direct limit is taken here with respect to continuation maps and
with respect to the partial order $\prec$ on $\cH(W;F)$ defined as
follows: $H\prec K$ if and only if $H(t,x)\le K(t,x)$ for all $(t,x)$. Note that in a cofinal
family the Hamiltonian necessarily goes to $+\infty$ on $F\cup
([1,\infty)\times \p^+W)$. Recall also that, in order to achieve
nondegeneracy of the $1$-periodic orbits, the Hamiltonian $H$
needs to be perturbed on $W$ where it is constant equal to
zero. Our convention is that we compute the direct limit using a
cofinal family for which the size of the perturbation goes to
Taking the direct limit in (<ref>) we obtain for $a<b<c$ the tautological
exact triangle
\begin{equation}\label{eq:taut2}
SH_*^{(a,b)}(W) \to SH_*^{(a,c)}(W) \to SH_*^{(b,c)}(W)
\to SH_*^{(a,b)}(W)[-1].
\end{equation}
We define six versions of symplectic homology groups of $W$
(with respect to the filling $F$):
SH_*(W)=\lim^{\longrightarrow}_{b\to\infty}\lim^{\longleftarrow}_{a\to -\infty} SH_*^{(a,b)}(W) \qquad \mbox{\sc (full symplectic homology)}
SH_*^{>0}(W)= \lim^{\longrightarrow}_{b\to\infty} \lim^{\longleftarrow}_{a\searrow 0} SH_*^{(a,b)}(W) \qquad \mbox{\sc (positive symplectic homology)}
SH_*^{\ge 0}(W)= \lim^{\longrightarrow}_{b\to\infty} \lim^{\longrightarrow}_{a\nearrow 0} SH_*^{(a,b)}(W) \qquad \mbox{\sc (non-negative symplectic homology)}
SH_*^{= 0}(W)= \lim^{\longleftarrow}_{b\searrow 0} \lim^{\longrightarrow}_{a\nearrow 0} SH_*^{(a,b)}(W) \qquad \mbox{\sc (zero-level symplectic homology)}
SH_*^{\le 0}(W)= \lim^{\longleftarrow}_{b\searrow 0} \lim^{\longleftarrow}_{a\to-\infty} SH_*^{(a,b)}(W) \qquad \mbox{\sc (non-positive symplectic homology)}
SH_*^{< 0}(W)= \lim^{\longrightarrow}_{b\nearrow 0} \lim^{\longleftarrow}_{a\to-\infty} SH_*^{(a,b)}(W) \qquad \mbox{\sc (negative symplectic homology)}
Since the actions of Reeb orbits are bounded away from
zero, the direct/inverse limits as $a$ (or $b$) goes to zero stabilize
for $a$ (respectively $b$) sufficiently close to zero, so they are not
actual limits. Note that the actual inverse limits as $a\to-\infty$ in these
definitions are always applied to finite dimensional vector spaces
when considering field coefficients.
This ensures that the inverse and direct limits preserve
exactness of sequences; see <cit.> for further discussion of the
order of limits, and also <cit.> for a discussion of
The geometric content of the definition is the following. Let $H$ be a
Hamiltonian as depicted in Figure <ref>, which is
constant and very positive on $F\setminus([\delta,1]\times \p F)$ with
$0<\delta<1$, which is linear of negative slope with respect to the
$r$-coordinate on $[\delta,1]\times\p F$, which vanishes on $W$, and
which is linear of positive slope with respect to the $r$-coordinate
on $[1,\infty[\times \p^+W$. The $1$-periodic orbits of $H$ fall in
four classes, denoted $F$ (orbits in the filling),
${I^-}$ (orbits that correspond to negatively parameterized
closed Reeb orbits on $\p^-W$), ${I^0}$ (constant orbits in
$W$), and ${I^+}$ (orbits that correspond to positively
parameterized closed Reeb orbits on $\p^+W$). As $\delta\to 0$ and
as the absolute values of the slopes go to $\infty$, Hamiltonians
of this type form a cofinal family in $\mathcal{H}(W;F)$. The
action of orbits in the class ${F}$ becomes very negative
and falls outside any fixed and finite action window $(a,b)$, so
that the homology groups $SH_*^{(a,b)}(W)$ take into account only
orbits of type ${I^{-0+}}$. Each flavour of symplectic
homology group $SH_*^{\heartsuit}(W)$,
$\heartsuit\in\{\varnothing, >0,\ge 0, =0, \le 0, <0\}$, with
$SH_*^\varnothing(W)$ as a notation for $SH_*(W)$, respectively
takes into account orbits in the class ${I^{-0+}}$,
${I^+}$, ${I^{0+}}$, ${I^{0}}$,
${I^{-0}}$, ${I^-}$ for arbitrarily large values of
the slope. As such, each of these symplectic homology groups
corresponds to a certain count of negatively parameterized closed
Reeb orbits on $\p^-W$, of constant orbits in $W$, and of
positively parameterized closed Reeb orbits on $\p^+W$.
Cofinal family of Hamiltonians for $SH_*^{\heartsuit}(W)$.
The next proposition will be proved as
Proposition <ref> below.
Each of the above six versions of symplectic homology is an invariant
of the Liouville homotopy type of the pair $(W;F)$.
The following computation is fundamental in applications.
Let $\dim\, W=2n$. Then we have a canonical isomorphism
SH_*^{=0}(W)\cong H^{n-*}(W).
Consider a Hamiltonian $K$ of the shape as in Figure <ref>.
Since $\wh W_F$ is symplectically aspherical, it follows
from <cit.> (see
also <cit.>) that if $K$ is sufficiently
$C^2$-small on $W$, then its Floer chain complex reduces to the Morse
cochain complex for an appropriate choice of almost complex structure.
Fix such a $K$ and denote by $c>0$ its constant value on the filling
$F$. Pick $\eps$ with $0<\eps<c$, so that the constant orbits in $F$
have action $-c<-\eps$. Since the Conley-Zehnder index of a critical
point is related to its Morse index by ${\rm CZ}=n-{\rm Morse}$, we
get a canonical isomorphism $FH_*^{(-\eps,\eps)}(K)\cong H^{n-*}(W)$.
Consider any other Hamiltonian $H$ of the shape as in Figure <ref>
with $K\leq H$. We choose $\eps$ smaller than the smallest action of a
closed Reeb orbit on $\p W$. Then all nonconstant orbits of $H$ have
action outside $(-\eps,\eps)$ and a monotone homotopy from $K$ to $H$
yields a continuation isomorphism $FH_*^{(-\eps,\eps)}(K)\stackrel{\cong}\to
FH_*^{(-\eps,\eps)}(H)$, which induces in the direct limit over $H$ a
canonical isomorphism $FH_*^{(-\eps,\eps)}(K)\stackrel{\cong}\to
SH_*^{(-\eps,\eps)}(W) = SH_*^{=0}(W)$.
If $W$ is a Liouville domain we have
SH_*^{<0}(W)=0,\qquad SH_*^{\le 0}(W)=SH_*^{=0}(W),\qquad SH_*^{\ge 0}(W)=SH_*(W),
and the group $SH_*^{>0}(W)$ coincides by definition with the group $SH_*^+(W)$ of <cit.>.
If $W$ is a Liouville cobordism with Liouville filling $F$ we have (by
a standard continuation argument)
SH_*^{>0}(W)\cong SH_*^{>0}(W_F).
The following “tautological" exact triangles hold for the symplectic homology groups of $W$:
\begin{equation*}
\scriptsize
\xymatrix
SH_*^{<0} \ar[rr] & &
SH_* \ar[dl] \\ & SH_*^{\ge 0} \ar[ul]^{[-1]}
\qquad
\xymatrix
SH_*^{\le 0} \ar[rr] & &
SH_* \ar[dl] \\ & SH_*^{> 0} \ar[ul]^{[-1]}
\end{equation*}
\begin{equation*}
\scriptsize
\xymatrix
SH_*^{< 0} \ar[rr] & &
SH_*^{\le 0} \ar[dl] \\ & SH_*^{= 0} \ar[ul]^{[-1]}
\qquad
\xymatrix
SH_*^{= 0} \ar[rr] & &
SH_*^{\ge 0} \ar[dl] \\ & SH_*^{> 0} \ar[ul]^{[-1]}
\end{equation*}
We prove the exactness of the triangle
\begin{equation}\label{eq:first-ex-tr}
SH_*^{\le 0}(W)\to SH_*(W)\to SH_*^{>0}(W) \to SH_*^{\le 0}(W)[-1]\,.
\end{equation}
The proofs for the other three triangles are similar and left to the reader.
Let $\varepsilon>0$ be smaller than the minimal period of a closed
characteristic on $\p^+W$. It follows from the definitions that
SH_*^{\le 0}(W) = \lim^{\longleftarrow}_{a\to-\infty} SH_*^{(a,\varepsilon)}(W)
SH_*^{>0}(W) = \lim^{\longrightarrow}_{b\to\infty} SH_*^{(\varepsilon,b)}(W).
For fixed $a,b\in\mathbb{R}$ such that
$-\infty<a<0<\varepsilon<b<\infty$ we have from (<ref>) an exact triangle
SH_*^{(a,\varepsilon)}(W)\to SH_*^{(a,b)}(W)\to SH_*^{(\varepsilon,b)}(W)\to SH_*^{(a,\varepsilon)}(W)[-1]\,.
All the terms in this exact triangle are finite dimensional vector spaces. The inverse limit functor is exact on directed systems consisting of finite dimensional vector spaces, and the direct limit functor is always exact. We then obtain (<ref>) by first taking the inverse limit on $a\to-\infty$, and then taking the direct limit on $b\to\infty$.
Symplectic homology groups relative to boundary components.
Let $A\subset \p W$ be a union of boundary components of $W$ and denote
A^\pm=A\cap \p^\pm W.
We further assume that $A^-$ is a union of boundaries of components of $F$. We refer to such an $A$ as an admissible subset of $\p W$.
One obvious choice is $A^-=\p ^-W$, which satisfies the assumption for any $F$. If each component of $F$ has connected boundary then one can take $A^-\subset \p^-W$ arbitrary. If $F$ consists of a single connected component then the only possible choices are $A^-=\p^- W$ or $A^-=\varnothing$. Note also that, if $A$ satisfies the assumption, then $A^c:=\p W\setminus A$ also does.
Let $F_{A^-}$ denote the filling of $(A^-,\alpha^-)$ consisting of the union of the components of $F$ with boundary contained in $A^-$. Denote
(\wh W_F\setminus W)_A= {\rm int}\,F_{A^-}\cup ((1,\infty)\times A^+),
so that
\wh W_F\setminus W = (\wh W_F\setminus W)_A \sqcup (\wh W_F\setminus W)_{A^c}.
Given real numbers $-\infty<a<b<\infty$, we define the filtered symplectic homology groups of $W$ relative to $A$ (with respect to the filling $F$) to be
\begin{equation} \label{eq:SH*abWA}
\lim^{\longrightarrow}_{\scriptsize \begin{array}{c} H\in\cH(W;F) \\ H\to\infty \mbox{ on } (\wh W_F\setminus W)_{A^c} \end{array}}
\lim^{\longleftarrow}_{\scriptsize \begin{array}{c} H\in\cH(W;F) \\ H\to-\infty \mbox{ on } (\wh W_F\setminus W)_A \end{array}}
\end{equation}
We define six flavors of symplectic homology groups of $W$ relative to $A$, or symplectic homology groups of the pair $(W,A)$,
SH_*^\heartsuit(W,A), \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\},
by the formulas in Definition <ref> with $SH_*^{(a,b)}(W)$ replaced by $SH_*^{(a,b)}(W,A)$.
The notation $SH_*^\heartsuit$ with $\heartsuit=\varnothing$ refers to $SH_*$.
We refer to Figure <ref> for an illustration of several significant cases of Hamiltonians used in the computation of relative symplectic homology groups. The case $A=\varnothing$ corresponds to Figure <ref>. In each case, in the limit the orbits that appear in the filling either fall below or fall above any fixed and finite action window, so that only orbits appearing near $W$ are taken into account. As an example, $SH_*(W,\p^-W)$ corresponds to a a certain count of positively parameterized closed Reeb orbits on $\p^-W$, of constant orbits in $W$, and of positively parameterized closed Reeb orbits on $\p^+W$. Similar interpretations hold for $SH_*(W,\p^+W)$, $SH_*(W,\p W)$, and also for all their $\heartsuit$-flavors. In Figure <ref> we encircled with a dashed line the region which contains the orbits that are taken into account. The mnemotechnic rule is the following:
To compute $SH_*^{\heartsuit}(W,A)$ one must use a family of Hamiltonians that go to $-\infty$ near $A$ and that go to $+\infty$ near $\p W\setminus A$.
Shape of Hamiltonians for $SH_*(W,A)$ with $A=\varnothing,\p W,\p^-W,\p^+W$.
Our notation is motivated by the following analogue of
Proposition <ref>, which is proved in the same way.
Let $\dim\, W=2n$ and $A\subset \p W$ be admissible. Then we have a canonical isomorphism
SH_*^{=0}(W,A)\cong H^{n-*}(W,A).
The tautological exact triangles described in Proposition <ref> also exist for the relative symplectic homology groups $SH_*^{\heartsuit}(W,A)$ (same proof). Also, the relative symplectic homology groups $SH_*^\heartsuit(W,A)$ are invariants of the Liouville homotopy type of the pair $(W,F)$ (see <ref>, compare Propositions <ref> and <ref>).
§.§ Symplectic homology groups of a pair of filled Liouville cobordisms
A Liouville pair, or pair of Liouville cobordisms, is a triple
$(W,V,\lambda)$ where $(W,\lambda)$ is a Liouville
cobordism and $V\subset W$ is a codimension $0$ submanifold with
boundary such that
* $(V,\lambda|_V)$ is a Liouville cobordism;
* $\overline{W\setminus V}$ is a disjoint union of two (possibly
empty) Liouville cobordisms $W^{bottom}$ and $W^{top}$ such that
W=W^{bottom}\circ V \circ W^{top}.
We fix a filling $F$ of $W$ and define $W_F$, $\wh W_F$ as above.
We define the class
\cH(W,V;F)
of admissible Hamiltonians on $(W,V)$ with respect to the filling $F$ to consist of elements $H:S^1\times \wh W_F\to \R$ such that $H\in\cH(\wh W_F)$ and $H= 0$ on $W\setminus V$ (see Figure <ref>).
Given real numbers $-\infty<a<b<\infty$, we define the action-filtered symplectic homology groups of $(W,V)$ (with respect to the filling $F$) to be
\begin{equation} \label{eq:SH*abWV}
\lim^{\longrightarrow}_{\scriptsize \begin{array}{c} H\in\cH(W,V;F) \\ H\to\infty \mbox{ on } (\wh W_F\setminus W) \end{array}}
\lim^{\longleftarrow}_{\scriptsize \begin{array}{c} H\in\cH(W,V;F) \\ H\to-\infty \mbox{ on } {\rm int}\,V \end{array}}
\end{equation}
We define six flavors of symplectic homology groups of the
Liouville pair $(W,V)$,
SH_*^\heartsuit(W,V), \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\},
by the formulas in Definition <ref> with $SH_*^{(a,b)}(W)$ replaced by $SH_*^{(a,b)}(W,V)$.
The notation $SH_*^\heartsuit$ with $\heartsuit=\varnothing$ refers to $SH_*$.
To describe the geometric content of the definition we consider a
cofinal family of Hamiltonians $H$ of the shape described in
Figure <ref>. Heuristically, each of the groups
$SH_*^\heartsuit(W,V)$ represents a certain count of negatively
parameterized closed Reeb orbits on $\p^-W$ and $\p^-V$, of constant
orbits in $\overline{W\setminus V}$, and of positively parameterized
closed Reeb orbits on $\p^+V$ and $\p^+W$, which correspond to
generators of type ${I^{-0+}}$ and ${III^{-0+}}$ in
Figure <ref>. However, unlike in the case of (relative)
symplectic homology groups for a single cobordism, it is not possible
to arrange the parameters of the Hamiltonians in the cofinal family so
that for a fixed and finite value of the action window $(a,b)$ the
group $FH_*^{(a,b)}(H)$ takes into account only orbits of types
${I^{-0+}}$ and ${III^{-0+}}$. Instead, we will use in
<ref> below an indirect argument relying on the
confinement lemmas in <ref> and on the
properties of continuation maps in order to prove an isomorphism
between $SH_*^\heartsuit(W,V)$ and $SH_*^\heartsuit(W^{bottom},\p^-V)\oplus
SH_*^\heartsuit(W^{top},\p^+V)$ (Theorem <ref>). There
we will also see (Corollary <ref>) that
Definition <ref> is a special case of
Definition <ref> by taking for $V$ a tubular neighbourhood
of a union of boundary components $A$.
The following three results generalize the corresponding ones for a single cobordism.
Each of the above six versions of symplectic homology is an invariant
of the Liouville homotopy type of the triple $(W,V,F)$.
See Proposition <ref> below.
Let $\dim\, W=2n$. Then we have a canonical isomorphism
SH_*^{=0}(W,V)\cong H^{n-*}(W,V).
The proof of Proposition <ref> does not carry over to this
situation because Hamiltonians as in Figure <ref> may have
nonconstant orbits of action zero of type $II^-$. Instead, we combine
the Excision Theorem <ref> with
Proposition <ref> and excision in singular cohomology to
obtain canonical isomorphisms
\begin{align*}
&\cong SH_*^{=0}(W^{bottom},\p^-V) \oplus SH_*^{=0}(W^{top},\p^+V) \cr
&\cong H^{n-*}(W^{bottom},\p^-V) \oplus H^{n-*}(W^{top},\p^+V) \cr
&\cong H^{n-*}(W,V).
\end{align*}
The proof of the following proposition is verbatim the same as the one
of Proposition <ref>.
Recall to this effect that we are using field coefficients, and note that $SH_*^{(a,b)}(W,V)$ is finite dimensional for any choice of parameters $-\infty<a<b<\infty$. This holds because in the nondegenerate case there are only a finite number of closed Reeb orbits on $\p(\overline{W\setminus V})$ with action smaller than $\max(|a|,|b|)$, and only these orbits contribute to the relevant Floer complex for the cofinal family of Hamiltonians described in <ref>.
The following tautological exact triangles hold for the symplectic homology groups of a pair $(W,V)$:
\begin{equation*}
\scriptsize
\xymatrix
SH_*^{<0} \ar[rr] & &
SH_* \ar[dl] \\ & SH_*^{\ge 0} \ar[ul]^{[-1]}
\qquad
\xymatrix
SH_*^{\le 0} \ar[rr] & &
SH_* \ar[dl] \\ & SH_*^{> 0} \ar[ul]^{[-1]}
\end{equation*}
\begin{equation*}
\scriptsize
\xymatrix
SH_*^{< 0} \ar[rr] & &
SH_*^{\le 0} \ar[dl] \\ & SH_*^{= 0} \ar[ul]^{[-1]}
\qquad
\xymatrix
SH_*^{= 0} \ar[rr] & &
SH_*^{\ge 0} \ar[dl] \\ & SH_*^{> 0} \ar[ul]^{[-1]}
\end{equation*}
§.§ Pairs of multilevel Liouville cobordisms with filling
As mentioned in the Introduction, according to our conventions for pairs of Liouville cobordisms the symplectic homology group $SH_*(W,\p W)$ cannot be interpreted as $SH_*(W,[0,1]\times \p W)$ in case $\p W$ has both negative and positive components. We explain in this section a further extension of the setup which removes this limitation.
Let $\ell\ge 0$ be an integer. A Liouville cobordism with $\ell$ levels is, in case $\ell\ge 1$, a disjoint union $W=W_1\sqcup W_2\sqcup\dots\sqcup W_\ell$ of Liouville cobordisms, called levels, and is the empty set if $\ell=0$. We think of $W_1$ as being the “bottom-most" level, and of $W_\ell$ as being the “top-most" level. Each $W_i$ may itself be disconnected. Our previous definition of Liouville cobordisms corresponds to the case $\ell=1$. We also refer to such a $W$ as being a multilevel Liouville cobordism.
Let $V$ and $W$ be two Liouville cobordisms with $\ell$ levels. We say that $V$ and $W$ can be interweaved if $\p^+V_i=\p^-W_i$ for $i=1,\dots,\ell$ and $\p^+W_i=\p^-V_{i+1}$ for $i=1,\dots,\ell-1$. The interweaving of $V$ and $W$, denoted $V\diamond W$, is the Liouville cobordism with one level $V_1\circ W_1\circ \dots\circ V_\ell\circ W_\ell$. We allow in the definition the bottom-most or the top-most level of $V$ or $W$ to be empty, and in that case the condition for interweaving $V$ and $W$ which involves that level has to be understood as being void. In the case of cobordisms with one level, interweaving specialises to composition. See Figure <ref>.
Interweaving of two multilevel cobordisms.
Given a Liouville cobordism $W$ with $\ell\ge 1$ levels, a Liouville filling for $W$ is a Liouville cobordism with $\ell$ levels $F=F_1\sqcup\dots\sqcup F_\ell$ such that $F_1$ is a nonempty Liouville domain and $F$ and $W$ can be interweaved. In the case $\ell=1$, this notion specialises to our previous notion of a Liouville filling.
Given a Liouville cobordism $W$ with one level, a Liouville sub-cobordism $V\subset W$ is a codimension $0$ submanifold such that with respect to the induced Liouville form $V$ and $V^c=\overline{W\setminus V}$ are multilevel Liouville cobordisms that can be interweaved. If $V$ has only one level then $(W,V)$ is a Liouville pair in the sense of <ref>.
Given a multilevel Liouville cobordism $W$, a Liouville sub-cobordism $V\subset W$Ęconsists of a collection of (possibly empty) multilevel Liouville sub-cobordisms, one for each of the levels of $W$. We speak in such a situation of a pair of multilevel Liouville cobordisms. In case $W$ has a filling, we speak of a pair of multilevel Liouville cobordisms with filling.
Let $(W,V)$ be a pair of multilevel Liouville cobordisms with filling $F$. Denote $W_F=F\diamond W$ and consider the symplectization $\wh W_F$. We define the class
\cH(W,V;F)
of admissible Hamiltonians on $(W,V)$ with respect to the filling $F$ to consist of elements $H:S^1\times \wh W_F\to \R$ such that $H\in\cH(\wh W_F)$ and $H= 0$ on $W\setminus V$ (see Figure <ref>).
Given real numbers $-\infty<a<b<\infty$, we define the action-filtered symplectic homology groups of $(W,V)$ (with respect to the filling $F$) to be
\begin{equation} \label{eq:SH*abWV-multilevel}
\lim^{\longrightarrow}_{\scriptsize \begin{array}{c} H\in\cH(W,V;F) \\ H\to\infty \mbox{ on } (\wh W_F\setminus W) \end{array}}
\lim^{\longleftarrow}_{\scriptsize \begin{array}{c} H\in\cH(W,V;F) \\ H\to-\infty \mbox{ on } {\rm int}\,V \end{array}}
\end{equation}
We define six flavors of symplectic homology groups of the
multilevel Liouville pair $(W,V)$,
SH_*^\heartsuit(W,V), \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\},
by the formulas in Definition <ref> with $SH_*^{(a,b)}(W)$ replaced by $SH_*^{(a,b)}(W,V)$.
The notation $SH_*^\heartsuit$ with $\heartsuit=\varnothing$ refers to $SH_*$.
The above definition obviously specialises to Definition <ref> in case $W$ is a filled Liouville cobordism with one level.
Hamiltonian in $\mathcal{H}(W,V;F)$ for a multilevel cobordism.
Within the paper we state and prove all the results for pairs of one level Liouville cobordisms with filling. However, all these results hold more generally for pairs $(W,V)$ of multilevel Liouville cobordisms with filling. The formulation of these more general statements is verbatim the same. The proofs are only superficially more involved: a repeated application of the Excision Theorem <ref> allows one to restrict to the case where $W$ is a one level cobordism with filling, and the case of a multilevel sub-cobordism $V$ is treated in exactly the same way as that of a one level sub-cobordism. For these reasons, we will not give in the sequel any more details regarding multilevel Liouville cobordisms and will restrict to one level pairs.
§ COHOMOLOGY AND DUALITY
§.§ Symplectic cohomology for a pair of filled Liouville cobordisms
We continue with the notation of the previous section. Our definition of symplectic cohomology for a pair of filled Liouville cobordisms
extends the one for Liouville domains
used in <cit.>.
The starting point of the definition is the dualization of the Floer
chain complex with coefficient field $\mathfrak{k}$. We denote
FC^*_{>a}(H) =\prod_{\scriptsize \begin{array}{c} x\in\cP(H)\\
A_H(x)>a\end{array}} \mathfrak{k}\cdot x.
The grading is given by the Conley-Zehnder index, and the differential
$\delta : FC^k_{>a}(H)\to FC^{k+1}_{>a}(H)$ is defined by
\delta x_- = \sum_{CZ(x_+)=CZ(x_-)+1} \#\cM(x_-,x_+;H,J)\cdot x_+.
The differential increases the action, so that $FC^*_{>b}(H)\subset FC^*_{>a}(H)$ is a subcomplex if $a<b$. We define filtered Floer cochain groups
We have a natural identification
FC^*_{(a,b)}(H)\cong FC_*^{(a,b)}(H)^\vee,\qquad \delta = \p^\vee,
where $FC_*^{(a,b)}(H)^\vee = \Hom_R(FC_*^{(a,b)}(H),R)$.
We have natural morphisms at filtered cochain level defined by shifting the action window
FC^*_{(a',b')}(H)\to FC^*_{(a,b)}(H),\qquad a\le a',\ b\le b'.
These morphisms are dual to the ones defined on Floer chain groups. Also, given admissible Hamiltonians $H_-\ge H_+$ and a decreasing homotopy from $H_-$ to $H_+$, we have filtered continuation maps which commute with the differentials
FC^*_{(a,b)}(H_-)\to FC^*_{(a,b)}(H_+).
These continuation maps are dual to the ones defined on Floer chain groups, and commute with the morphisms defined by shifting the action window. The homotopy type of the continuation maps does not depend on the choice of decreasing homotopy with fixed endpoints.
Let $W$ be a Liouville cobordism with filling $F$, and let $A\subset
\p W$ be an admissible union of boundary components as
in <ref>. Recall also the notation $A^c=\p W\setminus A$
and $(\wh W_F\setminus W)_A={\rm int}\,F_{A^-}\cup ((1,\infty)\times A^+)$, and
recall also the class $\cH(W;F)$ of admissible Hamiltonians
from <ref>. Let $-\infty<a<b<\infty$ be real numbers. We
define the filtered symplectic cohomology groups of $W$
relative to $A$ (with respect to the filling $F$) to be
\begin{equation} \label{eq:cohSH*abWA}
SH^*_{(a,b)}(W,A)= \lim^{\longrightarrow}_{\scriptsize \begin{array}{c} H\in\cH(W;F) \\ H\to-\infty \mbox{ on } (\wh W_F\setminus W)_{A} \end{array}}
\lim^{\longleftarrow}_{\scriptsize \begin{array}{c} H\in\cH(W;F) \\ H\to\infty \mbox{ on } (\wh W_F\setminus W)_{A^c} \end{array}}
\end{equation}
The mnemotechnic rule is the same as in the case of symplectic homology:
To compute $SH^*_{(a,b)}(W,A)$ one must use a family of Hamiltonians that go to $-\infty$ near $A$ and that go to $+\infty$ near $\p W\setminus A$.
We define six flavors of symplectic cohomology groups of $W$ relative to $A$, or symplectic cohomology groups of the pair $(W,A)$,
SH^*_\heartsuit(W,A),\qquad \heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\},
by the following formulas (the notation $SH^*_\varnothing$ refers to $SH^*)$:
SH^*(W,A)=\lim^{\longrightarrow}_{a\to-\infty}\lim^{\longleftarrow}_{b\to \infty} SH^*_{(a,b)}(W,A) \qquad \mbox{\sc (full symplectic cohomology)}
SH^*_{<0}(W,A)= \lim^{\longrightarrow}_{a\to-\infty} \lim^{\longleftarrow}_{b\nearrow 0} SH^*_{(a,b)}(W,A) \qquad \mbox{\sc (negative symplectic cohomology)}
SH^*_{\le 0}(W,A)= \lim^{\longrightarrow}_{a\to-\infty} \lim^{\longrightarrow}_{b\searrow 0} SH^*_{(a,b)}(W,A) \qquad \mbox{\sc (non-positive symplectic cohomology)}
SH^*_{= 0}(W,A)= \lim^{\longleftarrow}_{a\nearrow 0} \lim^{\longrightarrow}_{b\searrow 0} SH^*_{(a,b)}(W,A) \qquad \mbox{\sc (zero-level symplectic cohomology)}
SH^*_{\ge 0}(W,A)= \lim^{\longleftarrow}_{a\nearrow 0} \lim^{\longleftarrow}_{b\to\infty} SH^*_{(a,b)}(W,A) \qquad \mbox{\sc (non-negative symplectic cohomology)}
SH^*_{> 0}(W,A)= \lim^{\longrightarrow}_{a\searrow 0} \lim^{\longleftarrow}_{b\to\infty} SH^*_{(a,b)}(W,A) \qquad \mbox{\sc (positive symplectic cohomology)}
Let now $(W,V)$ be a pair of Liouville cobordisms with filling $F$ as in <ref>, and recall the class $\cH(W,V;F)$ of admissible Hamiltonians for the pair $(W,V)$ with respect to the filling $F$. Let $-\infty<a<b<\infty$ be real numbers. We define the filtered symplectic cohomology groups of $(W,V)$ (with respect to the filling $F$) to be
\begin{equation} \label{eq:cohSH*abWV}
\lim^{\longrightarrow}_{\scriptsize \begin{array}{c} H\in\cH(W,V;F) \\ H\to-\infty \mbox{ on } V \end{array}}
\lim^{\longleftarrow}_{\scriptsize \begin{array}{c} H\in\cH(W,V;F) \\ H\to\infty \mbox{ on } (\wh W_F\setminus W) \end{array}}
\end{equation}
We define six flavors of symplectic cohomology groups of the
Liouville pair $(W,V)$,
SH^*_\heartsuit(W,V), \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\},
by the formulas in Definition <ref> with $SH^*_{(a,b)}(W,A)$ replaced by $SH^*_{(a,b)}(W,V)$. The notation $SH^*_\varnothing$ refers to $SH^*$.
The discussion from <ref> regarding the geometric content of the definition holds for cohomology as well. The following proposition is proved similarly to Proposition <ref>.
Let $(W,V)$ be a pair of Liouville cobordisms with filling of
dimension $2n$. Then we have a canonical isomorphism
SH^*_{=0}(W,V)\cong H_{n-*}(W,V).
§.§ Poincaré duality
The differences and the similarities between symplectic homology and symplectic cohomology are
mainly dictated by the order in which we consider direct and inverse
limits. We illustrate this by the following theorem, which was one of
our guidelines for the definitions.
Let $W$ be a filled Liouville cobordism and $A\subset \p W$ be an
admissible union of connected components. Then we have a canonical isomorphism
SH_*^\heartsuit(W,A)\cong SH^{-*}_{-\heartsuit}(W,A^c).
Here the symbol $\heartsuit$ takes the values $\varnothing,>0,\ge 0, =0, \le 0, <0$, and $-\heartsuit$ is by convention equal to $\varnothing,< 0,\le 0,=0,\ge 0,> 0$, respectively.
Given a time-dependent $1$-periodic Hamiltonian $H:S^1\times\wh W\to\R$ we denote $\bar H:S^1\times\wh W\to \R$, $\bar H(t,x)=-H(-t,x)$. Given a time-dependent $1$-periodic family of almost complex structures $J=(J_t)_{t\in S^1}$ on $\wh W$, we denote $\bar J=(\bar J_t)$, $t\in S^1$ with $\bar J_t=J_{-t}$. Given a loop $x:S^1\to \wh W$, we denote $\bar x:S^1\to \wh W$, $\bar x(t)=x(-t)$. Given a cylinder $u:\R\times S^1\to \wh W$, we denote $\bar u:\R\times S^1\to \wh W$, $\bar u(s,t)=u(-s,-t)$.
The key to the proof of Poincaré duality for symplectic homology is the canonical isomorphism, which will be also referred to as Poincaré duality,
\begin{equation} \label{eq:PDchainlevel}
FC_*^{(a,b)}(H,J)\cong FC^{-*}_{(-b,-a)}(\bar H,\bar J),
\end{equation}
obtained by mapping each $1$-periodic orbit
$x$ of $H$ to the $1$-periodic orbit of $\bar H$ given by the oppositely parameterized loop $\bar x$, and
each Floer cylinder $u$ for $(H,J)$ to the cylinder $\bar u$, which is a Floer cylinder for $(\bar H,\bar J)$. Note that the positive and negative punctures
get interchanged when passing from $u$ to $\bar u$, so that a chain complex is transformed into a cochain complex. It is straightforward that $A_{\bar H}(\bar x)=-A_H(x)$. It is less straightforward, but true, that $CZ(\bar x)=-CZ(x)$. The proof follows from <cit.>, taking into account that the flows of $\bar H$ and $H$ satisfy the relation $\varphi^t_{\bar H}=\varphi^{-t}_H$. We refer to <cit.> for a discussion of this Poincaré duality isomorphism in the context of autonomous Hamiltonians, and for a precise statement of its compatibility with continuation maps.
The isomorphism (<ref>) directly implies a canonical isomorphism
\begin{equation} \label{eq:PDSHab}
SH_*^{(a,b)}(W,A)\cong SH^{-*}_{(-b,-a)}(W,A^c).
\end{equation}
To see this, note that the class of admissible Hamiltonians $\cH(W;F)$
is stable under the involution $H\mapsto \bar H$. It follows that we can present $SH^{-*}_{(-b,-a)}(W,A^c)$ as a first-inverse-then-direct limit on $FH^{-*}_{(-b,-a)}(\bar H)$ for $H\in \cH(W;F)$, whereas $SH_*^{(a,b)}(W,A)$ is presented as a first-inverse-then-direct limit on $FH_*^{(a,b)}(H)$. In view of (<ref>) it is enough to see that the inverse and direct limits in the definitions are taken over the same sets. Indeed, for $SH_*^{(a,b)}(W,A)$ the inverse limit is taken over Hamiltonians $H$ that go to $-\infty$ on $(\wh W_F\setminus W)_A$, which is equivalent to $\bar H$ going to $\infty$ on $(\wh W_F\setminus W)_A$, and this is precisely the directed set for the inverse limit in the definition of $SH^{-*}_{(-b,-a)}(W,A^c)$. A similar discussion holds for the direct limit.
The isomorphisms $SH_*^\heartsuit(W,A)\cong SH^{-*}_{-\heartsuit}(W,A^c)$ follow from (<ref>) and from the definitions. We analyse the case $\heartsuit=``>0"$ and leave the other cases to the reader. In the definition of $SH_*^{>0}(W,A)$ the inverse limit is taken over $a\searrow 0$ and the direct limit is taken over $b\to\infty$, which is equivalent to $-a\nearrow 0$ and $-b\to-\infty$. After relabelling $(-b,-a)=(a',b')$, this is the same as $b'\nearrow 0$ and $a'\to-\infty$, which corresponds to the definition of $SH^{-*}_{<0}(W,A^c)$.
§.§ Algebraic duality and universal coefficients
We discuss in this section the algebraic duality between homology and
cohomology in the symplectic setting that we consider. Recall that we use field coefficients.
The starting observation is that, given a degree $k$, real numbers
$a<b$, admissible Hamiltonians $H\le H'$, an admissible decreasing
homotopy $(H_s)$, $s\in \R$ connecting $H'$ to $H$, and a regular
homotopy of almost complex structures $(J_s)$, $s\in \R$ connecting an
almost complex structure $J'$ which is regular for $H'$ to an almost
complex structure $J$ which is regular for $H$, there are canonical
FC^k_{(a,b)}(H,J)\cong FC_k^{(a,b)}(H,J)^\vee,\qquad \sigma^k\cong (\sigma_k)^\vee,
where $\sigma_k:FC_k^{(a,b)}(H,J)\to FC_k^{(a,b)}(H',J')$,
$\sigma^k:FC^k_{(a,b)}(H',J')\to FC^k_{(a,b)}(H,J)$ are the
continuation maps induced by the homotopy $(H_s,J_s)$. These
identifications follow from the definitions and hold with arbitrary
We now turn to the relationship between $SH_*^{(a,b)}(W,V)$ and
$SH^*_{(a,b)}(W,V)$. Since we work in a finite action window $(a,b)$,
both the direct and the inverse limits in the definition of
$SH_*^{(a,b)}(W,V)$ and $SH^*_{(a,b)}(W,V)$ eventually stabilize, so
that we can compute these groups using only one suitable
Hamiltonian. The universal coefficient theorem then implies with
coefficients in a field $\mathfrak{k}$ the existence of a canonical
isomorphism (see for example <cit.>)
\begin{equation} \label{eq:dualitykab}
SH^k_{(a,b)}(W,V;\mathfrak{k})\cong SH_k^{(a,b)}(W,V;\mathfrak{k})^\vee.
\end{equation}
The issue of comparing $SH^k_\heartsuit(W,V)$ and $SH_k^\heartsuit(W,V)$ becomes therefore a purely algebraic one, as it amounts to comparing via duality the various double limits involved in Definitions <ref> and <ref> (see also Definitions <ref> and <ref>). The key property is the following: given a direct system of modules $M_\alpha$ and a module $N$ over some ground ring $R$, the natural map
\begin{equation}\label{eq:Homlim}
\Hom_R(\lim_{\longrightarrow} M_\alpha,N)\stackrel \simeq \longrightarrow \lim_{\longleftarrow} \Hom_R(M_\alpha,N)
\end{equation}
is an isomorphism. However, it is generally not true that, given an inverse system $M_\alpha$, the natural map
\Hom_R(\lim_{\longleftarrow} M_\alpha,N)\longleftarrow \lim_{\longrightarrow} \Hom_R(M_\alpha,N)
is an isomorphism (the two sets actually have different cardinalities
in general). In our situation, $N=R$ is the coefficient field
We omit in the sequel the field $\frak{k}$ from the notation.
Let $(W,V)$ be a pair of Liouville cobordisms with filling. Using
field coefficients we have canonical isomorphisms
SH^k_\heartsuit(W,V)\cong SH_k^\heartsuit(W,V)^\vee,\qquad \heartsuit\in \{>0,\ge 0,=0\}
SH_k^\heartsuit(W,V)\cong SH^k_\heartsuit(W,V)^\vee,\qquad
\heartsuit\in \{<0,\le 0,=0\}.
Assume first $\heartsuit\in \{>0,\ge 0,=0\}$. In all three cases, the
limit over $a\to 0$ in the definition of $SH_*^\heartsuit(W,V)$ and
$SH^*_\heartsuit(W,V)$ stabilizes, and the result follows
from (<ref>) and (<ref>) applied to the limit $b\to\infty$.
Assume now $\heartsuit\in \{<0,\le 0,=0\}$. In all three cases the
limit over $b\to 0$ in the definition of $SH_*^\heartsuit(W,V)$ and
$SH^*_\heartsuit(W,V)$ stabilizes, and the result follows again
from (<ref>) applied to the limit $a\to-\infty$, by
rewriting (<ref>) as
SH_k^{(a,b)}(W,V)\cong SH^k_{(a,b)}(W,V)^\vee.
This holds because the vector spaces which are involved are finite dimensional.
(a) Let $(W,V)$ be a pair of filled Liouville cobordisms with vanishing first Chern class. Suppose
that $\p V$ and $\p W$ carry only finitely many closed Reeb orbits of
any given degree. Then with field coefficients we have for all flavors
$\heartsuit$ canonical isomorphisms
SH^k_\heartsuit(W,V)\cong SH_k^\heartsuit(W,V)^\vee\qquad\text{and}\qquad
SH_k^\heartsuit(W,V)\cong SH^k_\heartsuit(W,V)^\vee.
(b) Let $W$ be a Liouville domain. Then with field coefficients we have
canonical isomorphisms
SH^k(W)\cong SH_k(W)^\vee.
Part (a) follows from the proof as Proposition <ref>,
using that all inverse limits remain finite dimensional. Part (b)
holds because for a Liouville domain we have $SH_k(W)=SH_k^{\ge 0}(W)$.
Proposition <ref> illustrates the fact
that the full symplectic homology and cohomology groups of a cobordism
or of a pair of cobordisms are of a mixed homological-cohomological
nature. This is due to the presence of both a direct and of an
inverse limit in the definitions. As such, the full version
$SH_*(W,V)$ does not satisfy in general any form of algebraic
duality. In fact, in Example <ref> below we construct a
Liouville cobordism $W$ for which in some degree $k$ (and with
$\Z_2$-coefficients) neither $SH^k(W)\cong SH_k(W)^\vee$ nor
$SH_k(W)\cong SH^k(W)^\vee$ holds.
§ HOMOLOGICAL ALGEBRA AND MAPPING CONES
§.§ Cones and distinguished triangles
Let $R$ be a ring. Let Ch denote the category of chain complexes of $R$-modules. The objects of this category are chain complexes of $R$-modules, and the morphisms are chain maps of degree $0$. Let Kom denote the category of chain complexes of $R$-modules up to homotopy. The objects are the same as the ones of Ch, and the morphisms are equivalence classes of degree $0$ chain maps with respect to the equivalence relation given by homotopy equivalence. We use homological $\mathbb Z$-grading, and we use the following notational conventions :
* given a morphism $X\longrightarrow Y$ in Kom, we use the notation $X\stackrel f\longrightarrow Y$ for a specific representative $f$ of this morphism. Thus $f$ is a morphism in Ch.
* all diagrams are understood to be commutative in Kom. If we specify representatives in Ch for the morphisms, we say that a diagram is strictly commutative if it commutes in Ch.
* we use the notation
\xymatrix
{\ar @{} [dr] |s X \ar[r]^f \ar[d]_\varphi & Y \ar[d]^\psi \\
X' \ar[r]_g & Y'
for a diagram in Ch which is commutative modulo a specified homotopy $s$, i.e. such that $\psi f - g\varphi = s\p_X + \p_{Y'}s$. In particular, the diagram
\xymatrix
{X \ar[r]^f \ar[d]_\varphi & Y \ar[d]^\psi \\
X' \ar[r]_g & Y'
is commutative in Kom.
* given a chain complex $X=\{(X_n),\p_X\}$ and $k\in\mathbb Z$, we define the shifted complex $X[k]$ by
X[k]_n=X_{n+k},\quad n\in\mathbb Z,\qquad \p_{X[k]}=(-1)^k\p_X.
Given a morphism $f:X\to Y$, we define $f[k]:X[k]\to Y[k]$ as $f[k]=f$.
Our conventions for cones and distinguished triangles follow the ones of Kashiwara and Schapira <cit.>, except that we use dual homological grading. Given a chain map $f:X \to Y$, we define its cone to be the chain complex
C(f)=Y\oplus X[-1],\qquad \p_{C(f)}=\left(\begin{array}{cc} \p_{Y} & f \\ 0 & \p_{X[-1]}\end{array}\right) = \left(\begin{array}{cc} \p_Y & f \\ 0 & -\p_X\end{array}\right)
We have in particular a short exact sequence of chain complexes
\begin{equation} \label{les:cone}
\xymatrix
0\ar[r] & Y \ar[r]^-{\alpha(f)} & C(f) \ar[r]^-{\beta(f)} & X[-1] \ar[r] & 0
\end{equation}
where $\alpha(f)=\left(\begin{array}{c} \mathrm{Id}_{Y} \\ 0 \end{array}\right)$ is the canonical inclusion, and $\beta(f)=\left(\begin{array}{cc} 0 & \mathrm{Id}_{X[-1]} \end{array}\right)$ is the canonical projection. For simplicity we abbreviate in the sequel the identity maps by $1$, e.g. we write $\alpha(f)=\left(\begin{array}{c} 1 \\ 0 \end{array}\right)$ and
$\beta(f)=\left(\begin{array}{cc} 0 & 1 \end{array}\right)$.
One of the key features of the cone construction is that the connecting homomorphism in the homology long exact sequence associated to the short exact sequence (<ref>) is equal to $f_*$, the morphism induced by $f$.
By definition, a triangle in Kom is a sequence of morphisms
\begin{equation} \label{les:triangle}
\xymatrix
X \ar[r]^-f & Y \ar[r]^-g & Z \ar[r]^-h & X[-1]
\end{equation}
A distinguished triangle is a triangle which is isomorphic in Kom to a triangle of the form
\begin{equation} \label{les:model}
\xymatrix
X \ar[r]^f & Y \ar[r]^-{\alpha(f)} & C(f) \ar[r]^-{\beta(f)} & X[-1]
\end{equation}
We call (<ref>) a model distinguished triangle.
It follows from the definition that a distinguished triangle (<ref>) induces a long exact sequence in homology
\begin{equation}\label{eq:hlescone}
\xymatrix
\cdots H_*(X) \ar[r]^-{f_*} & H_*(Y) \ar[r]^-{g_*} & H_*(Z) \ar[r]^-{h_*} & H_{*-1}(X) \ar[r]^-{f_*} & \cdots
\end{equation}
We shall often represent such a long exact sequence as
\xymatrix
{H(X) \ar[rr]^{f_*} && H(Y) \ar[dl]^{g_*} \\
& H(Z) \ar[ul]^-{h_*}_-{[-1]} &
We call such a diagram an exact triangle.
The above definition of the class of distinguished triangles makes Kom into a triangulated category in the sense of Verdier. This means that the class of distinguished triangles satisfies Verdier's axioms (TR0)–(TR5) (see for example <cit.>). One of the essential axioms is (TR3): a triangle (<ref>) is distinguished if and only if the triangle
\xymatrix
Y \ar[r]^-g & Z \ar[r]^-h & X[-1] \ar[r]^-{-f[-1]} & Y[-1]
is distinguished. This follows from Lemma <ref>(i) below, see also <cit.>.
Let $f:X\to Y$ be a morphism in Ch.
(i) <cit.> There exists a morphism in Ch
\Phi : X[-1]\to C(\alpha(f))
which is an isomorphism in Kom, with an explicit homotopy inverse in Ch denoted
\Psi:C(\alpha(f))\to X[-1],
and such that the diagram below commutes in Kom:
\xymatrix
Y \ar[r]^-{\alpha(f)} \ar@{=}[d] & C(f) \ar[r]^-{\beta(f)} \ar@{=}[d] & X[-1] \ar[r]^-{-f[-1]} \ar@<.5ex>[d]^{\Phi} &
Y[-1] \ar@{=}[d] \\
Y \ar[r]_-{\alpha(f)} & C(f) \ar[r]_-{\alpha(\alpha(f))} & C(\alpha(f)) \ar[r]_-{\beta(\alpha(f))} \ar@<.5ex>[u]^{\Psi} & Y[-1]
(ii) There exists a morphism in Ch
\tau:Y[-1]\to C(\beta(f))
which is an isomorphism in Kom, with an explicit homotopy inverse in Ch denoted
\sigma:C(\beta(f))\to Y[-1],
and such that the diagram below commutes in Kom
\xymatrix
C(f) \ar[r]^-{\beta(f)} \ar@{=}[d] & X[-1] \ar[r]^-{-f[-1]} \ar@{=}[d] & Y[-1] \ar@<.5ex>[d]^\tau \ar[r]^-{-\alpha(f)[-1]} & C(f)[-1] \ar@{=}[d] \\
C(f) \ar[r]_-{\beta(f)} & X[-1] \ar[r]_-{\alpha(\beta(f))[-1]} & C(\beta(f)) \ar@<.5ex>[u]^{\sigma} \ar[r]_-{\beta(\beta(f))} & C(f)[-1]
(i) (following <cit.>) Taking into account that $C(\alpha(f))=Y\oplus X[-1]\oplus Y[-1]$, we define in matrix form
\Phi=\left(\begin{array}{c} 0 \\ 1 \\ -f \end{array}\right),\qquad \Psi=\left(\begin{array}{ccc} 0 & 1 & 0 \end{array}\right).
(Here $1$ stands for $\mathrm{Id}_{X[-1]}$ according to our convention.) A direct verification shows that these are chain maps, and also that the third square in the diagram commutes in Ch, i.e. $\beta(\alpha(f))\Phi=-f[-1]$. Such verifications formally amount to elementary multiplications of matrices. For example:
\p_{C(\alpha(f))}\Phi =
{ \left(\begin{array}{ccc} \p_Y & f & 1 \\ 0 & \p_{X[-1]} & 0 \\ 0 & 0 & \p_{Y[-1]} \end{array}\right)}
{\left(\begin{array}{c} 0 \\ 1 \\ -f \end{array}\right)}
{\left(\begin{array}{c} 0 \\ \p_{X[-1]} \\ -\p_{Y[-1]}f \end{array}\right)}
\beta(\alpha(f))\Phi = \left(\begin{array}{ccc} 0 & 0 & 1 \end{array}\right) \left(\begin{array}{c} 0 \\ 1 \\ -f \end{array}\right) = -f.
The second square in the diagram is commutative in Kom. Indeed, direct verification shows that $\Psi\alpha(\alpha(f))=\beta(f)$. On the other hand, the maps $\Phi$ and $\Psi$ are homotopy inverses to each other. Indeed, direct verification shows that $\Psi\Phi=\mathrm{Id}_{X[-1]}$ and
\mathrm{Id}_{C(\alpha(f))}-\Phi\Psi = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & f & 1 \end{array}\right) = \p_{C(\alpha(f))}K + K\p_{C(\alpha(f))},
where $K:C(\alpha(f))\to C(\alpha(f))[1]$ is a homotopy given in matrix form by
K=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right).
(ii) Taking into account that $C(\beta(f))=X[-1]\oplus Y[-1]\oplus X[-2]$ we define in matrix form
\tau=\left(\begin{array}{c} 0 \\ -1 \\ 0\end{array}\right),\qquad \sigma=\left(\begin{array}{ccc} -f & -1 & 0 \end{array}\right).
Here $1$ stands for $\mathrm{Id}_{Y[-1]}$. Direct verification shows that these are chain maps, that $\beta(\beta(f))\tau=-\alpha(f)[-1]$ so that the third square is commutative in Ch, and that $\sigma\alpha(\beta(f))=-f[-1]$.
Commutativity in Kom of the second square follows again from the fact that $\sigma$ and $\tau$ are homotopy inverses to each other. Indeed, we have $\sigma\tau=\mathrm{Id}_{Y[-1]}$, whereas
\mathrm{Id}_{C(\beta(f))}-\tau\sigma = \left(\begin{array}{ccc} 1 & 0 & 0 \\ -f & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) = \p_{C(\beta(f))} L + L \p_{C(\beta(f))},
where $L:C(\beta(f))\to C(\beta(f))[1]$ is a homotopy defined in matrix form by
L = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right).
One consequence of Lemma <ref> (i.e. axiom (TR3)) is that a triangle
\xymatrix
X \ar[r]^-f & Y \ar[r]^-g & Z \ar[r]^-h & X[-1]
is distinguished if and only if the triangle
\xymatrix
X[-1] \ar[r]^-{-f[-1]} & Y[-1] \ar[r]^-{-g[-1]} & Z[-1] \ar[r]^-{-h[-1]} & X[-2]
is distinguished. The triangle
\xymatrix
X[-1] \ar[r]^-{f[-1]} & Y[-1] \ar[r]^-{g[-1]} & Z[-1] \ar[r]^-{h[-1]} & X[-2]
is in general not distinguished, but rather anti-distinguished in the sense of <cit.>. The class of distinguished triangles is distinct from that of anti-distinguished triangles, as explained to us by S. Guillermou.
We use Lemma <ref> in order to replace by cones in Kom the kernels and cokernels of certain maps in Ch.
\begin{equation} \label{les:1}
0\longrightarrow A \stackrel i \longrightarrow B \stackrel p \longrightarrow C \longrightarrow 0
\end{equation}
be a short exact sequence in Ch which is split as a short exact sequence of $R$-modules.
(i) Given a splitting $s:C\to B$, i.e. a degree $0$ map such that $ps=\mathrm{Id}_C$, there is a canonical chain map $f:C[1]\to A$ and there are canonical identifications in Ch
B= C(f),\qquad i= \alpha(f),\qquad p= \beta(f).
The maps
\Phi:C\stackrel \simeq \longrightarrow C(i), \qquad \tau:A[-1]\stackrel \simeq \longrightarrow C(p)
defined in (i) and (ii) of Lemma <ref> are isomorphisms in Kom and they determine isomorphisms of distinguished triangles
\xymatrix
A \ar[r]^-i \ar@{=}[d] & B \ar[r]^-p \ar@{=}[d] & C \ar[r]^-{-f[-1]} \ar[d]^{\Phi}_\simeq & A[-1] \ar@{=}[d] \\
A \ar[r]_-i & B \ar[r]_-{\alpha(i)} & C(i) \ar[r]_-{\beta(i)} & A[-1]
\xymatrix
B \ar[r]^-p \ar@{=}[d] & C \ar[r]^-{-f[-1]} \ar@{=}[d] & A[-1] \ar[r]^-{-i[-1]} \ar[d]^\tau_\simeq & B[-1] \ar@{=}[d] \\
B \ar[r]_{p} & C \ar[r]_-{\alpha(p)} & C(p) \ar[r]_-{\beta(p)} & B[-1]
In particular, the homology long exact sequences determined by the top and bottom line in each of the above diagrams are isomorphic.
(iii) Assume that the splitting $s:C\to B$ is a chain map. We then have an isomorphism in Kom
A\stackrel \simeq \longrightarrow C(s).
(The same holds if we assume that the splitting $s$ is homotopic to a chain map.)
For item (i) let $(i\ s):C(f)=A\oplus C\stackrel{\cong}\to B$ be the
isomorphism of $R$-modules induced by $s$. Since $p(\p_Bs-s\p_C)=0$
and $i$ is injective, we can define $f:C[1]\to A$ uniquely by
$if=\p_Bs-s\p_C$ and one checks that this map has the desired properties.
Item (ii) is simply a rephrasal of Lemma <ref>.
Item (iii) is a consequence of (ii) as follows. Let us write $s=\left(\begin{array}{c} \varphi \\ 1 \end{array}\right)$ with $\varphi:C\to A$. Viewing $B$ as the cone of $f$ as in (i), the condition that $s$ is a chain map translates into $\varphi\p_C=\p_A\varphi +f$. (This in turn can be reinterpreted as saying that $-\varphi$ is a chain homotopy between $f$ and $0$.)
We consider the map $\pi:B=A\oplus C\to A$ given by $\pi=\left(\begin{array}{cc} 1 & -\varphi \end{array}\right)$. Then $\pi$ is a chain map and $\ker\pi = \im s$, so that we have a split short exact sequence
0\longrightarrow C \stackrel s \longrightarrow B \stackrel \pi \longrightarrow A\longrightarrow 0
and we conclude using the first assertion in (ii).
The class of chain maps is closed under homotopies: if $s$ is homotopic to a chain map, then it is an actual chain map.
Remark. It is not true that a short exact sequence of complexes $0\to A\stackrel{i}\longrightarrow B\stackrel p \longrightarrow C\to 0$ can always be completed to a distinguished triangle $A\stackrel i \longrightarrow B \stackrel p \longrightarrow C \longrightarrow A[-1]$. Thus the splitting assumption in Lemma <ref> is necessary. Indeed, consider the example of the short exact sequence of $\Z$-modules
\xymatrix{0 \ar[r] & \Z \ar[r]^i_{\times 2} & \Z \ar[r]^p & \Z/2\ar[r] & 0}
where $p$ is the canonical projection and $i$ is multiplication by $2$, thought of as an exact sequence of chain complexes supported in degree $0$. The cone of $i$ is equal to $\Z$ in degrees $0$ and $1$, with differential $\left(\begin{array}{cc} 0 & \times 2\\ 0 & 0 \end{array}\right)$. The map $\left(\begin{array}{cc} p & 0 \end{array}\right):C(i)\to \Z/2$ is a quasi-isomorphism, yet $\Z/2$ is not homotopy equivalent to $C(i)$ since the only morphism $\Z/2\to C(i)$ is the zero map. This shows that the above short exact sequence cannot be completed to a distinguished triangle.
\xymatrix
{X \ar[r]^f \ar[d]_\varphi & Y \ar[d]^\psi \\
X' \ar[r]_g & Y'
be a commutative diagram in Kom.
This can be completed to a diagram whose rows and columns are distinguished triangles in
Kom and in which all squares are commutative (in Kom), except the bottom right square which is anti-commutative
\xymatrix
{X \ar[r]^f \ar[d]_\varphi & Y \ar[r] \ar[d]_\psi & Z \ar[r] \ar[d]_\chi & X[-1] \ar[d] \\
X' \ar[r]^g \ar[d] & Y' \ar[r] \ar[d] & Z' \ar[r] \ar[d] & X'[-1] \ar[d] \\
X'' \ar[r]^h \ar[d] & Y'' \ar[r] \ar[d] & Z'' \ar @{} [dr] |{-} \ar[r] \ar[d] & X''[-1] \ar[d] \\
X[-1] \ar[r] & Y[-1] \ar[r] & Z[-1] \ar[r] & X[-2]
This statement, attributed to Verdier, is proved in Beilinson, Bernstein, Deligne <cit.> by a repeated use of the octahedron axiom (TR5). This is also proved in <cit.> under the name “$3\times 3$ Lemma", where it is shown that it is actually equivalent to the octahedron axiom. The same statement appears as Exercise 10.2.6 in <cit.>. Our proof is more explicit and produces a diagram in which all the squares except the initial one and the bottom right one are commutative in Ch, and in which the bottom right square is anti-commutative in Ch. This result encompasses <cit.> and <cit.>. For completeness, we will reprove <cit.> as Lemma <ref> below as a consequence of Proposition <ref> (under an additional splitting assumption).
We start with the square
\xymatrix
{\ar @{} [dr] |s X \ar[r]^f \ar[d]_\varphi & Y \ar[d]^\psi \\
X' \ar[r]_g & Y'
which is commutative modulo the homotopy $s$, meaning in our notation that
\begin{equation} \label{eq:homotopy-s}
\psi f - g\varphi = s\p_X + \p_{Y'}s.
\end{equation}
We construct the grid diagram in the statement by a repeated use of the cone construction.
The first two lines and the first two columns are constructed as model distinguished triangles. More precisely, we define
Z=C(f)=Y\oplus X[-1],\qquad Z'=C(g)=Y'\oplus X'[-1], \qquad \chi=\left(\begin{array}{cc} \psi & s \\ 0 & \varphi \end{array}\right).
The condition that $\chi$ is a chain map is equivalent to equation (<ref>), and the second and third square formed by the first two lines are then commutative in Ch:
\xymatrix
{X \ar[r]^f \ar[d]_\varphi & Y \ar[r]^-{\alpha(f)} \ar[d]_\psi & Z \ar[r]^-{\beta(f)} \ar[d]_\chi & X[-1] \ar[d]_{\varphi[-1]} \\
X' \ar[r]^g & Y' \ar[r]^-{\alpha(g)} & Z' \ar[r]^-{\beta(g)} & X'[-1]
Similarly, we define
X''=C(\varphi)=X'\oplus X[-1],\qquad Y''=C(\psi)=Y'\oplus Y[-1],\qquad h= \left(\begin{array}{cc} g & -s \\ 0 & f \end{array}\right).
Again, the condition that $h$ is a chain map is equivalent to equation (<ref>) and the first two columns determine a diagram in which the second and third square are commutative in Ch:
\xymatrix
{X \ar[r]^f \ar[d]_\varphi & Y \ar[d]_\psi \\
X' \ar[r]^g \ar[d]_-{\alpha(\varphi)} & Y' \ar[d]_-{\alpha(\psi)} \\
X'' \ar[r]^h \ar[d]_-{\beta(\varphi)} & Y'' \ar[d]_-{\beta(\psi)} \\
X[-1] \ar[r]^{f[-1]} & Y[-1]
We define
We construct the third and fourth columns of the grid diagram as model distinguished triangles, and we are left to specify the morphisms $A,B,C,D$ below:
\xymatrix
{X \ar[r]^f \ar[d]_\varphi & Y \ar[r]^-{\alpha(f)} \ar[d]_\psi & C(f) \ar[r]^-{\beta(f)} \ar[d]_\chi & X[-1] \ar[d]^-{\varphi[-1]} \\
X' \ar[r]^g \ar[d]_-{\alpha(\varphi)} & Y' \ar[r]^-{\alpha(g)} \ar[d]_-{\alpha(\psi)} & C(g) \ar[r]^-{\beta(g)} \ar[d]_-{\alpha(\chi)} & X'[-1] \ar[d]^-{\alpha(\varphi[-1])} \\
C(\varphi) \ar[r]^h \ar[d]_-{\beta(\varphi)} & C(\psi) \ar@{.>}[r]^{A} \ar[d]_-{\beta(\psi)} & C(\chi) \ar@{.>}[r]^{B} \ar[d]_-{\beta(\chi)} & C(\varphi[-1]) \ar[d]^-{\beta(\varphi[-1])} \\
X[-1] \ar[r]^-{f[-1]} & Y[-1] \ar@{.>}[r]^{C} & C(f)[-1] \ar@{.>}[r]^{D} & X[-2]
The key point is that we have isomorphisms of chain complexes
\xymatrix
I:C(\chi) \ar[r]^-{\simeq} \ar@{=}[d] & C(h), \ar@{=}[d] \\
Y'\oplus X'[-1]\oplus Y[-1]\oplus X[-2] & Y'\oplus Y[-1] \oplus X'[-1] \oplus X[-2]
I:=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1
\end{array}\right)
\begin{equation} \label{eq:Jf}
\xymatrix
J(f):C(f)[-1] \ar[r]^-{\simeq} \ar@{=}[d] & C(f[-1]), \ar@{=}[d] \\
Y[-1]\oplus X[-2] & Y[-1]\oplus X[-2]
\end{equation}
J(f):=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right).
One checks directly that the maps $I$ and $J(f)$ commute with the differentials.
The third line in our diagram, involving the maps $A$ and $B$, is defined using the isomorphisms $I$ and $J(\varphi)$ from the model distinguished triangle associated to $h$, i.e. $A=I^{-1}\alpha(h)$, $B=J(\varphi)\beta(h)I$:
\xymatrix
{C(\varphi) \ar[r]^h \ar@{=}[d] & C(\psi) \ar@{.>}[r]^-{A} \ar@{=}[d] & C(\chi) \ar@{.>}[r]^-{B} \ar[d]_-{I}^-{\simeq} & C(\varphi[-1]) \\
C(\varphi) \ar[r]^h & C(\psi) \ar[r]^-{\alpha(h)} & C(h) \ar[r]^-{\beta(h)} & C(\varphi)[-1] \ar[u]^-{J(\varphi)}_-{\simeq}
In matrix form we have
A=\left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right),\qquad
B=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right).
The fourth line in our diagram, involving the maps $C$ and $D$, is defined using the isomorphism $J(f)$ from the model distinguished triangle associated to $f[-1]$, i.e. $C=J(f)^{-1}\alpha(f[-1])$, $D=\beta(f[-1])J(f)$:
\xymatrix
{X[-1] \ar[r]^-{f[-1]} \ar@{=}[d] & Y[-1] \ar@{.>}[r]^-{C} \ar@{=}[d] & C(f)[-1] \ar@{.>}[r]^-{D} \ar[d]_-{J(f)}^-{\simeq} & X[-2] \ar@{=}[d] \\
X[-1] \ar[r]^-{f[-1]} & Y[-1] \ar[r]^-{\alpha(f[-1])} & C(f[-1]) \ar[r]^-{\beta(f[-1])} & X[-2]
In matrix form we have
C=\left(\begin{array}{c} 1 \\ 0 \end{array}\right),\qquad D=\left(\begin{array}{cc} 0 & -1 \end{array}\right).
A direct check shows that
A\alpha(\psi)=\alpha(\chi)\alpha(g),\qquad B\alpha(\chi)=\alpha(\varphi[-1])\beta(g),\qquad
For later use, we recall Lemma 5.7 from <cit.> and show how it follows from Proposition <ref> under an additional assumption.
\begin{equation} \label{eq:fgh}
\xymatrix{
0\ar[r] & A \ar[r]^i \ar[d]^f & B \ar[r]^p \ar[d]^g & C \ar[r]
\ar[d]^h & 0 \\
0\ar[r] & A' \ar[r]^{i'} & B' \ar[r]^{p'} & C' \ar[r]
& 0
\end{equation}
be a morphism of short exact sequences of complexes.
We then have a diagram whose rows and columns are exact and in which all squares are commutative, except the bottom right one which is anti-commutative.
\xymatrix
{H_*(A) \ar[r]^{i_*} \ar[d]^{f_*} & H_*(B) \ar[r]^{p_*} \ar[d]^{g_*} & H_*(C) \ar[r] \ar[d]^{h_*} & H_{*-1}(A) \ar[d]^{f_*} \\
H_*(A') \ar[r]^{i'_*} \ar[d]^{\alpha(f)_*} & H_*(B') \ar[r]^{p'_*} \ar[d]^{\alpha(g)_*} & H_*(C') \ar[r] \ar[d]^{\alpha(h)_*} & H_{*-1}(A') \ar[d]^{\alpha(f)_*} \\
H_*(C(f)) \ar[r] \ar[d]^{\beta(f)_*} & H_*(C(g)) \ar[r] \ar[d]^{\beta(g)_*} & H_*(C(h)) \ar @{} [dr] |{-} \ar[r] \ar[d]^{\beta(h)_*} & H_{*-1}(C(f)) \ar[d]^{\beta(f)_*} \\
H_{*-1}(A) \ar[r]^{i_*} & H_{*-1}(B) \ar[r]^{p_*} & H_{*-1}(C) \ar[r] & H_{*-2}(A)
Up to changes in notation, this is exactly Lemma 5.7 in <cit.>.
To wrap up the story, we show here how this result follows from Proposition <ref> under the additional assumption that the short exact sequences are split as sequences of $R$-modules (this is always the case if $R$ is field or, more generally, if we work with chain complexes of free $R$-modules).
Choose splittings $s:C\to B$ and $s':C'\to B'$. By Lemma <ref>, these determine canonical chain maps $\varphi:C[1]\to A$ and $\varphi':C'[1]\to A'$, together with canonical identifications $B=C(\varphi)$, $i=\alpha(\varphi)$, $p=\beta(\varphi)$, $B'=C(\varphi')$, $i'=\alpha(\varphi')$, $p'=\beta(\varphi')$.
The map $g:B\to B'$ can then be identified with a map $C(\varphi)\to C(\varphi')$ written in matrix form as
g=\left(\begin{array}{cc}Ęf & t \\Ę0 & h \end{array}\right) : A\oplus C \to A'\oplus C'.
The condition that $g$ is a chain map is then equivalent to the three relations
f\p_A = \p_{A'} f,\qquad h\p_C = \p_{C'}h,\qquad f\varphi - \varphi'h=\p_{A'} t - t\p_C.
We interpret the last relation as $f\varphi-\varphi'h[1]=\p_{A'}t+t\p_{C[1]}$, which means that the square
\begin{equation}Ę\label{eq:square-t}
\xymatrix
{\ar @{} [dr] |t C[1] \ar[r]^\varphi \ar[d]_{h[1]} & A \ar[d]^f \\
C'[1] \ar[r]_{\varphi'} & A'
\end{equation}
is commutative up to a homotopy given by $t:C\to A'$. The initial diagram (<ref>) appears then as the horizontal extension of this commutative square in Kom to a map of distinguished triangles.
We now apply Proposition <ref> to the square (<ref>) in order to obtain the grid diagram
\xymatrix
{\ar @{} [dr] |t C[1] \ar[r]^\varphi \ar[d]_{h[1]} & A \ar[r]^i \ar[d]_f & B \ar[r]^p \ar[d]_g & C \ar[d]_h \\
C'[1] \ar[r]^{\varphi'} \ar[d] & A' \ar[r]^{i'} \ar[d] & B' \ar[r]^{p'} \ar[d] & C' \ar[d] \\
C(h[1]) \ar[r] \ar[d] & C(f) \ar[r] \ar[d] & C(g) \ar @{} [dr] |{-} \ar[r] \ar[d] & C(h) \ar[d] \\
C \ar[r] & A[-1] \ar[r] & B[-1] \ar[r] & C[-1]
The anti-commutativity of the bottom right corner can be traded for anti-commutativity of the bottom left corner by changing the sign of the two bottom middle vertical arrows. The grid diagram in the statement of the lemma is then obtained by passing to homology.
§.§ Uniqueness of the cone
We now spell out what is the additional piece of structure that is needed in order for the cone of a map to be uniquely and canonically defined up to homotopy.
(i) $\operatorname{Hom}$ complexes. Let $X,Y$ be chain complexes of $R$-modules and denote
\operatorname{Hom}_d(X,Y),\qquad d\in \Z
the $R$-module of $R$-linear maps of degree $d$. This is a chain complex with differential
\p:\operatorname{Hom}_d(X,Y)\to \operatorname{Hom}_{d-1}(X,Y),
\p\Phi=\p_Y\Phi - (-1)^{|\Phi|}\Phi\p_X,
where $|\Phi|=d$ denotes the degree of a map $\Phi\in\operatorname{Hom}_d(X,Y)$. The space of degree $d$ cycles
Z_d(X,Y)=\ker(\p:\operatorname{Hom}_d(X,Y)\to \operatorname{Hom}_{d-1}(X,Y))
is the space of degree $d$ chain maps $X\to Y$. Two degree $d$ chain maps are homologous, i.e. they differ by an element of
if and only if they are chain homotopic.
Remark/Notation. We denote a degree $d$ map $f$ from $X$ to $Y$ by
f:X\stackrel d\longrightarrow Y.
We do not use the notation $f:X\to Y[d]$, which we reserve for chain maps. This distinction is relevant in practice when using cones because the differential of the complex $Y[d]$ is not $\p_Y$, but $(-1)^d\p_Y$.
(ii) Chain maps between cones. Let
\xymatrix
{\ar @{} [dr] |s X \ar[r]^f \ar[d]_\varphi & Y \ar[d]^\psi \\
X' \ar[r]_g & Y'
be a diagram of degree $0$ chain maps which is commutative modulo a prescribed degree $1$ homotopy $s\in\operatorname{Hom}_1(X,Y')$, meaning that $\psi f - g\varphi = \p(s)$.
We have an induced chain map
\xymatrix
\chi_s={\left(\begin{array}{cc} \psi & s \\ 0 & \varphi[-1] \end{array}\right)} \qquad : & C(f)\ar[rrr]\ar@{=}[d] & & & C(g). \ar@{=}[d] \\
& Y\oplus X[-1] & & & Y'\oplus X'[-1]
The homotopy class of the map $\chi_s$ depends only on the equivalence class of the homotopy $s$ modulo $B_1(X,Y')$. Indeed, if $t\in\operatorname{Hom}_1(X,Y')$ is another map such that $\psi f - g\varphi = \p(t)$ then $s-t\in Z_1(X,Y')$. If
$s-t\in B_1(X,Y')$, meaning that
with $b\in\operatorname{Hom}_2(X,Y')$, then
\chi_s-\chi_t=\p\left(\begin{array}{cc} 0 & b \\ 0 & 0 \end{array}\right)\in B_0(C(f),C(g)),
meaning that $\chi_s$ and $\chi_t$ are chain homotopic.
(iii) Lifts of $B_0$ modulo $B_1$. Denote $B_1=B_1(X,Y)$, $Z_1=Z_1(X,Y)$, $\operatorname{Hom}_1=\operatorname{Hom}_1(X,Y)$. Let
V_1\subset \operatorname{Hom}_1
be a subspace such that $V_1\cap Z_1=B_1$ and $V_1+Z_1=\operatorname{Hom}_1$. Equivalently, $B_1\subset V_1$ is a subspace and $\p$ induces an isomorphism $V_1/B_1\stackrel\simeq\rightarrow B_0$. We call $V_1$ a linear lift of $B_0$ modulo $B_1$.
Let such a linear lift $V_1\subset \operatorname{Hom}_1(X,Y)$ be given. Given two homotopic maps $f,g\in \operatorname{Hom}_0(X,Y)$, i.e. $f-g=\p(s)$, we can assume without loss of generality that $s\in V_1$. The map $s$ is uniquely defined modulo $B_1$, which implies that the homotopy class of the map $\chi_s:C(f)\to C(g)$ is well-defined.
Thus, given a lift $V_1\subset \operatorname{Hom}_1(X,Y)$, the cone of any map $X\to Y$ is uniquely defined in Kom.
§.§ Directed, bi-directed, and doubly directed systems
We now explain a setup in which one can speak of limits of ordered systems of mapping cones. The motivation for the definitions to follow lies in the definition of symplectic homology as a direct/inverse limit over directed systems in which the morphisms are Floer continuation maps in Floer homology. To this effect, the reader my find it useful to refer to <ref> and <ref>.
We begin with a few definitions.
A directed set is a partially ordered set $(I,\prec)$ such that for any $i,j$ there exists $k$ with $i,j\prec k$. An inversely directed set is a partially ordered set $(I,\prec)$ such that for any $i,j$ there exists $\ell$ with $\ell\prec i,j$. Equivalently, we require that $I$ with the opposite order be a directed set. A bi-directed set is a partially ordered set $(I,\prec)$ which is both directed and inversely directed. Our typical example is $I=\R$.
A system in Kom indexed by $I$ is a collection of chain complexes $X(i)$, $i\in I$ together with chain maps $\varphi_i^j:X(i)\to X(j)$, $i\prec j$ such that
$\varphi_j^k \varphi_i^j=\varphi_i^k$ for $i\prec j\prec k$ and $\varphi_i^i=\mathrm{Id}_{X(i)}$ in Kom. More precisely, there exist maps $x_{ijk}\in \mathrm{Hom}_1(X(i),X(k))$, $i\prec j \prec k$ and $x_i\in \mathrm{Hom}_1(X(i),X(i))$ such that
\varphi_i^k-\varphi_j^k \varphi_i^j=\p(x_{ijk}),\qquad \mathrm{Id}_{X(i)}-\varphi_i^i=\p(x_i).
We speak of a directed system, of an inversely directed system, and of a bi-directed system if $(I,\prec)$ is a directed set, an inversely directed set, respectively a bi-directed set. We call the maps $\varphi_i^j$ structure maps.
More generally, let $(I^+,\prec)$ be a directed set and $(I^-,\prec)$
be an inversely directed set. A doubly directed set modelled on
$I^\pm$ is a subset $I\subset I^-\times I^+$ with the following two
* if $(i,j)\in I$ then $(i',j)\in I$ for all $i'\prec i$ and
$(i,j')\in I$ for all $j'\prec j$;
* for every $j\in I^+$ there exists $i\in I^-$ such that $(i,j)\in
Our typical example is $I^\pm=\R_\pm^*$ and $I=\{(a,b)\in
\R_-^*\times\R_+^*\, : \, a\le f(b)\}$, where $f:\R_+^*\to \R_-^*$ is
a decreasing function such that $f(b)\to-\infty$ as $b\to\infty$.
A doubly directed system in Kom indexed by the doubly directed set $I$ is a collection of chain complexes $X(i,j)$, $(i,j)\in I$ together with chain maps $\varphi_{i'j}^{ij}:X(i',j)\to X(i,j)$ for $i'\prec i$ and $\varphi_{ij}^{ij'}:X(i,j)\to X(i,j')$ for $j\prec j'$ with respect to which every $X(i,\cdot)$ is a directed system and every $X(\cdot,j)$ is an inversely directed system, and such that all diagrams
\begin{equation} \label{eq:diag-doubly-directed}
\xymatrix{
X(i',j) \ar[r] \ar[d] & X(i,j) \ar[d] \\
X(i',j') \ar[r] & X(i,j')
\end{equation}
are commutative in Kom, for any choice of indices such that $i'\prec i$, $j\prec j'$ and $(i,j),(i',j),(i,j'),(i',j')\in I$. We call the maps $\varphi_{i'j}^{ij}$ and $\varphi_{ij}^{ij'}$ structure maps.
Given a map of bi-directed systems or a map of doubly directed systems, which means a collection of chain maps indexed by the relevant indexing set which commute in Kom with the chain maps defining each of the systems, we are interested in understanding conditions under which the cone of that map is itself a bi-directed, respectively a doubly directed system. The two situations are similar, except for more cumbersome notation in the case of doubly directed systems since we need to work with two indexing variables $(i,j)$ rather than with just one index variable $i$. For this reason we shall focus in the sequel on bi-directed systems and indicate how the discussion adapts to doubly directed systems.
Let $\{X(i),\varphi_i^j\}$, $\{Y(i),\psi_i^j\}$ be two bi-directed systems in Kom with the same index set $I$. A map of bi-directed systems in Kom is a collection of chain maps $f_i:X(i)\to Y(i)$, $i\in I$ such that $\psi_i^jf_i$ and $f_j\varphi_i^j$ are homotopic for all $i\prec j$. Given $s_i^j\in\operatorname{Hom}_1(X(i),Y(j))$, $i\prec j$ such that $\psi_i^jf_i-f_j\varphi_i^j=\p(s_i^j)$, denote $\chi_i^j=\chi_{s_i^j}$. We then have a commutative diagram
\xymatrix
{\ar @{} [dr] |{s_i^j} X(i) \ar[r]^{f_i} \ar[d]_{\varphi_i^j} & Y(i) \ar[d]^{\psi_i^j} \ar[r] & C(f_i) \ar[d]^{\chi_i^j} \ar[r] & X(i)[-1] \ar[d] \\
X(j) \ar[r]_{f_j} & Y(j) \ar[r] & C(f_j) \ar[r] & X(j)[-1]
We are interested in finding conditions under which $\{C(f_i),\chi_i^j\}$ is a bi-directed system in Kom.
Let us consider the following condition:
(B) There exists a collection $\{b_{ijk}\}$, $i\prec j \prec k$ with $b_{ijk}\in\mathrm{Hom}_1(X(i),Y(k))$ such that
s_i^k-\psi_j^ks_i^j-s_j^k\varphi_i^j+f_k x_{ijk} - y_{ijk}f_i=\p(b_{ijk}),\qquad i,j,k.
Here it is understood that $\{x_{ijk}\}$, $\{y_{ijk}\}$ and $\{s_i^j\}$ are given as above.
A direct computation then shows that
\chi_i^k-\chi_j^k\chi_i^j=\p\left(\begin{array}{cc} y_{ijk} & b_{ijk} \\ 0 & -x_{ijk} \end{array}\right), \qquad i,j,k.
Indeed, the off-diagonal term on the left hand side is $s_i^k-\psi_j^ks_i^j-s_j^k\varphi_i^j$, while the off-diagonal term on the right hand side is $\p(b_{ijk})-f_k x_{ijk} + y_{ijk}f_i$.
Remark. Condition (B) is motivated both by the outcome of preliminary computations for bi-directed systems in Ch and by the example of Floer continuation maps discussed below.
Condition (B) is clearly independent of the choice of $\{s_i^j\}$, $\{x_{ijk}\}$, and $\{y_{ijk}\}$ up to homotopy. This motivates the following stronger condition, of a more intrinsic nature:
(C) We are given the data of collections of lifts of $B_0$ mod $B_1$:
\{X_i^j\subset \operatorname{Hom}_1(X(i),X(j))\},\qquad i\prec j,
\{Y_i^j\subset \operatorname{Hom}_1(Y(i),Y(j))\},\qquad i\prec j,
\{V_i^j\subset \operatorname{Hom}_1(X(i),Y(j))\},\qquad i\prec j
such that $(\psi_{j}^k)_*V_i^j\subset V_i^k$, $(\varphi_i^{j})^*V_j^k\subset V_i^k$, $(f_k)_*X_i^k\subset V_i^k$, and $(f_i)^*Y_i^k\subset V_i^k$.
We claim that
(C) \Longrightarrow (B).
For the proof we start by choosing $s_i^j\in V_i^j$, $x_{ijk}\in X_i^k$, $y_{ijk}\in Y_i^k$. We then remark that $-y_{ijk}f_i+s_i^k+f_kx_{ijk}$ and $\psi_j^ks_i^j+s_j^k\varphi_i^j$ are both contracting homotopies for $\psi_j^k\psi_i^jf_i-f_k\varphi_j^k\varphi_i^j$, so that their difference is a cycle. Now condition (C) implies that both these homotopies lie in $V_i^k$, which implies that their difference is a boundary $\p(b_{ijk})$.
Condition (B) implies that $\{C(f_i),\chi_i^j\}$ is a bi-directed system in Kom. The same holds in particular under condition (C).
We now indicate how the discussion adapts to the case of a map $\{f_{ij}:X(i,j)\to Y(i,j)\}$ between doubly directed systems indexed by the same doubly directed set $I$. Denote $\varphi_{i'j}^{ij},\varphi_{ij}^{ij'}$ the structure maps for $\{X(i,j)\}$, and denote $\psi_{i'j}^{ij},\psi_{ij}^{ij'}$ the structure maps for $\{Y(i,j)\}$. Denote $\sigma_{i'j}^{ij'}$, $\tau_{i'j}^{ij'}$ the homotopies that express the commutativity in Kom of the diagrams (<ref>):
\varphi_{ij}^{ij'}\varphi_{i'j}^{ij} - \varphi_{i'j'}^{ij'}\varphi_{i'j}^{i'j'}=\p(\sigma_{i'j}^{ij'}),\qquad
\psi_{ij}^{ij'}\psi_{i'j}^{ij} - \psi_{i'j'}^{ij'}\psi_{i'j}^{i'j'}=\p(\tau_{i'j}^{ij'}).
Denote $s_{i'j}^{ij}$ and $s_{ij}^{ij'}$ the homotopies that express the fact that $f_{\cdot j}$ and $f_{i\cdot}$ are maps of directed systems.
The analogue of condition (B) for doubly-directed systems is the following:
($\tilde{\mbox{B}}$) We require condition $\mathrm{(B)}$ to hold for each of the maps of directed systems
$f_{i\cdot}$ and $f_{\cdot j}$, and in addition we require that there exists a collection $\{B_{i'j}^{ij'}\}$ with $B_{i'j}^{ij'}\in \mathrm{Hom}_1(X(i',j),Y(i,j'))$ such that
\psi_{ij}^{ij'}s_{i'j}^{ij} + s_{ij}^{ij'}\varphi_{i'j}^{ij} - \psi_{i'j'}^{ij'}s_{i'j}^{i'j'}-s_{i'j'}^{ij'}\varphi_{i'j}^{i'j'} + f_{ij'}\sigma_{i'j}^{ij'}-\tau_{i'j}^{ij'}f_{i'j}=\p(B_{i'j}^{ij'}).
Similarly to the case of bi-directed systems, a direct computation shows that
\chi_{ij}^{ij'}\chi_{i'j}^{ij}-\chi_{i'j'}^{ij'}\chi_{i'j}^{i'j'}=\p\left(\begin{array}{cc} \tau_{i'j}^{ij'} & B_{i'j}^{ij'} \\ 0 & -\sigma_{i'j}^{ij'}\end{array}\right),
where $\chi_{ab}^{cd}:C(f_{ab})\to C(f_{cd})$ are the maps induced between cones, as before. It is important to note that condition ($\tilde{\mbox{B}}$) is of the same nature as condition (B), and the only difference between the two is that condition ($\tilde{\mbox{B}}$) takes into account the additional conditions of commutativity up to homotopy which are involved in the definition of a doubly directed system.
One can also phrase for doubly directed systems an analogue ($\tilde{\mbox{C}}$) of condition (C) for bi-directed systems, but we shall not need it and therefore we do not make it explicit.
Limiting objects. Let now the coefficient ring be a field $\mathfrak{k}$, and recall <cit.> that the inverse limit functor is exact on inversely directed systems consisting of finite dimensional vector spaces. Let $\{f_{ij}:X(i,j)\to Y(i,j)\}$ be a map of doubly directed systems, and assume that each $X(i,j)$ and $Y(i,j)$ has finite dimensional homology in each degree. Under condition ($\tilde{\mbox{B}}$) we obtain in the first-inverse-then-direct-limit a homology exact triangle
\xymatrix
{\lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i} H(X(i,j)) \ar[rr]^{\lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i} (f_{ij})_*} && \lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i} H(Y(i,j)) \ar[dl] \\
& \lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i} H(C(f_{ij})) \ar[ul]_-{[-1]} &
Remark. The following question is relevant. When is
\lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i}X(i,j) \longrightarrow
\lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i}Y(i,j) \longrightarrow
\lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i} C(f_{ij}) \longrightarrow
\lim\limits^{\longrightarrow}_{j}\lim\limits^{\longleftarrow}_{i} X(i,j)[-1]
a (model) distinguished triangle? This is related to exactness criteria for the inverse limit functor and to the so-called Mittag-Leffler condition, see for example <cit.> and the references therein.
§.§ Floer continuation maps
We now show how condition ($\tilde{\mbox{B}}$) above is satisfied in the case of Floer continuation maps for a doubly directed system of Hamiltonians. In order to streamline the discussion we shall actually treat the case of a directed system of Hamiltonians, the case of doubly directed systems being conceptually equivalent, except for the more complicated notation.
Higher continuation maps. Let $K\le L$ be two Hamiltonians and let $(FC(K),\p_K)$, $(FC(L),\p_L)$ be the Floer complexes for some choice of regular almost complex structures $J_K$ and $J_L$. An $s$-dependent Hamiltonian $H=H_s$, $s\in \R$ such that $H_s=L$ for $s\ll 0$, $H_s=K$ for $s\gg 0$, and $\p_sH\le 0$, together with an $s$-dependent almost complex structure interpolating between $J_L$ and $J_K$, determines a degree $0$ chain map
\varphi_H: FC(K)\to FC(L).
We refer to $H$ as a decreasing Hamiltonian homotopy (from $L$ to $K$), and to $\varphi_H$ as the associated continuation map.
Given two decreasing Hamiltonian homotopies $H^0$ and $H^1$ from $L$ to $K$, the choice of a homotopy $\{H^\lambda\}$, $\lambda\in [0,1]$ between the two, together with the choice of a homotopy of almost complex structures which we ignore from the notation, determines a degree $1$ map
\varphi_{\{H^\lambda\}}:FC(K)\stackrel {+1}\longrightarrow FC(L).
We refer to $\{H^\lambda\}$ as a homotopy of homotopies, or $1$-homotopy, and to $\varphi_{\{H^\lambda\}}$ as the associated degree $1$ continuation map. This is in general not a chain map. However, it is a chain homotopy between $\varphi_{H^0}$ and $\varphi_{H^1}$:
\varphi_{H^1}-\varphi_{H^0} =\p(\varphi_{\{H^\lambda\}}) = \p_K\varphi_{\{H^\lambda\}} + \varphi_{\{H^\lambda\}}\p_H.
We now go one step further. Given two $1$-homotopies $\{H_\mu^0\}$ and $\{H_\mu^1\}$, $\mu\in [0,1]$ the choice of a homotopy $\{H_\mu^\lambda\}$, $\lambda\in[0,1]$ connecting them, together with the choice of a homotopy of homotopies of almost complex structures which we ignore from the notation, determines a degree $2$ map
\varphi_{\{H_\mu^\lambda\}}:FC(K)\stackrel {+2}\longrightarrow FC(L).
We refer to $\{H_\mu^\lambda\}$ as a $2$-homotopy, and to $\varphi_{\{H_\mu^\lambda\}}$ as the associated degree $2$ continuation map. This is in general not a chain map. However, if $\{H^0_\mu\}$ and $\{H^1_\mu\}$ coincide at $\mu=0$ and at $\mu=1$, and if $\{H_\mu^\lambda\}$ is constant at $\mu=0$ and at $\mu=1$, the map $\varphi_{\{H_\mu^\lambda\}}$ is a contracting chain homotopy for $\varphi_{\{H_\mu^1\}}-\varphi_{\{H_\mu^0\}}$:
\varphi_{\{H_\mu^1\}}-\varphi_{\{H_\mu^0\}} = \p(\varphi_{\{H_\mu^\lambda\}}).
More generally, denote $I=[0,1]$ and, for $d\ge 0$, consider the $d$-dimensional cube $I^d$. (If $d=0$ then $I^d$ consists of a single point.) A generic pair $\{H_{s,z},J_{s,z}\}$, $z\in I^d$, $s\in\R$ consisting of an $I^d$-family of decreasing Hamiltonian homotopies from $L$ to $K$ and of an $I^d$-family of $s$-dependent almost complex structures which all coincide with $J_L$ for $s\ll 0$ and with $J_K$ for $s\gg 0$,
determines a map
\varphi_{\{H_{s,z},J_{s,z}\}}\in \operatorname{Hom}_d(FC(K),FC(L)).
This map is defined on a generator $x\in FC(K)$ by
x\mapsto \sum_{|x|-|y|=-d}\# \mathcal{M}(y,x;\{H_{s,z},J_{s,z}\})y
and then extended by linearity. Here $\mathcal{M}(y,x;\{H_{s,z},J_{s,z}\})$ denotes the moduli space of solutions to the Floer equation in the chosen $I^d$-family, asymptotic to $y$ at $-\infty$ and asymptotic to $x$ at $+\infty$.
In other words, the map $\varphi_{\{H_{s,z},J_{s,z}\}}$ counts index $-d$ solutions of the Floer equation within the $d$-dimensional family parameterized by $I^d$.
We refer to $\{H_{s,z},J_{s,z}\}$ as a $d$-homotopy, and to $\varphi_{\{H_{s,z},J_{s,z}\}}$ as the associated degree $d$ continuation map.
Let $\{H^0,J^0\}$ and $\{H^1,J^1\}$ be two $d$-homotopies which are equal on $\p I^d$. For any choice of a $(d+1)$-homotopy $\{H^\lambda,J^\lambda\}$, $\lambda\in [0,1]$ which interpolates between the two, and which is constant on $(\p I^d)\times I\subset I^d\times I=I^{d+1}$, the associated degree $d+1$ continuation map $\varphi_{\{H^\lambda,J^\lambda\}}$ is a contracting chain homotopy for $\varphi_{\{H^1,J^1\}}-\varphi_{\{H^0,J^0\}}$:
\varphi_{\{H^1,J^1\}}-\varphi_{\{H^0,J^0\}} = \p(\varphi_{\{H^\lambda,J^\lambda\}}).
We have thus proved the following
The difference between any two degree $d$ continuation maps determined by $d$-homotopies which coincide on $\p I^d$ is homotopic to zero. A contracting homotopy is provided by any degree $d+1$ continuation map determined by an interpolating $(d+1)$-homotopy which is constant on $(\p I^d)\times I\subset I^d\times I=I^{d+1}$.
This statement generalizes to higher homotopies the well-known fact that any two continuation maps in Floer theory are homotopic, so that the morphism that they induce in homology is independent of all choices. This last property is sometimes referred to as Floer homology being a connected simple system in the sense of Conley.
Directed systems of continuation maps.
Let $\{K_i\}$, $\{L_i\}$, $i\in I$ be two directed systems of Hamiltonians, meaning that $K_i\le K_j$ and $L_i\le L_j$ for $i\prec j$. Let $\{K_i^j\}$, $\{L_i^j\}$, $i\prec j$ be decreasing homotopies from $K_j$ to $K_i$, respectively from $L_j$ to $L_i$, yielding continuation maps $\varphi_i^j:FC(K_i)\to FC(K_j)$, $\psi_i^j:FC(L_i)\to FC(L_j)$. Then
\{FC(K_i),\varphi_i^j\},\qquad \{FC(L_i),\psi_i^j\}
are bi-directed systems in Kom.
Assume further that $K_i\le L_i$ for all $i$. Let $H_i$ be a decreasing homotopy from $L_i$ to $K_i$, yielding continuation maps $f_i:FC(K_i)\to FC(L_i)$. The collection $\{f_i\}$ is then a map of bi-directed systems in Kom.
Indeed, the maps $\psi_i^j f_i$ and $f_j\varphi_i^j$ are homotopic via a degree $1$ continuation map
s_i^j:FC(K_i)\stackrel{+1}\longrightarrow FC(L_j)
that is associated to a $1$-homotopy $\cH_i^j$ connecting $L_i^j\# H_i$ and $H_j\# K_i^j$. Here $\#$ denotes the gluing of Hamiltonians for a large enough value of the gluing parameter.
Similarly, the maps $\varphi_i^k$ and $\varphi_j^k\varphi_i^j$, respectively $\psi_i^k$ and $\psi_j^k\psi_i^j$, are homotopic via degree $1$ maps
x_{ijk}:FC(K_i)\stackrel{+1}\longrightarrow FC(K_k),\qquad y_{ijk}:FC(L_i)\stackrel{+1}\longrightarrow FC(L_k),
that are associated to $1$-homotopies $K_{ijk}$ connecting $K_i^k$ and $K_j^k\# K_i^j$, respectively $L_{ijk}$ connecting $L_i^k$ and $L_j^k\# L_i^j$.
We claim that condition (B) is satisfied in this setup. In view of Lemma <ref> it is enough to show that both $\psi_j^ks_i^j + s_j^k\varphi_i^j$ and $f_kx_{ijk}+s_i^k-y_{ijk}f_i$ are degree $1$ Floer continuation maps induced by $1$-homotopies parameterized by $\lambda\in [0,1]$ with the same endpoints $L_j^k\#L_i^j\#H_i$ at $\lambda=0$ and $H_k\# K_j^k\# K_i^j$ at $\lambda=1$. Consider the following diagram, where in each entry we have indicated a composition of Floer continuation maps and the $0$-homotopy which induces it, and where on each arrow we have indicated a homotopy between the target and source maps, together with the $1$-homotopy which induces it. The main point is that a concatenation of $1$-homotopies induces the sum of the corresponding degree $1$ maps, and the reversal of the direction of a $1$-homotopy induces minus the corresponding degree $1$ map. The composition of the bottom horizontal arrows is thus a degree $1$-continuation map which equals $\psi_j^ks_i^j+s_j^k\varphi_i^j$, while the composition of the other three arrows is a degree $1$ continuation map which equals $f_kx_{ijk}+s_i^k-y_{ijk}f_i$. The corresponding $1$-homotopies do have the same endpoints at $\lambda=0$ and $\lambda=1$, as expected.
\xymatrix
\psi_i^kf_i \ar[rr]^{s_i^k}_{\cH_i^k}& & f_k\varphi_i^k \\
L_i^k\# H_i \ar[dddddddd]_{y_{ijk}f_i}^{L_{ijk}\# H_i} & & {\color{black}H_k}\# K_i^k \ar[dddddddd]^{f_kx_{ijk}}_{H_k\#K_{ijk}} \\
& & \\
& & \\
& & \\
& & \\
& & \\
& & \\
& & \\
\psi_j^k\psi_i^jf_i \ar[r]^{\psi_j^ks_i^j}_{L_j^k\#\cH_i^j} & \psi_j^kf_j\varphi_i^j \ar[r]^{s_j^k\varphi_i^j}_{\cH_j^k\# K_i^j} & f_k\varphi_j^k\varphi_i^j \\
L_j^k\# L_i^j\# H_i & L_j^k\# H_j \# K_i^j & H_k\# K_j^k\# K_i^j
It follows from the results in Section <ref> that
the system
\{C(f_i),\chi_i^j\}
of cones $C(f_i)$ and induced maps $\chi_i^j:C(f_i)\to C(f_i)$ is a directed system in Kom. In particular the homotopy type of the maps $\chi_i^j$ does not depend on the choice of $1$-homotopies.
Similarly, for a doubly directed system of Hamiltonians we obtain a doubly directed system
\{C(f_{ij}),\chi_{ab}^{cd}\}
in Kom, together with the fact that the homotopy type of the maps $\chi_{ab}^{cd}$ does not depend on the choice of $1$-homotopies.
§ THE TRANSFER MAP AND HOMOTOPY INVARIANCE
Given a Liouville cobordism pair $(W,V)$ we construct in this section a transfer map
f_!^\heartsuit:SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ that is
invariant under homotopy of Liouville structures. This generalizes to cobordisms the transfer map defined
for Liouville domains by Viterbo in <cit.>. The whole
structure that we exhibit on symplectic homology is actually governed
by the underlying chain level map. Indeed, we prove
in <ref> that the shifted symplectic
homology groups of the pair $SH_*^\heartsuit(W,V)[-1]$ are isomorphic
to the homology of the cone of the chain level transfer map.
We recall that we use coefficients in a field $\mathfrak{k}$.
§.§ The transfer map
Let $(W,V)$ be a Liouville cobordism pair with filling $F$. Recall from <ref> the definition of the symplectic homology groups
where $\cH(W;F)$ is the class of Hamiltonians $H:S^1\times \wh W_F\to \R$ which are zero on $W$ and are linear of non-critical slope in the complement of $W_F$, and the meaning of the limits involving $a$ and $b$ is determined by the value of $\heartsuit$. In the previous formula the first direct limit is considered with respect to continuation maps $FH_*^{(a,b)}(H_+)\to FH_*^{(a,b)}(H_-)$ for $H_+\le H_-$ induced by non-increasing homotopies $H_s$, $s\in \R$ which are equal to $H_\pm$ for $s$ near $\pm\infty$.
The transfer map will be defined as a limit of a directed system of continuation maps. For that purpose the definition of $SH_*^\heartsuit(V)$, which involves Hamiltonians defined on $\wh V_{F\circ W^{bottom}}=F\circ W^{bottom}\circ V\circ [1,\infty)\times\p^+V$, needs to be recast in terms of Hamiltonians defined on $\wh W_F=F\circ W\circ [1,\infty)\times\p^+W$. The manifold $\wh W_F$ is the domain of the Hamiltonians involved in the definition of $SH_*^\heartsuit(W)$.
Denote by $\cH^W(V;F)$ the space of Hamiltonians $H:S^1\times \wh W_F\to\R$ such that $H\in\cH(\wh W_F)$ and $H=0$ on $V$.
For any two real numbers $-\infty < a < b <\infty$ we have
SH_*^{(a,b)}(V)= \lim^{\longrightarrow}_{H\in\cH^W(V;F)}ĘFH_*^{(a,b)}(H).
By definition we have
SH_*^{(a,b)}(V)= \lim^{\longrightarrow}_{H\in\cH(V;F)}ĘFH_*^{(a,b)}(H),
and we claim that the two limits are equal. Recall that the space $\cH(V;F)$ consists of Hamiltonians $H:\wh V_{F\circ W^{bottom}}\to\R$ which are linear outside a compact set and such that $H=0$ on $V$. The claim is a consequence of the existence of a special cofinal family in $\cH^W(V;F)$ constructed as follows. See Figure <ref>.
Consider a sequence $(\nu_k)$, $k\in\Z_-$ of positive real numbers such that $\nu_k\notin\mathrm{Spec}(\p^+V)$ and $\nu_k\to\infty$ as $k\to\infty$, and let $H^V_k:\wh V_{F\circ W^{bottom}}\to\R$ be a cofinal family in $\cH(V;F)$ such that $H^V_k(r,x)=\nu_k(r-1)$ on $[1,\infty)\times\p^+V$. Consider further sequences
(\eta_k),\quad (R_k),\quad (\tau_k),\qquad k\in\Z_+
such that
* $\eta_k>0$ is smaller than the distance from $\nu_k$ to $\mathrm{Spec}(\p^+V)$, and $\eta_k\to 0$ as $k\to\infty$;
* $R_k>\max(1,(\nu_k - a)/\eta_k)$;
* $\nu_k/4 <\tau_k<\nu_k/2$ and $\tau_k\notin\mathrm{Spec}(\p^+W)$.
Let $H_k:\wh W_F\to\R$ be a Hamiltonian which is equal to $H^V_k$ on $F\circ W^{bottom}\circ V \circ [1,R_k]\times\p^+V$, which is constant equal to $\nu_k(R_k-1)$ on $R_kW^{top}$, and which is equal to $\nu_k(R_k-1)+\tau_k(r-R_k)$ on $[R_k,\infty)\times\p^+W$. Here $R_kW^{top}$ stands for the image of $W^{top}$ by the flow of the Liouville vector field at time $\ln R_k$.
The Hamiltonian $H_k$ has three more groups of $1$-periodic orbits in addition to those of the Hamiltonian $H^V_k$:
($III^-$)orbits corresponding to positively parameterized closed Reeb orbits on $\p^+V=\p^-W^{top}$ and located near $R_k\p^+V$.
($III^0$)constants in $R_kW^{top}$.
($III^+$)orbits corresponding to positively parameterized closed Reeb orbits on $\p^+W=\p^+W^{top}$ and located near $R_k\p^+W^{top}$.
The orbits in group $III^0$ have action $-\nu_k(R_k-1)$, the maximal
action of an orbit in group $III^-$ is smaller than
$-\nu_k(R_k-1)+R_k(\nu_k-\eta_k)=\nu_k-R_k\eta_k$, and the maximal
action of an orbit in group $III^+$ is smaller than
$-\nu_k(R_k-1)+R_k\nu_k/2 = -\nu_k(R_k/2-1)$. The largest of these
actions is the one in group $III^-$, which however falls below the
action window $(a,b)$ due to the condition $R_k>\max(1,(\nu_k -
a)/\eta_k)$, so that the orbits contributing to the Floer complex in
the action window $(a,b)$ are the same for $H^V_k$ and for $H_k$.
Lemma <ref> for $s$-dependent Hamiltonians (decreasing
in $s$ outside $V_{F\circ W^{bottom}}$)
shows that the continuation Floer trajectories for the family $H^V_k$ and for the family $H_k$ stay within a neighborhood of $V_{F\circ W^{bottom}}$, where the two Hamiltonians coincide. These continuation Floer trajectories are therefore the same, and they define the same continuation maps in the two directed systems at hand. We obtain
SH_*^{(a,b)}(V) = \lim^{\longrightarrow}_{k\to\infty}FH_*^{(a,b)}(H^V_k) = \lim^{\longrightarrow}_{k\to\infty}FH_*^{(a,b)}(H_k).
Since $H_k$, $k\in\Z_-$ is a cofinal family in $\cH^W(V;F)$, the conclusion of the Lemma follows.
We obviously have $\cH(W;F)\subset \cH^W(V;F)$, and for each Hamiltonian $K$ in $\cH(W;F)$ there exists a Hamiltonian $H$ in $\cH^W(V;F)$ such that $K\le H$ (while the converse is not true). For any two such Hamiltonians we have continuation maps
f_{HK}^{(a,b)}:FC_*^{(a,b)}(K)\to FC_*^{(a,b)}(H)
induced by non-increasing homotopies which are linear at infinity, and these continuation maps define a morphism between the directed systems determined by $\cH(W;F)$ and $\cH^W(V;F)$.
The Viterbo transfer map in the action window $(a,b)$ is the limit continuation map
f_!^{(a,b)} : SH_*^{(a,b)}(W)\to SH_*^{(a,b)}(V), \qquad f_!^{(a,b)}
:= \lim^{\longrightarrow}_{\stackrel{K\le H}{K\in\cH(W;F),\, H\in\cH^W(V;F)}}f_{HK}^{(a,b)}.
By general properties of the continuation maps the Viterbo transfer maps $f_!^{(a,b)}$ fit into a doubly-directed system, inverse on $a$ and direct on $b$.
For $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ the Viterbo transfer map
f_!^\heartsuit:SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)
is defined as
f_!^\heartsuit = \lim_b \lim_a f_!^{(a,b)},
where the limits are inverse or direct according to the value of $\heartsuit$, as in Definition <ref>.
Hamiltonians for the definition of the transfer map.
Let $U\subset V\subset W$ be a triple of Liouville cobordisms with
filling. Let
$f_{VW}^\heartsuit$, $f_{UW}^\heartsuit$, $f_{UV}^\heartsuit$ be the transfer maps for the pairs $(W,V)$, $(W,U)$, and $(V,U)$ respectively, for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$. Then
f_{UW}^\heartsuit=f_{UV}^\heartsuit\circ f_{VW}^\heartsuit.
This is a direct consequence of the definition of the transfer map as a limit continuation map, together with functoriality of continuation maps. To see this, we recall the notation $W=W^{bottom}\circ V\circ W^{top}$ and $V=V^{bottom}\circ U \circ V^{top}$, and consider on $W$ the following three types of Hamiltonians, see Figures <ref> and <ref>:
* Hamiltonians $K$ which are admissible for $W$, and thus vanish on $W$ and are linear increasing towards $\p^+ W$.
* one step Hamiltonians $H$ which vanish on $V$, take a positive constant value on $W^{top}$, and are linear increasing towards $\p^+ V$ and $\p^+ W$.
* two step Hamiltonians $G$ which vanish on $U$, take a constant value on $V^{top}$, take a constant value on $W^{top}$, and are linear increasing towards $\p^+ U$, $\p^+ V$, and $\p^+ W$.
The transfer maps $f^\heartsuit_{VW}$ are defined above as limit
continuation maps induced by monotone homotopies from $K$ (at
$+\infty$) to $H$ (at $-\infty$). Similarly, the transfer maps
$f^\heartsuit_{UW}$ can be obtained as limit continuation maps induced
by monotone homotopies from $K$ (at $+\infty$) to $G$ (at $-\infty$),
and the transfer maps $f^\heartsuit_{UV}$ can be obtained as limit
continuation maps induced by monotone homotopies from $H$ (at
$+\infty$) to $G$ (at $-\infty$). We can choose the homotopies from
$K$ to $G$ to factor through $H$, so that they can be expressed as
concatenation of homotopies from $K$ to $H$, and from $H$ to $G$. The
composition of the continuation maps induced by each of these last two
homotopies is equal to the continuation map induced by the
concatenation of the two homotopies – this is what we call
functoriality of continuation maps – and the same property
holds in the limit. This proves $f_{UW}^\heartsuit=f_{UV}^\heartsuit\circ f_{VW}^\heartsuit$.
In the sequel we shall often drop the symbol $\heartsuit$ from the notation for the transfer map, and simply write $f_!$ instead of $f^{\heartsuit}_!$.
Hamiltonians for the proof of functoriality of the transfer map.
§.§ Homotopy invariance of the transfer map
Given a pair of Liouville cobordisms $(W,V)$ with filling, we denote the transfer map for a given Liouville structure $\lambda$ by
\xymatrix{
SH_*^\heartsuit(W;\lambda)\ar[r]^{f_{!,\lambda}} & SH_*^\heartsuit(V;\lambda).
Let $(W,V)$ be a pair of Liouville cobordisms with filling. Given a
homotopy of Liouville structures $\lambda_t$ on $W$, $t\in [0,1]$, there are induced isomorphisms $h_W:SH_*^\heartsuit(W;\lambda_0)\to SH_*^\heartsuit(W;\lambda_1)$, $h_V:SH_*^\heartsuit(V;\lambda_0)\to SH_*^\heartsuit(V;\lambda_1)$, and a commutative diagram
\scriptsize
\xymatrix
SH_*^\heartsuit(W;\lambda_0) \ar[r]^{f_{!,\lambda_0}} \ar[d]_\cong^{h_W} & SH_*^\heartsuit(V;\lambda_0) \ar[d]_\cong^{h_V} \\
SH_*^\heartsuit(W;\lambda_1) \ar[r]_{f_{!,\lambda_1}} & SH_*^\heartsuit(V;\lambda_1)
The isomorphisms $h_W$ and $h_V$ do not depend on the choice of homotopy $\lambda_t$ with fixed endpoints.
The homotopy invariance of the transfer
map under deformations of the Liouville structure which are constant
along the boundaries of $W$ and $V$ is a consequence of its
definition as a limit continuation map.
In particular, given a Liouville cobordism $W$ with two Liouville structures $\lambda$ and $\lambda'$ which coincide along $\p W$, the transfer map
SH_*^\heartsuit(W;\lambda)\to SH_*^\heartsuit(W;\lambda')
is an isomorphism.
The homotopy invariance in the general case is obtained using the functoriality of the transfer map, by a classical geometric construction which consists in attaching to $\p W$ topologically trivial cobordisms with Liouville structures that interpolate between any two given Liouville structures on the boundary of $W$, see <cit.>. A detailed argument is given in <cit.> in an $S^1$-equivariant setting.
That the isomorphisms $h_W$ and $h_V$ do not depend on the choice of homotopy $(\lambda_t)$, $t\in [0,1]$ is a consequence of the fact that any two such homotopies with the same endpoints are homotopic, together with the usual “homotopy of homotopies" argument in Floer theory (see also the discussion of Floer continuation maps at the end of <ref>).
§ EXCISION
Let $(W,V)$ be a pair of Liouville cobordisms and $F$ a filling of $W$, and define $W_F$, $\wh W_F$ as in <ref>. Recall the class $\cH(W,V;F)$ of admissible Hamiltonians defined in <ref>.
For $0<r_1<r_2$ and a subset $A\subset\wh W_F$, we denote by $[r_1,r_2]\times A=\phi_{[\log r_1,\log r_2]}(A)$ the image of $A$ under the Liouville flow $\phi_t$ on the time interval $[\log r_1,\log r_2]$. For parameters
\mu,\nu,\tau>0,\qquad 0<\delta,\eps<1
(that will be specified later), let $H\in\cH(W,V;F)$ be a “staircase Hamiltonian" on $\wh W_F$, defined up to smooth approximation as follows (see Figure <ref>):
* $H\equiv(1-\delta)\mu$ on $F\setminus(\delta,1]\times\p^-W$,
* $H$ is linear of slope $-\mu$ on $[\delta,1]\times\p^-W$,
* $H\equiv 0$ on $W^{bottom}$,
* $H$ is linear of slope $-\nu$ on $[1,1+\eps]\times\p^-V$,
* $H\equiv -\eps\nu$ on $V\setminus\bigl([1,1+\eps]\times\p^-V\cup [1-\eps,1]\times\p^+V\bigr)$,
* $H$ is linear of slope $\nu$ on $[1-\eps,1]\times\p^+V$,
* $H\equiv 0$ on $W^{top}$,
* $H$ is linear of slope $\tau$ on $[1,\infty)\times\p^+W$.
A smooth approximation of $H$ will thus be of the form
$H(r,y)=h(r)$ on $[0,\infty)\times\p^+W$ (and similarly near the other
boundary components of $W$ and $V$). Hence $1$-periodic orbits of $X_H$
on $\{r\}\times\p^+W$ correspond to Reeb orbits on $\p^+W$ of period
$h'(r)$, and their Hamiltonian action equals
We assume that $\mu,\nu,\nu,\tau$ do not lie in the action spectrum of
$\p^-W,\p^-V,\p^+V,\p^+W$, respectively. We denote by $\eta_\nu>0$
a positive real number smaller than the distance from
$\nu$ to the union of the action spectra of $\p^-V$ and $\p^+V$, and we define similarly $\eta_\mu,\eta_\tau>0$. The $1$-periodic orbits of $H$ fall into $11$ classes:
($F^0$) constants in $F\setminus ([\delta,1]\times \p F)$,
($F^+$) orbits corresponding to negatively parameterized
closed Reeb orbits on$\p F=\p^-W$ and located near $\delta\times\p^-W$,
($I^-$) orbits corresponding to negatively parameterized closed Reeb orbits on$\p^-W^{bottom}=\p^-W$ and located near $\p^-W$,
($I^0$) constants in $W^{bottom}$,
($I^+$) orbits corresponding to negatively parameterized closed Reeb orbits on$\p^+W^{bottom}=\p^-V$ and located near $\p^-V$,
($II^-$) orbits corresponding to negatively parameterized closed Reeb orbits on $\p^-V$ and located near $(1+\eps)\times\p^-V$,
($II^0$) constants in $V\setminus\bigl([1,1+\eps]\times\p^-V\cup [1-\eps,1]\times\p^+V\bigr)$,
($II^+$) orbits corresponding to positively parameterized closed Reeb orbits on $\p^+V$ and located near $(1-\eps)\times\p^+V$,
($III^-$) orbits corresponding to positively parameterized closed Reeb orbits on$\p^-W^{top}=\p^+V$ and located near $\p^+V$,
($III^0$) constants in $W^{top}$,
($III^+$) orbits corresponding to positively parameterized closed Reeb orbits on $\p^+W$ and located near $\p^+W^{top}=\p^+W$.
Hamiltonian in $\mathcal{H}(W,V;F)$.
Notational convention. For two classes of orbits $A,B$ we write $A\prec B$ if the homological Floer boundary operator maps no orbit from $A$ to an orbit from $B$. A priori, this relation is not transitive. However, when we write $A\prec B\prec C$ we also mean that $A\prec C$. We write $A<B$ if all orbits in $A$ have smaller action than all orbits in $B$. Note that $A<B$ implies $A\prec B$, and $A<B<C$ implies $A\prec B\prec C$.
Fix $a<b$. If the parameters $\mu,\nu,\tau,\delta,\eps$ above satisfy
\begin{equation}\label{eq:par}
(1-\delta)\mu>\min\{-a,\nu-\eta_\nu\} \quad\text{and}\quad \eps\nu > \min\{b,\tau-\eta_\tau\},
\end{equation}
and if we use an almost complex structure that is cylindrical and has a long enough neck
near $(1-2\varepsilon)\times \p^+V$,
then the four groups of orbits in the action interval $[a,b]$ satisfy
\begin{equation}\label{eq:order}
F \prec I \prec III\prec II\quad\text{and}\quad III\prec I.
\end{equation}
Moreover, within each group of orbits we have the relations
\begin{equation}\label{eq:order-within-groups}
\begin{gathered}
F^+\prec F^0, \qquad
I^+\prec I^-\prec I^0, \cr %\qquad I^0\prec I^-, \cr
II^-\prec II^0\prec II^+, \qquad %\qquad II^0\prec II^-\cr
III^0\prec III^-\prec III^+. %\qquad III^-\prec III^0.
\end{gathered}
\end{equation}
The combination of Lemmas <ref> and <ref> yields the relations
\begin{gather*}
F\prec I^-,\qquad F,I\prec II^{-+},\qquad F,I,II,III^{-0}\prec III^+, \cr
I^+\prec F,I^{-0},\qquad III^-\prec F,I,II.
\end{gather*}
For any choice of parameters, the actions satisfy
F^+<F^0,\qquad F,I^{-+} < I^0=III^0 < II^{0+},III^{-+},\qquad II^-<II^0<II^+.
We see that $F\prec I^{-0},II,III$. The remaining relation $F\prec I^+$ follows if the actions satisfy $F^0<I^+$, i.e., $-(1-\delta)\mu < \max\{a,-(\nu-\eta_\nu)\}$, which is the first condition in (<ref>). Next we see that $I\prec II,III$ and $III^-\prec I,II$. For the remaining relation $III^{0+}\prec I,II$ we arrange the actions to satisfy $III^+<II^0$, i.e., $\min\{b,\tau-\eta_\tau\}<\eps\nu$, which is the second condition in (<ref>). Then we have $III^0<III^+<II^0<II^+$. The relations $I^0\prec III^0$ and $III^0\prec I^0$ follow from monotonicity: there is an a priori strictly positive lower bound on the energy of trajectories traversing $V$, and this rules out trajectories running between $III^0$ and $I^0$ which after small Morse perturbation of $H$ have arbitrarily small energy. The remaining relation $III^{0+}\prec I,II^-$ now follows from Lemma <ref>, stretching the neck at the hypersurface $(1-2\eps)\times\p^+V$ where $H\equiv-\eps\nu$, and $\eps\nu$ is bigger than all actions in the groups $III^0$ and $III^+$. This proves (<ref>).
The relations in (<ref>) also follow from the preceding discussion.
Under the conditions of Lemma <ref>, the Floer boundary operator has upper triangular form if the periodic orbits are ordered by increasing action within each class and the classes are ordered (for example) as
F^+\prec F^0\prec I^+\prec I^-\prec I^0 \prec
III^0\prec III^-\prec III^+\prec
II^-\prec II^0\prec II^+.
Let us fix $a<0<b$ and $0<\delta,\eps<1$ and consider $\mu,\nu,\tau>0$ subject to the conditions
\begin{equation}\label{eq:par2}
\mu>-a/(1-\delta),\qquad \tau>b,\qquad \nu>\max\{-a,b/\eps\}.
\end{equation}
Note that these conditions allow us to make $\mu,\nu,\tau$ arbitrarily large, independently of each other. They ensure condition (<ref>) in Lemma <ref>. Moreover, the actions of all orbits in the classes $F,II^0,II^+$ lie outside the interval $[a,b]$. So the Floer chain complex can be written as
FC^{(a,b)} = FC^{(a,b)}_{III} \oplus FC^{(a,b)}_{I} \oplus FC^{(a,b)}_{II^-}
and with respect to this decomposition the Floer boundary operator has the form
\begin{equation}\label{eq:upper-triang}
\begin{pmatrix}* & 0 & *\\ 0 & * & *\\ 0 & 0 & *
\end{pmatrix}\;.
\end{equation}
Let us fix $\mu,\tau$ and consider $\nu<\nu'$ both satisfying (<ref>). We denote the corresponding Hamiltonians by $H_{\nu'}\leq H_\nu$ and consider the continuation maps
\phi_{\nu\nu'}:FC^{(a,b)}(H_{\nu'})\to FC^{(a,b)}(H_\nu)
induced by convex interpolation between $H_\nu$ and $H_{\nu'}$. These
continuation maps
may not have the upper triangular form (<ref>)
since the combination of Lemmas <ref> and <ref>
does not apply to the current homotopy situation. Therefore, we
decompose the above chain complex instead as
FC^{(a,b)} = FC^{(a,b)}_{III} \oplus FC^{(a,b)}_{I,II^-},
with differential written in upper triangular form as
$\left(\begin{array}{cc}Ę* & * \\Ę0 &
* \end{array}\right)$. The continuation maps $\phi_{\nu\nu'}$ have
upper triangular form with respect to this decomposition and we obtain
the commuting diagram with exact rows
\begin{equation}\label{eq:filt}
\xymatrix
0\ar[r] & FC^{(a,b)}_{III}(H_{\nu'}) \ar[r] \ar[d] & FC^{(a,b)}(H_{\nu'}) \ar[r] \ar[d] & FC^{(a,b)}_{I,II^-}(H_{\nu'}) \ar[r] \ar[d] & 0 \\
0\ar[r] & FC^{(a,b)}_{III}(H_{\nu}) \ar[r] & FC^{(a,b)}(H_{\nu}) \ar[r] & FC^{(a,b)}_{I,II^-}(H_{\nu}) \ar[r] & 0\;, \\
\end{equation}
where $FC^{(a,b)}_{I,II^-}$ denotes the quotient complex $FC^{(a,b)} / FC^{(a,b)}_{III}$.
\lim^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{III}(H_\nu) \cong SH^{(a,b)}(W^{top},\p^+V).
We consider a homotopy of Hamiltonians which on $V\cup W^{top}\cup [1,\infty)\times\p^+W$ is constant and which on $F\cup W^{bottom}$ is a convex interpolation between the Hamiltonian $H_\nu$ and the Hamiltonian $\ol H_\nu$ that is constant equal to $-\varepsilon\nu$. Since the homotopy is constant on the cobordism $V$, Lemma <ref> applies and shows that there is no interaction between the orbits in $III$ and the orbits appearing in $F\cup W^{bottom}$. The usual continuation argument then shows that the homology $FH^{(a,b)}_{III}$ is invariant during this homotopy. Since $\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty} FH^{(a,b)}_{III}(\ol H_\nu)=SH^{(a,b)}(W^{top},\p^+V)$ by definition, we obtain the desired isomorphism.
\lim^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I,II^-}(H_\nu) \cong SH^{(a,b)}(W^{bottom},\p^-V).
We consider a homotopy of Hamiltonians which on $F\cup W^{bottom}\cup V$ is constant and which on $W^{top}\cup [1,\infty)\times\p^+W$ is a convex interpolation between the Hamiltonian $H_\nu$ and the Hamiltonian $K_\nu$ that is constant equal to $-\varepsilon \nu$ on $V\cup W^{top}$ and is linear of slope $\tau$ (the same as the slope of $H_\nu$) on $[1,\infty)\times\p^+W$.
See Figure <ref>.
We have $FH^{(a,b)}(K_\nu) = FH^{(a,b)}_{I,II^-}(K_\nu)$ and so we have a well-defined continuation map $\phi^{HK}_\nu:FH^{(a,b)}_{I,II^-}(K_\nu)\to FH^{(a,b)}_{I,II^-}(H_\nu)$ obtained by composing the continuation map $FH^{(a,b)}(K_\nu)\to FH^{(a,b)}(H_\nu)$ with the map induced by projection $FH^{(a,b)}(H_\nu)\to FH^{(a,b)}_{I,II^-}(H_\nu)$. Since the homotopy is constant in the region $F\cup W^{bottom}\cup V$, which contains the orbits of type $I,II^-$, it follows that this continuation map is an isomorphism. Indeed, the generators of the two chain complexes are canonically identified and upon arranging them in increasing order by the action the continuation map at chain level has upper triangular form with $+1$ on the diagonal. (Note that we do not use at this point Lemma <ref>.)
For $\nu\leq\nu'$ we get commutative diagrams in which all maps are continuation morphisms
\xymatrix{
FH^{(a,b)}_{I,II^-}(H_\nu)Ę& \ar[l]^\cong_{\phi^{HK}_\nu} FH^{(a,b)}_{I,II^-}(K_\nu) \\Ę
FH^{(a,b)}_{I,II^-}(H_{\nu'}) \ar[u]^{\phi_{\nu\nu'}}Ę& FH^{(a,b)}_{I,II^-}(K_{\nu'}) \ar[l]^\cong_{\phi^{HK}_{\nu'}} \ar[u]_{\psi_{\nu\nu'}}\,.
Here $\psi_{\nu\nu'}:FH^{(a,b)}_{I,II^-}(K_{\nu'})\to FH^{(a,b)}_{I,II^-}(K_\nu)$ is the continuation map induced by a convex interpolation between $K_\nu$ and $K_{\nu'}$. As a consequence we have a canonical isomorphism
\begin{equation}Ę\label{eq:inverse-limit-H-K}
\xymatrix{
\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I,II^-}(H_\nu) & & \ar[ll]^\cong_{\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}\phi^{HK}_\nu}Ę\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I,II^-}(K_\nu).
\end{equation}
The complex $FC^{(a,b)}_{I,II^-}(K_\nu)$ can be decomposed as
\begin{equation}Ę\label{eq:decomposition-FCKnu}
FC^{(a,b)}_{I,II^-}(K_\nu)=FC^{(a,b)}_I(K_\nu)\oplus FC^{(a,b)}_{II^-}(K_\nu),
\end{equation}
with differential of upper triangular form $\left(\begin{array}{cc} * & * \\Ę0 & *\end{array}\right)$. Lemma <ref> below shows that this decomposition is preserved by the continuation maps $\psi_{\nu\nu'}$, which also have upper triangular form. (That this precise property could a priori fail for the Hamiltonians $H_\nu$ was the reason to deform them to the Hamiltonians $K_\nu$.) In particular, there is a well-defined inverse system of quotient homologies $FH^{(a,b)}_{II^-}(K_\nu)$, $\nu\to\infty$. Lemma <ref> below shows that the inverse limit of this system vanishes, and we thus obtain a canonical isomorphism
\begin{equation}Ę\label{eq:inverse-limit-K-I}
\xymatrix{
\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I}(K_\nu) \ar[rr]^\cong & & Ę\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I,II^-}(K_\nu),
\end{equation}
the map being induced in the limit by the inclusions $FC^{(a,b)}_I(K_\nu)\hookrightarrow FC^{(a,b)}_{I,II^-}(K_\nu)$.
We now prove the isomorphism
\begin{equation}Ę\label{eq:inverse-limit-K-I-SH}
\xymatrix{
\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I}(K_\nu) \cong SH^{(a,b)}(W^{bottom},\p^- V).
\end{equation}
The Floer trajectories which are involved in the definition of the
Floer differential for $FC^{(a,b)}_I(K_\nu)$ are contained in a
neighborhood of $F\cup W^{bottom}$ by Lemma <ref>. The
key point is that the Floer trajectories involved in the definition of
the continuation maps $FC^{(a,b)}_I(K_{\nu'})\to FC^{(a,b)}_I(K_\nu)$
are also contained in a neighborhood of $F\cup W^{bottom}$. For this
purpose we choose the Hamiltonians $K_\nu$ such that for $\nu'\ge \nu$
the Hamiltonian $K_{\nu'}$ coincides with $K_\nu$ on a neighborhood of
$F\cup W^{bottom}$ where the orbits in group $I$ for $K_\nu$ are
located. This ensures that the assumptions in the last paragraph of
Lemma <ref> are satisfied for the homotopy obtained by
convex interpolation between $K_\nu$ and $K_{\nu'}$. Denote $\ol
K_\nu$ the Hamiltonian defined on $F\cup W^{bottom}\cup
[1,\infty)\times\p^-V$ which is equal to $K_\nu$ on $F\cup W^{bottom}$
and linear of slope $-\nu$ (the same as the slope of $K_\nu$) on
$[1,\infty)\times\p^- V$. The previous argument then shows the equality
\xymatrix{
\lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I}(K_\nu) = \lim\limits^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{I}(\ol K_\nu),
and the right hand side is $SH^{(a,b)}(W^{bottom},\p^- V)$ by definition.
The conclusion of Lemma <ref> now follows by combining the isomorphisms (<ref>), (<ref>), and (<ref>).
The Hamiltonians $H_\nu$ and $K_\nu$.
The next lemma was used in the previous proof. We recall that $K_\nu$ denotes a Hamiltonian which coincides with $H_\nu$ on $F\cup W^{bottom}\cup V$, is constant equal to $-\varepsilon \nu$ on $V\cup W^{top}$, and is linear of slope $\tau$ (the same as the slope of $H_\nu$) on $[1,\infty)\times\p^+ W$. We choose the smoothings of the Hamiltonians $K_{\nu'}$ and $K_{\nu}$ to coincide up to a translation by $\epsilon(\nu'-\nu)$ in the region $II^-$ and for slopes in the interval $(-\nu+\eta_\nu, 0)$. We recall the decomposition (<ref>) of $FC^{(a,b)}_{I,II^-}(K_\nu)$, with respect to which the differential has upper triangular form.
The Floer continuation map $\psi_{\nu\nu'}:FC^{(a,b)}_{I,II^-}(K_{\nu'})\to FC^{(a,b)}_{I,II^-}(K_\nu)$
induced by a non-increasing $s$-dependent convex interpolation from $K_\nu$ at $-\infty$ to $K_{\nu'}$ at $+\infty$ has upper-triangular form with respect to the decompositions $FC^{(a,b)}_{I,II^-}=FC^{(a,b)}_I\oplus FC^{(a,b)}_{II^-}$ for $K_\nu$ and $K_{\nu'}$.
The only problematic relation is $I_{K_{\nu'}}\prec
II^-_{K_\nu}$. To prove it we use the fact that in the region $II^-$
the two Hamiltonians coincide up to a translation, so in this region
the homotopy is simply given by adding to the Hamiltonian $K_\nu$ some
function $\R\to [-\epsilon(\nu'-\nu),0]$ of $s$ with compactly supported
derivative. As such, the constant trajectories at the orbits in
$II^-_{K_\nu}$ solve the $s$-dependent continuation Floer equation.
Assume there exists a continuation Floer trajectory $u:\R\times S^1\to
\wh W_F$ from some orbit $x_+=\lim_{s\to+\infty}Ęu(s,\cdot)$ in
$I_{K_{\nu'}}$ to some orbit $x_-=\lim_{s\to-\infty}Ęu(s,\cdot)$ in
$II^-_{K_\nu}$. By Lemma <ref>, either $u$ is constant equal
to $x_-$ for very negative values of the parameter $s$, or there
exists $(s,t)\in\R\times S^1$ with $s$ very negative such that
In the first situation the Floer trajectory would need to be constant
equal to $x_-$ for all values of $s$ because of unique continuation
and the fact that the constant trajectory at $x_-$ solves the same
equation. This is a contradiction since $x_+\neq x_-$. In the second
situation we reach a contradiction using Lemma <ref>,
which we can apply in the $s$-independent case because the
homotopy is just given by a shift by a function of $s$ on $V\cup
The next lemma was used in the proof of Lemma <ref> as well. By Lemma <ref> we have a well-defined inverse system $FH^{(a,b)}_{II^-}(K_\nu)$, $\nu\to\infty$.
\lim^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_{II^-}(K_\nu) = 0.
For $\nu'>\nu$, generators of $FC^{(a,b)}_{II^-}(K_{\nu'})$ correspond to closed Reeb orbits $\gamma$ on $\p^-V$ with Hamiltonian action satisfying
A_{K_{\nu'}}(\gamma) = -(1+\eps)\Bigl(\int_\gamma\lambda\Bigr) + \eps\nu' \in (a,b).
Since this condition is equivalent to
A_{K_{\nu}}(\gamma) = -(1+\eps)\Bigl(\int_\gamma\lambda\Bigr) + \eps\nu \in (a+\eps(\nu-\nu'),b+\eps(\nu-\nu')),
we see that the same Reeb orbits also correspond to generators of the Floer chain group $FC^{(a+\eps(\nu-\nu'),b+\eps(\nu-\nu'))}_{II^-}(K_{\nu})$. Varying the slope continuously from $\nu'$ to $\nu$, we obtain a continuation isomorphism between these two groups fitting into the commuting diagram
\begin{equation*}
\xymatrix{
FH^{(a+\eps(\nu-\nu'),b+\eps(\nu-\nu'))}_{II^-}(K_{\nu}) \ar[r]^-{\cong} \ar[rd]^-{\pi} & FH^{(a,b)}_{II^-}(K_{\nu'}) \ar[d]^-{\psi_{\nu\nu'}} \\
& FH^{(a,b)}_{II^-}(K_{\nu}).
\end{equation*}
That the horizontal map is an isomorphism follows from the fact that
the Hamiltonian is deformed outside a compact set only by a global shift by a constant, and from the
fact that there are no orbits that cross the boundary of the moving
action window during the homotopy. The horizontal map can be expressed
as a composition of small-time continuation maps induced by homotopies
for fixed action windows, which are isomorphisms since each of these
homotopies can be followed backwards, and of tautological isomorphisms
given by shifting the action window by some small amount in the
complement of the action spectrum.
Now if $b+\eps(\nu-\nu')<a$, then the intervals $[a+\eps(\nu-\nu'),b+\eps(\nu-\nu')]$ and $[a,b]$ do not overlap and thus the projection $\pi$ vanishes in homology. Hence the Floer chain map $\psi_{\nu\nu'}$ vanishes whenever $\nu'-\nu>(b-a)/\eps$, from which the lemma follows.
Let $(W,V)$ be a pair of Liouville cobordisms with filling and consider parameters $-\infty<a<b<\infty$. There is a short exact sequence
0\to SH^{(a,b)}_*(W^{top},\p^+V)\to SH^{(a,b)}_*(W,V)\to SH^{(a,b)}_*(W^{bottom},\p^-V)\to 0.
Moreover, this short exact sequence splits canonically, so that we have a canonical isomorphism
SH^{(a,b)}_*(W,V)\cong SH^{(a,b)}_*(W^{top},\p^+V)\oplus SH^{(a,b)}_*(W^{bottom},\p^-V).
We fix the parameters $0<\delta,\eps<1$ and $\mu,\tau>0$ such that the first
two conditions in (<ref>) hold, and we work with the family of Hamiltonians $H_\nu=H_{\mu,\nu,\tau}$, $\nu\to\infty$ discussed above. Then
\lim^{\longleftarrow}_{\scriptsize \nu\to\infty}FH^{(a,b)}_*(H_\nu) \cong SH^{(a,b)}_*(W, V)
by definition. The short exact sequence of inverse
systems (<ref>) determines an inverse system of homology exact
triangles in which each term is a finite dimensional vector space. In
this case the inverse limit preserves exactness and we obtain using
Lemmas <ref> and <ref> an exact triangle
\xymatrix
SH^{(a,b)}_*(W^{top},\p^+V) \ar[rr]Ę& &
SH^{(a,b)}_*(W,V) \ar[dl] \\ & SH^{(a,b)}_*(W^{bottom},\p^-V) \ar[ul]^{[-1]}\;.
The proof of Lemma <ref> shows that each class in
$SH^{(a,b)}_*(W^{bottom},\p^-V)$ is represented by a sequence
(indexed by $\nu$ and representing an element of the inverse limit)
of classes in $FH^{(a,b)}_{I,II^-}(H_\nu)$ which are each
represented by a cycle that is a linear combination of orbits in
$I_{H_\nu}$. Indeed, the proof provides such a representative by a
cycle in $FC^{(a,b)}_I(K_\nu)$, and we have
$FC^{(a,b)}_I(K_\nu)=FC^{(a,b)}_I(H_\nu)$; on the other hand, since
$I_{H_\nu}\prec II^-_{H_\nu}$ as already seen
in (<ref>), this continues to be a cycle in
To prove the existence of the short exact sequence in the statement we use that the degree $-1$ connecting map $FH^{(a,b)}_{I,II^-}(H_\nu) \to FH^{(a,b)}_{III}(H_\nu)$ vanishes on elements of $I_{H_\nu}$ by (<ref>). Thus the connecting map in the above exact triangle vanishes, and the latter becomes the short exact sequence
0\to SH^{(a,b)}_*(W^{top},\p^+V)\to SH^{(a,b)}_*(W,V)\to SH^{(a,b)}_*(W^{bottom},\p^-V)\to 0.
To prove the existence of a canonical splitting for this exact
sequence we use again that $I\prec III$ for $H_\nu$. Thus a cycle in $FC^{(a,b)}_{I,II^-}(H_\nu)$ which is a linear combination of orbits in $I_{H_\nu}$ is canonically also a cycle in $FC^{(a,b)}(H_\nu)$. The splitting $SH^{(a,b)}_*(W^{bottom},\p^-V)\to SH^{(a,b)}_*(W,V)$ associates to each class, represented by a sequence of classes of cycles in $FC^{(a,b)}_{I,II^-}(H_\nu)$ which are linear combinations of orbits in $I_{H_\nu}$, the sequence of classes represented by the same cycles viewed in $FC^{(a,b)}(H_\nu)$. The latter represents indeed an element in the inverse limit of $FH^{(a,b)}(H_\nu)$, $\nu\to\infty$ because the continuation maps $\phi_{\nu\nu'}:FC^{(a,b)}(H_{\nu'})\to FC^{(a,b)}(H_\nu)$ preserve the relation $I\prec III$.
Taking limits over $a$ and $b$, Proposition <ref> implies
Let $(W,V)$ be a pair of Liouville cobordisms with filling.
Then for each flavour $\heartsuit$ we have canonical isomorphisms
SH_*^\heartsuit(W,V) \cong SH_*^\heartsuit(W^{bottom},\p^-V) \oplus SH_*^\heartsuit(W^{top},\p^+V).
In Proposition <ref> and Theorem <ref> we
allow $W^{bottom}$ or $W^{top}$ to be empty, in which case the
corresponding term is not present in the diagram. In particular,
taking $V$ to be a collar neighbourhood of some boundary components we
Given a Liouville cobordism $W$ and an admissible union of connected
components $A\subset \p W$, we have
SH_*^\heartsuit(W,A)\cong SH_*^\heartsuit(W,I\times A),
where $I\times A$ is a collar neighborhood of $A$ in $W$ which we view
as a trivial cobordism, so that $(W,I\times A)$ is a Liouville
This is the precise sense in which Definitions <ref>
and <ref> are compatible.
In order to make the excision theorem resemble the one in algebraic topology,
we introduce the following notion.
Liouville cobordism triple
$(W,V,U)$ consists of three Liouville cobordisms $U\subset V\subset W$
such that $(W,V)$ and $(V,U)$ are Liouville cobordism pairs.
A filling of a Liouville cobordism triple is a filling of $W$,
which induces fillings of $V$ and $U$ in the obvious way.
Then we have
Let $(W,V,U)$ be a
filled Liouville cobordism triple.
Then for each flavour $\heartsuit$ we have canonical isomorphisms
SH_*^\heartsuit(W,V)\cong SH_*^\heartsuit(\overline{W\setminus U},\overline{V\setminus U})\,.
Here if some boundary component $A$ of $V$ and $U$ coincides, then the
homology on the right hand side has to be understood relative to $A$.
(Alternatively, one can use Proposition <ref> below
to move $U$ into the interior of $V$ and avoid this situation.)
Also, if $\overline{W\setminus U}$ contains both a bottom and an upper
part then the right hand side has to be understood
according to Section <ref>
as a direct sum, as in the statement of Theorem <ref>.
Let us write
\ol{W\setminus V} = W^{bottom}\amalg W^{top},\qquad
\ol{V\setminus U} = V^{bottom}\amalg V^{top}.
\ol{W\setminus U} = (W^{bottom}\cup V^{bottom})\amalg (W^{top}\cup V^{top})
and we find
\begin{align*}
SH_*^\heartsuit(\ol{W\setminus U},\ol{V\setminus U})
&= SH_*^\heartsuit(W^{bottom}\cup V^{bottom},V^{bottom}) \oplus
SH_*^\heartsuit(W^{top}\cup V^{top},V^{top}) \cr
&\cong SH_*^\heartsuit(W^{bottom},\p^-V) \oplus
SH_*^\heartsuit(W^{top},\p^+V) \cr
&\cong SH_*^\heartsuit(W,V),
\end{align*}
where the first equality is the definition and the other two
isomorphisms follow from Theorem <ref>.
§ THE EXACT TRIANGLE OF A PAIR OF FILLED LIOUVILLE COBORDISMS
The main result of this section is
For each filled Liouville cobordism pair $(W,V)$ and $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ there exist exact triangles
\begin{equation*}
\xymatrix
SH_*^\heartsuit(W,V) \ar[rr] & &
SH_*^\heartsuit(W) \ar[dl] \\ & SH_*^\heartsuit(V) \ar[ul]^{[-1]}
\end{equation*}
\begin{equation*}
\xymatrix
SH^*_\heartsuit(W,V) \ar[dr]_{[+1]} & &
SH^*_\heartsuit(W) \ar[ll] \\ & SH^*_\heartsuit(V) \ar[ur]
\end{equation*}
These triangles are functorial with respect to inclusions of Liouville pairs.
This theorem will be proved in Section <ref> below.
§.§ Cofinal families of Hamiltonians
As a preparation, we now recast the definition of the symplectic homology groups $SH_*^\heartsuit(W)$, $SH_*^\heartsuit(V)$ and of the transfer map $f_!^\heartsuit:SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)$ at chain level in terms of some carefully chosen cofinal families of Hamiltonians. This will allow us to further express the relative symplectic homology groups $SH_*^\heartsuit(W,V)$ in terms of the cone construction.
Let $(W,V)$ be a Liouville pair with filling $F$.
Notational convention. Let $P$, $Q$ denote sets of $1$-periodic
orbits of a given Hamiltonian $H$. Recall that we write $Q<P$
if all the orbits in group $Q$ have strictly smaller action than all
the orbits in group $P$, and we write $Q\prec P$
if there is no Floer trajectory for $H$ asymptotic at the positive puncture to an orbit in $Q$ and asymptotic at the negative puncture to an orbit in $P$. This implies that the expression of the Floer boundary operator on any orbit in $Q$ does not contain any element in $P$. It is understood that the Floer equation involves some almost complex structure which is not specified in the notation.
Similarly, given two Hamiltonians $H_\pm$ and a homotopy $H_s$, $s\in \R$ equal to $H_\pm$ near $\pm\infty$, and given sets of $1$-periodic orbits $P_{H_\pm}$ for $H_\pm$, we write
P_{H_+}\prec P_{H_-}
if there is no Floer continuation trajectory for the homotopy $H_s$ asymptotic at the positive puncture to an orbit in $P_{H_+}$ and asymptotic at the negative puncture to an orbit in $P_{H_-}$. This implies that the expression of the Floer continuation map on any orbit in $P_{H_+}$ does not contain any element in $P_{H_-}$. Here it is again understood that the Floer continuation equation involves some almost complex structure which is not specified in the notation.
In the previous context, we write
if the $H_+$-action of any orbit in $P_{H_+}$ is smaller than the $H_-$-action of any orbit in $P_{H_-}$. This implies $P_{H_+}\prec P_{H_-}$ if $H_+\le H_-$ and the homotopy $H_s$ is non-increasing with respect to the $s$-variable.
Given $c\in \R$, we write
P_{H_+}< P_{H_-} - c
if the difference between the $H_+$-action of any orbit in $P_{H_+}$
and the $H_-$-action of any orbit in $P_{H_-}$ is smaller than $-c$.
Consider Hamiltonians $H_+\ge H_-$ and a homotopy $H_s$ which is a
convex interpolation between $H_+$ and $H_-$ given by a non-decreasing
$s$-dependent function, i.e., $H_s= H_-+f(s)(H_+-H_-)$ with $f:\R\to[0,1]$, $f'\ge
0$, $f=0$ near $-\infty$, $f=1$ near $+\infty$. Then $P_{H_+} <
P_{H_-} -\|H_+-H_-\|_\infty$ implies $P_{H_+}\prec P_{H_-}$.
If there is a continuation Floer trajectory $u:\R\times S^1\to \wh W_F$
solving $\p_s u + J_{s,t}(u)(\p_t u - X_{H_s}(u))=0$
with $\lim_{s\to\pm\infty}u(s,\cdot)=x_\pm(\cdot)$,
where $x_\pm$ are $1$-periodic orbits of $H_\pm$,
then we have
\begin{eqnarray*}
A_{H_+}(x_+)-A_{H_-}(x_-) & = & \int_{-\infty}^\infty \frac d{ds}A_{H_s}(u(s,\cdot))\, ds \\
& = & \int_{-\infty}^\infty dA_{H_s}(u(s,\cdot))\cdot \p_s u \ds - \int_{-\infty}^\infty \int_0^1 \p_s H_s(t,u(s,t))\, dt\, ds \\
& = & \int_{-\infty}^\infty\int_0^1 \|\p_s u(s,t)\|^2\, ds\, dt \\
& & -\int_{-\infty}^\infty\int_0^1 f'(s) \Bigl(H_+(t,u(s,t))-H_-(t,u(s,t))\Bigr)\, ds \, dt \\
& \ge & -\int_{-\infty}^\infty\int_0^1 f'(s) \sup_{t,x} \Bigl(H_+(t,x)-H_-(t,x)\Bigr) \, ds \, dt \\
& = & -\|H_+-H_-\|_\infty.
\end{eqnarray*}
Since the domain of definition of the Hamiltonians that we use in this paper is a noncompact manifold, it is appropriate to discuss the degree of applicability of the previous principle: it holds for compactly supported homotopies, so that $\|H_+-H_-\|_\infty$ is finite (and can be explicitly computed), but it also holds for non-compactly supported homotopies if one knows a priori that the continuation Floer trajectories are contained in a compact set, in which case it is enough to estimate $\|H_+-H_-\|_\infty$ on that compact set.
§.§.§ Hamiltonians for $SH_*^\heartsuit(W)$.
\mu,\tau>0
be such that $\mu\notin \mathrm{Spec}(\p^-W)$ and $\tau\notin \mathrm{Spec}(\p^+W)$. Denote by
the distance from $\mu$ to $\mathrm{Spec}(\p^- W)$ and let $\delta>0$
be such that
\begin{equation} \label{eq:deltamuetamu}
\delta\mu<\eta_\mu.
\end{equation}
We denote by
K_{\mu,\tau}=K_{\mu,\tau,\delta}:\wh W_F\to\R
the Hamiltonian which is defined up to smooth approximation as follows: it is constant equal to $\mu(1-\delta)$ on $F\setminus [\delta,1]\times\p F$, it is linear equal to $\mu(1-r)$ on $[\delta,1]\times\p F$, it is constant equal to $0$ on $W$, and it is linear equal to $\tau(r-1)$ on $[1,\infty)\times\p^+ W$. See Figure <ref>.
Hamiltonians $K_{\mu,\tau,\delta}$ for the definition $SH_*^\heartsuit(W)$.
A smooth approximation of $K_{\mu,\tau}$ will thus be of the form $K_{\mu,\tau}(r,y)=k(r)$ on $[1,\infty)\times\p^+W$ (and similarly near the negative boundary $\p^-W$). The $1$-periodic orbits of $X_{K_{\mu,\tau}}$ on $\{r\}\times \p^+W$ correspond to Reeb orbits on $\p^+W$ of period $k'(r)$, and their Hamiltonian action equals
Since we assumed that $\mu$ and $\tau$ are not equal to the period of a closed Reeb orbit on the respective boundaries of $W$, it follows that $K_{\mu,\tau}$ has no $1$-periodic orbits in the regions where it is linear.
The $1$-periodic orbits of the Hamiltonian $K_{\mu,\tau}$ naturally fall into 5 classes as follows:
($F^0$) constants in $F\setminus [\delta,1]\times\p F$.
($F^+$)Ęorbits corresponding to negatively parameterized closed Reeb orbits on $\p F=\p^-W$ and located near $\{\delta\}\times\p^-W$.
($I^-$)Ęorbits corresponding to negatively parameterized closed Reeb orbits on $\p^-W$ and located near $\p^-W$.
($I^0$)Ęconstants in $W$.
($I^+$)Ęorbits corresponding to positively parameterized closed Reeb orbits on $\p^+W$ and located near $\p^+W$.
We denote by $F$ the group of orbits $F^{0+}$, and by $I$ the group of orbits $I^{-0+}$. The maximal action of an orbit in group $F$ is $-\mu(1-\delta)=-\mu+\delta\mu$, while the minimal action of an orbit in group $I$ is $-\mu+\eta_\mu$. Condition (<ref>) implies
$F<I$, and in particular
F\prec I.
This last relation actually holds regardless of the choice of parameters by combining Lemmas <ref> and <ref> in order to prohibit trajectories from $F$ to $I^-$ with the relation $F<I^{0+}$ which prohibits trajectories from $F$ to $I^{0+}$. Alternatively, the relation $F\prec I^0$ is also a consequence of Lemma <ref>, while $F\prec I^+$ is also a consequence of Lemmas <ref> and <ref>.
Let now $(\mu_i)$, $i\in\Z_-$ and $(\tau_j)$, $j\in\Z_+$ be two
sequences which do not contain elements in
${\mathrm{Spec}}(\p^-W)\cup{\mathrm{Spec}}(\p^+W)$ and such that
$\mu_{i'}>\mu_i$ for $i'<i$ and $\tau_j<\tau_{j'}$ for $j<j'$. We
moreover require that $\mu_i\to\infty$ as $i\to-\infty$ and
$\tau_j\to\infty$ as $j\to\infty$. Choose a sequence $(\delta_i)$,
$i\in\Z_-$ of positive numbers such that $\delta_{i'}<\delta_i$ for
$i'<i$, such that $\delta_i\to 0$ as $i\to-\infty$, and such that
condition (<ref>) is satisfied:
\delta_i\mu_i<\eta_{\mu_i} \qquad \mbox{ for all } i\in\Z_-.
We denote
K_{i,j}:=K_{\mu_i,\tau_j,\delta_i},\qquad i\in \Z_-,\quad j\in\Z,
so that $K_{i',j}\ge K_{i,j}$ for $i'\le i$, and $K_{i,j}\le K_{i,j'}$ for $j\le j'$. We consider $FC_*(K_{i,j})$ as a doubly-directed system in Kom, inverse on $i\to-\infty$ and direct on $j\to\infty$, with maps
FC_*(K_{i',j})\to FC_*(K_{i,j}),\qquad i'\le i
induced by non-decreasing homotopies, and maps
FC_*(K_{i,j})\to FC_*(K_{i,j'}),\qquad j\le j'
induced by non-increasing homotopies.
Denote $FC_F(K_{i,j})$ the Floer subcomplex of $FC_*(K_{i,j})$ generated by orbits in the group $F$, and denote $FC_I(K_{i,j})$ the Floer quotient complex generated by orbits in the group $I$. The groups of orbits $I^-$, $I^0$, $I^+$ are ordered by action as $I^-<I^0<I^+$ within the group of orbits $I$, so that we have corresponding sub- and quotient complexes
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$, where $I^\heartsuit$ has the following meaning:
I^\varnothing=I, \quad I^{\le 0}=I^{-0}, \quad I^{>0}=I^+, \quad I^{<0}=I^-, \quad I^{=0}=I^0, \quad I^{\ge 0}=I^{0+}.
The homotopies that define the doubly-directed system $FC_*(K_{i,j})$, $i\in\Z_-$, $j\in\Z_+$ induce doubly-directed systems
FC_{I^\heartsuit}(K_{i,j}),\qquad i\in\Z_-,\quad j\in \Z_+,\quad \heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}.
Our choice of parameters ensures that
\begin{equation}\label{eq:FprecI}
F_{K_{i',j}}\prec I_{K_{i,j}},\qquad F_{K_{i,j}}\prec I_{K_{i,j'}}
\end{equation}
for $i'\le i$ and $j\le j'$. To prove these relations denote
$\mu'=\mu_{i'}$, $\tau'=\tau_{j'}$, $\delta'=\delta_{i'}$, and
similarly $\mu,\tau,\delta$ for the corresponding numbers not
decorated with primes. The first relation follows from
Lemma <ref> and the relation $F_{K_{i',j}}<I_{K_{i,j}} -
\|K_{i',j}-K_{i,j}\|_\infty$: the maximal action of an orbit in
$F_{K_{i',j}}$ is $-\mu'(1-\delta')$, the minimal action of an orbit
in $I_{K_{i,j}}$ is $-\mu+\eta_\mu$, and the maximal oscillation of
the homotopy is $\|K_{i',j}-K_{i,j}\|_\infty =
\mu'(1-\delta')-\mu(1-\delta)$; the desired relation then follows
from (<ref>). The second relation in (<ref>)
follows from $F_{K_{i,j}}<I_{K_{i,j'}}$ because in this case the
homotopy is non-increasing. Now we have already seen that
$F_{K_{i,j}}<I_{K_{i,j}}$, while the action of the orbits in
$I_{K_{i,j'}}$ is never smaller than the action of the orbits in
$I_{K_{i,j}}$. This proves the relations (<ref>). They imply
that we have a doubly-directed subsystem $FC_F(K_{i,j})$
and a doubly-directed quotient system $FC_I(K_{i,j})$, $i\in\Z_-$, $j\in\Z_+$.
To prove that we have doubly-directed systems $FC_{I^\heartsuit}(K_{i,j})$, $i\in\Z_-$, $j\in\Z_+$ for all values of $\heartsuit$ we need to show the relations
I^-_{K_{i',j}}\prec I^{0+}_{K_{i,j}} \quad \mbox{and }\quad I^{-0}_{K_{i',j}}\prec I^+_{K_{i,j}} \quad \mbox{ for }\quad i'\le i,
I^-_{K_{i,j}}\prec I^{0+}_{K_{i,j'}} \quad \mbox{and }\quad I^{-0}_{K_{i,j}}\prec I^+_{K_{i,j'}} \quad \mbox{ for }\quad j\le j'.
The last two relations follow from the fact that the non-increasing homotopies which induce maps $FC_*(K_{i,j})\to FC_*(K_{i,j'})$ for $j\le j'$ preserve the filtration by the action. In contrast, this argument cannot be used to prove the first two relations since non-decreasing homotopies typically do not preserve the action filtration. Instead we argue using the confinement lemmas in <ref>: the first relation follows from Lemma <ref>, and the second relation follows from Lemmas <ref> and <ref>.
We have isomorphisms
SH_*^\heartsuit(W)\cong \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i FH_{I^\heartsuit}(K_{i,j})
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$.
Recall that the slopes of $K_{i,j}$ are $-\mu_i$ and $\tau_j$, with
$-\mu_i<0<\tau_j$. We claim that
\begin{equation}\label{eq:tradeactionforI}
SH_*^{(-\mu_i,\tau_j)}(W)\cong FH_{I}(K_{i,j}).
\end{equation}
To prove (<ref>) recall that $SH_*^{(a,b)}(W)={\displaystyle\lim^{\longrightarrow}_K} \, FH_*^{(a,b)}(K)$, where $K$ ranges over the space $\cH(W;F)$ of admissible Hamiltonians on $\wh W_F$ with respect to the filling $F$ and the direct limit is considered with respect to non-increasing homotopies, see <ref>. Consider a decreasing sequence $i_k\to -\infty$ and an increasing sequence $j_k\to\infty$ as $k\to\infty$. The sequence of Hamiltonians $K_{i_k,j_k}$, $k\in\Z_+$ is then cofinal in $\cH(W;F)$ and we have $SH_*^{(a,b)}(W)={\displaystyle \lim^{\longrightarrow}_{k\to\infty}} \, FH_*^{(a,b)}(K_{i_k,j_k})$, where the direct limit is considered with respect to continuation maps $FH_*^{(a,b)}(K_{i_k,j_k})\to FH_*^{(a,b)}(K_{i_{k'},j_{k'}})$ induced by non-increasing homotopies. We can assume without loss of generality that $-\mu_{i_k}\le a$ and $\tau_{j_k}\ge b$. The smoothings of any such two Hamiltonians $K_{i_k,j_k}$ and $K_{i_{k'},j_{k'}}$, $k\le k'$ can be constructed so that they coincide in the neighborhood of $W$ where the periodic orbits in group $I$ for $K_{i_k,j_k}$ appear. As such, the continuation map $FC_*^{(a,b)}(K_{i_k,j_k})\to FC_*^{(a,b)}(K_{i_{k'},j_{k'}})$, which is upper triangular if we arrange the generators in increasing order of the action, has diagonal entries equal to $+1$ and is therefore an isomorphism. This proves that the canonical map $FH_*^{(a,b)}(K_{i_k,j_k})\to SH_*^{(a,b)}(W)$ is an isomorphism for all $k$ (such that $-\mu_{i_k}\le a$ and $\tau_{j_k}\ge b$).
The isomorphism (<ref>) is proved by considering the following three isomorphisms: we have $FH_I(K_{i,j})=FH_*^{(-\mu_i+\eta,\tau_j)}(K_{i,j})$ for any $\eta>0$ such that $\delta_i\mu_i<\eta<\eta_{\mu_i}$; we have $FH_*^{(-\mu_i+\eta,\tau_j)}(K_{i,j})\cong SH_*^{(-\mu_i+\eta,\tau_j)}(W)$ by the above; and we have
$SH_*^{(-\mu_i+\eta,\tau_j)}(W)\cong SH_*^{(-\mu_i,\tau_j)}(W)$ since there is no periodic Reeb orbit on $\p^-W$ with period in the interval $(\mu_i-\eta,\mu_i)$.
A variant of the same argument shows that, under the
isomorphism (<ref>), the continuation maps
$FH_I(K_{i',j})\to FH_I(K_{i,j})$, $i'\leq i$ and $FH_I(K_{i,j})\to
FH_I(K_{i,j'})$, $j\leq j'$ induced by a non-decreasing homotopy,
respectively by a non-increasing homotopy, coincide with the canonical
maps $SH_*^{(-\mu_{i'},\tau_j)}(W)\to SH_*^{(-\mu_i,\tau_j)}(W)$ and
$SH_*^{(-\mu_i,\tau_j)}(W)\to SH_*^{(\mu_i,\tau_{j'})}(W)$,
respectively. From this the conclusion of the lemma follows in the case $\heartsuit=\varnothing$.
The proof in the case $\heartsuit\neq\varnothing$ is similar in view of the isomorphisms
\begin{gather*}
SH_*^{(0^+,\tau_j)}(W)\cong FH_{I^{>0}}(K_{i,j}),\qquad
SH_*^{(0^-,\tau_j)}(W)\cong FH_{I^{\ge 0}}(K_{i,j}),\cr
SH_*^{(0^-,0^+)}(W)\cong FH_{I^{= 0}}(K_{i,j}),\qquad
SH_*^{(-\mu_i,0^+)}(W)\cong FH_{I^{\le 0}}(K_{i,j}),\cr
SH_*^{(-\mu_i,0^-)}(W)\cong FH_{I^{< 0}}(K_{i,j}).
\end{gather*}
Here $0^-$ and $0^+$ denote a negative, respectively a positive real number which is close enough to zero (with absolute value smaller than the minimal period of a closed Reeb orbit on $\p^-W$, respectively $\p^+W$).
§.§.§ Hamiltonians for $SH_*^\heartsuit(W,\p^\pm W)$.
We shall need in the sequel (Lemma <ref>) alternative descriptions of the homology groups $SH_*^\heartsuit(W,\p^\pm W)$ in the spirit of the previous section, which we now explain. We refer freely to the notation of <ref>.
Given $\mu,\tau>0$ such that $\mu\notin\mathrm{Spec}(\p^-W)$ and $\tau\notin\mathrm{Spec}(\p^+W)$, and given $\delta\in (0,1)$, we consider Hamiltonians $K^\pm=K^\pm_{\mu,\tau,\delta}:\wh W_F\to\R$ defined as follows:
* the Hamiltonian $K^-_{\mu,\tau,\delta}$ coincides with the Hamiltonian $K_{\mu,\tau,\delta}$ of <ref> on $W\cup[1,\infty)\times\p^+W$ and is equal to $-K_{\mu,\tau,\delta}$ on $F$. See Figure <ref>.
* the Hamiltonian $K^+_{\mu,\tau,\delta}$ coincides with the Hamiltonian $K_{\mu,\tau,\delta}$ on $F\cup W$ and is equal to $-K_{\mu,\tau,\delta}$ on $[1,\infty)\times\p^+W$. See Figure <ref>.
Hamiltonians $K^\pm$ for the definition of $SH_*^\heartsuit(W,\p^\pm W)$.
The $1$-periodic orbits of each of these Hamiltonians naturally fall into 5 groups, which we denote by $F^{0+},III^{-0+}$ for $K^-$, and by $F^{0+},I^{-0+}$ for $K^+$. We denote as usual by $\eta_\mu,\eta_\tau>0$ positive numbers smaller than the distance from $\mu$ to $\mathrm{Spec}(\p^- W)$, respectively smaller than the distance from $\tau$ to $\mathrm{Spec}(\p^+W)$. If the parameters are chosen such that
\begin{equation} \label{eq:another-equation}
\delta\mu<\eta_\mu \qquad \mbox{and}\qquad \mu-\eta_\mu>\tau-\eta_\tau
\end{equation}
then we have $F< I$ for $K^+$, respectively $III<F$ for $K^-$. We denote $III^{=0}=III^0$, $III^{>0}=III^{-+}$, and also $I^{=0}=I^0$, $I^{<0}=I^{-+}$.
This construction is well-behaved in families, just like the construction in the previous section.
Consider first an indexing parameter $j\in\Z_+$. We choose sequences $\mu_j\to\infty$, $\tau_j\to\infty$, $\delta_j\to 0$ as $j\to\infty$, such that $\mu_j\notin\mathrm{Spec}(\p^-W)$, $\tau_j\notin\mathrm{Spec}(\p^+W)$, such that $(\mu_j)$ and $(\tau_j)$ are increasing and $(\delta_j)$ is decreasing, and such that (<ref>) is satisfied for each $j$. We define $K^-_j=K^-_{\mu_j,\tau_j,\delta_j}$. Given $j\le j'$ we consider the interpolating homotopy from $K^-_j$ at $+\infty$ to $K^-_{j'}$ at $-\infty$
which is the concatenation of the following two monotone homotopies: first keep $K^-_j$ fixed on $W\cup[1,\infty)\times\p^+W$ and interpolate between $K^-_j$ and $K^-_{\mu_{j'},\tau_j,\delta_{j'}}$ on $F$, then keep the Hamiltonian fixed on $F\cup W$ and interpolate between $K^-_{\mu_{j'},\tau_j,\delta_{j'}}$ and $K^-_{j'}$ on $[1,\infty)\times \p^+W$. We claim that for such a homotopy we have
III_{K^-_j}\prec F_{K^-_{j'}}, \qquad III^{=0}_{K^-_j}\prec III^{>0}_{K^-_{j'}}.
The proof of the first relation uses Lemma <ref>. Since the
homotopy from $K^-_{j'}$ to $K^-_{j'}$ is non-increasing on
$[1,\infty)\times\p^+W$, the continuation Floer trajectories are
contained in $F\cup W$, where the gap between the Hamiltonians is
gap = \|(K^-_j - K^-_{j'}) \big|_{F\cup W}\|_\infty = \mu_{j'}(1-\delta_{j'})-\mu_j(1-\delta_j).
In view of Lemma <ref> it is enough to show that the maximal action of an orbit in $III_{K^-_j}$ is smaller than the minimal action of an orbit in $F_{K^-_{j'}}$ minus the $gap$. This is equivalent to the inequality $\mu_j-\eta_{\mu_j}<\mu_{j'}(1-\delta_{j'})-\big(\mu_{j'}(1-\delta_{j'})-\mu_j(1-\delta_j)\big)$, which is in turn equivalent to $\delta_j\mu_j<\eta_{\mu_j}$.
To prove the second relation we observe that the map induced by the homotopy is the composition of the maps induced by each of the monotone homotopies which constitute it. For the first homotopy, supported in $F$, there are no trajectories from $III^0_{K^-_j}$ to $III^-_{K^-_{\mu_{j'},\tau_j,\delta_{j'}}}$ by Lemma <ref>, and there are no trajectories from $III^0_{K^-_j}$ to $III^+_{K^-_{\mu_{j'},\tau_j,\delta_{j'}}}$ by Lemmas <ref> and <ref>. For the second homotopy, there are no trajectories from $III^0_{K^-_{\mu_{j'},\tau_j,\delta_{j'}}}$ to $III^{>0}_{K^-_{j'}}$ because the homotopy is non-increasing and $III^0_{K^-_{\mu_{j'},\tau_j,\delta_{j'}}}<III^{>0}_{K^-_{j'}}$. This proves the second relation. (Note that one could not argue here using the $gap$.)
As a consequence, we obtain well-defined directed systems in Kom
FC_{III^\heartsuit}(K^-_j),\qquad j\to\infty,\qquad \heartsuit\in\{\varnothing,=0,>0\}.
Consider now an indexing parameter $i\in\Z_-$. Given sequences $\mu_i\to\infty$, $\tau_i\to\infty$, $\delta_i\to 0$ as $i\to-\infty$, such that $\mu_i\notin\mathrm{Spec}(\p^- W)$, $\tau_i\notin\mathrm{Spec}(\p^+W)$, such that $(\mu_i)$ and $(\tau_i)$ are increasing with $|i|$ and $(\delta_i)$ is decreasing with $|i|$, and such that (<ref>) is satisfied for each $i$, we define $K^+_i=K^+_{\mu_i,\tau_i,\delta_i}$. Given $i'\le i$ the homotopy from $K^+_{i'}$ at $+\infty$ to $K^+_{i}$ at $-\infty$ defined as the concatenation of the two monotone homotopies from $K^+_{i'}$ to $K^+_{\mu_{i'},\tau_i,\delta_{i'}}$ and from $K^+_{\mu_{i'},\tau_i,\delta_{i'}}$ to $K^+_{i}$ is such that
F_{K^+_{i'}}\prec I_{K^+_i},\qquad I^{<0}_{K^+_{i'}}\prec I^{=0}_{K^+_i}.
The proof involves arguments entirely similar to the previous ones for the Hamiltonians $K^-$, hence we omit the details. We obtain well-defined inverse systems in Kom
FC_{I^\heartsuit}(K^+_i),\qquad i\to-\infty,\qquad \heartsuit\in\{\varnothing,<0,=0\}.
(a) For $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ we have isomorphisms
SH_*^\heartsuit(W,\p^-W) \cong \lim^{\longrightarrow}_j FH_{III^\heartsuit}(K^-_{j}).
(b) For $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ we have isomorphisms
SH_*^\heartsuit(W,\p^+W) \cong \lim^{\longleftarrow}_i FH_{I^\heartsuit}(K^+_{i}).
The proof is similar to the one of Lemma <ref>. For
part (a) observe first that the right hand side does not depend on the
choice of the family $K_j^-$ subject to conditions (<ref>).
We pick $\mu_j=\tau_j$ outside the action spectra of $\p^-W$ and
$\p^+W$ such that $\eta_{\mu_j}<\eta_{\tau_j}$, and then $\delta_j$
sufficiently small so that (<ref>) holds for all $j$.
Then a similar proof to that of equation (<ref>) yields
\begin{equation*}%\label{eq:tradeactionforI}
SH_*^{(-\infty,\tau_j)}(W,\p^-W)\cong FH_*^{(-\infty,\tau_j)}(K_j^-)\cong FH_{III}(K_j^-).
\end{equation*}
In the direct limit over $j$ we obtain part (a) for $\heartsuit=\varnothing$.
The cases $\heartsuit=''>0''$ and $\heartsuit=''=0''$ are proved
similarly, and the remaining cases are a formal consequence of these three.
The proof of part (b) is analogous, where now it suffices to treat the
cases $\heartsuit\in\{\varnothing,=0,<0\}$.
§.§.§ Hamiltonians for $SH_*^\heartsuit(V)$ inside $\wh W_F$.
Heuristically, the construction presented in this section can be viewed as the “gluing" of the three constructions presented in the two previous sections.
We consider a Liouville cobordism pair $(W,V)$ with filling $F$ and write $W=W^{bottom}\circ V\circ W^{top}$. Let
\mu, \quad \nu_\pm,\quad \tau>0
be such that $\mu\notin\mathrm{Spec}(\p^-W)$,
$\nu_\pm\notin\mathrm{Spec}(\p^\pm V)$,
$\tau\notin\mathrm{Spec}(\p^+W)$. Let $\eta_\mu$, $\eta_{\nu_\pm}$,
$\eta_\tau>0$ be positive real numbers smaller than $1/2$ and smaller
than the distances from $\mu$, $\nu_\pm$, $\tau$ to the corresponding
action spectra. Let
\delta,\epsilon\in (0,1),\quad R\in(1,\infty)
be such that
\begin{equation}\label{eq:Hmunutau-conditions-1}
\delta\mu<\eta_\mu,\quad \epsilon\nu_-<\eta_{\nu_-},\quad \nu_+<R\,\eta_{\nu_+},
\end{equation}
\begin{equation}\label{eq:Hmunutau-conditions-2}
%\nu_- - \eta_{\nu_-}<\mu(1-\delta),\quad \nu_+(R-1)\le \nu_-(1-\epsilon),\quad R(\tau-\eta_\tau)<\nu_+(R-1).
R(\tau-\eta_\tau)< R(\nu_+-\eta_{\nu_+}) < \nu_+(R-1)< \nu_--\eta_{\nu_-}<\mu-\eta_\mu.
\end{equation}
Note that the second inequality in (<ref>) is automatic in view of (<ref>).
We denote by
H_{\mu,\nu_\pm,\tau}=H_{\mu,\nu_\pm,\tau,\delta,\epsilon,R}:\wh W_F\to \R
the Hamiltonian which is defined up to smooth approximation as
follows: it is constant equal to $\epsilon\mu(1-\delta)+
\nu_-(1-\epsilon)$ on $F\setminus [\delta\epsilon,1]\times\p F$, it is
linear equal to $\mu(\epsilon-\delta\epsilon)+ \nu_-(1-\epsilon) +
\mu(\delta\epsilon-r)$ on $[\delta\epsilon,\epsilon]\times\p F$, it is
constant equal to $\nu_-(1-\epsilon)$ on $\epsilon W^{bottom}$, it is
linear equal to $\nu_-(1-\epsilon) + \nu_-(\epsilon-r)$ on
$[\epsilon,1]\times\p^-V$, it is constant equal to $0$ on $V$, it is
linear equal to $\nu_+(r-1)$ on $[1,R]\times \p^+V$, it is constant
equal to $\nu_+(R-1)$ on $R W^{top}$, and it is linear equal to
$\nu_+(R-1)+\tau(r-R)$ on $[R,\infty)\times\p^+ W$. See
Figure <ref>.
Hamiltonian adapted to the construction of the transfer map $SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)$.
The $1$-periodic orbits of the Hamiltonian $H_{\mu,\nu_\pm,\tau}$ fall into 11 classes as follows:
($F^0$) constants in $F\setminus ([\delta\epsilon,1]\times \p F)$,
($F^+$) orbits corresponding to negatively parameterized
closed Reeb orbits on$\p F=\p^-W$ and located near $\delta\epsilon\p^-W$,
($I^-$) orbits corresponding to negatively parameterized closed Reeb orbits on$\p^-W^{bottom}=\p^-W$ and located near $\epsilon\p^-W$,
($I^0$) constants in $\epsilon W^{bottom}$,
($I^+$) orbits corresponding to negatively parameterized closed Reeb orbits on$\p^+W^{bottom}=\p^-V$ and located near $\epsilon \p^-V$,
($II^-$) orbits corresponding to negatively parameterized closed Reeb orbits on $\p^-V$ and located near $\p^-V$,
($II^0$) constants in $V$,
($II^+$) orbits corresponding to positively parameterized closed Reeb orbits on $\p^+V$ and located near $\p^+V$,
($III^-$) orbits corresponding to positively parameterized closed Reeb orbits on$\p^-W^{top}=\p^+V$ and located near $R\p^+V$,
($III^0$) constants in $RW^{top}$,
($III^+$) orbits corresponding to positively parameterized closed Reeb orbits on $\p^+W$ and located near $R\p^+W^{top}=R\p^+W$.
We denote by $F$ the group of orbits $F^{0+}$, and by $J$ the group of orbits $J^{-0+}$ for $J=I,II,III$.
For the previous choices of parameters the above groups of orbits for $H_{\mu,\nu_\pm,\tau}$ are ordered as
F\prec I\prec III\prec II \quad\text{and}\quad III\prec I,
provided the almost complex structure is cylindrical and stretched enough on a collar neighborhood of $\p^+V$ in $V$.
The relation $F\prec I$ holds because $F<I$. Indeed, the maximal action of an orbit in $F$ equals $-\epsilon\mu(1-\delta)-\nu_-(1-\epsilon)$ (and is attained on $F^0$). The minimal action of an orbit in $I$ is larger than $-\nu_-(1-\epsilon) + \min(-\epsilon(\mu-\eta_\mu),-\epsilon(\nu_--\eta_{\nu_-}))$. The conclusion follows in view of $\delta\mu<\eta_\mu$ and $\mu(1-\delta)>\mu-\eta_\mu>\nu_--\eta_{\nu_-}$.
The relation $F\prec III$ holds because $F<III$. Indeed, the maximal action of an orbit in $F$ equals $-\epsilon\mu(1-\delta)-\nu_-(1-\epsilon)$. The minimal action of an orbit in $III$ is equal to $-\nu_+(R-1)$ (and is attained on $III^0$). The conclusion follows in view of $\nu_+(R-1)<\nu_--\eta_{\nu_-}<\nu_-(1-\epsilon)$.
The relation $F\prec II$ holds because $F<II$. The maximal action of an orbit in $F$ equals $-\epsilon\mu(1-\delta)-\nu_-(1-\epsilon)$. The minimal action of an orbit in $II$ is larger than $-\nu_-+\eta_{\nu_-}$. The conclusion follows in view of $\epsilon\nu_-<\eta_{\nu_-}$.
The relation $I\prec III$ holds because $I<III$, with the same proof as for $F<III$ taking into account that the maximal action of an orbit in $I$ equals $-\nu_-(1-\epsilon)$ (and is attained on $I^0$).
The relation $I\prec II$ holds because $I<II$, with the same proof as for $F<II$.
The relation $III\prec II$ is seen as follows. On the one hand we have $III<II^{0+}$. Indeed, the maximal action of an orbit in $III$ is smaller than $-\nu_+(R-1)+\max(R(\nu_+-\eta_{\nu_+}),R(\tau-\eta_\tau))$. The minimal action of an orbit in $II^{0+}$ equals $0$, and the conclusion follows in view of $R(\tau-\eta_\tau)<R(\nu_+-\eta_{\nu_+})<\nu_+(R-1)$. On the other hand we have $III\prec II^-$ by Lemma <ref> for an almost complex structure which is cylindrical and stretched enough within a collar neighborhood of $\p^+V$ in $V$.
The relation $III\prec I$ (and actually also $III\prec F$) follows also from Lemma <ref>.
Remark. Lemma <ref> should be compared to Lemma <ref>
which asserts the same ordering of groups of orbits. The latter concerns
the simpler Hamiltonians in Figure <ref> and its proof
crucially uses Lemmas <ref> and <ref>.
The former concerns the more complicated Hamiltonians in
Figure <ref> (with two additional parameters $\epsilon,R$) and its
proof uses only action estimates and Lemma <ref>.
This has the advantage that the ordering in Lemma <ref>
is preserved by continuation maps (see the proof of
Lemma <ref> below), whereas the one in
Lemma <ref> is not.
We now define a special cofinal family of Hamiltonians in $\cH^W(V;F)$ of the form above. Besides conditions (<ref>) and (<ref>), we will also need a finer relation, stated as (<ref>) below, which will be used in order to show that the continuation maps preserve the decomposition into groups of orbits given by Lemma <ref>. We will first choose the parameters $\nu_+$, $R$, $\tau$ in the region with positive slopes, and then choose the parameters $\nu_-$, $\epsilon$, $\mu$, $\delta$ in the region with negative slopes.
(a) Choice of the parameters in the region with positive slopes. We start with a sequence $(\nu_{+,j})$, $j\in\Z_+$ consisting of real numbers $\nu_{+,j}\ge 1$, which does not contain elements in $\mathrm{Spec}(\p^+V)$, such that $\nu_{+,j}<\nu_{+,j'}$ for $j<j'$, and such that $\nu_{+,j}\to\infty$ as $j\to\infty$.
We further consider a sequence $(\tau_j)$, $j\in\Z_+$ consisting of
positive real numbers such that $\tau_j\in(\nu_{+,j}/4,\nu_{+,j}/2)$,
which does not contain elements in $\mathrm{Spec}(\p^+W)$, and such that $\tau_j<\tau_{j'}$ for $j<j'$.
We choose the parameters $\eta_{\nu_{+,j}},\eta_{\tau_j}\in(0,1/2)$ such that they form monotone sequences which converge to $0$.
We then choose a sequence $(R_j)$, $j\in\Z_+$ consisting of numbers $R_j\ge 1$, such that $R_j<R_{j'}$ for $j<j'$ and $R_j\to\infty$, $j\to\infty$, and such that the last condition in (<ref>) is satisfied under the stronger form:
\begin{equation} \label{eq:Hmunutau-conditions-1-stronger}
R_j\eta_{\nu_{+,j}}>2\nu_{+,j} \qquad \mbox{Ęfor all } j\in\Z_+.
\end{equation}
(This stronger form of (<ref>) will be used in Lemma <ref>.)
The first two inequalities in (<ref>) are then satisfied.
(b) Choice of the parameters in the region with negative slopes. We start with a sequence $(\nu_{-,i})$, $i\in\Z_-$ consisting of real numbers $\nu_{-,i}\ge 1$, which does not contain elements in $\mathrm{Spec}(\p^-V)$, such that $\nu_{-,i'}>\nu_{-,i}$ for $i'<i$, and such that $\nu_{-,i}\to\infty$ as $i\to-\infty$. We choose the parameters $\eta_{\nu_{-,i}}\in(0,1/2)$ and such that they form a monotone sequence which converges to $0$. We require that the third inequality in (<ref>) is satisfied:
\nu_{+,j}(R_j-1)<\nu_{-,i}-\eta_{\nu_{-,i}}Ę\qquad \mbox{ for all }Ęi\le -j.
This last condition is implied by $\nu_{-,i}>\nu_{+,-i}(R_{-i}-1)+1/2$, $i\in\Z_-$, which provides an explicit recipe for the construction.
We choose a sequence $(\epsilon_i)$, $i\in\Z_-$ of positive numbers such that $\epsilon_{i'}<\epsilon_i$ for $i'<i$, such that $\epsilon_i\to 0$, $i\to-\infty$, and such that the second condition in (<ref>) is satisfied:
\epsilon_i\nu_{-,i}<\eta_{\nu_{-,i}} \qquad \mbox{Ęfor all } i\in\Z_-.
We also require that the sequence $1/\epsilon_i$ does not contain any element in $\mathrm{Spec}(\p^-W)$, which is a generic property.
We then consider two sequences $(\mu_i)$, $(\delta_i)$, $i\in\Z_-$ such that
\mu_i(1-\delta_i)=1/\epsilon_i
and which moreover satisfy the following conditions: the sequence $(\mu_i)$ consists of positive numbers and does not contain elements of $\mathrm{Spec}(\p^-W)$, we have $\mu_{i'}>\mu_i$ for $i'<i$ and $\mu_i\to\infty$, $i\to-\infty$; the sequence $(\delta_i)$ is such that $\delta_i\in(0,1)$ for all $i\in\Z_-$, we have $\delta_{i'}<\delta_i$ for $i'<i$ and $\delta_i\to 0$, $i\to-\infty$; the first condition in (<ref>) is satisfied:
\delta_i\mu_i<\eta_{\mu_i} \qquad \mbox{Ęfor all } i\in\Z_-.
Such sequences are easily constructed by choosing $\mu_i$ slightly larger than $1/\epsilon_i$ for all $i\in\Z_-$.
These conditions imply $\mu_i>1/\epsilon_i>\nu_{-,i}/\eta_{\nu_{-,i}}\ge 2\nu_{-,i}$ for all $i\in\Z_-$, so that the last inequality in (<ref>) is also satisfied since $\nu_{-,i}\ge 1$.
Remark. The previous choice of parameters ensures that
\begin{equation} \label{eq:special-equation}
\epsilon_{i'}\mu_{i'}(1-\delta_{i'}) = \epsilon_{i}\mu_{i}(1-\delta_{i})\qquad \mbox{Ęfor all }Ęi,i'\in\Z_-.
\end{equation}
This condition will simplify some arguments below.
Let now
H_{i,j}:=H_{\mu_i,\nu_{-,i},\nu_{+,j},\tau_j,\delta_i,\epsilon_i,R_j},\qquad i\in \Z_-,\quad j\in\Z, \quad i\le -j.
Then we have $H_{i',j}\ge H_{i,j}$ for $i'\le i$, and $H_{i,j}\le H_{i,j'}$ for $j\le j'$. We consider $FC_*(H_{i,j})$ as a doubly-directed system in Kom, inverse on $i\to-\infty$ and direct on $j\to\infty$, with maps
FC_*(H_{i',j})\to FC_*(H_{i,j}),\qquad i'\le i\le -j
induced by non-decreasing homotopies, and maps
FC_*(H_{i,j})\to FC_*(H_{i,j'}),\qquad j\le j',\quad i\le -j'
induced by non-increasing homotopies.
The choice of parameters ensures that for each $H_{i,j}$ the groups of
orbits are ordered as in Lemma <ref>.
Denote $FC_F(H_{i,j})$ the Floer subcomplex of $FC_*(H_{i,j})$ generated by orbits in the group $F$, denote $FC_{I,II,III}(H_{i,j})$ the Floer quotient complex generated by orbits in the groups $I,II,III$, and consider similarly $FC_{I,III}(H_{i,j})$ and $FC_{II}(H_{i,j})$. The groups of orbits $II^-$, $II^0$, $II^+$ are ordered by the action as $II^-<II^0<II^+$ within the group of orbits $II$, so that we have corresponding sub- and quotient complexes
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$, where $II^\heartsuit$ has the following meaning:
II^\varnothing=II,Ę \, II^{\le 0}=II^{-0}, Ę\, II^{>0}=II^+, Ę\, II^{<0}=II^-, Ę\, II^{=0}=II^0, Ę\, II^{\ge 0}=II^{0+}.
Similarly, we have orderings by the action $I^{-+}<I^0$ within the group $I$, and $III^0<III^{-+}$ within the group $III$, as well as orderings $I\prec III$ and $III\prec I$ from Lemma <ref>. We thus define $FC_{(I,III)^\heartsuit}(H_{i,j})$ for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ via
(I,III)^\varnothing=(I,III),\quad (I,III)^{\le 0}=(I,III^0),\quad (I,III)^{>0}=III^{-+},
(I,III)^{<0}=I^{-+},\quad (I,III)^{=0}=(I^0,III^0),\quad (I,III)^{\ge 0}=(I^0,III).
The homotopies that define the doubly-directed system $FC_*(H_{i,j})$ induce doubly-directed systems
FC_{II^\heartsuit}(H_{i,j}), \qquad FC_{I^\heartsuit}(H_{i,j}), \qquad FC_{III^\heartsuit}(H_{i,j})
\qquad \mbox{and }\qquad FC_{(I,III)^\heartsuit}(H_{i,j})
for $i\in\Z_-$, $j\in \Z_+$, $i\le -j$ and $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$.
(1) We consider first the continuation maps
FC_*(H_{i',j})\to FC_*(H_{i,j}),\qquad i'\le i \le j
induced by non-decreasing homotopies equal to $H_{i',j}$ near $+\infty$ and equal to $H_{i,j}$ near $-\infty$. The positive slopes $\nu_{+,j}$, $\tau_j$ are fixed, as well as the parameter $R_j$, and the homotopy is constant outside $F\circ W^{bottom}$.
Denote for simplicity $H=H_{i,j}$, $H'=H_{i',j}$, and $\nu_-=\nu_{-,i}$, $\nu_-'=\nu_{-,i'}$, $\epsilon=\epsilon_i$, $\epsilon'=\epsilon_{i'}$, $\mu=\mu_i$, $\mu'=\mu_{i'}$. The gap $\|H-H'\|_\infty$ between the two Hamiltonians is equal to the biggest value among $(1-\epsilon')\nu'_--(1-\epsilon)\nu_-$ (the difference of values in the region $I^0$) and $(1-\epsilon')\nu'_-+\epsilon'\mu'(1-\delta') - (1-\epsilon)\nu_--\epsilon\mu(1-\delta)$ (the difference of values in the region $F^0$). Condition (<ref>) ensures that these two values are equal, hence
gap:=\|H-H'\|_\infty = (1-\epsilon')\nu'_--(1-\epsilon)\nu_-\,.
In the sequel we will repeatedly apply Lemma <ref> (without
further mentioning it), which asserts that for two groups of orbits
$P_{H_+} < P_{H_-} - gap$ implies $P_{H_+}\prec P_{H_-}$.
We first prove that
F_{H'},I_{H'},III_{H'}\prec II_H,
so that we have induced maps $FC_{II}(H')\to FC_{II}(H)$. We have
$F^0_{H'}+gap < I^0_{H'}+gap <II^-_H$: the first inequality is obvious, and the second inequality is equivalent to $-(1-\epsilon)\nu_-<-\nu_-+\eta_{\nu_-}$, which is implied by $\epsilon\nu_-<\eta_{\nu_-}$. This ensures $F_{H'}\prec II_H$ and $I_{H'}\prec II_H$.
The condition $III_{H'}\prec II_H$ is proved in the same way as $III_H\prec II_H$, in view of $III_{H'}=III_H$.
We now prove that
F_{H'}\prec I_{H},III_H,
so that we also have induced maps $FC_{I,II,III}(H')\to FC_{I,II,III}(H)$ as well as $FC_{I,III}(H')\to FC_{I,III}(H)$. The relation $F_{H'}\prec I_H$ follows from $F^0_{H'}+gap <\min(I^-_H,I^+_H)$, which is $-\epsilon'(1-\delta')\mu'-(1-\epsilon)\nu_-<-(1-\epsilon)\nu_- + \min(-\epsilon(\mu-\eta_\mu),-\epsilon(\nu_--\eta_{\nu_-}) = -(1-\epsilon)\nu_- - \epsilon(\mu-\eta_\mu)$. This is equivalent to $-(1-\delta)\mu<-(\mu-\eta_\mu)$ in view of (<ref>), and holds in view of $\delta\mu<\eta_\mu$. The relation $F_{H'}\prec III_H$ follows from the previous one: indeed $I_H<III_H$, hence $F^0_{H'}+gap<III_H$.
We also have
III_{H'}\prec I_H \qquad \mbox{ and }ĘI_{H'}\prec III_H.
The first relation follows from Lemma <ref>. The second relation is a consequence of $I^0_{H'}+gap < III^0_H$, which is $-(1-\epsilon')\nu'_- + (1-\epsilon')\nu'_--(1-\epsilon)\nu_- < -\nu_+(R-1)$, which is equivalent to $\nu_+(R-1)<(1-\epsilon)\nu_-$ and is implied by (<ref>) and (<ref>). The continuation maps therefore preserve the decomposition $FC_{I,III}(H)=FC_I(H)\oplus FC_{III}(H)$.
We now prove that
II^-_{H'}\prec II^{0+}_H \qquad \mbox{Ęand }Ę\qquad II^{-0}_{H'}\prec II^+_H,
so that we have induced maps $FC_{II^\heartsuit}(H')\to
FC_{II^\heartsuit}(H)$ for all values of $\heartsuit$. The first
relation follows from Lemmas <ref>, <ref>,
and <ref>, while the last relation follows from
Lemmas <ref> and <ref>
(using $H'=H$ outside $F\circ W^{bottom}$).
Note that in this
situation we cannot argue using the action because the homotopy only
preserves the action filtration up to an error given by the $gap$, and
the latter can be arbitrarily large.
We now prove that
I^{-+}_{H'}\prec (I^0_H,III_H) \qquad \mbox{ and }Ę\qquad (I_{H'},III^0_{H'})\prec III^{-+}_H,
which implies that we have induced maps $FC_{(I,III)^\heartsuit}(H')\to FC_{(I,III)^\heartsuit}(H)$ for all values of $\heartsuit$.
In view of $I_{H'}\prec III_H$, the first relation is a consequence of
$I^{-+}_{H'}\prec I^0_H$, which is in turn implied by $I^{-+}_{H'}+gap
< I^0_H$. The latter is seen to hold as follows. Denote by
$T_{\p^-V}$, $T_{\p^-W}$ the minimal period of a closed Reeb orbit on
$\p^-V$, respectively on ${\p^-W}$, and set $T_-:=\min(T_{\p^-V},
T_{\p^-W})>0$. The desired inequality is implied by
$-(1-\epsilon')\nu'_--\epsilon'T_- + (1-\epsilon')\nu'_- -
(1-\epsilon)\nu_- < -(1-\epsilon)\nu_-$, which holds because
In view of $I_{H'}\prec III_H$, the second relation is a consequence
of $III^0_{H'}\prec III^{-+}_H$. The relation $III^0_{H'}\prec
III^+_H$ is a consequence of Lemmas <ref>
and <ref> in view of the fact that the homotopy is constant
outside $F\circ W^{bottom}$.
The relation $III^0_{H'}\prec III^-_H$ is a consequence of Lemma <ref>. Note that in both situations we cannot argue using the action because the homotopy only preserves the action filtration up to an error given by the $gap$, and the latter can be arbitrarily large.
(2) We now consider the continuation maps
FC_*(H_{i,j})\to FC_*(H_{i,j'}),\qquad j\le j',\qquad i\le -j
induced by non-increasing homotopies equal to $H_{i,j}$ near $+\infty$
and equal to $H_{i,j'}$ near $-\infty$. The negative slopes
$\nu_{-,i}$, $\mu_i$ are fixed, as well as the parameters
$\epsilon_i,\delta_i$, and the homotopy is constant on $F\circ
W^{bottom}\circ V$.
This situation is easier than the one in (1) because
here the continuation maps preserve the action filtration.
Denote again for simplicity $H=H_{i,j}$, $H'=H_{i,j'}$, and $\nu_+=\nu_{+,j}$, $\nu_+'=\nu_{+,j'}$, $R=R_j$, $R'=R_j'$, $\tau=\tau_j$, $\tau'=\tau_{j'}$.
The relations
F_H\prec I_{H'},II_{H'},III_{H'} \qquad \mbox{ and }Ę\qquad I_H\prec II_{H'}
follow as in Lemma <ref>. On the one hand we have
$I_{H'}=I_H$ and $II_{H'}^{-0}=II_H^{-0}$, so that $F_H\prec
I_{H'},II^{-0}_{H'}$ and $I_H\prec II^{-0}_{H'}$. On the other hand we
have $F^0_H<II^0_H=II^0_{H'}<II^+_{H'}$ and
$F^0_H=F^0_{H'}<III^0_{H'}<III^{-+}_{H'}$ for $i\le -j'$ which implies
$F_H\prec II^+_{H'},III_{H'}$. Finally, we also have
which implies $I_H\prec II_{H'}$.
The relation
III_H\prec II_{H'}
is proved as follows. We have $III_H\prec II^-_{H'}$ as in
Lemma <ref>, using Lemma <ref>. We have
$III^{0+}_H<II^{0+}_{H'}$ by (<ref>), namely
$R(\tau-\eta_\tau)<\nu_+(R-1)$. Finally we have
by (<ref>), namely $R\eta_{\nu_+}>\nu_+$.
The relation
III_H\prec I_{H'}
is proved as in Lemma <ref>, using Lemma <ref>.
The continuation map
FC_{II}(H)\to FC_{II}(H')
is induced by a non-increasing homotopy hence preserves the filtration by the action. As a consequence we obtain well-defined continuation maps
FC_{II^\heartsuit}(H)\to FC_{II^\heartsuit}(H')
for all values of $\heartsuit$.
Let us now prove that the continuation map
FC_{I,III}(H)\to FC_{I,III}(H')
induces maps
FC_{(I,III)^\heartsuit}(H)\to FC_{(I,III)^\heartsuit}(H')
for all values of $\heartsuit$. We need to show the relations
$I^{-+}_H\prec I^0_{H'},III_{H'}$ and $I_H,III^0_H\prec
III^{-+}_{H'}$. The first relation follows from
$I^{-+}_H<I^0_H=I^0_{H'}<III^0_{H'}<III^{-+}_{H'}$, where the middle
inequality is ensured
by (<ref>) and (<ref>), namely
The second relation follows from $I^0_H<III^0_{H'}<III^{-+}_{H'}$.
The above shows that we actually have non-interacting doubly-directed systems
FC_{I^\heartsuit}(H_{i,j})\qquad \mbox{and }\qquad FC_{III^\heartsuit}(H_{i,j})
for all values of $\heartsuit$ and Lemma <ref>
is proved.
We have isomorphisms
SH_*^\heartsuit(V)\cong \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i FH_{II^\heartsuit}(H_{i,j})
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$.
The proof is very much similar to that of Lemma <ref>.
Recalling that the slopes near $\p^\pm V$ for $H_{i,j}$ are $-\nu_{-,i}$ and $\nu_{+,j}$, the key identity is
\begin{equation} \label{eq:tradeactionforII}
SH_*^{(-\nu_{-,i},\nu_{+,j})}(V)\cong FH_{II}(H_{i,j}).
\end{equation}
To prove (<ref>) recall from
Lemma <ref> that $SH_*^{(a,b)}(V)$ can
be expressed as a direct limit over Hamiltonians in $H^W(V;F)$ of
Floer homology groups truncated in the action window $(a,b)$. In
particular, considering a decreasing sequence $i_k\to-\infty$ and an
increasing sequence $j_k\to\infty$ as $k\to\infty$
with $i_k\leq -j_k$, we have $SH_*^{(a,b)}(V)={\displaystyle
\lim^{\longrightarrow}_{k\to\infty}} \,
FH_*^{(a,b)}(H_{i_k,j_k})$. Here the direct limit is understood with
respect to continuation maps $FH_*^{(a,b)}(H_{i_k,j_k})\to
FH_*^{(a,b)}(H_{i_{k'},j_{k'}})$ induced by non-increasing homotopies.
We claim that for $k$ large enough such that $\nu_{+,j_k}\geq-a$
we have $FC_*^{(a,b)}(H_{i_k,j_k})=FC_{II}^{(a,b)}(H_{i_k,j_k})$. The
proof is similar to the proof of Lemma <ref>:
We need to show that the actions of orbits in groups $F$, $I$ and
$III$ are below $a$. For the groups $F$ and $I$ this is obvious. The
actions within group $III$ are ordered as $III^0<III^{-+}$. The maximal action of
the orbits in group $III^{-}$ is bounded above by
$-\nu_+(R-1)+R(\nu_+-\eta_{\nu_+}) = \nu_+-R\eta_{\nu_+} < -\nu_+\leq a$,
where we have dropped the index $j_k$ and the first inequality follows
from condition (<ref>). Similarly,
the maximal action of the orbits in group $III^+$ is bounded above by
$-\nu_+(R-1)+R(\tau-\eta_{\tau}) < -\nu_+(R-1)+R(\nu_+-\eta_{\nu_+})<a$,
where the first inequality follows from (<ref>)
and the second one from the one for group $III^-$.
Combining this with the previous paragraph we obtain
SH_*^{(a,b)}(V)={\displaystyle \lim^{\longrightarrow}_{k\to\infty}}
\, FH_{II}^{(a,b)}(H_{i_k,j_k}).
Assume now without loss of generality that $-\nu_{-,i_k}\le a$ and
$\nu_{+,j_k}\ge b$. The smoothings of any such two Hamiltonians
$H_{i_k,j_k}$ and $H_{i_{k'},j_{k'}}$, $k\le k'$ can be constructed so
that they coincide in the neighborhood of $V$ where the periodic
orbits in group $II$ for $H_{i_k,j_k}$ appear. As such, the
continuation map $FC_{II}^{(a,b)}(H_{i_k,j_k})\to
FC_{II}^{(a,b)}(H_{i_{k'},j_{k'}})$, which is upper triangular if we
arrange the generators in increasing order of the action, has diagonal
entries equal to $+1$ and is therefore an isomorphism. This proves
that we have a canonical isomorphism
$FH_{II}^{(a,b)}(H_{i_k,j_k})\cong SH_*^{(a,b)}(V)$ for all $k$ (such
that $-\nu_{-,i_k}\le a$ and $\nu_{+,j_k}\ge b$).
This implies (<ref>) by choosing $a=-\nu_{-,i}$ and $b=\nu_{+,j}$.
A variant of this argument shows that, under the isomorphism (<ref>), the continuation maps $FH_{II}(H_{i',j})\to FH_{II}(H_{i,j})$, $i'\leq i$ and $FH_{II}(H_{i,j})\to FH_{II}(H_{i,j'})$, $j\leq j'$ induced by a non-decreasing homotopy, respectively by a non-increasing homotopy, coincide with the canonical maps $SH_*^{(-\nu_{-,i'},\nu_{+,j})}(V)\to SH_*^{(-\nu_{-,i},\nu_{+,j})}(V)$ and $SH_*^{(-\nu_{-,i},\nu_{+,j})}(V)\to SH_*^{(-\nu_{-,i},\nu_{+,j'})}(V)$, respectively. The conclusion of the Lemma follows in the case $\heartsuit=\varnothing$.
The proof in the case $\heartsuit\neq\varnothing$ is similar, as in Lemma <ref>.
We have isomorphisms
SH_*^\heartsuit(W^{bottom},\p^+W^{bottom})\cong \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i FH_{I^\heartsuit}(H_{i,j})
SH_*^\heartsuit(W^{top},\p^-W^{top})\cong \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i FH_{III^\heartsuit}(H_{i,j})
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$.
(1) We prove the first isomorphism. Since the group of orbits $I$ is
located in the region where the Hamiltonians $H_{i,j}$ have negative
slope the direct limit over $j$ plays no role and we can assume
without loss of generality that $j=j_0$ is constant. The Floer
trajectories involved in the differential for $FC_I(H_{i,j})$ and
also the relevant continuation Floer trajectories are confined to a
neighborhood of $F\circ W^{bottom}$ by Lemma <ref>. We
can thus replace the Hamiltonians $H_i=H_{i,j_0}$ by Hamiltonians
$\wt H_i$ which coincide with $H_i$ in $F\circ W^{bottom}\circ V$
and are constant equal to $0$ on $V\circ
W^{top}\circ[1,\infty)\times\p^+W$. We can further shift these
Hamiltonians to $\ol H_i=\wt H_i-\nu_{-,i}(1-\epsilon_i)$ so that
the orbits in group $I$ lie on level $0$, and further replace $\ol
H_i$ by $\cH_i=\epsilon_i\ol H_i\circ \varphi_Z^{\ln
1/\epsilon_i}$, so that the orbits in group $I$ for $\cH_i$ lie
in a neighborhood of $W^{bottom}$, and the slopes of $\cH_i$ in
the linear regions are the same as the slopes of $\ol
H_i$. Finally, we can further replace the Hamiltonians $\cH_i$ by
$\wt \cH_i$ defined on $\wh W^{bottom}_F$ which coincide with
$\cH_i$ on $F\circ W^{bottom}$ and continue on
$[1,\infty)\times\p^+W^{bottom}$ linearly with the same slope
The resulting inverse system is cofinal and, by Lemma <ref>(b), it computes $SH_*^\heartsuit(W^{bottom},\p^+W^{bottom})$.
(2) We prove the second isomorphism. Since the group of orbits $III$
is located in the region where the Hamiltonians $H_{i,j}$ have
positive slope, the inverse limit over $i$ plays no role and we can
deform each Hamiltonian $H_{i,j}$ to a Hamiltonian $\wt H_j$ which
coincides with $H_{i,j}$ on $V\circ W^{top}\circ
[1,\infty)\times\p^+W$, and is constant equal to $0$ on $F\circ
W^{bottom}\circ V$. We can further shift these Hamiltonians to $\ol
H_j=\wt H_j-\nu_{+,j}(R_j-1)$ so that the orbits in group $III$ lie
on level $0$, and further replace $\ol H_j$ by $\cH_j=R_j\ol
H_j\circ \varphi_Z^{\ln 1/R_j}$, so that the orbits in group $III$
for $\cH_j$ lie in a neighborhood of $W^{top}$.
The resulting direct system is cofinal and, by Lemma <ref>(a), it computes $SH_*^\heartsuit(W^{top},\p^- W^{top})$.
Lemmas <ref> and <ref> imply
that for all flavors $\heartsuit$ we have isomorphisms
SH_*^\heartsuit(W^{top},\p^-W^{top})\oplus SH_*^\heartsuit(W^{bottom},\p^+W^{bottom})\cong \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i FH_{(I,III)^\heartsuit}(H_{i,j}).
On the other hand, by the Excision Theorem <ref> we have isomorphisms
SH_*^\heartsuit(W,V) \cong SH_*^\heartsuit(W^{bottom},\p^-V) \oplus SH_*^\heartsuit(W^{top},\p^+V).
Combining these isomorphisms we obtain
We have isomorphisms
SH_*^\heartsuit(W,V) \cong \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i FH_{(I,III)^\heartsuit}(H_{i,j})
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$.
§.§.§ The transfer map revisited
Consider again a Hamiltonian $H=H_{\mu,\nu_\pm,\tau}$ as in Figure <ref> above.
We associate to it a new Hamiltonian $L\leq H$ defined as follows: it
is constant equal to $\mu(\eps-\delta\eps) + \nu_-(1-\eps)$ on
$F\setminus[\delta\eps,1]\times\p F$, it is linear of slope $-\mu$ on
$[\delta\eps,\xi]\times\p F$, it is constant equal to $0$ on
$[\xi,1]\times\p F\cup W\cup[1,R]\times\p^+W$, and it is linear of
slope $\tau$ on $[R,\infty)\times\p^+W$. See Figure <ref>.
Hamiltonian $L$ for the construction of the transfer map.
Here the constant $\xi$ is determined by the construction and given by
\xi = \frac{\nu_-}{\mu}(1-\eps)+\eps \in (\eps,1).
The orbits of the Hamiltonian $L$ fall as usual into 5 groups
$F^{0+},I^{-0+}$ and we have $F<I^-<I^0<I^+$. Indeed, the smallest
action of an orbit in group $I^-$ is $-\xi(\mu-\eta_\mu)$,
whereas the largest action of an orbit in group $F$ is
$-\mu(\xi-\delta\epsilon)$, and we have
$-\mu(\xi-\delta\epsilon)<-\xi(\mu-\eta_\mu)$, which is equivalent to
$\mu\delta\eps<\xi\eta_\mu$, in view of $\mu\delta<\eta_\mu$ and
Consider now a Hamiltonian $K:=K_{\mu,\tau,\delta'}$ as in
Figure <ref>, with $\delta'\in(0,1)$ such that
$\mu\delta'<\eta_\mu$ and $\mu(1-\delta')>\mu(\xi-\delta\eps)$,
i.e. the maximal level of $K$ is larger than the maximal level of $L$.
We then have $L\leq K$, and the $1$-periodic
orbits of $L$ and $K$ are in canonical one-to-one correspondence.
The canonical one-to-one correspondence between $1$-periodic
orbits of $L$ and $K$ induces for all flavors $\heartsuit$ homotopy equivalences
FC_{I^\heartsuit}(L)\stackrel{\sim}\longrightarrow FC_{I^\heartsuit}(K).
Consider the non-increasing homotopy from $K$ to $L$
by moving the line segment of slope $-\mu$ to the left, moving the one of
slope $\tau$ to the right, and pushing down the level on region
$F$. The $1$-periodic orbits of the Hamiltonians
in this homotopy are in canonical one-to-one correspondence and the
actions of their orbit groups are always ordered as $F<I^-<I^0<I^+$.
Therefore, we can consider for each flavor $\heartsuit$ moving action
windows that single out the orbit group $I^\heartsuit$ such that no
action crosses the boundary of the action windows, so the continuation map is
a composition of small distance isomorphisms and therefore an isomorphism.
Each of these small distance isomorphisms is itself a continuation map and is induced by the canonical
correspondence between $1$-periodic orbits. To see this, argue by contradiction and use the fact that the constant homotopy at any given Hamiltonian induces the identity at chain level since the only index $0$ Floer trajectories in such a homotopy are the constant ones at any orbit. It follows that the global continuation map is itself induced by the canonical correspondence between $1$-periodic orbits.
Consider now a doubly-directed system $H_{i,j}$ as in Section <ref>.
Let $L_{i,j}$ and $K_{i,j}$ be the Hamiltonians associated
to $H_{i,j}$ as in the previous paragraph. We turn $L_{i,j}$ into a doubly
directed system in Kom by composing the continuation maps $FC(K_{i',j})\to
FC(K_{i,j})$ and $FC(K_{i,j})\to FC(K_{i,j'})$ with the canonical maps in
Lemma <ref> and their inverses. (Note that in general we do
not have $L_{i',j}\geq L_{i,j}$ for $i'\leq i\leq -j$.) Then all the
results for the system $K_{i,j}$ in <ref> carry over
to the system $L_{i,j}$.
Recall that $L_{i,j}\leq H_{i,j}$ and the orbits in group $F$ for
$L_{i,j}$ and $H_{i,j}$ coincide. Therefore, by Lemma <ref>
the actions of the orbit groups satisfy $F_{L_{i,j}}<
(I,II,III)_{H_{i,j}}$. We thus obtain induced chain maps
f_{i,j}:FC_{I}(L_{i,j})\to FC_{I,II,III}(H_{i,j}) \to FC_{II}(H_{i,j})
which define a morphism of doubly-directed systems in Kom. Here
the first map is the continuation map and the second one the
projection onto the quotient complex in view of Lemma <ref>.
Since these maps preserve the filtration by action, we also have
induced chain maps
f_{i,j}^\heartsuit:FC_{I^\heartsuit}(L_{i,j})\to FC_{II^\heartsuit}(H_{i,j})
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$, which define
morphisms of doubly-directed systems in Kom.
We denote $(f_{i,j}^\heartsuit)_*$ the maps induced in homology.
Under the isomorphisms of Lemmas <ref>,
<ref> and <ref> we have
f_!^\heartsuit = \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i \, (f_{i,j}^\heartsuit)_*,
where $f_!^\heartsuit: SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)$ is
the transfer map from Definition <ref>.
Recall from (<ref>) and Lemma <ref> the isomorphisms
SH_*^{(-\mu_i,\tau_j)}(W)\cong FH_{I}(K_{i,j})\cong FH_{I}(L_{i,j}).
Recall also from (<ref>) the isomorphism
SH_*^{(-\nu_{-,i},\nu_{+,j})}(V)\simeq FH_{II}(H_{i,j}).
Recall that $\mu_i\ge\nu_{-,i}$ and $\tau_j\le \nu_{+,j}$.
It follows from the proofs of Lemmas <ref>
and <ref> that the continuation map
$(f_{i,j})_*:FH_{I}(L_{i,j})\to FH_{II}(H_{i,j})$ coincides via the
above isomorphisms with the composition of the transfer map
SH_*^{(-\mu_i,\tau_j)}(V)$ with the canonical map given by enlarging/restricting
the action window $SH_*^{(-\mu_i,\tau_j)}(V)\to
SH_*^{(-\nu_{-,i},\nu_{+,j})}(V)$, i.e.
\xymatrix
SH_*^{(-\mu_i,\tau_j)}(W) \ar[rr]^{(f_{i,j})_*} \ar[dr]_{f_!^{(-\mu_i,\tau_j)}} & & SH_*^{(-\nu_{-,i},\nu_{+,j})}(V) \\
& SH_*^{(-\mu_i,\tau_j)}(V) \ar[ur] &
Since $-\nu_{-,i}\to -\infty$ as $i\to-\infty$ and $\tau_j\to+\infty$ as $j\to+\infty$, and since the continuation maps in the doubly-directed systems for $L_{i,j}$ and $H_{i,j}$ correspond under the previous isomorphisms to enlarging/restricting the action windows (Lemmas <ref> and <ref>), we obtain
f_! = \lim^{\longrightarrow}_j \lim^{\longleftarrow}_i \, (f_{i,j})_*.
This proves the lemma for $\heartsuit=\varnothing$. The proof for the
other values of $\heartsuit$ is entirely analogous.
§.§ Symplectic homology of a pair as a homological mapping cone
Let $f_{i,j}^\heartsuit$ be the chain maps
constructed in <ref>. The
discussion in <ref> shows that the cones
$C(f_{i,j}^\heartsuit)$ form a doubly-directed system, and we define
(compare with Corollary <ref>)
SH^{\heartsuit,cone}_*(W,V):=\lim^{\longrightarrow}_j \lim^{\longleftarrow}_i \, H_*(C(f_{i,j}^\heartsuit)).
The goal of this section is to prove the following proposition.
Let $(W,V)$ be a cobordism pair. Then we have an isomorphism
SH_*^{\heartsuit,cone}(W,V)\cong SH_*^\heartsuit(W,V)[-1]
for $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$.
In view of Corollary <ref> it will be enough to prove
\begin{equation}\label{eq:cone-lim}
\lim^{\longrightarrow}_j \lim^{\longleftarrow}_i \,H_*(C(f_{i,j}^\heartsuit))
= \lim^{\longrightarrow}_j\lim^{\longleftarrow}_i \, FH_{(I,III)^\heartsuit}(H_{i,j})[-1]
\end{equation}
for all values of $\heartsuit$.
We recall the notation $W=W^{bottom}\circ V\circ W^{top}$. Recall the
families of Hamiltonians $H_{i,j}$ and $L_{i,j}$ from <ref>.
For a fixed value of the double index $(i,j)$ we denote for
readability $H=H_{i,j}$ and $L=L_{i,j}$.
Let $\heartsuit=\varnothing$. We claim that any monotone homotopy from
$L$ to $H$ induces a homotopy equivalence
FC_{I}(L)\stackrel\sim \longrightarrow FC_{I,II,III}(H).
To see this, consider for $t\in[0,1]$ the non-increasing homotopy of
Hamiltonians $H^t$ as in Figure <ref> from $H^0=H$ to $H^1=L$.
Each $H^t$ has the shape considered in Section <ref> with parameters
\mu^t=\mu,\ \nu^t_-\in[0,\nu_-],\ \nu^t_+\in[0,\nu_+],\ \tau^t=\tau,\ \delta^t>0,\ \eps^t\in[\eps,\xi],\ R^t=R
\delta^t\eps^t=\delta\eps,\qquad \nu^t_-(1-\eps^t)=\mu(\xi-\eps^t).
Thus $\eps^t$ increases with $t$, while $\delta^t$ and $\nu^t_-$ decrease with $t$.
The actions of orbits in the regions $I$, $II$ and $III$ are
bounded below by $-\mu(\xi-\eps^t)-\eps^t(\mu-\eta_\mu)=-\mu\xi+\eps^t\eta_\mu$,
$-\wt\nu_-^t$ and $-\nu^t_+(R-1)$, respectively, all of which
increase with $t$. Here
$\wt\nu_-^t$ denotes $\nu_--\eta_{\nu_-}$ for $\nu_-^t\geq\nu_--\eta_{\nu_-}$ and $\nu_-^t$ otherwise.
Since the action of orbits in region $F$ is independent of $t$
and the actions satisfy $F<I,II,III$ for $t=0$, it follows that
$F<I,II,III$ holds for all $t\in[0,1]$.
Considering a moving action window separating the orbit group $F$ from
the groups $I,II,III$,
we see that the continuation map $FH_{I}(L)\to FH_{I,II,III}(H)$ is a
composition of small distance isomorphisms and thus an isomorphism.
This proves the claim.
From the commutative diagram
\xymatrix{
FC_I(L)\ar[rr]^f \ar[dr]^\sim_{h.e.}& & FC_{II}(H) \\
& FC_{I,II,III}(H)\ar[ur]_p &
in which $p$ is the projection induced by the ordering $I,III\prec II$, we infer by Lemma <ref>(ii) that we have an isomorphism in Kom
C(f)\cong C(p)\cong FC_{I,III}(H)[-1].
This isomorphism is compatible with continuation maps, and hence with the structure of a doubly-directed system.
In the first-inverse-then-direct limit this yields (<ref>) for $\heartsuit=\varnothing$.
Let $\heartsuit=``=0"$. The orbits of $L$ in the group $I^0$ are constants, and we separate them as $I^0=I^{0bottom}\sqcup I^{0V}\sqcup I^{0top}$, according to whether they lie in $W^{bottom}$, $V$, respectively $W^{top}$, with the orbits lying in $W^{bottom}\cup W^{top}$ forming a subcomplex, and the orbits lying in $V$ forming a quotient complex (this is achieved by perturbing $L$ along $W$ by a Morse function whose restriction to $V$ is smaller than its restriction to $W^{bottom}\cup W^{top}$). The Floer complex reduces to the Morse complex by symplectic asphericity <cit.>, and we therefore have canonical identifications $FC_{I^{0bottom}}(L)\equiv FC_{I^0}(H)$, $FC_{I^{0V}}(L)\equiv FC_{II^0}(H)$, and $FC_{I^{0top}}(L)\equiv FC_{III^0}(H)$.
The continuation map $f^{=0}:FC_{I^0}(L)\to FC_{II^0}(H)$ is identified with the projection $FC_{I^0}(L)\to FC_{I^{0V}}(L)$, and by Lemma <ref>(ii) we have an isomorphism in Kom
C(f^{=0})\cong FC_{I^{0bottom,0top}}(L)[-1] \equiv FC_{I^0,III^0}(H)[-1].
This identification is compatible with continuation maps, and hence with the structure of a doubly-directed system.
In the first-inverse-then-direct limit this yields (<ref>) for $\heartsuit=``=0"$.
Let $\heartsuit=``<0"$.
We denote $FC_{I^{0bottom}}(L)$ the complex generated by the critical
points of $L$ inside $W^{bottom}$, and we recall the canonical
identification $FC_{I^{0bottom}}(L)\simeq FC_{I^0}(H)$ which we
already discussed in the case $\heartsuit=``=0"$ above.
We claim that any monotone homotopy from $L$ to $H$ induces a homotopy
FC_{I^{-,0bottom}}(L)\stackrel\sim\longrightarrow FC_{I,II^-}(H).
To see this, consider the composition
g:FC_{I^{-,0bottom}}(L)\longrightarrow FC_{I,II^-,III}(H)\longrightarrow FC_{I,II^-}(H),
where the first map is the continuation map and the second one is the
quotient projection according to Lemma <ref>. Note that
the subcomplexes $FC_{I^{-,0bottom}}(L)$ and $FC_{I,II^-,III}(H)$
correspond to the negative action parts if we choose the perturbing
Morse functions to be positive on $W^{bottom}$ and negative on $V\cup W^{top}$.
Since the homotopy is constant on $V$, Lemma <ref> shows
that the Floer cylinders counted by the map $g$ lie entirely in $F\cup W^{bottom}$.
Therefore, the map $g$ agrees with the continuation map $FC^{<0}(\wt
L)\to FC^{<0}(\wt H)$, where $\wt L$, $\wt H$ are the Hamiltonians
that agree with $L$, $H$ on $F\cup W^{bottom}$ and are equal to zero on
$V\cup W^{top}$. The argument in the case $\heartsuit=\varnothing$,
setting the Hamiltonians $H^t$ also to zero on $V\cup W^{top}$, shows that
this map is a homotopy equivalence. This proves the claim.
Consider now the commutative diagram
\xymatrix{
FC_{I^{-,0bottom}}(L)\ar[rr]^\varphi \ar[dr]^\sim_{h.e.}& & FC_{II^-}(H) \\
& FC_{I,II^-}(H) \ar[ur]_p &
in which $p$ is the projection determined by the ordering $I\prec II^-$. It follows from Lemma <ref>(ii) that we have an isomorphism in Kom
C(\varphi)\cong C(p)\cong FC_{I}(H)[-1].
We then consider the diagram of short exact sequences of complexes
\xymatrix
FC_{I^{-}}(L)\ar[r] \ar[d]^{f^{<0}} & FC_{I^{-,0bottom}}(L) \ar[r] \ar[d]^\varphi & FC_{I^{0bottom}}(L) \ar[d]\\
FC_{II^-}(H) \ar@{=}[r] \ar[d] & FC_{II^{-}}(H) \ar[r] \ar[d] & 0 \ar[d] \\
C(f^{<0}) & C(\varphi)\cong FC_{I}(H)[-1] \ar[r]^{\cong \, proj[-1]} & C(0)\cong FC_{I^0}(H)[-1]
The top right square commutes up to homotopy by Proposition <ref> because the cone of the identity map on the second line is homotopic to zero.
The cone of $\varphi$ has been identified above, and the bottom right map induced between the cones is homotopic to the projection $FC_I(H)[-1]\stackrel{proj[-1]}\longrightarrow FC_{I^0}(H)[-1]$.
It then follows from Proposition <ref> and Lemma <ref>(ii) that we have isomorphisms in Kom
C(f^{<0})\cong C(proj[-1])[1]\cong C(proj) \cong FC_{I^{-+}}(H)[-1].
For the middle isomorphism see (<ref>). The identification
$C(f^{<0})\cong FC_{I^{-+}}(H)[-1]$ is compatible with continuation
maps, and hence with the structure of a doubly-directed system.
In the first-inverse-then-direct limit this yields (<ref>) for $\heartsuit=``<0"$.
Let $\heartsuit=``\ge 0"$. This is a consequence of the cases
$\heartsuit=\emptyset$ and $\heartsuit=``<0"$.
For this, we consider the diagram
\xymatrix
FC_{I^-}(L)\ar[r] \ar[d]^{f^{<0}} & FC_{I}(L)\ar[r] \ar[d]^{f} & FC_{I^{0+}}(L) \ar[d]^{f^{\ge 0}} \\
FC_{II^-}(H)\ar[r] \ar[d] & FC_{II}(H) \ar[r] \ar[d] & FC_{II^{0+}}(H) \\
C(f^{<0})\cong FC_{I^{-+}}(H)[-1] \ar[r]^{\cong\, incl[-1]} & C(f) \cong FC_{I,III}(H)[-1] &
The cones of $f^{<0}$ and $f$ have been identified above, and the map induced between the cones is homotopic to the inclusion $FC_{I^{-+}}(H)[-1] \stackrel{incl[-1]}\longrightarrow FC_{I,III}(H)[-1]$. It then follows from Proposition <ref> and Lemma <ref>(ii) that we have isomorphisms in Kom
C(f^{\ge 0})\cong C(incl[-1])\cong C(incl)[-1]\cong FC_{I^0,III}(H)[-1].
For the middle isomorphism see (<ref>). As before, the
identification $C(f^{\ge 0})\cong FC_{I^0,III}(H)[-1]$ is compatible
with continuation maps, and hence with the structure of a
doubly-directed system.
In the first-inverse-then-direct limit this yields (<ref>) for $\heartsuit=``\ge 0"$.
Let $\heartsuit=``>0"$. This is a consequence of the cases $\heartsuit=``=0"$ and $\heartsuit=``\ge 0"$. The proof is similar to that of the case $\heartsuit=``\ge 0"$.
Let $\heartsuit=``\le 0"$. This is a consequence of the cases $\heartsuit=``>0"$ and $\heartsuit=\varnothing$. The proof is similar to that of the case $\heartsuit=``\ge 0"$.
Remarks on the proof of Proposition <ref>. It is worth noting that we really needed to consider only three cases: $\heartsuit=\varnothing$, $\heartsuit=``=0"$, and $\heartsuit=``<0"$, the other three cases being in a sense formal consequences. As a matter of fact, given $\heartsuit=\varnothing$ and $\heartsuit=``=0"$, any of the four remaining cases suffices in order to deal with the other remaining three. A strategy that would have worked is to have considered the case $\heartsuit=``>0"$, i.e. work our way from the convex end throughout the cobordism (instead of starting from the concave end as in the proof). Should one wish to start with one of the cases $\heartsuit=``\le 0"$ or $\heartsuit=``\ge 0"$, an additional argument would be needed, related to excision, that would allow to decouple $I$ from $III^0$, respectively $I^0$ from $III$.
We can see a posteriori that the proof consists in a suitable iterative application of the following two elementary steps. (i) Identify suitable complexes for $L$ and $H$ which are homotopy equivalent via the continuation map. (ii) Embed the maps $f^\heartsuit$ whose cone we wish to compute inside grid diagrams of the type considered in Proposition <ref>, in which the other maps are either some of the homotopy equivalences of Step (i), or maps $f^\heartsuit$ whose cones have been already computed, or natural projections/inclusions for which the cones are known via Lemma <ref>.
§.§ The exact triangle of a pair
The homotopy invariance of the transfer map, together with the identification between the dynamical definition of the relative symplectic homology groups and the definition using cones given by Proposition <ref> implies that for any exact inclusion of pairs $(W,V)\stackrel{f}\longrightarrow (W',V')$ and for any $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ we have an induced transfer map
SH_*^\heartsuit(W',V')\stackrel{f_!}\longrightarrow SH_*^\heartsuit(W,V).
The following proposition establishes Theorem <ref> (the case of symplectic cohomology is completely analogous to that of symplectic homology).
Let $(W,V)$ be a cobordism pair for which we denote the inclusions $V\stackrel{i}\longrightarrow W\stackrel{j}\longrightarrow (W,V)$. Given $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ the following hold.
(i) For any Liouville structure $\lambda$ there exists an exact triangle
\begin{equation*}
\xymatrix
SH_*^\heartsuit(W,V;\lambda) \ar[rr]^-{j_!} & &
SH_*^\heartsuit(W;\lambda) \ar[dl]^-{i_!} \\ & SH_*^\heartsuit(V;\lambda) \ar[ul]^-\p_-{[-1]}
\end{equation*}
where the various symplectic homology groups are understood to be computed with respect to the Liouville structure $\lambda$.
(ii) Given a homotopy of Liouville structures $\lambda_t$, $t\in
[0,1]$, there are induced isomorphisms
SH_*^\heartsuit(V;\lambda_1)$, and
SH_*^\heartsuit(W,V;\lambda_1)$ which define a morphism between
the exact triangles in (i) corresponding to $\lambda_0$ and $\lambda_1$.
(iii) Given an exact inclusion of pairs $(W,V)\stackrel{f}\longrightarrow (W',V')$, the transfer maps $f_!:SH_*^\heartsuit(V')\to SH_*^\heartsuit(V)$, $f_!:SH_*^\heartsuit(W')\to SH_*^\heartsuit(W)$, and $f_!:SH_*^\heartsuit(W',V')\to SH_*^\heartsuit(W,V)$ define a morphism between the exact triangles of the pairs $(W',V')$ and $(W,V)$.
The existence of the exact triangle in (i) is a consequence of the tautological homology exact triangle of a cone (<ref>) and of the identification between $SH_*^\heartsuit(W,V)[-1]$ and $SH_*^{\heartsuit,cone}(W,V)$ proved in Proposition <ref>.
Part (ii) follows from the naturality of the homology exact triangle of a cone with respect to chain maps, and from the naturality of the absolute transfer map $SH_*^\heartsuit(W;\lambda)\to SH_*^\heartsuit(V;\lambda)$ with respect to homotopies of Liouville structures.
Part (iii) follows from the naturality of the homology exact triangle of a cone and from the functoriality of transfer maps (Proposition <ref>).
The Excision Theorem <ref> can also be reinterpreted using transfer maps. The proof uses the same kind of arguments as above and we shall omit it.
Given a Liouville cobordism triple $(W,V,U)$, denote the inclusion
(\overline{W\setminus U},\overline{V\setminus U})\stackrel{i}\longrightarrow (W,V).
The excision isomorphism in Theorem <ref> is induced by the transfer map $i_!$.
§.§ Exact triangle of a triple and Mayer-Vietoris exact triangle
Let $U\subset V\subset W$ be a triple of Liouville cobordisms with filling, meaning that $(V,U)$ and $(W,V)$ are pairs of Liouville cobordisms with filling, and denote the inclusions by $(V,U)\stackrel{i}\longrightarrow (W,U)\stackrel{j}{\longrightarrow} (W,V)$. For $\heartsuit\in\{\varnothing,\ge 0, >0,=0,\le 0,<0\}$ there exists an exact triangle
\begin{equation*}
\xymatrix
SH_*^{\heartsuit}(W,V) \ar[rr]^-{j_!} & &
SH_*^\heartsuit(W,U) \ar[dl]^-{i_!} \\ & SH_*^\heartsuit(V,U) \ar[ul]^-\p_-{[-1]}
\end{equation*}
which is functorial with respect to inclusions of triples, and which is invariant under homotopies of the Liouville structure that preserve the triple.
The proof is a formal consequence of the functorial properties of the long exact sequence of a pair. The proof of Theorem I.10.2 in <cit.> holds verbatim.
Let $U,V\subset W$ be Liouville cobordisms such that $W=U\cup V$ and $Z:=U\cap V$ is a Liouville cobordism such that
U=U^{bottom}\circ Z,\qquad V=Z\circ V^{top},\qquad W=U^{bottom}\circ Z\circ V^{top},
with $U^{bottom}=\overline{U\setminus Z}$, $V^{top}=\overline{V\setminus Z}$. We denote the inclusion maps by
\xymatrix
{& U \ar[dr]^-{j_U} & \\
Z \ar[ur]^-{i_U} \ar[dr]_-{i_V} & & W\\
& V \ar[ur]_-{j_V} &
There is a functorial Mayer-Vietoris exact triangle
\begin{equation*}
\scriptsize
\xymatrix
SH_*^\heartsuit(W) \ar[rr]^-{(j_{U!},j_{V!})} & &
SH_*^\heartsuit(U)\oplus SH_*^\heartsuit(V) \ar[dl]^-{i_{U!}-i_{V!}} \\ & SH_*^\heartsuit(Z) \ar[ul]_-{[-1]}^\delta
\end{equation*}
For $SH^{=0}$ this exact triangle is isomorphic to the Mayer-Vietoris exact triangle in singular cohomology.
Cobordisms for the Mayer-Vietoris theorem.
The Mayer-Vietoris exact triangle follows by a purely algebraic argument from the exact triangle of a pair and its naturality, and from the Excision Theorem <ref>. The idea is to consider the following commutative diagram.
\xymatrix{
& & SH^\heartsuit_{*-1}(W)
\ar@(d,u) @{<..} [ldd]_(.3){\color{black}\delta'}
& & \\
& SH^\heartsuit_{*-1}(V,Z) & SH^\heartsuit_{*-1}(W,U) \ar[u] \ar[l]_-{excision}^-\cong & & \\
SH^\heartsuit_{*-1}(U,Z) & SH^\heartsuit_*(Z) \ar[l] \ar[u] & SH^\heartsuit_*(U) \ar[l] \ar[u] & SH^\heartsuit_*(U,Z) \ar[l] &
SH^\heartsuit_{*+1}(Z) \ar[l] \\
SH^\heartsuit_{*-1}(W,V) \ar[u]_(.4){excision}^(.4)\cong & SH^\heartsuit_*(V) \ar[l] \ar[u] & SH^\heartsuit_*(W) \ar[l] \ar[u]
\ar@(r,l) @{<..} [urr]_(.3){\delta''}
\ar@(d,u) @{<..} [ldd]_(.3){\delta'}
& SH^\heartsuit_*(W,V) \ar[l] \ar[u]^(.4)\cong_(.4){excision} & \\
& SH^\heartsuit_*(V,Z) \ar[u] & SH^\heartsuit_*(W,U) \ar[l]_-{excision}^-\cong \ar[u] & & \\
& SH^\heartsuit_{*+1}(Z) \ar[u] &
The isomorphism $SH^\heartsuit_*(W,V)\stackrel\sim \longrightarrow SH^\heartsuit_*(U,Z)$ follows from the Excision Theorem <ref> for the exact triple $(W,V,V^{top})$. Similarly, we have an isomorphism
$SH^\heartsuit_*(W,U)\stackrel \sim\longrightarrow SH^\heartsuit_*(V,Z)$. The maps $\delta'$ and $\delta''$ are obtained by inverting the corresponding excision isomorphisms, and we actually have $\delta''=-\delta'$ by the “hexagonal lemma" of Eilenberg and Steenrod <cit.> which we recall below. We define the map $\delta$ in the statement of Theorem <ref> to be equal to $\delta''$, and a direct check by diagram chasing shows that the Mayer-Vietoris triangle is exact, see <cit.> for details.
<cit.> Consider the following diagram of groups and homomorphisms
\xymatrix
{& G_0 \ar[dl]_{\ell_1} \ar[dr]^{\ell_2} \ar[dd]_{i_0} & \\
G'_1 & & G'_2 \\
& G \ar[ul]_{j_1} \ar[ur]^{j_2} \ar[dd]_{j_0} & \\
G_2 \ar[uu]_\cong^{k_1} \ar[ur]^{i_2} \ar[dr]_{h_1} & & G_1 \ar[uu]^\cong_{k_2} \ar[ul]_{i_1} \ar[dl]^{h_2} \\
& G'_0 &
Assume that each triangle is commutative, that $k_1$ and $k_2$ are isomorphisms, that the two diagonal sequences are exact at $G$, and that $j_0i_0=0$. Then the two homomorphisms from $G_0$ to $G'_0$ obtained by skirting the sides of the hexagon differ in sign only.
The hexagonal lemma of Eilenberg and Steenrod is applied in the proof of Theorem <ref> to the following configuration.
\xymatrix
{& SH_{*+1}^\heartsuit(Z) \ar[dl]_{\ell_1} \ar[dr]^{\ell_2} \ar[dd]_{i_0} & \\
SH_*^\heartsuit(V,Z) & & SH_*^\heartsuit(U,Z) \\
& SH_*^\heartsuit(W,Z) \ar[ul]_{j_1} \ar[ur]^{j_2} \ar[dd]_{j_0} & \\
SH_*^\heartsuit(W,U) \ar[uu]_\cong^{k_1} \ar[ur]^{i_2} \ar[dr]_{h_1} & & SH_*^\heartsuit(W,V) \ar[uu]^\cong_{k_2} \ar[ul]_{i_1} \ar[dl]^{h_2} \\
& SH_*^\heartsuit(W) &
The vertical isomorphisms $k_1$ and $k_2$ are the excision isomorphisms. The connecting homomorphism $\delta$ in the Mayer-Vietoris exact sequence, or the homomorphism $\delta''$ in the notation of the proof of Theorem <ref>, is defined to be $h_2k_2^{-1}\ell_2$.
§.§ Compatibility between exact triangles
In this section we use the notation $(\heartsuit,\heartsuit',\heartsuit'/\heartsuit)$ for any one of the triples $(<0,\varnothing,\ge 0)$, $(\le 0,\varnothing,>0)$, $(<0,\le 0,=0)$, or $(=0,\ge 0,>0)$. To any such triple there corresponds a tautological exact triangle (see Propositions <ref> and <ref>)
\xymatrix
SH_*^{\heartsuit} \ar[rr] & &
SH_*^{\heartsuit'} \ar[dl] \\ & SH_*^{\heartsuit'/\heartsuit} \ar[ul]^{[-1]}
Let $(W,V)$ be a Liouville pair of cobordisms with filling. Let $(\heartsuit,\heartsuit',\heartsuit'/\heartsuit)$ be a triple as above.
(i) The transfer maps $f^\heartsuit_{WV}$, $f^{\heartsuit'}_{WV}$, and $f^{\heartsuit'/\heartsuit}_{WV}$ induce a morphism between the tautological exact triangles corresponding to $(\heartsuit,\heartsuit',\heartsuit'/\heartsuit)$ for $W$ and $V$.
(ii) The exact triangles of the pair $(W,V)$ for $\heartsuit,\heartsuit',\heartsuit'/\heartsuit$ determine “triangles of triangles" together with the corresponding tautological exact triangles. More precisely, upon expanding the exact triangles of a pair and the tautological ones into long exact sequences, we obtain the following diagram in which all squares are commutative, except the bottom right one which is anti-commutative
\xymatrix
SH_*^\heartsuit(W,V) \ar[r] \ar[d] & SH_*^\heartsuit(W) \ar[r]^{f^\heartsuit_!} \ar[d] & SH_*^\heartsuit(V) \ar[r] \ar[d] & SH_{*-1}^\heartsuit(W,V) \ar[d] \\
SH_*^{\heartsuit'}(W,V) \ar[r] \ar[d] & SH_*^{\heartsuit'}(W) \ar[r]^{f^{\heartsuit'}_!} \ar[d] & SH_*^{\heartsuit'}(V) \ar[r] \ar[d] & SH_{*-1}^{\heartsuit'}(W,V) \ar[d] \\
SH_*^{\heartsuit'/\heartsuit}(W,V) \ar[r] \ar[d] & SH_*^{\heartsuit'/\heartsuit}(W) \ar[r]^{f^{\heartsuit'/\heartsuit}_!} \ar[d] & SH_*^{\heartsuit'/\heartsuit}(V) \ar @{} [dr] |{-} \ar[r] \ar[d] & SH_{*-1}^{\heartsuit'/\heartsuit}(W,V) \ar[d] \\
SH_{*-1}^\heartsuit(W,V) \ar[r] & SH_{*-1}^\heartsuit(W) \ar[r]^{f^\heartsuit_!} & SH_{*-1}^\heartsuit(V) \ar[r] & SH_{*-2}^\heartsuit(W,V)
(iii) The exact triangle of a pair $(W,V)$ for $SH_*^{=0}$ is isomorphic to the exact triangle of the pair $(W,V)$ in singular cohomology $H^{n-*}$.
Assertion (i) follows from the fact that continuation maps induced by increasing homotopies respect the action filtration.
Assertion (ii) follows from Lemma <ref>, and from our identification of the relative symplectic homology groups with limit homology groups of mapping cones corresponding to chain level continuation maps (Proposition <ref>).
Lemma <ref> is applied to the following morphism between action filtration short exact sequences given by the chain level continuation maps:
\xymatrix
0 \ar[r] & FC_{I^\heartsuit}(K_{i,j}) \ar[r] \ar[d]^{f^\heartsuit_{i,j}} & FC_{I^{\heartsuit'}}(K_{i,j}) \ar[r] \ar[d]^{f^{\heartsuit'}_{i,j}} & FC_{I^{\heartsuit'/\heartsuit}}(K_{i,j}) \ar[r] \ar[d]^{f^{\heartsuit'/\heartsuit}_{i,j}} & 0 \\
0 \ar[r] & FC_{II^\heartsuit}(K_{i,j}) \ar[r] & FC_{II^{\heartsuit'}}(H_{i,j}) \ar[r] & FC_{II^{\heartsuit'/\heartsuit}}(H_{i,j}) \ar[r] & 0
Assertion (iii) is proved mutatis mutandis like <cit.>. We omit the details.
Finally, we prove the following compatibility between the tautological
exact triangles.
For every filled Liouville pair $(W,V)$ the four tautological
exact triangles fit into the commuting diagram
\xymatrix
& SH_{*+1}^{>0}(W,V) \ar@{=}[r] \ar[d] & SH_{*+1}^{>0}(W,V) \ar[d] & \\
SH_*^{<0}(W,V) \ar[r] \ar@{=}[d] & SH_*^{\leq 0}(W,V) \ar[r] \ar[d] &
SH_*^{=0}(W,V) \ar[r] \ar[d] & SH_{*-1}^{<0}(W,V) \ar@{=}[d] \\
SH_*^{<0}(W,V) \ar[r] & SH_*(W,V) \ar[r] \ar[d] & SH_*^{\geq
0}(W,V) \ar[r] \ar[d] & SH_{*-1}^{<0}(W,V)\\
& SH_*^{>0}(W,V) \ar@{=}[r] & SH_*^{>0}(W,V) &
Fix $\epsilon>0$ small enough. For any choice of real numbers $a,b$ such that $-\infty<a<-\epsilon<\epsilon<b<\infty$, and for any choice of admissible Hamiltonian and almost complex structure, we have a commutative diagram of short exact sequences
\xymatrix{
0\ar[r]& FC_*^{(a,-\epsilon)}\ar[r] \ar@{=}[d] & FC_*^{(a,\epsilon)}\ar[r]\ar[d] & FC_*^{(-\epsilon,\epsilon)}\ar[r]\ar[d] & 0 \\
0\ar[r]& FC_*^{(a,-\epsilon)}\ar[r] & FC_*^{(a,b)}\ar[r]& FC_*^{(-\epsilon,b)}\ar[r]& 0
in which the various maps are inclusions or projections. This induces a commutative diagram between the corresponding long exact sequences in homology, and by passing to the limit on the Hamiltonian and then on $a\to-\infty$, $b\to\infty$ as in Section <ref> we obtain the commutativity of the diagram formed by the two horizontal lines in the statement.
The commutativity of the diagram formed by the two vertical lines in the statement is proved analogously.
We conclude this subsection with a compatibility result between the
exact triangle of a triple and Poincaré duality.
For every triple $(W,V,U)$ of filled Liouville cobordisms and $\heartsuit\in\{\varnothing, >0,\ge
0, =0, \le 0, <0\}$ there exists a commuting diagram
\begin{equation*}%\label{eq:PD-triple}
\scriptsize
\xymatrix
SH_*^\heartsuit(W,V) \ar[r] \ar[d]^\cong_{exc} & SH_*^\heartsuit(W,U)
\ar[r] \ar[d]^\cong_{exc} & SH_*^\heartsuit(V,U) \ar[r] \ar[d]^\cong_{exc} &
SH_{*-1}^\heartsuit(W,V) \ar[d]^\cong_{exc} \\
SH_*^\heartsuit(W\setminus V,\p V) \ar[r] \ar[d]^\cong_{PD} &
SH_*^\heartsuit(W\setminus U,\p U)
\ar[r] \ar[d]^\cong_{PD} & SH_*^\heartsuit(V\setminus U,\p U) \ar[r] \ar[d]^\cong_{PD} &
SH_{*-1}^\heartsuit(W\setminus V,\p V) \ar[d]^\cong_{PD} \\
SH^{-*}_\heartsuit(W\setminus V,\p W) \ar[r] \ar@{=}[d] &
SH^{-*}_\heartsuit(W\setminus U,\p W)
\ar[r] \ar@{=}[d] & SH^{-*}_\heartsuit(V\setminus U,\p V) \ar[r] \ar[d]^\cong_{exc} &
SH^{1-*}_\heartsuit(W\setminus V,\p W) \ar@{=}[d] \\
SH^{-*}_{-\heartsuit}(W\setminus V,\p W) \ar[r] &
SH^{-*}_{-\heartsuit}(W\setminus U,\p W) \ar[r] &
SH^{1-*}_{-\heartsuit}(W\setminus U,W\setminus V) \ar[r] &
SH^{1-*}_{-\heartsuit}(W\setminus V,\p W) \\
\end{equation*}
where the first and last row are the long exact sequences of the
triples $(W,V,U)$ and $(W\setminus U,W\setminus V,\p W)$,
respectively, and the vertical arrows are the Poincaré duality
and excision isomorphisms from Theorem <ref> and
Theorem <ref>.
(The remaining horizontal maps are defined by this diagram.)
The conclusion follows directly
from the definition of the Poincaré
duality isomorphism in Theorem <ref> and the observation
that for a Hamiltonian $G$ as in Figure <ref> adapted to the
triple $(W,V,U)$, the Hamiltonian $-G$ is adapted to the triple
$(W\setminus U,W\setminus V,\p W)$.
Alternatively, one can reduce the general case by a purely algebraic argument to the case $U=\varnothing$, as in the proof of Proposition <ref>. The case $U=\varnothing$ is in turn treated by noting that for a Hamiltonian $H$ as in Figures <ref> or <ref> adapted to the
pair $(W,V)$, the Hamiltonian $-H$ is adapted to the triple
$(W,W\setminus V,\p W)$.
§.§ The exact triangle of a pair of Liouville domains revisited
The exact triangle
\begin{equation*}
\xymatrix
SH_*^\heartsuit(W,V) \ar[rr] & &
SH_*^\heartsuit(W) \ar[dl] \\ & SH_*^\heartsuit(V) \ar[ul]^-\p_-{[-1]}
\end{equation*}
can be established in a more direct way for a pair $(W,V)$ of Liouville domains since there is no need to first identify the symplectic homology of the pair with a homological mapping cone. Instead, one can argue directly on the chain complexes using truncation by the action. We find it instructive to spell out the argument. This proof is only apparently simpler: since the transfer maps induced by the inclusions $V\hookrightarrow W$ and $W\hookrightarrow (W,V)$ are only implicitly constructed, this proof would require additional arguments in order to incorporate it into the larger framework that we discuss in this paper, and these additional arguments would essentially amount to reinterpret this diagram in terms of transfer maps.
For a pair of Liouville domains we only need to consider three flavors $\heartsuit\in\{\varnothing,=0,>0\}$. We prove below the compatibility of the exact triangle of the pair with the tautological exact triangle relating these three flavors.
Let $V\subseteq W$ be an inclusion of Liouville domains and denote by
$\wh W$ the symplectic completion of $W$. Let $H=H_{\nu,\tau}$,
$\nu>0$, $\tau>0$ be a one step Hamiltonian on $\wh W$, defined up to
smooth approximation as follows (Figure <ref>):
* $H=0$ on $W\setminus V$,
* $H$ is linear of slope $\tau$ on $\wh W\setminus W$,
* $H$ is linear of slope $\nu$ on a collar $]\delta,1]\times \p V\subseteq V$ for some $0<\delta<1$,
* $H$ is constant equal to $-\nu(1-\delta)$ on the complement of this collar in $V$.
Hamiltonian for a pair of Liouville domains.
For $\nu$ and $\tau$ not lying in the action spectrum of $\p V$, respectively $\p W$, the $1$-periodic orbits of $H$ fall into five classes:
($II^0$) constants in the complement of the collar in $V$,
($II^+$) orbits corresponding to characteristics on $\p V$ and located in the region $\{\delta\}\times\p V$,
($III^-$) orbits corresponding to characteristics on $\p V$ and located in the region $\p V$,
($III^0$) constants in $W\setminus V$,
($III^+$) orbits corresponding to characteristics on $\p W$ and located in the region $\p W$.
Suitable choices of the parameters $\tau$ and $\delta$ as a function of $\nu$ ensure that the various classes of orbits are ordered according to the action as follows:
III^0 < III^-,III^+ < II^0 < II^+.
As $\nu\to \infty$ we can allow $\tau\to\infty$. In general we need to let $\delta\to 0$ if we wish to acquire $III^-<II^0$. However, by Lemmas <ref> and <ref> we have
III^-\prec II^0,II^+
for any fixed choice of $\delta>0$, independently of the choice of $\nu$.
Also, by Lemma <ref> we have
III^-\prec III^+,\qquad II^0,II^+\prec III^+.
The outcome is that for suitable choices of the parameters we have
III^0< III^-\prec II^0 < II^+ \prec III^+
III^0<III^-\prec III^+ <II^0<II^+.
Let $FC_{tot}$ be the total Floer complex for the Hamiltonian $H$. For a subset $\mathcal I\subset \{II^0,II^+,III^-,III^0,III^+\}$ denote by $FC_{\mathcal I}$ the complex generated by the orbits in the classes belonging to $\mathcal I$. For example, $FC_{III^-,III^0,III^+}$ stands for the subcomplex generated by the orbits in the classes $III^-,III^0,III^+$, and $FC_{III^-,III^+}$ stands for its quotient complex modulo $FC_{III^0}$ etc. We will also abbreviate $FC_{II}=FC_{II^0,II^+}$ and $FC_{III}=FC_{III^-,III^0,III^+}$.
Let us consider the following diagram whose first two rows and first two columns are exact
\xymatrix
& 0 \ar[d] & 0 \ar[d] & & \\
0\ar[r] & FC_{III^0} \ar[r] \ar[d] & FC_{II,III^-,III^0} \ar[r] \ar[d] & FC_{II,III^-} \ar[r] \ar[d]^p & 0 \\
0\ar[r] & FC_{III} \ar[r] \ar[d] & FC_{tot} \ar[r] \ar[d] & FC_{II} \ar[r] \ar@{.>}[d]^-f & 0 \\
& FC_{III^-,III^+} \ar[r]_q \ar[d] & FC_{III^+} \ar@{.>}[r]_-g \ar[d] & FC_{III^-}[-1] & \\
& 0 & 0 & &
Here the chain maps $f:FC_{II}\to FC_{III^-}[-1]$ and $g:FC_{III^+}\to FC_{III^-}[-1]$ are uniquely determined so that we have natural identifications
FC_{II,III^-}=C(f)[1], \quad p=\beta(f),\qquad \qquad FC_{III^-,III^+}=C(g)[1],\quad q=\beta(g).
Proposition <ref> and its proof ensure that the bottom right square in the above diagram is commutative in Kom, and moreover the diagram can be completed to a diagram in Kom whose lines and columns are distinguished triangles, and all of whose squares are commutative except the bottom-right one which is anti-commutative:
\begin{equation} \label{eq:grid-chain}
\xymatrix
FC_{III^0} \ar[r] \ar[d] & FC_{II,III^-,III^0} \ar[r] \ar[d] & FC_{II,III^-} \ar[r] \ar[d]^p & FC_{III^0}[-1] \ar[d] \\
FC_{III} \ar[r] \ar[d] & FC_{tot} \ar[r] \ar[d] & FC_{II} \ar[r] \ar@{.>}[d]^-f & FC_{III}[-1] \ar[d] \\
FC_{III^-,III^+} \ar[r]_q \ar[d] & FC_{III^+} \ar@{.>}[r]_-g \ar[d] & FC_{III^-}[-1] \ar@{} [dr] |{-} \ar[r] \ar[d] & FC_{III^-,III^+}[-1] \ar[d] \\
FC_{III^0}[-1] \ar[r] & FC_{II,III^-,III^0}[-1] \ar[r] & FC_{II,III^-}[-1] \ar[r] & FC_{III^0}[-2]
\end{equation}
Indeed, the term $FC_{III^-}[-1]$ is isomorphic in Kom to $C(p)[-1]$ on the one hand, and to $C(-q)[-1]$ on the other hand, and these two complexes are isomorphic as seen in the proof of Proposition <ref>.
We now remark that we have a homotopy equivalence that is well-defined up to homotopy
FC_{III^-}[-1]\cong FC_{II^+}.
This follows again from Proposition <ref>. For the proof we
consider a homotopy from a Hamiltonian $K=K_\tau$ which is zero on $W$
and coincides with $H_{\nu,\tau}$ outside $W$ to the Hamiltonian
$H$. We denote $FC_V(K)$ the subcomplex of $FC(K)$ generated by critical
points inside the domain $V$, so that the continuation map induces a
homotopy equivalence $FC_V(K)\simeq FC_{II,III^-}$. On the other hand we
have a canonical identification $FC_V(K)\equiv FC_{II^0}$, and a commutative
diagram up to homotopy
\xymatrix{
FC_V(K) \ar@{=}[r] \ar[d]_\simeq^{h.e.} & FC_{II^0} \ar[d]^{incl} \\
FC_{II,III^-} \ar[r]_{proj} & FC_{II}\;.
Then Proposition <ref> yields the desired homotopy equivalence $FC_{III^-}[-1]\cong FC_{II^+}$.
This chain homotopy equivalence provides one point of view on the vanishing of $SH_*(I\times \p V,\p^-(I\times \p V))$ proved in Proposition <ref>.
Diagram (<ref>) can now be used as a building block to prove the existence of a diagram with exact lines and columns and in which all squares are commutative except the one marked “$-$”, which is anti-commutative.
\begin{equation} \label{eq:grid-SH}
\xymatrix
H^{n-*}(W,V) \ar[r] \ar[d] & H^{n-*}(W) \ar[r] \ar[d] & H^{n-*}(V) \ar[r] \ar[d] & H^{n-*+1}(W,V) \ar[d] \\
SH_*(W,V) \ar[r] \ar[d] & SH_*(W) \ar[r] \ar[d] & SH_*(V) \ar[r] \ar[d] & SH_{*-1}(W,V) \ar[d] \\
SH_*^{>0}(W,V) \ar[r] \ar[d] & SH_*^{>0}(W) \ar[r] \ar[d] & SH_*^{>0}(V) \ar@{} [dr] |{-} \ar[r] \ar[d] & SH_{*-1}^{>0}(W,V) \ar[d] \\
H^{n-*+1}(W,V) \ar[r] & H^{n-*+1}(W) \ar[r] & H^{n-*+1}(V) \ar[r] & H^{n-*+2}(W,V)
\end{equation}
This grid diagram expresses the compatibility between the exact triangle of a pair of Liouville domains $(W,V)$ and the tautological exact triangle involving singular cohomology, symplectic homology, and positive symplectic homology. One relevant ingredient here is the chain homotopy equivalence $C^{III}[-1]\cong C^{II}$. The other ingredient is that all the above homological constructions are compatible with continuation maps and with direct limits.
§ VARIANTS OF SYMPLECTIC HOMOLOGY GROUPS
§.§ Rabinowitz-Floer homology
Given a pair of Liouville domains $(W,V)$, Rabinowitz-Floer
homology $RFH_*(\p V,W)$ was defined in <cit.>
as a Floer-type theory associated to the Rabinowitz action functional
\tilde A_H:\cL\wh W\times \R\to \R,\qquad \tilde A_H(\gamma,\eta)=A_{\eta H}(\gamma),
where $H:\wh W\to \R$ is a Hamiltonian such that $\p V=H^{-1}(0)$ is a regular level, $H|_V\le 0$, and $H|_{\wh W\setminus V}\ge 0$. The dynamical significance of Rabinowitz-Floer homology is that it counts leafwise intersection points of $\p V$ under Hamiltonian motions <cit.>, and one of its most useful properties is that Hamiltonian displaceability of $\p V$ (and hence of $V$) implies vanishing.
It was proved in <cit.> that $RFH_*(\p
V,W)$ does not depend on $W$, so we will denote it by $RFH(\p V)$ (it
does however depend on the filling $V$ of $\p V$). The main result
of <cit.> is that, with our current notation, we have an isomorphism
\begin{equation}\label{eq:RFH}
RFH_*(\p V)\cong SH_*(\p V),
\end{equation}
i.e. Rabinowitz-Floer homology is the symplectic homology of the
trivial cobordism over $\p V$. As such, Rabinowitz-Floer homology is naturally incorporated within the setup that we develop in this paper.
§.§ $S^1$-equivariant symplectic homologies
The circle $S^1=\R/\Z$ acts on the free loop space by shifting the
parametrisation. As such, one can define $S^1$-equivariant flavors of
symplectic homology groups. In the case of Liouville domains relevant
instances have been defined
in <cit.>. Following
Seidel <cit.> and <cit.>, the relevant structure
is that of an $S^1$-complex, meaning a $\Z$-graded chain
complex $(C_*,\p)$ together with a sequence of maps $\p_i:C_*\to
C_{*+2i-1}$, $i\ge 0$ such that $\p_0=\p$ and
\begin{equation} \label{eq:S1complex}
\sum_{i+j=k} \p_i\p_j=0
\end{equation}
for all $k\ge 0$. An $S^1$-complex for which $\p_i=0$ for $i\ge 2$ is
called a mixed complex in the literature on cyclic
homology. One should view $S^1$-complexes as being
$\infty$-mixed complexes, or mixed complexes up to
homotopy, see <cit.> and the references therein. Given a
Hamiltonian $H$ one can endow $FC_*^{(a,b)}(H)$ with the structure of an $S^1$-complex
that is canonical up to homotopy equivalence. Moreover, a
homotopy of Hamiltonians induces a morphism between the
$S^1$-complexes defined on the Floer chain groups at the endpoints.
Recall that we work with coefficients in a field $\mathfrak{k}$. Denote by $u$ a
formal variable of degree $-2$.
Given an $S^1$-complex $\cC=(C_*,\{\p_i\}_{i\ge 0})$ we define
following Jones <cit.> and Zhao <cit.>
the periodic cyclic chain complex
C_*[u,u^{-1}],\qquad \p_u = \sum_{i\ge 0}u^{i}\p_i,\qquad |u|=-2.
Here elements in $C_*[u,u^{-1}]$ of degree $k$ are by definition
Laurent polynomials $\sum_{j=-N}^N x_ju^j$ with $x_j\in
C_{k+2j}$. Then $\p_u^2=0$ as a consequence of (<ref>)
and the map $\p_u$ is $\mathfrak{k}[u]$-linear.
We consider
the sub/quotient complexes
C_*[u^{-1}] = C_*[u,u^{-1}]/uC_*[u]
with differential induced by $\p_u$ and the induced
$\mathfrak{k}[u]$-module structure. The homologies
\begin{gather*}
HC_*^{[u,u^{-1}]}(\cC) := H_*(C_*[u,u^{-1}]),\cr
HC_*(\cC) := HC_*^{[u^{-1}]}(\cC) := H_*(C_*[u^{-1}])
\end{gather*}
correspond to certain versions of the negative cyclic homology,
periodic (or Tate) cyclic homology,
respectively cyclic homology
of the $S^1$-complex $\cC$ in the literature. We will not use these
names but rather indicate in the notation which version of
(Laurent) polynomials we are using.
Due to the short exact sequence of complexes of $\mathfrak{k}[u]$-modules
0\to C_*[u]\to C_*[u,u^{-1}] \to C_*[u,u^{-1}]/C_*[u]\cong C_*[u^{-1}][-2]\to 0,
these homology groups fit into the fundamental exact triangle
\xymatrix
HC_*^{[u]}(\cC) \ar[rr] & &
HC_*^{[u,u^{-1}]}(\cC) \ar[dl]^{[-2]} \\ & HC_{*}(\cC) \ar[ul]^{[+1]}\;.
Given an $S^1$-space $X$, its singular chain complex with arbitrary
coefficients $C_*=(C_*(X),\p)$ carries the structure of a mixed complex
$\cC=(C_*,\p,\p_1)$ such that <cit.>
HC_*(\cC)\cong H_*^{S^1}(X).
Here $H_*^{S^1}(X)=H_*(X\times_{S^1}ES^1)$ is the
usual $S^1$-equivariant homology group of $X$ defined by the Borel construction.
The map $\p_1:C_*\to C_{*+1}$ is defined by inserting a suitable
representative of the fundamental class of the oriented circle $S^1$ into the first
argument of the composite map $C_*(S^1)\otimes
C_*(X)\stackrel{EZ}\longrightarrow C_*(S^1\times
X)\stackrel{\mu_*}\longrightarrow C_*(X)$, where $\mu:S^1\times X\to
X$ is the $S^1$-action and $EZ$ is the Eilenberg-Zilber equivalence,
explicitly described by the Eilenberg-McLane shuffle
map <cit.>. Define the homology groups
H_*^{[u,u^{-1}]}(X) = HC_*^{[u,u^{-1}]}(\cC),\qquad
H_*^{[u]}(X) = HC_*^{[u]}(\cC).
While these groups cannot be described as homology groups of a
topological space in the manner of $H_*^{S^1}(X)$ – they typically
have infinite support in the negative range – they are nevertheless
unavoidable should one wish to formulate duality. More precisely, let
us assume that $X$ is an oriented manifold of dimension $n$ with
boundary preserved by the $S^1$-action. Denoting by
$H^*_{S^1}(X)=H^*(X\times_{S^1} ES^1)$ the usual $S^1$-equivariant
cohomology groups, Poincaré duality in the
$S^1$-equivariant setting takes the form
H^i_{S^1}(X)\cong H_{n-i}^{[u]}(X,\p X).
More generally, dualizing the mixed complex structure on $C_*(X)$ and
changing the degree of $u$ to $+2$, one can define two other versions
$H^*_{[u,u^{-1}]}(X)$ and $H^*_{[u^{-1}]}(X)$ of $S^1$-equivariant
cohomology, with Poincaré duality isomorphisms
H^i_{[u,u^{-1}]}(X)\cong H_{n-i}^{[u,u^{-1}]}(X,\p X),\qquad
H^i_{[u^{-1}]}(X)\cong H_{n-i}^{[u^{-1}]}(X,\p X) = H_{n-i}^{S^1}(X).
See <cit.> for proofs of related statements. We
shall use below the following simple instance of duality: Consider an
oriented manifold $X$ of dimension $n$ with boundary viewed as an
$S^1$-space with trivial action. Then
H_i^{S^1}(X)=\bigoplus_{j\ge 0}H_{i-2j}(X)
\begin{equation} \label{eq:product-sum}
H^i_{[u^{-1}]}(X,\p X)=\prod_{j\ge 0}H^{i+2j}(X,\p X)=\bigoplus_{j\ge 0}H^{i+2j}(X,\p X),
\end{equation}
so that we indeed have $H_i^{S^1}(X)\cong H^{n-i}_{[u^{-1}]}(X,\p X)$
as a consequence of classical Poincaré duality.
In order to define $S^1$-equivariant symplectic homology and
cohomology groups, we use the structure of an $S^1$-complex on each
truncated Floer chain complex $\cC:=FC_*^{(a,b)}(H)$ and
cochain complex $\cC^\vee:=FC^*_{(a,b)}(H)$ constructed
in <cit.>. We set
\begin{gather*}
\end{gather*}
\begin{gather*}
\end{gather*}
and use these groups in formulas (<ref>),
(<ref>), (<ref>), (<ref>),
and (<ref>), as well as in Definitions <ref>,
<ref>, <ref>, <ref>,
and <ref>. The outcome for a pair $(W,V)$ of Liouville
cobordisms with filling are $S^1$-equivariant symplectic homology
SH_*^{S^1,\heartsuit}(W,V),\qquad SH_*^{[u,u^{-1}],\heartsuit}(W,V),\qquad SH_*^{[u],\heartsuit}(W,V),
and $S^1$-equivariant symplectic cohomology groups
SH^*_{S^1,\heartsuit}(W,V),\qquad SH^*_{[[u,u^{-1}]],\heartsuit}(W,V),\qquad S^*_{[[u^{-1}]],\heartsuit}(W,V),
with $\heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\}$ as usual.
The notation $[[u]]$ and $[[u,u^{-1}]]$ in the equivariant symplectic cohomology
groups is a reminder that, in the case of a Liouville domain, the inverse
limit in the definition leads in general to formal power series rather
than polynomials. It also indicates the analogy to the
$S^1$-equivariant cohomology groups defined by Jones and
Petrack <cit.>. Indeed, it is proved
in <cit.> that for a Liouville domain $W$ and with
rational coefficients the second group satisfies fixed point localization
\begin{equation}\label{eq:fixed}
SH^*_{[[u,u^{-1}]]}(W;\Q) \cong H_{n+*}(W,\p
\end{equation}
One can define several other potentially interesting versions of
$S^1$-equivariant symplectic homology by applying the direct/inverse
limit over the bounds of the action window $(a,b)$, the homology
functor, and the completions with respect to $u,u^{-1}$ in different
orders <cit.>. In particular, this gives rise to a version of
periodic/Tate symplectic cohomology of a Liouville domain that equals
the localization of $S^1$-equivariant cohomology and obeys
Goodwillie's theorem <cit.>. This can also serve as a motivation to phrase the theory of symplectic homology at chain level, see also the discussion of coefficients in the Introduction regarding this point.
The equivariant symplectic (co)homology groups are connected to each
other by fundamental exact triangles similar to the one for cyclic homology above, namely
\xymatrix
SH_*^{[u],\heartsuit} \ar[rr] & &
SH_*^{[u,u^{-1}],\heartsuit} , \ar[dl]^{[-2]} \\ & SH_{*}^{S^1,\heartsuit} \ar[ul]^{[+1]}
\qquad
\xymatrix
SH^*_{S^1,\heartsuit} \ar[rr] & &
SH^*_{[[u,u^{-1}]],\heartsuit} . \ar[dl]^{[+2]} \\ & SH^{*}_{[[u^{-1}]],\heartsuit} \ar[ul]^{[-1]}
The non-equivariant and equivariant theories are connected by Gysin exact triangles
\xymatrix
SH_*^{\heartsuit} \ar[rr] & &
SH_*^{S^1,\heartsuit} , \ar[dl]^{[-2]} \\ & SH_{*}^{S^1,\heartsuit} \ar[ul]^{[+1]}
\qquad
\xymatrix
SH^{*}_{S^1,\heartsuit} \ar[rr]^{[+2]} & &
SH^*_{S^1,\heartsuit}, \ar[dl] \\ & SH^*_{\heartsuit} \ar[ul]^{[-1]}
\xymatrix
SH_{*}^{[u],\heartsuit} \ar[rr]^{[-2]} & &
SH_*^{[u],\heartsuit} , \ar[dl] \\ & SH_*^{\heartsuit} \ar[ul]^{[+1]}
\qquad
\xymatrix
SH^{*}_{\heartsuit} \ar[rr] & &
SH^*_{[[u^{-1}]],\heartsuit}. \ar[dl]^{[+2]} \\ & SH^{*}_{[[u^{-1}]],\heartsuit} \ar[ul]^{[-1]}
By construction, all $S^1$-equivariant symplectic homology and cohomology groups
are modules over $\mathfrak{k}[u]$.
Moreover, the periodic versions are actually modules over the
larger ring $\mathfrak{k}[u,u^{-1}]$. In particular, this module
structure induces periodicity isomorphisms
SH_*^{[u,u^{-1}],\heartsuit}\cong SH_{*+2}^{[u,u^{-1}],\heartsuit},\qquad
SH^*_{[[u,u^{-1}]],\heartsuit}\cong SH^{*+2}_{[[u,u^{-1}]],\heartsuit}.
All the exact triangles above are obtained at the level of truncated
Floer homology by writing the complex that computes $HC_*^{[u,u^{-1}]}(\cC)$ as the
product total complex of a multicomplex of the form
\xymatrix{
& \ar[d] & \ar[d] & \ar[d] & \ar[d] & \ar@{.}[dl] \\
& C_3 \ar[l] \ar[d]^{\p} & C_2 \ar[l] \ar[d] & C_1 \ar[l]^-{\p_1} \ar[d]^{\p} & C_0 \ar[l]^-{\p_1} \ar@{-->}[ull]_(.3){\p_2} & \\
& C_2 \ar[l] \ar[d] & C_1 \ar[l] \ar[d]^\p & C_0 \ar[l]^{\p_1} \ar@{-->}[ull]^(.65){\p_2} \ar@{.>}[uulll]_(.2){\p_3} & \boldsymbol{u^{-1}} & \\
& C_1 \ar[l] \ar[d]^\p & C_0 \ar[l]^-{\p_1} \ar@{-->}[ull]_(.3){\p_2} & \boldsymbol{u^0} & & \\
& C_0 \ar[l] & \boldsymbol{u^{1}} & & & \\
\ar@{.}[ur] & & & & &
and considering natural subcomplexes/quotient
complexes <cit.>. The $[u^{-1}]$-complex sits on the right
half-plane with respect to the $0$-th column, the $[u]$-complex sits
on the left half-plane, and the non-equivariant theory
sits on the $0$-th column. For cohomology the arrows need to be
reversed. The resulting exact triangles for truncated Floer
(co)homology pass to the limit in symplectic (co)homology due to our
choice of order in the first-inverse-then-direct limit. Note that,
since for a given Hamiltonian $H$ and finite action window $(a,b)$
the complex $FC_*^{(a,b)}(H)$ has finite rank, it actually does not
matter whether we consider the product total complex or the direct sum
total complex to compute $HC_*^{[u,u^{-1}]}(\cC)$.
Here are some further properties of these symplectic (co)homology groups.
(1) At action level zero we have
SH_*^{S^1,=0}(W,V)\cong H^{n-*}_{[u^{-1}]}(W,V),\qquad SH_*^{[u],=0}(W,V)\cong H^{n-*}_{S^1}(W,V),
SH_*^{[u,u^{-1}],=0}(W,V)\cong H^{n-*}_{[u,u^{-1}]}(W,V).
In particular, for a Liouville domain $W$ of dimension $2n$ we have
SH_*^{S^1,=0}(W)\cong H^{n-*}_{[u]}(W)\cong H_{*+n}^{S^1}(W,\p W).
This formula appears already in <cit.>. We interpret in the Introduction this formula as a motivation for viewing the transfer maps as shriek maps.
(2) For a Liouville domain $W$, it is proved in <cit.>
that $SH_*^{S^1,>0}(W)$ is isomorphic over $\Q$ to linearized
contact homology of $\p W$ whenever the latter is defined, see
also <cit.> for applications.
(3) The arguments in <cit.> carry over to the setting of pairs of
Liouville cobordisms with filling in order to show that there is a
spectral sequence converging to $SH_*^{S^1,\heartsuit}(W,V)$ with
second page given by
$E^2=SH_*^\heartsuit(W,V)\otimes\mathfrak{k}[u^{-1}]$. In combination
with the Gysin exact triangle
this yields the fact that the non-equivariant symplectic homology of a
pair $(W,V)$ vanishes if and only if its $S^1$-equivariant symplectic
homology vanishes. The fixed point localization (<ref>) shows
that this is not true anymore for $SH^{[u,u^{-1}]}_*$.
(4) The above flavors of $S^1$-equivariant symplectic homology satisfy
Poincaré duality in the following general form: given a Liouville
cobordism $W$ and $A\subset \p W$ an admissible union of boundary components, for
any $\heartsuit\in\{\varnothing,>0,\ge 0, =0, \le 0, <0\}$ we have
SH_*^{S^1,\heartsuit}(W,A)\cong SH^{-*}_{[[u^{-1}]],-\heartsuit}(W,A^c),\qquad
SH_*^{[u],\heartsuit}(W,A)\cong SH^{-*}_{S^1,-\heartsuit}(W,A^c),
SH_*^{[u,u^{-1}],\heartsuit}(W,A)\cong SH^{-*}_{[[u,u^{-1}]],-\heartsuit}(W,A^c),
where the notation $-\heartsuit$ has the same meaning as
in <ref>. There are also algebraic dualities
over the ring $\mathfrak{k}[u]$ analogous to those in <cit.>
which pair $SH^*_{S^1,\heartsuit}$ with
$SH_*^{[u],\heartsuit}$, $SH^*_{[[u^{-1}]],\heartsuit}$ with
$SH_*^{S^1,\heartsuit}$, and $SH^*_{[[u,u^{-1}]],\heartsuit}$ with
Each of the these flavors of $S^1$-equivariant symplectic homology
groups obeys the same set of Eilenberg-Steenrod type axioms as their
nonequivariant counterparts. Transfer maps and invariance for the case of Liouville domains were previously discussed in <cit.>. Moreover, it follows from the
construction that the Gysin and fundamental exact triangles are
functorial with respect to the tautological exact triangles and also
with respect to the exact triangles of pairs, see
also <cit.> for a basic instance of this phenomenon.
§.§ Lagrangian symplectic homology, or wrapped Floer homology
Let $W$ be a Liouville cobordism.
An exact Lagrangian cobordism in $W$ or, for short, a Lagrangian cobordism, is an exact Lagrangian $L\subset W$ which intersects the boundary $\p W$ transversally along a Legendrian submanifold $\p L=L\cap \p W$. This means that $\lambda|_L$ is an exact $1$-form which vanishes when restricted to $\p L$. We denote $\p^\pm L=L\cap \p^\pm W$. Up to applying a Hamiltonian isotopy that fixes $\p W$ one can assume without loss of generality that $L$ is invariant under the Liouville flow near the boundary <cit.>. This means that near its negative or positive boundary we can identify $L$ via the Liouville flow with $[1,1+\epsilon]\times\p^-L$, respectively with $[1-\epsilon,1]\times \p^+L$. We interpret $L$ as a cobordism from $\p^+L$ to $\p^-L$. We refer to $\p^-L$ and $\p^+L$ as being the positive, respectively negative (Legendrian) boundary of $L$.
Let $F$ be a Liouville filling of $\p^-W$. An exact Lagrangian
filling of $\p^-L$ or, for short, a filling of
$\p^-L$, is a Lagrangian cobordism $F_L\subset F$ whose positive
Legendrian boundary is $\p^-L$ (and which has empty negative boundary).
One can associate to a Lagrangian cobordism $L$ with filling $F_L$ Lagrangian symplectic homology groups
SH_*^{\heartsuit}(L) , \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\}.
Similarly, given a pair of Lagrangian cobordisms $K\subset L$ inside a pair of Liouville cobordisms $V\subset W$, with Lagrangian filling $F_L$ inside a Liouville filling $F$, we define Lagrangian symplectic homology groups of the pair $(L,K)$:[ Not to be confused with the (wrapped) Lagrangian intersection Floer homology of a pair of Lagrangians.]
SH_*^{\heartsuit}(L,K) , \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\}.
These are “open string analogues" of the symplectic homology groups defined for the filled Liouville cobordism $W$, respectively for the pair of Liouville cobordisms $(W,V)$ with filling. They are defined using exactly the same shape of Hamiltonian as in the “closed string" case. Given such a Hamiltonian, the generators of the corresponding chain complexes are Hamiltonian chords with endpoints on $L$
\gamma:[0,1]\to W,\qquad \gamma(\{0,1\})\subset L,\qquad \dot\gamma = X_H\circ\gamma,
and the Floer differential counts strips with Lagrangian boundary
condition on $L$ which are finite energy solutions of the Floer
u:\R\times[0,1]\to W,\qquad u(\R\times\{0,1\})\subset L, \qquad
\p_su+J(u)(\p_t u - X_H\circ u)=0.
The theory is naturally defined over $\Z/2$, and an additional assumption on the Lagrangian is needed (e.g. relatively spin) in order to define the theory with more general coefficients.
Let $L$ be a Lagrangian cobordism inside a Liouville domain $W$, so
that $L$ has empty negative boundary and empty filling. The Lagrangian
symplectic homology group
$SH_*(L)$ coincides with the wrapped Floer homology group of $L$
introduced in <cit.>.
The Lagrangian symplectic homology group $SH_*^{>0}(L)$ is isomorphic
to the linearized Legendrian contact homology group of
$\p^+L$ <cit.>. The Lagrangian symplectic homology group
$SH_*^{=0}(L)$ is isomorphic to the singular cohomology group $H^{n-*}(L)$ of
$L$. The Lagrangian symplectic homology group of the trivial cobordism
$I\times \p^+L\subset I\times \p^+W$, with $I$ a closed interval in
$(0,\infty[$, is isomorphic to the Lagrangian Rabinowitz-Floer
homology group of $\p^+W$ <cit.>.
The Lagrangian symplectic homology groups obey the same formal
properties as their closed counterparts, reminiscent of the
Eilenberg-Steenrod axioms: functoriality, homotopy invariance, exact
triangle of a pair, excision. Also, the various flavors
$SH_*^\heartsuit(L,K)$ fit into tautological exact triangles, which
are compatible with the exact triangles of pairs. The proofs of all
these properties are word for word the same as for Liouville
cobordisms, using Lagrangian analogues of our confinement
lemmas <ref>, <ref>, <ref>,
see also <cit.>.
Open-closed theory. Let $(W,V)$ be a pair of
Liouville cobordisms with filling $F$, and $(L,K)\subset (W,V)$ be a
pair of Lagrangian cobordisms with filling $F_L$. One can define
open-closed symplectic homology groups
SH_*^{\heartsuit}((W,V),(L,K)) , \qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\}
by simultaneously taking into account closed Hamiltonian orbits in $W$
and Hamiltonian chords with endpoints on $L$, using the same shape of
Hamiltonians as in the closed or open setting (see
also <cit.>). These homology groups fit into exact
\begin{equation*}
\scriptsize
\xymatrix
SH_*^{\heartsuit}(W,V) \ar[rr] & &
SH_*^{\heartsuit}((W,V),(L,K)) \ar[dl] \\ & SH_*^{\heartsuit}(L,K) \ar[ul]^{[-1]}
\end{equation*}
and can be thought of as the homology groups of the cone of the
open-closed map, defined by the count of solutions of a Hamiltonian
Floer equation on a disk with one interior negative puncture and one
boundary positive puncture. The Eilenberg-Steenrod package holds in
this extended setup as well.
§ APPLICATIONS
§.§ Ubiquity of the exact triangle of a pair
A certain number of previous computations in the literature can be
reinterpreted from a unified point of view and generalized from our
(1) One of our original motivations for the definition of the
symplectic homology groups of a Liouville cobordism was the
exact triangle relating symplectic homology and Rabinowitz-Floer
homology <cit.>
\begin{equation*} \label{eq:SHRFH}
\xymatrix
SH^{-*}(V) \ar[rr] & & SH_*(V)
\ar[dl] \\ & RFH_*(\p V) \ar[ul]^{[-1]}
\end{equation*}
In view of Poincaré duality $SH^{-*}(V)\cong SH_*(V,\p V)$ and the
isomorphism (<ref>), this is just the exact triangle of the
pair $(V,\p V)$. See Theorem <ref> below for a
more detailed discussion of this triangle.
(2) The subcritical and critical handle attaching exact
triangles from <cit.> and <cit.>
are special instances of the exact triangle of a pair, see
Sections <ref> and <ref>
below. Moreover, the surgery exact triangles for linearized contact
homology appear as formal consequences of the corresponding triangles
for symplectic homology, via the relations between equivariant and
non-equivariant symplectic homologies; see
Section <ref> below.
(3) Let $L\subset V$ be an exact Lagrangian in a Liouville domain $V$
satisfying $SH_*(L)=0$. For example, by a straightforward adaptation
of the vanishing results in <cit.> this
is the case if the completion $\wh L$ is displaceable from $V$ in the
completion $\wh V$. Then the tautological sequence yields the
\begin{equation*}
SH_*^{>0}(L)\cong SH_{*-1}^{\leq 0}(L)\cong H^{n-*+1}(L),
\end{equation*}
which was previously conjectured by Seidel,
see <cit.>, and proved from a Legendrian
contact homology perspective by Dimitroglou
Rizell <cit.>. This isomorphism implies the
refinement of Arnold's chord conjecture given
in <cit.>, see Corollary <ref>
below. A combination of the tautological sequence with the exact
sequence of the pair $(L,\p L)$ and Poincaré duality yields the
Poincaré duality long exact sequence for Legendrian contact
homology in <cit.>
\begin{eqnarray*}\label{eq:EESduality}
\xymatrix{
H^{n-*}(\p L) \ar[rr] & & SH_{>0}^{-*+2}(\p L) \ar[dl] \\
& SH_*^{>0}(\p L) \ar[ul]^{[-1]}
\end{eqnarray*}
as well as its refinement in <cit.>
and <cit.>; see
Proposition <ref> below.
(4) The results of Chantraine, Dimitroglou Rizell, Ghiggini, and Golovko from <cit.> can also be reinterpreted from the perspective of the exact triangle of a pair. As an example, consider the following setup: $L$ is an exact Lagrangian cobordism, $\p^-L$ has an exact Lagrangian filling $F_L$, and we assume that $\wh{F_L\circ L}$ is displaceable from the Liouville domain which contains $F_L\circ L$ in the symplectic completion of the ambient exact symplectic manifold. Then $SH_*(F_L\circ L)=0$ and $SH_*(F_L)=0$ (cf. Theorems <ref> and <ref>), hence also $SH_*(L,\p^-L)=0$. The second long exact sequence in <cit.> is the exact triangle of the pair $(F_L\circ L,F_L)$ for $SH_{>0}^*$. The setup considered in <cit.> is that in which $L$ is a Lagrangian concordance, so that the transfer map $SH_{*}^{=0}(F_L\circ L)\stackrel \cong \longrightarrow SH_{*}^{=0}(F_L)$ is an isomorphism. In view of the commutative diagram given by the compatibility of tautological exact triangles with the exact triangle of the pair $(F_L\circ L,F_L)$,
\xymatrix{
SH_*^{>0}(F_L\circ L)\ar[r] \ar[d] & SH_*^{>0}(F_L) \ar[d] \\
SH_{*-1}^{=0}(F_L\circ L)\ar[r]^\cong & SH_{*-1}^{\color{black}=0}(F_L)
the vertical arrows being isomorphisms since $SH_*(F_L\circ L)$ and $SH_*(F_L)$ vanish, we obtain that the top transfer map is an isomorphism. This is the content of the main result of <cit.> in the case of linearized Legendrian contact homology, see also <cit.>. The more general bilinearized setup in <cit.> can be reinterpreted in a similar way.
This circle of ideas should be compared with the results of Biran and Cornea <cit.>, and also with the results of Dimitroglou Rizell and Golovko <cit.>.
§.§ Duality results
The following consequence of the long exact sequence of a pair and
Poincaré duality is proved in
<cit.>. For convenience, we provide
the short proof in our framework.
For a Liouville domain $V$ there is a commuting diagram with exact
upper row
\begin{equation}\label{eq:duality}
\xymatrix
\cdots SH^{-*}(V) \ar[d] \ar[r]^\phi & SH_*(V) \ar[r]^\psi & SH_*(\p V) \ar[r] &
SH^{1-*}(V) \cdots \\
H_{n+*}(V) \ar[r] & H^{n-*}(V) \ar[u].
\end{equation}
Here the horizontal maps come from the long exact sequences of
the pair $(V,\p V)$ in view of Poincaré duality $SH_*(V,\p V)\cong
SH^{-*}(V)$ and $H_{n+*}(V)\cong H^{n-*}(V,\p V)$, and the vertical
maps are given by the compositions
\begin{gather*}
SH^{-*}(V)\to SH^{-*}_{\leq 0}(V) = SH^{-*}_{=0}(V)\cong H_{n+*}(V), \cr
H^{n-*}(V) \cong SH_*^{=0}(V) = SH_*^{\leq 0}(V) \to SH_*(V).
\end{gather*}
Commutativity of the diagram (<ref>) follows from
commutativity of the diagram
\begin{equation*}
\scriptsize
\xymatrix
SH^{-*}(V) \ar[d] \ar[r]^-\cong & SH_*(V,\p V) \ar[d] \ar[r] & SH_*^{\ge 0}(V)=SH_*(V) \\
SH^{-*}_{\le 0}(V)=SH^{-*}_{=0}(V) \ar[d]^\cong \ar[r]^-\cong & SH_*^{=0}(V,\p V)=SH_*^{\ge 0}(V,\p V) \ar[d]^\cong \ar[r] \ar[ur] & SH_*^{=0}(V) \ar[u] \\
H_{n+*}(V) \ar[r]^-\cong & H^{n-*}(V,\p V) \ar[r] & H^{n-*}(V). \ar[u]^\cong
\end{equation*}
Here the left horizontal maps are Poincaré duality isomorphisms and
the lower right square commutes by
Proposition <ref>.
The commutativity of the upper right square can be interpreted as follows:
by definition of the symplectic homology groups, the composition of
the three maps around the upper square is obtained by considering a
Hamiltonian vanishing on $V$ and increasing its slope near $\p V$ from
large negative to small negative to small positive to large positive,
which yields the upper horizontal map.
Here is a computational application of the Poincaré Duality
Theorem <ref>, which will be needed for the discussion of
products in Section <ref>.
Let $W$ be a Liouville cobordism with Liouville filling $F$. Then we
have a canonical isomorphism
SH_*^{<0}(W)\cong SH_{>0}^{-*+1}(F).
We successively have
SH_*^{<0}(W)\cong SH_{*-1}^{<0}(F\cup W,W)\cong SH_{*-1}^{<0}(F,\p F) \cong SH^{-*+1}_{>0}(F).
The first isomorphism follows from the exact triangle of the pair
$(F\cup W,W)$ for $SH_*^{<0}$ (cf. <ref>)
taking into account that $SH_*^{<0}(F\cup W)=0$ because $F\cup W$ has
empty negative boundary. The second isomorphism is the Excision
Theorem <ref>. The third isomorphism is Poincaré
For further duality results we will need the following vanishing result.
Let $V$ be a Liouville domain. Then
SH_*^\heartsuit([0,1]\times\p V,0\times \p V)=0.
for $\heartsuit\in\{\varnothing, >0,\ge 0, =0, \le 0, <0\}$.
We are computing the symplectic homology group of a cobordism relative to the concave part of the boundary and therefore the relevant Floer complexes do not involve orbits with negative action. Thus $SH_*^{(a,b)}([0,1]\times\p V,0\times \p V)=SH_*^{(-\epsilon,b)}([0,1]\times\p V,0\times \p V)$ for all $a<0$, $b>0$ and $\epsilon>0$ smaller than the period of a closed Reeb orbit on $\p V$. In the definition of symplectic homology the inverse limit over $a\to-\infty$ therefore stabilizes and we have
$SH_*([0,1]\times\p V,0\times \p V)=\lim\limits^{\longrightarrow}_{b\to\infty} SH_*^{(-\epsilon,b)}([0,1]\times\p V,0\times \p V)$.
The point now is that $SH_*^{(-\epsilon,b)}([0,1]\times\p V,0\times \p V)=0$ for all $b>0$. Indeed, for $b>0$ not lying in the action spectrum of $\p V$, this homology group is computed using the Floer complex generated by closed orbits near $[0,1]\times\p V$ for a Hamiltonian which vanishes on $[0,1]\times \p V$, which has positive slope $b$ near $\{0,1\}\times \p V$, and which is constant in $V$ away from $[0,1]\times \p V$. But such a Hamiltonian can be deformed to one which has constant slope equal to $b$ all over $[0,1]\times \p V$ and for which the corresponding chain complex is zero. See Figure <ref>, in which the deformed Hamiltonian is drawn with a dashed line. The conclusion follows using the homotopy invariance of the homology under compactly supported deformations.
This proves $SH_*^{\geq 0}([0,1]\times\p V,0\times \p V)=0$.
Vanishing of $SH_*^{=0}([0,1]\times\p V,0\times \p V)$ follows from
vanishing of relative singular cohomology, and vanishing of
$SH_*^{>0}([0,1]\times\p V,0\times \p V)$ then follows from the
truncation exact triangle. Since there are no other versions to
consider, this proves the proposition.
Symplectic homology relative to the negative boundary for a trivial cobordism.
Let $V$ be a Liouville domain and denote by $SH_*(\p V)$ and $SH^*(\p
V)$ the symplectic (co)homology of the trivial cobordism
$[0,1]\times\p V$. Then there is a canonical isomorphism
SH_{*+1}(\p V)\cong SH^{-*}(\p V).
Denote $W=[0,1]\times \p V$ and consider the Liouville
cobordism triple $(W,\p W,\p^-W)$. Although this falls slightly off the setup that we considered for triples to the same extent that the symplectic homology group $SH_*(W,\p W)$ falls slightly off our setup for pairs, the exact triangle for the triple does hold, see also the discussion in the Introduction. In view of Proposition <ref>, this exact triangle writes
0 = SH_{*+1}(W,\p^-W)\to SH_{*+1}(\p^+W)\to SH_*(W,\p W) \to
SH_*(W,\p^-W) = 0.
The conclusion now follows from the canonical isomorphisms $SH_{*+1}(\p^+W)\cong SH_{*+1}(\p V)$ and
$SH_*(W,\p W)\cong SH^{-*}(W)\cong SH^{-*}(\p V)$, the last one being Poincaré duality.
For every Liouville domain $V$ there exist canonical isomorphisms
between the symplectic homology and cohomology groups of the trivial
cobordism over $\p V$,
PD: SH_*^\heartsuit(\p V)\stackrel{\cong}\longrightarrow
SH^{1-*}_{-\heartsuit}(\p V)
for $\heartsuit\in\{\varnothing, >0,\ge 0, =0, \le 0, <0\}$.
We consider the trivial cobordism $W=I\times\p V$ and apply
Proposition <ref> to the triple $(W,\p W,\p_+ W)$ to obtain
the commuting diagram
\begin{equation*}
\scriptsize
\xymatrix{
SH_*^\heartsuit(W,\p_+W) \ar[r] \ar@{=}[d] &
SH_*^\heartsuit(\p W,\p_+W) \ar[r]^\cong \ar[d]^\cong_{exc} &
SH_{*-1}^\heartsuit(W,\p W) \ar[r] \ar[d]^\cong_{PD} & SH_{*-1}^\heartsuit(W,\p_+W) \ar@{=}[d] \\
0 \ar[r] \ar@{=}[d] & SH_*^\heartsuit(W) \ar[r]^\cong \ar[d]^\cong_{PD} &
SH^{1-*}_\heartsuit(W) \ar[r] \ar[d]^\cong_{exc} & 0 \ar@{=}[d] \\
SH^{-*}_{-\heartsuit}(W,\p_-W) \ar[r] &
SH^{1-*}_{-\heartsuit}(W,\p W) \ar[r]^\cong &
SH^{1-*}_{-\heartsuit}(\p W,\p_-W) \ar[r] & SH^{-*}_{-\heartsuit}(W,\p_-W) \\
\end{equation*}
where the first and last row are the long exact sequences of the
triples $(W,\p W,\p_+W)$ and $(W,\p W,\p_+W)$,
respectively, and the vertical arrows are the Poincaré duality
and excision isomorphisms. The groups
$SH_*^\heartsuit(W,\p_+W)$ and $SH^{-*}_{-\heartsuit}(W,\p_-W)$ vanish
by Proposition <ref>.
The middle horizontal map defined by this diagram is the desired Poincaré duality
isomorphism from $SH_*^\heartsuit(\p V)=SH_*^\heartsuit(W)$ to
$SH^{1-*}_{-\heartsuit}(W)=SH^{1-*}_{-\heartsuit}(\p V)$.
For every Liouville domain $V$ and $\heartsuit\in\{\varnothing, >0,\ge
0, =0, \le 0, <0\}$ there exists a commuting diagram
\begin{equation}\label{eq:PD-LES}
\scriptsize
\xymatrix{
\dots SH_*^\heartsuit(V,\p V) \ar[r] \ar[d]^\cong_{PD} & SH_*^\heartsuit(V)
\ar[r] \ar[d]^\cong_{PD} & SH_*^\heartsuit(\p V) \ar[r] \ar[d]^\cong_{PD} &
SH_{*-1}^\heartsuit(V,\p V) \dots \ar[d]^\cong_{PD} \\
\dots SH^{-*}_{-\heartsuit}(V) \ar[r] & SH^{-*}_{-\heartsuit}(V,\p
V) \ar[r] & SH^{1-*}_{-\heartsuit}(\p V) \ar[r] & SH^{1-*}_{-\heartsuit}(V) \dots \\
\end{equation}
where the rows are the long exact sequences of the pair $(V,\p V)$
and the vertical arrows are the Poincaré duality
isomorphisms from Theorem <ref> (the third one) and
Theorem <ref> (the other ones).
Moreover, the Poincaré duality isomorphisms are compatible with
filtration exact sequences.
Denote by $W$ the trivial cobordism given by a collar neighborhood of
the boundary $\p V$ in $V$. Denote $U=\overline{V\setminus W}$, so
that $\p_+W=\p V$ and $\p _-W=\p U\simeq \p V$. Consider the following
\scriptsize
\xymatrix{
SH_*^\heartsuit(V,\p V) \ar[r] \ar[d]^\cong_{PD} & SH_*^\heartsuit(V)
\ar[r] \ar[d]^\cong_{PD} & SH_*^\heartsuit(\p V) \ar[r] \ar[d]^\cong_{PD} &
SH_{*-1}^\heartsuit(V,\p V) \ar[d]^\cong_{PD} \\
SH^{-*}_{-\heartsuit}(V) \ar[r] \ar[d]^\cong_{exc.}ĂŠ& SH^{-*}_{-\heartsuit}(V,\p
V) \ar[r] \ar@{=}[d] & SH^{-*}_{-\heartsuit}(V,U\cup \p V) \ar[r] \ar@{=}[d] & SH^{1-*}_{-\heartsuit}(V) \ar[d]^\cong_{exc.} \\
SH^{-*}_{-\heartsuit}(U\cup \p V,\p V) \ar[r]ĂŠ& SH^{-*}_{-\heartsuit}(V,\p
V) \ar[r] \ar@/_3pc/[ddrr] & SH^{-*}_{-\heartsuit}(V,U\cup \p V) \ar[r] & SH^{1-*}_{-\heartsuit}(U\cup \p V,\p V) \\
& & SH^{-*}_{-\heartsuit}(W,\p W) \ar[u]_\cong^{exc.}ĂŠ\ar[r]^\cong \ar[dr]ĂŠ& SH^{-*+1}_{-\heartsuit}ĂŠ(\p W,\p_+W)ĂŠ\ar[u]ĂŠ\\
& & & SH^{-*+1}_{-\heartsuit}(\p _-W) \ar[u]_\cong^{exc.}ĂŠ
The diagram is commutative. The first three rows with their vertical maps
correspond to the commutative diagram in Proposition <ref> applied to
the triple $(V,W,\varnothing)$, so the first and third rows are the long
exact sequences of the triples $(V,W,\emptyset)\cong(V,\p
V,\varnothing)$ and $(V,U\cup\p V,\p V)$, respectively.
The right bottom most square is commutative because the maps are induced by the inclusion of triples $(W,\p W,\p_+W)\hookrightarrow (V,U\cup \p V,\p V)$. The bottom right triangle is commutative by definition.
The third column vertical downward composition
SH_*^\heartsuit(\p V) \to SH^{-*}_{-\heartsuit}(V,U\cup \p V)\to SH^{-*}_{-\heartsuit}(W,\p W) \to SH^{-*+1}_{-\heartsuit}(\p_- W)\simeq SH^{-*+1}_{-\heartsuit}(\p V)
is the Poincaré duality isomorphism of
Theorem <ref> (by inspection of the diagram in its proof).
The bottom arrow composition
SH^{-*}_{-\heartsuit}(V,\p V)\to SH^{-*}_{-\heartsuit}(V,U\cup \p V) \to SH^{-*}_{-\heartsuit}(W,\p W)\to SH^{-*+1}_{-\heartsuit}(\p _-W)\simeq SH^{-*+1}_{-\heartsuit}(\p V)
is the connecting homomorphism in the cohomology long exact sequence of the pair $(V,\p V)$.
Finally, the fourth column vertical upward composition
SH^{-*+1}_{-\heartsuit}(\p V) \simeq SH^{-*+1}_{-\heartsuit}(\p_-W) \to SH^{-*+1}(\p W,\p _+W) \to SH^{-*+1}(U\cup \p V,\p V) \to SH^{-*+1}_{-\heartsuit}(V)
is the cohomology transfer map for the inclusion $\p V\hookrightarrow V$.
Ę Upon considering the triple $(W,\p W,\p_+W)$ in the proof of Theorem <ref> and the triple $(V,U\cup \p V,\p V)$ in the proof of Theorem <ref> we formally enter the setup of multilevel cobordisms discussed in <ref>. While we have not explicitly provided proofs for the excision theorem and for the existence of the homology long exact sequences of pairs/triples in that setup, the particular situations that we consider in Theorems <ref> and <ref> are the simplest possible and the proofs of those results clearly follow from the corresponding theorems for cobordisms with one level. See also the discussion at the end of <ref>.
Recall that at action zero symplectic homology specialises to singular
cohomology, $SH_*^{=0}(V) \cong H^{n-*}(V)$,
and similarly for the other versions. Therefore, we obtain
The commuting diagram in Theorem <ref> specialises at
action zero to
\begin{equation}\label{eq:PD-LES-zero}
\scriptsize
\xymatrix{
\dots H^{n-*}(V,\p V) \ar[r] \ar[d]^\cong_{PD} & H^{n-*}(V)
\ar[r] \ar[d]^\cong_{PD} & H^{n-*}(\p V) \ar[r] \ar[d]^\cong_{PD} &
H^{n-*+1}(V,\p V) \dots \ar[d]^\cong_{PD} \\
\dots H_{n+*}(V) \ar[r] & H_{n+*}(V,\p
V) \ar[r] & H_{n+*-1}(\p V) \ar[r] & H_{n+*-1}(V) \dots \\
\end{equation}
where the rows are the long exact sequences of the pair $(V,\p V)$
and the vertical arrows are the Poincaré duality
isomorphisms for the closed manifold $\p V$ (the third one) and the
manifold-with-boundary $V$ (the other ones). $\square$
We conclude this subsection with an example illustrating that full symplectic homology and
cohomology do not obey any kind of algebraic duality for general
Liouville cobordisms.
Let $V$ be the canonical Liouville filling of a Brieskorn manifold
$\{z\in\C^{n+1}\mid \sum_{j=0}^nz^{a_j}=0,\;|z|=1\}$ with $n\geq 3$
and integers $a_j\geq 2$ satisfying
$\sum_{j=0}^n\frac{1}{a_j}=1$. P. Uebele <cit.> has shown that
with $\Z_2$-coefficients its symplectic homology in degrees $n$ and
$1-n$ is an infinite direct sum
SH_k(V;\Z_2)\cong\bigoplus_{\N}\Z_2\quad\text{for $k=n$ and $k=1-n$}.
By algebraic duality, it follows that its symplectic cohomology in
these degrees is an infinite direct product
SH^k(V;\Z_2)\cong SH_k(V;\Z_2)^\vee\cong \prod_{\N}\Z_2\quad\text{for $k=n$ and $k=1-n$}.
In view of the exact sequence (<ref>) with the map $\phi$
of finite rank, $SH_k(\p V;\Z_2)$ agrees with $SH_k(V)\oplus
SH^{1-k}(V)$ up to an error of finite dimension, hence
SH_k(\p V;\Z_2) \cong \bigoplus_{\N}\Z_2\oplus \prod_{\N}\Z_2\quad\text{for $k=n$ and $k=1-n$}.
By Theorem <ref>, the symplectic cohomology groups in
these degrees are the same,
SH^k(\p V;\Z_2) \cong \bigoplus_{\N}\Z_2\oplus \prod_{\N}\Z_2\quad\text{for $k=n$ and $k=1-n$}.
Since the dual of the infinite direct product is not the infinite
direct sum, this shows that for $k=n,1-n$ neither $SH^k(\p
V;\Z_2)=SH_k(\p V;\Z_2)^\vee$ nor $SH_k(\p V;\Z_2)=SH^k(\p
§.§ Vanishing and finite dimensionality
In this subsection we give some conditions under which symplectic
homology groups are zero or finite dimensional. We begin with a simple
consequence of the duality sequence (<ref>).
For a Liouville domain $V$ the following hold using field coefficients:
(a) If one among $SH_n(V)$, $SH^{-n}(V)$, $SH_n(\p V)$, or $SH_n(V,\p V)$ vanishes, then
all of $SH_*(V)$, $SH^{-*}(V)$, $SH_*(\p V)$, and $SH_*(V,\p V)$ vanish.
(b) If one among $SH_*(V)$, $SH^*(V)$, $SH_*(\p V)$, or $SH_*(V,\p V)$ is finite
dimensional, then so are the other three.
Part (a) is Theorem 13.3 in <cit.>, except for the statement
that involves $SH_*(V,\p V)$, which is a consequence of Poincaré
duality. For part (b), in view of Poincaré duality $SH_*(V,\p
V)\cong SH^{-*}(V)$ we only need to deal with $SH_*(V)$, $SH^*(V)$,
and $SH_*(\p V)$.
Since $SH^k(V)\cong \Hom\bigr(SH_k(V),\frak{k}\bigr)$ in each degree,
$SH^*(V)$ is finite dimensional iff $SH_*(V)$ is. If both are finite
dimensional, then two out of three terms in the exact sequence (<ref>) are
finite dimensional, so the third term $SH_*(\p V)$ is finite
dimensional as well. Conversely, suppose that $\dim SH(\p V)<\infty$.
Then the map $\psi$ in (<ref>) has finite rank, as does the
map $\phi$ (because it factors through singular homology), and thus $\dim SH_*(V)<\infty$.
Alternatively, one could argue by contradiction: If $\dim SH(\p
V)<\infty$ and $SH_*(V)$, $SH^*(V)$ were infinite dimensional, then
the long exact sequence (<ref>) would imply $\dim SH_*(V) = \dim SH^*(V)$,
which is impossible by Remark <ref> below.
A $\frak{k}$-vector space is isomorphic to its dual space if and
only if it is finite dimensional (see <cit.> for a nice proof
– we thank I. Blechschmidt for pointing this out). Hence for a pair
of Liouville cobordisms with filling $(W,V)$ and using field
coefficients we obtain that $SH^k_\heartsuit(W,V)$ is isomorphic to
$SH_k^\heartsuit(W,V)$ for $\heartsuit\in\{<0\le 0,=0,\ge 0,>0\}$ if
and only if both vector spaces are finite dimensional.
We say that a subset of a symplectic manifold is displaceable if it can be displaced from itself by a compactly
supported Hamiltonian isotopy. It has been known for a while that
displaceability implies vanishing of Rabinowitz-Floer
homology <cit.> and symplectic
homology <cit.> of a Liouville domain. In the context of this
paper, these appear as special cases of the following general
vanishing result, whose proof is a straightforward adaptation of the
ones in <cit.> and <cit.>.
(a) Let $(W,V)$ be a Liouville cobordism pair with filling $F$ such that $V$ is
displaceable in the completion of $F\circ W$. Then $SH_*(V)=0$.
(b) Let $L\subset V$ be an exact Lagrangian in a Liouville domain $V$ whose
completion $\wh L$ is displaceable from $V$ in the completion
$\wh V$. Then $SH_*(L)=0$.
For example, the displaceability hypothesis in (a) is always satisfied if the
completion of $F\circ W$ is a subcritical Stein manifold, or more
generally the product of a Liouville manifold with $\C$.
(i) If in Theorem <ref>(a) the cobordism $V$ as well as its
filling $E=F\cup W^{bottom}$ are connected, then displaceability of $V$
implies displaceability of $E\cup V$ and the vanishing of $SH_*(V)$
follows from the vanishing of symplectic homology of the Liouville
domains $E$ and $E\cup V$.
(ii) In the situation of Theorem <ref>(a),
displaceability of $V$ implies that of $\p V$, so
we also have $SH_*(\p_\pm V)=SH_*(\p V)=0$ and (via exact sequences of
triples) $SH_*(V,\p_\pm V)=SH_*(V,\p V)=0$.
Another condition that ensures vanishing of $SH_*(V)$ is the vanishing
of $SH_*(W)$ for a pair $(W,V)$. This was observed for Liouville
domains by Ritter <cit.> as a consequence of the product
structure: vanishing of $SH_*(W)$ implies that its unit $1_W$
vanishes, hence so does its image $1_V$ under the transfer map
$SH_*(W)\to SH_*(V)$, which implies $SH_*(V)=0$. In view of
Theorem <ref>, the same argument proves
Let $(W,V)$ be a Liouville cobordism pair.
Then $SH_*(W)=0$ implies $SH_*(V)=0$. $\square$
Again, the hypothesis $SH_*(W)=0$ is satisfied if the
completion of $F\circ W$ is a subcritical Stein manifold, or more
generally the product of a Liouville manifold with $\C$. However,
there exist Liouville domains $W$ that are not of this type and still
have vanishing symplectic homology, e.g. flexible Stein
domains <cit.> as well as certain
non-flexible Stein
domains <cit.>.
Conversely, there exist many examples of Liouville pairs $(W,V)$ with
$V$ displaceable and $SH_*(W)\neq 0$. So neither of the two Vanishing
Theorems <ref> and <ref> implies the
§.§ Consequences of vanishing of symplectic homology
Suppose that $V$ is a Liouville domain with $SH_*(V)=0$. Then the
tautological sequence yields
\begin{equation}\label{eq:vanishing-hom}
SH_*^{>0}(V)\cong SH_{*-1}^{\leq 0}(V)\cong H^{n-*+1}(V)\neq 0.
\end{equation}
Similarly, if $L\subset V$ is an exact Lagrangian with $SH_*(L)=0$, then
\begin{equation}\label{eq:vanishing-hom-Lag}
SH_*^{>0}(L)\cong SH_{*-1}^{\leq 0}(L)\cong H^{n-*+1}(L)\neq 0.
\end{equation}
This has the following dynamical consequences <cit.>.
(a) Let $V$ be a Liouville domain with $SH_*(V)=0$ (e.g., this is the case
if $\p V$ is displaceable in $\wh V$). Then there exists at least one closed Reeb orbit.
(b) Let $L$ be an exact Lagrangian $L\subset V$ with
$SH_*(L)=0$ (e.g., this is the case if $\wh L$ is displaceable from
$V$ in $\wh V$). Then there exists at least one Reeb chord with boundary on $\p L$.
If all the Reeb chords are nondegenerate their number is bounded from below
by $\mathrm{rk}\,H_*(L)\geq \mathrm{rk}\,H_*(\p L)/2$.
The assertion in (a) follows immediately from (<ref>) because
$SH_*^{>0}(V)$ is generated by closed Reeb orbits. Similarly, the
first assertion in (b) follows from (<ref>).
The second assertion in (b) also follows
from (<ref>) because, if all Reeb chords are
nondegenerate, their number is bounded from below by $\mathrm{rk}\,
SH_*^{>0}(V)=\mathrm{rk}\, H^*(V)$. The estimate
$\mathrm{rk}\,H_*(V)\geq \mathrm{rk}\,H_*(\p V)/2$ follows readily
from the long exact sequence of the pair $(V,\p V)$ in singular
homology and Poincaré duality.
Vanishing of symplectic homology also implies the following refinement
of the duality sequence (<ref>).
(a) Let $V$ be a Liouville domain with $SH_*(V)=0$ (e.g., this is the case
if $\p V$ is displaceable in $\wh V$). Then there exists a commuting
diagram with exact rows
\begin{equation*}
\xymatrix{
\cdots H^{n-*}(\p V) \ar[r]^-\sigma \ar[d]^= & SH^{2-*}_{>0}(\p V) \ar[r]^-\tau &
SH_*^{>0}(\p V) \ar[r]^-\rho \ar[d]_g^\cong & H^{n-*+1}(\p V)\cdots \ar[d]^= \\
\cdots H^{n-*}(\p V) \ar[r]^-{\sigma_0} & H^{n-*+1}(V,\p V) \ar[r]^-{\tau_0} \ar[u]_f^\cong &
H^{n-*+1}(V) \ar[r]^-{\rho_0} & H^{n-*+1}(\p V)\cdots
\end{equation*}
(b) Let $L\subset V$ be an exact Lagrangian in a Liouville domain with
$SH_*(L)=0$ (e.g., this is the case if $\wh L$ is displaceable from
$V$ in $\wh V$). Then there exists a commuting diagram with exact rows
\begin{equation*}
\xymatrix{
\cdots H^{n-*}(\p L) \ar[r]^-\sigma \ar[d]^= & SH^{2-*}_{>0}(\p L) \ar[r]^-\tau &
SH_*^{>0}(\p L) \ar[r]^-\rho \ar[d]_g^\cong & H^{n-*+1}(\p L)\cdots \ar[d]^= \\
\cdots H^{n-*}(\p L) \ar[r]^-{\sigma_0} & H^{n-*+1}(L,\p L) \ar[r]^-{\tau_0} \ar[u]_f^\cong &
H^{n-*+1}(L) \ar[r]^-{\rho_0} & H^{n-*+1}(\p L)\cdots
\end{equation*}
For part (a) consider the commuting diagram whose columns are the exact sequences of
the pair $(V,\p V)$ and whose rows are the tautological sequences
\xymatrix
\cdots SH_*^{=0}(V) \ar[r] \ar[d] & SH_*^{\geq 0}(V) \ar[r] \ar[d] &
SH_*^{>0}(V) \ar[r] \ar[d] & SH_{*-1}^{=0}(V)\cdots \ar[d] \\
\cdots SH_*^{=0}(\p V) \ar[r] \ar[d] & SH_*^{\geq 0}(\p V) \ar[r] \ar[d] &
SH_*^{>0}(\p V) \ar[r] \ar[d] & SH_{*-1}^{=0}(\p V)\cdots \ar[d] \\
\cdots SH_{*-1}^{=0}(V,\p V) \ar[r] \ar[d] & SH_{*-1}^{\geq 0}(V,\p V) \ar[r] &
SH_{*-1}^{>0}(V,\p V) \ar[r] & SH_{*-2}^{=0}(V,\p V)\cdots \\
We replace the groups $SH_*^{=0}$ by the corresponding singular
cohomology groups, and insert $SH_*^{\geq 0}(V)=SH_*(V)=0$ (which holds by
hypothesis) and $SH_{*-1}^{>0}(V,\p V)=0$ (which always
holds). Moreover, we replace $SH_{*-1}^{\geq 0}(V,\p V)$ by the
isomorphic group $SH_{*-1}^{\geq 0}(V,\p V)\cong SH_{*-2}^{<0}(V,\p
V)\cong SH^{2-*}_{>0}(V) = SH^{2-*}_{>0}(\p V)$, where the first
isomorphism comes from the tautological sequence in view of $SH_*(V,\p
V)=0$ (which follows from the hypothesis $SH_*(V)=0$ via
Corollary <ref>) and the second one is
Poincaré duality. Then the diagram becomes
\xymatrix{
\cdots H^{n-*}(V) \ar[r] \ar[d] & 0 \ar[r] \ar[d] &
SH_*^{>0}(V) \ar[r]^\cong \ar[d]^\cong & H^{n-*+1}(V)\cdots \ar[d]^{\rho_0} \\
\cdots H^{n-*}(\p V) \ar[r] \ar[d]^{\sigma_0} \ar[rd]_\sigma & SH_*^{\geq 0}(\p V) \ar[r] \ar[d]^\cong &
SH_*^{>0}(\p V) \ar[r]_-\rho \ar[d] \ar[ru]_g^\cong & H^{n-*+1}(\p V)\cdots \ar[d] \\
\cdots H^{n-*+1}(V,\p V) \ar[r]_-f^-\cong \ar[d]^{\tau_0}
& SH^{2-*}_{>0}(\p V) \ar[r] \ar[ru]_\tau & 0 \ar[r] & H^{n-*+2}(V,\p V)\cdots \\
From this we read off the commuting diagram in
Proposition <ref>(a). Part (b) is proved analogously.
Corollary <ref>(b) and the upper long exact sequence in
Proposition <ref>(b) were proved
in <cit.> in the context of contact manifolds of
the form $P\times \R$ (compare also with <cit.>).
The commuting diagram in Proposition <ref>(b) appears
in <cit.>
and <cit.>.
§.§ Invariants of contact manifolds
We describe in this subsection how to obtain invariants of contact
manifolds from the various symplectic homology groups that we defined
in this paper.
Recall that a contact manifold with chosen contact form
$(M^{2n-1},\alpha)$ is called hypertight if it has no
contractible closed Reeb orbits. Following <cit.> we
call $(M,\alpha)$ index-positive if $\xi=\ker\alpha$ satisfies
$c_1(\xi)|_{\pi_2(M)}=0$ and either
* the Conley-Zehnder index of every contractible closed Reeb orbit
$\gamma$ in $M$ satisfies $\CZ(\gamma)+n-3>1$, or
* $(M,\alpha)$ admits a Liouville filling $F$ with
$c_1(F)|_{\pi_2(F)}=0$ such that $\CZ(\gamma)+n-3>0$ for every
closed Reeb orbit $\gamma$ in $M$ which is contractible in $F$.
We will call a (as always,
cooriented) contact manifold $(M,\xi)$ hypertight resp. index-positive if it admits a defining contact form with this
The following result follows in the index-positive case
(ii) from the arguments of <cit.>, as remarked
in <cit.>. For the hypertight case or the
index-positive case (i) see <cit.>. For another instance in the
$S^1$-equivariant case see <cit.>. We sketch below a short unified proof.
Given a Liouville cobordism $W$ whose negative boundary $\p^-W$ is
hypertight or index-positive, the symplectic homology groups
SH_*^{\heartsuit}(W) \mbox{\quad and \quad}
SH_*^{S^1,\heartsuit}(W), \qquad \heartsuit\in \{\varnothing,>0,\ge
0, =0, \le 0, <0\}
are defined, independent of the contact form $\alpha$ on $\p^-W$ in
the given class, and independent of the filling in case (ii).
We will discuss the case $SH_*^{\heartsuit}(W)$, the equivariant case being analogous.
In case (ii) we define $SH_*^{\heartsuit}(W)$ as the usual symplectic
homology group with respect to a filling $F$ in the given class. To
show independence of the filling, fix a finite action window $(a,b)$
and consider a Hamiltonian $H$ on the
completion $\wh{W}_F$ as in Figure <ref>. We perform
neck stretching as described in the proof of Lemma <ref>,
inserting cylindrical pieces $[-R_k,R_k]\times M$ with
$R_k\to\infty$, at the hypersurface $M:=\{\delta\}\times\p^-W$ where
$H\equiv c$ for a constant $c>-a$. We claim that for $k$
sufficiently large, Floer cylinders appearing in the differential between $1$-periodic orbits $x_\pm$ of
$H$ of types $I^-,I^0,I^+$ with action in $(a,b)$ do not enter the region $F\setminus[\delta,1]\times\p F$.
Then it follows that all these Floer cylinders can be viewed as lying in the
$2$-sided completion $\wh{W}$, so $FH_*^{(a,b)}(H)$ is independent of
the filling. By the same claim applied to continuation morphisms, we
deduce independence of the filling for the filtered symplectic
homology groups $SH_*^{(a,b)}(W)$ and the groups $SH_*^{\heartsuit}(W)$.
To prove the claim, we argue by contradiction and suppose that for all
$k$ there exist Floer cylinders $u_k$ as above entering $F\setminus[\delta,1]\times\p F$.
In the limit $k\to\infty$ they converge in the SFT sense to a broken
holomorphic curve $C$ with punctures asymptotic to closed Reeb orbits
on $M$. We first observe that $C$ can have only one component in $\wh{W}$.
This follows by the argument in the proof of Lemma <ref>:
Otherwise there would exist for large $k$ a separating loop $\delta_k$
on the domain $\R\times S^1$, winding around in the negative $S^1$-direction,
such that $u_k(\delta_k)$ is $C^1$-close to a (positively parameterized)
closed Reeb orbit $\gamma$ on $M$, and the resulting estimate
$A_H(x_-)\leq -c<a$ would contradict the condition $A_H(x_-)>a$.
It follows that $C$ consists of a Floer cylinder $C_+$ in $\wh{W}$ with
$p\geq 1$ negative punctures asymptotic to closed Reeb orbits $\gamma_i$ and
holomorphic planes $C_i$ in $\wh{F}$ asymptotic to $\gamma_i$. The
component $C_+$ belongs to a transversely cut out moduli space of
dimension at least $1$ (due to $\R$-translations in the domain), so
its Fredholm index satisfies $\ind(C_+)\geq 1$. On the other hand, its
index is given by
\ind(C_+) = \CZ(x_+)-\CZ(x_-)-\sum_{i=1}^p\bigl(\CZ(\gamma_i)+n-3\bigr),
which in view of $\CZ(x_+)-\CZ(x_-)=1$ for contributions to the Floer
differential yields
\sum_{i=1}^p\bigl(\CZ(\gamma_i)+n-3\bigr) \leq 0.
Since $\CZ(\gamma_i)+n-3>0$ and $p\geq 1$, this is a contradiction
and case (ii) is proved.
The proof in case (i) is very similar. We again consider $(a,b)$ and
$H$ as above, where $H$ is now defined on the $2$-sided completion
$\wh{W}$ rather than $\wh{W}_F$. We define the Floer differential for
$H$ by counting Floer cylinders between orbits $x_\pm$ in
$\wh{W}$. This is well-defined because SFT type breaking of Floer cylinders
at the negative end of $\wh{W}$ is ruled out by exactly the same
argument as in case (ii). In contrast to case (ii) where this was
automatic, we now must also show that the Floer differential squares
to zero. For this, we must rule out SFT type breaking of Floer cylinders
connecting orbits $x_\pm$ of index difference $2$. If such breaking
occurs the argument in case (ii) leads to $p\geq 1$ orbits $\gamma_i$
\sum_{i=1}^p\bigl(\CZ(\gamma_i)+n-3\bigr) \leq 1.
Under the stronger hypothesis $\CZ(\gamma_i)+n-3>1$ this is again a contradiction
and case (i) is proved.
This proposition leads to the definition of homological invariants of
hypertight or index-positive contact manifolds,
SH_*^{[S^1,] \heartsuit}(M,\xi) = SH_*^{[S^1,] \heartsuit}(I\times
M),\qquad \heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\},
where $I=[0,1]$ and $I\times M$ is the trivial Liouville cobordism.
Here the notation $SH_*^{[S^1,] \heartsuit}$ means that the symbol $S^1$
is optional.
In view of <cit.>, the group $SH_*(M,\xi)$ can be interpreted as the Rabinowitz-Floer homology group of $(M,\xi)$. A construction of Rabinowitz-Floer homology for hypertight contact manifolds has been recently carried out in <cit.>.
These contact invariants satisfy various functoriality relations, as
dictated by our functoriality relations for Liouville cobordisms. The
general picture is the following: Given a Liouville cobordism $W$
whose negative
boundary is hypertight or index-positive, we have maps
SH_*^{[S^1,] \heartsuit}(\p^-W) \longleftarrow SH_*^{[S^1,] \heartsuit}(W) \longrightarrow SH_*^{[S^1,] \heartsuit}(\p^+W)
determined by the embedding of trivial cobordisms
I\times \p^-W\subset W \supset I\times \p^+W.
Since $I\times \p^-W$ and $W$ share the same negative boundary we have an isomorphism $SH_*^{[S^1,] <0}(\p^-W) \stackrel \cong\longleftarrow SH_*^{[S^1,]<0}(W)$, and since $W$ and $I\times \p^+W$ share the same positive boundary we have an isomorphism $SH_*^{[S^1,] >0}(W)\stackrel \cong \longrightarrow SH_*^{[S^1,] >0}(\p^+W)$. In particular we obtain maps
SH_*^{[S^1,] >0}(\p^-W)\longleftarrow SH_*^{[S^1,] >0}(\p^+W)
SH_*^{[S^1,] <0}(\p^-W)\longrightarrow SH_*^{[S^1,] <0}(\p^+W).
In the equivariant case and under slightly different assumptions the
first of these two maps was previously constructed by Jean Gutt in
Such direct maps do not exist for the other versions $\heartsuit\in \{\varnothing,\ge 0, =0, \le 0\}$. In general the cobordism $W$ has to be interpreted as providing a correspondence, and this holds in particular for the case of Rabinowitz-Floer homology.
Invariants of Legendrian submanifolds.
Let $(M^{2n-1},\alpha)$ be a manifold with chosen contact form and
$\Lambda^{n-1}\subset M$ a Legendrian submanifold.
Extending the earlier definitions to the open case, we call $\Lambda$
hypertight if $(M,\alpha)$ is hypertight and $\Lambda$ has
no contractible Reeb chords. We call $\Lambda$ index-positive
if $(M,\alpha)$ is index-positive and in addition
* in case (i) the Maslov class of $\Lambda$ vanishes on $\pi_2(M,\Lambda)$
and every Reeb chord $c$ that is trivial in $\pi_1(M,\Lambda)$
satisfies $\CZ(c)>1$;
* in case (ii) $\Lambda$ admits an exact Lagrangian
filling $L\subset F$ in the filling $F$ whose Maslov class vanishes on
$\pi_2(F,L)$ such that $\CZ(c)>0$ for every
Reeb chord $c$ for $\Lambda$ that is trivial in $\pi_1(F,L)$.
We call a Legendrian submanifold in a contact manifold $(M,\xi)$ hypertight resp. index-positive if it admits a
defining contact form with this property.
The arguments given in the closed case adapt in a straightforward way
in order to define invariants of hypertight or index-positive
Legendrian submanifolds by
SH_*^{\heartsuit}(\Lambda)=SH_*^{\heartsuit}(I\times \Lambda), \qquad
\heartsuit\in \{\varnothing,>0,\ge 0, =0, \le 0, <0\}.
§.§ Subcritical handle attaching
In this subsection we compute the symplectic homology groups
corresponding to a subcritical handle in the sense of <cit.>, with coefficients in a field $\mathfrak{k}$.
Let $W^{2n}$ be a filled Liouville cobordism corresponding to a subcritical
handle of index $k<n$. Then
\begin{align*}
SH_*(W,\p^-W) =0, & \qquad SH_{*}(W,\p^+W)=0,\cr
SH_*^{=0}(W,\p^-W) &\cong SH_{-*}^{=0}(W,\p^+W)=\begin{cases}
\mathfrak{k} & *=n-k, \\ 0 & \text{else,} \end{cases}
\end{align*}
and the restriction maps induce isomorphisms
SH_*(W)\stackrel{\cong}\longrightarrow SH_*(\p^+W).
The vanishing of $SH_*(W,\p^-W)$ is proved in <cit.> with
arbitrary coefficients as a consequence of the following fact: for
each degree $i$ there exists $b_i>0$ such that
$SH_i^{(a,b)}(W,\p^-W)=0$ for any $a<0$ and $b\ge b_i$.
$SH_*(W,\p^-W)=SH_*^{\ge 0}(W,\p^-W)$, we can apply the algebraic
duality Proposition <ref> to obtain
$SH^*(W,\p^-W)=SH^*_{\ge 0}(W,\p^-W)=0$, which implies by Poincaré
duality $SH_{-*}(W,\p^+W)=0$.
Since $H^*(W,\p^-W)$ equals $\mathfrak{k}$ in degree $k$ and vanishes in all the
other degrees, we obtain
SH_*^{=0}(W,\p^-W) \cong H^{n-*}(W,\p^-W) = \begin{cases}
\mathfrak{k} & *=n-k, \\ 0 & \text{else}. \end{cases}
The remaining two isomorphisms follow from the long exact sequences
\begin{gather*}
0=SH_*(W,\p^-W) \to SH_*(W) \to SH_*(\p^-W) \to SH_{*-1}(W,\p^-W)
=0, \cr
0=SH_*(W,\p^+W) \to SH_*(W) \to SH_*(\p^+W) \to SH_{*-1}(W,\p^+W)
\end{gather*}
(a) From Proposition <ref> and the tautological sequence we can compute the remaining
relevant symplectic homology groups of the pair $(W,\p_\pm W)$, namely
SH_*^{>0}(W,\p^-W) \cong SH_{-*}^{<0}(W,\p^+W)=\begin{cases}
\mathfrak{k} & *=n-k+1, \\ 0 & \text{else}. \end{cases}
Note that the symplectic homology groups relative to one boundary
component only depend on the index $k$, whereas the group $SH_*(W)$
depends on the whole hypersurface $\p^-W$ and its filling.
(b) In view of (<ref>), the last statement in
Proposition <ref> gives in particular the isomorphism of
Rabinowitz Floer homology groups
RFH(\p^+W) \cong RFH(\p^-W).
(c) Suppose that $(W,V,U)$ is a Liouville cobordism triple such that $W\setminus
V$ is subcritical. Then Proposition <ref> implies
$SH_*(W,V)=0$, which together with the exact sequence of the triple (Proposition <ref>) yields the isomorphism
SH_*(W,U)\stackrel{\cong}\longrightarrow SH_*(V,U).
In particular, for $U=\emptyset$ we recover by induction the vanishing
of symplectic homology for subcritical Stein domains.
(d) The computation of Proposition <ref> is valid more generally with coefficients in an abelian group, but the proof uses filtered symplectic homology and a more general universal coefficients theorem.
Together with the exact triangle of a pair, these computations
provide a complete understanding of the behaviour of all the flavors of
non-equivariant symplectic homology groups under subcritical handle
attachment, as a consequence of the exact triangle of the pair
$(V\circ W,V)$, where $V$ is a Liouville domain.
The equivariant case is discussed in
Section <ref> below.
§.§ Critical handle attaching
Recall that we use coefficients in a field $\mathfrak{k}$.
In the previous section we saw that the key computation was that of $SH_*(W,\p^-W)$, and the key exact triangle was the exact triangle of the pair $(V',V)$, where $V$ is the filling of $\p^-W$ and $V'=V\circ W$ is the Liouville domain obtained after attaching the handle.
These same objects form the relevant structure in the case of a critical handle attachment.
Let $V$ be a Liouville domain, let
$\Lambda=\Lambda_1\sqcup\dots\sqcup\Lambda_\ell$ be a collection of
disjoint Legendrian spheres in $\p V$, denote by $W$ the cobordism
obtained by attaching $\ell$ critical handles (of index $n$) along
these spheres, and denote $V'=V\circ W$.
Bourgeois, Ekholm, and Eliashberg <cit.>
assert the existence of surgery exact triangles[Since at the time of
writing this article the proof of this result is not yet completed,
we formulate its consequences below as conjectures.]
\begin{equation} \label{eq:BEE}
\scriptsize
\xymatrix
L\H^{\text{Ho}}(\Lambda)_* \ar[rr] & &
SH_*(V') \ar[dl] \\ & SH_*(V) \ar[ul]^{[-1]}
\qquad
\xymatrix
L\H^{\text{Ho}+}(\Lambda)_* \ar[rr] & &
SH_*^{>0}(V') \ar[dl] \\ & SH_*^{>0}(V) \ar[ul]^{[-1]}
\end{equation}
in which $L\H_*^{\text{Ho}}(\Lambda)$ and
$L\H^{\text{Ho}+}(\Lambda)_*$ are homology groups of Legendrian
contact homology flavour, see also <cit.>
<cit.>. More precisely,
$L\H^{\text{Ho}+}(\Lambda)_*$ is defined as the homology of a complex
$LH^{\text{Ho}+}(\Lambda)_*$ whose generators are words in Reeb chords
on $\p V$ with endpoints on $\Lambda$, and whose differential counts
certain pseudo-holomorphic curves in the symplectization of $\p V$
with boundary on the conical Lagrangian $S\Lambda$ determined by
$\Lambda$, with a certain number of interior and boundary punctures at
which rigid pseudo-holomorphic planes in $\wh V$, respectively rigid
pseudo-holomorphic half-planes in $\wh V$ with boundary on $S\Lambda$
are attached (following the terminology
of <cit.> we call such curves anchored in $V$). The homology group $L\H_*^{\text{Ho}}(\Lambda)$
is defined as the cone of a map $LC^{\text{Ho}+}(\Lambda)_*\to
C^{n-*+1}$, where $C^{n-*+1}$ is the cohomological Morse complex for
the pair $(W,\p^-W)$, which has rank $\ell$ in degree $n-*+1=n$ and
vanishes otherwise, and with zero differential. This map counts curves
of the type taken into account in $LH^{\text{Ho}+}(\Lambda)_*$,
rigidified by imposing an intersection with an unstable manifold of a
critical point in $W$.
The exact sequence of the cone of a map reads in this case
\begin{equation} \label{eq:LHcone}
\xymatrix
H^{n-*}(W,\p^-W) \ar[rr] & & L\H^{\text{Ho}}(\Lambda)_*
\ar[dl] \\ & L\H^{\text{Ho}+}(\Lambda)_* \ar[ul]^{[-1]}
\end{equation}
The surgery exact triangles of Bourgeois, Ekholm, and Eliashberg can be
reinterpreted in our language as follows.
Let $W$ be a filled Liouville cobordism corresponding to attaching $\ell\ge 1$ critical
handles of index $k=n$ along a collection $\Lambda$ of disjoint Legendrian spheres. With field coefficients we have isomorphisms
SH_*^{>0}(W,\p^-W)\cong L\H^{\text{Ho}+}(\Lambda)_*,\qquad
SH_*(W,\p^-W)\cong L\H^{\text{Ho}}(\Lambda)_*
such that the following hold:
(i) the tautological exact triangle involving $SH_*^{=0}$, $SH_*$, and $SH_*^{>0}$ for the pair $(W,\p^-W)$ is isomorphic to (<ref>);
(ii) the exact triangles (<ref>) are isomorphic to the exact triangles of the pair $(V',V)$ for $SH_*$, respectively $SH_*^{>0}$.
Let us explain how this conjecture would follow from the surgery exact
triangle in <cit.>.
To establish the first two isomorphisms, the first step is to give a description of $SH_*(W,\p^-W)$ and $SH_*^{>0}(W,\p^-W)$ in terms of pseudo-holomorphic curves in a symplectization; this is similar to the description of $SH_*^{>0}(V)$ as a non-equivariant linearized contact homology group given in <cit.> and used in <cit.> as a definition of $SH_*^{>0}(V)$. The second step is to apply to this formulation of $SH_*^\heartsuit(W,\p^-W)$ with $\heartsuit=\{\varnothing,>0\}$ the methods of <cit.>. The proof of (i) is then straightforward, since $SH_*$ can naturally be expressed as the homology of a cone using the action filtration.
To prove (ii), the main step is to establish an isomorphism between the transfer map $SH_*^\heartsuit(V')\to SH_*^\heartsuit(V)$ and the map with the same source and target that appears in (<ref>) for $\heartsuit\in\{\varnothing,>0\}$. The latter map is described in terms of anchored pseudo-holomorphic curves in the symplectization of the cobordism $W$, and the proof of the isomorphism between these maps follows the same ideas as those in <cit.>, applied to the monotone homotopies which induce in the limit the transfer map. The claim in (ii) then follows from the results of <cit.> because, up to rotating a triangle, the groups $L\H^{\text{Ho}+}(\Lambda)_*$ and $L\H^{\text{Ho}}(\Lambda)_*$ can be expressed as homology groups of cones of such maps induced by cobordisms.
Following <cit.>, all the constructions that we describe in the setup of symplectic homology can be replicated in the language of symplectic field theory, or SFT (with the usual caveat regarding the analytical foundations of the latter). One outcome of this parallel is that our six flavors of symplectic homology provide some new linear SFT-type invariants (the group $SH_*(\p V)$ for $V$ a Liouville domain is the most prominent of these). It is a general fact that the Viterbo transfer maps for symplectic homology correspond to the well-known SFT cobordism maps.
As in the proof of Proposition <ref>,
Conjecture <ref> would imply
With coefficients in a field $\mathfrak{k}$ the following isomorphisms hold:
(i) $SH^{-*}(W,\p^+W)\cong L\H^{\text{Ho}}(\Lambda)_*$ and
SH_{-*}(W,\p^+W)\cong SH^*(W,\p^-W)\cong (L\H^{\text{Ho}}(\Lambda)_*)^\vee.
(ii) $SH^{-*}_{<0}(W,\p^+W)\cong L\H^{\text{Ho}+}(\Lambda)_*$ and
SH_{-*}^{<0}(W,\p^+W)\cong SH^*_{>0}(W,\p^-W)\cong (L\H^{\text{Ho}+}(\Lambda)_*)^\vee.
We also have the obvious
SH_*^{=0}(W,\p^-W) \cong SH_{-*}^{=0}(W,\p^+W)=\begin{cases}
\mathfrak{k} & *=0, \\ 0 & \text{else.} \end{cases}
Together with the long exact sequence of a pair, these computations
provide a theoretically complete understanding of the behaviour of all
the flavors of symplectic homology groups under critical handle
A particular case of interest is that of comparing $SH_*(\p^-W)$ and $SH_*(\p^+W)$. The answer does not take the form of a long exact sequence because these groups do not sit naturally in a long exact sequence of a pair. The best answer that one can give in such a generality is that we have a correspondence
SH_*(\p^-W)\longleftarrow SH_*(W)\longrightarrow SH_*(\p^+W)
in which the kernel and cokernel of each arrow can be described in terms of $SH_*(W,\p^-W)$, respectively $SH_*(W,\p^+W)$, which in turn are described in terms of the groups $L\H^{\text{Ho}}(\Lambda)$ as above, using the long exact sequences of the pairs $(W,\p^-W)$ and $(W,\p^+W)$.
This situation parallels the one encountered when comparing the singular cohomology groups of a manifold before and after surgery (in this case $\p^+W$ is obtained by surgery of index $n$ on $\p^-W$).
§.§ Handle attaching and $S^1$-equivariant symplectic
The discussion in <ref> and <ref> has $S^1$-equivariant analogues. We treat here only $S^1$-equivariant symplectic homology, since negative $S^1$-equivariant symplectic homology and also (negative) $S^1$-equivariant symplectic cohomology can be reduced to the former using Poincaré and algebraic duality.
Subcritical handle attaching.
Let $W$ be a Liouville cobordism corresponding to a subcritical
handle of index $k<n$. Then with $\mathfrak{k}$-coefficients we have
\begin{align*}
SH_*^{S^1}(W,\p_\pm W) &= 0,\cr
SH_*^{S^1,=0}(W,\p^-W) &=\begin{cases}
\mathfrak{k} & *=n-k+2\N, \\ 0 & \text{else,} \end{cases}, \cr
SH_{*}^{S^1,=0}(W,\p^+W) & =\begin{cases}
\mathfrak{k} & *=k-n+2\N, \\ 0 & \text{else,} \end{cases} \cr
SH_*^{S^1,>0}(W,\p^-W) &=\begin{cases}
\mathfrak{k} & *=n-k+1+2\N, \\ 0 & \text{else,} \end{cases}, \cr
SH_{*}^{S^1,<0}(W,\p^+W) & =\begin{cases}
\mathfrak{k} & *=k-n-1+2\N, \\ 0 & \text{else,} \end{cases}
\end{align*}
and the restriction maps induce isomorphisms
SH_*^{S^1}(W)\stackrel{\cong}\longrightarrow SH_*^{S^1}(\p^+W).
The vanishing of $SH_*^{S^1}(W,\p_\pm W)$ follows from that
of $SH_*(W,\p_\pm W)$ using the spectral sequence from non-equivariant
to equivariant symplectic homology. The statement concerning
$SH_*^{S^1,=0}(W,\p_\pm W)$ is a direct computation, using the fact
that the Floer complex reduces in low energy to the Morse complex, see
also <cit.>:
SH_*^{S^1,=0}(W,\p_\pm W) \cong H_{S^1}^{n-*}(W,\p_\pm W) \cong
H^{n-*}(W,\p_\pm W)\otimes\mathfrak{k}[u^{-1}].
The statement concerning $SH_*^{S^1,>0}(W,\p^- W)$ and $SH_*^{S^1,<0}(W,\p^+ W)$ follows from tautological exact triangles in view of the fact that, by definition, $SH_*^{S^1}(W,\p^- W)=SH_*^{S^1,\ge 0}(W,\p^- W)$ and $SH_*^{S^1}(W,\p^+ W)=SH_*^{S^1,\le 0}(W,\p^+ W)$. The last statement follows from the exact triangles of the pairs $(W,\p_\pm W)$.
Let $D^{2n}$ be the unit ball in $\R^{2n}$. Then $SH_*^{S^1}(D^{2n})=0$ and a direct computation, together with the tautological exact triangle, shows that
\[
SH_*^{S^1,=0}(W,\p^- W)\cong SH_*^{S^1,=0}(D^{2(n-k)})
\]
\[
SH_*^{S^1,>0}(W,\p^- W)\cong SH_*^{S^1,>0}(D^{2(n-k)}).
\]
These isomorphisms are not just algebraic or formal, but have the following geometric interpretation <cit.>: for any given finite action window there exists a Liouville structure on $W$ for which the periodic Reeb orbits on $\p^-W$ in the given action window survive to $\p^+W$, and the new periodic Reeb orbits which are created after handle attachment are in one-to-one bijective correspondence with the periodic Reeb orbits on the boundary of the symplectic reduction of the coisotropic cocore disk in the handle, which is a symplectic ball $D^{2(n-k)}$.
Let $V$ be a Liouville domain of dimension $2n$ and $V'$ be obtained from $V$ by attaching a subcritical handle of index $k<n$. We then have an exact triangle
\[
\xymatrix
SH_*^{S^1,>0}(D^{2(n-k)}) \ar[rr] & &
SH_*^{S^1,>0}(V') \ar[dl] \\ & SH_*^{S^1,>0}(V) \ar[ul]^{[-1]}
\]
in which the map $SH_*^{S^1,>0}(V') \to SH_*^{S^1,>0}(V)$ is the transfer map.
This is simply a reformulation of the exact triangle of the pair $(V',V)$, using excision and the computation of $SH_*^{S^1,>0}(W,\p^-W)$ above, with $W=\overline{V'\setminus V}$.
This statement can be interpreted as a subcritical surgery exact triangle for linearized contact homology in view of <cit.>. In that formulation, the case $k=1$ of contact connected sums was
proved using different methods by Bourgeois and van
Koert <cit.>. Also in that formulation, the exact triangle implies
Espina's formula <cit.> for the behaviour of
the mean Euler characteristic of linearized contact homology under
subcritical surgery. By induction over the handles, it yields
M.-L. Yau's formula for the linearized contact homology of subcritical
Stein manifolds <cit.>.
Critical handle attaching.
We restrict to rational coefficients, and recall the geometric setup
of section <ref>: $V\subset V'$ is a pair of
Liouville domains of dimension $2n$ such that $V'$ is obtained by
attaching $\ell\ge 1$ handles of index $n$ to $\p V$ along a
collection $\Lambda$ of $\ell$ disjoint embedded Legendrian
spheres. Following <cit.> we denote
$C\H(V)$ the linearized contact homology of $\p V$. One of the main
statements in <cit.> is the existence
of a surgery exact triangle
\begin{equation} \label{eq:BEES1}
\xymatrix
L\H^{\text{cyc}}(\Lambda)_* \ar[rr] & &
C\H(V'), \ar[dl] \\ & C\H(V) \ar[ul]^{[-1]}
\end{equation}
where $L\H^{\text{cyc}}(\Lambda)_*$ is a homology group of Legendrian contact homology flavour. More precisely, $L\H^{\text{cyc}}(\Lambda)_*$ is defined as the homology of a complex $LH^{\text{cyc}}(\Lambda)_*$ whose generators are cyclic words in Reeb chords on $\p V$ with endpoints on $\Lambda$, and whose differential counts certain pseudo-holomorphic curves in the symplectization of $\p V$, anchored in $V$, with boundary on the conical Lagrangian $S\Lambda$ determined by $\Lambda$.
This exact triangle can be reinterpreted in our language as follows.
Let $W$ be a Liouville cobordism corresponding to attaching $\ell\ge 1$ critical
handles of index $k=n$ along a collection $\Lambda$ of disjoint Legendrian spheres. With rational coefficients we have an isomorphism
SH_*^{S^1,>0}(W,\p^-W)\cong L\H^{\text{cyc}}(\Lambda)_*
such that the exact triangle (<ref>) is isomorphic to the exact triangle of the pair $(V',V)$ for $SH_*^{S^1,>0}$.
The proof should go along the same lines as the one of Conjecture <ref>, adding on top the isomorphism between $SH_*^{S^1,>0}(V)$ and $C\H(V)$ whenever the latter is defined <cit.>. There is also an $S^1$-equivariant counterpart of Conjecture <ref>(ii), which involves duality and hence the groups $SH_*^{[u],>0}$.
One can also give a Legendrian interpretation of $SH_*^{S^1}(W,\p^-W)$. This can be obtained either formally algebraically by computing ranks from the $S^1$-equivariant tautological exact triangle of the pair $(W,\p^-W)$ using the fact that $SH_*^{S^1,=0}(W,\p^-W)$ is supported in positive degrees, or geometrically along the lines of <cit.>, where a linearized contact homology counterpart of $SH_*^{S^1}(V)$ is defined.
§ PRODUCT STRUCTURES
§.§ TQFT operations on symplectic homology
As before, we use coefficients in a field $\mathfrak{k}$.
Recall from <cit.> the definition of TQFT operations on the
Floer homology of a Hamiltonian $H$ on a completed Liouville domain
$\wh V$. We freely use in this section the terminology therein, namely “negative punctures", “positive punctures", “cylindrical ends", “weights", see also <cit.>. Consider a punctured Riemann surface $S$ with $p$ negative
and $q$ positive punctures. Pick positive weights $A_i,B_j>0$ and a
$1$-form $\beta$ on $S$ with the following properties:
(i) $H\,d\beta\leq 0$;
(ii) $\beta=A_idt$ in cylindrical coordinates $(s,t)\in\R_-\times
S^1$ near the $i$-th negative puncture;
(iii) $\beta=B_jdt$ in cylindrical coordinates $(s,t)\in\R_+\times
S^1$ near the $j$-th positive puncture.
Note that $\beta$ and the weights are related by Stokes' theorem
\sum_{i=1}^pA_i - \sum_{j=1}^qB_j = -\int_Sd\beta.
Conversely, if the quantity on the left-hand side is nonnegative (zero,
nonpositive), then we find a $1$-form $\beta$ with properties (ii) and
(iii) such that $d\beta\leq 0$ ($=0$, $\geq0$). Thus we can arrange
conditions (i)–(iii) in the following situations:
(a) $H$ arbitrary, $d\beta\equiv 0$, $p,q\geq 1$;
(b) $H\geq 0$, $d\beta\leq 0$, $p\geq 1$;
(c) $H\leq 0$, $d\beta\geq 0$, $q\geq 1$.
Note that the condition $H\geq0$ is satisfied for admissible
Hamiltonians on a Liouville cobordism.
We consider maps $u:S\to\wh V$ that are holomorphic in the sense that
$(du-X_H\otimes\beta)^{0,1}=0$ and have finite energy
$E(u)=\frac{1}{2}\int_S|du-X_H\otimes\beta|^2{\rm vol}_S$. They
converge at the negative/positive punctures to $1$-periodic orbits
$x_i,y_j$ and satisfy the energy estimate
\begin{equation}\label{eq:energy}
0 \leq E(u) \leq \sum_{j=1}^qA_{B_jH}(y_j) -
\sum_{i=1}^pA_{A_iH}(x_i)
\end{equation}
(beware that our action is minus that in <cit.>).
The signed count of such holomorphic maps yields an operation
\psi_S:\bigotimes_{j=1}^qFH_*(B_jH) \to
\bigotimes_{i=1}^pFH_*(A_iH).
of degree $n(2-2g-p-q)$ which does not increase action. These
operations are graded commutative if degrees are shifted by $-n$ and
satisfy the usual TQFT composition rules. Let us pick
real numbers $a_j<b_j$, $j=1,\dots,q$ and $a_i'<b_i'$, $i=1,\dots,p$
\begin{equation}\label{eq:op-actions}
\sum_ia_i' = \max_j\Bigl(a_j+\sum_{j'\neq j}b_{j'}\Bigr),\qquad
b_i'=\sum_jb_j-\sum_{i'\neq i}a_{i'}'.
\end{equation}
Consider a term $x_1\otimes\cdots\otimes x_p$ appearing in
$\psi_S(y_1\otimes\cdots\otimes y_q)$. If $A_{B_jH}(y_j)\leq a_j$ for some $j$ and
$A_{B_{j'}H}(y_{j'})\leq b_{j'}$ for all $j'\neq j$, then the energy estimate
and the first condition in (<ref>) yield
\sum_{i=1}^pA_{A_iH}(x_i)\leq a_j+\sum_{j'\neq j}b_{j'} \leq
\sum_ia_i',
thus $A_{A_iH}(x_i)\leq a_i'$ for at least one $i$. This shows that
$\psi_S$ is well-defined as an operation
\psi_S:\bigotimes_{j=1}^qFH_*^{(a_j,b_j]}(B_jH) \to
\bigotimes_{i=1}^pFH_*^{(a_i',\infty)}(A_iH).
Similarly, if $A_{B_jH}(y_j)\leq b_j$ for all $j$ and
$A_{A_iH}(x_i)>a_i'$ for all $i$ (so that $a_1\otimes\cdots\otimes a_p\neq
0$ in the quotient space), then for each $i$ the energy estimate yields
A_{A_iH}(x_i)+\sum_{i'\neq i}a_{i'}' \leq
A_{A_iH}(x_i)+\sum_{i'\neq i}A_{A_iH}(x_{i'}) \leq \sum_jb_j,
thus $A_{A_iH}(x_i)\leq b_i'$ by the second condition
in (<ref>). It follows that $\psi_S$ induces an
operation on filtered Floer homology
\psi_S:\bigotimes_{j=1}^qFH_*^{(a_j,b_j]}(B_jH) \to
\bigotimes_{i=1}^pFH_*^{(a_i',b_i']}(A_iH).
To proceed further, let us first assume $p,q\geq 1$, so we are in case
(a) above. We specialise the choice of actions to $a_j=a$, $b_j=b$
for all $i$ and $a_i'=a'$, $b_i'=b'$ for all
$i$. Then (<ref>) becomes
\begin{equation}\label{eq:op-actions2}
pa' = a+(q-1)b,\qquad
b' = qb - (p-1)a',
\end{equation}
and under these conditions $\psi_S$ induces an operation
\psi_S:\bigotimes_{j=1}^qFH_*^{(a,b]}(B_jH) \to
\bigotimes_{i=1}^pFH_*^{(a',b']}(A_iH).
We now apply this to admissible Hamiltonians for a Liouville cobordism
$W$ relative to some admissible union $A$ of boundary components as in <ref>. The map $\psi_S$ is compatible with
continuation maps for $H\leq H'$ in the obvious way, and therefore
passes through the inverse and direct limit to define a map on
filtered symplectic homology
\psi_S:\bigotimes_{j=1}^qSH_*^{(a,b]}(W,A) \to
\bigotimes_{i=1}^pSH_*^{(a',b']}(W,A).
Let us first consider the case $p=1$. Then $a'\to-\infty$ and $b'=qb$
remains constant as $a\to-\infty$, so we can pass to the inverse limits
to obtain an operation
\psi_S:\bigotimes_{j=1}^qSH_*^{(-\infty,b]}(W,A) \to
In the direct limit as $b\to\infty$ this yields an operation
\psi_S:\bigotimes_{j=1}^qSH_*(W,A) \to SH_*(W,A).
Taking instead limits as $b\searrow 0$ and $b\nearrow 0$,
respectively, we see that this operation restricts to operations
\begin{align*}
\psi_S:\bigotimes_{j=1}^qSH_*^{\leq 0}(W,A) &\to SH_*^{\leq 0}(W,A), \cr
\psi_S:\bigotimes_{j=1}^qSH_*^{<0}(W,A) &\to SH_*^{<0}(W,A).
\end{align*}
In the case $p>1$ this procedure fails because $b'\to\infty$ as
$a\to-\infty$, so we cannot take the inverse limits $a,a'\to-\infty$
keeping $b,b'$ fixed. If all actions are nonnegative, as in the case
of a Liouville domain or a pair $(W,\p^-W)$, then there is no need to
take the inverse limit $a,a'\to-\infty$, but we can simply fix
$a,a'<0$ and take the direct limits $b,b'\to\infty$ to obtain
operations $\psi_S$ on all symplectic homology groups.
Next consider the case $q=0$, $p\geq 1$, which is possible for $H\geq
0$ (and thus $A=\emptyset$) according to case (b) above. Pick $a'\leq
0$ and consider the associated map
\psi_S:{\color{black}\mathfrak{k}} \to \bigotimes_{i=1}^pSH_*^{(a',\infty)}{\color{black}(W)},
with $\mathfrak{k}$ the ground field. For a nonzero term $x_1\otimes\cdots\otimes x_p$ appearing in
$\psi_S(1)$ we have $A_{A_iH}(x_i)>a'$ for all $i$, so the energy
estimate yields
A_{A_iH}(x_i)+(p-1)a' \leq
A_{A_iH}(x_i)+\sum_{i'\neq i}A_{A_iH}(x_{i'}) \leq 0,
thus $A_{A_iH}(x_i)\leq -(p-1)a'$. So we obtain a map
\psi_S:{\color{black}\mathfrak{k}} \to \bigotimes_{i=1}^pSH_*^{(a',-(p-1)a']}{\color{black}(W)}.
If $p=1$, then we take the inverse limit as $a'\to-\infty$ to
obtain the unit
\psi_S:{\color{black}\mathfrak{k}} \to SH_*^{\leq 0}{\color{black}(W)}.
If $p>1$, then we set $a'=0$ to obtain the operation
\psi_S:{\color{black}\mathfrak{k}} \to \bigotimes_{i=1}^pSH_*^{=0}{\color{black}(W)}.
So we have proved
For a filled Liouville cobordism $W$ and an admissible union $A$ of boundary
components, there exist operations
\psi_S:\bigotimes_{j=1}^qSH_*^{\heartsuit}(W,A) \to
\bigotimes_{i=1}^pSH_*^\heartsuit(W,A),\qquad
\heartsuit\in\{\emptyset,\leq 0,<0\}
of degree $n(2-2g-p-q)$ associated to punctured Riemann surfaces $S$
with $p$ negative and $q$ positive punctures, graded commutative if
degrees are shifted by $-n$ and satisfying the usual
TQFT composition rules, in each of the following situations:
* $\p^-W=A=\emptyset$, $p\geq 1$, $q\geq 0$;
* $A=\p^-W$, $p\geq 1$, $q\geq 1$;
* $A=\emptyset$, $p=1$, $q\geq 0$;
* $A$ arbitrary, $p=1$, $q\geq 1$.$\square$
As a consequence, we have
(a) For a filled Liouville cobordism $W$ and an admissible union $A$ of boundary
components, the pair-of-pants product on Floer homology induces a
product on $SH_*(W,A)$. The product has degree $-n$, and it is
associative and graded commutative when degrees are shifted by $-n$.
(b) The symplectic homology groups $SH_*^{\leq 0}(W,A)$ and
$SH_*^{<0}(W,A)$ also carry induced products which are compatible with
the tautological maps $SH_*^{<0}(W,A)\to SH_*^{\leq 0}(W,A)\to
SH_*(W,A)$. The image of the map $SH_*^{<0}(W,A)\to SH_*^{\leq
0}(W,A)$ is an ideal in $SH_*^{\leq 0}(W,A)$.
(c) The symplectic homology group $SH_*^{=0}(W,A)$ carries a product,
which coincides with the cup product in cohomology via the isomorphism
$SH_*^{=0}(W,A)\cong H^{n-*}(W,A)$. The map $SH_*^{\leq 0}(W,A)\to
SH_*^{=0}(W,A)$ is compatible with the product structures.
(d) In the case $A=\emptyset$, the products on $SH_*^{\le 0}(W)$,
$SH_*(W)$, and $SH_*^{=0}(W)$ have units, and the tautological maps
$SH_*^{\leq 0}(W)\to SH_*(W)$ and $SH_*^{\le 0}(W)\to SH_*^{=0}(W)$
are morphisms of rings with unit.
(e) For a filled Liouville cobordism pair $(W,V)$, the transfer map
$SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)$
is a morphism of rings for $\heartsuit\in \{<0,\le 0,\varnothing\}$,
and a morphism of rings with unit for $\heartsuit\in \{\le 0,\varnothing\}$.
Parts (a)–(d) follow directly from the preceding discussion, so it
remains to prove part (e). For this, fix a finite action interval
$(a,b)$ and consider two Hamiltonians $K\leq
H$ for the Liouville cobordism pair $(W,V)$ as in Figure <ref>.
Let us first describe more explicitly the transfer map from
Section <ref>. For this, let $\chi:\R\to[0,1]$ be a
smooth nondecreasing function with $\chi(s)=0$ for $s\leq 0$ and
$\chi(s)=1$ for $s\geq 1$ and define the $s$-dependent Hamiltonian
\wh H := \bigl(1-\chi(s)\bigr) H + \chi(s)K,
where $(s,t)$ are coordinates on the cylinder $\R\times S^1$. Then
$\p_s\wh H\leq 0$ and the count of Floer cylinders for $\wh H$ defines
a chain map $f:FC^{(a,b]}(K)\to FC^{(a,b]}(H)$.
Now we describe the products. Let $S$ be the Riemann sphere with two
positive punctures and one negative puncture. Let $\tau:S\to\R\times S^1$ be a
degree $2$ branched cover with $\tau(s,t)=(s,t)$ in cylindrical
coordinates $(s,t)\in[1,\infty)\times S^1$ near the positive punctures
and $\tau(s,t)=(s,t)$ in cylindrical coordinates
$(s,t)\in(-\infty,-1]\times S^1$ near the negative puncture.
We use the $1$-form $\beta:=\tau^*dt$ on $S$ (with $d\beta=0$) and
weights $B_1=B_2=1$ and $A_1=2$ at the positive/negative punctures to
define the pair-of-pants product
\mu_K:FC^{(a,b]}(K)\otimes FC^{(a,b]}(K)\to FC^{(a+b,2b]}(2K),
and similar $\mu_H$. Next, consider for $\sigma\in\R$ the function
$\chi_\sigma(s,t):=\chi(s-\sigma)$ and the Hamiltonian
\wh H_\sigma := (1-\chi_\sigma\circ\tau) H + \chi_\sigma K
depending on points $z\in S$. Since $H\,d\beta=0$ and
$d_zH\wedge\beta\leq 0$ as $2$-forms on $S$, the maximum principle
holds for the Floer equation of $\wh H_\sigma$ (see
e.g. <cit.>). It follows that the moduli spaces $\MM_\sigma(y_1,y_2;x_1)$
of pairs-of-pants for $\wh H_\sigma$ are compact modulo breaking,
where $y_1,y_2$ and $x_1$ are $1$-periodic orbits of $K$ and $2H$, respectively.
Considering for index $\CZ(y_1)+\CZ(y_2)-\CZ(x_1)-n=0$ the natural
compactifications of the $1$-dimensional moduli spaces
$\bigcup_{\sigma\in\R}\{\sigma\}\times\MM_\sigma(y_1,y_2;x_1)$, we
obtain the relation
\begin{equation}\label{eq:theta}
\mu_H(f\otimes f)-f_2\mu_K = \p_{2H}\theta-\theta\p_K.
\end{equation}
Here $\p_K$ and $\p_{2H}$ are the Floer boundary operators for $K$ and
$2H$, respectively, $f_2:FC^{(a,b]}(2K)\to FC^{(a,b]}(2H)$ is the chain map defined
by $2\wh H$, and
\theta:FC^{(a,b]}(K)\otimes FC^{(a,b]}(K)\to FC^{(a+b,2b]}(2H)
counts index $-1$ pairs-of-pants for $\wh H_\sigma$ occurring at
isolated values of $\sigma$.
Let us now choose $K,H$ such that the orbits in group $F$ for $K$ and
in groups $F,I,III^{0+}$ for $H$ have actions below $a$, so that
$FC^{(a,b]}(K)=FH^{(a,b]}_I(K)$ and $FC^{(a,b]}(H)=FH^{(a,b]}_{II,III^{-}}(H)$.
By Lemma <ref> and Lemma <ref>,
$FH^{(a,b]}_{III^{-}}(H)$ is a $2$-sided ideal for the product $\mu_H$,
so the latter passes to the quotient as a product
\mu_H:FC^{(a,b]}_{II}(H)\otimes FC^{(a,b]}_{II}(H)\to FC^{(a+b,2b]}_{II}(2H).
It follows that relation (<ref>) persists when we compose the
maps $f$ and $f_2,\theta$ with their projections to
$FC^{(a,b]}_{II}(H)$ and $FC^{(a,b]}_{II}(2H)$, respectively (keeping
the same notation for the new maps). Passing to homology and the
direct limit over $K,H$ we obtain the commuting diagram on filtered
symplectic homology
\begin{equation*}
\xymatrix{
SH^{(a,b]}(W)\otimes SH^{(a,b]}(W) \ar[r]^-{\mu_W} \ar[d]_{f\otimes
f} & SH^{(a+b,2b]}(W) \ar[d]^f\\
SH^{(a,b]}(V)\otimes SH^{(a,b]}(V) \ar[r]^-{\mu_V} & SH^{(a+b,2b]}(V)\,.
\end{equation*}
Passing to the limits $a\to-\infty$ and $b\nearrow 0$, $b\searrow 0$,
or $b\to\infty$, we conclude that the transfer map
$SH_*^\heartsuit(W)\to SH_*^\heartsuit(V)$ preserves the product for
$\heartsuit\in \{<0,\le 0,\varnothing\}$. A similar argument shows
that the transfer map preserves the unit for $\heartsuit\in \{\le
0,\varnothing\}$ and Theorem <ref> is proved.
In particular, Theorem <ref> provides a product of
degree $-n$ with unit and a coproduct of degree $-n$ (without counit) on
$SH_*(W)$ for every filled Liouville cobordism $W$. Applied to the
trivial cobordism, this yields via the isomorphism (<ref>) a
corresponding product and coproduct on Rabinowitz–Floer homology. We
refer to Uebele <cit.> and Appendix <ref> for a discussion of conditions
under which the product is defined in the absence of a filling if the
negative boundary is index-positive.
If $W$ is a Liouville cobordism with filling and $L\subset W$ is an
exact Lagrangian cobordism with filling, then the preceding discussion
shows that Lagrangian symplectic homology $SH_*^\heartsuit(L)$ is a
module over $SH_*^\heartsuit(W)$ for $\heartsuit\in \{<0,\le
0,\varnothing\}$, see also <cit.>.
Recall from <cit.> that, for a fixed admissible Hamiltonian
$H$, the pair-of-pants product defines a map
FH^{\leq b}(H)\otimes FH^{\leq b'}(H)\to FH^{\leq b+b'}(2H).
This product has degree $-n$, and it is associative and graded
commutative when degrees are shifted by $-n$. It is compatible with
continuation maps for $H\leq H'$ in the obvious way, and therefore
passes through the inverse and direct limit to a product
SH^{\leq b}(W,V)\otimes SH^{\leq b'}(W,V)\to SH^{\leq b+b'}(W,V).
For $a<b$ and $a'<b'$ it thus descends to a product
SH^{(a,b]}(W,V)\otimes SH^{(a',b']}(W,V)\to
Taking the inverse limit as $a,a'\to -\infty$, we obtain a product
SH^{(-\infty,b]}(W,V)\otimes SH^{(-\infty,b']}(W,V)\to
which in the direct limit as $b,b'\to\infty$ gives the desired product
SH(W,V)\otimes SH(W,V)\to SH(W,V).
Taking instead limits as $b,b'\searrow 0$ and $b,b'\nearrow 0$,
respectively, we see that this product restricts to products
\begin{align*}
SH^{\leq 0}(W,V)\otimes SH^{\leq 0}(W,V) &\to SH^{\leq 0}(W,V),\cr
SH^{<0}(W,V)\otimes SH^{<0}(W,V) &\to SH^{<0}(W,V).
\end{align*}
§.§ Dual operations
Combining Proposition <ref> with the Poincaré duality
isomorphism $S^*_{\heartsuit}(W,A)\cong SH_{-*}^{-\heartsuit}(W,A^c)$, we obtain
Consider a filled Liouville cobordism $W$ and an admissible union $A$ of boundary components. Then there exist operations
\psi_S:\bigotimes_{j=1}^qSH^*_{\heartsuit}(W,A) \to
\bigotimes_{i=1}^pSH^*_\heartsuit(W,A),\qquad
\heartsuit\in\{\emptyset,\geq 0,>0\}
of degree $-n(2-2g-p-q)$,
graded commutative if degrees are shifted by $n$ and satisfying the usual
TQFT composition rules, in the following situations:
* $\p^-W=\emptyset$, $A=\p^+W$, $p\geq 1$, $q\geq 0$;
* $A=\p^+W$, $p\geq 1$, $q\geq 1$;
* $A=\p W$, $p=1$, $q\geq 0$;
* $A$ arbitrary, $p=1$, $q\geq 1$.$\square$
Note that in Propositions <ref> and <ref> the
conditions on $p,q$ are the same, whereas $\heartsuit$ is replaced by
$-\heartsuit$ and $A$ by $A^c$.
Suppose now that the filled Liouville cobordism $W$ has vanishing first Chern class and that $\p W$ carries only finitely many closed Reeb
orbits of any given Conley-Zehnder index. Using field
coefficients Corollary <ref> yields canonical
isomorphisms $SH_k^\heartsuit(W,A)\cong SH^k_\heartsuit(W,A)^\vee$
for all $A$ and all flavors $\heartsuit$. The dualization of the
operations in Proposition <ref> then yields
Consider a filled Liouville cobordism $W$ with vanishing first Chern
class and an admissible union $A$ of boundary components.
Suppose that $\p W$ carries only finitely many closed Reeb
orbits of any given Conley-Zehnder index.
Then with field coefficients there exist operations (note the reversed
roles of $p$ and $q$)
\psi_S^\vee:\bigotimes_{i=1}^pSH_*^{\heartsuit}(W,A) \to
\bigotimes_{j=1}^qSH_*^\heartsuit(W,A),\qquad
\heartsuit\in\{\emptyset,\geq 0,>0\}
of degree $n(2-2g-p-q)$,
graded commutative if degrees are shifted by $-n$ and satisfying the usual
TQFT composition rules, in the following situations:
* $\p^-W=\emptyset$, $A=\p^+W$, $p\geq 1$, $q\geq 0$;
* $A=\p^+W$, $p\geq 1$, $q\geq 1$;
* $A=\p W$, $p=1$, $q\geq 0$;
* $A$ arbitrary, $p=1$, $q\geq 1$.$\square$
§.§ A coproduct on positive symplectic homology
Consider a Liouville cobordism $W$ filled by a Liouville domain
$V$. The choice of $W$ will be irrelevant, so we can take
e.g. $W=I\times \p V$. Proposition <ref>(iii) provides a product
of degree $-n$ on $SH_*^{<0}(W)$. In view of the isomorphism
$SH_*^{<0}(W) \cong SH_{>0}^{-*+1}(V)$ from
Proposition <ref>, this gives a product of
degree $n-1$ on the symplectic cohomology group $SH_{>0}^*(V)$.
Note that this cannot be the product arising from
Proposition <ref>(iv) (with $V$ in place of $W$ and
$A=\emptyset$) because the latter has degree $n$. Under the
finiteness hypothesis in Corollary <ref>, this gives a
coproduct of degree $1-n$ on the symplectic homology group
Following Seidel, there is another coproduct of degree $1-n$
on $SH_*^{>0}(V)$ obtained as a secondary operation in view of the
fact that the natural coproduct given by counting pairs of pants with
one input and two outputs vanishes, see also <cit.> for a generalization and <cit.> for a topological version of it. These two coproducts of
degree $1-n$ agree. The isomorphism between them is part of a larger picture related to Poincaré duality and will be the topic of another paper.
§ AN OBSTRUCTION TO SYMPLECTIC COBORDISMS
(JOINT WITH PETER ALBERS)
In this joint appendix we use the results of this paper to define an
obstruction to Liouville cobordisms between contact manifolds.
Consider a Liouville cobordism $W$ whose negative end $\p_-W$ is
hypertight, index-positive, or Liouville fillable. As explained in
Section <ref>, in these cases one can define symplectic
homology groups $SH_*^\heartsuit(W)$,
$\heartsuit\in\{\varnothing,\le 0,<0,=0,\ge 0, >0\}$ which will be
independent of a filling in the first two cases but may
depend on the filling in the Liouville fillable case.
We would like to show that vanishing of $SH_*(\p_+W)$ implies
vanishing of $SH_*(\p_-W)$. However, it is unclear how to deduce this
from the functoriality under cobordisms, which only gives correspondences
\xymatrix{
& SH_*^\heartsuit(W) \ar[dl]\ar[dr] & \\
SH_*^\heartsuit(\p_- W) & & SH_*^\heartsuit(\p_+ W).
Instead, we will consider the following property
(using coefficients in a field $\mathfrak{k}$).
A Liouville cobordism $W$ is called SAWC if $1_W$ is mapped to
zero under the map $H^0(W)\cong SH_n^{=0}(W)\to SH_n^{\geq 0}(W)$,
where $1_W$ is the unit in $H^0(W)$.
For a connected Liouville domain $W$, this agrees with the “Strong
Algebraic Weinstein conjecture” property of Viterbo <cit.>.
As usual, we define the SAWC property for $\p_\pm W$ via the trivial
cobordism $[0,1]\times\p_\pm W$, where $SH_*(\p_+W)$ is defined
with respect to the partial filling $W$.
Then this property is inherited under cobordisms:
Let $W$ be a Liouville cobordism with vanishing first Chern class whose negative end $\p_-W$ is
hypertight, index-positive, or Liouville fillable.
If $\p_+W$ is SAWC, then so are $W$ and $\p_-W$.
If the first Chern class of $W$ vanishes the symplectic homology groups $SH_*^\heartsuit$ are canonically graded in the component of constant loops. Consider thus the diagram with commutative squares and exact rows
\small
\xymatrix{
SH_{n+1}^{> 0}(\p_-W)\ar[r] & SH_n^{=0}(\p_-W)\simeq H^0(\p_-W) \ar[r]
& SH_n^{\ge 0}(\p_-W) \ar[r] & SH_n^{>0}(\p_-W)\\
SH_{n+1}^{> 0}(W)\ar[r]\ar[d]^\simeq\ar[u] & SH_n^{=0}(W)\simeq
H^0(W) \ar[r]\ar[d]^{\mbox{injective}}_{1_W\mapsto 1_{\p_+W}}\ar[u]^{1_W\mapsto 1_{\p_-W}} & SH_n^{\ge 0}(W)
\ar[r]\ar[d]^{\mbox{$\Rightarrow\ $injective}}\ar[u] & SH_n^{>0}(W)\ar[d]^\simeq\ar[u] \\
SH_{n+1}^{> 0}(\p_+W)\ar[r] & SH_n^{=0}(\p_+W)\simeq H^0(\p_+W)
\ar[r] & SH_n^{\ge 0}(\p_+W) \ar[r] & SH_n^{>0}(\p_+W).
The lower vertical arrows at the extremities are isomorphisms since
$W$ and $I\times\p_+W$ share the same positive boundary. The map
$H^0(W)\to H^0(\p_+W)$ is injective because every component of $W$ has
a positive boundary component. Injectivity of the vertical map
$SH_n^{\ge 0}(W)\to SH_n^{\ge 0}(\p_+W)$ then follows from the 5-lemma
as in <cit.>.
Suppose now that $1_{\p_+W}$ is sent to
zero by the map $H^0(\p_+W)\to SH_n^{\geq 0}(\p_+W)$. Then
commutativity of the lower middle square implies that $1_{W}$
goes to zero under the map $H^0(W)\to SH_n^{\geq 0}(W)$, and
commutativity of the upper middle square implies that $1_{\p_-W}$
goes to zero under the map $H^0(\p_-W)\to SH_n^{\geq 0}(\p_-W)$.
Note that Proposition <ref> uses the product
structure on singular cohomology but not on symplectic homology. Using
the latter we will now reformulate the SAWC condition. As observed
by Uebele in <cit.>, the pair-of-pants product $\cdot$ in
Section <ref> makes
$SH_*(W)$, $SH_*^{\leq 0}(W)$ and $SH^{=0}(W)$ unital graded commutative
rings for $W$ as in Proposition <ref>,
provided that in the index-positive case we require Ęthe following stronger
condition (called “product index-positivity” in <cit.>):
(i) $\pi_1(\p_- W)=1$, and
\begin{equation}Ę\label{eq:index}Ę
\CZ(\gamma)>3 \,\,\,Ę\text{Ęfor every closed Reeb orbit }
\gamma \text{ in } \p_-W,
\end{equation}
(ii) denoting $\xi_-$ the contact distribution on $\p_- W$, we have $2c_1(\xi_-)=0$ and there exists a trivialisation of the square of the canonical bundle $\Lambda_\C^{\max}\xi_-^{\otimes 2}$ such that, with respect to that trivialisation, all closed Reeb orbits $\gamma$ in $\p_-W$ satisfy (<ref>).
Remark. Examples in which (ii) is satisfied are unit cotangent bundles of spheres $S^n$ of dimension $n\ge 5$, and more generally Milnor fibers of $A_k$-singularities $\{z_0^k+z_1^2+\dots+z_n^2=0\}$ for $n\ge 5$, see <cit.> and also <cit.>.
The proof of this observation is similar to that of
Proposition <ref>. The new feature is that a
pair-of-pants with inputs $x_1,x_2$ and output $x_-$ might break into
a Floer cylinder $C_1$ connecting $x_1$ and $x_-$ with a negative puncture
asymptotic to a closed Reeb orbit $\gamma_1$, a Floer plane $C_2$
with input $x_2$ and a negative puncture at a closed Reeb orbit
$\gamma_2$, and a holomorphic cylinder with two positive punctures
asymptotic to $\gamma_1,\gamma_2$. The first two components are
regular, so their indices satisfy
\begin{align*}
\ind(C_1)&=\CZ(x_1)-\CZ(x_-)-\bigl(\CZ(\gamma_1)+n-3\bigr)\geq 0,\cr
\ind(C_2)&=\CZ(x_2)+n-\bigl(\CZ(\gamma_2)+n-3\bigr)\geq 0.
\end{align*}
When showing well-definedness of the product (resp. commutativity with
the boundary operator) we consider orbits satisfying
\CZ(x_1)+\CZ(x_2)-\CZ(x_-)-n = 0 \text{ (resp. $1$).}
Adding the two inequalities and inserting this relation yields
\bigl(\CZ(\gamma_1)-3\bigr)+\bigl(\CZ(\gamma_2)-3\bigr)\leq 0 \text{ (resp. $1$),}
contradicting condition (<ref>).
Let us fix a Liouville form $\lambda$ on $W$ and consider for $b\in\R$
the filtered symplectic homology groups $SH_*^{(-\infty,b)}(W)$ defined in
Section <ref> (which also exist under the above assumptions on
$W$). We define the spectral value of a class $\alpha\in
SH_*(W)$ by
c(\alpha) := \inf \{b\in\R \mid
\alpha\in\mathrm{im}(SH_*^{(-\infty,b)}(W)\to SH_*(W))\} \in[-\infty,\infty).
Here $c(\alpha)<\infty$ follows from the definition of
SH_*^{(-\infty,b)}(W)$. The fundamental inequality satisfied by
spectral values is
c(\alpha\cdot\beta)\le c(\alpha)+c(\beta),
as a consequence of the fact that the pair-of-pants product decreases
action (see inequality (<ref>) with $A_1=2$ and $B_1=B_2=1$).
The unit $1_W\in SH_n(W)$ plays a particular role. Indeed, we have
$c(1_W)\le 0$ since $SH_*^{\le 0}(W)\to SH_*(W)$ is a map of
rings with unit, but also
c(1_W)=c(1_W\cdot 1_W)\le 2c(1_W).
Thus either $c(1_W)= 0$ or $c(1_W)=-\infty$ (note that these
conditions are independent of the Liouville form $\lambda$). The condition $c(1_W)=-\infty$ is equivalent to the fact that the unit belongs to the image of the map $SH_n^{<0}(W)\to SH_n(W)$. In the latter case we also
obtain $c(\alpha)=-\infty$ for all $\alpha\in SH_*(W)$ since
$c(\alpha)\le c(1_W)+c(\alpha)$. This is in particular the case if
$SH_*(W)=0$, and the converse is also true. Indeed, assume $c(1_W)=-\infty$ and represent $1_W$ as the image of an element $\alpha^b\in SH_*^{(-\infty,b)}(W)$ for some $b<0$. By definition of the inverse limit, such an element is the equivalence class of a sequence $\alpha^b_n\in SH_*^{(-n,b)}(W)$ for $n>|b|$. We claim that each such element $\alpha^b_n$ is zero, hence $1_W=0$. Indeed, for any given $n$ we can choose $b'<-n$ and represent by assumption $1_W$ by an element $\beta^{b'}\in SH_*^{(-\infty,b')}(W)$, given by a sequence $\beta^{b'}_{n'}\in SH_*^{(-n',b')}(W)$ for $n'>|b'|$. But then $\alpha^b_n$ must be the image of $\beta^{b'}_{n'}$ under the map $SH_*^{(-n',b')}(W)\to SH_*^{(-n,b)}(W)$, which is zero for $b'<-n$.
We thus obtain:
Let $W$ be a Liouville cobordism whose negative end $\p_-W$ is
hypertight, Liouville fillable, or index-positive with the
stronger index condition (<ref>). Then $W$ is SAWC
if and only if $SH_*(W)=0$.
Proposition <ref> yields the commuting diagram
with exact rows and columns
\xymatrix
& SH_{n+1}^{>0}(W) \ar@{=}[r] \ar[d]^f & SH_{n+1}^{>0}(W) \ar[d]^g & \\
SH_n^{<0}(W) \ar[r]^h \ar@{=}[d] & SH_n^{\leq 0}(W) \ar[r]^i \ar[d]^j &
SH_n^{=0}(W) \ar[d]^k \\
SH_n^{<0}(W) \ar[r]^\ell & SH_n(W) \ar[r]^m & SH_n^{\geq
0}(W) \;,
where $i$ and $j$ are maps of unital rings. We will denote all units by
$1_W$. We prove that $W$ is SAWC if and only if $c(1_W)=-\infty$,
which by the discussion above is equivalent to $SH_*(W)=0$.
Suppose first that $c(1_W)=-\infty$, i.e. $1_W=\ell\alpha$ for some
$\alpha\in SH_n^{<0}(W)$. Then $1_W-h\alpha=f\beta$ for some $\beta\in
SH_{n+1}^{>0}(W)$, hence $1_W=i(1_W-h\alpha)=g\beta$ is mapped to
zero under $k$, which means that $W$ is SAWC. The converse
implication is proved similarly.
There is no Liouville cobordism $W$ with $\p_-W$ hypertight and
If $\p_-W$ is hypertight then the map $SH_n^{=0}(\p_-W)\to SH_n^{\geq
0}(\p_-W)$ is an isomorphism, so $\p_-W$ is not SAWC. On the other
hand, $SH_*(\p_+W)=0$ implies by Lemma <ref> that $\p_+W$ is
SAWC. This is impossible by Proposition <ref>.
Remark. In the statement of Corollary <ref> it is understood that $SH_*(\p_+W)$ is defined
with respect to the partial filling $W$.
There is no Liouville cobordism $W$ of dimension $2n\geq 4$ with vanishing first Chern class such that $\p_-W$ is hypertight and $\p_+W$ is fillable by a subcritical Stein manifold with vanishing first Chern class.
Let $F$ be such a subcritical filling of $\p_+W$. Denote ${^F}SH_*(\p_+W)$ the symplectic homology computed with respect to the filling $F$, and ${^W}SH_*(\p_+W)$ the symplectic homology computed with respect to the partial filling $W$. Since $SH_*(F)=0$, we also have ${^F}SH_*(\p_+W)=0$ by Corollary <ref>. On the other hand one can choose on $\p_+W$ a contact form so that all Conley-Zehnder indices of closed Reeb orbits are
$>1$ <cit.>, and therefore $>3-n$ provided that $n\geq 2$. It follows that the symplectic homology of $\p_+W$ is independent of the filling and therefore we also have ${^W}SH_*(\p_+W)=0$. The conclusion then follows from Corollary <ref>.
(1) Many examples of contact manifolds $M$ with $SH_*(M)=0$ arise as
boundaries of Liouville domains with vanishing symplectic homology,
e.g. subcritical or flexible Stein manifolds <cit.>.
(2) Examples of hypertight contact manifolds are the unit
cotangent bundles of Riemannian manifolds of nonpositive
curvature. Other examples are the $3$-torus $T^3$ with a Giroux
contact structure $\xi_k=\ker\bigl(\cos (ks) d\theta+\sin (ks) dt\bigr)$ and its higher-dimensional generalizations
$(T^2\times N,\xi_k)$ by Massot–Niederkrüger–Wendl <cit.>. The
latter are not strongly symplectically fillable (so in particular not
Liouville fillable) for $k\ge 2$. Therefore, it appears that
Corollary <ref> with $\p_-W$ one of these manifolds
cannot be obtained by more classical tools such as symplectic homology
of Liouville domains.
(3) Let us mention in the same vein the fact that there is no Liouville cobordism $W$ with $\p_-W$ hypertight and $\p_+W$ overtwisted. This is proved in the same way as non-fillability of overtwisted contact manifolds <cit.>, using filling by holomorphic discs in the symplectic manifold $(0,1)\times \p_-W \, \cup \, W$. However, this seems to fall outside the scope of our methods, while at the same time the case that we address in Corollary <ref> seems to fall outside the scope of the method of filling by holomorphic discs.
Ę(4) A contact manifold $(M,\xi)$ fails to satisfy the Weinstein conjecture if there exists a contact form whose Reeb vector field has no periodic orbit. In the simply connected case this is equivalent to the fact that $(M,\xi)$ is cobordant via a trivial cobordism to a hypertight contact manifold. Turning this around, $(M,\xi)$ satisfies the Weinstein conjecture if and only if it is not cobordant by a trivial Liouville cobordism to a hypertight manifold. From this perspective, obstructing the existence of Liouville cobordisms with hypertight negative end can be seen as a geometric generalisation of the Weinstein conjecture.
$'$ $'$
|
1511.00035
|
Institut Jean
Lamour, UMR 7198 CNRS - Université de Lorraine, BP 239 F-54506 Vandoeuvre-le-Nancy, France
Dipartimento di Fisica, Università di Roma-Tor Vergata, Roma, Italy
Earth, Planetary and Space Sciences, UCLA, Los Angeles, USA
Earth, Planetary and Space Sciences, UCLA, Los Angeles, USA
This paper discusses the transition to fast growth of the tearing instability in thin current sheets in the collisionless limit where electron inertia drives the reconnection process. It has been previously suggested that in resistive MHD there is a natural maximum aspect ratio (ratio of sheet length and breadth to thickness) which may be reached for current sheets with a macroscopic length $L$, the limit being provided by the fact that the tearing mode growth time becomes of the same order as the Alfvén time calculated on the macroscopic scale (Pucci and Velli (2014) <cit.>). For current sheets with a smaller aspect ratio than critical the normalized growth rate tends to zero with increasing Lundquist number $S$, while for current sheets with an aspect ratio greater than critical the growth rate diverges with $S$. Here we carry out a similar analysis but with electron inertia as the term violating magnetic flux conservation: previously found scalings of critical current sheet aspect ratios with the Lundquist number are generalized to include the dependence on the ratio $d_e^2/L^2$ where $d_e$ is the electron skin depth, and it is shown that there are limiting scalings which, as in the resistive case, result in reconnecting modes growing on ideal time scales. Finite Larmor Radius effects are then included and the rescaling argument at the basis of “ideal” reconnection is proposed to explain secondary fast reconnection regimes naturally appearing in numerical simulations of current sheet evolution.
§ INTRODUCTION
Magnetic reconnection is thought to be the mechanism underlying many explosive phenomena observed in both space and laboratory plasmas, ranging from magnetospheric substorms, to solar flares and coronal mass ejections, to the sawtooth crashes observed in tokamaks. The classic picture of reconnection involves current sheets, most often assumed to be planar-like and concentrated more narrowly in the third dimension. Often, a guide magnetic field lies within the current sheet itself, so that the actual three-dimensional field does not vanish in the sheet. Different models for reconnection occurring in such quasi-2D configurations have been developed, two prominent, different examples being the Sweet-Parker (SP) stationary reconnection scenario and the spontaneous reconnecting modes naturally developing due to the tearing instability of the current sheet itself. Biskamp (1986) <cit.> first pointed out the important role played by the current sheet aspect-ratio in determining whether a stationary reconnection configuration could be reached. He found, via numerical simulations, that the SP current sheet could become unstable to reconnecting modes once a critical value of the Lundquist number (estimated on the current sheet length or breadth, $L$) of about $S\simeq 10^4$ was exceeded. A detailed examination of the stability of the SP configuration led to the definition of the plasmoid-chain instability <cit.>, reminiscent of the plasmoid-induced reconnection concept and fractal reconnection models introduced by Shibata et al. (2001) <cit.>. Recently, Pucci and Velli (2014) <cit.> have pointed out that the divergence of the growth rate of the plasmoid chain instability in the limit of large Lundquist number within resistive MHD implies that current sheets should never elongate sufficiently to achieve the SP aspect ratio. They have shown that a critical aspect ratio separates slowly unstable current sheets (with growth rate scaling as a negative, fractional exponent of the Lundquist number) from violently unstable ones (growth rates scaling with a postive power of $S$). They dubbed the instability of the critically unstable current sheet “ideal tearing" (hereafter IT), because the growth rate, normalized to the Alfvén time along the sheet $L$, becomes of order unity, and independent of the Lundquist number itself.
The large predicted growth rates and the presence of critical values for dimensionless numbers such as current-sheet aspect ratio make the described instabilities good candidates to understand and model the mechanisms behind observed fast reconnection phenomena <cit.>. Indeed, to date there is no agreed theoretical explanation for the fast time scales over which reconnection events develop in nature, nor for their triggering, while evidence from both experiments and numerical simulations points to the importance of small scale formation and kinetic effects, <cit.> which are theoretically expected to lead to Alfvénic (or “ideal") reconnection in 3D configurations as well <cit.>. Moreover, numerical simulations of tearing mode instabilities have identified a secondary, nonlinear, increase of the reconnection rate, that has been sometimes interpreted in terms of a nascent plasmoid-unstable SP regime <cit.> or generically a secondary “explosive reconnection” regime <cit.>. A nonlinear increase of the reconnection rate on ideal, Alfvénic time-scales was also numerically measured by Yu. et al. (2014) <cit.> in simulations of low mode-number reconnection instabilities. Given the recent developments of the theory of large-aspect ratio current sheet instabilities, it is important to understand whether such augmented fast reconnection rates may indeed be interpreted as fast secondary instabilities of the nonlinearly generated current sheets stemming from the primary reconnection event. Specifically, given that kinetic and two-fluid effects easily become dominant compared to classical, collisional resistivity at small spatial scales, it seems timely to see whether and how such effects modify the transition to an IT regime.
The present paper focuses on the extension of the IT scaling arguments to weakly collisional regimes where reconnection is mediated by electron inertia effects, and on whether such generalized IT regimes might explain the nonlinear occurrence of fast exponentially growing reconnection rates. We will consider both the incompressible reduced MHD (RMHD $-$ see e.g. <cit.>) and electron MHD (EMHD <cit.>) frequency ranges, where the perturbations are dominated by Alfvén and whistler modes respectively. The formal similarity between RMHD and EMHD reconnection in slab geometry, previously discussed in <cit.>, allows a unified treatment for the onset of IT in an electron-inertia driven framework.
Electron inertia has long been considered the most promising alternative to standard resistive reconnection thanks to its greater weight with respect to resistivity in the generalized Ohm's law of quasi-collisionless plasmas <cit.>. Astrophysical and thermonuclear fusion plasmas are examples of such systems, since their particle mean free path tipically exceeds the characteristic hydrodynamic lengths by order(s) of magnitude. In general, interspecies collisions may be neglected with respect to inertial terms when the characteristic ion-electron collision frequency is negligible with respect to the inverse time scale of the phenomena considered <cit.>. The inertial slab RMHD regime we focus on here has indeed been widely used to model basic features of magnetic reconnection in tokamak devices, for which the strong guide field approximation, of which we consider the 2D-geometry limit, was first devised, as well as for astrophysical applications. In EMHD the neglect of collisional resistivity is even more justified, which is why EMHD reconnection is mostly studied in purely inertia-driven regimes (see <cit.> for a discussion of the transition from resistive to inertial EMHD). Because of the large characteristic frequencies involved, EMHD provides a natural framework for collisionless reconnection. The relation between the convection electron flow and the magnetic field, typical of the EMHD regime, plays a prominent role in explaining the quadrupolar structure of the out-of-plane magnetic field <cit.>, which is often recognized as a distinctive signature for the in situ detection of magnetospheric reconnection <cit.>. Rogers et al. (2001) <cit.> also adopted the incompressible, inertia-less, collisionless EMHD model to explain the opening-up of the reconnection layer in 2D simulations with no guide field. We finally note that the present paper does not cover the framework of the so-called Hall- or whistler- mediated reconnection (Appendix <ref>), especially relevant to the magnetopause environment <cit.>, and which is known to provide prominent examples of fast reconnection rates weakly dependent from both resistivity <cit.> and electron inertia <cit.>. This will be considered in future works.
The paper is structured as follows. In Sec.<ref> we summarize the re-scaling arguments leading to the concept of “ideal tearing". In Sec.<ref> we introduce the model equations for reconnection in the RMHD and EMHD regimes and the relevant dispersion relations (Sec.<ref>). In Sec.<ref> we extend the IT paradigm first to the inertial RMHD and EMHD reconnection regimes (Sec.<ref>) and then to include finite Larmor radius (FLR) effects (Sec.<ref>). We then discuss these results (Sec.<ref>) by comparing the role of inertia to that of resistivity in different natural and laboratory plasmas (Sec.<ref>), and by considering an application of the IT model to collisionless steady reconnecting current sheets (Sec.<ref>). Then, in Sec.<ref> we discuss how the re-scaling argument might explain explosive reconnection regimes nonlinearly observed in simulations of magnetic reconnection. Sec.<ref> provides a summary and conclusion and in the Appendix <ref> we recall the derivation of the model equations both from a two-fluid model and compared with the generalized Ohm's law (Sec.<ref>).
§ THE IDEAL TEARING MODEL
Consider a current sheet of length $L$ and thickness $a$. As MHD is scale-free, in the classical tearing mode theory it is customary to take the width $a$ as normalization length, since typically $L/a>1$ and $a$ is the only characteristic length defined by the (usually 1D) equilibrium profile. However, when dealing with thin sheets with $a$ arbitrarily small, the distinction between $L$ and $a$ becomes important, as the tearing mode growth rate is only small when measured with respect to the “ideal" Alfvén timescale based on $a$, but can become large when measured with respect to a macroscopic scale $L >>a$ (the basic idea behind the plasmoid instability and IT, detailed below). From now on, we will label quantities normalized to the scale $L$ with the apex “$*$”, using standard notation for non-dimensional quantities defined in terms of the (possibly microscopic) shear-scale $a$.
In this notation, the classical linear reconnecting mode on Harris-type current sheets has a maximal growth rate scaling as $\gamma_M \tau_{_A} \sim S^{-1/2}$ where the Lundquist number $S = aV_{_A}/\eta_m$ and $\tau_{_A} = a/V_{_A}$, with $V_{_A}$ the Alfvén speed based on the characteristic magnetic field strength far from the sheet. In the SP case, predicated on the renormalized Lundquist number $S^* = LV_{_A}/\eta_m$, one finds immediately that $\gamma_M \tau_{_A}^* = \gamma_M L/V_{_A} \sim {S^*}^{1/4}$, i.e. a growth rate which diverges with the macroscopic Lundquist number $S^*$. Pucci and Velli (2014) <cit.>, aiming to resolve this paradox, incompatible with the ideal MHD limit, studied large-aspect ratio current sheets with $L/a$ scaling as a positive fractional power of the Lundquist number $S^* = LV_{_A}/\eta_m \gg 1$. They showed that when a threshold $L/a\sim (S^*)^{\alpha}$ ($1/2>\alpha > 0$) is reached, the resistive tearing mode growth rate $\gamma_M \tau_{_A}^*$ becomes of order unity and independent of $S^*$. This regime was named “ideal tearing”, in contrast to the CT theory in which
the growth rates scale as a negative power of the $a$ -normalized Lundquist number $S$. The large aspect ratio limit allowed <cit.> to evaluate the characteristic CT reconnection rate through the fastest growing mode, from which the value
$\alpha=1/3$ was obtained, leading to the conclusion that SP current sheets should not form at large $S^*$ (different equilibrium profiles may induce small deviations from this value <cit.>). The renormalization in fact gives
\begin{equation}\label{eq:id_tearing_Harris}
\gamma_M \tau_{_A}^*\sim (S^*)^{-1/2}(L/a)^{3/2}\end{equation}
and the clock whose rate defines the reconnection speed enters this renormalized theory through $\tau_{_A}^*$ which depends itself on $L$, i.e., the clock set on the ideal scale $L$ results slower by a factor $a/L$ (or, as we shall see, $(a/L)^2$ in the EMHD regime) than the clock with which the reconnection rate is measured in the CT theory: it is thus always possible to find a critical exponent $\alpha>0$ such that $\gamma_M\tau_{_A}^*\simeq 1$ once the condition $(L/a)\sim (S^*)^{\alpha}$ is imposed. In other words, the tearing-mode theory, under the assumption of a current sheet whose aspect ratio scales as a power of the (small) non-ideal parameter $\varepsilon^*$ which allows reconnection, say $a/L\sim (\varepsilon^*)^\alpha$, can explain the transition to fast reconnection if the value of $\alpha$ is such that the growth rate of the instability is independent from $\varepsilon^*$ itself. Notice however that the IT criterion may be applied in principle to any reconnection unstable aspect ratio $L/a$, if $L$ is large enough with respect to $a$. It is e.g. the case of tearing unstable current sheets, nonlinearly developed by primary reconnection events, which we will consider later. We now consider how this happens once electron inertia first, and FLR-type effects second, are taken into account.
§ MODEL EQUATIONS
We restrict our analysis to a 2D system in the $(x, y)$ plane, and assume for simplicity an electron-proton plasma.
Consider the incompressible equations in slab-geometry.
We adopt the standard “poisson-bracket” representation $[f,g]\equiv\partial_x f\partial_y g-\partial_y f\partial_x g={\bf e}_z\cdot({\bm \nabla}f\times{\bm \nabla}g)$. The velocity stream functions $\varphi$ and $b$ are such that
${\bm U}_\perp=-{\bm\nabla}\varphi\times{\bf e}_z$ in RMHD and ${\bm u}^e_\perp=-{\bm\nabla}b\times{\bf e}_z$ in EMHD (see below), where “$\perp$” stands for components in the $(x, y)$ plane, and ${\bm u}^e$ and ${\bm U}$ are the electron and bulk plasma velocities, respectively. Analogously, the magnetic stream function $\psi$ is defined through ${\bm B}={\bm \nabla}\psi(x,y)\times{\bf e}_z+(B_0+b(x,y)){\bf e}_z$, with $B_0$ uniform in space. We assume an equilibrium in-plane magnetic field ${\bm B}_\perp^0=B_y^0(x/a){\bf e}_y$ with $B_y^0(x/a)= \partial_x \psi_0(x/a)$. Equilibrium quantities are labeled with “$0$”, and we introduce the fields $F\equiv \psi-d_e^2\nabla^2\psi$ and $W\equiv b-d_e^2\nabla^2b$. Here $d_e= c/\omega_{pe}$ is the electron-skin-depth.
Using $a$ as the reference length and characteristic quantities $B_\perp^0$ and $n_0$ for magnetic field and densities, the model equations may then be written in non-dimensional form either as:
\begin{equation}\label{eq:OhmRMHD}
\frac{\partial}{\partial t}F+[\varphi,\,F]=\rho_s^2[\nabla^2\varphi,\psi]+
\end{equation}
\begin{equation}\label{eq:RMHD_U}
\frac{\partial}{\partial t}\nabla^2\varphi+[\varphi,\,\nabla^2\varphi]=[\psi,\,\nabla^2\psi] +R^{-1}{\nabla}^4\varphi\, ,
\end{equation}
valid in the RMHD frequency range, or
\begin{equation}\label{eq:OhmEMHD}
\frac{\partial}{\partial t}F+[b,\,F]=S_{_{Emhd}}^{-1}{\nabla}^2\psi\end{equation}
\begin{equation}\label{eq:EMHD_U}
\frac{\partial}{\partial t}W+[b,\,W]=[\psi,\,\nabla^2\psi] +S_{_{Emhd}}^{-1}{\nabla}^2b\, ,
\end{equation}
valid in the EMHD frequency range.
In the above, time is normalized to $\tau_{{_A}}\equiv (a/ d_i)\Omega_{i}^{-1}$ in RMHD, where $\Omega_{i}$ is the ion cyclotron frequency and $d_i\equiv \sqrt{m_i} c/(\sqrt{m_e}\omega_{pe})$ is the ion-skin depth ($\omega_{pe}$ being the usual plasma frequency and with obvious notation for the masses);
in EMHD time is normalized to the inverse of the whistler frequency, $\tau_{{_W}}\equiv (a/d_e)^2\Omega_{e}^{-1}=(a/d_i)^2\Omega_{i}^{-1}$.
The other parameters on which the tearing reconnection rate depends are the ion-sound Larmor radius, also non-dimensionalized with $a$ i.e. $\rho_s\equiv c_{is}/a\Omega_i$, where $c_{is}$ is the ion sound speed, i.e. the thermal speed based on electron temperature and ion mass; $R\equiv (\nu_{ii}\tau_{_{A}})^{-1}$ (Reynold's number) with $\nu_{ii}$ the ion-ion viscosity; $S\equiv \tau_{_D}/\tau_{_{A}}$ (Alfvénic Lundquist number) and $S_{_{Emhd}}\equiv \tau_{_D}/\tau_{_{_W}}$ (EMHD Lundquist number) with $\tau_{_D} = 4\pi a^2/(\eta c^2)$ the resistive diffusion time ($\eta$ is the scalar resistivity). The physical meaning of the terms of Eqs.(<ref>-<ref>) and their relation to both the two-fluid model equations and the generalized Ohm's law are discussed in Appendix <ref>.
Note that, calling $L_{_{MHD}}$ and $L_{_{EMHD}}$ the normalization lengths in RMHD and EMHD,
the inequality
\begin{equation}
\frac{\tau_{_{W}}^*}{\tau_{_A}^*}= \left(\frac{L_{_{EMHD}}}{d_i}\right)
\left(\frac{L_{_{EMHD}}}{L_{_{MHD}}}\right)\ll 1,
\end{equation}
must hold since the characteristic quantities in EMHD must be much smaller than $d_i$ and those of RMHD much larger than $d_i$.
§.§ Linear dispersion relations
We now focus on the collisionless regimes, $S^{-1}=S^{-1}_{_{EMHD}}=0$; we will not consider viscous effects, whose role in MHD has been clarified recently by Tenerani et al. (2015)
<cit.>. In addition, to further simplify the analysis, we start by setting $\rho_s=0$ in Eqs.(<ref>)-(<ref>). Because of
the fact that both the (squared) electron skin depth and the Lundquist number weigh non ideal terms in Ohm's law which allow magnetic lines to reconnect (Appendix <ref>), and of other similarities which will be later discussed,
let us introduce for future use the notations $\varepsilon_d \equiv d_e^2$ and
$\varepsilon_{_S}\equiv S^{-1}$. Then, after re-scaling, we will write
\begin{equation}\label{eq:rescaling_epsilon}
\varepsilon_d^*=\varepsilon_d\left(\frac{a}{L}\right)^2,\qquad\qquad
\varepsilon_{_S}^*=\varepsilon_{_S}\left(\frac{a}{L}\right).
\end{equation}
After linearizing Eqs.(<ref>)-(<ref>) around an equilibrium $\psi_0(x/a)$ with perturbations of the form $\sim e^{iky+\gamma t}$, analytic approximations to the
dispersion relations in both RMHD and EMHD may be obtained by applying the boundary layer technique, as first shown by Furth et al. (1963) <cit.>.
Here we summarize the results valid in the two asymptotic regimes called large (LD) and small (SD) $\Delta'$, which respectively correspond to the internal kink and constant-$\psi$ orderings <cit.>. In RMHD such regimes are respectively defined by the conditions $\Delta'\delta > 1$ (LD) and $\Delta'\delta < 1$ (SD), where $\delta$ is the characteristic reconnection layer width. The inertial RMHD tearing dispersion relations become (see e.g. <cit.>):
\begin{equation}\label{eq:disp_RMHD}
\mbox{RMHD }\left\{\begin{array}{c}
\displaystyle{\gamma_{_{LD}}\tau_{_A}=k d_e}
\\
\displaystyle{\gamma_{_{SD}}\tau_{_A}= (C_1 \Delta')^{2} kd_e^3}
\\
\end{array}\right.,
\end{equation}
where $C_1\equiv \Gamma(1/4)/(2\pi\Gamma(3/4))\simeq 0.4709$.
In EMHD, where the LD limit corresponds more properly to the condition $\gamma_{LD}/k\sim$ constant, we consider the dispersion relations
\begin{equation}\label{eq:disp_EMHD}
\mbox{EMHD }\left\{\begin{array}{c}
\gamma_{_{LD}}\tau_{_W}=\displaystyle{
C_2k d_e^\frac{2}{3}} \\
\gamma_{_{SD}}\tau_{_W}=\displaystyle{ \left( C_1\Delta'\right)^2d_e^2} \\
\end{array}\right.,
\end{equation}
where $C_2\equiv(2\Gamma^4(3/4))^{-1/3}\simeq 0.6053$.
The $\gamma_{_{LD}}$ growth rate above, which is the one evaluated
by Attico et al. (2000) <cit.> starting from an equilibrium given by $\psi_0(x)=x/a$ for $-a<x<a$ and $\psi_0(x)=1$ for $|x|\geq a$, has been assumed as the prototype for the more general “LD” EMHD dispersion relation for a generic sheared, even, $\psi_0(x)$ profile. The reason is that this is the only available formula obtained for this wavelength regime, and, with the same equilibrium, the general $\gamma_{_{SD}}$ dispersion relation first computed in <cit.> and quoted in Eq.(<ref>), was exactly recovered.
For illustrative purposes in Fig.<ref> we show the scaling of the growth rate of a given unstable mode $\tilde{k}$ as a function of $\varepsilon_d$ in the RMHD regime. Notice that the whole range of regimes from SD to LD is spanned while varying the value of $d_e$ at given $k$. Indeed, since $\delta=\delta(k,\varepsilon_d)$, an interval in the $\varepsilon_d$ parameter space such that
$\Delta'(\tilde{k})\delta(\tilde{k},\varepsilon_d)$ is smaller (SD), equal ($\gamma_{_M}$, see Sec.<ref>), or greater (LD) than unity, always exists.
Scaling of $\gamma(\tilde{k})\tau_{_A}$ in the RMHD regime as a function of $d_e^2$ for a fixed $\tilde{k}$. At the increase (decrease) of $d_e$ the small (large) $\Delta'$ regime is progressively entered. Here $\tilde{k}=k_{_M}$ for $d_e^2\simeq 2\times 10^{-3}$ (lengths in units of $a$).
As a comment, note that almost ideal growth rates (saturating at $(\gamma_{_{LD}}^{_{EMHD}})^*\simeq 0.25 (\tau_{_W}^*)^{-1}$) were observed in numerical integrations of the EMHD linear system at $ 0.1\lesssim d_e < 1$ <cit.> for $L/a=2\pi$ and $k^*=k=1$. Such large values of $d_e$ are not unreasonable in the collisionless EMHD regime, because of the constraint $d_e\ll a\ll d_i$ (now in dimensional units), which must be fulfilled by the equilibrium shear length. With such large values of the reconnection parameter, we are outside the realm of the asymptotic/boundary layer analysis, but for EMHD this is to be expected, since characteristic EMHD scale lengths must satify $\ell$ fulfill $d_e \ll \ell$, or, given that $d_i/d_e\simeq 42Z$ for an ion charge $Z$, $d_e\ll \ell \ll 42 d_e Z$.
Similarly large growth rates are found in strongly resistive RMHD regimes $S^{-1}\gtrsim 0.01$, though these are normally of little interest. Discrepancies with analytical estimations from Eqs.(7)-(8), suggest that at $\varepsilon_d \sim 0.01$ or equivalently
$\varepsilon_S \sim 0.01$ the boundary layer approach to the linear tearing breaks down.
§ RESULTS
§.§ Transition to the inertial ideal regime
When $L/a\gg 1$, say, $L/a\gtrsim 20$ <cit.>, we can search for the fastest unstable mode $k_{_M}$ with corresponding growth rate $\gamma_{_M}$. As noticed by Battacharjee et al. (2009) <cit.>, the latter can be estimated by imposing the condition $\gamma_{_{LD}}(k_{_M})=\gamma_{_{SD}}(k_{_M})\equiv\gamma_{_M}$. Approximating $\Delta'(k_{_M})\simeq Kk_{_M}^{-p}$ where $K$ is a constant, from Eqs.(<ref>) and Eqs.(<ref>) we can estimate (see also <cit.>):
\begin{equation}\label{eq:max_RMHD}
\mbox{RMHD }\left\{\begin{array}{c}
\displaystyle{ k_{_M}\simeq \left( KC_1\right)^{\frac{1}{p}} d_{e}^{\frac{1}{p}}} \\
\displaystyle{\gamma_{_M}\tau_{A}\simeq \left( KC_1 \right) d_e^{\frac{1+p}{p}}},\\
\end{array}\right.
\end{equation}
\begin{equation}\label{eq:max_EMHD}
\mbox{EMHD }\left\{\begin{array}{c}
\displaystyle{ k_{_M}\simeq \left(\frac{K^2C_1^2}{C_2}\right)^{\frac{1}{1+2p}}
d_e^{\frac{4}{3(1+2p)}}} \\
\displaystyle{ \gamma_{_M}\tau_{W}\simeq
\left(KC_1C_2^p\right)^{\frac{2}{1+2p}} d_e^{\frac{2}{3}\frac{3+2p}{1+2p}}}.\\
\end{array}\right.
\end{equation}
Let us now apply the re-scaling argument to evaluate, from Eqs.(<ref>-<ref>) and from the definitions of $\tau_{_A}$ and $\tau_{_W}$, the scaling of the most unstable mode when lengths are normalized to $L$. Neglecting the numerical coefficients in the parentheses of Eqs.(<ref>-<ref>) we find in RMHD,
\begin{equation}\label{eq:RMHD_max_ideal}
\displaystyle{ k_{_M}^*\simeq
\quad
\displaystyle{ \gamma_{_M}\tau_{_A}^*\simeq
\end{equation}
and in EMHD
\begin{equation}\label{eq:EMHD_max_ideal}
\displaystyle{ k_{_M}^*\simeq
\quad
\displaystyle{ \gamma_{_M}\tau_{_W}^*\simeq
\left(\frac{L}{a}\right)^{\frac{12+16p}{3+6p}}}\, .
\end{equation}
In the RMHD regime it is easy to verify from the analytical estimates $\delta_{_{LD}}\sim d_e$ and $\delta_{_{SD}}\sim \Delta'd_e^2$ (see e.g. <cit.>) that the fastest growing mode satisfies the condition $\Delta'(k_{_M})\delta(k_{_M})\sim 1$. The characteristic width of the reconnection layer for the most unstable RMHD mode therefore becomes
\begin{equation}\label{eq:RMHD_delta}
\displaystyle{ \delta_{_M}\simeq d_e},
\end{equation}
which, after rescaling, reads $\displaystyle{ \delta_{_M}^*\simeq (\varepsilon_d^*)^{\frac{1}{2}} }$.
The condition for “ideal” tearing is set by searching for the value of $\alpha$ such that when $a/L\sim (\varepsilon_d^*)^{\alpha}$ with $\alpha>0$ $\gamma_{_M}^*$ becomes independent of $\varepsilon_d^*=\varepsilon_d a^2/L^2$.
Imposing this, we find the exponent $\alpha$ both in RMHD and EMHD, respectively,
\begin{equation}\label{eq:alpha}
\alpha_d^{_{RMHD}}=\frac{1+p}{2+4p},\qquad\quad
\alpha_d^{_{EMHD}}=\frac{3+2p}{12+16p}\, .
\end{equation}
In particular, for a Harris-pinch equilibrium, which has $p=1$, we find
\begin{equation}\label{eq:alpha_Harris}
\alpha_d^{_{RMHD}}=\frac{1}{3}, \qquad\quad\alpha_d^{_{EMHD}}=\frac{5}{28}
\simeq 0.1786.\end{equation}
A set of curves $\gamma({k})$ for different values of $d_e$ along the RMHD threshold condition $a/L=(\varepsilon_d^*)^{1/3}$ is plotted in Fig.<ref>a, while the corresponding graph for the EMHD regime is in Fig.<ref>b.
The independence of $\gamma_{_M}^*$ from $d_e$ and its value of order unity, namely $\simeq 0.39 (\tau_{_A}^*)^{-1}$ in RMHD and $\simeq 0.37 (\tau_{_W}^*)^{-1}$ in EMHD, is evidenced in both regimes. Referring to the example of the Harris-pinch profile and assuming for the EMHD the numerical threshold condition $a/L=(\varepsilon_d^*)^{\frac{3}{16}}$, we then deduce the scalings of the threshold current sheet widths $a$ with respect to $d_e$, which will be discussed in Sec.<ref>:
\begin{equation}\label{eq:a_vs_de}
\left( \frac{a}{d_e}\right)_{_{RMHD}}= \left( \frac{L}{d_e}\right)^{\frac{1}{3}},
\qquad \left( \frac{a}{d_e} \right)_{_{EMHD}}= \left( \frac{L}{d_e}\right)^{\frac{5}{8}}.
\end{equation}
RMHD (left frame) and EMHD (right frame) dispersion relations $\gamma^*=\gamma^*({k}^*,\varepsilon_d^*)$, computed for different values of $d_e$ and represented as functions of $k^*a^*$. For each curve an aspect ratio was chosen, satisfying the threshold condition for a Harris-pinch equilibrium, $a/L=(\varepsilon_d^*)^{1/3}$ in RMHD and $a/L=(\varepsilon_d^*)^{3/16}$ in EMHD. The maximum growth rate on each curve is independent from $d_e$ and of order unity with respect to the characteristic time: $\gamma_{_M}^*\tau_{_A}^*\simeq 0.39 $ in RMHD and $\gamma_{_M}^* \tau_{_W}^*\simeq 0.37$ in EMHD.
§.§ Kinetic effects in the transition to the inertial ideal tearing: FLR corrections
We now briefly consider the role played by other kinetic effects important at small spatial scales $a\ll L$ where the transition to “ideal” tearing takes place. Since ion-ion viscosity effects have been already discussed by Tenerani et al. (2015) <cit.> we focus on FLR effects, which enter in our set of equations through the so-called gyrofluid corrections, an example of which is provided by the $\rho_s$ term in Eqs.(<ref>-<ref>).
At small scales $\ell\ll L$ the fluid description formally breaks down, but it has been shown that gyrofluid RMHD models capture the essential physics of gyrokinetic reconnection <cit.>. A good agreement between our collisionless RMHD equations at $\rho_s\neq 0$ and a drift-kinetic model for magnetic reconnection was already pointed out <cit.>. The ion-sound Larmor radius was shown to increase the inertia-driven tearing reconnection rate both linearly <cit.> and nonlinearly <cit.>. Notice that the RMHD equations have been extended to include also ion FLR effects, $\rho_i\equiv v^{i}_{th}/\Omega_{ci}$ ($v_{th}^i$ being ion thermal velocity), related to the ion-sound Larmor radius by $\rho_s^2=\rho_i^2T_e/T_i$. These effects are usually introduced in RMHD equations, notably in Eq.(<ref>), by making some closure assumption on the ion kinetic response obtained from the transport equation. Different models are then available, but also those whose different Hamiltonian properties were compared by Welbroeck et al. (2009) <cit.>, were shown to provide numerical results in a remarkably good agreement <cit.>. Also notice that the isothermal assumption behind the definition of $\rho_s$ and $\rho_i$ has been shown to be in good agreement with the numerical results from gyrokinetic models for electrons, during the whole linear reconnection stage <cit.>. Interestingly, in a certain parameter range, the theoretically predicted scalings of tearing modes <cit.> display a symmetric dependence on the two FLR effects, as the latter enter in the dispersion relation as powers of $\rho_\tau^2=\rho_s^2+\rho_i^2$. Even if appreciable discrepancies from these predictions are seen as the ratio $\rho_\tau^2/d_e^2$ increases <cit.>, at $\Delta'd_e\gg \min{[1, (d_e/\rho_\tau)^{1/3}]}$ a good agreement is found <cit.>.
In the regime $\rho_\tau^2\gg d_e^2$, Comisso et al. (2013) <cit.> recently pointed out the existence of a maximum growth rate in the continuum spectrum limit (i.e. continuous $k$) of unstable tearing modes, corresponding, in our notation, to $k_{_M}$. The generalization of the result they obtained for the Harris-pinch case to generic equilibria, is obtained as described in Sec.<ref>, by starting from their Eqs.(26)-(27) instead of our Eqs.(<ref>). We find
\begin{equation}\label{eq:FLR_max}
\displaystyle{
\qquad
\displaystyle{ \gamma_{_M}\tau_{_A}\simeq
Then, applying the rescaling arguments, we obtain $\gamma_{_M}^{_{FLR}}\tau_{_A}^*\sim O(1)$ when
\begin{equation}\label{eq:FLR_threshold}
\frac{a}{L}\sim (\varepsilon_d^*)^{\frac{2+p}{6+12p}}
We then see that, depending on the value of the ratio $d_e/\rho_\tau$, the inclusion of FLR corrections may imply an even larger critical aspect ratio for the transition to “ideal” tearing, with respect to the cold-plasma limit. Indeed, if we now assume $\rho_\tau\simeq A d_e$ and we compare the threshold condition of
Eq.(<ref>) with that of Eq.(<ref>) for the RMHD, we see that, with obvious notation, the two are related through $(a/L)_{_{FLR}}\sim A^{1/3}(a/L)_{_{RMHD}}$. Since usually $A>1$ (e.g., tipically $A\sim 10$ in tokamak plasmas and it may be even larger in the magnetosphere this implies a broadening of the ideally unstable current sheet with respect to the cold plasma case, when kinetic effects are taken in account.
§ DISCUSSION
§.§ Collisionless ideal tearing in space, solar and laboratory plasmas
In order to discuss the relevance of electron inertia and resistivity in various natural and laboratory environments where low-collision reconnection occurs, different plasma parameters, including $ \varepsilon_S^*$ and $\varepsilon_d^* $, are shown in Table <ref>.
We recall that the condition for purely collisionless reconnection ($S^{-1}=0$) is given by $\gamma_d\tau_{_A}\varepsilon_d\gg \varepsilon_{_S}$, with $\gamma_d$ reconnection rate of the sheer inertia-driven regime. Notice that this condition becomes less critical when approaching the ideal regime ($a/L\ll 1$), where $\gamma_d^*\rightarrow 1$ because of the rescaling, which in RMHD implies $\varepsilon_{_S}/\varepsilon_d=(a/L)\varepsilon_{_S}^*/\varepsilon_d^* $
(Eqs.(<ref>)): if $\varepsilon_{_S}^*/\varepsilon_d^*\ll 1$, then we can assume the IT model applied to large aspect-ratio current sheets as essentially inertia-driven. This means, for example, that the magnetotail is in an essentially inertia-dominated tearing regime. On the other hand, fusion devices, for which $a\simeq L$, may operate in conditions in which the resistive contribution to tearing reconnection is not negligible even if $\varepsilon_{_S}^*/\varepsilon_d^*\sim
\varepsilon_{_S}/\varepsilon_d\sim 10^{-2}-10^{-3}$, because of the smallness of $\gamma_d\tau_{_A}^*$, which remains of the same order of $\gamma_d\tau_{_A}\ll 1 $.
This is because electron inertia, $\varepsilon_d$, enters in the the dispersion relation of tearing modes with a less favorable scaling with respect to resistivity, $\varepsilon_{_S}$. For practical purposes, at $a/L\sim 1$ the collisionless regime is essentially inertial if $\varepsilon_{_S}$ is sufficiently small ($\varepsilon_{_S}^{-1}\lesssim 10^{-8}$) and $\varepsilon_d$ is at least $3-5$ orders of magnitude larger than $\varepsilon_{_S}$.
The case in which the inertial $\gamma_d$ may dominate over the resistive $\gamma_{_S}$, is exemplified in Figs.<ref>, where some examples of the inertial-resistive growth rate are represented, for which only an implicit analytical expression for $\gamma$ is available (see e.g. Eq.(16) of <cit.>). The dispersion relations displayed are obtained by numerical integration of the linearized Eqs.(<ref>-<ref>). These examples show that no appreciable differences in the inertial-resistive growth rates with $\varepsilon_{_S}=10^{-8}$ are observed between $\varepsilon_d=10^{-8}$ and $\varepsilon_d=10^{-5}$. At higher values of $S^{-1}$, both the inertial and the resistive contributions to the inertial-resistive growth rate become appreciable, and for $S^{-1}\gtrsim 10^{-6}$ the resistive contribution to the growth rate is relevant even for $d_e$ approaching unity.
Dispersion relations $\gamma$ vs. $k$ for different values of $d_e$ and for $S^{-1}=10^{-8}$ (upper panel) and $S^{-1}=0$ (lower panel). Some orders of magnitude of separation between the purely inertial and purely resistive growth rates (tipically about $3-4$, at least) are needed in order for $\varepsilon_S$ to be really negligible.
§.§ Ideal tearing and stability of steady-state reconnecting current sheets in the collisionless regime
Both in MHD <cit.> and in EMHD <cit.>, the reconnection rate of a steady state current sheet has been evaluated in the collisionless regime, as a generalization of the classic Sweet-Parker configuration. In both[Notice that Wesson's result <cit.> was specialized to a geometric configuration corresponding to the $m=1$ mode in a tokamak, but his reasoning is easily adapted to the standard planar sheet configuration.] cases the same scaling in $\varepsilon_d$ of the stationary Sweet-Parker-like reconnection rate $\gamma_{_{SP}}$ was obtained with respect to the respective normalization times,
$(\tau_{_{SP}}^{_{EMHD}})^{-1}\tau_{_W}^*\sim (\varepsilon_d^*)^{1/2} $ and $(\tau_{_{SP}}^{_{RMHD}})^{-1}\tau_{_A}^*\sim (\varepsilon_d^*)^{1/2} $. This implies that both in collisionless RMHD and EMHD, the aspect ratio scaling of a steady current sheet of length $L$ is $(a/L)_{_{SP}}\sim (\varepsilon_d^*)^{1/2}$. By comparing the scaling of this ratio with the threshold conditions for the onset of “ideal” tearing (Eqs.(<ref>)) the same qualititative behavior, though with different scalings, is evidenced in both RMHD and EMHD. In RMHD the width of the steady reconnecting layer corresponds to a much thinner current sheet than that which is unstable to ideal tearing: at a given length $L$, the collisionless Sweet-Parker sheet width, $a_{_{SP}}$, is related to the ideal-tearing unstable one, $a_{_{IT}}$, by the relation $a_{_{SP}}^*\simeq (a_{_{IT}}^*)^{(1+2p)/(1+p)}$. Using the same reasoning we can estimate from Eq.(<ref>) $a_{_{SP}}^*\simeq (a_{_{IT}}^*)^{(6+8p)/(3+2p)}$ for EMHD. If we now neglect the effect of the flow along the neutral line on the growth rate (cfr. also <cit.> for why flows may be neglected), this means that both in RMHD and EMHD a collisionless Sweet-Parker-type current sheet is always unstable on ideal time scales.
§.§ “Secondary” ideal tearing and “explosive reconnection”
The rescaling argument at the basis of ideal tearing may thus provide a fairly general paradigm to describe explosive growth rate increases observed in the nonlinear stage of simulations of reconnection at $L/a$ not much larger than unity <cit.>, when an $X$-point collapses into two $Y$-points and the current sheet between the two becomes tearing unstable, eventually leading to the so-called plasmoid-chain instability. During this stage, even before an ideal growth rate is achieved, a secondary growth rate may be measured, which is arbitrarily large (possibly up to the inverse macroscopic time scale, in the ideal tearing limit).
Let be $L_{_Y}$ the length and $a_{_Y}$ the width of a secondary current-sheet between two $Y$-points, generated in the nonlinear stage of the tearing of a current sheet with aspect ratio $a/L$.
Focusing on the dynamics of this secondary current sheet, the CT growth rates would refer lengths to $a_{_Y}$, whereas we now need to label with “$\tilde{...}$” the quantities normalized to $L_{_Y}$, since the latter plays the role of macroscopic length for the secondary dynamics (cfr. Sec.<ref>). Even when we consider a primary tearing mode with $L/a\gtrsim 1$, the secondary current sheet develops with a much smaller thickness (corresponding to the singular layer thickness of the original tearing instability) so that we can assume the most unstable tearing mode to be destabilized:
accounting for FLR effects, the re-normalized, most unstable,
tearing mode growth rate on the secondary current sheet (cfr. Sec.<ref>) therefore becomes dependent from the ratio $L_{_Y}/a_{_Y}$,
\begin{equation}\label{eq:secondary_FLR}
{\gamma}_{_M}^{_{FLR}}\tilde{\tau}_{_A}\sim \tilde{\varepsilon}_d^{\frac{2+p}{6p}}\tilde{\rho}_{\tau}^{\frac{1+2p}{3p}}
\left(\frac{L_{_Y}}{a_{_Y}}\right)^{\frac{1+2p}{p}}.\end{equation}
Analogously, we can rewrite
the correspective for the resistive inviscid and viscous, high-Prandtl number RMHD regimes, respectively discussed in <cit.> and <cit.>,
\begin{equation}\label{eq:secondary_S}
\sim \tilde{\varepsilon}_{_S}^{\frac{1+p}{1+3p}}
\left(\frac{L_{_Y}}{a_{_Y}}\right)^{\frac{2+4p}{1+3p}},
\end{equation}
\begin{equation}\label{eq:secondary_S_visc}
\tilde{\varepsilon}_{_S}^{\frac{1+2p}{1+3p}}
\tilde{R}^{\frac{p}{1+3p}}
\left(\frac{L_{_Y}}{a_{_Y}}\right)^{2}.
\end{equation}
The occurrence of a secondary, ideal tearing mode developing as a consequence of a primary tearing in a large aspect ratio current sheet in the resitive RMHD regime was first numerically evidenced by Landi et al. (2015) <cit.> and discussed in depth in <cit.>.
Let us now focus on a primary tearing mode of a small aspect ratio current sheet, assuming for simplicity $a\sim L$. In this case the primary reconnection rate can not be estimated with that of the most unstable mode $\gamma_{_M}$ but the specific LD or SD regime in which the unstable wave-number falls must be taken in account, instead. Let us now compare such primary reconnection rate to the secondary one, as estimated from Eqs.(<ref>-<ref>). We immediately recognize that, even before the ideal tearing threshold is reached, the re-scaling argument predicts an increase in the growth rate, measured with respect to the primary mode macroscopic scale $L$, by some positive power of $(L_{_Y}/a_{_Y})>1$ times some positive power of $(L/L_{_Y})>1$. Comparing Eqs.(<ref>-<ref>) we see that, for equal equilibrium profiles (same $p\geq 1$), such an increase is relatively more important in the inertia driven-FLR regime.
To fix the ideas with a quantitative example, consider the RMHD-FLR regime supposing the primary reconnection to develop on a current sheet described by the equilibrium used by Comisso et al. (2013) <cit.>, assuming an aspect ratio so close to unity that a single primary mode $m_0$ (i.e. $k_0=2\pi m_0/L$) is excited in the SD, constant-$\psi$ regime, in the whole range of parameters in which $d_e$ and $\rho_\tau$ are varied ($\Delta'\rho_\tau^{\frac{1}{3}}d_e^{\frac{2}{3}} < 1$). The primary tearing mode (see e.g. Eq.(27) of <cit.>), once rescaled to $L$, grows with $\gamma_I\tau_{_A}^*\simeq k_0^*(\varepsilon_d^*)^\frac{1}{2}\rho_\tau^*(\Delta')^*(L/a)$, with some $(\Delta'(k_0^*))^*$ of order unity. For the secondary mode we may now use Eq.(<ref>). Assuming for simplicity (but with no loss of generality) that the secondary current sheet resembles a Harris-pinch profile to specify some value of $p$ (here $p=1$), the secondary growth rate, expressed again in terms of the scale $L$, is given by Eq.(<ref>) opportunely rescaled, $\gamma_{II}\tau_{_A}^*\sim (\varepsilon_d^*)^\frac{1}{2}\rho_\tau^* (L/a_{_Y})^3$. A dominant increase of the reconnection rate is therefore provided by the ratio $L/a_{_Y}\gg 1$. In particular, in this example we obtain $\gamma_{II}^*/\gamma_{I}^*\sim (a/a_{_Y})(L/a_{_Y})^2(k_0^*\Delta'^*)^{-1}$.
Of course, a more detailed analysis would be required to verify whether the re-scaling argument summarized by Eqs.(<ref>-<ref>) and the corresponding threshold conditions for the ideal tearing suffice to explain the explosive reconnection regimes observed in the above mentioned numerical studies. However, the qualitative considerations about the scalings provided in Fig.(3) of <cit.> and in Fig.(2) of <cit.> seem encouraging. Because of the normalization assumed in these articles, the increase of the growth rates with decreasing plasma $\beta$ implies for the linear growth rate a scaling $\gamma_I\tau_{_A}^* \sim d_e$ and for the nonlinear one a scaling $\gamma_{II}\tau_{_A}^*\sim d_e^0$ at fixed $\rho_s$, thus suggesting (cfr. Eq.(<ref>) and Eq.(<ref>) for $p=1$) that an ideal tearing regime was observed in the nonlinear stage of the simulations discussed by Biancalani et al. (2012) <cit.>. Future studies will elucidate whether the explosive reconnection predicted by Eq.(<ref>) and that studied in <cit.> are effectively the same phenomenon.
To conclude this Section, we finally notice that, provided the ratio $L_{_Y}/a_{_Y}$ is large enough to destabilize a most unstable mode $\gamma_{_M}$ (which is typical for secondary current sheets developed from the collapse of an $X$-point in the resistive regime <cit.>) the measured growth rate would be that of an exponentially growing instability, which in the resistive regime has the same scaling with $S$ as the Sweet-Parker reconnection rate (i.e. $\sim S^{-1/2}$).
§ SUMMARY
We have extended the analysis of <cit.> to collisionless regimes, both in RMHD and EMHD, by providing the scaling threshold values $a/L\sim(d_e^2/L^2)^{\alpha}$ at which a current sheet with $L/a\gtrsim 20$ reconnects on the ideal macroscopic times of the model. For the Harris-pinch equilibrium profile the exponents measured after numerical solution of the eigenvalue problem are $\alpha_d^{_{RMHD}}=1/3$ and $\alpha_d^{_{EMHD}}\simeq 3/16$, in excellent agreement with the analytical estimations obtained by starting from the SD and LD dispersion relations. In RMHD, FLR corrections typically reduce the width of the critical aspect ratio for the transition to “ideal” tearing. In the parameter range $\Delta'd_e\gg \min[1,(d_e/\rho_\tau)^{1/3}]$ and for the Harris-pinch case, such an aspect ratio becomes $(a/L)_{_{FLR}}\sim (\varepsilon_d^*)^{1/6}(\rho_\tau^*)^{1/3}$, instead of $(a/L)_{_{d}}\sim (\varepsilon_d^*)^{1/3}$ in the $\rho_s=0$ limit. Since this implies a broadening of the critical reconnection current layer by a factor $(d_e^*)^{-1/3}(\rho_\tau^*)^{1/3}\sim A^{1/3}$, when $\rho_\tau\simeq A d_e$ with $A>1$, as it is usually the case, FLR effects are expected to correspondingly lower the instability threshold.
The collisionless IT model has been applied to discuss the instability of steady collisionless reconnecting current sheets, which, just as in the resistive case, should not be observable as they become unstable to inertia-driven tearing modes on ideal time-scales. We notice however that the threshold current sheet to the IT, found to be thinner in RMHD than in EMHD (Eqs.(<ref>)), leaves the open question of how the Alfvénic and whistler-dominated frequency regimes relate to the Hall-MHD framework, which in principle encompasses both as two of its limits, opposite one to each other (see Appendix <ref>).
We have also pointed out the relevance and importance of inertia-driven vs. resistive reconnection: the condition $S^{-1}\gg d_e^2\gamma$ provides a stringent constraint on when resistivity may be neglected which is often overlooked, for example, when applying Vlasov models of reconnection to tokamak plasmas.
We have finally discussed how the rescaling argument at the basis of the IT model may explain the “explosive” reconnection rate increase observed during the nonlinear stage of primary reconnection events, as secondary elongated current sheets are generated during the collapse of an $X$-point <cit.>. The IT regime may thus be in principle achieved also during secondary reconnection events involving the thin, elongated current layers nonlinearly generated by classical tearing processes <cit.> or in kinetic turbulence <cit.>. Notice that large aspect ratio current-layers are generally expected to develop because of the “exponentiation” of neighboring magnetic field lines <cit.>), and evidence of such exponential thinning of current sheets was recently provided, in the coronal heating context, by the numerical 3D simulations of <cit.>. This model provides therefore a promising key to interpretate reconnection rates, which both in laboratory and astrophysics are observed to be orders of magnitude faster than what is predicted by the CT theory. The simplicity of the rescaling argument at the basis of the IT model should not betray its non trivial reach. The dominant trend of recent research on magnetic reconnection, aiming at predicting almost ideal reconnection rates, focuses indeed on the role played by kinetic processes and secondary instabilities, whereas the model first considered by <cit.> has the appealing feature of relying on simple and well known results.
§ DISCUSSION OF THE MODEL EQUATIONS
Eqs.(<ref>-<ref>) are derived with different approximations from the
electron and ion momentum equations, which we write here below, again non-dimensionalized using $a$ and $\tau_{_A}$ (and the electric field normalized to a fraction $V_{_A}/c$ of the reference magnetic field):
\begin{equation}\label{eq:electron}
\left( \frac{\partial {\bm u}_e}{\partial t}+{\bm u}_e
\cdot{\bm\nabla}{\bm u}_e\right)=
-d_i\left({\bm E}+{\bf u}_e\times {\bm B}
- \frac{\bm J}{S}\right)-\rho_s^2\frac{{\bm\nabla}\cdot{\bm\Pi}_e}{n_e}
\end{equation}
\begin{equation}\label{eq:ion}
\left(\frac{\partial {\bm u}_i}{\partial t}+{\bm u}_i
\cdot{\bm\nabla}{\bm u}_i\right)=
d_i\left({\bm E}+{\bf u}_i\times {\bm B}
- \frac{\bm J}{S}\right)-{\rho_s^2}\frac{{\bm\nabla}\cdot{\bm\Pi}_i}{n_i}
\end{equation}
Here the kinetic pressure has been normalized to a reference value $P_0$ for the electron plasma pressure. This explains the weight $\rho_s^2 $ in front of the ion pressure force in Eq.(<ref>), even though the ion thermal Larmor radius is $\rho_i= (T_i/T_e)^{1/2}\rho_s$.
As discussed in <cit.> for the purely collisionless regime, Eqs.(<ref>-<ref>) may be indeed obtained, under appropriate approximations and closures for the pressure tensors (and after re-normalization to $ \tau_{{_W}}$ for the EMHD equations), from Eqs.(<ref>-<ref>) coupled with Maxwell's equations using quasi-neutrality, $n_e=n_i$. Such an approach is essentially the one via which electron inertia effects were first included in reconnection models in the full MHD <cit.>) and RMHD frameworks <cit.>. Within this approach, inclusion of resistive diffusion $S^{-1}$ is straightforward, and the perpendicular ion-ion viscosity too can be retained in the form given in Eq.(<ref>) if the hypothesis of a strong guide field is also assumed (for a recent discussion see <cit.>). Derivation of the EMHD equations follows simply from Eqs.(<ref>)-(<ref>), since ion dynamics is completely neglected <cit.>.
It can be verified that both Eq.(<ref>) and Eq.(<ref>) represent the $z$-component of electron momentum equation (Eq.<ref>) in the RMHD and EMHD regime respectively, $\psi$ and $-\nabla^2\psi$ expressing the $z$ component of the vector potential ${\bm A}$ and of the electron current density ${\bm J}$.
In RMHD, the $\rho_s^2$ contribution on the r.h.s. of Eq.(<ref>) expresses thermal effects related to electron compressibility along the magnetic field lines (see e.g. <cit.>): in the usual, strong guide field limit, $b$ is completely neglected since is ordered $b\sim \epsilon^2$ with $\epsilon\equiv |\nabla\psi|/B_z\ll 1$, and to leading order ($\sim\epsilon$) both ${\bm u}_e$ and ${\bm u}_i$ are given by the incompressible ${\bm E}\times{\bm B}$-drift velocity. As consequence, the stream function $\varphi$ corresponds to the normalized electrostatic potential while the $\rho_s^2$ term appears in the electron momentum equation as a result of the diamagnetic corrections to ${\bm u}_e$ in the Lorentz force and the $z$ component of the gyrotropic electron pressure tensor <cit.>. For this reason this term is considered to be an FLR-type contribution. However, the cancellation between the diamagnetic drift contribution to the $z$-component of ${\bm u}_e\cdot\nabla{\bm u}_{e}$ and the $z$-component of the gyrotropic pressure tensor is required in the derivation only if we do not order $\rho_s$ and $d_e$ with respect to $\epsilon$; in that case Eqs.(<ref>-<ref>) contain terms up to the second order in $\epsilon$. If instead we remember that in the slab, strong guide field, RMHD ordering, $\beta_e\sim\epsilon$ and that $\rho_s^2=\beta_ed_i^2/2$, then we may order $\rho_s^2\sim d_e^2\sim\epsilon$. This is sufficient to re-obtain Eqs.(<ref>-<ref>) even by assuming a scalar electron pressure tensor, if we disregard any contribution of order $\epsilon^4$ or higher, since from ${\bm u}_{e,\perp}\simeq {\bm E}\times{\bm B}/B^2 +{\bm \nabla}P_e\times{\bm B}/(eB^2)$ we would obtain $({\bm u}_e\times{\bm B})\cdot{\bm e}_z=[\varphi-\rho_s^2 U, \psi]$; our equations will now retain terms up to $\epsilon^3$.
In EMHD, instead, the convection velocity field (i.e. ${\bm u}_\perp^e$) appearing in the second term of Eq.(<ref>) is due to the magnetic field component $b$, since the current density is carried by electrons only, which drive the dynamics through ${\bm u}_e\propto{\bm J}\propto{\bm\nabla}\times{\bm B}$ in the incompressible regime that we consider here. As a consequence, $b$ acts as a stream function for the in-plane electron dynamics, and resistivity, when included, enters also in the equivalent of the vorticity equation. For the same reason, the in-plane components of electron momentum equation, taken in the polytropic, incompressible limit, completely close the system of EMHD equations: Eq.(<ref>) is the $z$-component of the rotational of Eq.(<ref>), and the field $W$ is proportional to the $z$-component of the electron generalized vorticity, defined by the curl of the electron fluid canonical momentum ${\bm\nabla}\times({\bm u}_e + e{\bm A}/(m_ec))$.
The EMHD equation for the electron generalized vorticity is mirrored in RMHD by the equation for the fluid vorticity alone (Eq.(<ref>)), of which $\nabla^2\varphi$ represents the $z$-component (see also <cit.>). This happens because in the Alfvénic frequency range the plasma moves at the bulk velocity ${\bm U}\simeq {\bm u}_i+O(m_e/m_i)$): Eq.(<ref>) is therefore the curl of Eq.(<ref>), under the assumption of incompressibility, which allows expression of the perpendicular fluid velocity in terms of the stream function $\varphi$. If the plasma fluid is assumed to be incompressible but without imposing the strong guide field condition, this function can not be interpeted as the electrostatic potential. With no guide field however a separate analysis would be required to include $\rho_s^2$-type contributions. The delicate point about the applicability of Eqs.(<ref>)-(<ref>) lies indeed in the validity of the incompressibility assumption and in its relationship with the ordering of the parallel fluctuations of the magnetic field, which weighs the importance of Hall's term in Ohm's law, discussed below.
§.§ Comparison with the generalized Ohm's law and Hall's term
Since reconnection models are usually discussed in relation to the non-ideal terms in Ohm's law rather than in the framework of the full two-fluid equations for ions and electrons, it is worth to make here reference also to the generalized Ohm's law, written with respect to the average plasma velocity ${\bm U}$. The standard text-book form obtained by combining Eqs.(<ref>)-(<ref>) (see e.g. <cit.>, p.91) while neglecting $O(m_e/m_i)$ corrections, reads, after normalizing again lengths to $a$ and times to $\tau_{_A}$,
$${\bm E}+{\bm U}\times {\bm B}=
d_i\frac{{\bm J}\times {\bm B}}{n} + S^{-1}{\bm J}
\begin{equation}\label{eq:Ohm}
+\frac{d_e^2}{n}\left\{\frac{\partial {\bm J}}{\partial t} +{\bm\nabla}\cdot\left(
{\bm U}{\bm J}+ {\bm J}{\bm U}-d_i\frac{{\bm J}{\bm J}}{n} \right)\right\}-\frac{\rho_s^2}{d_i}\frac{{\bm\nabla}\cdot{\bm \Pi}_e}{n}\, .
\end{equation}
Here $n=n_e=n_i$ is the average plasma density and ${\bm\Pi}_e$ is the electron pressure tensor of Eq.(<ref>), measured in the electron rest frame. The ion pressure tensor contribution is neglected since it is $O(m_e/m_i)$ smaller when the temperatures of the two species are comparable. Note that it has been recently shown by Kimura et al. (2014) <cit.> that the (often neglected) term ${\bm \nabla }\cdot({\bm J}{\bm J}/n)$ is necessary to respect energy conservation of the 1-fluid system in the collisionless limit ($S^{-1}=0$).
The generalized Ohm's law is essentially the rewriting of the electron momentum equation with respect to ${\bm U}$ and ${\bm J}$, that replace ${\bm u}_e$. We then recognize the essential difference between the dynamics of the bulk plasma and of the magnetic induction, and the role that the Hall-term ${\bm J}\times{\bm B}$ has in this: while the plasma always moves at the fluid velocity of ions, the magnetic induction evolves (with the rotational of Eq.(<ref>)) as dragged by the fluid velocity of the electrons, ${\bm u}_e=({\bm U}-d_i{\bm J}/n)$. In particular, the term ${\bm u}_e\times {\bm B}$ describes the convection of magnetic field lines by the electron fluid in the collisionless limit neglecting electron inertia. As well known <cit.>, the RMHD and EMHD sets of equations for slab reconnection without electron temperature effects may be therefore seen as two extreme limits with respect to the Hall term ($d_i$-term), in Ohm's law: the RMHD regime described by Eqs.(<ref>-<ref>) at $\rho_s=0$ corresponds to neglecting Hall's term entirely, whereas the EMHD framework is recovered when the fluid dynamics is restricted to electrons only (${\bm U}\simeq{\bm u}_i\simeq 0$), that is at scales $\ell\ll d_i$ and $\Omega_i\lesssim \omega\ll \Omega_e$, so that Eq.(<ref>) becomes the only relevant equation for our fluid system. It is however interesting to remark that in the strong guide field ordering, both ions and electrons in-plane velocities are equal at the leading order in $\epsilon$ to the ${\bm E}\times{\bm B}$-drift. By direct comparison of the $z$-component of ${\bm u}_e\times{\bm B}=({\bm U}-d_i{\bm J}/n)\times{\bm B}$ with $[\varphi-\rho_s^2U,\psi]$ (cfr. previous Section), is immediate to recognize that the Hall term survives in the ordering with $\rho_s^2\sim \epsilon$ through the diamagnetic-drift contribution to ${\bm u}_{e,\perp}$, $\rho_s^2[U,\psi]=d_i({\bm J}\times {\bm B})\cdot{\bm e}_z/n$. This expresses the balance between kinetic and magnetic pressure forces not only at equilibrium but also for the perturbations.
We conclude by recalling that, when Hall's term is retained while still considering the bulk plasma response to field evolution (i.e. ion momentum equation is not neglected, so that ${\bm J}\neq-ne{\bm u}_e$), an intermediate regime is entered, which is sometimes called “Hall-mediated reconnection” (HMR) or even “whistler mediated reconnection[ Not to be here confused with the EMHD regime of Eqs.(<ref>)-(<ref>), though there has been some ambiguous notation for different regimes in the past. Also note that, in some works, what we here name (resistive) HMR was even refered to as the “collisionless reconnection” regime (see e.g. <cit.>), due to the weak dependence from $S$ found in the Hall-dominated reconnection rate (see e.g. <cit.>). ]” <cit.>. These regimes are not of concern in this paper, since they can not be recovered in the framework of two-field models. The decoupling of ion and electron motions at the ion inertial scale (i.e. for $\ell\lesssim d_i$) requires more than two scalar fields to be retained to account for two-fluid effects (also notice that Eqs.(<ref>)-(<ref>) do not contain $d_i$ as a characteristic scale length). As discussed by Fruchtman et al. (1993) <cit.>, first, and more recently by Bian et al. (2007) <cit.> and Hosseinpur et al. (2009) <cit.>, Hall term effects are retained by relating the magnitude of $b$, as generated by Hall's term in Eq.(<ref>), to the compressible component of ${\bm U}_\perp$, absent in our incompressible model. By Helmholtz decomposition, this should enter through an irrotational contribution, ${\bm U}_\perp={\bm\nabla}\varphi\times {\bm e}_z+{\bm \nabla}\chi$, related to the scalar field $\chi$; in turn, the components $u_z^e$ and $U_z$ should also be retained. This immediately highlights the most delicate point concerning the ${\bm J}\times{\bm B}$ term in Ohm's law, already pointed out at the end of the previous Section: due to the direct relation between $b$ and $\chi$, the (in)compressibility assumption plays a major role in determining the extent of Hall physics retained in the model. Remarkably, if $\partial_z=0$, the in-plane incompressibility ${\bm\nabla}\cdot{\bm U}_\perp=0$ is admitted both in the ${\bm E}\times{\bm B}$-drift regime of the low-$\beta$ limit, where $b$ is neglected with respect to the strong guide field, and in the high-$\beta$ limit, where the large kinetic (electron) pressure implies the smallness of both ${\bm\nabla}\cdot{\bm U}=0$ and ${\bm\nabla}\cdot{\bm u}^e=0$.
The authors are grateful to Francesco Pegoraro for discussions and comments.
DDS is in debt with Maurizio Ottaviani for many interesting discussions,
and in particular for having pointed out the possible importance of the ideal tearing during the non-linear stage of primary reconnection instabilities, and with Alessandro Biancalani
for discussions about the explosive reconnection regime and for having kindly provided details about the numerical simulations performed in <cit.>. This research was partially
supported by the joint training PhD program in Astronomy, Astrophysics
and Space Science between the University of Rome “Tor Vergata” and “Sapienza”.
Characteristic plasma parameters of magnetized plasma environments where MHD reconnection may occur. Physical quantities are expressed in cgs units and temperatures are expressed in $eV$. For magnetotail reconnection parameters, typical conditions in the plasma sheet during a substorm growth phase have been considered. For the tokamak devices the value are estimated from design (ITER) or measurements (JET) near to the $q=1$ surface, whose circumference on a poloidal section gives an estimation of the typical, reconnecting current sheet length, $L$. Source for the parameters, as labelled in the Table's third row, are: <cit.> (I); <cit.> (II); <cit.> (III); <cit.> (IV).
( $\sim 1R_\odot$ )
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$B$ $10-100$ $ 10^{-4}$ $5.68\times 10^4$ $3.45\times 10^4$ $(1-3)\times 10^{2}$
$T_e$ $86$ $ 10^3-10^4$ $2\times 10^4$ $3\times 10^3$ $5-15$
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$\varepsilon_d^*\equiv (d_e/L)^2$ $ 10^{-19}- 10^{-16}$
$10^{-9}-10^{-6}$ $10^{-9}$ $10^{-5}$ $10^{-5}-10^{4}$
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1511.00271
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ListNet is a well-known listwise learning to rank model and has gained much attention in recent years. A particular problem of ListNet, however, is the high computation complexity in model training, mainly due to the large number of object permutations involved in computing the gradients. This paper proposes a stochastic ListNet approach which computes the gradient within a bounded permutation subset. It significantly reduces the computation complexity of model training and allows extension to Top-k models, which is impossible with the conventional implementation based on full-set permutations. Meanwhile, the new approach utilizes partial ranking information of human labels, which helps improve model quality. Our experiments demonstrated that the stochastic ListNet method indeed leads to better ranking performance and speeds up the model training remarkably.
§ INTRODUCTION
Learning to rank aims to learn a model to re-rank a list of objects, e.g., candidate documents in document retrieval. Recent studies show that listwise learning delivers better performance in general than traditional pairwise learning <cit.>, partly attributed to its capability of learning human-labelled scores as a full rank list. A potential disadvantage of listwise learning, however, is the high computation complexity in model training, which is mainly caused by the large number of permutations of the objects to rank.
A typical listwise learning method is the ListNet model proposed by cao2007learning. This model has been utilized to tackle many ranking problems, e.g. modeling the hiring behavior in online labor markets <cit.>, ranking sentences in document summarization <cit.>, improving detection of musical concepts <cit.> and ranking the results in video search <cit.>. Basically, ListNet implements the rank function as a neural network (NN), with the objective function set to be the cross entropy between two probability distributions over the object permutations, one derived from the human-labelled scores and the other derived from the model prediction (network output). In order to deal with the high computation complexity associated with the large number of permutations, cao2007learning proposed a Top-k approach, which clusters the permutations by the first $k$ objects, so the number of distinct probabilities that need to evaluate in model training reduces from $n!$ to $\frac{n!}{(n-k)!}$, where $n$ is the number of objects in the list.
To ensure efficiency, $k=1$ was selected in the seminal paper <cit.> and in the open source implementation of RankLib <cit.>. This Top-1 approach is a harsh approximation to the full listwise learning and may constrain the power of the ListNet method.
We therefore seek to extend the Top-1 approximation to Top-k ( k $>$ 1) models.
The major obstacle for the Top-k extension is the large number of permutations, or more precisely, permutation classes in the Top-k setting. A key idea of this paper is that the rank information involved in the permutation classes is highly redundant and so a small number such permutation classes are sufficient to convey the rank information required to train the model. Meanwhile, the partial rank information associated with the subset of permutation classes may
represent more detailed knowledge for model training, leading to better ListNet models.
Based on these two conjectures, we propose a stochastic ListNet method, which samples a subset of the permutation classes (object lists) in model training
and based on this subset to train the ListNet model. Three methods are proposed to conduct the sampling. In the uniform distribution method, the candidate objects are selected following a uniform distribution; in the fixed distribution method, the candidate objects are selected following a distribution derived from the human-labeled scores; in the adaptive distribution method, the candidates are selected following a distribution defined by the rank function, i.e., the neural network output. Experimental results demonstrated that the stochastic ListNet method can significantly reduce the computation cost in model training. In fact, if the size of the permutation subset is fixed, the computation complexity is bounded, which allows training Top-k models where $k$ is large. Meanwhile, better performance was obtained with the stochastic ListNet approach, probably due to the learning of partial rank information.
The contributions of the paper are three-fold: (1) proposes a stochastic ListNet method that significantly reduces the training complexity and delivers better ranking performance; (2) investigates Top-k models based on the stochastic ListNet, and studies the impact of a large $k$; (3) provides an open source
implementation based on RankLib.
The rest of the paper is organized as follows. Section <ref> introduces some related works, and Section <ref> presents the stochastic ListNet method. Section <ref> presents the experiments, and the paper is concluded by Section <ref>.
§ RELATED WORK
This work is an extension of the Top-k ListNet method proposed by cao2007learning. The novelty is that we propose a stochastic learing method which not only speeds up the model training but also produces stronger models. The code is based on the Top-1 ListNet implementation of RankLib <cit.>.
Another related work is the SVM-based pairwise learning to rank model based on stochastic gradient descent (SGD) <cit.>. In this approach,
training instances (queries) are selected randomly and for each query, a number of object pairs are sampled from the object list.
These pairs are used to train the SVM model. In the stochastic ListNet method proposed in this paper, the randomly selected training samples are
permutation classes (object lists) rather than pairs of objects, and a set of object lists rather than a single pair forms a training sample.
§ METHODS
§.§ Review of ListNet
The ListNet approach proposed by cao2007learning trains a neural network which predicts the scores $z^{(i)}$ of a list of candidate objects $x^{(i)}$ given a query $q^{(i)}$, formulated by $z^{(i)} = f_w(x^{(i)})$, where $f_w$ stands for the scoring function defined by the NN. The objective function is given by:
\begin{eqnarray}
\nonumber
\mathcal{L} &=& \sum_i \mathcal{L}({y}^{(i)},{z}^{(i)}) \\
\label{eq:cost}
&=& \sum_i \sum_{\forall g \in \mathscr{G}_k}P_{y^{(i)}}(g)log(P_{z^{(i)}}(g))
\end{eqnarray}
where $y^{(i)}$ denotes the human-labelled scores, and $\mathscr{G}_k$ is the set of permutation classes defined by:
\begin{eqnarray}
\nonumber
\mathscr{G}_k &=& \{\mathscr{G}_k(j_1,j_2,...,j_k) | j_t = 1,2,...,n, \\
&& \ s.t. \ j_u \ne j_v \ \ \ for \ \ \forall u \ne v\}
\label{eq:g}
\end{eqnarray}
where $n$ is the number of candidate objects, $j_t$ is the object ranked at the $t$-th position, and $\mathscr{G}_k(j_1,j_2,...,j_k)$ is a permutation class which involves all the permutations whose first $k$ objects are exactly $(j_1,j_2,...,j_k)$. Following cao2007learning, the probability of $\mathscr{G}_k(j_1,j_2,...,j_k)$ can be computed by:
\begin{equation}
\label{eq:pg}
P_{s}(\mathscr{G}(j_1,j_2,...,j_k)) = \prod_{t=1}^{k} \frac{e^{s_{j_{t}}}} {\sum_{l=t}^{n} e^{s_{j_l}}}.
\end{equation}
where $s_{j_{t}} $ is the score of object at position $j_t(t = 1, 2,,, k)$ at a certain permutation. By this definition of permutation probability, Eq. (<ref>) defines a cross entropy between the distributions over permutations (precisely, permutation classes) derived from the human-labelled scores and the NN-predicted scores. Therefore, optimizing the objective function Eq. (<ref>) with respect to the NN model $f_w$ leads to a scoring function that approximates the human-labelled ranking.
§.§ Stochastic Top-k ListNet
A particular difficulty of the Top-k ListNet method is that it requires very demanding computation in model training.
Refer to Eq. (<ref>), the permutation set $\mathscr{G}_k$ involves $\frac{n!}{(n-k)!}$ members,
and for each member, computing its probability involves $\frac{(2n-k+1)k}{2}$ summations plus $k$
multiplications and divisions. To let the algorithm practical, $k$=$1$ was selected in <cit.>,
as well as the public toolkit RankLib <cit.>. Although this is a good solution and reduces
computation dramatically, we argue that this approach largely buries the power of ListNet. In fact,
setting $k$=$1$ effectively marginalizes all the probabilities over the candidate objects of a
permutation class except the top one. By this approximation, Eq. (<ref>) reduces to a softmax over the
candidate objects, which means that it actually focuses on how the probabilities are distributed
over individual objects, rather than how the probabilities are distributed
over object lists. This potentially loses much rank information involved in the
human labels.
Another disadvantage of the Top-1 model is that it learns the rank information
of the full list, but ignores the rank information of partial sequences,
which may lead to ineffective learning.
As an example, considering an object list where the score of the most relevant object is much higher
than the scores of others, then the learning is dominated by the highest score, and largely throws
away the rank information conveyed by the scores of other objects. It would be quite helpful if
the rank information involved in partial sequences of the candidate objects can be learned. Top-k
models place distributions over object lists (in length $k$), and so can learn partial
sequences of objects.
We are interested in how to learn Top-k ($k$ $>$ 1) models while keeping the computation tractable.
To achieve the goal, we propose a stochastic ListNet approach, which
samples a small set of the Top-k permutation classes (object lists), and train the Top-k model based on this
small set instead of the full set of permutation classes. As a comparison, the full set of
permutation classes of the Top-k model is $\frac{n!}{(n-k)!}$, which is
computationally prohibitive if $k$ $>$ 1. With stochastic ListNet, a subset of the permutation classes
that involves only $l$ members are randomly selected. Training the Top-k model based on
this subset greatly reduces the computation cost, even with a large $k$. In fact, the subset approach
imposes a bound of the computation cost that is largely
determined by the the size of the subset ($l$), while independent of the total number of objects
$n$ and the model order $k$.
Interestingly, the stochastic approach offers not only quick learning, but also a chance of learning
partial ranks. This is obvious because only a subset of the object lists are selected in model training,
and so the rank information involved in the subset of the permutation classes can be learned. With the Top-1
model, partial ranks reduces to partial sequences since each object list involves only one object. As we
have discussed, learning partial sequences is an advantage of Top-k models with $k > 1$. This means that
stochastic Top-1 ListNet possesses some advantages of Top-k ListNet, while the computation cost is much lower.
§.§ Sampling methods for stochastic ListNet
The training process of stochastic ListNet starts from sampling $l$ permutation classes, or object lists. For each object list,
$k$ objects are sampled following a particular distribution.
As mentioned in Section <ref>, three distributions are studied in this paper: uniform
distribution, fixed distribution and adaptive distribution. They are presented as follows.
Uniform distribution sampling:
In this method, all the $k$ objects of a particular object list are sampled with an equal probability. This sampling method is simple but biased
towards irrelevant candidates, since there are much more irrelevant objects than relevant ones in the training data. A re-sampling
approach is proposed to remedy the bias, as will be discussed in Section <ref>.
Fixed distribution sampling:
In this method, the objects are sampled following a distribution proportional to the human-labelled scores. For instance, in the LETOR
dataset that is used in this study, each candidate object (document) is labelled as 2 (very relevant), 1 (relevant) or 0 (irrelevant). These scores are normalized by
softmax and are used as the probability distribution when sampling objects. Because the probabilities of relevant objects are larger than those of irrelevant
objects, more relevant objects would be selected by this sampling approach in model training.
Adaptive distribution sampling:
The fixed distribution sampling mentioned above relies on human-labelled scores, which may be impacted by label errors. Moreover, the absolute
values of human labels are not good measures of object relevance. To solve these problems, we choose the outputs of the `current' neural
network as the relevance scores, and sample the objects according to these scores. Note that the network
outputs are natural measures of object relevance based on the present ranking model. As the model (the neural network) keeps updated during
model training, the relevance scores are accordingly changed. In each iteration, the relevance scores are
re-calculated, and the sampling is based on the new scores in the next iteration.
§.§ Gradients with linear networks
cao2007learning optimized the ListNet model by gradient descent. For each query, the learn rule is formuated by:
\[
w = w - \eta \Delta w
\]
where $\eta$ is the learning rate, and $w$ denotes the parameters of the model $f_w$. $\Delta w$ denotes the gradient and it can be computed as follows:
\[
\Delta w = \sum_{\forall g \in \mathscr{G}_k} \frac{\partial P_{z^{(i)}(f_w)}(g)}{\partial w} \frac{P_{y^{(i)}}(g)}{P_{z^{(i)}(f_w)}(g)}.
\]
For simplicity, a linear NN model was used by cao2007learning. This has been adopted in our study as well, written by $z^{(i)} = f_w (x^{(i)}_j) = w^T x^{(i)}_j$, where $x^{(i)}_j$ denotes the feature vector of the $j$-th object of the $i$-th query. In the case of the Top-1 model, it shows that:
\begin{eqnarray}
\nonumber
\Delta w= \sum_j [\sigma(z^{(i)},j) - \sigma(y^{(i)},j)] x^{(i)}_j
\end{eqnarray}
where $\sigma(s,j)$ is the $j$-th value of the softmax function of the score vector $s$, given by:
\[
\sigma(s^{(i)},j) = \frac{e^{s^{(i)}_j}} {\sum_{t=1}^{n^{(i)}} e^{s^{(i)}_t} }.
\]
In the case of the Top-k model, the gradient(Derivative of cross entropy between $P_{z^{(i)}}$ and $P_{y^{(i)}}$ when $k$ $>=$ 2) is a bit complex, but still manageable:
\begin{equation}
\begin{aligned}
\label{eq:topk}
& \Delta w = \sum_{g \in \mathscr{G}_k} [(\prod_{t=1}^{k} \hat{\sigma}(y^{(i)}, t))\cdot \\
& (\sum_{f=1}^{k} \{ x^{(i)}_{j_f} - \sum_{v=f}^{n^{(i)}} \hat{\sigma}(z^{(i)}, v) x^{(i)}_{j_v} \})]
\end{aligned}
\end{equation}
where $\hat{\sigma}(\cdot)$ defines a `partial' softmax(The ˇ®partial softmaxˇŻ means that the $\sigma(s,f)$ has a similar form as softmax, however when computing the value for each f, the denominator is not the summation from 1 to n, instead a ˇ®partial sequenceˇŻ from f to n.), given by:
\[
\hat{\sigma}(s^{(i)}, f) = \frac{e^{s^{(i)}_{j_f}}} {\sum_{t=f}^{n^{(i)}} e^{s^{(i)}_{j_t}}}.
\]
§.§ Stochastic Top-k ListNet algorithm
We present the stochastic Top-k ListNet algorithm, by employing the techniques described above. The gradient descent (GD) approach is adopted. All the training samples are processed sequentially in an iteration. The training runs several iterations until the convergence criterion is reach. Another
detail is that the learning rate is multiplied by $0.1$ whenever the objective function is worse than the previous iteration. The procedure is illustrated in Algorithm <ref>, where $\mathcal{L}(t)$ denotes value of the objective function after the $t$-th iteration.
Stochastic Top-k ListNet
$\mathscr{D} = \{(q^{(1)}, x^{(1)}, y^{(1)}), ..., (q^{(m)}, x^{(m)}, y^{(m)}) \}$: training data
T: number of iterations
$\eta$: learning rate
Randomly initialize $w$
$t=1$ to T
$i=1$ to m
select the $i$-th training instance $(q^{(i)},x^{(i)},y^{(i)}) \in \mathscr{D}$
Sample the permutation classes $\mathscr{G}_k $
Compute $\Delta w$ according to Eq. (<ref>)
Update $f_w$: $w = w - \eta \Delta w$
$\mathcal{L}(t) < \mathcal{L}(t-1)$
$\eta = 0.1 \eta$
§ EXPERIMENTS
§.§ Data
The proposed stochastic Top-k ListNet method is tested on the document retrieval task based on the
MQ2008 dataset of LETOR 4.0 <cit.>. This database was released in early 2007 and has been widely used in learning to
rank studies. It contains queries and corresponding candidate documents. The human-labelled scores
are among three values $\{0, 1, 2\}$, representing little, medium, and strong relevance
between queries and candidate documents, respectively. The training set, validation set and test data all
contain $784$ queries. The document features used in this study include term frequency, inverse document frequency, BM25,
and language model scores for IR. Some new features proposed recently are also included, such as
HostRank, feature propagation, and topical PageRank.
§.§ Experiment Setup
In our experiments, we consider Top-k models where k = 1, 2, 3, and 4.
Although any $k$ is possible with the proposed stochastic ListNet, we will show that simply increasing the
model order $k$ does not improve performance. The P@1 and P@10 performance is used as the evaluation metric.
Specially, for all the three distribution sampling methods, the sampling process involves two steps: pre-selection
and re-sampling. The pre-selection step samples a list of documents
following three distributions mentioned above, and in the re-sampling step, document lists including more
relevant documents are retained with
a higher probability. For example, denoting the pre-selected document list by ($v_1$,$v_2$,...,$v_k$)
where $k$ is the length of the list, and denoting the corresponding human-labelled
scores by ($s_1$, $s_2$,...,$s_k$), the probability that the list is retained is given by
\[
\frac{\sum_{i=1}^{k} s_i}{kS}
\]
where $S$ is the maximum value of the human-labelled scores, which is $2$ in our case.
The re-sampling approach is designed to encourage document lists containing more relevant documents,
which is the most important for the uniform distribution sampling.
In stochastic Top-k ListNet, the learning rate is set as $10^{-3}$ for $k$ = 1, and
$10^{-5}$ for $k$ $>$ 1. These values are set to achieve the best
performance on the validation set. Another important parameter of the stochastic Top-k ListNet
approach is the number of samples of the document lists (or the size of subset of permutation classes selected),
denoted by $l$. Various settings of $l$ are experimented with in this study. To eliminate randomness
in the results, all the experiments are repeated $20$ times and the averaged performance is reported.
§.§ Experimental results
The P@1 performance on the test data with the Top-1 ListNet utilizing the three sampling approaches. The size of the permutation subset varies from 50 to 500.
The P@1 performance on the test data with the Top-2 ListNet utilizing the three sampling approaches. The size of the permutation subset varies from 5 to 500.
The P@1 performance on the test data with the Top-3 ListNet utilizing the three sampling approaches. The size of the permutation subset varies from 5 to 500.
The P@1 performance on the test data with the Top-4 ListNet utilizing the three sampling approaches. The size of the permutation subset varies from 5 to 500.
The P@1 results on the test dataset with different orders of Top-k ListNet are reported in Figure <ref> to Figure <ref>.
In each figure, the number of document lists varies from $5$ to $500$. For comparison, the results with the conventional
ListNet are also presented. Note that the re-sampling approach was not applied to the Top-1 model as we found it caused performance
reduction. This is perhaps because the sampling space is small with the Top-1 model, and so re-sampling tends to cause
over-emphasis on relevant documents.
From these results, we first observe that stochastic ListNet with either fixed or adaptive distribution
sampling tends to outperform the conventional ListNet approach, particularly with a large $k$. This confirms our argument
that rank information can be learned from a subset of the permutation classes that are randomly selected, and the partial rank learning
can lead to even better performance than the full rank learning, the case of conventional ListNet. This is an interesting result and
demonstrates the stochastic ListNet is both faster and better than the conventional ListNet. It is also seen
that the adaptive distribution sampling performs slightly better than the fixed distribution sampling.
This is not surprising as the adaptive distribution sampling uses a more reasonable relevance score (neural network output)
to balance relevant and irrelevant documents. The uniform distribution sampling performs a little worse than the other two sampling methods, probably
caused by the less informative uniform distribution.
Another observation is that in all the four figures, the performance of the stochastic ListNet
methods increases with more samples of the object lists. However if there are too many samples,
the performance starts to decrease. This can be explained by the fact that the sampling prefers
relevant documents which are more informative.
A larger sample set often includes more informative documents; however if the set is too large,
many irrelevant documents will be selected and the performance is reduced. In the case that
the number of samples is very large ($500$ for example for Top-1), the stochastic ListNet falls back to the conventional ListNet,
and their performance becomes similar.
Comparing the results with different $k$, it can be seen that a larger $k$ leads to a better performance
with stochastic ListNet. This confirms that high-order Top-k models can learn more ranking information. However,
this is not necessarily the case with the conventional ListNet. For example, the Top-2 model does not offer
better performance than the Top-1 model. This is perhaps because high-order Top-k models consider a large number
of document lists and most of them are not informative, which leads to ineffective learning. Remind that
the conventional ListNet is a special case of the stochastic ListNet with a very large sample set,
and we have discussed that an over large sample set actually reduces performance.
3c|P@1 3c|P@10
Model Top-k Sampling Time (s) Train Val. Test Train Val. Test
C-ListNet k=1 - 2.509 0.4101 0.4107 0.4119 0.2684 0.2684 0.2676
S-ListNet k=1 UDS 0.753 0.4097 0.4106 0.4120 0.2680 0.2683 0.2676
S-ListNet k=1 FDS 0.391 0.4094 0.4090 0.4127 0.2679 0.2681 0.2676
S-ListNet k=1 ADS 0.375 0.4102 0.4097 0.4121 0.2680 0.2682 0.2677
C-ListNet k=2 - 2275.5 0.4119 0.4043 0.4043 0.2678 0.2674 0.2674
S-ListNet k=2 UDS 2.898 0.4140 0.4143 0.4130 0.2682 0.2686 0.2681
S-ListNet k=2 FDS 2.410 0.4145 0.4144 0.4164 0.2684 0.2688 0.2684
S-ListNet k=2 ADS 2.013 0.4162 0.4168 0.4145 0.2686 0.2689 0.2687
S-ListNet k=3 UDS 4.358 0.4167 0.4204 0.4152 0.2686 0.2681 0.2680
S-ListNet k=3 FDS 3.997 0.4137 0.4205 0.4131 0.2687 0.2695 0.2685
S-ListNet k=3 ADS 3.483 0.4184 0.4196 0.4177 0.2692 0.2697 0.2689
S-ListNet k=4 UDS 6.161 0.4145 0.4226 0.4104 0.2686 0.2694 0.2687
S-ListNet k=4 FDS 5.773 0.4145 0.4232 0.4150 0.2690 0.2695 0.2686
S-ListNet k=4 ADS 4.358 0.4149 0.4247 0.4164 0.2692 0.2700 0.2689
Averaged training time (in seconds), P@1 and P@10 on training, validation (Val.) and test data with different Top-k methods. `C-ListNet' stands for conventional ListNet, `S-ListNet' stands for stochastic ListNet.
The averaged training time and the performance in precession are presented in Table <ref>. For precession, both P@1
and P@10 results are reported, though we focus on P@1 since it is more concerned for applications such as QA.
Note that for stochastic ListNet, the optimal number of samples (document lists) has been selected according to the P@1 performance on the validate set.
From these results, it can be seen that the conventional Top-1 ListNet is rather fast, however
the Top-2 model is thousands of times slower. With $k > 2$, the training time
becomes prohibitive and so they are not listed in the Table. This is expected since the conventional ListNet considers
the full set of permutations which is a huge number with a large $k$. With the stochastic ListNet,
the training time is dramatically reduced. Even with a large $k$, the computation cost is still manageable,
because the computation is mostly determined by the number of object lists, rather than the value of $k$.
When comparing the three sampling methods, it can be found the convergence speed of the uniform distribution approach
is the slowest, probably due to the ineffective selection for relevant documents.
The adaptive distribution sampling is the fastest, probably attributed to the collaborative update
of the model and the distribution.
As for the P@1 performance, the stochastic ListNet method generally outperforms its non-stochastic
counterpart, particularly with the adaptive distribution
sampling. For example, the best P@1 results obtained on the test data with the stochastic Top-1
ListNet is $0.4127$, which outperforms the conventional
Top-1 ListNet ($0.4119$). This advantage of stochastic ListNet, as we argued, is largely attributed to its
capability of learning partial rank information with samples of partial sequences of the rank list.
Comparing the results with different $k$ values, it can be seen that a larger $k$ tends to offer better P@1
performance on the training set, with either the conventional ListNet or the stochastic ListNet.
For example, with the conventional ListNet, the results are $0.4101$ vs. $0.4119$ with the Top-1 and Top-2
models respectively. However, the performance
gap is rather marginal, and the advantage with the large $k$ does not propagate to the results on the test data
(as has been seen in Figure <ref> and Figure <ref>).
This indicates that for the conventional ListNet, the Top-1 model is not the only choice in the sense of
computation complexity, but also the best choice in the sense of P@1 performance.
For stochastic ListNet, the performance improves with $k$ increases. In contrast to the conventional ListNet,
this improvement propagates to the results on the test data. For example, with the adaptive distribution
sampling, the P@1 results on the training set are $0.4102$ vs. $0.4184$ with the Top-1 and Top-3
models respectively, and the results on the test data are $0.4121$ vs. $0.4177$ respectively.
The P@1 performance on the test data with the stochastic Top-k ListNet approach, where $k$ varies from $1$ to $100$.
Nevertheless, the P@1 performance improvement with a large $k$ is rather marginal, and an over large $k$
simply reduces the performance. To make it clear, we vary the value of $k$ from $1$ to $100$ and plot the
P@1 results in Figure <ref>). It can be seen that larger $k$ ($> 4$) does not
offer any merit but causes performance instability, particularly with the adaptive sampling approach.
As we have discussed, with the stochastic ListNet,
partial rank information can be learned with simple Top-k models, even the Top-1 model.
This capability of partial rank learning with simple models reduces the necessity of employing
complex Top-k models. This is a highly valuable conclusion, and it suggests that a simple Top-1 or Top-2
model is sufficient for the ListNet method, if the stochastic method is applied. Considering
the trade-off between computation cost and model strength, we recommend stochastic Top-2 ListNet
which delivers better P@1 performance than the Top-1 model consistently, with sufficiently fast computing.
If more computation is affordable, stochastic Top-3 ListNet can be used to obtain better performance.
Finally, we highlight that the conclusions obtained from the P@1 results and the P@10 results perfectly match.
In fact, the P@10 results look more consistent between training and test data, and the advantage of the stochastic
approach seems more clear, particularly with the adaptive sampling. This is not surprising as the optimization
goal of ListNet is essentially to form a good rank that involves multiple candidates, and so P@10 is apt to measure
the superiority of a better rank approach.
§ DISCUSSION
An interesting observation with the stochastic ListNet approach is that sampling more relevant documents improves performance.
This can be explained by the data imbalance between relevant and irrelevant documents, i.e., there are much
more irrelevant documents than relevant documents in the training data. This imbalance leads to biased models
that tend to classify all documents as irrelevant.
The re-sampling approach can be regarded as a way of balancing the two classes, and the fixed
and adaptive distribution sampling can be regarded as another way to achieve the goal. Note that in the fixed distribution sampling,
the distribution is solely dependent on the human-labeled scores. These scores are good measures of the rank of relevance but not good
measures of the relevance itself. A possible way to solve this problem is to learn a scoring function that maps
human-labelled scores to more reasonable measures of document relevance, though we took a different way
that employs the network outputs as the relevance measures,
which is what the adaptive distribution sampling method does. Note that the network output is a natural
measure of document relevance,
so the adaptive distribution sampling works the best in our experiments.
Another related issue is the harsh labelling of the AM2008 dataset. In this dataset, documents are labelled
by only three values $\{0,1,2\}$, which is rather imprecise and the rank information is very limited. This harsh labeling is
another reason why the uniform distribution sampling does not work: by uniform distribution sampling, there is a large probability that the
sampled object lists involve documents that are all labelled by $0$. This leads to an inefficient learning. Another consequence of
the harsh labeling is that the power of complicated ranking models is largely constrained. For example, with the Top-k ($k > 1$)
ListNet model, many of the $k$ documents in a candidate list are labelled as the same score, resulting in limited rank information
for the Top-k model to learn. This is why Top-k models did not exhibit much superiority to the Top-1 model in our
experiments. We argue that top-k models would provide more contributions with
more thorough labels (e.g., scores in real values). This is an ongoing research of our group.
Finally, we highlight that the stochastic approach is not limited to the ListNet model, but any model for listwise learning. It is well known that listwise
learning outperforms pairwise learning, due to it is capability of learning full ranks <cit.>. However
learning full ranks requires unaffordable computation and so is infeasible in practice, even with the Top-k approximation. Our work demonstrated
that learning full ranks can be approximated by learning partial ranks, and a limited number samples of such partial ranks is sufficient
to convey the rank information. This stochastic learning is very fast, and even delivers better performance. It can be regarded as a general
framework that treats both the pair-wise learning and the full rank learning as two special cases. In fact, if the set of partial ranks involves
all the permutation classes, it reduces to the conventional listwise learning, and if the set of partial ranks involves all object pairs,
it resembles the pairwise learning. A wide range of listwise learning methods can benefit from the idea of stochastic learning provided
in this paper.
§ CONCLUSION
This paper proposed a stochastic ListNet method to speed up the training of ListNet models and improve the ranking performance. The basic idea is to
approximate the full rank learning by learning a small number of partial ranks. Three sampling approaches were proposed to select the partial
ranks, and Top-k ListNet models with various complexity ($k$ values) were investigated.
Our preliminary results on the MQ2008 dataset confirmed that the stochastic ListNet approach can dramatically speeds up the model training,
and more interestingly, it can produce better ranking performance than the conventional ListNet. Especially, the adaptive distribution
sampling method delivered the best P@1 performance. An appealing observation is that the simple Top-2 model is very effective and more
complex Top-k models seem not very necessary, considering the trade-off between training complexity and model strength. This
observation, however, is purely based on the MQ2008 dataset. As have been discussed, more
detailed human labels may require more complex models, for which the stochastic method proposed in this paper is essential to conduct
the model training. For the future work, we plan to study Top-k ListNet models with other databases and apply the stochastic
learning approach to other listwise learning to rank methods.
§ ACKNOWLEDGMENTS
This research was supported by the National Science Foundation
of China (NSFC) under the project No. 61371136, and the
MESTDC PhD Foundation Project No. 20130002120011. It
was also supported by Sinovoice and Pachira.
|
1511.00562
|
Raptor code ensembles with linear random outer codes in a fixed-rate setting are considered. An expression for the average distance spectrum is derived and
this expression is used to obtain the asymptotic exponent of the weight distribution.
The asymptotic growth rate analysis is then exploited to develop a necessary and sufficient condition under which the fixed-rate Raptor code ensemble exhibits a strictly positive typical minimum distance. The condition involves the rate of the outer code, the rate of the inner fixed-rate LT code and the LT code degree distribution. Additionally, it is shown that for ensembles fulfilling this condition, the minimum distance of a code randomly drawn from the ensemble has a linear growth with the block length. The analytical results can be used to make accurate predictions of the performance of finite length Raptor codes. These results are particularly useful for fixed-rate Raptor codes under ML erasure decoding, whose performance is driven by their weight distribution.
Fountain codes, Raptor codes, erasure correction, maximum likelihood decoding.
WEweight enumerator
WEFweight enumerator function
IOWEFinput output weight enumerator function
IOWEinput output weight enumerator
LTLuby Transform
BPbelief propagation
MLmaximum likelihood
MDSmaximum distance separable
LDPClow density parity check
i.i.d.independent and identically distributed
CERcodeword error rate
BECBinary Erasure Channel
ARQautomatic repeat request
TCPtransmission control protocol
LDPCLow-density parity-check
§ INTRODUCTION
channels, the first example of which was introduced in <cit.>, have attracted an increasing attention in the last decades. Originally regarded as purely theoretical channels, they turned out to be a very good abstraction model for the transmission of data over the Internet, where packets get lost randomly due to, for example, buffer overflows at intermediate routers. Erasure channels also find applications in wireless and satellite channels where deep fading events can cause the loss of one or multiple packets.
Traditionally, ARQ mechanisms have been used in order to achieve reliable communication. A good example is the TCP that is used for data transmission over the Internet. ARQ relies on feedback from the receiver and retransmissions and it is known to perform poorly when the delay between transmitter and receiver is high or when multiple receivers are present (reliable multicasting). An early work on erasure coding is <cit.>, where Reed-Solomon codes and (dense) linear random codes are proposed. However, those techniques become impractical due to their complexity already for small block lengths. More recently Tornado codes were proposed for transmission over erasure channels <cit.>. Tornado codes have linear encoding and decoding complexities (under BP decoding). However, the encoding and decoding complexities are proportional to their block lengths and not their dimension. Hence, they are not suitable for low rate applications such as reliable multicasting in which the transmitter needs to adapt its code rate to the user with the worst channel (highest erasure probability). LDPC codes have also been proposed for use over erasure channels <cit.> and they have been proved to be practical in several scenarios even under ML decoding.
Fountain codes <cit.> are erasure codes potentially able to generate an endless amount of encoded symbols. They find application in contexts where the channel erasure rate is not known a priori.
The first class of practical fountain codes, LT codes, was introduced in <cit.> together with an iterative BP decoding algorithm that is efficient when the number of input symbols $k$ is large. One of the shortcomings of LT codes is that in order to have a low probability of unsuccessful decoding, the encoding cost per output symbol has to be $\mathcal O \left(\ln(k)\right)$. Raptor codes were introduced in <cit.> <cit.> as an evolution of LT codes. They were also independently proposed in <cit.>, where they are referred to as online codes. Raptor codes consist of a serial concatenation of an outer code $\mathcal C$ (usually called precode) with an inner LT code. The LT code design can thus be relaxed requiring only the recovery of a fraction $1-\gamma$ of the input symbols with $\gamma$ small. This can be achieved with linear encoding complexity. The outer code is responsible for recovering the remaining fraction of input symbols, $\gamma$. If the outer code $\mathcal C$ is linear-time encodable, then the Raptor code has a linear encoding complexity, $\mathcal O\left( k \right)$, and therefore the overall encoding cost per output symbol is constant with respect to $k$. If BP decoding is used, the decoding complexity is also linear in the dimension $k$ and not in the blocklegth $n$, as it is the case for LDPC and Tornado codes. This leads to a constant decoding cost per symbol, regardless of the blocklength (i.e., of the rate). Furthermore, in <cit.> it was shown that Raptor codes under BP decoding are universally capacity-achieving on the binary erasure channel.
Most of the works on LT and Raptor codes consider BP decoding which has a good performance for very large input blocks ($k$ at least in the order of a few tens of thousands symbols). Often in practice, smaller blocks are used. For example, for the Raptor codes standardized in <cit.> and <cit.> the recommended values of $k$ range from $1024$ to $8192$. For these input block lengths, the performance under BP decoding degrades considerably. In this context, an efficient ML decoding algorithm in the form of inactivation decoding <cit.> may be used in place of BP. Some recent works have studied the decoding complexity of Raptor and LT codes under inactivation decoding <cit.>.
In <cit.> lower bounds on the distance and error exponent are derived for a concatenated scheme with random outer code and a fixed inner code.
In <cit.> it is shown how the rank profile of the constraint matrix of a Raptor code depends on the rank profile of the outer code parity check matrix and the generator matrix of the LT code.
In <cit.> upper and lower bounds on the bit error probability of LT and Raptor codes under ML decoding are derived. The outer codes there considered in this work are picked from a linear ensemble in which the elements of the parity check matrix are independently set to one with a given probability $p<1/2$.
This work is extended in <cit.>, where upper and lower bounds on the codeword error probability of LT codes under ML decoding are developed. Another extension of this work is <cit.> where a pseudo upper bound on the performance of Raptor codes under ML decoding is derived under the assumption that the number of erasures correctable by the outer code is small. Hence, this approximation holds only if the rate of the outer code is sufficiently high.
In <cit.> lower and upper bounds on the probability of successful decoding of LT codes under ML decoding as a function of the receiver overhead are derived, while corresponding bounds are developed in <cit.> for Raptor codes. In <cit.> finite length protograph-based Raptor-like LDPC codes are proposed for the AWGN channel.
Despite their rateless capability, Raptor codes represent an excellent solution also for fixed-rate communication schemes requiring powerful erasure correction capabilities with low decoding complexity. It is not surprising that Raptor codes are used in a fixed-rate setting by some existing communication systems (see, e.g., <cit.>). In this context, the performance under ML erasure decoding is determined by the distance properties of the fixed-rate Raptor code ensemble.
In contrast to <cit.>, in this work we consider Raptor codes in a fixed-rate setting analyzing their distance properties. In particular, we focus on the case where the outer code is picked from the linear random code ensemble. The choice of this ensemble is not arbitrary. The outer code used by the R10 Raptor code, the most widespread version of binary Raptor codes (see <cit.>), is a concatenation of two systematic codes, the first being a high-rate regular LDPC code and the second a pseudo-random code characterized by a dense parity check matrix. The outer codes of R10 Raptor codes were designed to behave as codes drawn from the linear random ensemble in terms of rank properties, but allowing a fast algorithm for matrix-vector multiplication <cit.>. Thus,
the ensemble we analyze may be seen as a simple model for practical Raptor codes with outer codes specifically designed to mimic the behavior of linear random codes. This model has the advantage to make the analytical investigation tractable. Moreover, although it is simple, the results obtained using this model allow predicting the behavior of binary Raptor codes in the standards rather accurately, as illustrated by simulation results in this paper.
For the considered Raptor ensemble we develop a necessary and sufficient condition to guarantee a strictly positive normalized typical minimum distance, that involves the degree distribution of the inner fixed-rate LT code, its rate, and the rate of the outer code. It identifies a positive normalized typical minimum distance region on the $(\ri,\ro)$ plane, where $\ri$ and $\ro$ are the inner and outer code rates. This can be used as an instrument for fixed-rate Raptor code desing. In particular, for a given overall rate $\rate$ of the fixed-rate Raptor ensemble, it allows to identify the smallest fraction of $\rate$ that has to be assigned to the outer code to obtain good distance properties. A necessary condition is also derived which, beyond the inner/outer code rates, depends on the average output degree only.
Finally we show how the analytical results presented in this paper may be used to predict the performance of finite length fixed-rate Raptor codes. This work extends the earlier conference paper <cit.>. [ In this paper we provide full proofs of all the results developed in <cit.>. More in detail, rigorous proofs of the growth rate expression (Theorem 2) and of the positive distance region (Theorem 3) are provided, together with both new results on the distance properties of the considered fixed-rate Raptor codes (Theorem 4 and Theorem 5) and performance curves of finite length codes obtained via software simulations.]
The rest of the paper is organized as follows. In Section <ref> we introduce the main definitions. Section <ref> provides the derivation of the average weight distribution of the Raptor code ensemble considered and the associated growth rate.
Section <ref> provides necessary and sufficient conditions for a linear growth of the minimum distance with the block length (positive normalized typical minimum distance).
Numerical results are presented in Section <ref>. The conclusions follow in Section <ref>.
§ PRELIMINARIES
We consider fixed-rate Raptor code ensembles based on the encoder structure depicted in Figure <ref>. The encoder is given by a serial concatenation of an $(h,k)$ outer code with an $(n,h)$ inner fixed-rate LT code. We denote by $\vecu$ the outer encoder input, and by $\vecU$ the corresponding random vector. Similarly, $\vecv$ and $\vecx$ denote the input and the output of the fixed-rate LT encoder, with $\vecV$ and $\vecX$ being the corresponding random vectors. The vectors $\vecu$, $\vecv$, and $\vecx$ are composed by $k$, $h$, and $n$ symbols respectively. The symbols of $\vecu$ are referred to as source symbols, whereas the symbols of $\vecv$ and $\vecx$ are referred to as intermediate and output symbols, respectively.
A Raptor code consists of a serial concatenation of an (outer) linear block code with an LT code.
We restrict ourselves to symbols belonging to $\field$. We denote by $\hw(\veca)$ the Hamming weight of a binary vector $\veca$. For a generic LT output symbol $x_i$, $\deg (x_i)$ denotes the output symbol degree, i.e., the number of intermediate symbols that are added (in $\mathbb F_2$) to produce $x_i$.
We will denote by $\ro=k/h$, $\ri=h/n$, and $\rate=k/n=\ro \ri$ the rates of the outer, inner LT codes.
We consider the ensemble of Raptor codes $\ensemble(\oensemble,\Omega, \ri, \ro, n)$ obtained by a serial concatenation of an outer code in the $\left(\ri n,\ro\ri n\right)$ binary linear random block code ensemble $\oensemble$, with all possible realizations of an $\left(n,\ri n\right)$ fixed-rate LT code with output degree distribution $\Omega= \{ \Omega_1, \Omega_2,\Omega_3, \ldots, \Omega_{\dmax}\}$, where $\Omega_i$ is the probability of having an output symbols of degree $i$. We also denote by $\bar \Omega$ the average output degree, $
\bar \Omega = \sum_i i\Omega_i$.
Picking randomly one code in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$ is performed by randomly drawing the parity-check matrix of the linear random outer code and the low density generator matrix of the fixed-rate LT encoder. The parity-check matrix of the outer code is obtained by drawing $(h-k)h$ i.i.d. Bernoulli uniform random variables. The generator matrix of the fixed-rate LT encoder is generated by independently drawing $n$ degrees $i$ according to the probability mass function (p.m.f.) $\Omega$ and, for each such degree $i$, by choosing uniformly at random $i$ distinct symbols out of the $h$ intermediate ones.
We make use of the notion of exponential equivalence <cit.>. Two real-valued positive sequences $a(n)$ and $b(n)$ are said to be exponentially equivalent, writing $a(n)~\doteq~b(n)$, when
\begin{equation}\label{eq:asymp_eq}
\lim_{n \to \infty} \frac{1}{n} \log_2 \frac{a(n)}{b(n)}=0.
\end{equation}
If $a(n)$ and $b(n)$ are exponentially equivalent, then
\begin{align}
\lim_{n \to \infty} \frac{1}{n} \log_2 a(n) = \lim_{n \to \infty} \frac{1}{n} \log_2 b(n).
\end{align}
Given two pairs of reals $(x_1,y_1)$ and $(x_2,y_2)$, we write $(x_1,y_1) \succeq (x_2,y_2)$ if $x_1 \geq x_2$ and $y_1 \geq y_2$.
§ DISTANCE SPECTRUM OF FIXED-RATE RAPTOR CODE ENSEMBLES
In this section we characterize the expected WE of a fixed-rate Raptor code picked randomly in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$. An expression for the expected WE is first obtained. Then, the asymptotic exponent of the WE is analyzed.
Let $A_\d$ be the expected multiplicity of codewords of weight $$̣ for a code picked randomly in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$. For $d\geq1$ we have
\begin{align}\label{eq:WEF_Raptor}
A_\d = \binom {n}{\d} 2^{-h (1-\ro)} \sum_{\l=1}^h \binom{h}{\l} \pl^\d (1-\pl)^{n-\d}
\end{align}
\begin{align}\label{eq:pl_finite}
\pl &= \sum_{j=1}^{\dmax} \Omega_j \sum_{\substack{i=\max(1,\l+j-h)\\ i~\mathrm{odd}}}^{ \min (\l,j)} \frac{ \binom{j}{i} \binom{h-j}{\l-i} } { \binom{h}{\l}} \\
&= \sum_{j=1}^{\dmax} \Omega_j \sum_{\substack{i=\max(1,\l+j-h)\\ i~\mathrm{odd}}}^{ \min (\l,j)} \frac{ \binom{\l}{i} \binom{h-\l}{j-i} } { \binom{h}{j}} \, .
\end{align}
For a serially concatenated code we have
\begin{align}
A_\d = \sum_{\l=1}^{h} \frac{\weo_{\l} \wei_{\l,\d}}{ \binom {h} {\l}}
\label{eq:we_serial}
\end{align}
where $\weo_{\l}$ is the average WE of the outer code, and $\wei_{\l,\d}$ is the average IOWE of the inner fixed-rate LT code. The average WE of an $(h,k)$ linear random code is known to be <cit.>
\begin{align}
\weo_{\l} = \binom{h}{l} 2^{-h (1-\ro)}.
\label{eq:wef_random}
\end{align}
We now focus on the average IOWE of the fixed-rate LT code. Let us denote by $l$ the Hamming weight of the input word to the LT encoder and let us denote by $\pjl$ the probability that any of the $n$ output bits of the LT encoder takes the value $1$ given that the Hamming weight of the intermediate word is $\l$ and the degree of the LT code output symbol is $j$, i.e.,
\[
\pjl:=\Pr\{X_i=1|\hw(\vecV)=\l,\deg(X_i)=j\}
\]
for any $i\in \{1,\dots,n\}$. This probability may be expressed as
\begin{align}
\pjl =
\sum_{\substack{i=\max(1,\l+j-h)\\ i~\textrm{odd}}} ^{ \min (\l,j)} \frac{ \binom{j}{i} \binom{h-j}{\l-i} } { \binom{h}{\l} } \,= \sum_{\substack{i=\max(1,\l+j-h)\\ i~\textrm{odd}}} ^{ \min (\l,j)} \frac{ \binom{\l}{i} \binom{h-\l}{j-i} } { \binom{h}{j}}
\label{eq:p_j_l}
\end{align}
Removing the conditioning on $j$ we obtain $\pl$, the probability of any of the $n$ output bits of the fixed-rate LT encoder taking value $1$ given a Hamming weight $l$ for the intermediate word, i.e.,
\[
\pl:=\Pr\{X_i=1|\hw(\vecV)=l\}
\]
for any $i\in \{1,\dots,n\}$.
We have
\begin{align}
\pl = \sum_{j=1}^{\dmax} \Omega_j \pjl.
\label{eq:p_l}
\end{align}
Since the output bits are generated by the LT encoder independently of each other, the Hamming weight of the LT codeword conditioned to an intermediate word of weight $l$ is a binomially distributed random variable with parameters $n$ and $\pl$. Hence, we may write
\begin{align}
\Pr\{\hw(\vecX) = \d | \hw(\vecV) = \l\}
=\binom {n}{\d} \pl^\d (1-\pl)^{n-\d}.\label{eq:distr_weight_LT}
\end{align}
The average IOWE of a LT code may now be easily calculated multiplying (<ref>) by the number of weight-$\l$ intermediate words, yielding
\begin{align}
\wei_{\l,\d}= \binom {h}{\l} \binom {n}{\d} \pl^\d (1-\pl)^{n-\d}.
\label{eq:iowef_lt}
\end{align}
Making use of (<ref>), (<ref>) and (<ref>), we obtain (<ref>).
As opposed to $A_d$ with $d \geq 1$, whose expression is given by (1), the expected number of codewords of weight $0$, $A_0$, is given by
\begin{align*}
A_0 &= 1 + \sum_{l=1}^h \frac{\weo_{\l} \wei_{\l,0}}{ \binom {h} {\l}} \\
&= 1 + 2^{-n \ri (1 - \ro) } \sum_{l=1}^h \binom {h} {\l} (1 - \pl)^n \, .
\end{align*}
An expected number of weight-$0$ codewords larger than one is related to the fact that we have a nonzero probability that the $h \times n$ generator matrix of the fixed-rate LT code is not full-rank. This matrix, in fact, is generated “online” in the standard way for LT encoding, i.e., by drawing $n$ i.i.d. discrete random variables with p.m.f. $\Omega$, representing the weights of the $n$ columns. For each such column, the corresponding `$1$' entries are placed in random positions. It will be shown in Section <ref>, Theorem <ref>, that if the $(\ri,\ro)$ pair belongs to the region there called “positive normalized typical minimum distance region”, the expected number $A_0$ of zero weight codewords approaches $1$ (exponentially) as $n$ increases.
Next we compute the asymptotic exponent (growth rate) of the weight distribution for the ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$, that is the ensemble $\msr{C}(\oensemble,\Omega, \ri, \ro, n)$ in the limit where $n$ tends to infinity for constant $\ri$ and $\ro$.
Hereafter, we denote the normalized output weight of the Raptor encoder by $\nd = \d/n$ and the normalized output weight of the outer code (input weight to the LT encoder) by $\nl = \l/h$. The growth rate is defined as
\begin{align}
\G(\nd) = \lim_{n \to \infty} \frac{1}{n} \log_2 \we_{\nd n} \, . \label{eq:growth_rate_def}
\end{align}
The asymptotic exponent of the weight distribution of the fixed-rate Raptor code ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$ is given by
\begin{align}\label{eq:growth_rate}
\G(\nd) = \Hb(\nd) - \ri (1-\ro) + \fmax(\nd)
\end{align}
\begin{align}\label{eq:max}
\fmax(\nd) := \max_{ \nl \in \mathscr D_{\nl}} \f(\nd, \nl),
\end{align}
being $\f(\nd, \nl)$ and $\mathscr D_{\nl}$ defined as follows,
\begin{align}\label{eq:f}
\f(\nd, \nl) := \ri \Hb(\nl) + \nd \log_2 \npnl + (1- \nd) \log_2 \left(1 - \npnl\right) \, ,
\end{align}
\begin{align}
\mathscr D_{\nl} = \left\{ \begin{array}{cl} (0,1) & \textrm{if } \Omega_j = 0 \textrm{ for all even } j\\
(0,1] & \textrm{otherwise} \, , \end{array} \right.
\end{align}
with $\npnl$ defined as
\begin{align}
\npnl := \frac{1}{2} \sum_{j=1}^{\dmax} \Omega_j \left[ 1-\left( 1-2\nl\right)^j \right].
\label{eq_npnl}
\end{align}
Let us define $\mathbb N^*_h = \{1,2,\dots,h\}$. From (<ref>) we have
\begin{align} \label{eq:proof_G}
& \frac{1}{n} \log_2 A_{\delta n} \notag \\
% &= \frac{1}{n} \log_2 \left( {n \choose d} 2^{-n \ri (1- \ro)} \sum_{l=1}^h {h \choose l} \pl^d (1-\pl)^{n-d} \right) \\
&=\frac{1}{n} \log_2 {n \choose \delta n} - \ri (1- \ro) + \frac{1}{n} \log_2 \sum_{l=1}^h {h \choose l} \pl^d (1-\pl)^{n-d} \notag \\
&\stackrel{\mathrm{(a)}}{\leq} \Hb(\delta) -\frac{1}{2n} \log_2 \left(2 \pi n \delta (1-\delta)\right) - \ri (1- \ro) \\
&+ \frac{1}{n} \log_2 \sum_{l=1}^h {h \choose l} \pl^d (1-\pl)^{n-d} \notag \\
&\stackrel{\mathrm{(b)}}{\leq} \Hb(\delta) -\frac{1}{2n} \log_2 \left(2 \pi n \delta (1-\delta)\right) - \ri (1- \ro) \\
&+ \frac{1}{n}\log_2 ( \ri n) %\notag \\
+ \frac{1}{n} \log_2 \max_{l \in \mathbb N^*_{h-1}} \left\{ {h \choose l} \pl^d (1-\pl)^{n-d} \right\} \notag \\
&\stackrel{\mathrm{(c)}}{\leq} \Hb(\delta) -\frac{1}{2n} \log_2 (2 \pi n \delta (1-\delta)) - \ri (1- \ro) + \frac{1}{n}\log_2 ( \ri n) \notag \\
&+ \max_{l \in \mathbb N^*_{h-1}} \left\{ \ri \Hb\left(\frac{l}{h}\right) - \frac{1}{2n}\log_2\left(2 \pi \ri n\frac{l}{h}\left(1-\frac{l}{h}\right)\right) \right.\\
& + \delta \log_2 \pl + (1-\delta) \log_2(1-\pl) \Big\} \notag \\
&= \Hb(\delta) -\frac{1}{2n} \log_2 (2 \pi n \delta (1-\delta)) - \ri (1- \ro) + \frac{1}{n}\log_2 ( \ri n) \notag \\
&+ \max_{\nl \in \left\{\frac{1}{\ri n},\dots,\frac{\ri n -1}{\ri n}\right\}} \left\{ \ri \Hb\left(\nl\right) - \frac{1}{2n}\log_2\left(2 \pi \ri n \nl \left(1- \nl\right)\right) \right.\\
&+ \delta \log_2 p_{\ri n \lambda} + (1-\delta) \log_2(1-p_{\ri n \lambda}) \Big\}
\end{align}
Inequality $\mathrm{(a)}$ follows from the well-known tight bound <cit.>
\begin{align}
{n \choose \sigma n} \leq \frac{2^{n \Hb(\sigma)}}{\sqrt{2 \pi n \sigma (1-\sigma)}}, \qquad 0<\sigma<1
\label{eq:gallagher_upper}
\end{align}
while $\mathrm{(b)}$ from
\begin{align}
\sum_{l=1}^h {h \choose l} \pl^d (1-\pl)^{n-d} \leq h \max_{l \in \mathbb N^*_h} {h \choose l} \pl^d (1-\pl)^{n-d}
\label{eq:proof_G_summ}
\end{align}
and from the fact that the maximum cannot be taken for $l=h$ for large enough $h=\ri n$, hence for large enough $n$ (as shown next). Inequality $\mathrm{(c)}$ is due again to (<ref>), to $\log_2(\cdot)$ being a monotonically increasing function, and to $1/n$ being a scaling factor not altering the result of the maximization with respect to $l$.
That the maximum is not taken for $l=h$, for large enough $h$, may be proved as follows. By direct calculation of (<ref>) for $l=h$ and $l=h-1$ it is easy to show that we have
\begin{align*}
\p_h = \sum_{\substack{j=1\\ j~\textrm{odd}}}^{d_{\max}} \Omega_j ~~ \mathrm{and}
\qquad \p_{h-1} = \sum_{\substack{j=1\\ j~\textrm{odd}}}^{d_{\max}} \frac{h-j}{h} \Omega_j + \sum_{\substack{j=1\\ j~\textrm{even}}}^{d_{\max}} \frac{j}{h} \Omega_j \, .
\end{align*}
Since $\p_{h-1}/\p_h \rightarrow 1$ for increasing $h$, there exists $h_0(\Omega)$ such that
\begin{align}
h\, \p_{h-1}^d (1-\p_{h-1})^{n-d} > \p_h^d (1-\p_h)^{n-d}
\end{align}
for all $h>h_0(\Omega)$. Hence, for all such values of $h$ the maximum cannot be taken at $l=h$.
Next, by defining
\begin{align}
\hat{\lambda}_n &= \mathop{\mathrm{argmax}}_{\lambda \in \left\{\frac{1}{\ri n}, \frac{2}{\ri n}, \dots, \frac{\ri n-1}{\ri n} \right\} } \Big\{ \ri H_b(\lambda) - \frac{1}{2n}\log_2(2 \pi \ri n\lambda(1-\lambda)) \\
& + \delta \log_2 \p_{\ri n \lambda} + (1-\delta) \log_2(1-\p_{\ri n \lambda}) \Big\}
\label{eq:lambda_hat}
\end{align}
the right-hand side of (<ref>) may be recast as
\begin{align}
&\Hb(\delta) -\frac{1}{2n} \log_2 \left(2 \pi n \delta (1-\delta)\right) - \ri (1- \ro) + \frac{1}{n}\log_2 ( \ri n) \notag \\
& + \ri \Hb(\hat{\lambda}_n) - \frac{1}{{2n}}\log_2(2 \pi \ri n\hat{\lambda}_n(1-\hat{\lambda}_n)) \\
&+ \delta \log_2 p_{\ri n \hat{\lambda}_n} + (1-\delta) \log_2(1-p_{\ri n \hat{\lambda}_n}) \, .
\end{align}
The two terms $\frac{1}{2n} \log_2 (2 \pi n \delta (1-\delta))$ and $\frac{1}{n} \log_2 (\ri n)$ in the last expression converge to zero as $n \rightarrow \infty$. Moreover, also the term $\frac{1}{{2n}}\log_2 (2 \pi \ri n\hat{\nl}_n(1-\hat{\nl}_n))$ converges to zero regardless of the behavior of the sequence $\hat{\lambda}_n$. In fact, it is easy to check that the term $\frac{1}{{2n}}\log_2(2 \pi \ri n\hat{\lambda}_n(1-\hat{\lambda}_n))$ converges to zero in the limiting cases $\hat{\lambda}_n=\frac{1}{\ri n}$ $\forall n$ and $\hat{\lambda}_n = \frac{\ri n-1}{\ri n}$ $\forall n$, so it does in all other cases.
Developing the right hand side of (<ref>) further, for large enough $n$, we have
\begin{align}\label{eq:proof_G_2}
& \Hb(\delta) -\frac{1}{2n} \log_2 (2 \pi n \delta (1-\delta)) - \ri (1- \ro) + \frac{1}{n}\log_2 ( \ri n) \notag \\
&+ \max_{\nl \in \left\{\frac{1}{\ri n},\dots,\frac{\ri n -1}{\ri n}\right\}} \left\{ \ri \Hb\left(\nl\right) - \frac{1}{2n}\log_2\left(2 \pi \ri n \nl \left(1- \nl\right)\right) \right.\\
&+ \delta \log_2 p_{\ri n \lambda} + (1-\delta) \log_2(1-p_{\ri n \lambda}) \bigg\} \notag \\
& \stackrel{\mathrm{(d)}}{\leq} \Hb(\delta) -\frac{1}{2n} \log_2 (2 \pi n \delta (1-\delta)) - \ri (1- \ro) + \frac{1}{n}\log_2 ( \ri n) \notag \\
& + \sup_{\nl \in \mathbb Q \cap (0,1)} \bigg\{ \ri \Hb\left(\nl\right) - \frac{1}{2n}\log_2\left(2 \pi \ri n \nl \left(1- \nl\right)\right) \\
&+ \delta \log_2 \left(\npnl + \frac{K}{n} \right) + (1-\delta) \log_2 \left(1-\npnl + \frac{K}{n} \right) \bigg\} \notag \\
& \stackrel{\mathrm{(e)}}{=} \Hb(\delta) -\frac{1}{2n} \log_2 (2 \pi n \delta (1-\delta)) - \ri (1- \ro) + \frac{1}{n}\log_2 ( \ri n) \notag \\
& + \sup_{\nl \in (0,1)} \bigg\{ \ri \Hb\left(\nl\right) - \frac{1}{2n}\log_2\left(2 \pi \ri n \nl \left(1- \nl\right)\right) \\
&+ \delta \log_2 \left(\npnl + \frac{K}{n} \right) + (1-\delta) \log_2 \left(1-\npnl + \frac{K}{n} \right) \bigg\} \\
&:= \Gamma_n (\nd).
\end{align}
where $\mathbb Q$ is the set of rational numbers. Inequality $\mathrm{(d)}$ follows from the fact that, as it can be shown, $|\npnl - p_{\ri n \lambda}|<K/n$ (uniformly in $\lambda$) for large enough $n$ and from the fact that the supremum over $\mathbb Q \cap (0,1)$ upper bounds the maximum over the finite set $\left\{ \frac{1}{\ri n}, \dots, \frac{\ri n - 1}{\ri n} \right\}$. Equality $\mathrm{(e)}$ is due to the density of $\mathbb Q$. In equality $\mathrm{(e)}$, the function of $\nl$ being maximized is regarded as a function over the real interval $(0,1)$ (i.e., $\lambda$ is regarded as a real parameter).
The upper bound (<ref>) on $\frac{1}{n} \log_2 A_{\nd n}$ is valid for any finite but large enough $n$. If we now let $n$ tend to infinity, all inequalities $\mathrm{(a)}$–$\mathrm{(d)}$ are satisfied with equality. In particular: for $\mathrm{(a)}$ this follows from the well-known exponential equivalence ${n \choose \nd n} \doteq 2^{n \Hb(\nd)}$; for $\mathrm{(b)}$ from the exponential equivalence $\sum_l 2^{n f(l)} \doteq \max_l 2^{n f(l)}$; for $\mathrm{(c)}$ from ${\ri n \choose \hat{\nl}_n \ri n} \doteq 2^{n \Hb(\hat{\nl}_n)}$ (due to $\frac{1}{{2n}}\log_2 (2 \pi \ri n\hat{\lambda}_n(1-\hat{\lambda}_n))$ vanishing for large $n$); for $\mathrm{(d)}$ from the fact that, asymptotically in $n$, applying the definition of limit we can show that the maximum over the set $\left\{\frac{1}{\ri n}, \dots, \frac{\ri n -1}{\ri n} \right\}$ upper bounds the supremum over $\mathbb Q \cap (0,1)$ (while at the same time being upper bounded by it for any $n$). The expression of $\npnl$ is obtained by assuming $n$ tending to $\infty$ using the expression of $\pl$. Alternatively, the same expression is obtained by assuming $n$ tending to $\infty$ and letting an output symbol of degree $i$ choose its $i$ neighbors with replacement.
By letting $n$ tend to infinity and by cancelling all vanishing terms, we finally obtain the statement. Note that we can replace the supremum by a maximum over $\mathscr D_{\nl}$ as this maximum is always well-defined.[In fact, for any $\nd \in [0,1]$ the function $\f(\nd, \nl)$ diverges to $-\infty$ as $\nl \rightarrow 0^+$. Moreover, it diverges to $-\infty$ as $\nl \rightarrow 1^-$ if $\Omega_j=0$ for all even $j$ and converges as $\nl \rightarrow 1^-$ otherwise. Finally, for all $\delta \in [0,1]$ it is continuous for all $\nl \in \mathscr D_{\nl}$.]
The next two lemmas, which will be useful in the sequel, characterize the derivative of the growth rate function. For the sake of clarity, we use the notation $\np(\nl)$ instead of $\np_{\nl}$.
The derivative of the growth rate of the weight distribution of a fixed-rate Raptor code ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$ is given by
\begin{equation}
G'(\nd) = \log_2 \frac{1-\nd}{\nd} + \log_2 \frac{\np(\nlo)}{1-\np(\nlo)} \, \nonumber
\end{equation}
\begin{align}\label{eq:lo_def}
\nlo(\nd) := \argmax_{\nl \in D_{\nl}} \left\{ \f(\nd, \nl) \right\} \, .
\end{align}
Let us rewrite the expression of $G(\nd)$ in (<ref>) as ${\G(\nd)=\Hb(\nd) - \ri(1-\ro) + \f(\nd,\nlo(\nd))}$. We must have
\begin{equation}\label{eq:critical_point}
\frac{\partial \f}{\partial \nl} (\nd, \nlo) = 0 \, .
\end{equation}
Taking the derivative with respect to $\nd$, after elementary algebraic manipulation we obtain
\begin{align*}
G'(\nd) &= \log_2\frac{1-\nd}{\nd} + \log_2 \frac{\np(\nlo)}{1-\np(\nlo)} + \frac{\partial \f}{\partial \nl} (\nd, \nlo) \, \frac{\mathrm d \nlo}{\mathrm d \nd}
\end{align*}
which, applying (<ref>), yields the statement.
For all $0< \nd < 1/2$, the derivative of the growth rate of the weight distribution of a fixed-rate Raptor code ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$ fulfills
\[
\]
By imposing $G'(\nd)=0$, from Lemma <ref> we obtain
\frac{1-\nd}{\nd} = \frac{1-\varrho(\lambda_0)}{\varrho(\lambda_0)}
which implies $\nd = \varrho(\lambda_0)$ since the function $(1-x)/x$ is monotonically decreasing for $x \in (0,1)$. Next, due to the definition of $\lambda_0$ in (<ref>) we know that the partial derivative $\partial \mathsf f(\nd,\lambda) / \partial \lambda$ must be zero when calculated for $\lambda=\lambda_0$. The expression of this partial derivative is
\frac{\partial \mathsf f}{\partial \lambda}(\nd,\lambda) = \mathsf r_{\mathsf i} \log_2 \frac{1 - \lambda}{\lambda} + \frac{\varrho'(\lambda)}{\log 2} \cdot \frac{\nd - \varrho(\lambda)}{\varrho(\lambda)(1-\varrho(\lambda))} \,
so we obtain
\mathsf r_{\mathsf i} \log_2 \frac{1 - \lambda_0}{\lambda_0} + \frac{\varrho'(\lambda_0)}{\log 2} \cdot \frac{\nd - \varrho(\lambda_0)}{\varrho(\lambda_0)(1-\varrho(\lambda_0))} = 0 \, .
As shown above, for any $\nd$ such that $G'(\nd)=0$ we have $\nd=\varrho(\lambda_0)$. Substituting in the latter equation we obtain $\lambda_0=1/2$ which implies $\nd=\varrho(1/2)=1/2$. Therefore, the only value of $\nd$ such that $G'(\nd)=0$ is $\nd=1/2$. Due to continuity of $G'(\nd)$ and to the fact that $G'(\nd) \rightarrow +\infty$ as $\nd \rightarrow 0^+$ (as shown in Subsection B of Appendix <ref>) we conclude that $G'(\nd)>0$ for all $0 < \nd < 1/2$.
The normalized typical minimum distance of an ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$ is the real number
\begin{align*}
\dmint := \begin{cases}
0 & \text{if } \lim_{\nd \to 0^+} \G(\nd) \geq 0 \\
\inf \{ \nd>0 : \G(\nd) > 0 \} & \text{otherwise.}
\end{cases}
\end{align*}
Fig. <ref> shows $\G(\nd)$ for the ensemble $\msr{C}_{\infty}(\oensemble,\Omega^{(1)}, \ri, \ro)$, where $\Omega^{(1)}$ is the output degree distribution used in the standards <cit.>, <cit.> (see details in Table <ref>) and $\ro=0.99$ for three different $\ri$ values. The growth rate, $\G(\nd)$, of a linear random code ensemble with rate $\rate=0.99$ is also shown. It can be observed how the curve for $\ri = 0.95$ does not cross the $x$-axis, the curve for $\ri = 0.88$ has $\dmint=0$ and the curve for $\ri=0.8$ has $\dmint=0.0005$.
Growth rate vs. normalized output weight $\nd$. The continuous line shows the growth rate of a linear random code with rate $\rate=0.99$. The dot-dashed, dashed, and dotted lines show the growth rates $\G(\nd)$ of the ensemble $\msr{C}_{\infty}(\oensemble,\Omega^{(1)}, \ri, \ro=0.99)$ for $\ri=0.95$, $0.88$ and $0.8$, respectively.
Fig. <ref> shows the overall rate $\rate$ of the Raptor code ensemble $\msr{C}_{\infty}(\oensemble,\Omega^{(1)}, \ri~=~\rate/\ro, \ro)$ versus the normalized typical minimum distance $\dmint$. It can be observed how, for constant overall rate $\rate$, $\dmint$ increases as the outer code rate $\ro$ decreases. It also can be observed how decreasing $\ro$ allows to get closer to the asymptotic Gilbert-Varshamov bound.
Overall rate $\rate$ vs. the normalized typical minimum distance $\dmint$. The continuous line represents the asymptotic Gilbert-Varshamov bound. The markers represent Raptor codes ensembles $\msr{C}_{\infty}(\oensemble,\Omega^{(2)}, \ri=\rate/ \ro, \ro)$ with different outer code rates, $\ro$.
§ TYPICAL DISTANCE RATE REGIONS
In this section we aim at determining under which conditions the ensemble $\msr{C}_{\infty}(\oensemble,\Omega,\ri,\ro)$ exhibits good normalized typical distance properties. More specifically, given a distribution $\Omega$ and an overall rate $\rate$, we are interested in the allocation of the rate between the outer code and the fixed-rate LT code to achieve a strictly positive normalized typical minimum distance.
[Positive normalized typical minimum distance region] We define the positive normalized typical minimum distance region of an ensemble $\msr{C}_{\infty}(\oensemble,\Omega,\ri,\ro)$ as the set $\region$ of code rate pairs $\left( \ri, \ro \right)$ for which the ensemble possesses a positive normalized typical minimum distance. Formally:
\begin{align}
\region:=\left\{(\ri,\ro) \succeq (0,0) | \dmint(\Omega, \ri,\ro)> 0 \right\}
\nonumber
\end{align}
where we have used the notation $\dmint= \dmint(\Omega, \ri,\ro)$ to emphasize the dependence on $\Omega$, $\ri$ and $\ro$.
The positive normalized typical distance region for an LT output degree distribution $\Omega$ is developed in the following theorem.
The region $\region$ is given by
\begin{align}
\region &:=\left\{\left( \ri, \ro \right) \succeq (0,0) | \ri (1-\ro) \right.\\
& \quad \quad \quad> \max_{ \nl \in \mathscr D_{\nl}} \left\{\ri \Hb(\nl) + \log_2 \left(1 - \npnl\right)\right\}
\bigg\}\, .
\label{eq:theorem_region}
\end{align}
See Appendix <ref>.
The next two theorems characterize the distance properties of a fixed-rate Raptor code with linear random outer code picked randomly in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$ with $(\ri, \ro)$ belonging to $\region$.
Let the random variable $\D$ be the minimum nonzero Hamming weight in the code book of a fixed-rate Raptor code picked randomly in an ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$. If $(\ri, \ro) \in \region$ then
\begin{align*}
\lim_{n \rightarrow \infty} \Pr \{ \D \leq \nd n \} = 0
\end{align*}
exponentially in $n$, for all $0 < \nd < \dmint$.
It is well known that this probability can be upper bounded via union bound as
\begin{align}
\label{eq:pr_dmin}
\Pr\{ \D \leq \nd n\} & \leq \sum_{w=1}^{\nd n} A_w.
\end{align}
We will start by proving that the sequence $A_{\d}$ is non-decreasing for $\d < n/2$ and sufficiently large $n$. As $n \rightarrow \infty$, the expression $\frac{1}{n} \log_2 \frac{A_{\nd n}}{A_{\nd n -1}}$ converges to $\Gamma_n (\nd) - \Gamma_n (\nd -\frac{1}{n})$, being $\Gamma_n(\nd)$ given in (<ref>). From Lemma <ref> we know that $\G'(\nd)> 0$ for $0 < \nd < 1/2$. As $n \rightarrow \infty$, from Theorem <ref> we have $\Gamma_n(\nd) \rightarrow \G(\nd)$. Hence, for sufficiently large $n$, $\Gamma_n (\nd) \geq \Gamma_n (\nd -\frac{1}{n})$, and $A_{\d}$ is non decreasing.
We can now write
\begin{align}
\Pr\{ \D \leq \nd n\} \leq \nd n A_{\nd n} \leq\nd n 2^{n \Gamma_n(\nd) }\, ,
\end{align}
where we have used $A_{\nd n}\leq 2^{n \Gamma_n(\nd) }$, being $\Gamma_n(\nd)$ given in (<ref>).
As $n \rightarrow \infty$ we have $\Gamma_n(\nd) \rightarrow \G(\nd)$. Moreover, $\G(\nd)<0$ for all $0 < \nd < \dmint$, provided $(\ri,\ro) \in \region$. Hence, $\Pr\{ \D \leq \nd n\}$ tends to $0$ exponentially on $n$.
As from Theorem <ref>, we have an exponential decay of the probability to find codewords with weight less than $\dmint n$ when the $(\ri, \ro)$ pair belongs to the region $\region$. Such an exponential decay shall be attributed to the presence of the linear random outer code characterized by a dense parity-check matrix, which makes the growth rate function monotonically increasing for the values of $\nd$ for which it is negative.
As a comparison, for LDPC code ensembles characterized by a positive normalized typical minimum distance, the growth rate function starts from $\G(0)=0$ with negative derivative, reaches a minimum, and then increases to cross the $x$-axis. In this case, for $\nd < \dmint$ the sum in the upper bound is dominated by those terms corresponding to small values of $w$, yielding either a polynomial decay (as for Gallager's codes <cit.> ) or even $\Pr \{ D \leq \nd n \}$ tending to a constant (as it is for irregular unstructured LDPC ensembles <cit.>).
Let the random variable $\Z$ be the multiplicity of codewords of weight zero in the code book of a fixed-rate Raptor code picked randomly in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$. If $(\ri, \ro) \in \region$ then
\begin{align*}
\Pr \{ \Z > 1 \} \rightarrow 0 \quad \textrm{as } n\rightarrow \infty \, .
\end{align*}
In order to prove the statement we have to show that the probability measure of any event $\{ \Z = t \}$ with $t \in \mathbb N \setminus \{0, 1\}$ vanishes as $n \rightarrow \infty$. We start by analyzing the behavior of $\mathbb E[\Z]=A_0$, whose expression is $\mathbb E[\Z] = 1 + 2^{-n \ri (1 - \ro) } \sum_{l=1}^h \binom {h} {\l} (1 - \pl)^n$. Using an argument analogous to the one adopted in the proof of Theorem <ref>, for large enough $n$ we have
\begin{align*}
\frac{1}{n} \log_2 \left( 2^{-n \ri (1 - \ro) } \sum_{l=1}^h \binom {h} {\l} (1 - \pl)^n \right) \leq \Xi_n
\end{align*}
\begin{align*}
\Xi_n &:= - \ri (1 - \ro) + \frac{1}{n}\log_2(\ri n) + \sup_{\nl \in (0,1)} \Big\{ \ri \Hb(\nl)\\
& - \frac{1}{2n} \log_2 \left( 2 \pi \ri n \nl ( 1 - \nl) \right)
+ \log_2 (1 - \np_{\nl} + K/n) \Big\} \, .
\end{align*}
Therefore we can upper bound $\mathbb E[\Z]$ as $\mathbb E [\Z] \leq 1 + 2^{n \Xi_n}$ which, if $(\ri,\ro) \in \region$, implies $\mathbb E[\Z] \rightarrow 1$ exponentially as $n \rightarrow \infty$ due to $\Xi_n \rightarrow \G(0)$ and $\G(0)<0$.[It is worth noting that $\G(\nd)$ is right-continuous at $\nd=0$. This follows from the expression of $\G(\nd)$ proved in Theorem <ref> and from the fact that $\fmax(\nd)$ is right-continuous at $\nd=0$ as shown in the proof of Theorem <ref>.] Next, it is easy to show that $\mathbb E[\Z] \geq 1$ and, via linear programming, that the minimum is attained if and only if $\Pr \{\Z=1\}=1$ and $\Pr\{\Z=t\}=0$ for all $t \in \mathbb N \setminus \{0,1\}$. Since in the limit as $n \rightarrow \infty$ of $\mathbb E[\Z] \rightarrow 1$, we necessarily have a vanishing probability measure for any event $\{ \Z = t \}$ with $t \in \mathbb N \setminus \{0, 1\}$.
From Theorem <ref> and Theorem <ref>, a fixed-rate Raptor code picked randomly in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$ is characterized with probability approaching $1$ as $n \rightarrow \infty$ by a minimum distance at least equal to $\dmint n$ and by an encoding function whose kernel only includes the all-zero length $k$ message (hence bijective).
In the following we introduce an outer region to $\region$ that only depends on the average output degree.
The positive normalized typical minimum distance region $\region$ of a fixed-rate Raptor code ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$ fulfills $\region \subseteq \outer$, where
\begin{equation}
\outer := \left\{(\ri,\ro) \succeq (0,0) | \ri \leq \min \left( \phi(\ro), \frac{1}{\ro}\right) \right\}
\label{eq:outer_bound_2}
\end{equation}
\[
\phi(\ro)=
\begin{cases}
\frac{\bar \Omega \log_2 (1/\ro)}{\Hb(1-\ro) -(1-\ro)} \qquad 1 > \ro> \ro^* \\
1/\ro \qquad \qquad \qquad otherwise
\end{cases},
\]
being $\ro^*$ the only root of $\Hb(1-\ro) -(1-\ro)$ in $\ro \in (0,1)$, numerically $\ro^* \approx 0.22709$.
See Appendix <ref>.
Degree distributions $\Omega^{(1)}$, defined in <cit.> and $\Omega^{(2)}$, defined in <cit.>
Degree $\Omega^{(1)}$ $\Omega^{(2)}$ 0pt2.6ex [-0.9ex]0pt0pt
${1}$ 0.0098 0.0048
${2}$ 0.4590 0.4965
${3}$ 0.2110 0.1669
${4}$ 0.1134 0.0734
${5}$ 0.0822
${8}$ 0.0575
${9}$ 0.0360
${10}$ 0.1113
${11}$ 0.0799
${18}$ 0.0012
${19}$ 0.0543
${40}$ 0.0156
${65}$ 0.0182
${66}$ 0.0091
$\bar \Omega$ 4.6314 5.825 0pt2.6ex [-0.9ex]0pt0pt
In Fig. <ref> we show the positive normalized typical minimum distance region, $\region$ for $\Omega^{(1)}$ and $\Omega^{(2)}$ (see Table <ref>) together with their outer bound $\outer$. It can be observed how the outer bound is tight in both cases except for inner codes rates close to $\ri=1$. The figure also shows several isorate curves, along which the rate of the Raptor code $\rate=\ri~\ro$ is constant. For example, in order to have a positive normalized typical minimum distance and an overall rate $\rate=0.95$, the figure shows that the rate of the ouer code must lay below $\ro<0.978$ for both distributions. Let us assume we want to design a fixed-rate Raptor code, with degree distribution $\Omega^{(1)}$ or $\Omega^{(2)}$, overall rate $\rate=0.95$ and for a given length $n$, which we assume to be large. Different choices for $\ri$ and $\ro$ are possible. If $\ro$ is not chosen as $\ro<0.978$ the average minimum distance of the ensemble will not grow linearly on $n$. Hence, many codes in the ensemble will exhibit high error floors even under ML erasure decoding.
Positive growth rate region. The solid lines with black markers represent the positive growth-rate $\region$ and the dashed lines with white markers represents its outer bound $\outer$. The gray dashed lines represent isorate curves for different rates $\rate$.
§ FINITE-LENGTH RESULTS
In this section experimental results are presented to validate the analytical results obtained in the previous sections. By means of examples we illustrate how the developed results can be used to make accurate statements about the performance of fixed-rate Raptor code ensembles in the finite length regime.
Furthermore, we provide some results that show a tradeoff between performance and decoding complexity.
Finally we present some simulation results that show that the results obtained for linear random outer codes are a fair approximation for the results obtained with the standard R10 Raptor outer code (see <cit.>).
§.§ Results for Linear Random Outer Codes
In this section we will consider Raptor code ensembles $\ensemble(\oensemble,\Omega^{(1)}, \ri, \ro, n)$ for different values of $\ri$, $\ro$, and $n$ but keeping the overall rate of the Raptor code constant to $\rate=0.9014$. Fig. <ref> shows the boundary of $\region$ and $\outer$ for LT distribution $\Omega^{(1)}$ together with an isorate curve for $\rate=0.9014$. The markers along the isorate curve in the figure represent the two different combinations of $\ri$ and $\ro$ that will be considered in this section. The first point ($\ri=0.9155$, $\ro=0.9846$), marked with an asterisk, is inside but very close to the boundary of $\region$ for $\Omega^{(1)}$. We will refer to ensembles corresponding to this point as bad ensembles. The second point, ($\ri=0.9718$, $\ro=0.9275$) marked with a triangle, is inside and quite far from the boundary of $\region$ for $\Omega^{(1)}$. We will refer to ensembles corresponding to this point as good ensembles.
growth rate region
minimum distance
In the upper figure the solid and dashed lines represent the positive growth rate region of $\Omega^{(1)}$ its outer bound. The dashed-dotted line represents the isorate curve for $\rate=0.9014$ and the markers represent two different points along the isorate curve
with the same rate $\rate$ but different values of $\ri$ and $\ro$.
The lower figure shows the typical minimum distance $\dmintt$ as a function of the blocklength $n$ for ensembles with $\ro= 0.9275$ and $\ro=0.9846$ and $\rate=0.9014$. The markers represent $\dmintt$ whereas the lines represent $\dmint n$.
In order to analyze ensembles of finite length Raptor codes it is useful to introduce a notion of minimum distance for finite length.
The typical minimum distance, $\dmintt$ of an ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$ is defined as the integer number
\begin{align*}
\dmintt := \begin{cases}
0 & \mkern-48mu \text{if } A_0 > 1 + 1/2 \\
\max \{ d\geq0 : \left(\sum_{i=0}^{d} \we_i -1\right) < 1/2 \} & \text{otherwise.}
\end{cases}
\end{align*}
This definition will come in handy when we expurgate Raptor code ensembles. In fact, at least half of the codes in the ensemble will have a minimum distance of $\dmintt$ or larger. The equivalent of $\dmintt$ in the asymptotic regime is $\dmint$, the (asymptotic) normalized minimum distance of the ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$. For sufficiently large $n$ one expects that $\dmintt$ converges to $\dmint n$. Fig. <ref> shows $\dmintt$ and $\dmint n$ as a function of the blocklength $n$. It can be observed how the good ensemble has a larger typical minimum distance than the bad ensemble. In fact for all values of $n$ shown in Fig. <ref> $\dmintt=0$ for the bad ensemble. We can also see how already for small values of $n$ the $\dmintt$ and $\dmint n$ are very similar. Hence, the result of our asymptotic analysis of the minimum distance holds already for small values of $n$.
The expression of the average weight enumerator in Theorem <ref> can be used in order to upper bound the average CER over a BEC with erasure probability $\epsilon$, <cit.>. However, the upper bound proposed in <cit.> needs to be slightly modified to take into account codewords of weight $0$. We have
\begin{align}
\label{eq:bound_Gavg}
&\Bbb{E}_{\ensemble(\oensemble,\Omega, \ri, \ro, n)} \left[P_B(\epsilon)\right]\leq
P^{(\mathsf S)}_{B}(n,k,\epsilon) \nonumber \\
& + \sum_{e=1}^{n-k} {n \choose e} \epsilon^e (1-\epsilon)^{n-e} \min \left\{1, \sum_{w=1}^e {e \choose w} \frac{\we_w}{{n \choose w}}\right\} + \we_0-1
\end{align}
where $P^{(\mathsf S)}_{B}(n,k,\epsilon)$ is the Singleton bound
\begin{equation}\label{eq:bound_S}
P^{(\mathsf S)}_{B}(n,k,\epsilon)= \sum_{e=n-k+1}^n {n \choose e} \epsilon^e (1-\epsilon)^{n-e}.
\end{equation}
Considering Raptor codes in a fixed-rate setting also allows us to expurgate Raptor code ensembles as it was done in <cit.> for LDPC code ensembles. Let us consider an integer
$\ds \geq 0$ so that
\begin{align}
\label{eq:pr_d_min_ex}
\Pr\{ \dmin \leq \ds\} & \leq \sum_{w=0}^{\ds} A_w -1 = \theta < 1/2.
\end{align}
We can define the expurgated ensemble $\ensemble^{\text{ex}}(\oensemble,\Omega, \ri, \ro, n, \ds)$ as the ensemble of codes in the ensemble $\ensemble(\oensemble,\Omega, \ri, \ro, n)$ whose minimum distance is $\dmin >\ds$. The expurgated ensemble will contain a fraction at least $1 - \theta>1/2$ of the codes in the original ensemble. From <cit.> it is known that the average WE of the expurgated ensemble can be upper bounded by:
\begin{align*}
\we^{\text{ex}}_d
\begin{cases}
\leq 2 \we_d & \text{if } d > \ds \\
= 0 & \text{if } 1 \leq d \leq\ds \\
\end{cases}
\end{align*}
For each ensemble considered in this section $6000$ codes[For clarity of presentation only 300 codes are shown in the figures.] were selected randomly from the ensemble. For each code Monte Carlo simulations over a BEC were performed until $40$ errors were collected or a maximum of $10^5$ codewords were simulated. We remark that the objective here was not so much characterizing the performance of every single code but rather to characterize the average performance of the ensemble.
Fig. <ref> shows the CER vs the erasure probability $\epsilon$ for two ensembles with $\rate=0.9014$ and $k=128$ that have different outer code rates, $\ro=0.9275$ (good ensemble) and $\ro=0.9846$ (bad ensemble). The good ensemble is characterized by a typical minimum distance $\dmintt=2$ whereas the bad ensemble is characterized by $\dmintt=0$ (cf. Fig. <ref>).
For the two ensembles the upper bound in (<ref>) holds for average CER. However, the performance of the codes in the ensemble shows a high dispersion due to the short blocklength ($n=142$). In fact in both ensembles there are codes with minimum distance equal to zero which have CER$=1$ (around $1\%$ for the good ensemble and $30\%$ for the bad ensemble).
Comparing Fig. <ref> and Fig. <ref> one can easily see how the fraction of codes performing close to the random coding bound is larger in the good ensemble than in the bad ensemble. For the good ensemble Fig. <ref> shows also an upper bound on the average CER for the expurgated ensemble with $\ds=1$, that has a lower error floor.
For the bad ensemble no expurgated ensemble can be defined (no $\ds\geq 0$ exists that leads to $\theta<1/2$ in (<ref>)).
$\ro=0.9275$, $\rate=0.9014$
$\ro=0.9846$, $\rate=0.9014$
Codeword error rate CER vs erasure probability $\epsilon$ for two ensembles with $\rate=0.9014$ and $k=128$ but different values of $\ro$. The solid, dashed and dot-dashed lines represent respectively the Singleton bound, the Berlekamp random coding bound and the upper bound in. The dotted line represents the upper bound for the expurgated ensemble for $\ds=1$. The markers represent the average CER of the ensemble and the thin gray curves represent the performance of the different codes in the ensemble, both obtained through Monte Carlo simulations.
Fig. <ref> shows the CER vs $\epsilon$ for two ensembles using the same outer code rates as in Fig. <ref> but this time for $k=256$. It can be observed how the CER shows somewhat less dispersion than for $k=128$. If we compare Fig. <ref> and Fig. <ref> we can see how for the good ensemble ($\ro=0.9275$) the error floor is much lower for $k=256$ than for $k=128$, due to an increase in the typical minimum distance. In fact, whereas for $k=128$ there were some codes with minimum distance zero for $256$ we did not find any code with minimum distance zero out of the $6000$ codes which were simulated. For the good ensemble it is possible again to considerably lower the error floor by expurgation.
However, comparing Fig. <ref> and Fig. <ref> we can see how the error floor is approximately the same for $k=128$ and $k=256$, because in both cases the typical minimum distance is zero.
$\ro=0.9275$, $\rate=0.9014$
$\ro=0.9846$, $\rate=0.9014$
Codeword error rate CER vs erasure probability $\epsilon$ for two ensembles with $\rate=0.9014$ and $k=256$ but different values of $\ro$. The solid, dashed and dot-dashed lines represent respectively the Singleton bound, the Berlekamp random coding bound and the upper bound. The dotted line represents the upper bound for the expurgated ensemble for $\ds=2$. The markers represent the average CER of the ensemble and the thin gray curves represent the performance of the different codes in the ensemble, both obtained through Monte Carlo simulations.
So far we have only considered the CER performance under ML decoding. In practical systems one needs to consider decoding complexity as well.
When inactivation decoding is used the decoding throughput is largely determined by the number of inactivations needed for decoding <cit.>, since the decoding complexity is cubic in the number of inactivations. Fig. <ref> shows the averaged number of inactivations needed for ensembles of Raptor codes with output degree distribution section.
It can be observed how the good ensembles ($\ro=0.9275$) need more inactivations than bad ensembles ($\ro=0.9846$). Hence, the better CER performance obtained by using an outer code with lower rate comes at the cost of a higher decoding complexity.
Average number of inactivations vs. erasure probability of the channel. The dot markers stand for an outer code rate $\ro=0.9275$ and the square markers for $\ro=0.9846$. The solid line stands for $k=128$ and the dashed line for $k=256$.
§.§ Comparison with Raptor Codes with a Standard R10 Outer Code
In this section we illustrate by means of a numerical example how the results obtained for linear random outer code closely approximate the results with the standard R10 Raptor outer code <cit.><cit.>. We consider Raptor codes with an LT degree distribution $\Omega(x) = 0.0098 x + 0.4590 x^2 + 0.2110 x^3 + 0.1134 x^4 + 0.2068 x^5$. Fig. <ref> shows the positive growth rate region for such a degree distribution (assuming a linear random outer code) and the three different rate points, two of which are inside the region $\region$ while the third one lays outside. The $(\ri,\ro)$ rate pairs for the three points are specified in the figure caption.
Fig. <ref> shows the average CER obtained through Monte Carlo simulations for the ensembles of Raptor codes with $k=1024$, output degree distribution $\Omega(x)$ and two different outer codes, the standard R10 outer code and a linear random outer code. For the three rate points considered the average CER using the standard outer code and a linear random outer code are very close.
As it can be observed, the error floor behavior of the Raptor code ensemble with R10 outer code is in agreement with the position of the corresponding point on the $(\ri,\ro)$ plane with respect to the $\region$ region, although this region is obtained using the simple linear random outer code model. For rate points inside $\region$ the error floor is much lower, and it tends to become lower the further the point is from the boundary of $\region$.
positive growth rate region
CER results
The upper figure shows the positive growth rate region for the degree distribution $\Omega(x) = 0.0098 x + 0.4590 x^2 + 0.2110 x^3 + 0.1134 x^4 + 0.2068 x^5$. The markers represent three different rate points all of them with $\ro=1024/1096$ but with different inner code rates, $\ri=1096/1100$, $\ri=1096/1205$ and $\ri=1096/1250$. The lower figure shows the average CER for Raptor code ensembles using $\Omega(x)$ as output degree distribution and two different outer codes, the standard outer code of R10 Raptor codes and a linear random outer code, (l.r.) in the legend.
§ CONCLUSIONS
In this work we have considered ensembles of binary fixed-rate Raptor codes which use linear random codes as outer codes. We have derived the expression of the average WE of an ensemble and the expression of the growth rate of the WE as functions of the rate of the outer code and the rate and degree distribution of the inner LT code.
Based on these expressions we are able to determine necessary and sufficient conditions to have Raptor code ensembles with a positive typical minimum distance. A simple necessary condition has been developed too, which only requires (besides the inner and outer code rates) the knowledge of the average output degree. Simulation results have been presented that demonstrate the applicability of the theoretical results obtained for finite length Raptor codes. Moreover, simulation results have been presented that show that the performance of Raptor codes with linear random outer codes is close to that of Raptor codes with the standard outer code of R10 Raptor codes. Thus, we speculate that the results obtained for Raptor codes with linear random outer codes hold as first approximation for standard R10 Raptor codes.
The work presented in this paper helps to understand the behavior of fixed-rate Raptor codes under ML decoding and can be used to design Raptor codes for ML decoding. Despite the fact that only the fixed-rate setting has been considered, we speculate that Raptor code ensembles with a good fixed-rate performance will have also a good performance in a rateless setting.
Although only binary Raptor codes have been considered, the authors believe that the work can be extended to non-binary Raptor codes with a limited effort.
§ PROOF OF THEOREM <REF>
We will first prove that for all $(\ri,\ro)$ pairs in $\region$ we have a
positive normalized typical minimum distance. Then we will prove that this is not possible for any other $(\ri,\ro)$ pair.
§.§ Proof of Sufficiency
A sufficient condition for a positive normalized typical minimum distance is
\begin{equation}
\lim_{\nd \to 0^+} \G(\nd) < 0
\end{equation}
which, from Theorem <ref>, is equivalent to
\begin{equation}
\ri (1-\ro) > \lim_{\nd \to 0^+} \max_{ \nl \in \mathscr D_{\nl}} \f(\nd, \nl).
\label{eq:lim_G}
\end{equation}
As done in Lemma <ref> and Lemma <ref>, let us use the notation $\np(\nl)= \npnl$ to emphasize the dependence on $\nl$. We now show that
\begin{align} \label{eq:lim_max}
\lim_{\nd \to 0^+} \max_{ \nl \in \mathscr D_{\nl}} \f(\nd, \nl) &= \max_{ \nl \in \mathscr D_{\nl}} \lim_{\nd \to 0^+} \f(\nd, \nl) %\\
= \max_{ \nl \in \mathscr D_{\nl}} \left[ \ri \Hb(\nl) + \log_2 \left(1 - \np(\nl) \right) \right]
\end{align}
that is we can invert maximization with respect to $\lambda$ and limit as $\delta \rightarrow 0^+$, so that the region $\region$ in (<ref>) is obtained.
This fact is proved by simply showing that
\begin{align}
\lim_{\nd \rightarrow 0^+} \fmax(\nd) = \fmax(0),
\end{align}
that is the function $\fmax(\nd)=\max_{ \nl \in \mathscr D_{\nl}} \f(\nd,\nl)$ is right-continuous at $\nd = 0$. It suffices to show
\begin{align}\label{eq:max_a_b}
\fmax(\nd) = \max_{\nl \in (a, b)} \f(\nd, \nl)
\end{align}
where $(a,b)$ is an interval independent of $\nd \in [0,\frac{1}{2})$ and such that the function
\begin{align}
\log_2 \np(\nl)-\log_2(1-\np(\nl))
\end{align}
is bounded over it, i.e.,
\sup_{\nl \in (a,b)} \left| \log_2 \npnl - \log_2(1-\npnl) \right| = K \, .
In fact, under these conditions we have uniform convergence of $\f(\nd,\nl)$ to $\f(0,\nl)$ in the interval $(a,b)$ as $\nd \rightarrow 0^+$, namely,
\begin{align}\label{eq:uniform_convergence}
\f(0,\nl) - K \nd \leq \f(\nd,\nl) \leq \f(0,\nl) + K \nd, \\
\quad \quad \quad \forall \lambda \textrm{ s.t. } a < \nl < b \,
\,
\end{align}
The second inequality in (<ref>) implies $\fmax(\nd) \leq \fmax(0) + K \nd$. Moreover, denoting by $\hat{\nl} \in (a,b)$ the maximizing $\nl$, we have
\fmax(0) - K \nd = \f(0,\hat{\nl}) - K \nd \leq \f(\nd,\hat{\nl})
which implies $\fmax(0) - K \nd \leq \fmax(\nd)$. So we have
\fmax(0) - K \nd \leq \fmax(\nd) \leq \fmax(0) + K \nd
which yields $\lim_{\nd \rightarrow 0^+} \fmax(\nd) = \fmax(0)$ as desired.
Next, we prove (<ref>). We first observe that
in the case $\Omega_j=0$ for all even $j$ (in which case $\np(\nl)$ is strictly increasing) by direct computation we have $\partial\, \f(\nd,\nl) / \partial \nl < 0$ for all $0 \leq \nd < 1/2$ and for all $1/2 \leq \nl < 1$. Hence in this case we can take $b=1/2$. In all of the other cases there exists $\xi$ such that $\np(\nl) \leq \xi < 1$ for all $0 < \nl < 1$ and we can take $b=1$. The existence of $0 < a < 1/2$ (independent of $0 \leq \nd < 1/2$) such that the maximum is not taken for all $0 < \nl \leq a$ is proved as follows. Denoting $c = \log_2 e$ and $\np'(\nl)=\mathrm d \np(\nl) / \mathrm d \nl$, we have
\begin{align*}
\frac{\partial\, \f(\nd,\nl)}{\partial \nl} = \ri \log_2(1-\nl) - \ri \log_2 \nl \\
+ c\, \nd \, \frac{\np'(\nl)}{\np(\nl)} - c\,(1-\nd) \frac{\np'(\nl)}{1-\np(\nl)} \, .
\end{align*}
Since $0 < \np'(\nl) < +\infty$ for all $0 < \nl \leq 1/2$ and since
\begin{align*}
\lim_{\nl \rightarrow 0^+} \ri (1-\np(\nl)) (\log_2 (1-\nl) - \log_2 \nl) = + \infty \, ,
\end{align*}
there exists $a > 0$ such that
\begin{align*}
\ri(1-\np(\nl)) (\log_2 (1-\nl) -\log_2 \nl) > c\, \np'(\nl), \\
\quad \quad \textrm{for all } 0 < \nl < a \, .
\end{align*}
This latter inequality implies
\begin{align*}
\ri(1-\np(\nl)) (\log_2 (1-\nl) -\log_2 \nl) > c\, \np'(\nl) - \nd \frac{c\, \np'(\nl)}{\np(\nl)}, \\
\qquad \textrm{for all } 0 < \nl < a
\end{align*}
uniformly with respect to $\nd \in [0,1/2)$, which is equivalent to $\partial\, \f(\nd,\nl) /\partial \nl > 0 $ for all $0 < \nl < a$, independently of $\nd \in [0,1/2)$. Therefore the maximum cannot be taken between $0$ and $a$, with $a$ independent of $\nd \in [0,1/2)$.
§.§ Proof of Necessity
So far we have proved that the condition on $(\ri,\ro)$ expressed by Theorem <ref> is sufficient to have a positive normalized typical minimum distance. Now we need to show that this condition is also necessary. We need to prove that for the ensemble $\msr{C}_{\infty}(\oensemble,\Omega, \ri, \ro)$ all rate pairs $(\ri,\ro)$ such that $\lim_{\nd \rightarrow 0^+}\G(\nd)=0$, the derivative of the growth rate at $0$ is positive, $\lim_{\nd \rightarrow 0^+} G'(\nd) > 0$.
According to Lemma <ref> the expression of $G'(\nd)$ corresponds to
\[
G'(\nd) = \log_2 \frac{1-\nd}{\nd} + \log_2 \frac{\np(\nlo)}{1-\np(\nlo)} \, .
\]
Hence, since $G'(\nd)$ is the sum of two terms the first of which diverges to $+\infty$ as $\nd \rightarrow 0^+$, a necessary condition for the derivative to be negative is that the second term diverges to $-\infty$, i.e., $\lim_{\nd \rightarrow 0^+}\np(\nlo)=0$. This case is analyzed in the following lemma.
If $\np (\nl)=0$ then $\nl \in \{ 0, 1 \}$ in case the LT distribution $\Omega$ is such that $\Omega_j=0$ for all odd $j$, and $\nl=0$ for any other LT distribution $\Omega$.
Let us recall that $\np(\nl)$ is the probability that the LT enconder picks an odd number of nonzero intermediate bits (with replacement) given that the intermediate codeword has Hamming weight $\nl h$. If $\Omega_j > 0$ for at least one odd $j$, then the only case in which a zero LT encoded bit is generated with probability $1$ is the one in which the intermediate word is the all-zero sequence. If $\Omega_j=0$ for all odd $j$, there is also another case in which a nonzero bit is output by the LT encoder with probability $1$, i.e., the case in which the intermediate word is the all-one word.
Consider now a pair $(\ri,\ro)$ such that $\lim_{\nd \rightarrow 0^+}\G(\nd)=0$. For a fixed-rate Raptor code ensemble corresponding to this pair, we have a positive typical minimum distance if and only if $\lim_{\nd \rightarrow 0^+} G'(\nd)<0$.
By Lemma <ref> this implies $\lim_{\nd \rightarrow 0^+} \nlo(\nd)=0$ when $\Omega_j>0$ for at least one odd $j$. It implies either $\lim_{\nd \rightarrow 0^+} \nlo(\nd)=0$ or $\lim_{\nd \rightarrow 0^+} \nlo(\nd)=1$ otherwise. That $\nlo(\nd)$ cannot converge to $0$ follows from the proof of sufficiency (as shown, the maximum for $\delta \in [0,1/2)$ is taken for $\lambda > a >0$). To complete the proof we now show that, in the case where $\Omega_j=0$ for all odd $j$, assuming $\lim_{\nd \rightarrow 0^+} \nlo(\nd)=1$ leads to a contradiction.
In case $\Omega_j=0$ for all odd $j$, a Taylor series for $\np(\nl)$ around $\nl=1$ is $\np(\nl) = \bar \Omega (1-\nl) + o(\nl)$. Assuming $\lim_{\nd \rightarrow 0^+} \nlo(\nd)=1$, we consider the left-hand side of (<ref>) and calculate its limit as $\nd \rightarrow 0^+$. We obtain
\begin{align*}
& \lim_{\nd\rightarrow 0^+} \frac{\partial \f}{\partial \nl} (\nd, \nlo) \\
&= \ri \lim_{\nlo\rightarrow 1^-} \log_2 \frac{1-\nlo}{\nlo} \\
&+ \lim_{\nd \rightarrow 0^+} \! \left( \frac{\nd}{\log 2} \, \frac{\np'(\nlo)}{\np(\nlo)} - \frac{1-\nd}{\log 2} \, \frac{\np'(\nlo)}{1-\np(\nlo)} \right) \\
&= \ri \lim_{\nlo\rightarrow 1^-} \log_2 \frac{1-\nlo}{\nlo} + \frac{1}{\log 2} \lim_{\nd \rightarrow 0^+} \frac{\np'(\nlo)(\nd-\np(\nlo))}{\np(\nlo)(1-\np(\nlo))} \\
&= \ri \lim_{\nlo\rightarrow 1^-} \log_2 \frac{1-\nlo}{\nlo} + \frac{1}{\log 2} \lim_{\nd \rightarrow 0^+} \frac{\bar \Omega (1- \nlo) - \nd}{1-\nlo}
\end{align*}
where the last equality follows from the above-stated Taylor series. According to (<ref>), the last expression must be equal to zero, a constraint which requires the second limit to diverge to $+\infty$ (as the first limit diverges to $-\infty$). This, however, cannot be fulfilled in any case when $\nd$ converge to zero and $\nlo$ to one. In fact, using standard Landau notation, when $1-\nlo = \Theta(\nd)$ or $\nd = \mathrm o(1-\nlo)$ the second limit converges, while when $1-\nlo = \mathrm o(\nd)$ it diverges to $-\infty$.
§ PROOF OF THEOREM <REF>
The proof consists of deriving a lower bound for $\G(\nd)$ and evaluating it for $\nd \to 0^+$. To derive a lower bound for $\G(\nd)$ we
first derive a lower bound for $\we_{\nd}$. Observing (<ref>) we see how $\we_{\nd}$ is obtained as a summation over all possible intermediate Hamming weights. A lower bound to $\we_{\nd}$ can be obtained by limiting the summation to the term $\nls=1-\ro$ yielding to
\begin{align}
\we_{\nd n} &\geq \frac{\weo_{\nls h} \wei_{\nls h,\nd n}}{ \binom {h} {\nls h}} %\nonumber\\
= \weo_{\nls h} \Q_{\nd n,\nls h}\label{eq:appC:truncation}
\end{align}
where we have introduced
\[
\Q_{\nd n,\nl h}:=\frac{\wei_{\nl h,\nd n}}{ \binom {h} {\nl h}}
\]
representing the probability that the inner encoder outputs a codeword with Hamming weight $\nd n$ given that the encoder input has weight $\nl h$.
Hence, we can write
\begin{align}
\G(\nd) &\geq \lim_{n \to \infty} \frac{1}{n} \log_2 \weo_{\nls h} \Q_{\nd n,\nls h} \nonumber \\
& = \lim_{n \to \infty} \frac{1}{n} \log_2 \weo_{\nls h} + \lim_{n \to \infty} \frac{1}{n} \log_2 \Q_{\nd n,\nls h} \\
&= \ri \left(\Hb(\nls) - (1-\ro) \right) + \lim_{n \to \infty} \frac{1}{n} \log_2 \Q_{\nd n,\nls h}
\label{eq:app_upper_G}
\end{align}
We shall now lower bound $\lim_{\nd \to 0^+} \Q_{\nd n,\nl h}$. We denote by
\[
\]
Note that $q_{j,\l}=1-\pjl$. We have that
\begin{align}
\lim_{\nd \to 0^+} \Q_{\nd n,\nl h} & = \left(\sum_j \Omega_j q_{j,\nl h}\right)^n %\nonumber \\
\geq \left(\sum_j \Omega_j \underline{q}_{j,\nl h}\right)^n
\end{align}
with $\underline{q}_{j,\l}\leq q_{j,\l}$ being the probability that the $j$ intermediate symbols selected to encoder $X_i$ are all zero.
For large $h$, we have
\[
\underline{q}_{j,\l}=\left( 1- \frac{\l}{h}\right)^j.
\]
Denoting by $\underline{q}_{\l}=\sum_j \Omega_j \underline{q}_{j,\l}$, we have by Jensen's inequality
\[
\underline{q}_{\l} \geq \left( 1- \frac{\l}{h}\right)^{\avgd}.
\]
We have thus that
\begin{equation}
\lim_{\nd \to 0^+} \Q_{\nd n,\nl h} \geq \left( 1- \nl\right)^{n\avgd}. \label{eq:appC:Qbound}
\end{equation}
Replacing (<ref>) in (<ref>) and recalling that $h=n\ri$ we get
\begin{align}
\G(\nd) &\geq \ri \left(\Hb(\nls) - (1-\ro) \right) + \lim_{n \to \infty} \frac{1}{n} \log_2 \left( 1- \nls\right)^{n\avgd} \\
&= \ri \left(\Hb(\nls) - (1-\ro) \right) + \avgd \log_2 \left( 1- \nls\right) \\
&= \ri \left(\Hb(1-\ro) - (1-\ro) \right) + \avgd \log_2 \ro
\end{align}
If we now impose the $\G(\nd)=0$ we obtain:
\[
\phi(\ro)= \frac{\bar \Omega \log_2 (1/\ro)}{\Hb(1-\ro) -(1-\ro)} \, .
\]
This expression is only valid when the denominator is negative, that is, for $1>\ro>\ro^*$, being $\ro^*$ the only root of the denominator in $\ro \in (0,1)$, whose approximate numerical value is $\ro^* \approx 0.22709$.
§ ACKNOWLEDGEMENTS
The authors would like to to thank Prof. Massimo Cicognani for the useful discussions about the proof of Theorem 3.
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1511.00412
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This paper examines the verification of stability, a control requirement, over discrete control systems represented as Simulink diagrams, using different model checking approaches and tools.
Model checking comprises the (exhaustive) exploration of a model of a system, to determine if a requirement is satisfied.
If that is not the case, examples of the requirement's violation within the system's model are provided, as witnesses.
These examples are potentially complementary to previous work on automatic theorem proving, when a system is not proven to be stable, but no proof of instability can be provided.
We experimentally evaluated the suitability of four model checking approaches to verify stability on a set of benchmarks including linear and nonlinear, controlled and uncontrolled, discrete systems, via Lyapunov's second method or Lyapunov's direct method.
Our study included symbolic, bounded, statistical and hybrid model checking, through the open-source tools NuSMV, UCLID, S-TaLiRo and SpaceEx, respectively.
Our experiments and results provide an insight on the strengths and limitations of these model checking approaches for the verification of control requirements for discrete systems at Simulink level.
We found that statistical model checking with S-TaLiRo is the most suitable option to complement our previous work on automatic theorem proving.
§ INTRODUCTION
The verification of control systems is a timely need, especially for complex autonomous systems interacting with people, such as autonomous cars, or service robots.
Verification processes allow gaining confidence and gathering evidence that the designed systems work according to functional requirements <cit.> or that they are dependable.
Requirements are grouped into safety (“nothing bad ever happens”) or liveness (“something good eventually happens”).
Control systems need to be verified at all phases in the design process from conceptual ideas to code implementation.
Along these phases, different requirements need to be considered for verification.
Whereas at the initial and mathematical abstract design phases, theoretical control and functional requirements such as stability or robustness need to be verified, further requirements such as absence of runtime errors (arrays out of bounds, variable overflows) and floating-point issues arise at code implementation, the lowest levels.
It is important to: (a) verify requirements early, at the right level of abstraction and optimal design-to-implementation phase; and (b) follow a “design and implementation for verification” philosophy, to facilitate the use of verification techniques and tools.
As control systems differ according to their target systems (e.g., linear, nonlinear, stochastic, noisy, partially observable), there is a growing need to understand what existing verification techniques can deliver for the verification of these systems.
In our work, we have focused on the verification of discrete control systems modelled at Simulink level from difference equations, assumed to lay in between a pen-and-paper theoretical design phase, and a code implementation derived automatically through MATLAB or developed by hand.
We look into the use and combination of well established verification techniques (testing, theorem proving, model checking), control systems theory, and accomplished and supported tools, targeting these discrete systems.
Previously, we investigated the combination of two verification techniques, theorem proving and numerical tests <cit.>, to verify control requirements such as stability in the most automatic manner possible, and “for all possible variable values and initial conditions”.
Theorem proving allows computing a mathematical proof of the requirements in a symbolic and static manner, i.e., without running simulations.
However, if the theorem proving fails to compute a proof, no additional information is provided, making debugging difficult.
In contrast, other verification techniques such as model checking and testing provide witnesses or evidence of a violation of the requirements.
Nevertheless, both are computationally expensive, as model checking exhaustively explores a model, and testing might require a large number of simulations to achieve an acceptable level of coverage of the system.
Transparency in the systems to observe relevant variables and parameters, and the formulation of the requirement at the right abstraction level (e.g., referring to elements in a Simulink model) and in a quantifiable manner, if possible, are paramount in “design for verification”.
Hence, control requirements in natural language, like “stability”, require translation into a metric in terms of a system's variables and parameters.
The translation process requires specialist knowledge, such as control systems theory, and a degree of control over the implementation of a Simulink model.
In our previous work <cit.>, we made use of assertions in the form of Simulink blocks, to express the requirements to verify at the Simulink diagram level.
We incorporated Lyapunov functions into the Simulink diagrams, to assess stability.
Although Lyapunov functions can be computed through well established procedures, more than one might be suitable for the same system.
Automated verification procedures, such as model checking or automatic theorem proving, would help to establish, according to the proposed Lyapunov functions, if a system is stable or that no stability guarantees can be provided.
The latter can be caused by a proposed function that does not behave as a Lyapunov function, or if the system is indeed unstable.
Translating the system into a suitable model for model checking is a challenging task, as some model checking tools only function with finite-state transition or hybrid models.
In this paper we experimentally investigated the feasibility to verify stability for discrete systems using model checking.
Model checking has diversified to handle systems with many states and continuous components (e.g., hybrid systems). We set out to compare four different model checking approaches in a systematic manner.
Additionally, we sought to find out whether model checking can complement previously proposed theorem proving methodologies, such as the ones proposed in <cit.>, by providing evidence of requirement satisfaction or violation in the form of witnesses.
Our study included state-of-the-art symbolic <cit.>, bounded (BMC) <cit.>, statistical (sampling-based) <cit.> and hybrid <cit.> model checking approaches, through representative tools (model checkers): NuSMV[http://nusmv.fbk.eu/], UCLID[http://uclid.eecs.berkeley.edu/], S-TaLiRo[https://sites.google.com/a/asu.edu/s-taliro/s-taliro], and SpaceEx[http://spaceex.imag.fr/], respectively.
These tools were chosen due to their announced compatibility with Simulink, where applicable; also, they are well maintained and user friendly.
We determined the advantages and limitations of each one of these model checking approaches, with respect to the verification of stability requirements for linear scalar, linear multi-variable and nonlinear multi-variable discrete systems, all modelled as Simulink diagrams with basic blocks.
Applied quantitative and qualitative performance criteria comprised: correctness of the translation semantics from Simulink to the input language of the model checker, time to pre-process a model into a suitable representation (if needed), time to check a requirement, and amount of additional user analysis to specify checking parameters such as simulation time or loop iterations.
As in our previous work, we incorporated Lyapunov functions into the Simulink diagrams, formulating the Temporal Logic properties <cit.> to verify through model checking in terms of Lyapunov functions and their characteristics.
For the linear systems, we employed Lyapunov's second method to determine stability, as in <cit.>; for the nonlinear system, we employed Lyapunov's direct method.
We encoded suitable models for verification into the input languages of the model checkers, based on existent translation semantics.
In the models, we attempted to balance the number of states in the models on one hand, to avoid a state-space explosion, and expressing the continuity of the state-space of the discrete systems on the other.
The paper proceeds with an overview of related work on verification of Simulink diagrams, and verification of the stability control requirement in Section <ref>.
We then present different case studies used as benchmarks (Section <ref>), followed by a brief introduction of the main features of the model checking tools (Section <ref>) that were employed to verify stability in each case study.
Section <ref> presents the comparative results of the verification experiments.
Section <ref> concludes the paper and gives and outlook towards future work.
§ RELATED WORK
Verification techniques include testing, model checking and theorem proving.
In practice, combinations of these techniques are used to verify complex real-life systems, departing from the “one technique fits all” paradigm.
The presence of signals and parameters theoretically in the domain of the real numbers, corresponding to a continuous real world, leads to state-space explosion problems in the computational mechanisms of some of these techniques.
In testing, inputs are applied to a system to stimulate actions and reactions, and outputs are observed to determine if the requirements are satisfied.
The selection of inputs (test cases) needs to thoroughly explore the system's state space, whilst targeting its interesting regions (i.e., “covering” the system).
Simulink is an ideal tool for testing models of control systems in simulation. Test generation systematically samples the state space of variables and parameters, e.g., through automated search <cit.>.
Theorem proving or deductive verification <cit.> is a static verification technique that involves finding a mathematical proof of a requirement, through the application of axioms, lemmas and inference rules.
A proof can be computed automatically via Satisfiability Modulo Theory (SMT) solvers or Satisfiability (SAT) solvers, or interactively (with user guidance), which requires a great degree of domain knowledge and expertise.
A description of the system and requirements in Propositional, First-Order or Higher-Order Logic is required, along with any other relevant mathematical theory (e.g., sets, linear algebra).
These definitions and additional information are normally encoded by hand into “theories”, as required by case studies; they can be reused once embedded in the theorem proving tools.
Theorem proving has been employed to verify functional equivalence between Simulink diagrams and auto-generated code (e.g., <cit.>), for data type checks (e.g., <cit.>) and to verify high-level requirements including stability (e.g., <cit.>).
Model checking is the exhaustive traversal of a finite-state model of a system (i.e., all the states and state transitions in the model are explored) to check for requirements defined as properties in a variety of different Temporal Logics <cit.>.
Hence, most model checking variants require a discrete or hybrid model that is decidable.
If a property is found to be false, a counterexample is returned, comprising a sequence of states or a trace (according to the valid state transitions in the model).
Computing a decidable model and reducing the state-space to avoid state-space explosion issues, have motivated the shift from explicit-state model checking, i.e., enumerating and traversing all possible states, to symbolic (grouping states in compacted Binary Decision Diagrams or BDDs), bounded (exploring up to $k$ transitions in the model) <cit.>, and statistical (sampling the model's state space) <cit.> model checking.
Probabilistic model checking tools <cit.> suit stochastic models such as Discrete Time Markov Chains.
Specialist hybrid model checking tools – for hybrid models comprising both discrete and continuous transitions, such as switched systems – make use of geometrical methods to approximate the explored state space of the continuous transitions <cit.>.
Hybrid model checkers (and other verification techniques such as theorem proving) commonly restrict the continuous components to ordinary differential equations (ODE) with linear or affine forms.
Reduction of the models can be achieved by systematic abstractions (e.g., bisimulations), or symmetry reduction techniques.
The absence of runtime errors (or low-level requirements) such as overflows or arrays out of bounds for fixed data widths, was verified in <cit.> using model checking for Simulink diagrams. Other tools, such as Mathwork's Polyspace, translate the Simulink diagrams into code before checking for runtime errors.
Higher-level requirements in terms of safety and liveness have been verified directly in the Simulink models (e.g., the Prover Plug-In® or CheckMate for hybrid systems <cit.>), after translating the models (or parts of them) into the language of a specific model checker <cit.>, or after translating the Simulink diagrams into code <cit.>.
Since model checking is based on the exploration of finite-state decidable models, which implies discretization and abstraction processes over the original systems, formalized “translation” processes are highly desirable.
We explored available translation semantics from Simulink to NuSMV <cit.> and to UCLID <cit.>.
The computation of decidable models goes in hand with developing sound automated translation procedures.
This leads to further considerations on the pros and cons of translating and verifying the Simulink models as code, potentially with runtime issues having been introduced in the process, versus considering them as mathematical control system models and to verify the absence of fundamental design flaws before runtime issues are being introduced on the way to code generation.
Control systems requirements such as stability via Lyapunov methods have been verified mostly through theorem proving by directly posing the problems in mathematical terms (e.g., inequalities of region intersections as in <cit.>) for continuous systems, or over controllers implemented in code <cit.>, closer to discrete systems.
From a theoretical control systems perspective, model checking has been applied, via Lyapunov methods, to verify stability for particular types of continuous <cit.> or hybrid systems <cit.>.
For practitioners, however, it is important to understand whether any model checking approach would be suitable also for more generic discrete systems, linear and nonlinear. Our paper aims to provide an insight into this.
§ SYSTEM EXAMPLES
In this paper we verified stability control requirements over Simulink diagrams through model checking, to evaluate how model checking variants compare and if they could be used to complement theorem proving by providing witnesses when no proof can be found.
We chose textbook case studies from control systems theory: linear scalar, linear multi-variable, and non linear multi-variable discrete systems.
The stability requirement was parametrized in terms of the Simulink diagrams' components through the application of Lyapunov theory, as proposed previously in <cit.>, and summarized next.
§.§ Lyapunov's Second Method
Linear systems have a single equilibrium point, nonlinear systems have multiple equilibrium points.
An equilibrium point is stable if the system's state trajectories, starting from any initial point close to the equilibrium point, remain close to it.
An equilibrium point is asymptotically stable if it is stable and the trajectories move towards the equilibrium point as the time $t \rightarrow \infty$.
A Lyapunov function, $V(\mathbf{x}(k))$ for a discrete system (with variables $\mathbf{x}(k)$) is a function such that:
* $V(\mathbf{x}(k))>0, \forall \mathbf{x}(k) \neq \mathbf{0}$, and $V(\mathbf{x}(k))=0$ if $\mathbf{x}(k)=\mathbf{0}$ (at the equilibrium point).
* $V(\mathbf{x}(k))-V(\mathbf{x}(k-1))< 0, \forall \mathbf{x}(k) \neq \mathbf{0}$, and $V(\mathbf{x}(k))-V(\mathbf{x}(k-1))= 0$ if $\mathbf{x}(k)=\mathbf{0}$.
A discrete system is asymptotically stable at the equilibrium point if and only if there exists a Lyapunov function.
For linear and hybrid systems, a candidate Lyapunov function, with $\mathbf{P}$ a positive definite matrix, is
\begin{equation}\label{lyapunovfunction}
\end{equation}
This function can be computed from solving a relevant Lyapunov's equation or a set of equations (for hybrid systems).
For a nonlinear system, Lyapunov's second method can be applied after linearizing around each one of the equilibrium points, for all the resulting linear systems.
Alternatively, a specific Lyapunov function can be proposed (Lyapunov's direct method).
Although we can compute single Lyapunov functions given established procedures, we can propose other Lyapunov functions that might be compatible with the system.
Furthermore, the translation of the system for which we designed the Lyapunov functions, from pen-and-paper into a Simulink diagram (or code), might be incorrect.
Thus, automated procedures to verify stability are greatly desirable, to help ensure that the designs satisfy their control requirements.
§.§ Linear Time Invariant Discrete Systems
Three discrete systems were chosen: two simple uncontrolled loops, and a controlled system. We proposed suitable Lyapunov functions for each system, to facilitate the verification of stability.
Multiplication loop Inspired by the example in <cit.>, shown in Fig. <ref> and defined as
\begin{equation}
\end{equation}
A Lyapunov function for this system is
\begin{equation}
V(x) = x^2.
\end{equation}
Linear single-variable system with Lyapunov function
Multi-variable loop Example from <cit.>, shown in Fig. <ref>, and defined as
\begin{equation}
\mathbf{x}(k+1)=\mathbf{A}\mathbf{x}(k).
\end{equation}
A Lyapunov function was proposed according to (<ref>), with $\mathbf{P}=\mathbf{I}$.
Linear multi-variable system with Lyapunov function
Multi-variable controlled system Example from <cit.>, shown in Fig. <ref>. In state-space equation form, defined as
\begin{equation}
\mathbf{x}(k+1)=\mathbf{A}\mathbf{x}(k)+\mathbf{Bu}(k) \label{system:loop},
\end{equation}
with matrices
\begin{eqnarray}
\mathbf{A}&=&\left[ \begin{array}{cc}
1.5&0.5\\ 0.5&1 \end{array}\right], \ \ \mathbf{B}=\left[ \begin{array}{r}
2\\ 0 \end{array} \right], \nonumber
\end{eqnarray}
and a feedback controller for stability,
\begin{equation}\label{controller}
\mathbf{u}(k)=-\mathbf{K} \mathbf{x}(k), \ \ \mathbf{K}=\left[\begin{array}{cc} 1.15&0.57\end{array}\right],
\end{equation}
by pole placement with desired poles $[0.8,0.3;0 -0.6]$. A Lyapunov function was proposed according to (<ref>), computed from a Lyapunov's discrete equation,
\begin{equation}\label{disclyap}
\mathbf{(A-BK)}^{\mathrm{T}}\mathbf{P(A-BK)}-\mathbf{P}=-\mathbf{I},
\end{equation}
\begin{equation}
\mathbf{P}= \left[ \begin{array}{cc}
2.26&1.50\\ 1.50&4.06 \end{array}\right].\nonumber
\end{equation}
Controlled linear multi-variable system with Lyapunov function
These systems represent infinite loops computationally. If the systems are stable, fixpoints can be derived for the system variables, in the equilibrium points. If the systems are unstable, the system variables will grow without bounds, a challenge for automated verification tools such as model checking, in terms of state-space explosion, data representation (overflows) and decidability (procedures to exhaustively explore a system's model might not terminate).
§.§ Nonlinear Discrete System
The selected nonlinear discrete system is shown in Fig. <ref>, defined as
\begin{eqnarray}
x_1(k+1) &=& \frac{x_2(k)}{1+x_2^2(k)} \nonumber \\
x_2(k+1) &=& \frac{x_1(k)}{1+x_2^2(k)}.
\end{eqnarray}
A Lyapunov function was proposed,
\begin{equation}
V(\mathbf{x}(k))=x_1^2(k) + x_2^2(k),
\end{equation}
with the difference
\begin{equation}
V(\mathbf{x}(k))-V(\mathbf{x}(k-1)) = V(\mathbf{x}(k))\left[\frac{1}{[1+x_2^2(k)]^2}-1 \right] \leq 0. \nonumber
\end{equation}
Nonlinear multi-variable system with Lyapunov function
Computationally, nonlinear system loops employ arithmetic operations that might lead to errors, such as divisions by zero, in automated verification processes. This is added to the aforementioned state-space explosion issues.
§ SELECTED MODEL CHECKING TOOLS
Specific model checking tools, corresponding to different model checking approaches, were chosen following the criteria of:
(a) widespread usage within the verification community;
(b) good support, to ensure fully functioning tools and, thus, high productivity;
(c) user friendliness, providing guides and examples; and
(d) previous application for the verification of Simulink diagrams.
§.§ NuSMV
NuSMV, a symbolic model checker, was originally developed for the verification of hardware designs.
A finite-state machine (FSM) describes the model, in terms of states and their transitions.
The transition model is transformed into a Boolean function, which is encoded into a BDD, a data structure developed for compressed Boolean function representations <cit.>.
Consequently, the encoded BDD structures are not efficient if the transitions are dictated by complex arithmetic operations, such as a series of multiplications in a control system loop.
The requirements to verify can be encoded into Linear Temporal Logic (LTL) or Computation Tree Logic (CTL) <cit.>.
Also, NuSMV can perform BMC, by specifying a maximum number of transitions to explore in the model.
NuSMV's syntax includes signed and unsigned bit-vectors, and matrix data types (arrays of arrays of bit-vectors).
Translation semantics from Simulink to NuSMV have been proposed in <cit.>.
§.§ UCLID
UCLID was also originally developed for the verification of hardware designs.
Its syntax is similar to NuSMV's, although it does not include matrix data types and the arithmetic operations have a more limited functionality (e.g., the division operation only allows integers to the power of 2 as denominators).
The models have a FSM form, but they are not encoded as BDDs.
UCLID performs BMC, by “simulating” the FSM for a specified number of transitions, and checking a specified logical-mathematical expression at each step or once all steps have finished.
Translation semantics from Simulink to UCLID have been proposed in <cit.>.
§.§ S-Taliro
S-TaLiRo is categorized within statistical model checking, since sample traces or state sequences are extracted from the model to determine if a Metric Temporal Logic (MTL) property is true or false in all these samples <cit.>.
Its goal is finding (sequences of) inputs in a Simulink diagram that satisfy (or falsify) a requirement in terms of outputs in the diagram, or the verification of the requirement according to a range of initial system parameters (e.g., initial states $\mathbf{x}(0)$).
Furthermore, S-TaLiRo provides a metric on how well a sample trace (system trajectory) satisfies the property, denominated “robustness”.
Hence, exploration methods looking for inputs that provide the best robustness value, can be applied in an automated manner, e.g., simulated annealing, cross-entropy, genetic algorithms and uniform random sampling.
S-TaLiRo operates directly in MATLAB/Simulink, and numerous examples verifying performance requirements for complex control systems are provided.
§.§ SpaceEx
SpaceEx specializes in analysis of hybrid systems with piecewise affine, non-deterministic dynamics, through computationally efficient reachability algorithms for the continuous transitions of the hybrid systems.
In particular, SpaceEx computes “flowpipe” approximations of sets of reachable states.
The hybrid system models to verify are constructed through a graphic interface, and saved in an xml format.
The restrictions on the continuous components of the hybrid systems, to piecewise affine dynamics, is shared by other tools for hybrid systems, such as the CheckMate model checker, and the theorem prover Keymaera <cit.>.
Consequently, we hypothesised that this tool, and other similar ones, would not be suitable to model and consequently verify the stability requirements of the discrete systems in Section <ref>.
§ EXPERIMENTS AND RESULTS
For each system described in Section <ref>, a model in the model checker's input language was developed and its verification attempted.
All models are available online[<https://github.com/riveras/model_checking>] together with the results obtained.
For all the linear systems, stable and unstable system parameters were applied to assess the correctness of the verification results.
We evaluated the different model checking approaches according to the following criteria: (a) time to pre-process a model into a BDD representation, which is a critical aspect for symbolic model-checking; (b) total checking time, i.e., time used to verify a requirement; (c) amount of additional user input to specify parameters such as initial values, simulation time or loop iterations; and (d) correctness of the translation semantics from Simulink to the input language of the model checker, if translation is necessary.
§.§ Experiments in NuSMV
FSMs were developed manually, according to the semantics proposed in <cit.>, for NuSMV version 2.4.1.
On each FSM, the state variables transition sequentially according to each one of the operation blocks in the Simulink diagram, starting from the delay.
The state variables in the model were represented by 8 bits, to reduce the state space at the cost of inaccuracy and representation.
We adjusted the basic operations to represent basic floating point numbers, also at great accuracy cost.
In the verification stage, initial parameters of $x=2$, $\mathbf{x}=[1;1]$ were applied for the scalar and matrix systems, respectively, signifying the verification of stability only for a single system's trajectory.
For the first two scalar and matrix loops, parameters $a=\{0.9,1.9\}$ and $\mathbf{A}=\{[0.5 \quad 0; 0 \quad 0.5],[1.5 \quad 0; 0 \quad 1.5]\}$ were used for the stable and unstable versions. $K=[10.1,6]$ was employed for the unstable controlled system.
No unstable version of the nonlinear system was verified.
An LTL formula specified the stability requirement, over the Lyapunov function's difference, .
Results are shown in Table <ref>, where YES denotes the compilation and the verification taking place, and NO denotes the failure to compile in less than two hours. T indicates the property is true, and F the property is false. Where the model compilation process succeeded, the size of the state space is indicated.
Experiments in NuSMV
EXAMPLE 3c|STABLE SYSTEM 3cUNSTABLE SYSTEM
2c|Comp.? Verif.? 2c|Comp.? Verif.?
Scalar loop YES $2^{120}$ YES: T YES $2^{120}$ YES: F
Matrix loop YES $2^{160}$ YES: T NO – NO
Controller NO – NO NO – NO
Nonlinear YES $2^{176}$ YES: T – – –
Time to pre-process a model into a BDD
The results in Table <ref> show the impressive size of the state-space, for relatively “simple systems”, even with the reduced bit-vector size.
NuSMV struggled to compute BDDs for systems that loop continuously without a fixpoint, such as the one of the unstable matrix linear system, within a reasonable time threshold. Consequently, verification cannot take place.
This state-space explosion for unstable and the controlled loop is caused by the variables overflowing, as no related flags or added constraints were implemented to stop the infinite loops.
Verification time
The main overhead was caused by the building of the models into BDDs.
When models were built, the verification time was less than one hour, although counterexamples took longer to compute when the property was found to be false.
Parameters to specify
Ideally, as many initial state conditions as possible should be verified (i.e. “all possible states”), representing different system trajectories.
This process can be automated through scripting to control the models and NuSMV, although the chosen initial conditions will always be constrained by the bit-vector size.
Nevertheless, this “sampling” of the initial conditions is not exhaustive over the state space.
Correctness of the translation semantics
The semantics in <cit.> do not specify how to deal with floating-point operations and non-integer data, these are at the core of the semantics of Simulink.
The provided arithmetic operations in NuSMV are not equivalent to the floating-point ones in Simulink.
Furthermore, their semantics offer no guidance on how to correctly represent the functionality of a loop that increases continuously, as the bit-vectors would overflow without any implemented constraints.
Overall, this approach is more suited to verify control systems at code implementation level, as proposed in <cit.>, providing adequate semantics (e.g., floating-point) and system loop constraints (to avoid overflows when variables reach their limits) are added, to emulate the system loop properties more closely within fixed width data types.
Nonetheless, the state-space sizes in the computation of BDDs are still an obstacle.
§.§ Experiments in UCLID
We manually adjusted the FSM models developed for NuSMV, according to UCLID's syntax.
Following the semantics in <cit.>, a transition in the FSM is an unrolling of the whole system loop, going through all the serial block operations in the Simulink diagrams at once.
The arithmetic division operation in the nonlinear system was approximated to 0, as the provided operators do not allow divisions with variable denominators as in NuSMV.
We ran UCLID version 3.1, in a Fedora 6 Virtual Machine.
In the verification stage, the parameters for the stable and unstable systems were $a=\{0.9,12.9\}$ for the scalar loop, $\mathbf{A}=\{[0.5 \quad 0; 0 \quad 0.5],[13.5 \quad 0; 0 \quad 13.5]\}$ for the matrix loop, and $K=[11,6]$ for the unstable controlled system.
The same initial values of NuSMV were used for the state variables, and a bit-vector size of 16 for all variables.
Iteration bounds of $k=\{10,20,40,80\}$ were explored, i.e., the models were unrolled up to 80 times from the specified initial states.
We verified the requirement through the computation of the expression at each exploration step (“simulation”) of the model.
Table <ref> shows the results when checking the expression, for the different $k$ bounds. F indicates the expression was not true in at least one of the checks, and T indicates the expression was always true.
Experiments in UCLID
EXAMPLE 4c|STABLE SYSTEM 4cUNSTABLE SYSTEM
$k$ bound 10 20 40 80 10 20 40 80
Scalar loop T T T T F F F F
Matrix loop T T T T F F F F
Controller F F F F F F F F
Nonlinear T T T T – – – –
Verification time
The verification process took seconds, as the models are unrolled iteratively according to the specified steps on-the-fly.
Unfortunately, no counterexamples were provided when the expression checks failed.
Parameters to specify
We specified initial values, as for NuSMV, and bounds on the number of exploration steps.
It is not clear how to chose a suitable number of steps. However, this information is critical for verification since too small a number may lead to false positives if the expressions failed (i.e. are falsified) in the future.
Additionally, the expression checks at each execution step have to be encoded by hand explicitly, whereas in other model checkers this is done automatically by indicating a property to check.
Correctness of the translation semantics
As in NuSMV, the semantics in <cit.> do not consider floating-point operations, nor overflows.
The impact of the lack of floating point built-in support is evident in the controlled system, where high precision multiplication operations are needed for an accurate computation of the requirement's expression value.
Furthermore, the available arithmetic operations have limited functionality and there is no matrix data type, compared to NuSMV.
This approach is computationally less expensive than using NuSMV, allowing larger bit-vector sizes, although more limited in terms of arithmetic operations and data representation.
Extending the operational functionalities and implementing some overflow constraints, this approach would be more suitable to the verification of control systems code, by exploring the loops for $k$ iterations, for both high-level functional and runtime requirements.
Pre- and post-conditions to avoid overflows and underflows would enable sound verification “for all possible representable values” of variables in a system loop, within bounded ranges and a fixed bit-vector widths.
§.§ Experiments in S-TaLiRo
For these experiments, we modified the Simulink diagrams presented in Section <ref> by adding output probes to measure the Lyapunov function's difference over time, as required by the tool, S-TaLiRo version 1.61, running in MATLAB/Simulink version R2013a.
For verification, ranges of $x=[-10 \quad 10]$ and $\mathbf{x}=[-10 \quad 10;-10 \quad 10]$ were specified for the initial state variable values, to be sampled by the tool.
We used the same parameters for $a$ and $\mathbf{A}$ in the linear systems as in the NuSMV experiments, and an unstable controller of $-K$.
The MTL properties to falsify were, for stable systems, “the Lyapunov function's difference is eventually $>0$”; and, for unstable systems, “the Lyapunov function's difference is always $<=0$”.
We allowed 100 tests for different input samples, for three of the offered exploration methods to find traces that falsify the properties: simulated annealing (SA), cross-entropy (CE), and uniform random (UR). The cross-entropy method did not run successfully for the stable systems.
The experiment configuration settings, initial or input value ranges, and MTL properties and exploration methods were specified via MATLAB scripts.
The results are shown in Table <ref>, where T indicates a falsifying trace was not found (the system might not be unstable), and F indicates a falsifying trace was found (the system is unstable).
Experiments in S-TaLiRo
EXAMPLE 3c|STABLE SYSTEM 3cUNSTABLE SYSTEM
Sampling SA CE UR SA CE UR
Scalar loop T – T F F F
Matrix loop T – T F F F
Controller T – T F F F
Nonlinear T – T – – –
Verification time
The verification process took less than a minute in total, for the specified number of samples and diagram simulation time per sample.
This time is expected to increase if more samples and larger value ranges are introduced.
Parameters to specify
S-TaLiRo requires specifying either ranges for the initial values of the state variables, or ranges for the inputs (if any), the number of samples, the simulation time per run (sample), and the exploration method, to mention some of the most important parameters.
Some of these parameters can be intuitively tuned, but others depend more on understanding of the tool's functionality and previous knowledge.
Correctness of the translation semantics
The “translation” process from our Simulink diagrams involves adding input and output block probes, to indicate which are the input signals to sample, and which are the output signals that the property to verify refers to.
Overall, this approach is the most straightforward to use over our Simulink diagrams, since no translation is needed, and the amount of additional specifiable parameters is reduced.
Additionally, S-TaLiRo allowed to cover more of the possible initial state variable values in an automated manner, compared to having to change the initial values manually in the models used in UCLID and NuSMV.
Nevertheless, this approach is not complete for the verification of stability in general, since it does not offer a proof in the case a system is stable – i.e., the results for the MTL property of “eventually unstable” for a stable system only indicate that the system is not unstable within the sampled initial state values and simulated time.
Alternatively, it could be the case that the requirement of “always stable” is not falsified in unstable systems, if the initial state values and simulation time interval are of a trajectory that appears to converge to an equilibrium point.
Statistical model checking through S-TaLiRo allows the verification of all types of systems (e.g., continuous, discrete, linear, nonlinear, stochastic and delayed) as long as they are modelled in Simulink.
Although, this approach cannot substitute the strength of computing a proof of a requirement “for all possible variable values and initial states”, from theorem proving (e.g., in <cit.>), it is suitable as a method to search for evidence that a requirement is not satisfied.
§.§ Experiments in SpaceEx
We attempted to construct models for the systems in Section <ref>, but the syntax did not allow expressing difference equations and nonlinear terms, including the Lyapunov functions.
In contrast, continuous systems similar to the discrete ones, such as
\begin{equation}
\dot{x}=-0.5x,\nonumber
\end{equation}
were easily constructed.
Stability can only be verified, for continuous or hybrid systems, through computing the reachability of the models, given initial state conditions, $x(0)$, specified as single values or intervals. If the system is stable, the system's reachability is bounded by these initial conditions, and the system converges (reaches a fixpoint) to the equilibrium point in $x=0$. If the system is unstable, the system's state variables diverge and no fixpoint is reached.
The same concepts extend to multi-variable linear continuous systems.
SpaceEx has the potential to be extended to include discrete systems and their respective “flowpipe” algorithms, once developed.
§ CONCLUSIONS AND FUTURE WORK
We explored the verification of stability for discrete systems' designs in Simulink, through different model checking variants and corresponding state-of-the-art tools.
We aimed to find how well these different model checking variants are suited to this particular problem domain, how compatible they are with respect to Simulink diagrams as a system modelling language, what are their limitations in practice, and if any of them would substitute or complement our previously developed theorem proving approach <cit.>, by providing examples as evidence of requirement violation.
We explored four model checking variants through related tools, symbolic, bounded, statistical and hybrid, verifying stability based on Lyapunov methods for discrete linear and nonlinear systems.
Our experiments and results provide an insight on the strengths and limitations of these model checking approaches with respect to the verification of stability of discrete systems.
We found that statistical model checking through S-TaLiRo is the most suitable option to complement our previous work on automatic theorem proving.
This same approach is the most compatible with Simulink diagrams of systems described as difference equations, compared to symbolic model checking with NuSMV or bounded model checking with UCLID – based on available translation semantics from Simulink to NuSMV and UCLID–, and hybrid model checking with SpaceEx.
In the future, we will incorporate model checking into our automatic theorem proving methodology <cit.>. This will allow us to return evidence as proof (in the form of counterexamples) when the systems are not stable or do not satisfy other control or performance requirements, thus enhancing the usability of the approach and facilitating debug.
Our evaluation shows that S-TaLiRo is the ideal candidate for this extension.
Instead of developing individual tools to verify the same requirement over different types of systems, software platforms could be extended to recognize the system's characteristics and apply relevant algorithmic variants, according to sound theoretical frameworks.
MATLAB/Simulink could be used as platform to encode standardized models (i.e., through the same graphical language), connected to theorem provers, external constraint, satisfiability and optimization solvers.
The computation of Lyapunov functions for all kinds of systems remains a research challenge.
A plausible alternative is to further explore model checking related computational techniques, such as reachability approximations or BMC combined with SMT solvers as in <cit.>, to find “stable” regions of the state space <cit.>, or regions that satisfy other performance requirements.
Other alternatives point towards the use of statistical model checking or sampling and search methods as in S-TaLiRo, combined with optimization problems, to compute Lypaunov functions and barrier certificates <cit.> for larger sets of types of systems.
We will be exploring such alternatives in the future.
§ ACKNOWLEDGEMENTS
The work presented in this paper was supported by the EPSRC grant EP/J01205X/1 RIVERAS: Robust Integrated Verification of Autonomous Systems.
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1511.00356
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High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Dresden High Magnetic Field Laboratory (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Experimental details. Single crystals of PdCrO$_2$ of typical dimensions 1 x 0.4 x 0.2 mm$^3$ were grown by a flux method, as described in Ref. <cit.>. The experiments were carried out at the High Field Magnetic Laboratory (HFML) in Nijmegen using a specially built sample-holder designed to be incorporated into one of the He-4 flow cryostats available at HFML. The cryostat themselves are designed for measurements in the temperature range between 2 K and 100 K. The observed thermopower suppression in magnetic field motivated us to extend the operating range of this system to around 130 K, more than three times $T_N$ of PdCr$O_2$. The sample holder has a one-heater-two-thermometers configuration with RuO$_2$ bare chip resistors used for thermometry. The thermoelectric voltage was measured with phosphor-bronze lead wires for the majority of the field sweeps because of their low background contribution. Exceptionally, in the search for Shubnikov-de-Haas oscillations in the thermopower, gold wires were used for their smaller resistance values. The thermoelectric voltage was captured using a low-noise analog nanovoltmeter connected to twisted pairs of signal wires leading to the cold point of the holder in order to inhibit various noise sources. In all measurements reported in the main text, the magnetic field is oriented perpendicular to the thermal gradient and to the highly conducting planes.
(Color online). (a) Magnetic field dependence of the in-plane resistivity $\Delta \rho_{ab}(B)/\rho_0$ $(\rho_0$ = $\rho_{ab}(B=0))$ of PdCrO$_2$ in the temperature range from 1.4 K to 150 K in the field up to $B$ = 30 T perpendicular to the $ab$-plane. (b) The magnetoresistivity data shown in (a) on an expanded scale in order to emphasize the high-$T$ behavior.
High-field magnetoresistivity in PdCrO$_2$. Fig. <ref> shows the magnetoresistivity (MR) $\rho_{ab}(B)$ =$\rho_0$ ($\rho_0 $ = $\rho_{ab}(B$ = 0)), recorded on a second crystal at fixed temperatures between 1.4 K and 150 K. In contrast to the magnetothermopower (MTP) behavior, the size of the MR decreases sharply with increasing temperature (see Fig. <ref>). In the temperature range 95 K $\leq T \leq$ 130 K where the suppression of the thermopower reaches 100 % in a field of 30 T, the MR is less than 5 %, compared with 300 % at $T$ = 1.4 K.
The presented magnetoresistivity data was obtained as a part of the study published in Ref. <cit.>. There, the large transverse MR was reported together with a large longitudinal MR, indicating a significant coupling between itinerant electrons with the Cr spins.
(Color online). Temperature dependence of the magnetoresistivity of PdCrO$_2$ presented in Fig. <ref> for constant values of magnetic field of 10 T, 20 T and 30 T. The antiferromagnetic ordering at $T_N$ is indicated by the dashed line.
Determination of the effective mass of the $\alpha$ pocket. Below $T_N$, where the Cr$^{3+}$ spins order, $S_{ab}/T$ increases as the conduction electrons undergo FS reconstruction <cit.>. The existence of small pockets is confirmed by the observation of QO in the high-field thermopower below $T_N$. The observation of QO here illustrates not only the quality of the single crystals used in this study, but also the high sensitivity and stability of our optimized set-up.
The $T$-dependence of the QO amplitude in $S_{ab}/T$ is presented in Fig. <ref>a. Figure <ref>b shows the Lifshitz-Kosevich (L-K) fit to the thermopower quantum oscillation amplitude determined over the field range 24-32 T. The fit to the L-K formula $A/T = [\sinh (am^{*}T/B)]^{-1}$ results in an effective mass of $m^* = 0.31(3) m_0$, where $m_0$ is the free electron mass.
(Color online).
(a) The QO seen in $S_{ab}/T$ show suppression in amplitude with increasing temperature which satisfies the Lifshitz-Kosevich formalism <cit.>. (b) Temperature dependence of the thermopower oscillation amplitude $A/T$. The L-K fit is shown while the fit uncertainty is indicated with dotted lines. The $T$-error bars originate from a large applied thermal gradient, which is necessary for the detection of the quantum oscillations.
$S(T)$ vs. magnetic field
In the main text, Fig. 1 emphasizes the drop in the magnitude of the field-suppression of thermopower as the temperature is lowered and the magnetic phase is approached. The peak in $S_{ab}$ around 15 K for $B$ = 30 T, at first glance unconventional, is in fact an artefact of the plotting and the scatter in the data. This is most clearly seen in the plot of $S/T$ (Fig. 3 of original manuscript) where any putative feature at 15 K is lost. Fig. <ref> shows the evolution of $S(T)$ for different field strengths. Here, it can be seen clearly that the peak at 15 K is indeed a by-product of the scatter in the data and, at $B$ = 30 T, is an consequence of the marked enhancement in $S/T$ with decreasing temperature on the one hand, and the thermodynamic requirement for the thermopower to vanish in the zero temperature limit on the other.
(Color online). $S_{ab}$ as a function of temperature for values of magnetic field ranging from zero to $B$ = 30 T.
Thermopower vs. neutron scattering.
The role of spin fluctuations in PdCrO$_2$ is underlined in Fig. <ref> with a comparison of neutron scattering data from Ref. <cit.> with the thermopower $S_{ab}/T$ data shown in Fig. 3 of the main text. The scattering of itinerant electrons on short-range magnetic order above $T_N$ results in a linear $S/T$ behavior on a logarithmic temperature scale. Below $T_N$, formation of the long-range magnetic order results in a stronger peak intensity of neutron scattering which increases with lowering temperature. Coincidentally, the thermopower coefficient $S/T$ changes its slope at $T_N$, confirming a strong interaction of itinerant electrons with magnetic order.
The possible role of magnon-electron interactions in the thermopower of PdCrO$_2$. For the case of purely elastic scattering of electrons the thermoelectric power $S$ is expressed by Mott-Jones formula (see, e.g. Ref. <cit.>):
\begin{equation}\label{eq:Ssigma}
S\sigma = \frac{1}{e}\int dE \sigma(E) \frac{E-\mu}{T} \frac {\partial f}{\partial E}
\end{equation}
where $\mu$ is chemical potential, $f(E)$ is the Fermi function, $\sigma (E)$ is the conductivity at zero temperature and $\mu = E$ and $\sigma = -\int dE \sigma (E) \frac{\partial f}{\partial E}$ is a conductivity at a given temperature. If a typical energy scale of the dependence of $\sigma(E)$, $E_0$, is much larger than the temperature $T$, thermoelectric power is estimated as $S\approx \frac{k_B^{2}T}{eE_0}$, that is small (in a factor $\frac {k_B T}{E_0}$ smaller than a classical value $\frac{k_B}{e}$) and linear in temperature. Exceptions can be in the cases where electron density of states has unusually sharp energy dependence near the Fermi energy $E_F = \mu(T=0)$ or for Kondo systems <cit.>. The former option is not applicable to PdCrO$_2$ according to the electronic structure calculations <cit.>. The latter case would assume the formation of the Kondo lattice state. Although this can not be completely excluded it seems less probable for a 4$d$ system. Also, the experimental data on heat capacity and magnetic susceptibility do not support this scenario <cit.>.
(Color online). Temperature dependence of the neutron diffraction peak intensity at the $q=(\frac{1}{3},\frac{1}{3},1)$ position, as reported in Ref. <cit.>. The data is compared with the thermopower $S_{ab}/T$ data on a semilogarithmic scale presented in Fig. 3 of the main text.
It is known that when inelastic scattering mechanisms are relevant such as electron-phonon scattering, the nonequilibrium effects in the scatterer subsystem such as phonon drag can be of crucial importance for the thermoelectric power <cit.>. It is natural to assume that in magnetic systems magnons (spin waves) can play a similar role (for the case of ferromagnets, see e.g. References <cit.>. Keeping in mind that the measured $S$ (see Fig. 1) is essentially nonlinear in temperature and very strongly dependent on magnetic field (except the case of low temperatures) it seems to be natural to attribute these features to magnon drag. The latter is relevant if the magnon-electron scattering is, at least, comparable (or more important) than the scattering on magnons by defects. This assumption looks very reasonable for PdCrO$_2$ which is a system with a low level of defects.
PdCrO$_2$ is a quasi-two-dimensional itinerant-electron antiferromagnet with the Neel temperature $T_N =$ 37.5 K much smaller than a typical energy of exchange interactions $J$, of the order of paramagnetic Curie-Weiss temperature $\Theta_W =$ 500 K <cit.>. Importantly, a short-range magnetic order remains strong enough, with a correlation length $\xi \gg a$ ($a$ is the lattice period) up to the temperatures $T \approx J \gg T_N$ (see Fig. <ref>). As was shown in Ref. <cit.>, character of electron-magnon interactions in such a case does not change essentially at $T = T_N$ and remains basically the same assuming that $\xi > k_F^{-1}$ which in the case of metals corresponds to $T < J$. Therefore, we can use until these temperatures a theory of electron-magnon interaction in itinerant-electron antiferromagnets developed in Ref. <cit.>.
In isotropic antiferromagnets, in the absence of magnetic field, the magnon frequency $\omega_{\vec{q}}$ tends to zero linearly for the wave vector $\vec{q} \rightarrow 0$ and $\vec{q} \rightarrow \vec{Q}$ where $\vec{Q}$ is the antiferromagnetic order wave vector; for the theory specifically describing magnons in a triangular lattice with the nearest-neighbor antiferromagnetic interactions please refer to Ref. <cit.>. There is a dramatic difference in the character of electron-magnon interactions with these two types of soft magnons <cit.>: for $\vec{q} \rightarrow 0$, the scattering probability with the momentum-transfer $\vec{q}$, $W_{vec{q}} \propto \omega_{\vec{q}} \propto q$ vanishes (identically to the scattering of electrons by acoustic phonons <cit.>) whereas for $\vec{q} \rightarrow \vec{Q}$ it is formally divergent: $W_{\vec{q}} \propto 1/\omega_{\vec{q}} \propto |\vec{q}-\vec{Q}|^{-1}$. These arguments indicate that the latter class of scattering processes is much more relevant. Simultaneously, this process requires scattering to overcome the antiferromagnetic energy gap $\Delta$ and therefore there is a threshold in temperature: such strong electron-magnon interaction becomes relevant at $T > T^{*} \approx (\Delta/E_F)J$. We believe that these arguments provide a qualitative understanding of the MTP behavior shown in Fig. 1.
The Hamiltonian of antiferromagnet in the spin-wave temperature region ($T \ll J$) is formally equivalent to that for electron-phonon interaction (with different matrix elements). To estimate the drag contribution $S_g$ to the thermoelectric power one can use the expression for the phonon case <cit.>:
\begin{equation}\label{eq:Sg}
S_g=\frac{1}{|e|\left\langle (\nu^e_{\vec{k},\chi})^2\right\rangle}\sum_{\vec{q}} \frac{\partial N_{\vec{q}}}{\partial T}\alpha_{\vec{q}}\nu^m_{\vec{q}\chi} \left\langle \nu^{\epsilon}_{\vec{k}+\vec{q},\chi}-\nu^{\epsilon}_{\vec{k},\chi} \right\rangle
\end{equation}
where $N_{\vec{q}}=\left[\exp(\hbar\omega_{\vec{q}}/k_BT)-1) \right]^{-1}$ is Bose distribution function, $\nu^m_{\vec{q} \chi}=\partial \omega_{\vec{q}}/\partial q_{\chi}$ is magnon group velocity along current propagation $x$-direction, $\nu^{\epsilon}_{\vec{k}, \chi}$ is electron group velocity in the same direction, angular brackets means the average over the Fermi surface for the vector $\vec{k}$, $\alpha_{\vec{q}}$ is the rate of scattering of magnon with the wave vector $\vec{q}$ by electrons normalized to the total probability of scattering of this phonon due to all interaction where this magnon is involved (namely, with defects, sample boundaries, and magnon-magnon interactions). For example, if magnons are in almost ballistic regime, one can estimate $\alpha_{\vec{q}}\approx W_{\vec{q}}L/\nu^m_{\vec{q}}$, where $L$ is the sample size.
Below $T^*$, where “singular” electron-magnon scattering processes with $\vec{q} \rightarrow \vec{Q}$ are forbidden, the contribution from small $q$ is dominant. One has $\alpha_{\vec{q}}\propto q$, $\nu^m_{\vec{q}\chi}\left\langle \nu^{\epsilon}_{\vec{k}+\vec{q},\chi}-\nu^{\epsilon}_{\vec{k},\chi}\right\rangle \propto q$, and $S_g \propto \frac{\partial}{\partial T}\sum_{\vec q}N_{\vec{q}}q^2 \propto T^3$. In this regime, the Mott-Jones contribution <cit.> is dominant, thermoelectric power is roughly proportional to temperature and weakly dependent on magnetic field.
Above $T^*$, where “singular” electron-magnon scattering processes with $\vec{q} \rightarrow \vec{Q}$ are allowed, the contribution from small $q$ is dominant. One has $\alpha_{\vec{q}}\propto|\vec{q}-\vec{Q}|^{-1}$, $\nu^m_{\vec{Q},\chi}\left\langle \nu^{\epsilon}_{\vec{k}+\vec{Q},\chi}-\nu^{\epsilon}_{\vec{k},\chi}\right\rangle$ is constant, and $S_g\propto \frac{1}{T^2}\sum_{|\vec{q}-\vec{Q}|>q^*} \frac{\omega_{\vec{q}}N_{\vec{q}}(1+N_{\vec{q}})}{\omega_{\vec{q}}} \propto \ln T/T^*$, where $q^* \approx \Delta/\hbar\nu_{F}$ is the infrared cut-off wave vector <cit.>. In this regime, the drag contribution is dominant. Indeed, temperature dependence of thermoelectric power in the absence of magnetic field at relatively high temperatures is roughly logarithmic. One can estimate $T^*$ as 20-50 K which seems to be a reasonable estimate for $J \approx$ 150-500 K, $\Delta/E_F\leq$ 0.1. Indeed, from the electronic structure calculations for ferromagnetic phase <cit.> the splitting between spin-up and spin-down Pd states $\Delta \leq$ 1 eV, $E_F \approx$ 8 eV.
The drag contribution to the $S$ is strongly dependent on the spin-wave spectrum which is very sensitive to magnetic field <cit.>. This may explain qualitatively the dramatic growth of magnetothermoelectric power at higher temperatures.
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1511.00356
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High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Dresden High Magnetic Field Laboratory (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands
Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525, AJ Nijmegen, Netherlands
We report the temperature $T$ and magnetic field $H$ dependence of the thermopower $S$ of an itinerant triangular antiferromagnet PdCrO$_2$ in high magnetic fields up to 32 T. In the paramagnetic phase, the zero-field thermopower is positive with a value typical of good metals with a high carrier density. In marked contrast to typical metals, however, $S$ decreases rapidly with increasing magnetic field, approaching zero at the maximum field scale for $T >$ 70 K. We argue here that this profound change in the thermoelectric response derives from the strong interaction of the 4$d$ correlated electrons of the Pd ions with the short-range spin correlations of the Cr$^{3+}$ spins that persist beyond the Néel ordering temperature due to the combined effects of geometrical frustration and low dimensionality.
72.15.-v, 72.15.Jf, 75.10.Jm, 75.47.-m
The interplay between itinerant electrons and even simple magnetic structures can lead to spectacular effects, the giant magnetoresistance seen in magnetic multilayers being arguably the most prominent example <cit.>. In geometrically frustrated magnets, complex spin textures that couple to the conduction electrons create an altogether different landscape, where short-range correlations are expected to play a major role. Moreover, since magnetism in metals can be destabilized much more readily than in insulators, magnetic frustration in metallic systems offers a rich playground to search for the emergence of novel transport phenomena. Notable recent examples include the unconventional anomalous Hall effect (AHE) observed in magnetic pyrochlores <cit.> and the suppression of thermopower in a longitudinal magnetic field in the layered Curie-Weiss metal Na$_x$CoO$_2$ <cit.>.
Despite their obvious potential for new physics, metallic frustrated magnets have been noticeably less studied than their insulating counterparts, largely due to the fact that such materials are rare. Of particular interest are materials in which the conduction electrons and magnetic moments arise from different subsystems. In this context, the quasi-two-dimensional (quasi-2D) antiferromagnet PdCrO$_2$ <cit.> is somewhat unique. PdCrO$_2$ has a delafossite crystal structure with layers of Pd ions arranged in a triangular lattice stacked between magnetic edge-sharing CrO$_6$ octahedra. The latter contains Cr$^{3+}$ ions with localized (Mott insulating) 3/2 spins which order in the 120$^\circ$ antiferromagnetic (AFM) structure below $T_N =$ 37.5 K <cit.>. The frustration parameter $f$, defined as an absolute ratio of the Weiss temperature $\Theta_W$ and the ordering temperature $T_N$, is around 13 for PdCrO$_2$, indicating a high level of frustration <cit.>. According to band structure calculations <cit.>, angle-resolved photoemission <cit.> and quantum oscillation (QO) studies <cit.>, the Fermi surface (FS) of PdCrO$_2$, in the paramagnetic (PM) phase, is identical to that of the nonmagnetic analog PdCoO$_2$, and thus is derived uniquely from the 4$d$ electrons on the Pd site. PdCrO$_2$ also draws special interest because it too exhibits an unconventional AHE, i.e. one that does not scale with its magnetization <cit.>.
Here, we report the discovery of a new feature in the transport properties of PdCrO$_2$, namely a strong magnetothermopower (MTP) at elevated temperatures. In a transverse field, $S$ exhibits a marked decrease which for $T >$ 70 K, reaches 100 % of the zero-field value at $\mu_0H =$ 30 T. The suppression is reminiscent of that first reported in Na$_x$CoO$_2$ ($x =$ 0.7) <cit.> and attributed to a lifting of the spin degeneracy of the large spin entropy of the mobile Co$^{4+}$ spins that gives rise to its enhanced thermopower <cit.>. We argue here however, that the suppression of $S(B)$ in PdCrO$_2$ is distinct from that observed in Na$_x$CoO$_2$ and signifies instead a novel magnon drag contribution to the thermopower that persists far beyond $T_N$ due to the highly frustrated short-range spin correlations on the Cr sublattice.
Single crystals of PdCrO$_2$ of typical dimensions 1$\times$0.4$\times$0.2 mm$^3$ were grown by a flux method, as described in Refs. <cit.>. Details of our thermoelectric measurements can be found in the Supplemental Material (SM) <cit.>. In all measurements reported here, the magnetic field is oriented perpendicular to the thermal gradient and to the highly conducting planes.
(Color online). Temperature dependence of the in-plane thermopower $S_{ab}$ of PdCrO$_2$ at zero and high magnetic field $B$ $\parallel$ $c$ shows an almost complete field suppression for $T >$ 70 K. The magnetic ordering transition at $T = T_N$ is indicated by a vertical dashed line.
The key finding of our study is the effect of a magnetic field on the thermoelectric response of PdCrO$_2$. This is summarized in Fig. <ref> where the in-plane thermopower $S_{ab}(T)$ in zero field (solids black squares) is compared with that obtained in an applied field of 30 T (solid red circles) (see Supplemental Material for $S_{ab}(T)$ data at intermediate fields <cit.>). In zero field, $S_{ab}$ has a small positive value less than 2 $\mu$V/K — typical of good metals — while the sign and order of magnitude of $S_{ab}$ at $T =$ 130 K are similar to those found in PdCoO$_2$ <cit.>. In a large magnetic field, as summarized in Fig. <ref>, the thermopower is almost completely suppressed for $T >$ 70 K. Lowering the temperature towards the magnetically ordered phase reduces the magnitude of the field-suppression of $S_{ab}$.
Such a large suppression of $S$ is not expected in a conventional metal, where the field has a negligible effect on the relative spin-up and spin-down populations of electrons and their entropic current <cit.>. In strongly correlated electron systems, however, the spin degrees of freedom can give a large contribution to $S$ according to Heikes formula $S = \mu/eT = S_E/e$, where $\mu$ is the chemical potential, and $S_E$ is the entropy per charge carrier <cit.>. Since $S_E$ depends on the spin and configuration degeneracies, the spin entropy term can raise $S$ to order $k_B/e$ and subsequently be suppressed to zero in a magnetic field by a lifting of the spin degeneracy. Such an effect was observed first in Na$_x$CoO$_2$ <cit.> where the field dependence of $S(B)$ for different $T$ was found to be consistent with the variation of the residual spin entropy $S_E(B,T)$ for noninteracting spins in a magnetic field.
(Color online). (a) Magnetic field dependence of $S_{ab}$ as a function of temperature. The suppression of the normalized thermopower by an out-of-plane magnetic field decreases with lowering temperature, as indicated by the arrow. (b) $T$ dependence of the normalized field change of $S_{ab}$ in an applied field of 30 T. Line is a guide to the eye. The inset shows the comparison of a modeled spin entropy in field from Ref. <cit.> (dashed curve) and the normalized $S_{ab}(B)$ for $T >$ 70 K. While the form of the suppression is consistent with the model, its magnitude clearly does not scale with $\mu_0H/k_BT$.
The field dependence of $S_{ab}(B)$ in PdCrO$_2$, normalized to its zero-field value, is shown in Fig. <ref>a for constant temperature field sweeps over a range of temperatures 1.2 K $\leq T \leq$ 130 K. As shown in the inset of Fig. <ref>b, the form of the field suppression is qualitatively similar to that found in Na$_x$CoO$_2$ <cit.>. Indeed, since both transition-metal oxides form a layered triangular lattice of localized, but frustrated magnetic moments which interact with the conducting $d$ electrons, it might be tempting to assign the same origin to the field suppression of $S(B)$ in both cases. Closer inspection, however, reveals some important differences between the compounds and their thermoelectric response that suggest otherwise.
First, in Na$_x$CoO$_2$ ($x$ = 2/3), one third of the Co ions are in a Co$^{4+}$ configuration, giving rise to a band of mobile but AFM-coupled charges (with spin $s = 1/2$) moving through a magnetically inert background of $s =$ 0 moments localized on the Co$^{3+}$ sites. Wang et al. showed that the spin entropy term associated with these mobile spin excitations accounts for almost all of $S$ at 2 K and a dominant fraction at 300 K <cit.>. In PdCrO$_2$, on the other hand, the sea of conduction electrons is comprised uniquely from the 4$d$/5$s$ states of the Pd ions, as shown convincingly in recent QO studies <cit.>, while the Cr$^{3+}$ states are (Mott) insulating and therefore cannot, by themselves, respond to a thermal gradient. Thus, while there is significant spin entropy in the system, it does not contribute to the thermopower of PdCrO$_2$, and correspondingly, there is no enhancement in $S_{ab}$.
Second, the temperature evolution of the suppression is markedly different in the two compounds. In Na$_x$CoO$_2$, the relative suppression of $S$ in field grows with decreasing $T$ as the contribution of the spin entropy term becomes ever more dominant. In PdCrO$_2$, by contrast, the field suppression becomes more pronounced with increasing temperature. Hence, the $H/T$ scaling observed in Na$_x$CoO$_2$ <cit.> fails in PdCrO$_2$ [see inset to Fig. <ref>(b)]. As illustrated in the main panel of Fig. <ref>(b), where the ratio $S(B =$ 30 T)/$S(B =$ 0 T) is plotted, a total suppression of $S_{ab}$ is only observed for $T \geq$ 2$T_N$. Below this temperature scale, $S$(30 T) remains finite and grows in magnitude with decreasing temperature.
Large magnetothermopower is often observed when the zero-field thermopower itself has an enhanced value, e.g. as found in semiconductors, and is commonly attributed to the effects of a magnetic field on the respective mobilities of the electron and hole carriers <cit.>. A sizeable MTP is also found in systems exhibiting a large magnetoresistance <cit.>. Neither of these scenarios apply to PdCrO$_2$. As mentioned above, the zero-field thermoelectric response in PdCrO$_2$ is that of a single-band metal with a high carrier density. Moreover, at these elevated temperatures where the suppression of the thermopower is most complete, the in-plane magnetoresistance of PdCrO$_2$ is very small, of order 5$\%$ or less in a field of 30 T <cit.>. Thus, it would appear that the large MTP in PdCrO$_2$ stems from a different origin. Before discussing this in more detail, however, we first complete the summary of our experimental findings, some of which have an important bearing on this discussion.
(Color online). $S_{ab}/T$ as a function of temperature, plotted on a semilogarithmic scale, in zero field and at $B$ = 30 T. The dashed line is a logarithmic fit to $S_{ab}/T$ above $T_N$ that is associated with the scattering of itinerant electrons on spin fluctuations with a wave vector $Q$ <cit.>, as indicated in the Fermi surface model shown in the right inset <cit.>. Below $T_N$, the enhancement is attributed to a dominant contribution from the $\alpha$ pocket in the reconstructed zone, labeled in the left inset <cit.>.
Figure <ref> shows the variation of $S_{ab}/T$ in PdCrO$_2$ between 1 K and 130 K, both in 0 T and 30 T, on a semilogarithmic scale. Over the entire temperature range, $S_{ab}/T$ in zero field increases with decreasing temperature. Above $T_N$, $S_{ab}/T$ appears to exhibit a logarithmic enhancement which we attribute to the scattering of electrons on short-range magnetic correlations in the PM phase that are also presumed to be responsible for changes in the zero-field <cit.> or low-field <cit.> transport properties. The established FS topology of PdCrO$_2$ makes the electron states highly sensitive to scattering processes, either from magnons or spin fluctuations, associated with the AFM wave vector $Q$ (illustrated in the right inset of Fig. <ref>) <cit.>. In an isotropic antiferromagnet, the scattering probability $W_q$ with momentum-transfer $q$, while vanishing for $q \rightarrow 0$, is formally divergent for $q \rightarrow Q$, $W_q \propto 1/\omega_q \propto |q-Q|^{-1}$ <cit.>. Above a threshold temperature $T^{*} \sim$ 20–50 K (see Ref. <cit.> and below for a fuller description), “singular” scattering off such FS hot spots can lead to a contribution to $S \propto \ln T^{*} / T$ <cit.>, qualitatively consistent with what is found here in zero field above $T_N$ (Fig. <ref>).
At lower temperatures, where the Cr$^{3+}$ spins order, $S_{ab}/T$ increases more rapidly as the conduction electrons finally undergo FS reconstruction <cit.> (note the upward deviation from the dashed line in Fig. <ref> below $T_N$). The existence of small pockets is confirmed by the observation of QO in the high-field thermopower below $T_N$, shown in Fig. <ref>. Both the QO frequency $F$ = 710 T and the obtained mass $m^{*} =$ 0.31(3) $m_e$ <cit.> agree with those previously attributed to the small $\alpha$ pocket in the reconstructed zone <cit.>. Assuming that the $\alpha$ pocket is the dominant contribution to $S_{ab}$ in the low-$T$ limit (given that it has the smallest $T_F$ = 3000 K), we use a simple Drude model to estimate $S_{ab}/T$ = 140 nV/K , comparable to the measured value at 1.2 K of 200 nV/K. Thus, we can attribute the additional enhancement of $S_{ab}/T$ below $T_N$ to FS reconstruction. In contrast to the thermoelectric response of other systems (such as the parent pnictide BaFe$_2$As$_2$ <cit.>) that undergo a magnetic zone folding, the change in $S/T$ here is very gradual.
(Color online). Slow quantum oscillations in $S_{ab}$ (obtained by subtracting a polynomial background) and in magnetic torque (obtained previously <cit.>) originating from a small electron pocket in the reconstructed Fermi surface.
Let us now turn to discuss the origin of the unusual field dependence of $S_{ab}$ in PdCrO$_2$. The observed FS reconstruction indicates strong coupling between the itinerant electrons and the local moments on the Cr$^{3+}$ sublattice. According to specific heat and susceptibility data <cit.>, there is a broad region in temperature above $T_N$, extending up to 150 K (i.e. 4$T_N$), in which short-range correlations among the frustrated spins persist. This is also confirmed by the observation of diffuse magnetic scattering in Refs. <cit.>, discussed in more detail in the Supplemental Material. Such an extended range of critical behavior is a feature of quasi-2D triangular <cit.> and kagome <cit.> lattice magnets.
Short-range magnetic order in the PM phase, with a correlation length $\xi \gg a$, the lattice parameter, can in principle persist up to a temperature scale of the AFM exchange energy, i.e., $T \sim J \gg T_N$. As was shown in Ref. <cit.>, the character of electron-magnon interactions does not change markedly at $T_N$ assuming that $\xi \gg k_F^{-1}$ which for metals is the same as $T < J$. Below a threshold temperature $T^{*} \sim (\Delta / E_F)J \sim$ 20–50 K, where $\Delta$ is the AFM gap and $E_F$ the Fermi energy, the thermopower will be dominated by the usual diffusive term that is weakly dependent on magnetic field. Above $T^{*}$, however, the strong electron-magnon interaction with $q \rightarrow Q$ will become relevant <cit.>, giving rise to a magnon-drag contribution $S_g$ to the total thermopower, which can become significant provided that the magnon-electron scattering is comparable to the scattering of magnons by defects, which seems a reasonable assumption in a clean, stoichiometric PdCrO$_2$. The magnon drag term will be strongly dependent on the spin-wave spectrum which is also very sensitive to magnetic field <cit.>. According to the theory described in Ref. <cit.>, AFM magnons become unstable with respect to two-magnon decay processes at high enough magnetic fields. This effect leads to a strong suppression of magnon drag and its contribution to the thermoelectric power which explains qualitatively the dramatic growth of the MTP at higher temperatures. This picture should be contrasted with the typical magnon-drag scenario proposed in AFM and FM metals where a strong MTP is only seen below the magnetic transition and is absent above it <cit.>. The persistence of such a term in PdCrO$_2$ above $T_N$ is then a direct consequence of the highly frustrated nature of the magnetic order arising from the combined effects of geometrical frustration and low dimensionality.
Another interesting possibility for the large field suppression of $S$ at elevated temperatures is a reduction of the scattering amplitudes related to the interaction between the itinerant electrons and the emergent spin chirality. At temperatures above $T_N$, the fluctuating Cr spins are easily aligned in a field, forming new spin textures which give rise, owing to their triangular arrangement, to a finite scalar spin chirality. Indeed, analysis of electron spin resonance experiments in PdCrO$_2$ have shown evidence for spin relaxation processes involving $Z_2$ vortices associated with chiral fluctuations of the 120$^\circ$ spin structure extending up to 300 K <cit.>. Moreover, the unconventional AHE observed in PdCrO$_2$ above $T_N$ has been attributed to a strong coupling of the itinerant electrons to the emergent field-induced spin chirality <cit.>. The evolution of the field-induced suppression of the thermopower in PdCrO$_2$ suggests that both phenomena may be linked to the same physics, since below $T_N$, where the 120$^\circ$ spin structure becomes increasingly more resilient to an applied field, both the MTP and AHE are correspondingly reduced. The form of the in-plane resistivity in PdCrO$_2$ above $T_N$ has also been attributed to magnetic scattering off short-range magnetic correlations among the frustrated spins <cit.>. It is therefore tempting to attribute the suppression of $S_{ab}(B)$ to a similar effect. For the effect to grow with increasing temperature, however, either the coupling of the conduction electrons to the underlying spin textures would have to become stronger, or the induced chirality more pronounced as $T$ increases.
In summary, we have uncovered a marked suppression of the in-plane thermopower of the metallic frustrated antiferromagnet PdCrO$_2$ in high magnetic fields up to 32 tesla. Certain features of the thermoelectric response suggest that this suppression is in the metallic, rather than in the spin entropic contribution to $S_{ab}$. The temperature evolution of the suppression, in particular, implies a dominant magnon-drag contribution that persists far beyond $T_N$ due to the thermally-robust interaction between the conduction electrons and the short-range magnetic correlations.
The authors thank C. Proust and N. Shannon for stimulating discussions and acknowledge the support of the HFML-RU/FOM and HLD/HZDR, members of the European Magnetic Field Laboratory (EMFL). The work at POSTECH was supported by the NRF through SRC (Grant No. 2011-0030785), Max Planck POSTECH/KOREA Research Initiative (Grant No. 2011-0031558) Programs, and also by IBS (No. IBSR014-D1-2014-a02). A portion of this work was performed with the help of Dr. E. S. Choi at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490, the State of Florida, and the U.S. Department of Energy.
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In our previous papers
we showed that the Hamilton–Jacobi problem
can be regarded as a way to describe a given dynamics on a
phase space manifold
in terms of a family of dynamics on a lower-dimensional manifold.
We also showed how constants of the motion
help to solve the Hamilton–Jacobi equation.
Here we want to delve into this interpretation
by considering the most general case:
a dynamical system on a manifold
that is described in terms of a family of dynamics
(`slicing vector fields')
on lower-dimensional manifolds.
We identify the relevant geometric structures
that lead from this decomposition of the dynamics
to the classical Hamilton–Jacobi theory,
by considering special cases like fibred manifolds
and Hamiltonian dynamics,
in the symplectic framework and the Poisson one.
We also show how a set of functions on a tangent bundle
can determine a second-order dynamics for which they are
constants of the motion.
Key words:
Hamilton–Jacobi equation, slicing vector field,
complete solution, constant of the motion.
MSC 2010: 70H20, 70G45
§ INTRODUCTION
Hamilton–Jacobi theory originated with Hamilton
to deal with what nowadays is called Hamiltonian optics,
i.e.to describe the ray propagation of light,
and with Jacobi who was interested
in devising a procedure to integrate equations of motions
when they are given in canonical form.
In Jacobi's own words:
“After we have reduced the problems of mechanics to
the integration of a nonlinear first order
partial differential equation, we must concern
ourselves with the integration of the same,
i.e., with the search for a complete solution”
Hadamard <cit.>
and Volterra <cit.>
derived the Hamilton–Jacobi equations
by considering the short-wave limit of wave equations.
It was this association which paved the way for de Broglie
to introduce the relation
p\,\d x-H\, \d t = \hbar\,(k\,\d x-\omega\,\d t)
relating wave concepts with particle concepts <cit.>.
Using this analogy,
Schrödinger proposed the evolutionary equation for wave mechanics,
opening the route to a formalism
able to describe physical phenomena at atomic scale.
(A geometrical description of the quantum–to–classical transition
on space–time was elaborated by Synge
Concerning the role of solutions to the Hamilton–Jacobi equation,
providing a family of solutions for Hamilton's equations,
Dirac wrote
“The family does not have any importance
from the point of view of Newtonian mechanics;
but it is a family which corresponds to one state of motion
in the quantum theory,
so presumably the family has some deep significance in nature,
not yet properly understood”.
These general comments are aimed at contextualizing
the role of the Hamilton–Jacobi theory in theoretical physics.
To enter the
raison d'être of the present paper,
let us recall how Hamilton–Jacobi theory is usually dealt with
in textbooks and works on analytical mechanics
Hamilton–Jacobi theory is usually considered
when dealing with canonical transformations
to define them by means of generating functions.
Specifically, by using canonical coordinates,
say $(p,q;t)$ and $(\bar p,\bar q;t)$,
one looks for a function
$S \colon Q \times Q \times \R \to \R$
such that
p\,\d q-H\,\d t
\bar p\,\d \bar q-K\,\d t+\d S(q,\bar q;t)
\,,
with $H$ and $K$ Hamiltonian functions on phase space.
The associated transformation is defined by means of
the implicit equations
p = \frac{\partial S}{\partial q}
\,,\quad
\bar p = -\frac{\partial S}{\partial \bar q}
\,,\quad
K-H = \frac{\partial S}{\partial t}
\,,
and this canonical transformation,
if it exists,
converts the Hamiltonian system described by $H$
into the one described by $K$.
By further requiring that $K$ is a constant
or that it is a function depending only on $\bar p$,
one relates the original system
to another one which is completely integrable,
and therefore integrable by quadratures.
The short-wave limit point of view
starts from a second order partial differential equation
of hyperbolic type and derives what is known
as the eikonal equation
\left(\frac{\partial S}{\partial x}\right)^2+
\left(\frac{\partial S}{\partial y}\right)^2+
\left(\frac{\partial S}{\partial z}\right)^2
with $n$ denoting the refractive index
The function $S$ is usually called the
eikonal function
or the characteristic function.
As a matter of fact,
Hamilton introduced two functions,
$S(t,x,y,z)$, called the principal function,
and putting $W(x,y,z)-t\, E=S(t,x,y,z)$,
$W$ was called the characteristic function <cit.>.
From the point of view of Jacobi
the integration of Hamilton's equations is achieved
by solving first the first-order differential equation
on configuration space
\frac{\d q^j}{\d t}
\left. \frac{\partial H}{\partial p_j} \right|%
_{p_j=\frac{\partial S}{\partial q^j}}
\,;
then, setting
p_j=\frac{\partial S}{\partial q^j}(t,q^j(t))
\,,
one finds a full solution of Hamilton's equations
with initial condition $(q^j(0), p_j(0), t=0)$.
Thus, from this point of view,
the Hamilton–Jacobi equation is instrumental
to define a
family of first–order differential equations
configuration space
whose solutions will eventually produce solutions
for the Hamilton equations on phase space.
In the first-order differential equation
(fode, in the sequel)
\frac{\d q^j}{\d t}
\left. \frac{\partial H}{\partial p_j} \right|%
_{p_j=\frac{\partial S}{\partial q^j}}
one changes the values
of the arbitrary constants appearing
in a complete integral function $S$
and obtains a family of fodes.
The solutions of each one of these equations
are the solutions alluded to by Dirac
and correspond to a given $S$,
related to the phase of the wave function in quantum mechanics.
Therefore a complete solution to the Hamilton–Jacobi equation
gives rise to a family of first-order differential equations
on the configuration space,
say $Q$,
which are sufficient to recover
all the solutions to Hamilton's equations on $\Tan^*Q$.
From the geometrical point of view,
a complete solution amounts to
an invariant foliation of $\Tan^*Q$,
with leaves diffeomorphic to $Q$
and transverse to fibres of the cotangent bundle projection.
A family of first–order differential equations is obtained
by restricting the Hamiltonian vector field
to each leaf of the invariant foliation.
From all that we have said about Hamilton–Jacobi theory
it is clear that we may identify
two main aspects in the Hamilton–Jacobi theory.
The first one is to solve a fode in a manifold $P$
(usually $\Tan^*Q$)
by solving an associated family of
fode's on a lower dimensional manifold $Q$;
when all the solutions may be found in this manner, the family is
said to be complete.
The second one consists of finding this complete family by solving
an associated PDE for a single function $S$,
this would be the analog of the eikonal equation.
To analyse these problems
we introduce a general scheme by means of
a vector field $Z$ on a manifold $P$,
along with a fibration $P\to M$.
We consider all the integral curves of $Z$ on $P$
and project them onto $M$.
Having all these curves on $M$, we would like to `group' them
into coherent sets of integral curves for vector fields on $M$.
In other terms, we would like to put together
all those integral curves of $Z$ which may be obtained as
integral curves of a certain vector field $X$ on $M$.
If all integral curves of $Z$ may be grouped into families
such that each family, after projection,
arises as integral curves of a vector field $X$ on $M$,
we say that the family of vector fields $X$
is a
complete slicing
of the dynamics $Z$,
or that it is a complete solution to the
generalized Hamilton–Jacobi problem.
This paper deals mostly with the first aspect,
i.e., to solve a differential equation on $P$ by means of
a familly of differential equations on $Q$.
A similar problem, i.e.going from trajectories to vector fields on $M$,
was discussed in
It is shown there that, in this generality,
by no means the problem will have solutions.
Thus, the existence of a family of vector fields on $M$
sufficient to reproduce all integral curves of $Z$ on $P$
will put quite strong conditions on $Z$.
We have already remarked that
for Hamiltonian systems on $P=\Tan^*Q$
the existence of the family
would require $Z$ to be a completely integrable system.
Of course a kind of inverse problem could be posed:
given a family of vector fields on $M$,
is it possible to find a vector field $Z$ on $P$
such that it would be possible to represent
the whole family of integral curves of the various vector fields on $M$
as projections of integral curves of the vector field $Z$ on $P$?
Let us stress that these problems would arise
in particular physical problems
like motion of particles with internal structure and
in general in problems with
restricted allowed Cauchy data,
for instance, gauge terms.
It would also occur in quantum mechanics
when we consider a composite system and
we would like to describe it
in terms of the evolution of subsystems
(entanglement would be an obstruction to the solution of
the posed inverse problem).
A single case where the inverse problem has a nice solution
is provided by a second order vector field on $\Tan Q$,
completely determined by a suitable family of
functionally independent constants of the motion,
as we show at the end of the paper.
To pin-point the geometrical contents
of the standard Hamilton–Jacobi equation,
first we shall consider the usual Hamilton–Jacobi
theory from a more geometric point of view.
In the usual approach
and $\pi \colon P\to Q$ is the usual cotangent bundle projection.
The dynamical vector field $\Gamma=Z$
solves the equation $\mathrm{i}_{\Gamma}\omega=\d H$,
$\omega$ is the canonical symplectic structure in $\Tan^*Q$ and
$H$ is the Hamiltonian function.
By using the symplectic potential for $\omega$,
say $\omega=-\d \theta_0$,
we define a vector field $\Delta$,
\theta_0$,
which represents the
linear structure along the fibers,
and the Hamilton–Jacobi equation for $S$ becomes
(\d S)^*\theta_0=\d S
\,,\quad
(\d S)^*H=E
\,,
where $E$ is a `parameter'.
When $S$ is a complete integral,
we have that
$\d S \colon Q \times N \to \Tan^*Q$
is a diffeomorphism for `most initial conditions'
for $\Gamma$.
It provides a $\dim Q$-foliation of $\Tan^*Q$
(or some open dense submanifold of it)
transversal to the fibers.
The vector field $\Gamma$, restricted to each leaf,
being tangent to it,
defines a vector field
which projects onto a vector field $X$ defined on $Q$.
There would be a vector field $X$ for each leaf.
In this manner the invariant foliation
defines a family of first-order differential equations on $Q$,
each one of them being the projection
of the restriction of $\Gamma$ to the invariant leaf.
This means that $\Gamma$ may be replaced by
the family of vector fields that we obtain
by restricting $\Gamma$ to a family of leaves transversal to the fibres.
Thus the issue becomes
how to find an invariant foliation transversal to fibres.
These and other intrinsic considerations about
the Hamilton–Jacobi equation can be found in
In addition, in
<cit.> a general geometric framework for the Hamilton–Jacobi theory
was presented
and the Hamilton–Jacobi equation in the
Lagrangian and in the Hamiltonian formalisms was formulated
for autonomous and non-autonomous mechanics,
recovering the usual Hamilton–Jacobi equation as a special case
in this generalized framework.
The relationship between the Hamilton–Jacobi equation
and some geometric structures of mechanics were analyzed also in
A similar generalization of the Hamilton–Jacobi formalism
was outlined in
Later on, these geometric frameworks were used to develop the
Hamilton–Jacobi theory in many different situations.
Thus, in
this is done for holonomic and non-holonomic mechanical systems,
in <cit.>
the theory is extended for singular systems,
in <cit.> and
for geometric mechanics on Lie algebroids and almost-Poisson manifolds
in <cit.>
for control theory,
in <cit.>
for different formulations of classical field theories
(and in <cit.> for partial differential equations in general),
and in
for higher order dynamical systems and higher-order field theories.
Finally, the geometric discretization of the Hamilton–Jacobi equation
is also considered in
In particular, in our previous papers
we saw that the Hamilton–Jacobi problem
can be regarded as a way
to describe a given dynamics on a phase space manifold
in terms of a family of dynamics on a lower-dimensional manifold.
Moreover, we saw that the existence of many constants of the motion
for the given dynamics helps to solve the Hamilton–Jacobi problem.
The aim of this paper is
to look more deeply into
this interpretation
by considering the most general case and
identifying what are the relevant geometric structures.
We should remark that our framework allows to handle
dynamical vector fields
which cannot be handled with classical approaches to
Hamilton–Jacobi equation.
For instance,
suppose we have a completely integrable Hamiltonian system
given by a Hamiltonian vector field $Z_H$;
its Hamilton–Jacobi equation has a complete solution,
and therefore we have a complete slicing of the dynamics.
Then consider a new dynamics given by
$Z' = f\,Z_H$,
where $f$ is a generic function
—this leads to a reparametrization of the integral curves.
Our procedure allows to construct a complete slicing for $Z'$,
although $Z'$ may not be Hamiltonian.
(An instance where this reparametrization may be required
is when $Z_H$ is not a complete vector field.)
The paper is organized as follows:
In section 2 we present the general concepts and results needed to state
a more general framework for the Hamilton–Jacobi problem.
The study of constants of the motion and complete solutions and
their relationship for this general setting is done in section 3,
by introducing the concept of slicing vector fields
and complete slicings.
Section 4 is devoted to discuss some particular situations deriving from
this general framework, such as Hamiltonian systems defined on
symplectic and Poisson manifolds.
The slicing problem is discussed again in section 5 in the case
where the dynamical system, either general or Hamiltonian,
is defined on a generic fibered manifold.
Finally, in section 6
we show how our previous results in
<cit.> are recovered form here,
and we also study how the knowledge of enough constants of the motion
determines a second-order dynamics.
Along the work,
different examples are also introduced to illustrate our results.
All the manifolds and maps are assumed to be $\Cinfty$.
§ DYNAMICAL SYSTEMS, INVARIANT SUBMANIFOLDS AND CONSTANTS OF THE MOTION
§.§ Dynamical systems
A dynamical system is a pair $(P,Z)$
given by a manifold $P$ and a vector field $Z$ on $P$.
This defines a (first-order, autonomous) differential equation
on $P$,
$\gamma' = Z \comp \gamma$,
for a path $\gamma \colon I \to P$.
This dynamics may possess several features.
For the purposes of this work we are especially interested in
invariant submanifolds and constants of the motion.
A submanifold $M \subset P$ is said to be invariant by $Z$
when the flow of $Z$ leaves $M$ locally invariant,
or, in other words,
when every integral curve of $Z$ meeting $M$ is
contained in $M$ at least for some time
(if $M$ is not closed then this integral curve may
eventually leave it).
These conditions are equivalent to saying that
$Z$ is tangent to $M$.
The preceding definition is applicable to regular submanifolds
but also to immersed submanifolds.
A particular instance of invariant submanifolds
is provided by constants of the motion.
In its most elementary form
a constant of the motion for $Z$ is a function
$f \colon P \to \R$
such that,
along every integral curve $\gamma$ of $Z$,
the function $f \comp \gamma$ is constant.
This is equivalent to saying that
the Lie derivative of $f$ with respect to $Z$ is zero,
$\Lie_Z f = 0$.
In the same way one can consider
a vector-valued constant of the motion
$F \colon P \to \R^n$,
whose components are scalar constants of the motion,
or, more generally,
a manifold-valued map
$F \colon P \to N$
such that for every integral curve $\gamma$
the map $F \comp \gamma$ is constant.
If $c \in N$,
then the closed subset $F^{-1}(c) \subset P$
is clearly invariant by $Z$.
So, those of the sets $F^{-1}(c)$ that are not empty
constitute a partition of $P$.
In some cases we can ensure that they are also submanifolds,
for instance when $F$ is a submersion.
In this case, constants of the motion provide a whole family
of invariant submanifolds.
Of course, not all invariant submanifolds are levels sets of
constants of the motion.
A very simple example is given by the planar system
$\dot x = -y + x(1\!-\!x^2\!-\!y^2)$,
$\dot y = x + y(1\!-\!x^2\!-\!y^2)$,
that reads in polar coordinates
$\dot r = r(1\!-\!r^2)$,
$\dot\phi = 1$;
it has an equilibrium point (the origin),
a limit cycle ($r=1$),
and no nontrivial global constants of the motion.
More interesting examples are provided by Liénard's equation
and the particular case given by
van der Pol's equation.
For instance, the system
$\dot x = -y+x \,\sin(x^2+y^2)$,
$\dot y = x+y \,\sin(x^2+y^2)$,
has a countable number of limit cycles.
§.§ A general framework for the Hamilton–Jacobi theory:
slicing vector fields
One of the distinctive facts of the Hamilton–Jacobi equation
is that it allows to describe the dynamics given by the
Hamilton equation on the cotangent bundle
in terms of a family of first-order dynamics on the configuration
(as for instance in
According to this general principle,
to describe the dynamics $Z$ on $P$
in terms of other dynamics on lower-dimensional manifolds,
we consider another
manifold $M$,
a vector field $X$ on $M$,
and a map $\alpha \colon M \to P$.
The following diagram captures the situation:
\xymatrix{
*++{\Tan M} \ar[r]^{\Tan \alpha} \ar[d]_{} &
*++{\Tan P} \ar[d]_{}
\\
*++{M} \ar[r]^{\alpha} \ar@/^3mm/[u]^{X} &
*++{P} \ar@/_3mm/[u]_{Z}
What can be said about the relation between $X$, $\alpha$ and $Z$?
The following results are well-known:
Given the preceding data,
the following properties are equivalent:
0pt plus 1pt
* For every integral curve $\xi$ of $X$,
$\zeta = \alpha \comp \xi$ is an integral curve of $Z$.
* $X$ and $Z$ are $\alpha$-related
($X \relat{\alpha} Z$),
that is to say,
αX = Z α ,
Suppose moreover that $\alpha$ is an injective immersion,
thus inducing a diffeomorphism
$\alpha_\cc \colon M \to \alpha(M)$
of $M$ with an immersed submanifold
$\alpha(M) \subset P$.
Then the preceding properties are also equivalent to
0pt plus 1pt
* $Z$ is tangent to $\alpha(M)$,
and, if $Z_\cc$
is the restriction of $Z$ to $\alpha(M)$,
$X$ is given by the pullback
$X = \alpha_\cc^*(Z_\cc)$.
In this case,
the map
$\xi \mapsto \alpha \comp \xi$
is a bijection between integral curves of $X$
and integral curves of $Z$ passing through $\alpha(M)$.
When these conditions hold,
we can regard $X$ as a `partial dynamics',
or a `slice' of the dynamics given by $Z$.
if we knew enough of these slices,
we could recover the whole dynamics of $Z$.
Given a dynamical system $(P,Z)$,
we will call a
slicing of it
a triple
satisfying the
slicing equation (<ref>).
When $\alpha$ is an immersion
the vector field $X$, if it exists,
is uniquely determined by $\alpha$ and $Z$;
so, in this case, we can speak of $(M,\alpha)$
being a solution of the slicing equation for $(P,Z)$.
This hypothesis
will hold in many applications,
in particular for the sections $\alpha$ of a bundle $P \to M$
(as a matter of fact, they are embeddings).
As we will see later on in this paper,
equation (<ref>) may be thought of as a
generalisation of the
Hamilton–Jacobi equation.
One of our main purposes is
to identify the precise conditions
that take us from the slicing equation to
the Hamilton–Jacobi equation.
Coordinate expression
Let us express equation (<ref>) in coordinates.
Consider coordinates
$(x^i)$ in $M$,
$(z^k)$ in $P$,
and use them to express
the map
$\alpha(x) = (a^k(x))$
the vector fields
$X = X^i \,\tanvec{x^i}$,
$Z = Z^k \,\tanvec{z^k}$.
Then the difference
\Tan \alpha \comp X - Z \comp \alpha
(x^i) \mapsto
\left(
a^k(x) ,
\derpar{a^k}{x^i} \, X^i -
\right)
\,,
and so
$(M,\alpha,X)$ is a solution of the slicing equation iff
\derpar{a^k}{x^i} \, X^i(x) =
Z^k (\alpha(x))
§.§ Gauge freedom of the solutions
The notion of a slicing of $Z$ has a certain
`gauge freedom',
in the sense that with a given solution $(M,\alpha,X)$
there exist many associated solutions that are
equivalent to it:
if $\varphi \colon M' \to M$ is a diffeomorphism
$(M',\alpha \comp \varphi,\varphi^*(X))$
is also a solution of the slicing equation.
There are two situations where this freedom can be easily removed.
One, to be studied later on,
occurs when $P$ is assumed to be fibred over a manifold
and one only deals with maps $\alpha$
that are sections of this projection.
The other one is provided by invariant submanifolds of $P$.
Indeed, this is an immediate consequence of proposition 1:
$P_\cc \subset P$
be a regular submanifold.
The canonical inclusion
$j \colon P_\cc \hookrightarrow P$
is a solution of the slicing equation
$Z$ is tangent to $P_\cc$.
Every other solution given by an embedding $\alpha$
with $\alpha(M) = P_\cc$
is equivalent to it.
§ CONSTANTS OF THE MOTION AND COMPLETE SOLUTIONS
§.§ Constants of the motion
We still deal with our dynamical system $(P,Z)$.
A (generalized)
constant of the motion
of it is a map
$F \colon P \to N$ into another manifold $N$
satisfying the following property:
for any integral curve
$\zeta \colon I \to P$ of $Z$,
$F \comp \zeta$ is constant.
We consider the isotropic harmonic oscillator
with two degrees of freedom
(with phase space $\R^4$),
\begin{array}{rcl} \dot x&=& -y\\
\dot y&=&x.
\end{array}
All its integral curves are a foliation of
$\R^4 -\{0\} \cong \S^3 \times \R^+$ onto
$\R^3 -\{0\} \cong \S^2 \times \R^+$
and the projection
$\R^4 -\{0\} \to \R^3 -\{0\}$
(Kustaanheimo–Stiefel map),
or $\S^3 \to \S^2$,
is a constant of the motion.
We have several characterisations of this property:
The following properties are equivalent:
0pt plus 1pt
* $F$ is a (manifold valued) constant of the motion.
* Each integral curve $\eta$ of $Z$ is contained in a level set
$F^{-1}(c)$ of $F$.
* $Z$ is $F$-related with the zero vector field of $N$:
($Z \relat{F} 0$).
Suppose moreover that $F$ is a submersion
(thus $\Ker \Tan F \subset \Tan P$
is an integrable tangent subbundle
whose associated foliation has as leaves the
level sets $F^{-1}(c)$,
which are closed submanifolds of $P$).
Then the preceding properties are also equivalent to
0pt plus 1pt
* $Z$ takes its values in $\Ker \Tan F$.
* $Z$ is tangent to every level set $F^{-1}(c)$.
The following diagram summarizes the situation:
\xymatrix{
*++{\Tan P} \ar[r]^{\Tan F} \ar[d]_{} &
*++{\Tan N} \ar[d]_{}
\\
*++{I} \ar[r]^{\eta} &
*++{P} \ar[r]^{F} \ar@/^3mm/[u]^{Z} &
*++{N} \ar@/_3mm/[u]_{0}
The tangency of $Z$ to a certain submanifold
shows up in propositions 1 and 2.
This comparison suggests that
a constant of the motion
is related to a whole family of solutions of
the slicing equation,
as we are going to show.
§.§ Complete solutions
A single solution
$\alpha \colon M \to P$, $X \colon M \to \Tan M$,
of the slicing equation
allows to describe the integral curves of $Z$
contained in $\alpha(M) \subset P$.
To describe all of its integral curves
we need a complete solution.
This can be defined as a family of solutions indexed by some
parameter space $N$.
Given a dynamical system $(P,Z)$, a
complete slicing
of it is given by
* a map
\overline\alpha \colon M \times N \to P
* a vector field
\overline X \colon M \times N \to \Tan M
along the projection
$M \times N \to M$
(that is, smooth families of
$\alpha_c \equiv \overline\alpha(\cdot,c) \colon M \to P$
and vector fields
$X_c \equiv \overline X(\cdot,c) \colon M \to \Tan M$,
both indexed by the points $c \in N$)
such that:
* $\overline\alpha$ is surjective
(or at least its image is an open dense subset),
* for each $c \in N$,
the map
$\alpha_c %\equiv \overline\alpha(\cdot,c)
\colon M \to P$
and the vector field
$X_c %\equiv \overline X(\cdot,c)
\colon M \to \Tan M$
constitute a slicing of $Z$.
\xymatrix{
*++{\Tan M \times N} \ar[r]^{\Tan_1 \overline\alpha} \ar[d]_{} &
*++{\Tan P} \ar[d]_{}
\\
*++{M \times N} \ar[r]^{\overline\alpha} \ar@/^3mm/[u]^{\overline X} &
*++{P} \ar@/_3mm/[u]_{Z}
Since (almost) every $z \in P$ is the image by $\overline\alpha$ of a point
$(x,c) \in M \times N$,
the integral curve of $Z$ through $z$
can be described as the integral curve of $X_c$ through $x$
by means of the map $\alpha_c$.
When each $\alpha_c$ is an immersion
(for instance, when $\overline\alpha$ is a diffeomorphism)
the vector fields $X_c$ are determined by the $\alpha_c$,
so in this case we do not need to specify $\overline X$
to define the complete solution.
The simplest example of a solution of the slicing equation
for a vector field $Z$ is just given by its integral
curves $\alpha \colon I \to P$.
Indeed, consider the following diagram:
\xymatrix{
*++{\Tan I} \ar[r]^{\Tan \alpha} \ar[d]_{} &
*++{\Tan P} \ar[d]_{}
\\
*++{I} \ar[r]^{\alpha} \ar@/^3mm/[u]^{{\d \over \d t}}
\ar[ru]^{\alpha'}
*++{P} \ar@/_3mm/[u]_{Z}
The commutativity of its upper triangle is
the definition of the velocity $\alpha'$,
whereas the commutativity of the lower one
is the assertion that $\alpha$ being an integral curve of $Z$.
When this holds,
${\d \over \d t} \relat{\alpha} Z$,
which means that $\alpha$ is a solution of the
slicing equation for $Z$.
Let $z \in P$ be a noncritical point of $Z$.
Then one can build a
local complete slicing
around $z$.
To this end, consider a hypersurface
$N \subset P$ containing $z$,
and such that $Z(z)$ is transversal to $N$.
Then the restriction of the flow $F$ of $Z$
to a smaller product $I_\cc \times N_\cc$
gives a diffeomorphism
$F_\cc \colon I_\cc \times N_\cc \to P_\cc$
with an open neighbourhood
$P_\cc$ of $z$,
such that
${\partial \over \partial t} \relat{F_\cc} Z$.
So, $F_\cc$ with ${\partial \over \partial t}$
is a complete slicing for $Z$
restricted to $P_\cc$.
Indeed, this is the usual procedure to prove
the straightening theorem for vector fields.
§.§ Local existence of complete slicings
The preceding example can be extended
to prove a general existence theorem for complete slicings.
Indeed, we are going to prove that,
under some regularity conditions,
any given slicing
can be locally embedded in a regular local complete slicing.
Let $(P,Z)$ be a dynamical system,
and $z_\cc \in P$ a noncritical point of $Z$.
Let $(M,\alpha,X)$ be a solution
of the slicing equation for $Z$,
with $z_\cc = \alpha(x_\cc)$,
and such that
$\alpha$ is an immersion at $x_\cc$.
There exist
an open neighbourhood $M_\cc$ of $x_\cc$,
an open neighbourhood $N_\cc$ of 0 in $\R^n$
(where $n = \dim P - \dim M$),
a diffeomorphism
$\overline\alpha \colon M_\cc \times N_\cc \to P_\cc$
with an open neighbourhood $P_\cc$ of $z_\cc$,
such that
0pt plus 1pt
* $\overline\alpha$ is a complete slicing
for $Z|_{P_\cc}$,
* $\overline\alpha(\cdot,0) = \alpha|_{M_\cc}$.
Since the result is a local one,
and every immersion is locally an embedding,
the gauge freedom of the solutions of the slicing equation
allows us to suppose that $M$ is a regular submanifold of $P$
and that $\alpha$ is the inclusion.
The hypothesis is that $Z$ is tangent to $M$.
The proof of the straightening theorem for vector fields
can be adapted
to construct coordinates
$(z_1,\ldots,z_m,\ldots,z_p)$ around $z_\cc$
such that
$M$ is locally described by
$z_{m+1} = \ldots = z_p = 0$,
and that
$Z = \tanvec{z_1}$.
in a small product $M_\cc \times N_\cc$,
where the right-hand side is expressed in terms of
these coordinates.
In a small neighbourhood of $(x_\cc,0)$
this is a diffeomorphism,
and for every $s \in N_\cc$
the vector field $Z$ is tangent to the submanifold
Therefore $\overline{\alpha}$ is a complete slicing of $Z$.
§.§ Relation between complete slicings,
constants of the motion and connections
Now we are going to see that,
under some regularity hypotheses,
there is a close relationship between
complete slicings and constants of the motion.
Let $(P,Z)$ be a dynamical system,
\overline\alpha \colon M \times N \to P
a diffeomorphism.
is a complete slicing
for $Z$
$F = \pr_2 \comp \overline\alpha^{-1} \colon P \to N$
is a constant of the motion for $Z$.
\xymatrix{
*++{P} \ar[r]^{F} &
\\
*++{M \times N} \ar[u]^{\overline\alpha\,}
\ar[ru]_{\mathrm{pr}_2}
If $\overline\alpha$ is a complete slicing,
for each $c \in N$,
$\overline\alpha$ restricts to a map
$M \times \{c\} \to \alpha_c(M)$
which is a diffeomorphism,
and all the integral curves of $Z$ in $\alpha_c(M)$
correspond to a common value of $c$.
This means the map
$F = \pr_2 \comp \overline\alpha^{-1} \colon P \to N$
is a constant of the motion.
from $F = \pr_2 \comp \overline\alpha^{-1}$
we have that, for every $c$,
$F(\overline\alpha(x,c)) = c$,
or $\alpha_c(M) \subset F^{-1}(c)$.
Both submanifolds have the same dimension,
since $F$ is a constant of the motion,
$Z$ is tangent to
therefore $Z$ is tangent to
which proves that
the $\alpha_c$ are solutions to slicing equation for $Z$.
This result shows that there is a bijection
between complete slicings and constants of the motion,
but these are being assumed to satisfy a very strong
regularity condition,
which essentially requires that
all the level sets $F^{-1}(c)$
are diffeomorphic to a common manifold $M$,
in such a way that gluing the collection of diffeomorphisms
$M \to F^{-1}(c)$
yields a diffeomorphism
$\overline\alpha \colon M \times N \to P$.
Of course,
these conditions are very restrictive,
but in practice they may hold in a generic way.
We will see this in some examples.
Consider the manifold $\R^2$ with the radial vector field
$Z = z_1 \,\tanvec{z_1} + z_2 \,\tanvec{z_2}$,
whose integral curves are
the equilibrium at the origin and
the paths $\zeta(t) = e^t (a_1,a_2)$,
$(a_1,a_2) \neq (0,0)$,
running along the half-lines from the origin.
To illustrate the preceding theorem
we have to exclude the origin:
$P = \R^2-\{0\}$.
The map $F \colon P \to N = \S^1$ given by
$F(z) = z/\|z\|$
is clearly a constant of the motion for $Z$.
Its level sets $F^{-1}(u)$ (for $u \in \S^1$)
are diffeomorphic to the real line $M = \R$;
for instance, by
$\alpha_u \colon \R \to P$,
$\alpha_u(x) = e^x u$.
All together yield a diffeomorphism
$\overline\alpha \colon \R \times \S^1 \to \R^2-\{0\}$:
$\overline\alpha(x,u) = e^x u$.
This is a complete solution of the slicing equation for $Z$.
The corresponding vector fields on $M$ are
$X_u = \tanvec{x}$.
The relationship between slicings and constants of the motion
is lost when we do not consider complete slicings.
A solution of the slicing equation doesn't need
to preserve any given constant of the motion,
and the preservation of a constant of the motion
does not guarantee that
a map is a slicing of the dynamics.
The simplest way to show all this is by an example.
We consider the manifold
$P = \R^3$,
with coordinates $(x,y,z)$,
and the simple dynamics given by the vector field
$Z = \tanvec{x}$.
The function $F = z$ is obviously a constant of the motion
with values in $\R$.
The map $\alpha \colon \R^2 \to \R^3$
given by $\alpha(u,v) = (u,v,0)$,
$F \comp \alpha = 0$, constant.
On the other hand,
$\bar \alpha(u,v) = (u,0,v)$
satisfies $(F \comp \bar\alpha) (u,v) = v$,
not constant.
Both $\alpha$ and $\bar\alpha$
are solutions of the slicing equation for $(P,Z)$,
since $Z$ is tangent both to the planes
$\alpha(\R^2)$ and $\bar \alpha(\R^2)$.
Now consider
$\beta \colon \R \to \R^3$
given by
$\beta(v) = (0,v,0)$.
Obviously $F \comp \beta = 0$
but $\beta$ is not a solution of the slicing equation
since $Z$ is not tangent to the line $\beta(\R)$.
§.§ Invariant foliations
The notion of complete solution is close to that of invariant
Roughly speaking,
a foliation of $P$ consists in describing it
as the disjoint union of immersed submanifolds.
This defines an integrable tangent distribution on $P$,
and conversely.
The leaves of the foliation
are solutions of the slicing equation for $Z$
$Z$ is tangent to the foliation
(or, in other words, if the foliation is invariant by the flow
of $Z$).
iff $Z$ is a section of the associated tangent distribution.
So, if $\overline\alpha \colon M \times N \to P$
is a complete slicing,
and with every partial map $\alpha_c$ an immersion,
then the submanifolds $\alpha_c(M)$ are a foliation of $P$
invariant by $Z$.
However, not every invariant foliation
can be defined by a global diffeomorphism in this way.
Consider the `irrational linear flow' on the
2–dimensional torus $\mathbb{T}^2$:
$\dot x = 1$, $\dot y = ry$,
with $r$ an irrational number.
Its integral curves are dense immersions $\R \to \mathbb{T}^2$.
These immersed submanifolds constitute a foliation
of the torus invariant by the flow.
there is no diffeomorphism $\R \times N \to \mathbb{T}^2$,
as well as no nontrivial constants of the motion.
In the usual Hamilton–Jacobi theory
a family of vector fields is usually determined
by solving an associated partial differential equation
of first order.
This requires the use of a skew-symmetric $(0,2)$-tensor field
which relates a vector field, say $Z$, with a 1-form.
The skew-symmetry ensures that
the contraction of $Z$ with the corresponding 1-form
identically vanishes.
§ SLICING OF HAMILTONIAN SYSTEMS
In the standard Hamilton–Jacobi theory
the skew-symmetric $(0,2)$-tensor is assumed to be
the natural symplectic structure of the cotangent bundle.
The classical Hamilton–Jacobi theory makes an essential use
of a symplectic structure.
In view of this,
we still consider the most general slicing problem
but now for a Hamiltonian system.
Thus $P$ is endowed with a symplectic form $\omega$,
which defines a vector bundle isomorphism
$\widehat \omega \colon \Tan P \to \Tan^*P$;
and $Z = Z_H$ is the Hamiltonian vector field
of a Hamiltonian function $H \colon P \to \R$:
$Z = \widehat\omega^{-1} \comp \d H$.
Consider a Hamiltonian dynamical system
and $Z = Z_H$ its Hamiltonian dynamical vector field.
Let $\alpha \colon M \to P$ be a map,
and $X$ an arbitrary vector field on $M$.
We have the following relations:
{}^{t}(\Tan \alpha)
\comp
\widehat\omega
\comp
\Tan \alpha \comp X
i_X \alpha^*(\omega)
\,,
{}^{t}(\Tan \alpha)
\comp
\widehat\omega
\comp
Z \comp \alpha
\d \,\alpha^*(H)
\,,
where all the vector bundle sections and maps
are understood to be over the base space $M$.
These relations are expressed in the following diagram
(we insist that, since we have to work with the transpose morphism
${}^{t}(\Tan \alpha)$,
all the involved vector bundles are considered
over the base space $M$):
\xymatrix{
*++{\Tan M} \ar[d]_{} \ar[r]^{\Tan \alpha \quad}
*++{M \times_\alpha \Tan P} \ar[r]^{\widehat\omega}
*++{M \times_\alpha \Tan^*P} \ar[r]^{\quad {}^{t}(\Tan \alpha)}
\\
\ar[ru]^{\Tan \alpha \comp X \;}_{\; Z \comp \alpha}
\ar@/_2mm/[urrr]^{ i_X \alpha\!^*(\omega) \quad}_{\qquad \d \,\alpha\!^*(H)}
The map
\Tan \alpha \comp X
is a vector field along $\alpha$,
$\widehat\omega \comp \Tan \alpha \comp X$
is a differential 1-form along $\alpha$,
and finally
its composition with the transpose morphism
${}^{t}(\Tan \alpha)$
(along $M$),
{}^{t}(\Tan \alpha)
\comp
\widehat\omega
\comp
\Tan \alpha \comp X
is the differential 1-form on $M$
i_X \alpha\!^*(\omega)
{}^{t}(\Tan \alpha)
\comp
\widehat\omega
\comp
\Tan \alpha
\widehat{\alpha^*(\omega)}
On the other hand,
since $Z$ is the Hamiltonian vector field of $H$,
\widehat\omega \comp Z \comp \alpha
\d H \comp \alpha
a differential 1-form along $\alpha$,
and its composition with the transpose morphism
${}^{t}(\Tan \alpha)$
is just de pull-back by $\alpha$ of $\d H$,
{}^{t}(\Tan \alpha)
\comp
\widehat\omega
\comp
Z \comp \alpha
\alpha\!^*(\d H)
\d \,\alpha\!^*(H)
With the preceding notations, if
$(M,\alpha,X)$ is a solution of the slicing equation
for $(P,Z)$,
\Tan \alpha \comp X - Z \comp \alpha = 0
i_X α^*(ω) - α^*(H) = 0
From the preceding lemma we have
ω(αX - Z α)
i_X α^*(ω) - α^*(H)
Notice by the way that,
if $\alpha\!^*(\omega)$ were a symplectic form on $M$,
then equation (<ref>) would mean that
$X$ is the Hamiltonian vector field associated with
the Hamiltonian function $\alpha\!^*(H)$.
Coordinate expressions
It is interesting to reproduce the proof of the previous equations
in coordinates.
Again we have local charts
$(x^i)$ in $M$,
$(z^k)$ in $P$,
and use them to express
$\alpha(x) = (a^k(x))$
$X = X^i \,\tanvec{x^i}$.
The symplectic form reads
\omega =
\frac12 \,\omega_{k\ell} \, \d z^k \wedge \d z^\ell
$\Omega = (\omega_{k\ell})$
is skew-symmetric.
The matrix of $\widehat\omega$ is $\Omega^\top$.
And the Hamiltonian vector field
Z = Z_H =
\derpar{H}{z^\ell} \, \omega^{\ell k} \tanvec{z^k}
where $(\omega^{k\ell}) = \Omega^{-1}$.
\Tan \alpha \comp X - Z \comp \alpha
in coordinates reads
(x^i) \mapsto
\left(
a^k(x) ;
\derpar{a^k}{x^i} X^i -
\derpar{H}{z^\ell}(\alpha(x)) \, \omega^{\ell k}(\alpha(x))
\right)
\,.
\alpha^*(\omega) =
\frac12 \, \omega_{k\ell}(\alpha(x)) \,
\derpar{a^k}{x^i} \, \derpar{a^\ell}{x^j} \,
\d x^i \! \wedge \d x^j
i_X \alpha^*(\omega) =
X^i \,
\omega_{k\ell}(\alpha(x)) \,
\derpar{a^k}{x^i} \derpar{a^\ell}{x^j} \,
\d x^j
\d \alpha^*(H) =
\derpar{H}{z^k}(\alpha(x)) \, \derpar{a^k}{x^j} \, \d x^j
so that
i_X \alpha^*(\omega) - \d \,\alpha^*(H)
\left(
X^i \,
\omega_{k\ell}(\alpha(x)) \,
\derpar{a^k}{x^i} \derpar{a^\ell}{x^j}
\derpar{H}{z^k}(\alpha(x)) \derpar{a^k}{x^j}
\right)
\d x^j
\,.
We see that multiplying the local expression of
\Tan \alpha \comp X - Z \comp \alpha
and then by
we obtain the local expression of
i_X \alpha^*(\omega) - \d \,\alpha^*(H)
In general the morphism ${}^{t}(\Tan \alpha)$ is not bijective,
therefore the implication in the previous proposition
cannot be inverted.
This is easily seen in an example.
Consider a Hamiltonian system $(P,\omega,H)$.
Let $\alpha \colon I \to P$ be any path that
is not a solution of the Hamilton's equation,
but such that $H \comp \alpha = \mathrm{const}$,
and consider the vector field $X = {\d \over \d t}$
on $I \subset \R$.
i_X \alpha^*(\omega) - \d \,\alpha^*(H)
vanishes trivially,
whereas, of course,
\Tan \alpha \comp X - Z \comp \alpha =
\alpha' - Z \comp \alpha \neq 0
The preceding proposition gives a link between
the slicing problem
and the usual formulation of the
Hamilton–Jacobi equation, $\d\,\alpha^*(H) = 0$.
However, we need to revert the direct implication,
and this can be done in some cases,
as it was already shown in our paper <cit.>.
There are at least two ways for doing this,
according to whether we have an isotropy condition, as below,
or a fibred structure, as in the next section.
§.§ Isotropic and Lagrangian embeddings
In this subsection we are going to study slicings
satisfying a geometric property with respect to the
symplectic form.
First we need to recall that
a submanifold $M \subset P$ is called
isotropic, coisotropic or Lagrangian <cit.>
when all the tangent spaces at each point are,
which means:
* isotropic:
$\Tan_zM \subset (\Tan_zM)^\bot$;
* coisotropic:
$(\Tan_zM)^\bot \subset \Tan_zM$;
* Lagrangian:
isotropic and coisotropic:
$\Tan_zM = (\Tan_zM)^\bot$.
(Here the orthogonality is taken with respect to the symplectic form.)
An important type of solutions $\alpha \colon M \to P$
of the slicing equation for $Z$ satisfy
the condition
α^*(ω) = 0
When $\alpha$ has constant rank
this condition means that, locally, the image
$\alpha(M) \subset P$
is an isotropic submanifold.
When $\alpha$ is an immersion this requires that
$\dim M \leq \frac12 \dim P$.
Of course, in this case the preceding proposition takes a simpler
a solution of the slicing problem
\d \,\alpha^*(H) = 0
whereas its converse is false,
as is also shown by the same preceding example.
To go further, we need a couple of lemmas.
The notation $F^\circ \subset E^*$ denotes the annihilator
of a vector subspace $F \subset E$.
Suppose that $\alpha$ is an embedding,
so that $P_0 = \alpha(M) \subset P$ is a submanifold.
* $\alpha^*(\omega) = 0$
$\widehat\omega(\Tan P_0) \subset (\Tan P_0)^\circ$,
i.e., $P_0 \subset P$ is an isotropic submanifold.
* $\widehat\omega(\Tan P_0) = (\Tan P_0)^\circ$
$\widehat\omega(\Tan P_0) \subset (\Tan P_0)^\circ$
and $\dim P = 2 \dim M$,
i.e., $P_0 \subset P$ is a Lagrangian submanifold.
The first statement is a consequence of the fact that
is essentially the restriction of $\omega$
to tangent vectors to $\alpha(M)$;
the second one is a matter of dimension counting: $m = p-m$.
If $\alpha$ is an embedding with $\alpha(M) = P_0$ then
$\Ker {}^{t}(\Tan \alpha) = (\Tan P_0)^\circ$.
Basic linear algebra applied to $\Tan_x\alpha$ for every $x \in M$.
Consider the following diagram, which contains all of these objects:
\xymatrix{
*++{M \!\times_\alpha\! \Tan P_0}
\ar@{^{(}->}[d]
*++{(M \!\times_\alpha\! \Tan P_0)^\circ}
\ar@{^{(}->}[d]
\\
*++{M \times_\alpha \Tan P} \ar[r]^{\widehat\omega}
*++{M \times_\alpha \Tan^*P} \ar[r]^{\quad {}^{t}(\Tan \alpha)}
\\
\ar[u]_{Z \comp \alpha}
\ar[ur]_{\d H \comp \alpha}
\ar@/_3mm/[urr]_{\qquad \alpha\!^*(\d H) \,=\, \d \,\alpha\!^*(H)}
Let $(P,\omega,H)$ be a symplectic Hamiltonian system,
with Hamiltonian vector field $Z$,
and let $\alpha \colon M \to P$ be an embedding.
If $\alpha$ is a solution of the slicing equation (<ref>)
(that is, $Z$ is tangent to $\alpha(M)$)
and satisfies the isotropy condition
($\alpha^*(\omega) = 0$)
then $\alpha$ satisfies
α^*(H) = 0
Conversely, if $\alpha$ satisfies this equation
and the Lagrangianity condition
($\alpha^*(\omega) = 0$ and $\dim P = 2 \dim M$)
then it is a solution of the slicing equation.
We have already proved the direct implication.
Conversely, if $\alpha\!^*(\d H)$ is zero then
$\d H \comp \alpha$ takes its values in the kernel,
which is
$(M \!\times_\alpha\! \Tan P_0)^\circ$.
When the Lagrangianity condition
$\widehat\omega(\Tan P_0) = (\Tan P_0)^\circ$
holds we conclude that
$Z \comp \alpha$ is a section of $\Tan P_0$,
or, in other words,
that $Z$ is tangent to $P_0$,
which is one of the ways of saying that $\alpha$ is slicing of $Z$.
So for Lagrangian embeddings
to solve the slicing equation
is equivalent to solving equation (<ref>).
We call these solutions Lagrangian slicings of $Z$.
§.§ Constants of the motion and involutivity
In the preceding section we have observed
the close relationship between complete slicings
and constants of the motion.
So, consider a submersion $F \colon P \to \R^n$,
with level sets $P_c \equiv F^{-1}(c)$.
The functions $F^i$ are in involution,
all the level sets $P_c$
are coisotropic submanifolds of $P$.
When $\dim P = 2n$ this means that the $P_c$ are
Lagrangian submanifolds.
The proof is easy, see
a complete slicing given by
$n$ constants of the motion in involution
has coisotropic leaves,
and if $\dim P = 2n$
then the leaves are Lagrangian submanifolds,
and conversely.
As all our preliminary analysis has been made without
the help of a $(0,2)$-tensor field,
it is clear that when the vector field $Z$ allows
for alternative invariant skew symmetric $(0,2)$-tensor fields,
it is possible to consider alternative cotangent bundle structures
on $P$ and therefore different projections.
We can consider the isotropic harmonic oscllator and if we write,
in coordinates
$(x,p) \in \R^2$,
x\,\cos \alpha + P\, \sin\alpha = q
\,, \quad
P\,\cos \alpha -x\, \sin\alpha=p
\,,
we have that
$\d q \wedge \d p = \d x \wedge \d P$,
$\d (p\,\d q) = \d(P\,\d x)$.
The fibering vector fields
$p \,\partial/\partial p$ and
$P \,\partial/\partial P$
are diffeomorphically related but induce
alternative cotangent bundle structures on $\mathbb{R}^2$
§.§ Local existence of complete Lagrangian slicings
In the preceding section we have proved a
local existence theorem for complete slicings.
Now we are going to prove a similar result
in the Hamiltonian framework,
for solutions satisfying the Lagrangianity condition.
See also
Let $(P,\omega,H)$ be a symplectic Hamiltonian system,
with Hamiltonian vector field $Z$,
and $z_\cc \in P$ a noncritical point of $H$.
There exists a Lagrangian slicing of $Z$
passing through $z_\cc$.
Indeed, this slicing is contained in
a local complete Lagrangian slicing of $Z$.
By applying the Carathéodory–Jacobi–Lie theorem
—see for instance
$H$ can be included in a set of local Darboux coordinates
centered at $z_\cc$.
Then $Z = \derpar{}{q^1}$
is tangent to the Lagrangian submanifolds of $P$ defined by
$p_1 = c_1$, … , $p_n = c_n$.
These submanifolds constitute the complete slicing we sought.
§.§ Poisson Hamiltonian systems
The preceding argument can be adapted to the Poisson case.
Let $P$ be a manifold endowed with an almost-Poisson tensor field
$\Lambda$, that is to say,
a section of $\mathsf{\Lambda}^2 \Tan P$.
This defines a vector bundle morphism
\widehat\Lambda \colon \Tan^*P \to \Tan P
\left\langle \beta , \widehat\Lambda(\alpha) \right\rangle
\Lambda(\alpha,\beta)
The image of this morphism,
$C = \mathrm{Im} \widehat\Lambda \subset \Tan P$,
is called the
characteristic tangent distribution
of $\Lambda$.
If $\Lambda$ has constant rank then $C$ is a vector subbundle.
We will need a generalisation of the concept of
Lagrangian submanifold to the Poisson case.
A submanifold $P_0 \subset P$ of an almost-Poisson manifold
is called Lagrangian
\widehat\Lambda
\left(
(P_0 \times_{P_0} \Tan P_0)^\circ
\right)
\Tan P_0 \cap (P_0 \times_{P_0} C)
\,.
The almost-Poisson tensor field also defines an almost-Poisson bracket
\{f,g\} = \Lambda (\d f ,\d g)
which is skew-symmetric
and a derivation on each of its arguments
(it does not necessarily satisfy the Jacobi identity
unless the Schouten bracket vanishes, i.e.,
Suppose that we have a Hamiltonian function
$H \colon P \to \R$,
which defines a Hamiltonian vector field
$Z = Z_H = \widehat\Lambda \comp \d H$
and the corresponding Hamiltonian dynamics.
We want to study the slicing problem for $(P,Z)$.
As before, we consider the elements in this diagram,
but notice that $\Lambda$ may be degenerate:
\xymatrix{
*++{M \!\times_\alpha\! \Tan P_0}
\ar@{^{(}->}[d]
*++{(M \!\times_\alpha\! \Tan P_0)^\circ}
\ar@{^{(}->}[d]
\\
*++{M \!\times_\alpha\! \Tan P}
*++{M \!\times_\alpha\! \Tan^*P}
\ar[l]_{\widehat\Lambda}
\ar[r]^{\qquad {}^t(\Tan \alpha) \quad}
\\
\ar@/^2mm/[ul]^{Z \comp \alpha}
\ar[u]_{\d H \comp \alpha}
\ar@/_2mm/[ur]_{\quad \alpha\!^*(\d H) \,=\, \d \,\alpha\!^*(H)}
Let $E$ be a finite-dimensional vector space,
$E^*$ its dual space,
$E_0 \subset E$ a vector subspace,
$\lambda \colon E^* \to E$ a linear map,
$\delta \in E^*$ a covector.
Denote by
$E_0^\circ \subset E^*$
the annihilator of $E_0$
and by
${}^t\lambda \colon E^* \to E$
the transpose map of $\lambda$.
$\lambda(\delta) \in E_0$
$\delta \in ({}^t\lambda(E_0^\circ))^\circ$.
If moreover $\lambda$ is
symmetric or skew-symmetric
(${}^t \lambda = \pm\lambda$),
$\lambda(\delta) \in E_0$
$\delta \in (\lambda(E_0^\circ))^\circ$.
Let $(P,\Lambda,H)$ be an almost-Poisson Hamiltonian system,
with Hamiltonian vector field $Z$.
Let $\alpha \colon M \to P$ be an embedding
with image
$\alpha(M) = P_0$.
Then $\alpha$ is a solution of the slicing equation
Ḥ α is a section of
(M ×_αP_0)^∘)
Suppose that $P_0 \subset P$ is a Lagrangian submanifold.
Then $\alpha$ is a solution of the slicing equation
Ḥ α is a section of
(M ×_αP_0)^∘+ (M ×_αΛ)
that is to say,
is a section of
^t(α) (Λ)
The first statement is a consequence of the lemma.
As for the second statement,
being $P_0$ Lagrangian means that
\widehat\Lambda\left(
(P_0 \times_{P_0} \Tan P_0)^\circ
\right)
\Tan P_0 \cap (P_0 \times_{P_0} C)
When restricted this to $\alpha$ and with the annihilator we have
\left(
\widehat\Lambda\left(
(M \times_\alpha \Tan P_0)^\circ
\right)
\right)^\circ
\left(
(M \times_\alpha \Tan P_0) \cap (M \times_\alpha C)
\right)^\circ
(M \times_\alpha \Tan P_0)^\circ + (M \times_\alpha C)^\circ
and remember that
C^\circ = \Ker {}^t\widehat \Lambda = \Ker \widehat \Lambda
The symplectic case is obtained when $C = \Tan P$,
or equivalently when $\Ker \widehat \Lambda = \{0\}$.
Then for the Lagrangian case
the last statement in the theorem means that
$\alpha$ is a slicing iff
$\alpha\!^*(\d H) = 0$,
as was already given by theorem 3.
$P = \R^3$
with coordinates $(x,y,z)$
and the Poisson structure given by the Poisson bracket
\{f,g\} =
z \left(
\derpar{f}{y} \derpar{g}{x} - \derpar{f}{x} \derpar{g}{y}
\right)
this is the Lie–Poisson structure constructed
from the Heisenberg Lie algebra
The Hamiltonian function
H = \frac12 z(x^2+y^2)
defines the Hamiltonian vector field
Z =
z^2 \left( -y \derpar{}{x} + x \derpar{}{y} \right)
We have two constants of the motion,
${x^2+y^2}$ and $z$.
Excluding the $z$-axis,
all their level sets are diffeomorphic to the unit circle;
parametrising the circle with the natural angle,
these diffeomorphisms read
$\alpha_{r,c}(\phi) = (r \cos\phi, r\sin\phi, c)$.
It is easily checked that
$\alpha_{r,c}^*(\d H) = 0$.
Since for $c \neq 0$
all these level sets are Lagrangian submanifolds,
we conclude from the preceding theorem
that the $\alpha_{r,c}$ constitute
a complete Lagrangian slicing for $Z$
on the open set given by $z \neq 0$.
In general this situation prevails for Poisson manifolds
and we have to consider Casimir functions and constants of
the motion in involution.
Casimir functions identify `parameters'
(like mass, spin, charge, isospin, coloured charge)
while the constants of the motion identify the decomposition
into vector fields on lower dimensional submanifolds
§ SLICING IN FIBRED MANIFOLDS
In this section we consider a dynamical system $(P,Z)$,
where the manifold $P$ is fibred over another manifold,
that is to say,
we work in a
fibre bundle
$\pi \colon P \to M$.
We consider the slicing problem as before:
\xymatrix{
*++{\Tan M} \ar[r]^{\Tan \alpha} \ar[d]_{} &
*++{\Tan P} \ar[d]_{}
\\
*++{M} \ar[r]^{\alpha} \ar@/^3mm/[u]^{X} &
*++{P} \ar@/^3mm/[l]^{\pi} \ar@/_3mm/[u]_{Z}
but only for sections of $\pi$,
that is to say,
for maps
$\alpha \colon M \to P$
such that $\pi \comp \alpha = \mathrm{Id}_M$.
For this problem there is not a `gauge freedom' as mentioned
in section 2:
the submanifold $\alpha(M) \subset P$
cannot be expressed as the image of any other section.
Since $\alpha$ is an embedding,
we know that equation (<ref>) determines $X$.
Nevertheless, composing this equation with the tangent map
$\Tan \pi$,
we can give an explicit formula for $X$:
X = πZ α .
So, from now on we assume that $X$ is defined by this equation
from $\alpha$.
In this case, proposition 1 adopts the following form:
A section $\alpha$ of $\pi \colon P \to M$
is a solution of the slicing equation for $(P,Z)$
απZ α= Z α ;
that is to say,
if $\Tan \alpha \comp \Tan \pi \comp Z$ agrees with $Z$
on the submanifold $\alpha(M)$.
If $\alpha$ is a slicing section,
the vector field along $\alpha$
defined as
$\Tan \alpha \comp \Tan \pi \comp Z \comp \alpha - Z \comp \alpha$
is $\pi$–vertical.
Remember that the vertical subbundle of $\Tan P$
\mathrm{V} P = \Ker \Tan \pi
Its fibres are
\mathrm{V}_z P =
\Ker \Tan_z \pi \subset
\Tan_zP$
and are naturally identified with the tangent spaces to the fibres
of $\pi$.
Application of $\Tan \pi$ to
$\Tan \alpha \comp \Tan \pi \comp Z \comp \alpha - Z \comp \alpha$
yields immediately zero since $\alpha$ is a section of $\pi$.
§.§ Sections, projectors, and connections
If $\alpha$ is a section of $P$,
let us have a look at the composition
$\Tan \alpha \comp \Tan \pi$.
At a given point $z = \alpha(x) \in P$,
$\Tan_z(\alpha \comp \pi) \colon \Tan_zP \to \Tan_zP$
is an endomorphism,
and since $\Tan \pi \comp \Tan \alpha$ is the identity,
we note that
\Tan_z(\alpha \comp \pi) \comp \Tan_z(\alpha \comp \pi) =
\Tan_z(\alpha \comp \pi)
therefore it is a projector in $\Tan_zP$.
Since $\Tan_x\alpha$ is injective, it is clear that
\Ker \Tan_z(\alpha \comp \pi) =
\Ker \Tan_z \pi =
\mathrm{V}_z P
\,.
\mathrm{Im} \Tan_z(\alpha \comp \pi) =
\Tan_{z}\alpha(M)
is a complementary subspace to $\mathrm{V}_z P$.
So we can write, for every $x \in M$, a direct sum decomposition
\Tan_{\alpha(x)} P =
\Ver_{\alpha(x)}P \,\oplus\, \Tan_{\alpha(x)} \alpha(M)
\,.
This can be written globally in the pull-back vector bundle:
M \!\times_\alpha\! \Tan P =
M \!\times_\alpha\! \Ver P
\,\oplus\,
M \!\times_\alpha\! \Tan \,\alpha(M)
\,.
Now suppose that we have not only a section
but a family of non overlapping sections covering the whole
manifold $P$;
this can be defined by a diffeomorphism
$\overline\alpha \colon M \times N \to P$,
where each $\alpha_c = \overline\alpha(\cdot,c)$ is a section of $P$,
but this diffeomorphism could as well be defined on open sets of $P$.
The preceding study can be performed at every point $z \in P$,
therefore the family $\overline\alpha$ defines a
horizontal subbundle,
that is,
a vector subbundle $H \subset \Tan P$
complementary to the vertical subbundle
$\Ver P \subset \Tan P$.
A horizontal subbundle of $\Tan P$ is also called
a (nonlinear) connection on the bundle $P$.
This horizontal subbundle is obviously integrable,
its integral manifolds being given by the embeddings $\alpha_c$.
if a connection on the bundle
$P \to M$
has integrable horizontal subbundle
(which amounts to saying that its curvature vanishes,
its integral manifolds are locally the images of sections of
the bundle.
§.§ Complete solutions and connections
Still working with the diffeomorphism
$\overline\alpha \colon M \times N \to P$,
when is it a complete solution of the
slicing equation for sections?
In addition to defining an integrable horizontal subbundle,
$Z$ has to be tangent to it.
Therefore, locally,
complete solutions of the slicing equation
are equivalent to
connections on $\pi \colon P \to M$,
with zero curvature,
and invariant by $Z$.
§.§ The Hamiltonian case on a fibred manifold
Here we consider both a bundle structure and a Hamiltonian structure
on $P$.
So, $\pi \colon P \to M$ is a fibre bundle
and $(P,\omega)$ is a symplectic manifold,
and $Z = Z_H$ is a Hamiltonian vector field
(with Hamiltonian function $H$).
Let $\alpha$ be a section of $\pi$,
and let us determine if it is a slicing section for $Z$.
We wish to give a kind of converse to
proposition <ref>,
which relates
\Tan \alpha \comp X - Z \comp \alpha
i_X \alpha\!^*(\omega) - \d \,\alpha\!^*(H)
(where $X$ is given by
$X = \Tan \pi \comp Z \comp \alpha$.)
\xymatrix{
*++{\Tan M} \ar[d]_{} \ar[r]^{\Tan \alpha \quad}
*++{M \!\times_\alpha\! \Tan P} \ar[r]^{\widehat\omega}
*++{M \!\times_\alpha\! \Tan^*P} \ar[r]^{\quad {}^{t}(\Tan \alpha)}
\\
\ar[ru]_{\; \Tan \alpha \comp X - Z \comp \alpha}
\ar@/_3mm/[urrr]_{\qquad i_X \alpha\!^*(\omega) - \d \,\alpha\!^*(H)}
In this diagram
$\widehat \omega$ is bijective,
and, as we have already noted, the problem is that
${}^t(\Tan \alpha)$
is not injective,
$\Ker {}^t(\Tan \alpha) = (M \!\times_\alpha\! \Tan P_0)^\circ$.
However, we have also noted that
$\Tan \alpha \comp X - Z \comp \alpha$
is $\pi$–vertical.
Therefore we only need to impose the injectivity of
the restriction of
${}^t(\Tan \alpha) \comp \widehat\omega$
to the subbundle
$M \!\times_\alpha\! \Ver P$,
and this is equivalent to saying that
\widehat\omega ( M \!\times_\alpha\! \Ver P )
\cap
(M \!\times_\alpha\! \Tan P_0)^\circ
\{0\}
\,.
With the preceding hypotheses,
the following conditions are equivalent:
0pt plus 1pt
* The fibres of $\pi \colon P \to M$
are isotropic submanifolds (with respect to $\omega$).
* For every couple of vertical vectors
$w_z,w'_z \in \Ver_zP \subset \Tan_zP$
one has
$\omega(w_z,w'_z) = 0$.
* $\widehat\omega(\Ver P) \subset (\Ver P)^\circ$.
The equivalence of the first two is due to the fact that
the vertical vectors are those that are tangent to the fibres.
If $\alpha$ is a section of $P$
and the fibres are isotropic
\widehat\omega ( M \!\times_\alpha\! \Ver P )
\cap
(M \!\times_\alpha\! \Tan P_0)^\circ
\{0\}
The vertical+horizontal decomposition yields
M \!\times_\alpha\! \Tan^*P =
(M \!\times_\alpha\! \Ver P)^\circ \oplus
(M \!\times_\alpha\! P_0)^\circ
Let $(P,\omega,H)$ be a Hamiltonian system
on a fibre bundle $\pi \colon P \to M$.
Let $\alpha \colon M \to P$
be a section of $\pi$,
and define its associated vector field
$X = \Tan \pi \comp Z \comp \alpha$.
Suppose that the fibres of $\pi$ are isotropic.
Then $\alpha$ is a slicing section
i_X \alpha^*(\omega) - \d \,\alpha^*(H) = 0
\,.
As we have just shown, the isotropy condition
implies that
${}^t(\Tan \alpha) \comp \widehat\omega$
is injective when applied to vertical vectors.
i_X \alpha^*(\omega) - \d \,\alpha^*(H)
is zero
$\Tan \alpha \comp X - Z \comp \alpha$
also is.
Coordinate expressions
Let's understand the proof of the theorem
on the light of coordinates.
We use coordinates
$(x^i)$ in $M$
and adapted coordinates
$(x^i,y^\mu)$ in $P$.
The section takes the form
$\alpha(x) = (x,a^\mu(x))$
and its tangent map is represented by the matrix
\left(
I \atop A
\right)
where $A$ is the jacobian matrix of the $a^\mu$.
The symplectic form $\omega$ is represented by a skew-symmetric matrix
\Omega =
\small
\left( \begin{array}{cc}
\Omega_b & N
\\
-N^\top & \Omega_f
\end{array} \right)
The matrix of $\widehat\omega$ is $\Omega^\top$.
Then the linear map
{}^t(\Tan_x\alpha) \comp \widehat \omega_z
is represented by the matrix
\left( \begin{array}{cc}
\Omega_b^\top + A^\top N^\top
-N + A^\top \Omega_f^\top
\end{array} \right)
and its restriction to the vertical subspace by its second block,
-N + A^\top \Omega_f^\top
\,.
the fibres are isotropic iff $\Omega_f = 0$,
and since $\Omega$ is nondegenerate
$N$ has to have maximal rank and be injective.
So, the only vertical vector sent to 0 by this map is 0.
In the preceding section we have already obtained
the equation for the Lagrangian slicings.
We can combine theorems 3 and 6 in this way:
For a Hamiltonian system $(P,\omega,H)$ fibred over $M$,
with isotropic fibres,
let $\alpha \colon M \to P$ be a section with
isotropic image.
Then $\alpha$ is a solution of the slicing problem
\d \,\alpha^*(H) = 0
\,.
The isotropy of the fibres requires
$\dim M \geq \dim P \,/\,2$
and the isotropy of $\alpha(M)$ requires
$\dim M \leq \dim P \,/\,2$.
$\dim M = \dim P \,/\,2$,
which in particular means that $\alpha(M) \subset P$
is a Lagrangian submanifold
and then application of theorem 2 yields the desired result.
the isotropy of the image means
$\alpha^*(\omega) = 0$
and one can apply theorem 4 at once.
The isotropy of the fibres is necessary to prove this result,
as shown by the following example.
$P = \R^4$,
with coordinates
with the usual symplectic form
$\omega = \d x \wedge \d p_x + \d y \wedge \d p_y$,
and the Hamiltonian of the isotropic double harmonic oscillator
$H = \frac12( x^2 + p_x^2 + y^2 + p_y^2)$;
its Hamiltonian vector field is
$Z =
p_x \,\tanvec{x} - x \,\tanvec{p_x} + p_y \,\tanvec{y} - y \,\tanvec{p_y}
Consider the trivial fibre bundle
$\pi \colon P \to M$
given by
$M = \R$,
with projection
$\pi(x,p_x,y,p_y) = x$.
Of course, since $M$ is 1-dimensional
any section $\alpha$ of $\pi$ satisfies $\alpha^*(\omega)=0$.
Then consider the local section
$\alpha(x) = (x,x,\sqrt{c^2-x^2},\sqrt{c^2-x^2})$.
It satisfies
$H \comp \alpha = c^2 = \mathrm{const}$,
but one easily checks that it is not a slicing section.
The point is that the fibres of $\pi$ are not isotropic
—they cannot be since they are 3-dimensional submanifolds
of a 4-dimensional symplectic manifold.
§ LAGRANGIAN AND HAMILTONIAN FORMALISMS
In this section we study some features specific to the dynamics
on tangent and cotangent bundles,
and in particular to Lagrangian and Hamiltonian formalisms.
First, notice that
the results of the preceding section
apply directly to a canonical Hamiltonian system
whith $\Tan^* Q$ endowed with its
vector bundle structure
$\pi \colon \Tan^*Q \to Q$
and its canonical symplectic form $\omega$.
The dynamical vector field $Z$ is the symplectic gradient
$Z_H$ of the Hamiltonian function $H$.
Then we consider the slicing equation $X \sim_\alpha Z$
for a section $\alpha$ of $P$, that is to say,
a differential 1-form on $Q$.
we can compute the slicing vector field $X$,
which in this case turns out to be
$X = \FD H \comp \alpha$,
$\FD H \colon \Tan^*Q \to \Tan Q$
is the fibre derivative of $H$.
Now, notice that the fibres of $\Tan^*Q$,
that is to say,
the cotangent spaces $\Tan_q^*Q$,
are isotropic submanifolds of the cotangent bundle
with respect to its canonical symplectic structure.
So, we are under the hypotheses of
theorem <ref>
and its corollary,
which give a special form for the slicing equation.
In particular,
the classical Hamilton–Jacobi equation
is nothing but
the slicing equation for a closed 1-form $\alpha$.
This means that $\alpha$ is locally exact,
$\alpha = \d W$,
and the slicing equation has the well-known form
H Ẉ = const
The same applies to the Lagrangian formulation
of mechanics
when it is defined by a regular Lagrangian function
$L \colon \Tan Q \to \R$.
In this case the fibred manifold is $P = \Tan Q$;
now we don't have a canonical symplectic form,
but the 2-form $\omega_L$ defined from the Lagrangian.
The Hamiltonian vector field is the symplectic gradient
of the energy $E_L$
Then all proceeds as in the Hamiltonian case.
Within this framework we recover some of our previous results.
In fact
theorem <ref>
has, as particular cases,
theorems 1 and 2 in our paper
corresponding to
the Lagrangian and
the Hamiltonian
formulations, respectively.
In the same way, corollary 3 corresponds to
propositions 3 and 7 of the same paper.
There it is also proved (theorem 3) the equivalence between
the Hamilton–Jacobi theories
for the Lagrangian and the Hamiltonian dynamics
for regular systems.
The relationship between constants of the motion and complete slicings
(theorem <ref>)
was also established for these particular cases in
§.§ Determination of a second-order dynamics
from constants of the motion
Suppose we have a foliation
of a manifold $P$.
A vector field
$Z$ on $P$
tangent to the foliation
defines a vector field
$X_c$ on every leaf $M_c$ of the foliation.
a vector field $X_c$ on every $M_c$ defines a vector field
$Z$ on $P$
(though a priori one cannot guarantee it to be continuous).
This is what happens when we have a complete slicing
of a dynamics $(P,Z)$, as discussed in section 3.
Now, suppose that the hypotheses of
theorem <ref>
are satisfied,
so that the complete slicing is equivalent to a
(manifold-valued) constant of the motion $F \colon P \to N$.
Then it could seem that
the dynamics $Z$ is determined by $F$.
But of course this is not true:
the conditions of the theorem assume that $Z$ is already given,
otherwise the vector fields $X_c$ could not be determined.
However, there is a very special instance
where the knowledge of some constants of the motion
suffices to determine the dynamics.
Recall that
a vector field $Z$ defined on the tangent bundle $\Tan M$
of a manifold
is said to satisfy the second-order condition when
its integral curves are the velocities of their projections
to the base space $M$.
It is easily proved that this is equivalent to saying that,
besides being a section of the tangent bundle of $\Tan M$,
$\tau_{\Tan M} \colon \Tan(\Tan M) \to \Tan M$,
$Z$ is also a section of the other vector bundle structure
of $\Tan (\Tan M)$,
the one given by
$\Tan\tau_{M} \colon \Tan(\Tan M) \to \Tan M$.
In brief, this means that
$\Tan \tau_M \circ Z = \mathrm{Id}$.
Consider a dynamical system
$P \subset \Tan M$
is an open subset projecting over $M$.
is a slicing of $Z$ by a section $\alpha$ of $P$,
and $Z$ satisfies the second-order condition,
$X = \alpha$.
suppose we have a complete slicing
of $Z$ by sections $\alpha_c$ of $P$.
If $X_c=\alpha_c$ for every $c$,
$Z$ satisfies the second-order condition.
If $Z$ satisfies this condition,
then the slicing equation
$\Tan \alpha \circ X = Z \circ \alpha$,
composed with $\Tan \tau_M$,
yields $X = \alpha$.
Conversely, when $X = \alpha$
the slicing equation reads
$\Tan \alpha \circ \alpha = Z \circ \alpha$,
and composition with $\Tan \tau_M$ yields
$\alpha = \Tan \tau_M \circ Z \circ \alpha$.
This means that $Z$ satisfies the second-order condition
on every point of $\alpha(M)$.
$P \subset \Tan M$
be an open subset projecting onto $M$.
Suppose we have $m=\dim M$ functions
$f^\alpha \colon P \to \R$
whose fibre derivatives
$\mathcal{F}f^\alpha \colon P \to \Tan^*M$
are linearly independent at each point.
Then around any point $v \in P$
there exists a unique local vector field $Z$,
satisfying the second-order condition,
and for which the $f^\alpha$ are constants of the motion.
Put $F = (f^1,\ldots,f^m) \colon P \to \R^m$.
For every $c \in \R^m$ we have a submanifold
$P_c = F^{-1}(c) \subset P$.
The hypotheses imply that the restriction of the projection to
this submanifold,
$\tau|_{P_c} \colon P_c \to M$,
is a diffeomorphism in a neighbourhood of any point $v \in P_c$.
$\alpha_c \colon M \to P_c$
be its inverse.
This $\alpha_c$ is also a vector field on $M$,
so it defines a vector field
and this satisfies the second-order condition by the preceding lemma.
All of these together yield $Z$.
To complete the proof we need to show that $Z$ is smooth,
and we will do this by an explicit computation in coordinates.
Let $v_\circ \in P$ be an arbitrary point,
and use natural coordinates $(q^i,v^i)$ around it.
Z = v^i \derpar{}{q^i} + Z^i(q,v) \derpar{}{v^i}
Imposing that the $f^\alpha$ are constants of the motion for $Z$
we obtain
\Lie_Z f^\alpha =
\derpar{f^\alpha}{q^i} v^i +
\derpar{f^\alpha}{v^i} Z^i =
\,.
The linear independence of the fibre derivatives means
in coordinates that the matrix
\left(
\derpar{f^\alpha}{v^i}
\right)
is invertible.
Hence, we determine the last coefficients of $Z$ as
Z^i = -
\bigg(\left(
\derpar{f}{v}
\right)^{-1}
\bigg)^{\!\!i}_{\beta}
\, \derpar{f^\beta}{q^j}
\, v^j
\,.
\vadjust{\kern -7mm}
It is known that from a vector field $X$ on $M$
one can construct its canonical lift $X^{\Tan}$
to the tangent bundle.
This vector field does not satisfy the second-order condition
in the whole $\Tan M$,
but in the points of $X(M)$ it does.
Indeed, in the first part of the preceding proof,
what we are defining is
$Z|_{X(M)} = X^{\Tan}|_{X(M)}$,
where $\alpha \equiv X$.
Since we have a whole family of $\alpha$'s covering the whole space,
the vector field $Z$ constructed in this way satisfies
the second-order condition at every point.
We will use the free particle to show that working in the
Lagrangian or in the Hamiltonian formalisms
is philosophically different.
If we take
$P = \Tan^* \R^n$
and the Hamiltonian
$H = \frac12 \left( p_1^2 + \ldots + p_n^2 \right)$
then the dynamical vector field is
$Z = \sum p_i \,\derpar{}{q_i}$,
and its constants of the motion are the functions
But notice that these functions are constants of the motion
for $H$ and also for any Hamiltonian of the form
the corresponding dynamical vector field is
$Z = \sum \derpar{H}{p_i} \,\derpar{}{q_i}$.
Now take
$P = \Tan \R^n$
and consider the functions
$f_i = v_i$.
Following the preceding theorem,
we can look for a second-order vector field $Z$
having the $f_i$ as constants of the motion.
There is a unique such a vector field,
and it is
$Z = \sum v_i \,\derpar{}{q_i}$.
By the way, the same vector field would be obtained
if one considered, instead of the $v_i$,
any set of $m$ independent functions $f_i(v)$.
§.§ Acknowledgments
JFC and EM
acknowledge the financial support of
the Ministerio de Economía y Competitividad (Spain) project
the DGA (Aragon) project
DGA E24/1.
XG, MCML and NRR
acknowledge the financial support of the
Ministerio de Ciencia e Innovación (Spain) projects
MTM2014–54855–P and MTM2011–22585.
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|
1511.00424
|
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, Karnataka, 560012, India.
Leibniz-Institute for Solid State and Materials Research Dresden, P.O.Box 270116, D-01171 Dresden, Germany.
CSIR-National Physical Laboratory, New Delhi 110012, India.
Van der Waals-Zeeman Institute, IoP, University of Amsterdam, NL-1098 XH, Amsterdam, The Netherlands.
Institute f$\ddot{u}$r Theoretische Physik III, Ruhr-Universit$\ddot{a}$t Bochum, D-44801 Bochum, Germany.
Helmholtz-Zentrum Berlin, Albert-Eistein-Str. 15, D-12489 Berlin, Germany.
Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA94025, USA
Van der Waals-Zeeman Institute, IoP, University of Amsterdam, NL-1098 XH, Amsterdam, The Netherlands
Institute f$\ddot{u}$r Theoretische Physik III, Ruhr-Universit$\ddot{a}$t Bochum, D-44801 Bochum, Germany.
Institute of Physics, Kazan (Volga Region) Federal University, 420008 Kazan, Russian Federation.
Using angle-resolved photoemission spectroscopy we have studied the low-energy electronic structure and the Fermi surface topology of Fe$_{1+y}$Te$_{1-x}$Se$_x$ superconductors. Similar to the known iron pnictides we observe hole pockets at the center and electron pockets at the corner of the Brillouin zone (BZ). However, on a finer level, the electronic structure around the $\Gamma$- and $Z$-points in $k$-space is substantially different from other iron pnictides, in that we observe two hole pockets at the $\Gamma$-point, and more interestingly only one hole pocket is seen at the $Z$-point, whereas in $1111$-, $111$-, and $122$-type compounds, three hole pockets could be readily found at the zone center. Another major difference noted in the Fe$_{1+y}$Te$_{1-x}$Se$_x$ superconductors is that the top of innermost hole-like band moves away from the Fermi level to higher binding energy on going from $\Gamma$ to $Z$, quite opposite to the iron pnictides. The polarization dependence of the observed features was used to aid the attribution of the orbital character of the observed bands. Photon energy dependent measurements suggest a weak $k_z$ dispersion for the outer hole pocket and a moderate $k_z$ dispersion for the inner hole pocket. By evaluating the momentum and energy dependent spectral widths, the single-particle self-energy was extracted and interestingly this shows a pronounced non-Fermi liquid behaviour for these compounds. The experimental observations are discussed in context of electronic band structure calculations and models for the self-energy such as the spin-fermion model and the marginal-Fermi-liquid.
§ INTRODUCTION
The present consensus for the normal state of the high-$T_c$ iron-based superconductors is that they show strange metallic character <cit.> near a quantum critical point (QCP) <cit.>, that is reached by either charge carrier doping, chemical pressure or by applying mechanical pressure to the parent compound. <cit.> This strange metallic character is attributed to strong antiferromagnetic spin fluctuations, originating from interband scattering between the hole and electron pockets located in the center and corner of the Brillouin zone, respectively. <cit.> Comparing the iron pnictides and the iron chalcogenide systems, FeTe and FeSe, the latter have been suggested to possess stronger many-body correlation effects near the Fermi level from density functional theory (DFT) plus dynamic mean-field theory (DMFT) calculations. <cit.> This conclusion is supported further by transport measurements <cit.> and photoemission <cit.> experiments. More recent theory work has also argued that for the iron chalcogenide systems, electron correlations lead to bad-metal behavior, despite the intermediate values of the Hubbard repulsion U and Hund's rule coupling J. <cit.>
Strong interest in the iron chalcogenides has been rekindled recently due to the spectroscopic observation of superconducting energy gaps at and above the boiling point of liquid nitrogen for single unit-cell thin films of FeSe on SrTiO$_3$ substrates. <cit.> These systems are now the record-holders for highest $T_c$ in the Fe-based superconductors. Recently, ARPES data have been modelled to extract theoretical parameters suggesting that coupling of a SrTiO$_3$ phonon can significantly enhance the magnetism-driven pairing energy for the electrons in the single unit cell thick film of FeSe. <cit.>
Also of interest have been the recent and ongoing discussions as to whether a Fermi liquid ground state is the appropriate description for optimally n-type (electron) doped BaFe$_2$As$_2$. DMFT calculations argue for canonical Fermi liquid character when the Ba122 compound is optimally doped with electrons, while optimal hole doping leads to strong band renormalization near the Fermi level and thus to non-Fermi liquid character. <cit.> Recent optical experiments on n-doped BaFe$_2$As$_2$ would seem to offer partial support for this, <cit.> but other experimental data from transport, <cit.> thermal properties, <cit.> NMR, <cit.> quantum oscillations <cit.> and photoemission measurements <cit.> suggest non-Fermi liquid character near the quantum critical point in the BaFe$_2$As$_2$ system when doped with charge carriers of either sign or upon applying chemical pressure.
Thus, given the backdrop of new data and insights into novel, high temperature pairing phenomena in the iron chalcogenides, and the ongoing, lively discussions as to the Fermi liquid (or not) behavior in the iron pnictides, it is of great interest to examine the iron chalcogenides from the point of view of Fermi liquid theory and how strong electron correlations make themselves felt. Indeed, one theory report suggests non-Fermi liquid behaviour also for the chalcogenides. <cit.>
There are various angle-resolved photoelectron spectroscopy (ARPES) studies <cit.> which indicate that the Fe chalcogenides show strong electron correlation effects. It should be noted that all these studies of correlation effects have been carried out in the $\Gamma-M-X$ plane in 3D $k$-space. As yet, no report has been made of if and how the picture changes upon variation of the $k_z$ value in these compounds, and most experimental studies have inferred the impact of electronic correlation from the renormalisation of the band structure (band velocity).
In this paper we present electronic structure studies of Fe$_{1+y}$Te$_{1-x}$Se$_x$ superconductors using and combination angle-resolved photoelectron spectroscopy (ARPES) and DFT calculations.
We compare the experimental results with our DFT calculations, as well as with other existing experimental and theoretical reports on these systems. <cit.>
Our ARPES data enable attribution of the orbital character of the bands involved (by exploiting photon polarization) and we explicitly examine the role of $k_z$ (by variation of the photon energy) for the hole pocket states along the $\Gamma-Z$ ($k$) direction. The data suggest weak $k_z$ dispersion for one hole pocket, while a moderate $k_z$ dispersion is observed for the other hole pocket at the Brillouin zone center, a result which is consistent with the DFT calculations.
In agreement with previous reports, <cit.> the hole pockets display a mass renormalization ($m^*/m_b$) of 2-4 at higher binding energies. The experimental data are also analyzed with respect to a possible $k_z$ dependence of the mass renormalization and the Fermi velocity ($v_F$).
The ARPES data have also been fitted so as to enable estimation of the imaginary part of the self-energy ($\Im\Sigma$). Our results suggest a departure from canonical Fermi-liquid behaviour for the quasi-particles near the zone center.
In particular, the imaginary part of the self-energy is linear in energy for the inner hole pocket, whose band top generates a van Hove singularity (vHs) near the Fermi level.
This linear-in-energy self-energy is shown to be well described using a marginal-Fermi-liquid theory (MFL) <cit.> approach with a coupling constant ($\lambda$) of 1.5.
§ EXPERIMENTAL DETAILS
ARPES provides information on the energy and momentum dependent spectral function. <cit.> By detecting the emitted photoelectrons at various angles one can extract the in-plane ($k_x-k_y$ plane) electronic structure, while by changing the photon energy it is possible to derive the $k_z$ dependent electronic structure. Using polarized photons, due to the matrix element effects, it is possible to obtain information on the orbital character of the detected bands.
Single crystals of Fe$_{1.068}$Te$_{1-x}$Se$_x$ ($x$ = 0.36 and 0.46) were grown in Amsterdam by the Bridgman technique using self-flux. The crystals show superconducting transitions at $T_c$ $\approx$ 11 K and 15 K with $x$=0.36 and 0.46, respectively. Further elemental analysis on these single crystals are reported elsewhere, as are data showing them to possess simple, high quality and non-reconstructed cleavage surfaces. <cit.>.
Another set of high quality of Fe$_{1+y}$Te$_{0.5}$Se$_{0.5}$ (y 1$\%$) single crystals were grown in NPL, Delhi using the self-flux growth technique. These crystals showed a $T_c$ of 14 K. The elemental analysis of these crystal is reported elsewhere. <cit.>
ARPES measurements were carried out in BESSY II (Helmholtz Zentrum Berlin) synchrotron radiation facility at the UE112-PGM2b beam line using the "1$^3$-ARPES" end station equipped with SCIENTA R4000 analyzer. <cit.> The total energy resolution was set between 5 and 10 meV, depending on the applied photon energy. Samples were cleaved $\textit{in situ}$ at a sample temperature lower than 20 K. All the measurements were carried out at a sample temperature $T\approx$1 K.
§ CALCULATIONS
To understand the experimental data we have performed a theoretical analysis of the electronic band structure of FeSe, following Ref. Eschrig2009. Using a three-dimensional tight-binding parametrization of the LDA (local density approximation) band structure, we computed the Fermi velocity variation along the $k_z$ direction for the three hole pockets near the center of the Brillouin zone.
The hole pockets which possess mostly ${xz}$ and ${yz}$ character, mixed with ${x^2-y^2}$, demonstrate a weak $k_z$ variation of the Fermi velocity, while the variation is stronger for the hole pocket which has an admixture of $z^2$ orbital character. We expect this variation to be further enhanced by the effects of short-range electronic correlations, which are not included in our LDA-based calculations.
§ RESULTS
§.§ ARPES data: Fermi surfaces and band dispersions
(Color online) ARPES spectra of Fe$_{1.068}$Te$_{0.54}$Se$_{0.46}$ measured with an excitation energy h$\nu$=75 eV using $p$-polarized light. (a) is the Fermi surface map. The light polarization vector ($\vec{\varepsilon}$) is displayed on the figure. Panels (b), (d) and (f) show energy distribution maps (EDMs) taken from cuts #1, #2 and #3, which are overlaid on the Fermi surface map. Panel (c) shows the energy distribution curves (EDCs) from the EDM shown in (b). Panels (e) and (g) contain the second derivatives of the EDMs shown in (d) and (f), respectively. The sample temperature was 1K.
(Color online) ARPES spectra of Fe$_{1.068}$Te$_{0.54}$Se$_{0.46}$ measured with an excitation energy h$\nu$=75 eV using $s$-polarized light. (a) is the Fermi surface map. The light polarization vector ($\vec{\varepsilon}$) is displayed on the figure. Panels (b), (d) and (f) show energy distribution maps (EDMs) taken from the cuts #1, #2 and #3, which are overlaid on the Fermi surface map. Panel (c) shows the energy distribution curves (EDCs) from the EDM shown in (b). Panels (e) and (g) contain the second derivatives of the EDMs shown in (d) and (f), respectively. The sample temperature was 1K.
Figure <ref> shows the ARPES spectra of the Fe$_{1.068}$Te$_{0.54}$Se$_{0.46}$ superconductor, recorded along the $\Gamma-X$ high symmetry line using $p$-polarized light with an excitation energy h$\nu$=75 eV. The Fermi surface (FS) map shown in Fig. <ref> (a) results from integration over an energy window of 10 meV centered at the Fermi level ($E_F$). In Fig. <ref> (b), we show an I(k,E) image (EDM or energy distribution map), taken along the cut #1 as shown on the FS map in panel (a). Similarly, Figs. <ref> (d) and (f) depict EDMs along cuts #2 (through $\Gamma$) and #3 (through Z), respectively. Fig. <ref> (c) shows energy dispersion curves (EDCs) taken from the EDM shown in Fig. <ref> (b).
The data shown in panels (e) and (g) of Fig.<ref> are the second derivative of the EDMs shown in Figs. <ref> (d) and (f), respectively.
The data shown in Figs. <ref> (b-d) clearly show the existence of two hole-like bands, which we label $\alpha_1$ and $\alpha_2$, at the center of the Brillouin zone.
The band $\alpha_1$ disperses strongly towards $E_F$ but does not cross it, forming a van Hove singularity near the Fermi level, consistent with the iron pnictide superconductors. <cit.> The $\alpha_2$ hole-pocket crosses $E_F$ at a Fermi wavevector ($k_F$) of 0.15$\pm$0.02 $\AA^{-1}$.
At $Z$, the high symmetry point is reached at a larger polar angle, and we observe a band having weak spectral weight crossing $E_F$ at a $k_F$=0.16$\pm$0.02 $\AA^{-1}$. This observation of hole pockets at the zone center is in keeping with previous reports on these compounds. <cit.>
Following an analysis of the measurement geometry and polarization dependent selection rules laid out in detail in Ref. Fink2009, it can be concluded that the even parity ${xz}$, ${xy}$, and $z^2$ states are visible using $p$-polarized light as used in Fig. <ref>.
From the DFT calculations reported in detail later in the paper, it transpired that the third, $\Gamma$-centered hole pocket, $\alpha_3$, that we were unable to distinguish in the present data has mainly $xy$ orbital character.
Therefore we assign the bands $\alpha_1$ and $\alpha_2$, detected using $p$-polarized light to have mainly $xz$ and $z^2$ orbital character.
Figure <ref> depicts analogous data to Fig. <ref> but now recorded using $s$-polarized light. In Fig. <ref> we could again resolve two bands at the zone center: $\alpha_1$ and $\alpha_2$.
As in the data shown in Fig. 1, the $\alpha_1$ disperses strongly towards $E_F$, and the $\alpha_2$ feature crosses the Fermi level at a momentum vector of $k_F$=0.15$\pm$0.02 $\AA^{-1}$.
In contrast to the data shown in Fig. <ref>, we did not observe any spectral weight at the $Z$-point for the $s$-polarized case.
In this measurement geometry, $s$-polarized light would be expected to detect bands having ${x^2-y^2}$ and ${yz}$ orbital characters. As we know that the ${x^2-y^2}$ states are located far below the Fermi level at the zone center, <cit.> we exclude these states from further discussion. Hence, the bands $\alpha_1$ and $\alpha_2$ shown in Fig. <ref> have predominantly only $yz$ orbital character.
From Figs. <ref> and <ref> it is clear that the spectral intensity of the hole pocket $\alpha_2$ at the zone center is elongated in the $k_{y}$ direction when probed with $p$-polarized light and is elongated in the $k_{x}$ direction when measured using $s$-polarized light. This observation suggests that the orbital contribution to the $\alpha_2$ Fermi sheet is directional, i.e., in the $k_y$ direction the FS sheet has predominantly $xz$ orbital character and in the $k_x$ direction it is predominantly of $yz$ character. This observation is in very good agreement with the predictions made in Ref. Graser2009. Note here that the orbital contribution to the $\alpha_1$ Fermi sheet will be the other way round, meaning that in the $k_y$ direction this FS sheet has predominantly $yz$ character and in the $k_x$ direction it is predominantly composed by the $xz$ character, as reported in Ref. Graser2009 . The directional orbital contribution to this Fermi surface is predicted by theory for the iron pnictide compounds, but in experimental data, the presence of an $xy$ hole pocket with circular energy contours does cast some doubt on this, when viewed from the perspective of the ARPES data of the iron pnictide system.
In Figs. <ref> (d) and (f), a broad spectral feature labelled $\gamma$ can be seen at a binding energy E$_B$=0.35 eV that is not seen when the experiment is conducted with $s$-polarized light.
A very similar band dispersion has been observed experimentally in BaFe$_2$As$_2$, <cit.> but at the greater binding energy E$_B$=0.6 eV, and is ascribed to the band formed by the ${z^2}$ states, thus we follow this
attribution here also for the Fe chalcogenide. <cit.>
Within this picture, the band $\gamma$ is shifted almost 250 meV towards the Fermi level compared to Ba122, <cit.>, indicating a different hybridization between the Fe 3$d$ states and the chalcogenide 4$p$ states in these compounds compared to the 122 materials. <cit.> This conclusion is also consistent with the earlier ARPES data on stoichiometric and non-stoichiometric Fe chalcogenide and related compounds, as well with DFT calculations. <cit.>
ARPES data from Fe$_{1.068}$Te$_{0.64}$Se$_{0.36}$ measured with an excitation energy h$\nu$=88 eV using $s$-polarized light. Panel (a) shows the Fermi surface map. On the figure, the light polarization vector ($\vec{\varepsilon}$) is displayed. Panels (b), (c) and (f) show the energy distribution maps (EDMs) taken from the cuts #1, #2 and #3, respectively, as shown overlaid on the FS map. Panels (d), (e) and (g) are the second derivatives of (b), (c) and (f), respectively. (h) and (i) show energy dispersive curves from EDMs in (b) and (c), respectively.
Figure <ref> shows ARPES data from Fe$_{1.068}$Te$_{0.64}$Se$_{0.36}$ measured with an excitation energy of h$\nu$=88 eV using $s$-polarized light. In Fig. <ref>(a) we show the FS map extracted from integrating over an energy window of 10 meV centred at $E_F$, in which hole pockets at the zone center and an electron pocket at the zone corner are seen, similar to the data from the crystals with x=0.46. Figs. <ref>(b) and <ref>(c) show EDMs taken along the cuts $\#$1 and $\#$2, respectively. From these EDMs, two hole-like bands, $\alpha_1$ and $\alpha_2$, can be resolved at the zone center. The band $\alpha_1$ disperses strongly towards $E_F$ but does not cross it, while $\alpha_2$ crosses $E_F$ at a Fermi vector ($k_F$) 0.15$\pm$0.02 $\AA^{-1}$. Fig. <ref>(f) shows the EDM resulting from cut $\#$3 in which an electron-like band we label $\beta_1$ can be seen at the zone corner. The second derivative of the EDM from panel (f) is shown in Fig. <ref>(g). Both the raw data and the second derivative show that the bottom of the electron pocket is close to $E_F$, indicating that the electron pocket is shallow, as has also seen in other iron-based superconductors. <cit.>
(Color online) Photon energy dependent data taken from Fe$_{1.068}$Te$_{0.54}$Se$_{0.46}$ to reveal the $k_z$ dependence of the electronic structure at the zone center measured using $p$-polarized light. Panel (a) shows the $k_y$, $k_z$ Fermi surface map extracted over an integration window of 10 meV centred at $E_F$, with the high symmetry points in $k_z$ marked. Panel (b) shows a stack-plot of momentum dispersion curves (MDCs) sampling different $k_z$, together with the results of a fit using two Lorentzian functions near the $Z$-point and using three Lorentzian functions near the $\Gamma$-point. The black circles overlaid on the MDCs in panel (b) represent the peak positions of the $\alpha_2$ band.
Next we show ARPES measurements performed to reveal information on the $k_z$ dependent electronic structure. For this, photon energy dependent ARPES spectra were recorded for $k_{||}$ near the zone center, with photon energies ranging from h$\nu$=63 to 117 eV in steps of 3 eV. Data were recorded using $p$-polarized light along the $\Gamma$-$X$ high symmetry line. Fig. <ref>(a) depicts the Fermi surface map in the $k_y-k_z$ plane. Figure <ref>(b) shows momentum distribution curves as a function of photon energy, fitted with two or three Lorentzian functions. The peak positions of the $\alpha_2$ band extracted from the fits are shown by the black cirlces on the MDCs. The high symmetry points $\Gamma$ (h$\nu$ = 96 eV) and $Z$ (h$\nu$ = 81 and 114 eV) have been identified using the formula
\begin{eqnarray}
k_{\bot} = \sqrt{\frac{2m_e}{\hbar ^2} [E_{kin} cos^2\theta+V_0]}\hspace{1 mm}
\label{eq1},
\end{eqnarray}
where the inner potential, $V_0$, has been taken to be 15$\pm$2 eV. <cit.>
Figs. <ref> (a)-(c) show the EDMs taken along the $\Gamma$-$X$ high symmetry line at $k_z$ = 0, 0.5 and 1 in units of $\pi/c$, where $c$ is the c-axis lattice parameter. These data were measured using h$\nu$ = 96, 90 and 81 eV, respectively, and with $p$-polarized light. Superimposed in black on panels <ref> (a)-(c) are the dispersion relations of the hole-like band, $\alpha_2$, estimated from the fit to the MDC curves using two Lorentzian functions. The white dashed lines represent a parabolic fit to the black, MDC-derived curve.
Similarly, Figs. <ref> (d)-(f) show analogous EDMs recorded at $k_z$ = 0, 0.5 and 1 ($\pi/c$), measured using the photon energies h$\nu$ = 96, 90 and 81 eV, with $s$-polarized light. Superimposed in black on panels <ref> (d)-(f) are the dispersion relations of the hole-like band, $\alpha_1$, estimated from the fit to the MDC curves using two Lorentzian functions. The white dashed lines again represent a parabolic fit to the black, MDC-derived curve.
Figs. <ref> (g)-(i) show the hole-like band dispersions from the DFT band structure calculations along the $\Gamma - X$ high symmetry direction in $k_{||}$ for $k_z$ = 0, 0.5 and 1 ($\pi/c$), respectively.
In panels (g)-(i) the dashed-curves are hole-like bands from the calculations, while the red/blue solid lines are the results of the parabolic fit to the experimental bands corresponding to $\alpha_1$/$\alpha_2$. The Fermi level of the calculated bands is shifted such that the Fermi wavevector of the $\alpha_2$ hole pocket matches that seen in experiment. In this way it is easier to calculate the renormalization of the bands. However, this method may lead to discrepancy in estimating the renormalization of the $\alpha_1$ band (which does not cross $E_F$). We will discuss this point in detail in the next section.
§.§ Spectral functions and self-energies
§.§.§ Theory
ARPES provides an experimental window on the single particle spectral function, $A(E,k)$, and with a complex self-energy $\Sigma(E,k)=\Re\Sigma(E,k)+i\Im\Sigma(E,k)$ it is given by
\begin{eqnarray}
A(E,k) = -\dfrac{1}{\pi} \dfrac{\Im\Sigma}{(E_k-\epsilon(k)-\Re\Sigma)^2+(\Im\Sigma)^2},
\label{eq2}
\end{eqnarray}
where the real part of self-energy $\Re\Sigma(E,k)$ can be extracted by subtracting the bare-band dispersion $\epsilon(k)$ from the experimentally determined, renormalized band dispersion ($E_k$): $\Re\Sigma(E,k)=E_k-\epsilon(k)$.
The imaginary part of the self-energy can be extracted from the momentum widths of the experimental band features $\Delta_k$ and the bare-band velocity $v_k$, is given by
\begin{eqnarray}
\Im\Sigma(E,k)=\Delta_k v_k,
\label{eq2x}
\end{eqnarray}
in which $\Delta_k$ is the half-width half maximum of the momentum distribution curve.
On the other hand, one can also calculate the imaginary part of the self-energy using the scattering rate $S(E)=\Delta_k v^*_k$ and mass renormalization ($m^*/m_b$) using the expression
\begin{eqnarray}
\Im\Sigma(E) = S(E)\dfrac{m^*}{m_b}.
\label{eq3}
\end{eqnarray}
Here $v^*_k$ is the renormalized velocity and it is assumed that $\Im\Sigma(E)$ depends only weakly on the momentum $k$. $m^*$ is the effective mass estimated from the experimental band structure and $m_b$ is the bare-band mass estimated from the calculated band structure.
There are several theoretical approaches to describe non-Fermi liquid behaviour of the single-particle spectral function that can be observed in ARPES. For example, using purely phenomenological ansatz, marginal Fermi liquid theory (MFL) <cit.> gives:
\begin{eqnarray}
\Sigma(E)^{MFL} = \dfrac{1}{2}[\lambda_{MFL}E\ln(\dfrac{E_c}{u})-i\pi\lambda_{MFL}u],
\label{eq4}
\end{eqnarray}
which is often used in fitting the ARPES data of high-$T_c$ cuprates. <cit.> Here $u = max(|E|, k_BT)$, where $k_BT$ is the thermal energy. $E_c$ is the cutoff energy, which in a first approximation corresponds to the width of the conduction band. <cit.> Note here that in context of the marginal-Fermi liquid theory, the scattering rate can be expressed as $S(E)=\alpha+\beta E$, where $\alpha$ represent the elastic electron-impurity scattering processes and $\beta$ represents the electron-electron inelastic scattering.
On comparing Eqs. <ref> and <ref>, and considering the linear dependence of $S(E)$ on the energy, we can then calculate the electron coupling constant using the formula
\begin{eqnarray}
\lambda_{MFL} = \dfrac{2}{\pi}\dfrac{m^*}{m_b}\beta.
\label{eq5}
\end{eqnarray}
This marginal-Fermi liquid behaviour naturally emerges in microscopic theories near the quantum critical point in 3D systems. However obtaining this behavior in 2D systems remains problematic.
Another scenario for non-Fermi liquid behaviour is based on the idea that the dominant interaction in the cuprates is between the fermions and their low-energy collective spin excitations. In this scenario, the non-Fermi liquid behavior in the normal state is associated with the proximity to a critical point, but this point now separates paramagnetic and antiferromagnetically ordered phases.
It has been shown in the past <cit.> that in this case the self-energy can be written as
\begin{equation}
\Sigma^{sf}(E)=\lambda_{sf}\frac{2E}{1+\sqrt{1-i|\frac{E}{\omega_{sf}}|}}
\label{eq6}
\end{equation}
At small energies, $E<<\omega_{sf}$, the system displays Fermi-liquid behavior but is non-Fermi-liquid-like for intermediate and frequencies well above $\omega_{sf}$
§.§.§ Application of the theory to the ARPES data
In Fig. <ref> we show the spectral width analysis of the data measured on the Fe$_{1+y}$Te$_{0.5}$Se$_{0.5}$ sample. From Fig. <ref> (c) it is clear that energy dependent scattering rate obtained near the zone center suggests a non-Fermi liquid behaviour for the quasiparticles populating the $\alpha_1$ band, specifically a marginal-Fermi liquid type behaviour. As an example, if Eq. <ref> (spin-fluctuation, SF) is applied to extract the self-energy of the hole pocket near the $\Gamma$ point, we find that the expression is able to give a very good agreement to the data, as shown in Fig.<ref>. On the other hand, the qualifier is that for the spin-fluctuation theory, an unrealistically large value of $\lambda_{sf} \sim 7$ with $\omega_{sf}=30~meV$ is required to get this good fit. We note, however, that the SF-theory expression used here does refer to the single band case, while in the multiband situation relevant for Fe$_{1+y}$Te$_{1-x}$Se$_x$, the quasiparticle linewidth is determined by the sum of intraband and interband interactions and therefore the absolute numbers for $\lambda_{sf}$ inferred from the single-band theory should be taken with caution. In addition, spin-fluctuation theory predicts Fermi-liquid behaviour at energies well below $\lambda_{sf}$ (here $\approx$ 15 meV), a behavior that is not resolved in these data at present. Due to these facts, we chose in the following to concentrate on the MFL expression ( Eq. <ref>) for the analysis of data, without specifying the microscopic origin of its self-energy. Given the MFL picture, the electron coupling constant $\lambda_{MFL}$ extracted is 1.5 near the zone center.
This value matches well with the MFL coupling constant, $\lambda_{MFL}$=1.6, extracted from ARPES data recorded from doped BaFe$_2$As$_2$ and NaFeAs iron pnictides. <cit.>
(Color online) ARPES data taken on Fe$_{1.068}$Te$_{0.54}$Se$_{0.46}$. EDMs shown in (a)-(c) are measured at the $k_z$ values indicated (in units of $\pi/c$), using $p$-polarized light, and show the dispersive $\alpha_2$ band. Panels (d)-(f) show analogous EDMs, but measured using $s$-polarized light, and show the $\alpha_1$ band. In all panels (a)-(f), the black dotted curves result from a fit to the MDC's using a pair of Lorentzian functions, and the thin white dashed curves shows parabolae fitted to the black dotted dispersion curves. From these parabolae, the effective mass, $m^*$, can be determined experimentally. Panels (g)-(i) show the results of DFT band structure calculations performed on the parent FeSe compound, <cit.> and the dashed lines show the pair of hole-like bands predicted for each $k_z$ value. The parabolic fits to the experimental band dispersions corresponding to $\alpha_1$ and $\alpha_2$ are shown in panels (g)-(i) as red and blue solid lines, respectively. Panel (j) depicts the $k_z$ dependence (probed via changing the photon energy) of the mass renormalization ($m^*/m_b$) for the $\alpha_1$ (red) and $\alpha_2$ (blue) bands and Fermi velocity (upper curve, $v_F$) for the $\alpha_2$ band. Panel (k) shows the calculated $k_z$ dependence of the Fermi velocity ($v_F$) for the three hole-like bands estimated from the DFT calculations.
(Color online) EDM shown in panel (a) is taken from the Fe$_{1+y}$Te$_{0.5}$Se$_{0.5}$ sample measured using $s$- polarized light with a photon energy of 46 eV, which corresponds to $k_z$=0. Black curves in panel (b) are the experimental band dispersions extracted from fitting Lorentzian functions to the momentum dispersive curves from the data shown in panel (a). The red curve shows a fit to the experimental hole-like band using a 4$^{th}$ order $E-k$ dispersion relation.
The black curve in panel (c) is the energy dependent scattering rate extracted from the data shown in (a) and the red curve shows the result of a fit using marginal-Fermi-liquid theory.
§ DISCUSSION
§.§ Fe non-stoichiometry
Three well-resolved hole pockets around $\Gamma$ have been reported in ARPES data from an iron-stoichiometric FeTe$_{0.42}$Se$_{0.58}$ <cit.> superconductor, while the data presented here from our non-Fe-stoichiometric Fe$_{1.068}$Te$_{1-x}$Se$_{x}$ ($x$=0.36 and 0.46) superconductors only contain two hole pockets at the zone center.
This difference matches with other published data on non-Fe-stoichiometric Fe$_{1.03}$Te$_{0.7}$Se$_{0.3}$ <cit.> and Fe$_{1.03}$Te$_{0.94}$Se$_{0.6}$ <cit.> compounds, in which only two hole pockets were observed in ARPES at the zone center.
We did pick up three hole pockets from data (not shown) measured on the close to Fe-stoichiometric Fe$_{1+y}$Te$_{0.5}$Se$_{0.5}$ (y 1$\%$) sample, a result consistent with data from the Fe-stoichiometric FeTe$_{0.56}$Se$_{0.44}$ compound. <cit.>.
One recent ARPES report on the stoichiometric FeTe$_{0.56}$Se$_{0.44}$ suggested that upon increasing the sample temperature the hole pocket with $xy$ character completely loses its spectral weight, while the other two pockets ($xz/yz$ and $z^2$) maintain their itinerant character also at higher temperature. <cit.> This was explained as the evolution to an orbitally-selective Mott-insulator at higher temperature. The data presented here are measured at a sample temperature close to 1K, and yet the third hole pocket around $\Gamma$ is already missing in the case of Fe$_{1.068}$Te$_{1-x}$Se$_{x}$ compounds.
The FeSe and FeTe systems and their doped variants display complex and rich defect chemistry. For example, in Ref. Chen2014, ordering of Fe vacancies in $\beta$-Fe$_{1-x}$Se is argued to lead to a non-superconducting, 'parent' phase of the FeSe superconductors. In Ref. Wang2015, K$_2$Fe$_4$Se$_5$ is argued to be an Fe vacancy-ordered non-superconducting parent compound to the high-T$_c$ K-intercalated FeSe superconductors.
Thus, the issue of off-stoichiometry in these systems is central to their electronic structure and ground-state properties.
Comparing the electronic structure between stoichiometric and non-stoichiometric compounds it can be seen that already an iron excess of only 3$\%$ - irrespective of the amount of Se doping - is enough to lead to the third hole pocket at the zone center being barely resolvable. <cit.>
The absence of the third hole pocket (that one which has dominant $xy$ character) could be linked to its Mott-insulating character due to the interaction with the local magnetic moment of the excess iron. As the because of which the spectral weight of $xy$ band is totally lost compared to the $xz/yz$ bands, <cit.> the interaction between the itinerant electrons and the local magnetic moment of excess iron would seem to have more effect on the in-plane $xy$ band compared to the $xz/yz$ bands which possess more out of plane character.
A theoretical study suggested that each excess iron atom provides an additional electron to the system in these compounds, <cit.> which could be expected to give rise to a rigid-band-type shift of the Fermi level. This kind of behaviour has been seen on electron doping in the 122 iron pnictide systems. <cit.>
From a comparison of our 11 ARPES data with those of Ref. Nakayama2010 and Ref. Tamai2010, we notice that the $\gamma$ band [as seen in Fig. <ref> (d-g)] has a constant binding energy of 0.35 eV, irrespective of the amount of the excess Fe present in the composition. This would argue against a simple rigid-band-type scenario for the excess iron in the 11 compounds.
§.§ Isovalent Se,Te substitution
Isovalent substitution generally induces an additional crystal field potential to the system, and therefore, could lead to changes in the electronic structure as has been seen in the iron pnictide 122 system (BaFe$_2$As$_2$) on P substitution for As <cit.> or Ru for Fe. <cit.>
In our present study, Se substitution at the Te site is also isovalent doping that could lead to a crystal field splitting of the Fe $3d$ orbitals. Hence, one may expect changes in the electronic structure of Fe$_{1.068}$Te$_{1-x}$Se$_{x}$ with varying Se doping concentration.
However, we did not observe noticeable changes for $x$ varying between $x$=0.36 and 0.46 (see Figs. <ref>, <ref> and <ref>).
In the case of the iron pnictides, we have seen that the isovalent substitution of P for As in the Ba122 system leads to changes in the electronic structure even for a substitution as small as 5$\%$. <cit.>
A recent report on the iron chalcogenides offers a solution to this apparent discrepancy, as it communicates that Se doping mainly affects the band of $xy$ character, leaving the other two hole-like bands ($xz/yz$ and $z^2$ at the zone center) mostly unchanged. <cit.>
As already discussed above, our ARPES data show only two hole-like bands around the zone center, and our polarisation analysis attributes these to the $xz/yz$ and $z^2$ related bands, so the arguments of Ref. Liu2015 also fit our data well.
§.§ Orbital ordering
A directional orbital contribution to the hole pockets in the iron pnictides has been proposed by Graser $et~al$., in their itinerant picture of the electronic structure of these systems. <cit.>
In the present study of the 11 system, a directional orbital contribution to the hole pockets could clearly be observed [see Fig. <ref> and Fig. <ref>]. What the implications are of this orbital ordering in $k$-space for superconductivity is not clear at present. Intraorbital interactions between hole and electron pockets have been argued to be advantageous for iron-based superconductivity, over interorbital interactions. <cit.> In this context the orbital ordering of the Fermi sheets that contribute to the Cooper pairs at both the center and corner of the Brillouin zone is certainly an asset for high-$T_c$ superconductivity, in addition to considerations involving Fermi surface nesting.
(Color online) Imaginary part of self-energy ($\Im\Sigma$) is plotted as a function of energy below $E_F$. The different lines compare $\Im\Sigma$ from marginal-Fermi-liquid theory and spin-fluctuation theory. Black curve is the experimental data and red solid curve is a MFL fit to the experimental data shown in Fig. <ref>(c). Blue and green dashed curves are the SF-theory simulations using Eq. <ref>, and differ with respect to their coupling constant ($\lambda_{sf}$) and characteristic energy ($\omega_{sf}$).
§.§ Dependence of the electronic structure on $k_z$
§.§.§ Orbital character
Next, we discuss the $k_z$ dependence of the electronic structure at the zone center. In 122 systems, we earlier reported a strong $k_z$ Fermi surface warping at the zone center due to the transformation of orbital character from $xz/yz$ to $z^2$ while going from $\Gamma$ to $Z$. <cit.> In contrast, Fe$_{1.068}$Te$_{0.54}$Se$_{0.46}$ shows only a weak $k_z$ warping along the $\Gamma-Z$ direction (see Fig. <ref>), a situation also picked up on in Ref. Starowicz2013. This can be linked to the absence of an orbital character switch from $xz/yz$ to $z^2$ in the 11 compounds, as opposed to the 122 systems. This conclusion is supported by the observation of $k_z$ dependent band dispersion of the $\alpha_1$ band [see Figs. <ref> (d)-(f)], attributed here with the help of the DFT calculations to the $xz/yz$ and $z^2$ orbital character.
From Figs. <ref> (d)-(f) it can clearly be seen that the $\alpha_1$ band just touches $E_F$ at the $\Gamma$-point and then disperses away from the Fermi level towards higher binding energy while approaching the $Z$-point. Therefore, the $z^2$ orbital does not contribute to the Fermi surface at the $Z$ point, meaning that the states seen in the $k_z$ map measured near the zone center have solely $xz/yz$ character [see Fig. <ref> (a)].
These observations are in good agreement with the minimal orbital theory of iron-based superconductors, <cit.> which stresses not only the simple crystal structure of the iron chalcogenide superconductors but also their simple low-energy electronic structure.
Two further interesting points can be noted here: (a) in this 11-compound, only a single band exists at the Fermi surface at the $Z$-point which could contribute to superconductivity, whereas in 122 systems all three bands are present and (b) the top of the $\alpha_1$ band shifts towards higher binding energy in the present system while going from $Z$ to $\Gamma$, whereas it shifts towards lower binding energy in the 122 systems while going from $Z$ to $\Gamma$.
§.§.§ Mass renormalization
From the estimation of mass renormalization as a function of photon energy shown in Fig. <ref>(j), it can be seen that the $\alpha_2$ band retains a value of $m^*/m_b$ $\approx$ 1.8$\pm$0.3 for all $k_z$ values probed.
This is in contrast to the case for the $\alpha_1$ band, which shows strong variation in the mass renormalization from $m^*/m_b$ $\approx$ 1.5$\pm$0.4 to 5.2$\pm$1 in the region for which $k_z$=0.5 (h$\nu$ = 87 and 90 eV).
We note that a $m^*/m_b$ value of just under two is shared by both bands close to $\Gamma$ (h$\nu$ = 96 eV).
As mentioned previously in the results section, the $\alpha_2$ band from DFT was shifted so as to match the experimental $k_F$ for this band. This could not be done for the $\alpha_1$ band, and the resultant uncertainty in the fidelity of the energy location of the top of this band in the DFT could contribute to the observed strong variation in the mass renormalization for $\alpha_1$.
There is good consistency between the mass renormalization and the calculated Fermi velocity for the band $\alpha_1$. Fig. <ref>(k), which changes in Fermi velocity from 0.65 eVÅ at the $Z$-point to greater than 0.9 mid-way to $\Gamma$ and finally takes a value of 0.85 eVÅ at the $\Gamma$-point itself.
In contrast, the DFT predicts a $k_z$ independent Fermi velocity of $v_F$=0.5$\pm$0.1 eVÅ for the $\alpha_2$ band, and this is not only quantitatively consistent with the experimental data that give a $k_z$-independent $v_F$=0.4$\pm$0.1 eVÅ [see Fig. <ref>(j)], but also consistent with the $k_z$ independent mass renormalization for this band.
We note here that, on the whole, the mass renormalizations we observe for both hole pockets are consistent with the values of $m^*/m_b$ $\approx$ 2-4 reported in Refs.Xia2009a,Liu2015.
Closing the discussion on the effective mass, we emphasize that in the light of the calculations reported in Ref. Fink2015, the moderate mass enhancements seen here of between 2 and 4 occur only at higher binding energies, i.e. well away from the chemical potential.
In the case where a flat band lies close to the Fermi level yielding a van Hove singularity and there is an imminent Lifshitz transition, then a dramatic increase in the mass enhancement occurs within the marginal-Fermi liquid model, which directly follows from the linear-in-energy dependence of imaginary part of the self-energy ($\Im\Sigma$). This means that when calculating the real part of self-energy ($\Re\Sigma$) via a Kramers Kronig transformation of $\Im\Sigma$, the low-energy logarithmic increase of $\Re\Sigma$ leads to a very flat band and to strong mass enhancements of order 10 near the chemical potential. <cit.>
§.§.§ Quantum criticality and energy dependent scattering rates
Quantum criticality in the iron-based superconductors is part of the current consensus as regards the understanding of high-$T_c$ superconductivity in these materials. A quantum critical point in these compounds has been observed experimentally <cit.> and predicted theoretically. <cit.> Quantum criticality in iron-based superconductors is rooted to short range spin-fluctuations active across an interband nesting vector ($\pi$,0). Near the quantum critical point, the system switches from being a Fermi liquid to displaying marginal Fermi liquid behavior. This means that the imaginary part of the self energy has a linear dependence on the energy, <cit.> which is significantly different from the quadratic energy dependence observed in conventional Fermi liquids. In the present case this has been systematically studied for the 11 system.
Earlier DMFT calculations suggested a crossover from Fermi liquid to a non-Fermi liquid character in the case of BaFe$_2$As$_2$ at optimal hole doping given sufficiently high sample temperatures.<cit.> No such behaviour has been predicted with temperature for electron doping in the 122 materials, and recent optics data show Fermi liquid behavior in the bulk of annealed, electron doped Ba122 crystals. <cit.>
On the contrary, a recent ARPES study on various 122 and 111 systems doped with charge carriers and with isovalent substitution into the parent compound unambiguously shows a non-Fermi-liquid character near a regime of optimal charge doping or substitution. <cit.>
In the ARPES data presented here from the iron chalcogenide Fe$_{1+y}$Te$_{0.5}$Se$_{0.5}$ system, a non-Fermi-liquid behaviour of the quasiparticles was found for the band $\alpha_1$ near the zone center by extraction of the scattering rates as a function of the binding energy [see Fig. <ref>]. Specifically, we found a linear energy dependency of the scattering rate on binding energy, resembling the behaviour of a marginal-Fermi-liquid. Given the discussion above, It is relevant to note here that the top of the $\alpha_1$ band is very close to the Fermi level and will yield a van-Hove-singularity-like peak in the density of states. in close proximity to a van Hove singularity near the Fermi level. Following the argumentation of Ref. Fink2015, the presence of a van Hove singularity would induce non-Fermi-liquid behaviour for the quasiparticles.
The data presented here, therefore, can be taken to provide evidence for the importance of such phenomena in high-$T_c$ superconductors of iron parentage, besides the well-known case of the copper-oxides. <cit.>
§ CONCLUSIONS
In conclusion, using angle-resolved photoelectron spectroscopy (ARPES), we have studied the electronic structure of Fe$_{1+y}$Te$_{1-x}$Se$_x$ superconductors.
From polarization-dependent measurements we disentangled the orbital character of the detected bands that are formed mainly by the combination of ${xz}$, ${yz}$ and $z^2$ states in the vicinity of the Fermi level.
We observed that the presence of excess Fe does not shift the bands in a rigid-band manner in these compounds.
The $k_z$ dependent band structure suggests weak Fermi surface warping along the $\Gamma - Z$ direction for the $\alpha_2$ band, while the $\alpha_1$ hole-like band that does not cross the Fermi level shows a moderate $k_z$ dispersion.
The mass enhancement factor ($m^*/m_b$) was not observed to change significantly from $\Gamma$ to $Z$ for the $\alpha_2$ band, but a dramatic change in $m^*/m_b$ was seen for the $\alpha_1$ band close to $k_z$=0.5 in units of $\pi/c$.
Despite this, near the $\Gamma$- and $Z$-points, both the $\alpha_1$ and $\alpha_2$ bands show the same mass enhancement factor within the range $m^*/m_b$=1.8$\pm$0.2. The observation of a $k_z$-independent Fermi velocity ($v_F$) for the $\alpha_2$ hole pocket is consistent with our DFT calculations.
We go on to show that the experimentally obtained imaginary part of the self-energy can be compared with both the marginal-Fermi-liquid and spin-fluctuation theoretical scenarios.
The spin-fluctuation theory give a reasonable agreement to the data but with unrealistic parameters ($\lambda_{sf}$=7 and $\omega$=30 meV).
The marginal-Fermi-liquid approach fitted the self-energy data for the $\alpha_1$ band well, yielding a coupling constant, $\lambda_{MFL}$=1.5, which is in close agreement with analogous constants derived for doped BaFe$_2$As$_2$ and NaFeAs iron pnictides. <cit.>
We discuss that the observed non-Fermi-liquid behaviour for the quasiparticles near the zone center in the 11 compounds could follow from the proximity of a van Hove singularity due to the $\alpha_1$ band to the Fermi level, thus making a direct link between the existence of a near $E_F$ van Hove singularity, non-Fermi-liquid behavior and high-$T_c$ superconductivity in iron-based compounds.
§ ACKNOWLEDGEMENTS
T.S. acknowledges support by the Department of Science and Technology (DST) through INSPIRE-Faculty program (Grant number: IFA-14 PH-86). T.S. thanks D. D. Sarma for his enormous support in I.I.Sc. J.F. and I.E. acknowledge support by the German Research Foundation (DFG) through the priority program SPP1458. This work is a part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). I.E. acknowledges the support by a Kazan (Volga Region) Federal University grant targeted at strengthening the university's competitiveness in the global research and educational environment. The authors from CSIR-NPL would like to acknowledge financial support from the Govt. of India through the DAE-SRC outstanding researcher award scheme.
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1511.00577
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This paper presents an efficient hardware design approach for list successive cancellation (LSC) decoding of polar codes.
By applying path-overlapping scheme, the $l$ instances of $(l>1)$ successive cancellation (SC) decoder for LSC with list size $l$ can be cut down to only one.
This results in a dramatic reduction of the hardware complexity without any decoding performance loss.
We also develop novel approaches to reduce the latency associated with the pipeline scheme.
Simulation results show that with proposed design approach the hardware efficiency is increased significantly over the recently proposed LSC decoders.
§ INTRODUCTION
Recently, polar codes <cit.> have received significant attention due to its capability to achieve the capacity of binary-input memoryless symmetric channels with low-complexity encoding and decoding schemes.
E. Arikan in <cit.> presents a recursive cancellation way to successively accomplish decoding; and this method is referred to successive cancellation (SC).
Also, N. Hussami et al. in <cit.> shows that the belief propagation (BP) can be applied as decoding algorithm.
However, the decoding performances of both SC and BP are inferior to that of low density parity check (LDPC) codes.
In order to make polar codes more competitive, the list SC (LSC) decoding algorithm is presented in <cit.>.
By exploiting a larger range in the codeword tree, LSC significantly improves the decoding performance.
Attracted by the potentials of LSC, a number of relevant hardware designs have been explored.
In <cit.>, hardware LSC architectures of list sizes two and four are proposed with pointer memory technique, which can avoid the high complexity of likelihood copying.
In <cit.>, a hardware efficient architecture of LSC concatenated with cyclic redundancy check (CRC) is presented.
In <cit.>, a hardware architecture of sub-optimal version of LSC decoding is introduced.
In <cit.>, a LSC with multi-bit decision is discussed, which significantly reduces the decoding latency, and the corresponding hardware architecture is presented.
All of aforementioned designs are using $l$ duplications of SC decoder for LSC decoder with list size $l$.
Consequently, compared with SC decoder, the complexity of LSC increases from $n{\log} n$ to $l\cdot n{\log} n$, where $n$ and $l$ are the length of codeword and list size, respectively.
However, such complexity increasing makes all current existing LSC architectures are impractical for decoders with large list size.
This paper presents a hardware design approach for LSC decoding using path-overlapping to maximize hardware efficiency for optimal energy utility.
Instead of using $l$ copies of SC decoder for LSC decoder, only one SC decoder used in our design.
The calculations associated with each path occur simultaneously in the same decoder by carefully arranging the hardware configuration and scheduling of SC decoding.
We arrange the LLR calculations of each path instantiated to occupy the decoder hardware stages serially in a streamlined fashion.
This yields a significant reduction of hardware complexity.
We also analyze and mitigate the latency overhead incurred in the path-overlapping scheme.
Three approaches developed to reduce this overhead are: multi-decision LSC decoding, path-LLR-compute-ahead scheme and adaptive LSC decoding.
The simulation results show that with proposed approach, the widely proposed LSC decoder can achieve a significantly higher hardware efficiency.
For instance, for LSC decoder with code length $n=1024$ and list size $l=4$, at least 50% hardware efficiency improvement achieved with proposed design approach, and the maximum improvement is up to around 130%.
This paper is organized as follows.
The relative background is reviewed in section <ref>.
In following, the proposed approach is described in section <ref>.
After that, the hardware efficiency performance and relevant analysis are presented in section <ref>.
Finally, this paper is concluded in section <ref>.
§ BACKGROUND
§.§ Polar Code
As introduced in <cit.>, a polar code is constructed by successively performing channel polarization.
Mathematically, polar codes are linear block codes of length $N = 2^n$.
The transmitted codeword ${\bm{x}}\triangleq {(x_1,x_2,\cdots,x_N)}$ is computed by $\bm{x}=\bm{u}\bm{G}$ where $\bm{G=F^{\otimes m}}$, and $\bm{F^{\otimes m}}$ is the $m$-th Kronecker power of
$\bm{F} =
\begin{bmatrix}
\end{bmatrix}
Each row of $G$ is corresponding to an equivalent polarizing channel.
For an $(N,k)$ polar code, $k$ bits that carry source information in $\bm{u}$ are called information bits.
They are transmitted via the most $k$ reliable channels.
While the rest $N-k$ bits, called frozen bits, are set to zeros and are placed at the least $N-k$ reliable channels.
An example of LSC decoding with list size 4 for $(8,4)$ polar code from codeword tree aspect
Polar codes can be decoded by recursively applying successive cancellation to estimate $\hat{u}_i$ using the channel output $y_{0}^{N-1}$ and the previously estimated bits $\hat{u}_{0}^{i-1}$.
The calculation starts from channel output to the codeword, and is computed stage by stage.
Polar code with length $n$ has ${\log} _{2}n$ stages. The previously estimated bits for intermediate stages are called partial sum.
This decoding process of polar code can be regarded as the path searching in the code tree.
SC decoding reserves only one survival path every layer.
If multiple paths are reserved in every layer, it is LSC decoding. The more paths survive, the higher chances the correct codeword can be found.
Fig. <ref> shows an example of LSC decoding with list size 4 for $(8,4)$ polar code from codeword tree aspect.
§.§ Conventional architecture of LSC
For the LSC algorithm, every information bit can derive two candidate paths, which are used to represent the decision of bit as $0$ or $1$.
Each path has its own path metric which is corresponding to its survival probability.
When performing the LSC decoding, $l$ paths are expanded to $2l$ paths for each estimated information bits.
Then the metrics of $2l$ paths are calculated to decide the $l$ survivals.
All the corresponding inner log likelihood ratios (LLRs) and partial sum of the reserved paths need to be kept along with $l$ paths as well.
Finally, the $l$ paths are fed back to SC decoders and do all the steps again and again until the last information bit is decoded.
Although all the LSC designs mentioned in Section <ref> have differences at some details, the main architecture are similar.
Typically, for a LSC decoder, it has $l$ copies of SC decoders and one metrics computation units (MCU), one sorting module and three memory banks with respect to path metrics, current survival paths and LLRs and partial sums.
The SC decoder consists of multiple processing units (PUs) with a tree architecture which consumes most of hardware resources.
Such duplications of SC decoder yield a significant hardware redundancy of LSC decoder design. In our proposed design, we are trying to avoid such unnecessary redundancy.
§ PROPOSED APPROACH
The architecture of proposed design
In this section we present our path-overlapping approach and discuss how performance optimization is carried out.
Fig. <ref> shows the architecture of proposed approach and the examples of the modified architecture of SC decoders associated with the list sizes two and four.
Since the duplications of SC decoder involves the most hardware complexity, we removed all the copies and kept only one SC decoder.
However, this modification of architecture does not mean that we just simply change parallel computing to a single-threaded lazy serial approach that computes one path at a time.
Instead, every path is computed simultaneously in the decoding threads by judiciously utilizing the decoder hardware as follows: The processing timing of each path is overlapped with others in the pipeline arrangement.
The architecture of SC decoder is modified to support this new paradigm.
Since modifications are made only on architecture and scheduling plan, no decoding performance gain loss or change is incurred.
The sorting module, MCU, and related memory components are compatible with other LSC decoders, and the partial sum generator is scheduled a similar way to be compatible with the path-overlapping SC decoder.
Thus we do not discuss that in this paper.
In the next subsections, the details of the scheme and the specific SC decoder are discussed.
§.§ Path-Overlapping Scheme and Relevant Analysis
Decoding schedule of the path-overlapping scheme for $(8,4)$ polar code with path_2 list size = 2 and path_4 list size = 4
Simultaneous processing approach is already presented in some SC decoders, and it is used for multiple frames in order to increase the throughput <cit.>.
The SC decoder with tree architecture consists of multiple processing unites (PU) arranged like a binary tree.
For every clock cycle, only one stage of PUs in the tree is activated.
The basic idea of simultaneous processing approach is activating multiple decoding stages in one clock cycle by feeding in several frames in pipeline. This means that each frame comes into the decoder with one clock cycle delay.
Stemming from above idea, we realize that the duplications of SC decoder in conventional LSC decoder is unnecessary. All the paths can be fed into the same decoder in pipelined fashion. Different stages in the single SC decoder can process different paths simultaneously.
Computations of successive paths are overlapped in temporal with only one clock cycle delay.
However, the decoding scheme is not exactly the same as multiple frames overlapping SC decoder.
Fig. <ref> and Fig. <ref> show the decoding schedule of two and four path-overlapping scheme, respectively.
The number means current activated stages, and the duplicated stage is marked with gray.
According to <cit.>, if a SC decoder is with $l$ path-overlapping scheme, where $l\leq(2^i-1)$, it can be constructed by duplicating $(2^{i-1}-1)$ stages, where the index starts from the information bits side with respect to the tree architecture.
The duplication plan is also presented in Fig. <ref>.
Noticeably in Fig. <ref> there is only one duplication of stage one, which is not the same as what presented in Fig. <ref>.
This is because the number of copies in Fig <ref> are the minimum requirement for all the case. The actual requirement is decided by the code length and rate. Fig. <ref> is just a certain case only one stage duplication is needed for four path-overlapping scheme.
Such architecture significantly reduces hardware complexity.
Another advantage of proposed approach is that it can reduce the critical path length of decoder.
Usually, the critical path lies in the sorting block.
For conventional LSC decoder, the sorting block is composed of staged combination logic.
Even for very small list size, e. g. list = 4, the critical path is much longer than any other module.
With proposed approach, since each path metrics comes with pipeline arrangement, naturally, the sorting block is designed as a pipeline module which has a shorter critical path than that of combination logic for the same list size. This means, by applying proposed approach, LSC decoder can run at a much higher frequency.
Although proposed approach can achieve a higher frequency compared with the conventional LSC decoder, there are some additional clock cycles introduced.
These consist of two parts.
The first part is the path pipeline latency $L_p$. Since all the paths are fed into decoder with one clock cycle delay, for the LSC with list size $l$, $L_p~=~(l-1)$.
The second part is path waiting latency $L_w$. After the number of path extending to the maximum, the pipeline processing has to suspend when estimating the newly generated information bit since the decoder needs to wait for all the paths to finish before commencing metric sorting and LLR copying.
This waiting period is referred to as pipeline stalling.
The waiting time is equal to $L_p$.
Thus, for the list size $l$ LSC with respect to $(n,k)$ polar code, $L_w~=~(k-{\log} _{2}l-1)\cdot (l-1)$.
Thus, the total latency overhead introduced by path-overlapping scheme $L_m$ can be calculated by:
\begin{equation}\label{TotalD}
L_m~=~L_w+L_p~=~(k-{\log} _{2}l)\cdot (l-1).
\end{equation}
This design approach can be applied to any current existing LSC decoders.
It significantly reduce the hardware complexity by eliminating redundant instances, and it incurs few additional clock cycles to achieve the improvement.
Thus, it is difficult to evaluate such design approach merely in term of the usage of hardware resource or the latency.
Thus we introduce the hardware efficiency (HE) metric which is noted as $e$ to measure the performance of proposed approach. The $e$ is defined as: $e~=~Throughput/Area$.
From Eq. (<ref>), we can tell that the latency overhead would significantly aggregate with either list size or code rate, which can significantly diminish the $e$. In order to achieve a high $e$ with proposed approach, the latency overhead must be reduced to an acceptable level.
In the next sections, we will present three approaches aimed at decreasing the latency overhead.
§.§ Latency Reduction via Multi-Decision List SC Decoding
The first part of Eq. (<ref>) corresponds to the path waiting latency.
For every instance of estimating an information bit, the pipeline processing has to suspend until all the paths finish calculations. This provides an observation that if the times of estimating the information bit can be reduced, the $L_w$ will decrease significantly.
Multi-decision is an approach of estimating $m$ bits $(m>1)$ instead of just one at the same time.
It helps to reduce the number of estimations. Many approaches can be regarded as multi-decision <cit.> <cit.> <cit.> <cit.>.
Generally, they can be classified into two types.
The first type is referred to as regular mutil-decision decoder; it estimates $m$ bits $(m>0)$ every time.
Most of current multi-decision decoders belong to this type <cit.> <cit.>.
The second type is called irregular mutil-decision decoders; the number of bits estimated every time is not fixed.
Currently, only the list fast-SSC decoder <cit.> belong to this type.
It simplifies the SC decoding by finding certain pattern in the codewords.
Such subcodes with certain pattern also refer to constituent codes.
The number of bits estimated every time is corresponding to the size of constituent code.
Besides, the distribution of constituent codes irregularly change along with code rate.
latency overhead for different scheme
For path-overlapping LSC decoder with mutil-decision, $L_m$ can be further reduced to $L_m~=~\alpha\cdot (l-1)$.
For $m$ bits regular mutil-decision, $\alpha~=~\lceil (k-{\log} _{2}l)/m \rceil. $
For irregular mutil-decision, $\alpha~=~S-{\log} _{2}l$ where $S$ is the total number of constituent codes which irregularly changes along with code rate.
Fig. <ref> shows the latency overhead of different schemes for LSC decoder with code length $n=1024$ and list size $l=4$.
We can see that all the mutil-decision schemes can significantly reduce latency overhead, and as increasing of code rate, the irregular mutil-decision scheme can still keep a very low latency overhead.
§.§ Latency Reduction via Path-LLR-Compute-Ahead Scheme
decoding schedule of path-LLR-compute-ahead scheme
Besides reducing the number of estimations, the other approach to decrease latency overhead is by avoiding the pipeline stalling.
This can be done via path-LLR-compute-ahead scheme (PLCAS).
Fig. <ref> shows this decoding schedule.
A single bar means the decoding process between estimations of two successive information bits.
When pipeline stalling happens in one path, instead of waiting, current path can do a pre-estimated between two candidates ($0$ and $1$) which it solely generates without suspension.
The pipeline processing continues with the one with larger metrics and keeps the other to compared with the next coming paths.
If more suitable paths are found later, the previous computed ones are discarded.
With this scheme, the $L_m$ for the best case is equal to pipeline latency $L_p$, which means the entire processing is handled without any stalling, and the $L_m$ for the worst case is equal to simple path-overlapping scheme.
§.§ Latency Reduction via Adaptive LSC Decoding
In Eq. (<ref>), the second part of the formulation is equal to the $L_p$.
It is determined by the number of paths set in the pipeline.
This makes the latency overhead increas linearly with respect to the list size $l$.
If we can decrease the value, the latency overhead can be significantly reduced.
Typically, $L_p$ is fixed for a LSC with given length. However, by applying adaptive LSC algorithm <cit.>, the $L_p$ is allowed to change on the fly according to current metrics of each path.
The list size would decrease along the decoding processing, which also means the latency overhead would get reduction.
In <cit.>, basic hardware architecture is also proposed.
Even though the list size would decrease along the decoding processing, the architecture proposed in <cit.> still needs $l$ copies of SC decoder for its initial status.
The usage of hardware resource is same as regular LSC decoder.
Proposed approach can exploit the metric of adaptive LSC decoder via cutting down the unnecessary hardware complexity.
With proposed approach there is no redundant hardware even when the list size decrease.
Such property allows adaptive LSC decoder to benefit more in term of $e$.
This will be shown in section. <ref>.
§ PERFORMANCE AND ANALYSIS
Fig. <ref> shows the improvement of $e$ with proposed design approach for widely proposed LSC decoders with code length $n=1024$ and list size $l=4$.
The x-axis is the rate of polar code, and the y-axis is the ratio of $e$ with proposed approach over $e$ with ordinary approach. The $e$ with ordinary approach for a given LSC decoder has a consistent value.
We apply proposed approach to four types of LSC decoder.
They are conventional LSC decoder which also is regarded as 1-bit decision LSC decoder, 4-bit decision LSC decoder, irregular multi-bit decision decoder and the adaptive LSC decoder.
We also calculated the upper and lower bound of the $e$ improvement with PLCAS.
These simulations are based on the decoders described in <cit.>, <cit.>, <cit.> and <cit.>, the related synthesis results and the analysis we made in the previous sections.
the improvement of $e$ with proposed design approach
In Fig. <ref>, all the curves are beyond the ratio of one, which means with the proposed approach, all the decoders are able to achieve a better hardware efficiency.
According to curve 1 and curve 2, the hardware efficiency of regular decision decoder, 1-bit and 4-bit decision decoder, is decreasing alone with the code rate increasing.
This is because the latency overhead is larger at higher code rate.
Besides, the regular multi-bit (4-bit) decoder achieves more improvement of $e$ than that of conventional (1-bit) decoder, which is due to the latency reduction as we described in section <ref>. This can easily derive that for $n-bit$-decision regular decoder, the bigger the $n$, the more the improvement of $e$ can be achieved with proposed approach.
Curve 1 and 5 indicate the range of the $e$ improvement with PLCAS. The actual value depends on the channel outputs and channel quality.
According to curve 4 and curve 1, we can tell that the adaptive LSC help proposed approach to dramatically increase the hardware efficiency. Such increasing benefits from the decreasing of latency overhead as we analyze in section <ref>.
Another very interesting phenomenon is about the improvement of irregular multi-bit decision (list fast-SSC decoder). The gain of $e$ does not change too much with code rate varying. This is because the latency overhead of irregular multi-bit decision decoder does not linearly change along with coder rate.
The average improvement of irregular multi-bit decision is less than that of regular one.
This is due to the inherent latency of irregular LSC decoder is already very low <cit.>.
Noticeably all the improvements are calculated based on the assumption that the maximum frequency of decoder with proposed approach or ordinary approach are the same. However, according to the analysis in section <ref>, the maximum frequency of decoder with proposed approach should be higher, which indicates that the improvements of $e$ in Fig. <ref> should be even more in practice.
Additionally, all the approaches mentioned above are not conflicting with each other. Using multiple approaches together can further increase the hardware efficiency.
The above mentioned properties indicate that proposed approach can measurably contain the hardware complexity associated with large scale LSC decoder implementation.
§ CONCLUSION
This paper presents a novel design approach to improve the hardware efficiency of LSC decoder via path-overlapping scheme. The details of design approach and three strategies to reduce the latency overhead are also presented. The numerical results show that the conventionally used LSC decoders can significantly achieve a higher hardware efficiency using the proposed approach.
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1511.00376
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1511.00153
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[Electronic address: ][email protected]
Yerevan Physics Institute, 0036 Yerevan, Armenia
[Electronic address: ][email protected]
Albert Einstein Center for Fundamental Physics, Institute
for Theoretical Physics,
University of Bern, CH-3012 Bern,
[Electronic address: ][email protected]
Department of Mathematical Sciences, University of Liverpool,
L69 3BX Liverpool, United Kingdom
In this analysis, we present the contribution associated with the
chromomagnetic dipole operator ${\cal O}_8$
to the double differential decay width $d\Gamma/(ds_1 ds_2)$ for the inclusive
process $\bar{B} \to X_s \gamma \gamma$.
The kinematical variables $s_1$ and $s_2$ are defined as
$s_i=(p_b - q_i)^2/m_b^2$, where $p_b$, $q_1$, $q_2$ are the momenta
of $b$-quark and two photons. This contribution (taken at tree level) is of
order $\alpha_s$, like the recently calculated QCD corrections to the
contribution of the operator ${\cal O}_7$.
In order to regulate possible collinear singularities of one of the photons
with the strange quark,
we introduce a nonzero mass $m_{s}$ for the strange quark.
Our results are obtained for exact $m_{s}$, which we interpret as
a constituent mass being varied between 400 and 600 $\rm MeV$.
Numerically it turns out that the effect of the (${\cal O}_8,{\cal O}_8$) contribution to the branching ratio of $\bar{B} \to X_s
\gamma \gamma$ does not exceed $+0.1\, \%$ for any kinematically allowed value of our physical cutoff parameter $c$,
confirming the expected suppression of this contribution relative to the QCD
corrections to $d\Gamma_{77}/(ds_1 \, ds_2)$
13.20.He, 12.38.Bx
LTH 1061
§ INTRODUCTION
Inclusive rare $B$-meson decays are known to be a unique source of indirect information about
physics at scales of several hundred GeV. In the standard model (SM) all these processes
proceed through loop diagrams and thus are relatively suppressed. In the extensions
of the SM the contributions stemming from the diagrams with “new”
particles in the loops can be comparable or even larger than the contribution from
the SM. Thus getting experimental information on rare decays puts strong
constraints on the extensions of the SM or can even lead to a
disagreement with the SM predictions, providing evidence for some "new physics”.
To make a rigorous comparison between experiment and theory, precise
SM calculations for the (differential) decay rates are mandatory.
While the
branching ratios for $\bar{B} \to X_s \gamma$ <cit.>
and $\bar{B} \to X_s \ell^+
\ell^-$ are known today even to
next-to-next-to-leading logarithmic (NNLL) precision (for reviews, see
<cit.> and <cit.> for recent updated predictions on radiative decay modes of $B$ meson),
other branching ratios, like the one for $\bar{B} \to X_s \gamma
\gamma$ discussed in this paper, were only known to leading logarithmic
(LL) precision in the SM
As the process $\bar{B} \to X_s \gamma \gamma$ is expected to be measured at the
planned Super $B$-factory in Japan (SuperKEKB)
<cit.>, we recently completed
first steps towards a next-to-leading logarithmic (NLL) result for this decay
<cit.>, by working out QCD corrections to the numerically
important (${\cal O}_7$, ${\cal O}_7$) contribution.
In this paper, we go one step further and provide the self-interference
contribution to $\bar{B} \to X_s \gamma \gamma$ stemming from the chromomagnetic dipole operator ${\cal O}_8$
which starts at order $\alpha_s$. Although a naive estimate suggests that this
contribution is suppressed by a factor of $|C_{8}^{{\rm eff}}
Q_{d}/C_{7}^{{\rm eff}}|^{2} \sim 1/36$ relative to the QCD corrections to the (${\cal O}_7$, ${\cal O}_7$)
interference, a more detailed investigation is in order: In both cases
(${\cal O}_7$ and ${\cal O}_8$), one of the two photons can be emitted from
the strange quark in a collinear way, leading to contributions involving
$\log(m_{s}/m_{b})$ terms.[We interpret $m_s$ to be a constituent
mass, varying it between 400 and 600 MeV.] Concerning the other photon, the
two cases differ, however. Unlike in the ${\cal O}_7$, the second photon can also be emitted
from the $s$-quark in the ${\cal O}_8$ case. While a fully collinear emission
of both photons is excluded by our cuts (see later), a leftover enhancement effect could
still apply in the ${\cal O}_8$ case and thereby milder the naive suppression
factor. As the average energies of the two photons are not very high, there
might be a second effect related to the different infrared structure
($1/E_{\gamma}$-terms) of the two cases, which also potentially milders the naive suppression
factor given above. We feel that these considerations motivate a detailed
evaluation of the $({\cal O}_8,{\cal O}_8)$-interference contribution.
The starting point of our calculation is the effective Hamiltonian,
obtained by integrating out the heavy particles in the SM, leading to
H_eff = - 4 G_F/√(2) V_ts^⋆V_tb
∑_i=1^8 C_i(μ) O_i(μ) ,
where we use the operator basis introduced in <cit.>:
O_1 =
(s̅_L γ_μT^a c_L)
(c̅_L γ^μT_a b_L) ,
O_2 =
(s̅_L γ_μc_L)
(c̅_L γ^μb_L) ,
O_3 =
(s̅_L γ_μb_L)
(q̅ γ^μq) ,
O_4 =
(s̅_L γ_μT^a b_L)
(q̅ γ^μT_a q) ,
O_5 =
(s̅_L γ_μγ_νγ_ρb_L)
(q̅ γ^μγ^νγ^ρq) ,
O_6 =
(s̅_L γ_μγ_νγ_ρT^a b_L)
(q̅ γ^μγ^νγ^ρT_a q) ,
O_7 =
e/16π^2 [
s̅ σ^μν ( m̅_b R + m̅_s L ) F_μν b ] ,
O_8 =
g_s/16π^2 [
s̅ σ^μν ( m̅_b R + m̅_s L ) T^a G^a_μν b ] .
The symbols $T^a$ ($a=1,8$) denote the $SU(3)$ color generators;
$g_s$ and $e$ denote the strong and electromagnetic coupling constants.
In opbasis,
$\bar{m}_b$ and $\bar{m}_s$ are the running $b$ and $s$-quark masses
in the $\MS$-scheme at the renormalization scale $\mu$.
We keep the exact dependence on the strange-quark mass in our calculation.
Further, as we are not interested in CP-violation effects in the present paper, we
exploited the unitarity of the Cabibbo–Kobayashi–Maskawa (CKM) matrix and
neglected $V_{ub} V_{us}^*$ (as $V_{ub} V_{us}^* \ll V_{tb} V_{ts}^* $) when writing
While the Wilson coefficients $C_i(\mu)$ appearing in Heff
have been known to sufficient precision at the low scale $\mu \sim m_b$
for a long time (see e.g. the reviews <cit.>
and references therein), the matrix elements
$\langle s \gamma \gamma|{\cal O}_i|b\rangle$ and
$\langle s \gamma \gamma \, g|{\cal O}_i|b\rangle$,
which in a NLL calculation are needed to order
$g_s^2$ and $g_s$, respectively, are only partially known now (see
<cit.> for the details of the provided
and <cit.> for a recent summary). Calculating the
$({\cal O}_i,{\cal O}_j)$-interference contributions for the
differential distributions at order
$\alpha_s$ is in many respects of similar complexity as the
calculation of the photon energy spectrum in $\bar{B} \to X_s \gamma$
at order $\alpha_s^2$
needed for the NNLL computation. There, the individual
interference contributions, which all involve extensive calculations, were
published in separate papers, sometimes even by two independent groups
(see e.g. <cit.>).
It therefore cannot be expected that the NLL results for the
differential distributions related to $\bar{B} \to X_s \gamma \gamma$ are
given in a single paper. As a next step in the NLL enterprise, we
derive in the present paper the $({\cal O}_8,{\cal O}_8)$-interference
contribution (which starts at order $\alpha_s$) to the double
differential decay width $d\Gamma/(ds_1 ds_2)$.
The variables $s_1$
and $s_2$ are defined as $s_i=(p_b-q_i)^2/m_b^2$, where $p_b$
and $q_i$ denote the four-momenta of the $b$-quark and the two
At order $\alpha_s$ there are only contributions to $d\Gamma_{88}/(ds_1 ds_2)$
with four particles ($s$-quark, two photons and a gluon) in the final state.
These contributions correspond to specific cuts of the $b$-quark
self energy at order $\alpha^2 \times \alpha_s$, involving twice the
operator ${\cal O}_8$. As there are additional cuts, which contain for
example only one photon, our observable cannot be obtained using the
optical theorem, i.e., by taking the absorptive part of the $b$-quark
self energy at three loops. We therefore calculate the mentioned
contributions with four particles in the final state individually.
When calculating the contribution of ${\cal O}_8$ to $d\Gamma/(ds_1 ds_2)$,
we restrict ourselves (as in refs. <cit.>) to the region in the $(s_1,s_2)$-plane
which is also accessible to three body decays $b \to s \gamma \gamma$
(associated e.g. with the tree-level contribution of ${\cal O}_7$), i.e.,
s_1 > x_4 ;
s_2 > x_4 ;
s_1 + s_2 < 1+ x_4 ;
s_1 s_2 > x_4 ,
where $x_4=(m_s/m_b)^2$.
The energies $E_1$ and $E_2$ in the rest frame of the $b$-quark of the two photons are
related to $s_1$ and $s_2$ in a simple way: $s_i=1-2 \, E_i/m_b$.
As the energies $E_i$ of the photons have to be away from zero in order to
be observed, the values of $s_1$ and $s_2$ should be considered to be
smaller than one. Furthermore, in order to see two separate photons, their invariant mass should
also be away from zero. All these requirements can be implemented in terms of one
physical cut parameter $c$ ($c>0$), by demanding[The normalized
invariant mass squared $s=(q_1+q_2)/m_b^2$ of the two photons can be written
as $s=1-s_1-s_2+s_3$, where $s_3$ is the normalized hadronic mass squared.]
s_1≥c , s_2 ≥c , 1-s_1-s_2 ≥c .
The kinematical region in the $(s_1,s_2)$-plane, which we take into account in this
paper, therefore corresponds to the intersection of the regions given in eqs.
(<ref>) and (<ref>). For explicit formulas representing this
intersection, we refer to the appendix.
Imposing these cuts, the photons do not become soft in our case, while one of
them can become collinear with the strange quark.
This implies that in the final result a single logarithm of $m_{s}$ survives.
The only source for such $\log(m_s)$ terms in our result is the mentioned
collinear emission of the photons from the $s$ quark. In particular, we emphasize that the
$({\cal O}_8,{\cal O}_8)$-contribution to the double differential decay width
does not become singular when the gluon and the strange quark become
collinear, since the gluon is
emitted from the effective operator ${\cal O}_8$ directly and therefore there
is no propagator denominator of the form $(p_s + p_g)^2$ which could become singular. In addition,
soft-gluon related singularities also do not appear in this case (the matrix
element associated with ${\cal O}_8$ even goes to zero when the gluon energy
tends to $0$). The absence of singularities generated by soft and/or
collinear gluons is related to the fact that concerning QCD our
observable (i.e. the triple or double differential decay width), based on the
full effective Hamiltonian, is fully inclusive and therefore nonsingular.
We also stressed this fact in <cit.>, where
the $({\cal O}_7,{\cal O}_7)$-contribution was worked out. In this case
there were gluon induced singularities in the virtual and bremsstrahlung
corrections, but they canceled when combined
as a consequence of the Kinoshita-Lee-Nauenberg (KLN) theorem. This means that the origin of
$\log(m_s)$ terms is from collinear photon emission only. Note that concerning QED our
observable is not fully inclusive, because we want to observe exactly two
photons in the final state; therefore $\log(m_s)$ terms remain.
A further remark on the numerical $m_s$-dependence is in order: The $({\cal O}_7,{\cal O}_7)$-contribution
to the double differential decay width starts at order $\alpha_s^0$. This
leading contribution does not contain $\log(m_s)$ terms when applying the
kinematical cuts discussed above. Only at order $\alpha_s^1$
terms $\sim \log(m_s)$ appear, because one of the photons can become collinear with the
strange quark. As a consequence, we expect the relative $m_s$-dependence of the
$({\cal O}_7,{\cal O}_7)$ contribution to be smaller than the corresponding dependence of the
$({\cal O}_8,{\cal O}_8)$ contribution, because the latter only starts at order
$\alpha_s^1$. In other words the $m_s$-dependence of the complete double
differential decay width will be smaller than the one which is only based on
the $({\cal O}_8,{\cal O}_8)$ contribution discussed in this paper.
The main goal of this paper is to work out $d\Gamma_{88}/(ds_1 ds_2)$ as a further
ingredient towards a systematic NLL prediction for the decay rate of $\bar{B} \to
X_s \gamma \gamma$.
For similar analysis for the case of $\bar{B} \to X_s \gamma$, one can see
e.g. <cit.>.
In this regard, we employ in our calculation a finite
strange-quark mass $m_s$ which we interpret to be of
constituent type in the numerics. This approach
has also been adopted previously, e.g. by Kaminski et al. in
<cit.> and Asatrian and Greub in <cit.>. The experience gained in these references shows that
the constituent mass approach gives
results which are similar to those when using fragmentation functions
<cit.>. Therefore, we believe that this method is sufficient
to obtain an estimate of the $({\cal O}_8,{\cal O}_8)$-interference
contribution. While the fragmentation approach seems better from the
theoretical point of view, it is not clear that it leads to
better final results in practice, because the fragmentation
functions (for $s\to\gamma$ or $g\to\gamma$) suffer from experimental
uncertainties, as pointed out in <cit.>. An alternative could be to look at the version with
“isolated photons” a la Frixione <cit.> which corresponds,
however, to a slightly different observable. Such an approach is beyond the
scope of the present paper and is left for future studies.
Before moving to the detailed organization of our paper, we should
mention that the inclusive double radiative process $\bar{B} \to X_s \gamma
\gamma$ has also been explored in several extensions of the SM
<cit.>. Also
the corresponding exclusive modes, $B_s \to \gamma \gamma$
and $B\to K \gamma \gamma$, have been
examined before, both in the SM
<cit.> and
in its extensions
We should add that the long-distance resonant effects were
also discussed in the literature (see e.g. <cit.> and the
references therein).
Finally, the effects of photon emission from the spectator quark in
the $B$-meson were discussed in <cit.>.
The remainder of this paper is organized as follows. In section <ref>
the calculation of the $({\cal O}_8,{\cal O}_8)$-contribution to the double
differential decay width $d\Gamma/(ds_1 ds_2)$ is presented.
To regulate the configurations where photons are emitted from the $s$ quark in a
collinear way, a finite strange-quark mass $m_{s}$ is introduced.
This way the collinear singularities manifest themselves as $\log(m_{s})$
terms in our final result, which reflects the feature for the photons having hadronic substructure.
In section <ref> we illustrate the numerical impact of the
$({\cal O}_8,{\cal O}_8)$-contribution to the double differential width and the
total decay width (depending on a kinematical cut).
The main text of our paper ends with a short summary in section
<ref>. In the appendix <ref>, we give the explicit formulas defining the four-particle phase-space region considered in this paper together with the explicit
expressions for the master integrals (MIs) appearing in our calculation.
§ $({\CAL O}_8,{\CAL O}_8)$ CONTRIBUTION TO THE DOUBLE DIFFERENTIAL SPECTRUM
$D\GAMMA/(DS_1 DS_2)$ AT ${\CAL O}(\ALPHA_{S})$
On the first line the diagrams defining the
${\cal O}_8$ contribution to $b \to s g \gamma \gamma$ are shown
at the amplitude level. The crosses in the graphs stand
for the possible emission places of the gluon (emerging from the operator ${\cal O}_8$).
On the second line the contribution to the decay width corresponding
to the interference of diagram 1 with diagram 4 is illustrated. This sample
interference diagram gives rise to
$\log\left( m_{s}/m_{b}\right)$ terms due to collinear configurations of one of the photons with the $s$ quark.
We now turn to the calculation of the ${\cal O}_8$ self-interference contribution to the decay width for $\bar{B} \to X_s \gamma
\gamma$, which is based on the partonic process $b \to s g \gamma
\gamma$, where $g$ denotes a gluon. Although this is only a tree-level computation at order $\alpha_s$,
it is quite complicated because of the four particles in the final state, one
of them being massive (the strange quark).
Before going into detail, we mention that the kinematical range of
the variables $s_1=(p_b-q_1)^2/m_b^2$ and $s_2=(p_b-q_2)^2/m_b^2$ is larger in
the $1 \to 4$ process considered in this section than the range given in eq:cutsA, which corresponds
to the $1 \to 3$ process $b \to s \gamma \gamma$.
Nevertheless, we restrict ourselves to the range which corresponds to the
intersection of the regions given in eq:cutsA and eq:cuts, as
we also did in <cit.> when considering
virtual and bremsstrahlung corrections to the ${\cal O}_7$-contribution. For explicit formulas of the considered $(s_1,s_2)$-region, we refer to
eq:PScases in the appendix.
The diagrams defining the ${\cal O}_8$ contribution at the
amplitude level are shown in the first line of
Fig. <ref>. The amplitude squared, needed to get the (double differential) decay width, can be written
as a sum of interferences of the different diagrams shown on the first line in
Fig. <ref>. One such interference is shown on the
second line of the same figure. The four-particle final state is
described by five independent kinematical variables; $s_1$ and $s_2$ are just
two of them.
In the present paper, we worked out in a first step the triple differential
spectrum $d\Gamma_{88}/(ds_1 \, ds_2\, ds_{3})$, where
$s_{3}=(p_{s}+p_{g})^{2}/m_{b}^{2}$ is the normalized hadronic mass squared
and $p_{g}$ is the final state gluon momentum. At this level, we computed
the resulting MIs numerically for exact $m_{s}$ (see
section <ref> for their explicit expressions). To get the double
differential spectrum $d\Gamma_{88}/(ds_1 \, ds_2)$ we then integrated over
$s_{3}$ in its range $s_{3}\, \in \, \left[ m_{s}^{2}/m_{b}^{2},\,
Last, as the various steps of the calculation are similar to those in
Ref. <cit.>, we refer to section $7$ of that paper for more
details on the techniques applied. Also, we refer to appendix B of
Ref. <cit.> for a useful
parametrization of the four-particle phase-space for the case where one of the
particles is massive, which is based on the work in
Ref. <cit.>.
Parameter Value
$\rm BR_{sl}^{exp}$ $0.1049$
$\rm m_{b}$ $4.8$ GeV
$\rm m_{c}/m_{b}$ $0.29$
$\rm G_{F}$ $1.16637\times10^{-5}$ GeV$^{-2}$
$\rm V_{cb} $ $0.04$
$\rm V_{tb} V_{ts}^* $ $0.04$
$\rm {\alpha_{\rm (em)}}^{-1}$ $137$
$C_{8,eff}^{0}(\mu)$ $\alpha_s(\mu)$
$\mu= M_W$ $-0.09739$ $0.1213$
$\mu=2 \, m_{b}$ $-0.13516$ $0.1818$
$\mu=m_{b}$ $-0.14905$ $0.2175$
$\mu=m_{b}/2$ $-0.16529$ $0.2714$
Upper: Relevant input parameters used in this paper. Lower: The Wilson coefficient $C_{8,eff}(\mu)$ and $\alpha_s(\mu)$ at
different values of the renormalization scale $\mu$.
§ NUMERICAL ILLUSTRATIONS
In the previous section we described the calculation for the $({\cal O}_8,{\cal O}_8)$ contribution to the double differential decay
width for $\bar{B} \to X_s \gamma \gamma$ at NLL precision.
The Wilson coefficient $C_{8,eff}(\mu)$ at the low scale[At NLL
precision, $C_{8,eff}(\mu)$ is needed only up to order $\alpha_s^{0}$,
because the square of the matrix element $\langle s g \gamma \gamma| {\cal O}_8|
b \rangle$ starts at order $\alpha_s^1$.
Furthermore, for our current purpose we identify the
$\overline{\mbox{MS}}$ mass $\bar{m}_b(\mu)$
with the corresponding pole mass.]
C_8,eff(μ) = C_8,eff^0(μ_b)
has been known for a long time (see Ref. <cit.> and
references therein).
Numerical values for
the input parameters and for this Wilson coefficient at various values
for the scale $\mu$, together with the
numerical values of $\alpha_s(\mu)$, are given in upper and lower panels of Table <ref>, respectively.
To stress that the $({\cal O}_8,{\cal O}_8)$-contribution to $d\Gamma/(ds_1 ds_2)$ only
starts at the NLL level, we write
dΓ_88/ds_1 ds_2 = dΓ_88^(1)/ds_1 ds_2
where $d\Gamma_{88}^{(1)}/(ds_1 ds_2)$ has the form
\begin{eqnarray}
\frac{d\Gamma_{88}^{(1)}}{ds_1 \, ds_2} &&\,=\,
\frac{ \alpha^2 \, \bar{m}_b^2(\mu) \, m_{b}^3 \, |C_{8,eff}(\mu)|^2 \, G_F^2 \,
|V_{tb} V_{ts}^*|^2 \, Q_d^4}{1024 \, \pi^5} \, \, \, \, \nonumber \\
\times \,\frac{\alpha_s}{4 \pi} \, C_{F}\, \,\kappa^{(1)}_{88}(s_{1},s_{2},m_{s}/m_{b}) \, .
\label{eq:double88}
\end{eqnarray}
The function $\kappa^{(1)}_{88}(s_{1},s_{2},m_{s}/m_{b})$, which encodes the
dependence on $s_1$, $s_2$ and on $m_s/m_b$, is too lengthy to be displayed
explicitly. We note that we will keep the exact $m_s$ dependence in our
$d\Gamma_{88}/(ds_1 \, ds_2)$ [as given in eqs. (<ref>) and (<ref>)] as a
function of $s_1$ for
$s_{2}$ fixed at $0.2$, $\mu=m_{b}/2$ and $m_{s}$ varied between
$400$ and $600$ MeV. The blue(top), yellow(middle) and
red(bottom) lines show the width when choosing $m_{s}$ to be $400$,
$500$ and $600$ MeV, respectively.
In Table <ref>, the impact of $\frac{d\Gamma_{88}^{(1)}}{ds_1 \, ds_2}$ on the branching ratio for $\bar{B} \to X_s \gamma \gamma$ is presented
for various choices of $m_{s}$, $c$ and the scale $\mu$. It is seen that this contribution is much
smaller than the corresponding numbers for the $({\cal O}_7$, ${\cal O}_7)$
contribution (see Table 4 of Ref. <cit.> for comparison).
To obtain the values for the branching ratio in Table <ref> as a function of the cutoff parameter $c$ defined in
eq:cuts, we integrate the double differential spectrum over the
corresponding ranges in $s_{1}$ and $s_{2}$ ${\left[\rm see~\eq{eq:PScases}\right]}$, divide by
the semileptonic decay width and multiply with the measured semileptonic
branching ratio. For illustrative purposes, it is sufficient to take the lowest order
formula for the semileptonic decay width [see e.g. Eq. (6.2) in Ref. <cit.>].
In Fig. <ref> we plot $d\Gamma_{88}/(ds_1 \, ds_2)$, calculated
in this paper, as a function of $s_1$, while $s_2$ is kept fixed at
$s_{2}=0.2$. The renormalization scale is chosen to be $\mu=m_{b}/2$ and
$m_{s}$ is varied between $400$ and $600$ MeV. This figure shows that
$d\Gamma_{88}/(ds_1 \, ds_2)$ is orders of magnitude smaller in size
than $d\Gamma_{77}/(ds_1 \, ds_2)$ (for comparison see Fig. 7
of Ref. <cit.> which is an extended analysis of the work in
Ref. <cit.>). For other choices of the scale $\mu$, the
behavior of the spectrum is similar, but even smaller in size.
Relative shift ($ {\rm \frac{ Br[\bar{B}\to X_s\gamma \gamma
]_{c}^{88} }{ Br[\bar{B}\to X_s\gamma \gamma ] _{c}} } $) of
the branching ratio for ${\rm \bar{B}\to X_s\gamma \gamma }$ (in percent)
due to the (${\cal O}_8$ , ${\cal O}_8$) contribution as a function
of the cut parameter $c$ for $\mu=m_{b}/2$. The blue(top), yellow(middle)
and red(bottom) lines show the relative shifts when setting $m_{s}=400$ MeV,
$500$ MeV and $600$ MeV, respectively. For other choices of the scale $\mu$
the relative change is even smaller.
In Fig. <ref> we investigate the numerical impact of the
(${\cal O}_8$ , ${\cal O}_8$) contribution on the branching ratio
of $\bar{B}\to X_s\gamma \gamma$ (see the discussion in the
third paragraph of the introduction). More precisely,
we worked out the relative shift
Br[B̅→X_sγγ]_c^88 / Br[B̅→X_sγγ] _c
of the branching ratio due to the (${\cal O}_8$, ${\cal O}_8$) contribution,
as a function of the kinematical cut parameter $c$.
Fig. <ref> clearly shows that this contribution is below
$0.1\%$ in the full $(s_1,s_2)$-range considered in this paper.
We mention that in $\bar{B} \to X_s \gamma$ the situation concerning the
${\cal O}_8$ contribution is different. As pointed out in refs.
<cit.>, in this decay mode the contribution
of ${\cal O}_8$ is non-negligible, in particular, for values of $E_\gamma <
1.1$ GeV. On the other hand, in the double radiative decay, the effects
described in the references just mentioned are also present in the ${\cal O}_7$
contribution; as a consequence the effect of the ${\cal O}_8$ contribution stays small
in the full phase space.
§ CONCLUDING REMARKS
In the present work we calculated the set of the $O(\alpha_s)$
corrections to the decay process $\bar{B} \to X_s \gamma \gamma$
originating from diagrams involving the chromomagnetic
dipole operator ${\cal O}_8$.
To perform this calculation, it was necessary to work out diagrams
with four
particles ($s$ quark, two photons and a gluon) in the final state.
From the technical point of view, the calculation was made possible
by the use of the Laporta algorithm <cit.> to identify the
needed master integrals. We then solved the resulting MIs numerically, keeping
the exact dependence on the strange-quark mass $m_{s}$, which we varied
between 400 and 600 $\rm MeV$ in the numerical illustrations.
We conclude that the numerical impact of the self-interference contribution of the chromomagnetic dipole operator ${\cal O}_8$ to the decay
rate is minor when compared to the self-interference effect of the electromagnetic dipole operator ${\cal O}_7$.
Acknowledgments —
H.M.A. acknowledges support from the State Committee of Science of
Armenia Program Grant No. 13-1c153 and Volkswagen Stiftung Program Grant No. 86426.
C.G. acknowledges the support from the Swiss National Science Foundation.
A.K. acknowledges the support from the United Kingdom Science and Technology Facilities
Council (STFC) under Grant No. ST/L000431/1.
A.K. thanks Martin Gorbahn for numerous useful discussions.
6c| Branching ratios for $\bar{B}\to X_s\gamma \gamma$
3c||$c=1/50$ 3c| $c=1/100$
$\mu=\,$$m_{b}/2$ $\mu=$ $m_{b}$ $\mu=$ $2m_{b}$ $\mu=$ $m_{b}/2$ $\mu=$ $m_{b}$ $\mu=$ $2m_{b}$
${\rm NLL_{1}}~$ 1.57 1.03 0.71 1.79 1.17 0.80
${\rm NLL_{2}}$ 0.96 0.63 0.43 1.09 0.71 0.49
${\rm NLL_{3}}$ 0.59 0.39 0.27 0.67 0.44 0.30
Branching ratios (in units of $10^{-11}$) for $\bar{B}\to X_s\gamma \gamma$ when only
considering the (${\cal O}_8$ , ${\cal O}_8$) contribution calculated in
this paper. The left half of the table corresponds to the results when
choosing $c=1/50$, while in the right half $c$ is set to be $c=1/100$. The
rows labeled with ${\rm NLL_{1}}$, ${\rm NLL_{2}}$ and ${\rm NLL_{3}}$ give
the result of this specific NLL contribution when setting $m_{s}=400$ MeV, $m_{s}=500$ MeV and $m_{s}=600$ MeV, respectively. See the text for details.
§ APPENDIX
In this appendix, we give the explicit formulas defining the four-particle
phase-space region considered in this paper as a result of the intersection
of regions given in eq:cutsA and eq:cuts. Further, we give the
explicit forms of the master integrals appearing in our calculation of the
$({\cal O}_8,{\cal O}_8)$-contribution to the decay width for $\bar{B} \to X_s
\gamma \gamma$.
§.§ Explicit formulas for the range in the $(s_1,s_2)$-plane
The kinematical conditions on the phase-space variables $s_1$ and $s_2$, as implicitly
formulated in eq:cutsA and eq:cuts, can easily be converted to
explicit ranges. There are the following three cases (using $x_4=m_s^2/m_b^2$):
(i) if x_4 ≤c^2
c<s_1<1-2 c ; c<s_2<1-s_1-c
(ii) if c^2< x_4 < c (1-2 c)
c < s_1 < x_4 / c ; x_4 / s_1 <s_2<1-s_1-c
x_4 / c < s_1 < 1-2 c ; c < s_2<1-s_1-c
(iii) if x_4 ≥c (1-2 c)
s_1^- <s_1 < s_1^+ ; x_4 / s_1 < s_2 < 1-s_1-c
s_1^±= (1-c ± √((1-c)^2-4 x_4 ) )/2 .
Case (ii) is understood to be the sum of the two possibilities written in
eq:PScases. Further, it can be seen from the same equation
that if one puts $m_{s}=0$, one would simply end up with case (i), as previously considered in <cit.>.
The shaded area shows the $(s_1,s_2)$ phase-space region for the case $c^2\geq x_4$.
As an example, in Fig. <ref> we give the geometrical representation of case (i) of eq:PScases.
§.§ Explicit forms for the Master Integrals
In a first step, we managed to write the triple differential decay width
$d\Gamma_{88}/(ds_1 ds_2 ds_3)$ as a linear combination
of five independent MIs.
The full four-particle phase space can be parametrized in terms of five independent
variables. According to the procedure described in Appendix B.2 of
Ref. <cit.>, three of the five variables can be chosen to be
$s_1$, $s_2$ and $s_3$. The MIs are therefore given in terms of
integrals over two variables called $\lambda_4$ and $\lambda_5$, running
in the interval $[0,1]$.
Since we regulated possible collinear singularities by keeping $m_{s}$ exact
and since soft photons are excluded by the cuts imposed through eq:cuts,
we can work in $d=4$ dimensions; this considerably simplifies the expressions
in Appendix B.2 of Ref. <cit.>.
The MIs, defined at the level of the triple differential decay width, depend on $s_1,s_2,s_3$ and $x_4$.
We denote them by $B^{\nu_1\nu_2}_{{\rm set}_{i}}(s_1,s_2,s_3,x_4)$, where ${\nu_1,\nu_2}$ stand for the powers of the propagators in the MIs
and $i$ defines the set (propagator structure) where they belong. Our parametrized MIs are of the form ($\lambda_{4,5} \in [0,1]$)
B_set_i^ν_1ν_2(s_1,s_2,s_3,x_4) =
N_ps ∫_λ_4 ∫_λ_5 dλ_4dλ_5 P_1,i^-ν_1 P_2,i^-ν_2 / √( (1-λ_5) λ_5 )
where $ {\cal N}_{\rm ps} $ is the phase-space factor with ${\cal N}_{\rm ps}
= \frac{s_{3}-x_4}{ 2048 \pi^{6} \, s_{3} }$, and the propagators $P_{1,i},
P_{2,i}$ are understood to be expressed in terms of the integration variables
$\lambda_4,\lambda_5$ and the variables $s_1$, $s_2$, $s_3$, following the parametrization used in
Based on these considerations, we have the following expressions for the MIs:
set_1 :
P_1=(p_g-p_b+q_1)^2-m_s^2, P_2=(p_g-p_b)^2-m_s^2
B^00_set_1 = s_3-x_4/ 2048 π^5 s_3
B^10_set_1 = ∫_λ_4 ∫_λ_5 dλ_4dλ_5 I^10_ set_1(λ_4 ,λ_5 ) / √((1-λ_5) λ_5)
= log( s_3/x_4 )/ 2048 π^5 (s_1-s_3)
B^01_set_1 = ∫_λ_4 ∫_λ_5 dλ_4dλ_5 I^01_ set_1(λ_4 ,λ_5 ) / √((1-λ_5) λ_5)
B^11_set_1 = ∫_λ_4 ∫_λ_5 dλ_4dλ_5 I^11_ set_1(λ_4 ,λ_5 ) / √((1-λ_5) λ_5)
set_2 :
P_1=(p_g-p_b+q_1)^2-m_s^2, P_2=(p_g-p_b+q_2)^2-m_s^2
B^11_set_2 = ∫_λ_4 ∫_λ_5 dλ_4dλ_5 I^11_ set_2(λ_4 ,λ_5 ) / √((1-λ_5) λ_5)
where the respective integrands explicitly read
I^10_ set_1 = N_ps s_3/ (s_1-s_3) (s_3 (1-λ_4)+x_4 λ_4) ,
I^01_ set_1 = N_ps s_1 (s_1-s_3) s_3 [ s_1 { s_3 ( s_1 + (s_2-2) s_3 . . .
. . . - (s_1+s_2-s_3) x_4 + x_4) - (s_1 (s_1+s_2) . . .
. . . -(s_1-s_2+2) s_3) λ_4 (s_3-x_4) } .
. -2 f_root (s_1-1) (s_1-s_3) (2 λ_5-1)
(s_3-x_4) ]^-1 ,
I^11_ set_1 = - N_ps s_1 s_3^2 [ (s_3 (λ_4-1)-x_4 λ_4) .
. { s_1 (s_3
( s_1 + (s_2-2) s_3 -(s_1+s_2-s_3) x_4 + x_4) . . .
. . . -( s_1
(s_1+s_2) - (s_1-s_2+2) s_3) λ_4 (s_3-x_4)) . .
. . -2 f_root (s_1-1) (s_1-s_3) (2 λ_5-1)
(s_3-x_4) } ]^-1 ,
I^11_ set_2 = - N_ps s_1 s_3^2 [ (λ_4 (x_4-s_3)+s_3) { 2 (2 λ_5-1) . .
. . (s_1-1) (s_1-s_3) f_root (s_3-x_4) + s_1 (λ_4 (s_3 (s_3-s_2 . . . . .
. . . . . + 2) - s_1 (s_2+s_3))
(x_4-s_3) + s_3 ((s_1+s_2-s_3 . . . . .
. . . . . -1) x_4 -s_1 s_2+s_3)) } ]^-1 ;
f_root = √(s_1^2 (s_1+s_2-s_3-1) (s_1 s_2-s_3) s_3 (λ_4-1)
λ_4/(s_1-1)^2 (s_1-s_3)^2) .
In eq:MIs, the integrations involved in $ B^{00}_{{\rm set}_{1}}$ were trivial to
perform. For $B^{10}_{{\rm set}_{1}}$, an analytical solution is possible,
using the differential equation (DE) method.
For the remaining MIs, as the corresponding integrands
$I^{\nu_{1}\nu_{2}}_{ {\rm set}_i}({\lambda_4} ,{\lambda_5})$ develop
complicated structures, we performed these integrations numerically for exact
As can be understood from their propagator structures, two of the MIs, $B^{01}_{{\rm set}_{1}}$ and $B^{11}_{{\rm set}_{2}}$, are symmetric under the exchange of $s_{1} \leftrightarrow s_{2}$.
|
1511.00205
|
ise]Alex Olshevskyack
[ise]Department of ISE, University of Illinois at Urbana-Champaign, USA
[ack]This research was supported by NSF grant ECCS-1351684.
We prove two bounds showing that if the eigenvalues of a matrix are clustered in a region of the complex plane then the corresponding discrete-time linear system requires significant energy to control. A curious feature of one of our bounds is that the dependence on the region is via its logarithmic capacity, which is a measure of how well a unit of mass may be spread out over the region to minimize a logarithmic potential.
§ INTRODUCTION
We will consider discrete-time linear systems
\begin{equation} \label{maineq} x(t+1) = A x(t) + B u(t), \end{equation}
where $A \in \C^{n \times n}$ and $B \in \C^{n \times k}$. Our goal is to understand the relation between the locations of the eigenvalues of $A$ within the complex plane and the energy needed to steer Eq. (<ref>) by choosing the input $u(t)$. We will prove two bounds to the effect that if the eigenvalues of $A$ are clustered, then Eq. (<ref>) is “difficult to control” in the sense of requiring large inputs to steer between states.
Our work is related to a growing body of literature investigating the control properties of large-scale systems. A strand of this literature, to which this paper belongs, is to identify the fundamental limitations for controlling such networks <cit.>. A common concern is control difficulty when $n$ (the number of states) is large; it has been experimentally observed that in some scenarios the minimum control energy grows exponentially as a function of $n$ <cit.>. In this note, we will study how eigenvalue locations of $A$ in the complex plane can sometimes be a de-facto obstacle to efficient control for systems with many states.
It is textbook material that the energy needed to steer a linear system is related to the smallest eigenvalue of the controllability Gramian, and we spell this out before describing the problem and our results. Given initial state $x_0$ and final state $x_{\rm f}$, we let $\E( A, B, x_0 \rightarrow x_{\rm f}, t) $ be the minimal energy $\sum_{i=0}^{t-1} ||u(i)||_2^2$ among all inputs which result in $x(t)=x_{\rm f}$ starting from $x(0) = x_0$. We then use this notion to define the “difficulty of controllability” of a linear system by considering the worst-case energy needed to move from the origin to a point on the unit sphere, i.e.,
\[ \E(A,B, t) := \sup_{||y||_2 = 1} ~\E(A,B, 0 \rightarrow y, t). \]
We will allow both sides to be infinite if there is a vector $y$ on the unit sphere which cannot be reached with any choice of $u(0), \ldots, u(t-1)$. This is not the only way to formalize the difficulty of controllability of a linear system (for example, one might also consider the expected energy to move to a random point on the unit sphere) but it is among the most natural. Defining the $t$-step controllability Gramian as
\begin{equation} \label{gramian} W(t) := \sum_{i=0}^t A^i B B^* (A^*)^i, \end{equation}
basic linear algebra then gives that
\[ \mathcal{E}(A,B,t) = \frac{1}{\lambda_{\rm min}(W(t-1))}, \]
where $\lambda_{\rm min}(W(t-1))$ is the smallest eigenvalue of the nonnegative definite matrix $W(t-1)$.
Thus the question of how difficult a system is to control (in a certain worst-case sense) reduces to the analysis of the smallest eigenvalue of the controllability Gramian. The study of that eigenvalue is the subject of this note.
We are motivated by a recent result of <cit.> which showed that if $A$ is a diagonalizable matrix with $m$ eigenvalues within the circle $\X = \{ z ~|~ |z| \leq \mu < 1\}$, then the smallest eigenvalue of $W(t)$ for any $t$ is upper bounded by a product of two terms one of which is $\mu^{2 (\lceil m/k \rceil -1) }/(1-\mu^2)$ (where recall $k$ is the number of columns of $B$). In other words, if $m$ is large enough compared to $k$ and $\mu$ is not close to $1$, then the smallest eigenvalue of $W(t)$ is exponentially close to zero.
Our goal here is to produce similar results for other sets $\X$, especially those which are not contained within the interior of the unit circle. In this case we will not be able to obtain bounds on $\lambda_{\rm min}(W(t))$ for all $t$, but we will be able to upper bound this eigenvalue for some concrete choices of time $t$.
§.§ Related work
As already mentioned, the main motivating work for the present paper is <cit.>, which was the first (to our knowledge) to obtain results connecting eigenvalue clustering to lower bounds on control energy. Follow-up work included <cit.>, which studied the relation between the difficulty of controllability and the propagation of inputs by the system in various directions, and <cit.> which studied connections to measures of centrality such as PageRank.
The existence of small eigenvalues of the discrete-time controllability Gramian appears to have not attracted significant attention in the existing control literature beyond the above papers. In continuous time, results on the condition number (which is the ratio of the largest and smallest eigenvalue of the Gramian) in the case when $A$ is stable have been derived <cit.>, as well as more general results on ratios of eigenvalues <cit.>.
Our work is also related to a series of recent preprints analyzing properties of eigenvalues of the Gramian of linear <cit.> and bilinear <cit.> systems. These papers studied the efficiency of algorithms for placement of sensors and actuators, as well as the underlying properties of the Gramian that allow for efficient approximation algorithms.
§.§ Our results
This note has two main results. The first concerns control energy at the first time the system can become controllable, namely at time
$$\tm := \lceil n/k \rceil - 1,$$
where $\lceil x \rceil$ is the smallest integer which is at least $x$ and $k$, recall, is the number of columns of $B$. It is immediate that if $t<\tm$, then $W(t)$ is singular because there are not enough columns for the controllability matrix to be full-rank. Our first result shows that if the eigenvalues of $A$ lie in a set with logarithmic capacity (to be formally defined later) smaller than one, then $W(\tm)$ has an eigenvalue upper bounded by something that decays to zero exponentially fast in $\tm$.
Our second result considers $A$ which are Hermitian with $m$ eigenvalues which are stable (i.e, which lie in $[-1,1]$). Roughly speaking, we show that if $t = O \left( \left( \frac{m}{k} \right)^{2-\epsilon} \right)$ for[We use the standard $O$-notation, i.e., the statement $f=O(g)$ where $f$ and $g$ are positive quantities denotes the existence of a constant $C$ such that $f \leq C g$.] some $\epsilon>0$, the controllability Gramian $W(t)$ has an eigenvalue which is upper bounded by a quantity that goes to zero as $m/k \rightarrow +\infty$. For example, if $t= O \left( \left( m/k \right)^{3/2} \right)$, our result gives the bound $\lambda_{\rm min} \left( W(t) \right) = O \left( (m/k)^{3/2} e^{-\sqrt{m/k}} \right)$ in this case.
The formal statements of these results are a little involved and will be given within the body of this paper. We conclude our summary by illustrating their use on some simple examples. Suppose $x(t+1) = A x(t) + b u(t)$ where $A \in \C^{n \times n}$ is diagonalizable as $V A V^{-1} = D$ and $b \in \R^{n \times 1}$ is a vector of unit norm. Then:
* If the eigenvalues of $A$ are contained within any equilateral triangle of side length $2$ in the complex plane, then there is some $n_0$ such that for all $n \geq n_0$, we have
\begin{equation} \label{equilateral} \lambda_{\rm min} \left( W(n-1) \right) \leq ||V||_2^2 ||V^{-1}||_2^2 \cdot 0.133^n. \end{equation}
By contrast, if all the eigenvalues are contained within a circle in the complex plane of the very same area as this equilateral triangle, our methods give the bound
\begin{equation} \lambda_{\rm min} \left( W(n-1) \right) \leq ||V||_2^2 ||V^{-1}||_2^2 \cdot 0.552^n, \label{circle} \end{equation}
once again for $n$ large enough.
* Suppose $n=10,000$ and $A$ is a Hermitian matrix at least half of whose eigenvalues are stable.
It turns out that this is enough information to conclude that
\begin{eqnarray} \lambda_{\rm min}(W(250,000)) & \leq& 1.03 \times 10^{-37} \label{first-sym} \\
\lambda_{\rm min}(W(1,000,000)) & \leq& 1.58 \times 10^{-4} \label{second-sym}
\end{eqnarray}
In other words, the presence of many stable modes appears to be a significant obstacle to the efficient control of Hermitian systems.
§.§ Organization of this paper
We conclude the introduction with Section <ref> which introduces some notation as well as some background facts which we will draw on throughout this paper. Section <ref> is dedicated to proving our first main result, namely the bound on $W(\tm)$ in terms of logarithmic capacity. Section <ref> proves our second main result, which bounds the eigenvalues of $W(t)$ for a range of times $t$ in the special case when the matrix $A$ is Hermitian with many stable eigenvalues. We end with some concluding remarks in Section <ref>.
§.§ Notation and background
We first describe some notation that we will use for the remainder of the paper. We use the standard notation
$o_l(1)$ to denote any function of $l$ that goes to zero as $l \rightarrow +\infty$. Given a matrix $V$, its condition number is defined as ${\rm cond}(V) := ||V||_2 ||V^{-1}||_2$. The Frobenius norm of $V$ is denoted as $||V||_F$. As is standard, $V^*$ will denote the conjugate transpose of $V$ and $\overline{V}$ will denote its (elementwise) complex conjugate. We will use $\mathcal{P}_{j}$ to denote the set of univariate polynomials with complex coefficients of degree at most $j$, and $\mathcal{P}_j'$ to denote the set of monic[Meaning the coefficient in front of the highest power is one.] univariate polynomials with complex coefficients of degree $j$.
Given a compact set $K$ in the complex plane, let $\mu^K$ be the set of probability measures on $K$, i.e., the set of Borel measures $\mu$ supported on $K$ which satisfy $\mu(K)=1$. We then define $I(K)$, called the logarithmic energy of the
set $K$, as
\[ I(K) := \sup_{\mu \in \mu^K} ~\int_{K \times K} \log |z-w| ~ d\mu(z) d\mu(w). \]
The logarithmic capacity is then defined as
\[ {\rm cap}(K) := e^{I(K)}. \]
Logarithmic capacity comes up in our results due to its connection with polynomial approximation, which we now describe. Given a set $\X \subset \C$, we define
\begin{equation} \label{endef} {\rm Err}(l, \X) := \min_{p \in \mathcal{P}_{l-1}} \max_{z \in X} \left| z^l - p(z) \right|. \end{equation}
In other words, ${\rm Err}(l,\X)$ is the best possible approximation error of the
function $z^l$ by a polynomial of degree $l-1$ over $\X$. The following statement is part of the “fundamental theorem of potential theory” from <cit.>:
\[ \lim_{l \rightarrow \infty} \left( {\rm Err}(l,\X) \right)^{1/l} = {\rm cap}(\X). \]
As long as ${\rm cap}(\X) > 0$, we may re-arrange this as
\begin{equation} \label{cap} {\rm Err}(l,\X) = \left[ \left( 1 + o_l(1) \right) {\rm cap}(\X) \right]^l. \end{equation}
In other words,
the logarithmic capacity of a region determines the growth (or decay) of the polynomial approximation of $z^l$ over that region by polynomials of
lower degree.
§ CONTROL ENERGY AT TIME $\TM$ AND LOGARITHMIC CAPACITY
We begin with a statement of first our main result. Informally, the
theorem says that if the eigenvalues of the matrix $A$ lie in the set $\X$, then $\lambda_{\rm min}(W(\tm))$ should scale roughly as $\sim \left( {\rm cap}(\X)^2 \right)^{\tm}$. The formal statement is given next.
Suppose that $A$ is diagonalizable matrix with $VAV^{-1} = D$ where $D$ is diagonal and with all of its eigenvalues within a set $\X \subset \C$ with $\cap > 0$. We then have,
\begin{align*} \lambda_{\rm min} \left( W \left( \tm \right) \right) & \leq {\rm cond}^2(V) ~ ||B||_F^2 \\ & ~~~~\left( \left(1 + o_{\tm}(1) \right) {\rm cap}^{2} (\X) \right)^{t_{\rm min} } \end{align*}
We briefly sketch some intuition for this statement. Observe that if $n>k$ and the (diagonalizable) matrix $A$ only has one eigenvalue (necessarily of multiplicity $n$), then an easy implication of the Popov-Belevitch-Hautus theorem is that the system is uncontrollable. It is therefore reasonable to guess that this fact should have a quantitative extension, namely some statement to the effect that if all the eigenvalues are close together then the controllability Grammian is close to singular. One formalization of this is precisely the above theorem.
Moreover, diagonalizability is crucial for any such bound; consider choosing $A$ to be the lower-shift matrix and $b=e_{1}$. Indeed, for this pair $A,b$, we have that the set $\{0\}$ contains the (single) eigenvalue of the system while the controllability Grammian $W(t)$ has all of its eigenvalues equal to $1$ whenever $t \geq \tm$. Thus for non-diagonalizable systems, having eigenvalues contained in a small set does not translate into good bounds on control energy. Furthermore, by considering diagonalizable approximations to this system, we see any bound we derive cannot depend solely on eigenvalues, and must somehow “blow up” whenever the system approaches this non-diagonalizable system, explaining the presence of the ${\rm cond}^2(V)$ term.
Our first task is to prove the theorem; later in this section, we will show how to use this result to obtain more “concrete” estimates on $\lambda_{\rm min}(W(\tm))$. The idea of the proof is to argue that, in the definition of $W(\tm)$ in Eq. (<ref>), the final $k$ terms have, in an appropriately defined sense, small contributions and consequently $W(\tm)$ is close to singularity. This is a technique we have taken from the papers <cit.> which we here combine with the arguments of <cit.>.
From the definition of the controllability Gramian, we will then have
\begin{align*} V W(t) V^{*} & = \sum_{i=0}^{t} \left( V A^i V^{-1} \right) \left(V B \right) \\ &~~~~~~~~~~~~ \left( B^* V^{*} \right) \left( V^{-*} (A^*)^i V^* \right). \end{align*}
Introducing the notation $Q(t) := V W(t) V^{*}$ and $Z := VB$, we can rewrite this as
\[ Q(t) = \sum_{i=0}^{t} D^i Z Z^* \overline{D}^i. \]
We will now derive an upper bound on the smallest eigenvalue of $Q(\tm)$, which we will later translate into a bound
on the smallest eigenvalue of $W(\tm)$. To analyze the smallest eigenvalue of $Q(\tm)$, we introduce the following notation.
We define $z_1, \ldots, z_k$ be the columns of the matrix $Z$, and we define $S(t)$ to be the matrix
\[ S(t) := [Z, ~DZ, \ldots, ~D^{t} Z ]. \]
Observe that $S(t) \in \C^{n \times (t+1)k}$ and has the columns $D^j z_i$ as $j$ runs over $j=0, \ldots, t$ and $i$ runs over $i=1, \ldots, k$. Moreover,
\begin{equation} \label{sdef} S(t) S(t)^* = Q(t). \end{equation}
\[ \lambda_n(Q(t)) = \sigma_n^2(S(t)). \]
Inspecting the last equation, we see that our goal of $\lambda_{\rm min}(Q(\tm))$ can be achieved by upper bounding the smallest singular value of $S(\tm)$. We will do so by arguing that $S(\tm)$ is close to singular.
The argument proceeds as follows. We define the subspace
$${\mathcal S}_i^j := {\rm span} \left(z_i, ~D z_i, \ldots, ~D^{j} z_i \right).$$
We adopt the notation of $P_{\Y}$ to denote the matrix which projects onto the subspace $\Y$. Now the projection of $D^{j+1} z_i$ onto the subspace ${\mathcal S}_i^j$ is naturally some linear combination of the
vectors which span ${\mathcal S}_i^j$, i.e.,
\begin{align*} P_{\mathcal{S}_i^j} D^{j+1} z_i & = \alpha_j D^j z_i + \cdots + \alpha_1 D z_i + \alpha_0 z_i,
\end{align*}
for some (possibly complex) coefficients $\alpha_j, \ldots, \alpha_0$. Since $P_{\Y^\perp} (y) = y - P_{\Y}(y)$, we have that
\begin{align*} P_{(\mathcal{S}_i^j)^\perp} D^{j+1} z_i & = D^{j+1} z_i - \alpha_j D^j z_i \cdots - \alpha_1 D z_i - \alpha_0 z_i,
\end{align*}
which we may write as
$P_{(S_i^j)^\perp} D^{j+1} z_i = p(D) z_i$, where
\[ p(t) := t^{j+1} - \alpha_j t^j - \cdots - \alpha_1 t - \alpha_0. \]
Note that $p(t)$ is a monic polynomial of degree $j+1$, i.e., $p(t) \in \mathcal{P}_{j+1}'$.
Now since the projection operator onto the subspace $\mathcal{X}$ maps every point to the closest point in $\mathcal{X}$, we have that
\begin{equation} \label{minchar} \left| \left| P_{(S_i^j)^\perp} D^{j+1} z_i \right| \right|_2 = \min_{p \in \mathcal{P}_{j+1}'} \left| \left| p \left(D \right) z_i \right| \right|_2. \end{equation}
Indeed, if Eq. (<ref>) did not hold, then there would be a point in ${\mathcal S}_i^j$ which is closer to $D^{j+1} z_i$ than
$P_{\mathcal{S}_i^j} (D^{j+1} z_i)$, which cannot be.
Using the fact that entries of $D$ lie in the set $\X$ and recalling the definition of ${\rm Err}(j,\X)$ from Eq. (<ref>), we have
\begin{equation} \left| \left| P_{(S_i^j)^\perp} D^{j+1} z_i \right| \right|_2 \leq {\rm Err}(j+1,\X) ||z_i||_2. \label{Eappears}
\end{equation}
Recall that our goal is to argue that $S(\tm)$ is close to a singular matrix, and to this end we use the last inequality as follows. We define $L(\tm)$ to be the matrix
\[ L(\tm) := [Z, ~~DZ, \ldots, ~~D^{\tm-1} Z, ~~P_{S_i^{\tm -1}} D^{\tm} Z] \]
In other words, the definition of $L(\tm)$ is identical to $S(\tm)$ with the exception of the final $k$ columns, which are now multiplied by the
projection matrix $P_{S_i^{\tm -1}}$.
Using Eq. (<ref>), we can conclude that $L(\tm)$ and $S(\tm)$ are not too far apart:
\begin{align} ||S(\tm) - L(\tm)||_2^2 & \leq ||S(\tm) - L(\tm)||_{\rm F}^2 \\ & \leq \sum_{i=1}^k \left| \left| P_{\left( {\mathcal S}_i^{\tm - 1}\right)^\perp} D^{\tm} z_i \right| \right|_2^2 \nonumber \\
%& = \left| \left| \sum_{i=1}^k P_{{\mathcal S}_i^{\left( \lceil \frac{m}{k} \rceil - 1} \right) ^\perp} D(m)^{\lceil \frac{m}{k} \rceil} z_i(m) z_i(m)^T D(m)^{\lceil \frac{m}{k} \rceil} P_{{\mathcal %S}_{\lceil \frac{m}{k} \rceil -1}^\perp}^T \right| \right|_2^2 \nonumber \\
& \leq \left( {\rm Err}(\tm, \X) \right)^2 \sum_{i=1}^k ||z_i||_2^2 \nonumber \\
%& \leq \left( {\rm Err}(\tm , \X) \right)^2 ~ ||V||_2^2 \sum_{i=1}^k ||b_i||_2^2 \nonumber \\[1ex]
& \leq \left( {\rm Err}(\tm, \X) \right)^2 ~||V||_2^2 ~ ||B||_F^2, \label{sbound} \end{align}
On the other hand,
\begin{equation} \label{rankbound} {\rm rank}(L(\tm)) \leq k \tm < k \frac{n}{k} = n.
\end{equation}
Next we can use the standard interpretation of the singular values in terms of the distance to the
closest low-rank matrix, i.e.,
\[ \sigma_n^2(S(\tm)) \leq \inf_{{\rm rank} ~M < n} ~||S(\tm) - M||_2^2, \]
see e.g., Section 7.4.2 of <cit.>. Putting this together with
Eq. (<ref>) and Eq. (<ref>), we obtain
\begin{equation} \label{sigmabound} \sigma_n^2(S(\tm)) \leq \left( {\rm Err}(\tm, \X) \right)^2 ~||V||_2^2 ~ ||B||_F^2.
\end{equation}
We now put together the various estimates we have derived. The first step is bound $\lambda_n(W(\tm))$ in terms of $\lambda_n(Q(\tm))$:
\begin{align*} \lambda_n(W(\tm)) & = \lambda_n (V^{-1} Q(\tm) V^{-*}) \\[0.5ex]
& = \min_{x^* x = 1} ~x^* V^{-1} Q(\tm) V^{-*} x \\[0.5ex]
& = \min_{y = V^{-*} x, x^*x=1} ~~ y^* Q(\tm) y
\end{align*}
Observe, however, that since $Q(\tm)$ is Hermitian, $y^* Q(\tm) y$ is nonnegative; furthermore, as $x$ ranges over the unit sphere in $\C^n$, $V^{-*} x$ ranges over
a nondegenerate ellipsoid whose every element has norm at most $||V^{-*}||_2$. Therefore,
\begin{align} \lambda_n(W(\tm)) & \leq \min_{||y||_2 = ||V^{-*}||_2} ~ y^* Q(\tm) y \nonumber \\[0.5ex]
& \leq ||V^{-*}||_2^2 ~\lambda_{n} \left( Q(\tm) \right) \nonumber \nonumber \\[0.5ex]
& = ||V^{-1}||_2^2 ~\sigma_n^2(S(\tm)) \nonumber \nonumber \\[0.5ex]
& = ||V^{-1}||_2^2 ~ ||V||_2^2 ~\left( {\rm Err}(\tm, \X) \right)^2 ||B||_F^2 \nonumber \\[0.5ex]
& = {\rm cond}^2(V) ~\left( {\rm Err}(\tm, \X) \right)^2 ||B||_F^2 \label{approxform}
\end{align}
Combining the last equation with Eq. (<ref>) (which bounds $ {\rm Err}(\tm, \X) $ in terms of $\cap$) completes the proof.
§.§ Some concrete estimates
The logarithmic capacities of many different sets have been worked out in the literature, and we can use them to spell out some
concrete forms of Theorem <ref>.
Let us begin by revisiting the example we have previously mentioned in Section <ref>. Suppose that $A \in \C^{n \times n}$ has all of its eigenvalues in an equilateral triangle $\X$ of length $l$; further, suppose $b={\bf e}_1$. Then it is known that <cit.>,
$$\cap = \frac{(\Gamma(1/3))^2}{4 \pi^2} l \approx 0.18\cdot l,$$
so that
Theorem <ref> specializes to the bound
\[ \lambda_{\rm min} \left( W \left( n-1 \right) \right) \leq {\rm cond}^2(V) \left( \left(1 + o_{n}(1) \right) \cdot 0.18^2 \cdot l^2 \right)^{ n}
\]
Plugging in $l=2$ and using the fact that $o_{n}(1)$ becomes arbitrarily small for large enough $n$, we obtain Eq. (<ref>) from Section <ref>. We may view this last equation as a somewhat more concrete form of Theorem <ref>.
Along the same lines, the following proposition collects a number of estimates of the logarithmic capacity from the literature.
Suppose that $\X$ is:
* ...ellipse with semi-axes $a$ and $b$. Then <cit.>, $\cap = (a+b)/2$.
* ...a rectifiable curve length $l$. Then <cit.>, $\cap \leq l/4$. In particular, if $X = [a,b]$ then <cit.>, $\cap = (b-a)/4$.
* ...two intervals on the real line of the form $[-b,-a] \cup [a,b]$. Then <cit.>,
$$\cap = \frac{\sqrt{b^2-a^2}}{2}.$$
* ...a half disk of radius $r$. Then <cit.>, $\cap = 4r/3^{3/2}$.
* ...square with side $l$. Then <cit.>,
$$\cap = \frac{(\Gamma(1/4))^2}{4\pi^2} l \approx 0.59 l.$$
* ...a regular $n$-gon with side $h$. Then <cit.>,
$$ \cap = \frac{\Gamma(1/n)}{2^{1+2/n} \pi^{1/2} \Gamma(1/2 + 1/n)} h. $$
* ...a compact set of diameter $D$. Then (<cit.>, quoted in <cit.>) $\cap \leq D/2$.
One may use this proposition to come up with statements similar to Eq. (<ref>) for other sets of the complex plane. For example, one can use the
first bullet of the proposition on the logarithmic capacity of the circle to obtain Eq. (<ref>) from Section <ref>; we omit the details of the calculation.
§ A LOWER BOUND ON CONTROL ENERGY FOR HERMITIAN SYSTEMS WITH STABLE EIGENVALUES
In this section we will consider the case of Hermitian matrices $A$ with $m$ eigenvalues within the interval $[-1,1]$. We will be able to obtain
stronger results in this case, in particular showing that $W(t)$ has an eigenvalue close to zero until $t$ is nearly quadratic in $m/k$. A formal statement
of this is in the following theorem.
Suppose $A$ is a Hermitian matrix with $m$ stable eigenvalues where $m > 2k$. Let $q$ be any nonnegative number and set
$$t_{\rm quad} := \frac{(\lceil m/k \rceil - 2)^2}{q}. $$
Then if $t \leq t_{\rm quad} $ we have
\[ \lambda_n(W(t)) \leq 4 \frac{\left( \lceil m/k \rceil - 2 \right)^2}{q} e^{-q} ||B||_F^2. \]
We remark that by plugging in $m=5000$, $k=1$, $||B||_F=1$, and $q=99$, we can obtain Eq. (<ref>), which was discussed in the introduction. The next Eq. (<ref>) can be obtained by plugging in $q=24$ with the same
values of $m$, $k$, and $||B||_F$. We also remark that the theorem holds for any nonnegative $q$ and one can, for example, plug in $q = \sqrt{m/k}$ to obtain the bound
$\lambda_{\rm min} \left(W (t) \right) \leq O \left( (m/k)^{3/2} e^{-\sqrt{m/k}} \right)$ when $t = O \left( (m/k)^{3/2} \right)$ (we mentioned this inequality
in the introduction).
To prove Theorem <ref>, we will need to derive some additional properties of polynomial approximations over the interval $[-1,1]$. For integers $n,m$ satisfying $n \geq m \geq 1$, we define
\[ \Phi_{n,m} := \min_{p_m \in { \Pi}_m} \max_{|x| \leq 1} \left| x^n - p_m(x) \right|, \]
where $\Pi_m$ is the set of polynomials of degree $m$ with real coefficients. In other words, $\Phi_{n,m}$ is the error involved in approximating the function $x^n$ over $[-1,1]$ by an $m$'th degree polynomial in $x$.
It has been observed in <cit.> that when $n \geq m \geq cn$ for some fixed $c$, the quantity $\Phi_{n,m}$ decays exponentially in $n$. The following lemma gives an explicit estimate of the decay rate.
\[ \Phi_{n,m} \leq 2 e^{-m^2/(2n)} \]
Just like the corresponding proof in <cit.>, our starting point is the well-known identity
\begin{equation} \label{x-exp} x^n = \frac{1}{2^{n-1}} \sum_{i=0}^{\lfloor n/2 \rfloor} {n \choose i} T_{n - 2i}(x) \frac{\delta_{i,n}}{2} \end{equation}
where $T_k(x)$ the Chebyshev polynomial of degree $k$, and $\delta_{i,n}$ equals $1$ if $i=n/2$ and $2$ otherwise (for a reference, see Eq. (2.14) in <cit.>).
Let $i'$ be the first integer such that $n - 2i' \leq m$, i.e., $i' = \lceil (n-m)/2 \rceil$. Then, to approximate $x^n$ by a polynomial of degree at most $m$, we might try the polynomial
\[ p_{n,m} = \frac{1}{2^{n-1}} \sum_{i=i'}^{\lfloor n/2 \rfloor} {n \choose i} T_{n - 2i} \frac{\delta_{i,n}}{2}, \]
which, note, has degree at most $m$. As a consequence of Eq. (<ref>) as well as the fact that $|T_k(x)| \leq 1$ for all $x \in [-1,1]$, this leads to the bound
\begin{eqnarray*} \Phi_{n,m} \leq \frac{1}{2^{n-1}} \sum_{i =0, \ldots, i'-1 } {n \choose i}
= 2 \left( \frac{1}{2^{n}} \sum_{i =0}^{i'-1} {n \choose i} \right)
\end{eqnarray*}
The expression in big parentheses is the probability that a fair coin lands on heads at most $i' - 1$ times
out of $n$ tosses. We use the following form of Hoeffding's inequality
\[ \frac{1}{2^{n}} \sum_{i =0, \ldots, (1/2-\epsilon) n} {n \choose i} \leq e^{-2 \epsilon^2 n}, \]
for a formal reference see Section B.4 in <cit.>. Since $i' -1 \leq n/2$, we may apply this form of the Hoeffding bound to obtain
\begin{align*} \Phi_{n,m} & \leq 2 e^{-2n (1/2 - ( i' - 1 )/n)^2 } \\
& = 2 e^{-2n (1/2 - ( \lceil (n-m)/2 \rceil - 1)/(n) )^2} \\
& \leq 2 e^{-2n(\frac{1}{2} - \frac{(n-m)/2}{n})^2} \\
& \leq 2 e^{-m^2/(2n)}
\end{align*}
Let $q$ be any nonnegative number. We have that
if $m \geq 1$,
$$\sum_{n=m}^{\lfloor m^2/q \rfloor} \Phi_{n,m}^2 \leq 4 \frac{m^2}{q} e^{- q}. $$
Indeed, by Lemma <ref>,
\begin{align*} \sum_{n=m}^{\lfloor m^2/q \rfloor} \Phi_{n,m}^2 & \leq \sum_{n=m}^{\lfloor m^2/q \rfloor} \left( 2 e^{-m^2/(2n)} \right)^2 \\[1.5ex]
& \leq 4 \sum_{n=m}^{\lfloor m^2/q \rfloor} e^{-m^2/n}
\end{align*}
The number of terms in the last sum is upper bounded by $m^2/q$, and each term is upper bounded by
$$e^{-\frac{m^2}{m^2/q}} = e^{-q}.$$
This immediately implies
the lemma.
With this lemma in hand, we are now ready to prove Theorem <ref>.
As in the proof of Theorem <ref>, let
$$V A V^{-1} = D$$
be the diagonalization of $A$; here $V$ is unitary since $A$ is Hermitian, and we will assume without loss of generality that $d_{11}, d_{22}, \ldots, d_{mm} \in [-1,1]$.
Let $Q(t) = V W(t) V^*$ be the same as defined earlier. As a consequence of the fact that $V$ is unitary, we have that
$$\lambda_n(W(t)) = \lambda_n(Q(t)).$$
We define $D(m)$ to be the principal $m \times m$ submatrix[That is, the submatrix obtained by taking the first $m$ rows and first $m$ columns.] of the diagonal matrix $D$, let $z_i(m)$ be the vector obtained by taking the first $m$ entries of the vector[Recall that $Z=VB$ and $z_1, \ldots, z_k$ are the columns of the matrix $Z$.] $z_i$,
and let $S(m,t)$ be the matrix with columns $D(m)^j z_i(m)$ for $i = 1, \ldots, k$ and $j=0, \ldots, t$. As before, $S(t)$ is the matrix with columns $D^j z_i$ with $i,j$ running
over the same ranges; recall that $Q(t) = S(t) S(t)^*$. Finally, abusing notation slightly, we will now use $S_i^j$ to denote the span of the vectors $z_i(m), D(m) z_i(m), \ldots, D(m)^j z_i(m)$.
Using the Courant-Fischer theorem, we obtain
\begin{align} \lambda_{\rm min} (Q(t)) & = \min_{x \in \C^n, ~x^*x=1} ~x^* S(t) S(t)^* x \nonumber \\[1.5ex]
& \leq \min_{\substack{x \in \C^n, ~x^* x = 1 \\ x_{m+1}=x_{m+2} = \cdots = x_n = 0}} ~~~~ x^* S(t) S(t)^* x \nonumber \\[1.5ex]
& = \min_{x \in \C^m, ~x^* x = 1} ~~ x^* S(m, t) S(m, t)^* x \nonumber \\[1.5ex]
& = \sigma_{m}^2(S(m,t)). \label{sigmalambda}
\end{align}
We thus have
\begin{equation} \label{lams} \lambda_n(W(t)) \leq \sigma_m^2(S(m,t)). \end{equation}
We now turn to the problem of upper bounding the $m$'th smallest singular value of $S(m,t)$, which we will do by arguing, along the lines of the proof of Theorem <ref>, that $S(m,t)$ is well-approximated by a matrix whose rank is strictly less than $m$.
Indeed, define $t'=\lceil m/k \rceil - 1$. We may as well assume that $t \geq t'$ since otherwise there is nothing to prove. Define further the matrix $\widehat{L}(m,t)$ as having its first $k t'$ columns being $D(m)^j z_i(m)$, $i=1, \ldots, k$ and $j = 0, \ldots, t' -1$; however,
its final $k(t+1-t')$ columns will be the
vectors $P_{{\mathcal S}_i^{t' - 1}} D(m)^{j} z_i(m)$, $i=1, \ldots, k$ and $j=t', \ldots, t$. Note that $\widehat{L}(m,t)$ is roughly analogous to the matrix $L(m,\tm)$ defined in the proof of Theorem <ref>.
As in the proof of Theorem <ref>, we have that
\[ {\rm rank} (\widehat{L}(m,t)) \leq k t' < k \frac{m}{k} = m. \]
We will next argue that $\widehat{L}(m,t)$ and $S(m,t)$ are close to each other. Indeed, we have that for all $l \geq j$,
\begin{align*} \left| \left| P_{(S_i^j)^\perp} D(m)^{l} z_i(m) \right| \right|_2 & = \min_{p \in {\cal P}_j} \left| \left| \left( D(m)^l - p(D(m)) \right) z_i(m) \right| \right|_2 \\
& \leq \Phi_{k,j} ||z_i(m)||_2,
\end{align*}
where the first step follows from definition of projection and the second step used the fact that all diagonal entries of $D(m)$ belong to $[-1,1]$. Thus,
\begin{align*} ||S(m,t) - \widehat{L}(m, t)||_2^2 & \leq ||S(m,t) - \widehat{L}(m, t)||_{\rm F}^2 \\ & \leq \sum_{i=1}^k \sum_{l=t'}^{t} \left| \left| P_{\left({\mathcal S}_i^{t' - 1}\right)^\perp} D(m)^{l} z_i(m) \right| \right|_2^2 \nonumber \\
& \leq \sum_{i=1}^k ||z_i(m)||_2^2 \sum_{l=t'}^{\lfloor t_{\rm quad} \rfloor} \Phi_{l, t'-1}^2
\end{align*}
where we used the fact that $t$ is an integer and $t \leq t_{\rm quad}$ in the last inequality. Note that we may as well begin the second sum above at $l=t'-1$. Observe that by
definition $t_{\rm quad} = (t'-1)^2/q$, and thus we can use Lemma <ref> to bound the sum. Furthermore, due to the assumption $m>2k$ we have $t'-1 \geq 1$. Thus applying Lemma <ref>, we obtain
\[ ||S(m,t) - \widehat{L}(m, t)||_2^2 \leq 4 \frac{(\lceil m/k \rceil - 2)^2}{q } e^{-q} ||B||_F^2. \]
Putting it all together: this inequality along with ${\rm rank} (\widehat{L}(m,t)) < m$ implies that the right-hand side is an upper bound on $\sigma_m^2(S(m,t))$, and via Eq. (<ref>) it is also an upper bound on $\lambda_n(W(t))$.
§ CONCLUSION
We have proven two bounds to the effect that if the eigenvalues of the matrix $A$ are clustered, the linear system of Eq. (<ref>) is difficult to control. Our work points the way to a number of open questions.
First, it is unclear whether it is possible to extend the bounds of Theorem <ref> for $t$ that are much greater than $\tm$. The antecedent work <cit.> bounded the eigenvalues of $W(t)$ for all $t$ under the assumption of clustering within the interior of the unit circle, and it is at present unclear if Theorem <ref> can be extended in such a way. Second, we conjecture that the bounds of Theorem <ref> are essentially tight; specifically, we conjecture that for all $n$ there exists a symmetric matrix in $\R^{n \times n}$ with all of its eigenvalues in $[-1,1]$ and with $\lambda_{\rm min}(W(t)) \geq 1$ for all $t \geq \Omega(n^2 \log^k n)$ for some $k$.
More broadly, the main contribution of this note is to highlight the apparently subtle connection between eigenvalue clustering and control energy (recall the contrast between the bounds we discussed in Section <ref>). An open problem is to understand this connection better.
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1511.00278
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Condensed Matter and Magnet Science Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
ISSP, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Joseph Henry Laboratory and Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Joseph Henry Laboratory and Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Joseph Henry Laboratory and Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Joseph Henry Laboratory and Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
ISSP, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan
Joseph Henry Laboratory and Department of Physics,
Princeton University, Princeton, New Jersey 08544, USA
Condensed Matter and Magnet Science Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Topological superconductors host new states of quantum matter which show a pairing gap in the bulk and gapless surface states providing a platform to realize Majorana fermions. Recently, alkaline-earth metal Sr intercalated Bi$_2$Se$_3$ has been reported to show superconductivity with a T$_c$ $\sim$ 3 K and a large shielding fraction.
Here we report systematic normal state electronic structure studies of Sr$_{0.06}$Bi$_2$Se$_3$ (T$_c$ $\sim$ 2.5 K) by performing photoemission spectroscopy. Using angle-resolved photoemission spectroscopy (ARPES), we observe a quantum well confined two-dimensional (2D) state coexisting with a topological surface state in Sr$_{0.06}$Bi$_2$Se$_3$. Furthermore, our time-resolved ARPES reveals
the relaxation dynamics showing different decay mechanism between the excited topological surface states and the two-dimensional states. Our experimental observation is understood by considering the intra-band scattering for topological surface states and an additional electron phonon scattering for the 2D states, which is responsible for the superconductivity. Our first-principles calculations agree with the more effective scattering and a shorter lifetime of the 2D states. Our results will be helpful in understanding low temperature superconducting states of these topological materials.
The discovery of three-dimensional (3D) topological insulators (TIs) in bismuth-based semiconductors
has attracted considerable amount of research interest in condensed matter physics <cit.>. In these materials, the bulk has a full energy gap whereas the surface possesses an odd number of Dirac-cone electronic states,
where the spin of the surface electrons is locked to their linear momentum <cit.>.
Such unique properties make these materials alluring not only for studying various fundamental phenomena in condensed matter physics and particle physics, but also provide high potential for applications ranging from spintronics to quantum computation <cit.>.
More specifically, it has been recently predicted that the long-sought-out Majorana fermions can be realized in the interface between a topological insulator and a superconductor <cit.>. There has been a great effort in condensed matter physics to realize the Majorana fermion quasiparticle states associated with topological superconductivity.
Although the experimental realizations of topological superconductors in real materials have remained
considerably limited, the search for the bulk topological superconductors is an ongoing project in condensed-matter physics. To date, possible topological superconductivity has been suggested in Cu-intercalated
Bi$_2$Se$_3$ (Cu$_x$Bi$_2$Se$_3$) <cit.>, highly-pressurized Bi$_2$Te$_3$ and Sb$_2$Te$_3$ <cit.>, and Bi$_2$X$_3$ (X= Se, Te) thin films grown on superconducting substrates <cit.>. After the discovery of superconductivity in Cu$_x$Bi$_2$Se$_3$, an enormous amount of research effort was devoted to realize the first bulk topological superconductor <cit.>. However, a clear signature of topological superconductivity in this material remains still elusive mainly due to the relatively low superconducting volume fraction of the sample. Recently, it was reported that the intercalation of an alkaline-earth metal Sr in the well-studied topological insulator
Bi$_2$Se$_3$ (Sr$_x$Bi$_2$Se$_3$) shows a superconducting state with T$_c$ $\sim$ 3.0 K and a large superconducting volume fraction ($\sim$ 90%), providing a more ideal platform to realize a topological superconductor <cit.>. However, a detailed electronic structure of this new compound has not been reported.
A detailed systematic high-resolution angle-resolved photoemission spectroscopy (ARPES) study is needed to investigate the nature of the surface states in the normal state of Sr intercalated Bi$_2$Se$_3$.
Such a characterization is a necessary first step towards understanding the relationship between superconductivity and the topological properties of this new compound.
Furthermore, the occurrence of superconductivity (SC) in topological insulators is a subject of strong current interest, because of the theoretical predictions for the possible observation of exotic and unexplored excitations. ARPES and time-resolved ARPES (TRARPES) have provided detailed pictures of the topological surface states, and appear as the most appropriate tools to reveal the unusual features of topological superconductivity.
Here, we report the normal ARPES and time-resolved ARPES investigation of the detailed electronic structure of Sr intercalated Bi$_2$Se$_3$. Using normal ARPES, we observe the coexistence of the two-dimensional (2D) quantum well states and single Dirac cone topological surface states present in this system with the Dirac point located around 450 meV below the Fermi level. More importantly, by using time-resolved ARPES (TRARPES), we observe different relaxation mechanisms for the 2D state and topological surface state. The observed effect can be understood by considering an additional scattering term for 2D states. Our theoretical calculations show the more effective scattering and a shorter lifetime of the 2D states, which is consistence with the experimental observation.
Our results provide critical knowledge necessary for realizing and understanding the topological superconductivity in this system which helps to demonstrate the Majorana fermion associated with topological superconductivity.
Transport characterization of Sr$_{0.06}$Bi$_2$Se$_3$
We start our discussion by presenting transport characterizations of the samples used in our spectroscopy measurements. Fig. 1a shows the temperature dependent in-plane resistivity of Sr$_{0.06}$Bi$_2$Se$_3$.
The onset of superconducting transition is about T$_c \sim$ 2.5 K. The observation of a narrow superconducting transition width suggests that the samples used in our measurements are of high quality (see the inset of Fig. 1a for picture of a sample used for measurements). Fig. 1b shows the temperature dependent magnetic susceptibility measured with a magnetic field parallel to the in-plane of the sample. The shielding volume fraction at 0.5 K is estimated to be about 90$\%$.
The high superconducting volume fraction of these samples provides a new and better opportunity to study topological superconductivity. Furthermore, results obtained from transport characterization are consistent with recent reports <cit.>.
Observation of 2D states in Sr$_{0.06}$Bi$_2$Se$_3$
Fig. 2a shows the ARPES measured dispersion map of Sr$_{0.06}$Bi$_2$Se$_3$ using normal ARPES setup with photon energies of 22 eV and 24 eV. A sharp V-shaped topological surface states is observed, where the Dirac point is located about 450 meV below the Fermi level. Most importantly, we observe the coexistence of the topological surface states and a 2D states on the surface of the Sr$_{0.06}$Bi$_2$Se$_3$ <cit.>. These 2D states originate from the bulk band-bending near the surface due to the Sr-intercalation. The left panel of Fig. 2b shows the ARPES spectra measured in a pump probe setup using a 6 eV photon source in the absence of a pump pulse. Using this experimental setup, the coexisting topological surface states and 2D states are also observed.
From these two sets of experimental results, it is clear that the presence of the 2D quantum well state is a generic property of the Sr intercalated prototypical topological insulator Bi$_2$Se$_3$.
Carrier dynamics of Sr$_{0.06}$Bi$_2$Se$_3$
Now we turn our focus onto the TRARPES results. We note that our results constitute the first pump-probe ARPES
in a system with coexisting topological surface states and 2D states.
Figs. 2b(right panel) and Fig. 3a-b show TRARPES spectra of the Sr$_{0.06}$Bi$_2$Se$_3$ sample at the representative delay time ($d.t.$) values before and after the pump pulse.
Without a pump pulse, the chemical potential of the Sr$_{0.06}$Bi$_2$Se$_3$ sample cuts the topological surface states and 2D quantum well states, which shows its bulk $n$-type metallic character.
In order to systematically study how the excited electronic states relax, we present the transient ARPES spectra as a function of $d.t.$ in Fig. 3c. For a metallic sample of Sr$_{0.06}$Bi$_2$Se$_3$, the population of the excited topological surface states and 2D quantum well state relaxes within $5$ ps. It is consistent with the short optical life-time of a few picoseconds ($10^{-11}$-$10^{-12}$ s) for the Dirac surface states reported previously for bulk metallic samples <cit.>. The short life-time is expected because
these experiments were performed with bulk metallic TI samples, which means both the surface states and the bulk conduction bands are present at the chemical potential.
We note that the spread of the intensity into the unoccupied side is less pronounced than those observed
in less metallic TIs. In the latter case, the spread is visible as far as 1 eV above $E_F$.
Furthermore, the rise of intensity is instantaneous, or resolution limited, and is strongly contrasted to
the cases of less metallic TIs. In the latter case, topological surface states are indirectly populated and shows delayed filling of $\gtrsim$0.5 ps.
Most importantly, we observe a different decay mechanism for the excited topological surface states than the 2D state (see Fig. 3c-d). If the scattering of the carriers in both the topological surface states and 2D states was strong and of similar nature, then the decay profile of the two should overlap.
In order to get an insight into the decay mechanism, we use simple single and double exponential decay fitting functions (see Fig. 3d). When fitted with the double exponential decay function, both Dirac quasiparticle populations L-SS and R-SS, integrated as shown in Fig. 3b, produce almost identical results, with both components of the exponential fitting giving the same amplitude and decay constants for both components, and only a slightly difference between L and R parts, as shown in Table below. We conclude that only one decay constant is present in Dirac quasiparticle band and we use a single exponential decay function here. However, the 2D surface state does not converge well when fitted with one decay channel. It shows a short-time decay channel followed by almost the same, longer decay as in Dirac bands, and hence requires a two-component exponential fit.
The values obtained from the fitting (see Fig. 3d) are shown in the following Table:
Band Decay constants (ps) r$^2$ Amplitude (A)
L-SS $\tau_1 =\tau_2$=1.58 0.99 A=1.382
R-SS $\tau_1 =\tau_2$ =1.69 0.99 A=1.347
2D state $\tau_1$=0.70 and $\tau_2$=1.46 0.99 A1=0.33, A2=1.14
The performed fitting indicates that within the Dirac bands we are looking at a simple, one-component scattering mechanism, while in the 2D surface state part, there are two channels. One of those channels is fast, not present in the Dirac bands, and
the second one is slow and similar to the mechanism found in the Dirac bands.
The latter indicates that there is a weak inter-band coupling that synchronizes the decay at $\gtrsim$1 ps between the Dirac and 2D bands.
To explain the observed difference in life-time, we first ascribe the picosecond decay time to the coupling of surface Dirac quasiparticles and the Sr-doping induced 2D state to the surface phonons. In this time regime, the scattering of intra-band quasiparticles off the phonon modes is the dominant contribution to quasiparticle lifetime. Theoretically, the decay time is then written as follows:
$1/\tau = \pi \sum_{\mu} \vert g_{\mu} \vert^2 [f_{BE}(\Omega_\mu) + f_{FD}(\Omega_\mu)][N(\Omega_\mu) + N(-\Omega_\mu)]$, where $\Omega_\mu$ is the phonon mode from a branch $\mu$, $g_{\mu}$ is the coupling strength between this mode and quasiparticles,
$f_{BE}(E) = 1/(e^{E/k_{B}T}-1)$ and $f_{FD}(E)=1/(1+e^{E/k_{B}T})$ are the Bose-Einstein and Fermi-Dirac distribution function, $N(E)$ is the quasiparticle density of states. Based on our photoemission spectroscopy measurement, a schematic drawing of surface density of states for the Dirac quasiparticles and the 2D states are shown in Fig. 3e. The above equation shows clearly that for a given phonon mode energy, the decay time is inversely proportional to the quasiparticle density of state (DOS) at this energy $\Omega_\mu$ and its imaging $-\Omega_\mu$.
Experiments on pure Bi$_2$Se$_3$ <cit.> have revealed that the surface phonon modes mostly strongly coupled to the surface Dirac quasiparticles are located at near 7.4 meV, the location of which and its imaging are marked with dashed lines in Fig. 3e. Our measurements indicate that the 2D surface states are located between the Fermi level and roughly 200 meV below E$_{F}$. Based on these observations we propose that the phonon scattering channel which can be linked to the intra-band scattering is the same for all bands, while the 2D states are open to an additional scattering mechanism, possibly related to local vibration modes around the Sr - dopants.
Theoretically, we have performed first-principles electronic structure calculations to the end member SrBi$_2$Se$_3$ in the R$\bar{3}$m structure. Although this structure phase has much higher concentration of Sr, which may affect the electronic states near the Fermi energy, it should give a reasonable approximation for the local vibrational properties of Sr dopant along the $c$-axis. The energy of this vibrational mode of Sr is found to be 14.12 meV and is schematically marked with the thick magenta lines in Fig. 3e. The coupling of this mode to the 2D electronic states, through the hybridization of the 6p$_z$ orbital of Bi ion with the 5s orbitals of Sr ion, leads to more effective scattering and a shorter lifetime of the 2D states.
In order to reveal the mechanism of superconductivity, we consider the electron-phonon coupling ($\lambda$) in Bi$_2$Se$_3$ determined from ARPES experiments ranging from 0.08 <cit.> to 0.25 <cit.>. ARPES approach represents the electron aspect of the electron-phonon coupling, incorporating
the quasiparticle renormalization effect contributed from all relevant phonon modes in one measured slope of temperature dependence.
The inelastic helium-atom surface scattering measurements <cit.> was recently used to look at the coupling purely from the phonon perspective for a surface-phonon branch, and the value found was 0.43, greater than those estimated by ARPES. The value of 0.43 represents a lower bound on the value of the actual coupling and is used here. Using the McMillan formula <cit.> of the Migdal-Eliashberg theory <cit.>, a value of 0.1 for Coulomb pseudopotential and a value of 185K for Debye temperature <cit.> we obtain T$_c$=1K for $\lambda$ = 0.43, and T$_c$=2.5K for $\lambda$ = 0.54, in agreement with experimental T$_c$ of around 2.5K and the estimate of $\lambda$ with inelastic helium-atom surface scattering measurements <cit.>. For completeness, we also calculate the simple BCS value, which does not include the retardation effect of the actual electron-phonon interactions as described by the Migdal-Eliashberg theory. The BCS value is 9K for the same set of parameters, which still within a factor of 4 from the experimental value of T$_c$.
The result obtained from Migdal-Eliashberg theory is in better-than-expected agreement with experimental value of T$_c$. We interpret this finding as evidence for phonon-mediated mechanism of superconductivity in Sr-doped Bi$_2$Se$_3$.
Sr intercalated Bi$_2$Se$_3$ system serves a platform to study the interconnection between topology and superconductivity.
It provides a natural interface between a spin-polarized topological surface state and superconductivity. Majorana fermions are predicted to occur at the surface of a topological insulator in proximity of superconductivity <cit.>, and may also occur at such a natural interface.
The strong spin splitting of the TI allows for p-wave Cooper pairing on
the surface of this compound in spite of the s-wave superconductivity in the
bulk. We note that because the T$_c$ of Sr$_{0.06}$Bi$_2$Se$_3$ is large enough to study the zero-biased peak (ZBP) by
scanning tunneling microscopy (STM), this compound is a promising candidate
to get insights of the interaction between superconductivity and
topological surface states.
In conclusion, by using ARPES and TRARPES we study the normal state properties of the Sr intercalated Bi$_2$Se$_3$. We find the coexistence of the topological surface states and 2D quantum well states at the surface of Sr$_{0.06}$Bi$_2$Se$_3$. Furthermore, using TRARPES, we find the different decay mechanisms for the excited TI surface states and 2D quantum wells, which can be understood by considering the different scattering mechanisms.
Our first-principles calculations show the more effective scattering leading to a shorter lifetime of the 2D states, which is consistence with the experimental observation.
Our systematic study will be helpful in understanding the topological superconducting properties of this material, which helps to realize
the properties of Majorana fermion quasiparticle states associated with topological superconductivity.
Sample growth and characterization
Single crystalline samples of Sr-intercalated Bi$_2$Se$_3$ topological insulator (Sr$_{0.06}$Bi$_2$Se$_3$) used in our measurements were grown using the Bridgman method and characterized by transport methods, which is detailed elsewhere <cit.>.
Spectroscopic measurements
The TRARPES setup at the Institute for Solid State Physics (ISSP) in the University of Tokyo consisted of an amplified Ti:sapphire laser system delivering h$\nu$= 1.47 eV pulses of 170-fs duration with 250-kHz repetition and a hemispherical analyzer <cit.>. A portion of the laser was converted into h$\nu$= 5.9 eV probing pulses using two non-linear crystals, $\beta-$BaB$_2$O$_4$ (BBO), and the time delay ($d.t.$) from the pump was controlled by a delay stage.
The diameters of the pump and probe beams were 0.5 and 0.3 mm, respectively, and the time resolution of the pump-and-probe measurement was 250 fs.
The pump and probe pulses were $s$ and $p$-polarized, respectively. The probe intensity was lowered so that the space-charge induced shift of the spectrum is less than 5 meV. The TrARPES spectra were recorded with the energy resolution of about 15 meV. Multi-photon photoelectrons due to the
pump pulse were not observed in the dataset presented herein.
The base pressure of the photoemission chamber was 5$\times$10$^{-11}$ Torr.
With this TrARPES setup, our Sr$_{0.06}$Bi$_2$Se$_3$ samples were cleaved and measured at low temperature ($\sim$ 8 K).
Normal ARPES measurements were performed at the SIS-HRPES end-station of the Swiss Light Source with a Scienta R4000 hemispherical electron analyzer. The minimum sample temperature was 17 K and the pressure during measurements was better than 5$\times$10$^{-11}$ mbar. The energy and angular resolution were set to be better than 20 meV and 0.2$^{\circ}$ for the measurements with the synchrotron beamline.
Electronic structure calculations
Our calculations are carried out within the density functional theory as implemented in the ab initio package VASP <cit.>. The Perdew-Burke-Ernzerhof exchange-correlation functional <cit.> in the projector augmented plane-wave potential format <cit.> is used and the spin-orbit coupling is taken into account. The first Brillouin zone is sampled with 8$\times$ 8$\times$1 k-points using the Monkhorst-Pack grid.
The vibration analysis of Sr atoms is done after a full structure relaxation with the converged lattice parameters a=b=4.535 Å and c=31.38 Å. By taking into the concentration difference of Sr, these lattice constants agree reasonably well with those Sr$_x$Bi$_2$Se$_3$ in the dilute limit <cit.>.
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M.N. is supported by LANL LDRD Program. T.D. is supported by Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences, and by NSF IR/D program.
The work at Princeton and synchrotron X-ray-based measurements are supported by the Office of Basic Energy Sciences, US Department of Energy (grants DE-FG-02-05ER46200, AC03-76SF00098 and DE-FG02-07ER46352).
S.S. and Y.I. at ISSP in University of Tokyo acknowledge support from KAKENHI Grants number
23740256 and 2474021. J.-X. Zhu is supported by the DOE Office of Basic Energy Sciences.
We thank Plumb Nicholas Clark for beamline assistance at the SLS, PSI.
Correspondence and requests for materials should be addressed to M.N. (Email: [email protected]).
Transport characterizations of Sr$_{0.06}$Bi$_2$Se$_3$.
a, Resistivity vs temperature of Sr$_{0.06}$Bi$_2$Se$_3$. Inset shows the picture of a sample used for measurements. b, Temperature dependence of magnetic susceptibility for the Sr$_{0.06}$Bi$_2$Se$_3$ sample measured with applied magnetic field (10 Oe) parallel to the in-plane of the sample. The shielding volume fraction estimated from the zero-field cooling (ZFC) process is about 90$\%$. The inset shows the magnetic field dependence of magnetization measured at 2.0 K.
Observation of 2D states in Sr$_{0.06}$Bi$_2$Se$_3$. a, Dispersion map of Sr$_{0.06}$Bi$_2$Se$_3$ along the $\bar{\textrm{K}}$-$\bar\Gamma$-$\bar{\textrm{K}}$ high-symmetry direction recored by using incident photon energy of 22 eV and 24 eV at a temperature of 17 K. This dataset was obtained with normal ARPES setup. b, Dispersion maps measured in TRARPES setup with 6 eV laser source. Left panel shows the ARPES band dispersion of Sr$_{0.06}$Bi$_2$Se$_3$ at a negative time delay along the $\bar{\textrm{K}}$-$\bar\Gamma$-$\bar{\textrm{K}}$ high-symmetry direction. Right panel shows the dispersion maps for positive time delay. At both experimental setup, the 2D quantum well states are observed.
Relaxation mechanism.
a, TRARPES images of Sr$_{0.06}$Bi$_2$Se$_3$ before and after the pump pulse.
Top panels show band dispersions obtained with the difference to the image before pumped, and bottom panels show the time-evolution sprectra. b, ARPES band dispersion of Sr$_{0.06}$Bi$_2$Se$_3$ at a positive time delay along the $\bar{\textrm{K}}$-$\bar\Gamma$-$\bar{\textrm{K}}$ high-symmetry direction. The blue rectangles represent the integration window of transient photoemission intensity for Dirac surface state and 2D states.
c, Ultrafast evolution of the population of 2D quantum well states (black curve) and surface states (red for left surface state and blue for right surface state) for Sr$_{0.06}$Bi$_2$Se$_3$. 2D quantum-well states relax faster than topological surface states. d, Exponential fitting for the decay curves; fitting parameters are given in the Table. We only consider the decay of the carriers. The rise of the carriers are almost identical within the time-resolution.
e, Schematic view of the decay mechanism, where blue line represents the density of Dirac quasiparticles and red line corresponds to the density of surface 2D state, while dashed grey lines indicate the energy of the phonon mode. The surface phonon modes mostly strongly coupled to the surface Dirac quasiparticles are located at near 7.4 meV, marked with dashed lines. The thick magenta lines represent the vibrational mode of Sr, which is found to be about 14.12 meV.
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1511.00448
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§ INTRODUCTION
In these proceedings we study non-linear electroweak (EW) effective theories including a light Higgs, which we will denote as the electroweak chiral
Lagrangian with a light Higgs (ECLh).
In Ref. <cit.> we have computed the next-to-leading order (NLO) corrections induced by scalar boson
(Higgs $h$ and EW Goldstones $\omega^a$) one-loop diagrams. These contributions provide the one-loop
corrections to the amplitude that grow with the energy as $p^4$, as these particles
are the only ones that couple derivatively
in the lowest-order (LO) effective Lagrangian <cit.>.
We used the background field method and heat-kernel expansion
to extract the ultraviolet divergences of the theory at NLO, i.e.,
$\cO(p^4)$, where $p$ is the effective field theory (EFT) expansion parameter and refers to any infrared (IR) scale of the EFT
–external momenta or masses of the particles in the EFT–.
Many beyond Standard Model (BSM) scenarios show this non-linear realization,
which are typically
strongly coupled theories with composite states <cit.>.
A common feature in non-linear EFT's is that one-loop corrections are formally
of the same order as tree-level contributions from higher dimension
operators <cit.>.
Phenomenologically, these two types of corrections are of a similar size,
provided the scale of non-linearity
$\Lambda_{\rm non-lin}$ that suppresses the $h$ and $\omega^a$ loops and the composite resonance masses $M_R$
are similar <cit.>:
\mbox{NLO tree diagrams} \quad \sim \quad \mbox{NLO loop diagrams .}
§ LOW-ENERGY LAGRANGIAN AND CHIRAL COUNTING
The low-energy theory is given by the usual ingredients <cit.>:
* EFT particle content: the singlet Higgs field $h$,
the non-linearly realized triplet of EW Goldstones $\omega^a$ and
the SM gauge bosons and fermions.
* EFT symmetries: we based our analysis on the symmetry pattern
of the SM scalar sector $\mG=SU(2)_L\times SU(2)_R$,
which breaks down spontaneously into the custodial subgroup $\mH=SU(2)_{L+R}$.
The subgroup $SU(2)_L\times U(1)_Y\in \mG$ is gauged.
When fermions are included in the theory $\mG$ must be enlarged to
$\mG= SU(2)_L\times SU(2)_R\times U(1)_{\rm B-L} \supset SU(2)_L\times U(1)_Y$ and
$\mH= SU(2)_{L+R}\times U(1)_{\rm B-L}\supset U(1)_{\rm EM}$ <cit.>,
with ${\rm B}$ and ${\rm L}$ the baryon and lepton numbers, respectively.
* Locality: The underlying theory may contain non-local exchanges of heavy states.
Nevertheless, in the low-energy limit
the effective action is provided by an expansion of local operators.
In the case of non-linear Lagrangians the classification of the EFT operators
in terms of their canonical dimension is not appropriate and what really ponders
the importance of an operator
in an observable is their “chiral” scaling $p^{\hat{d}}$ in terms of the infrared scales
$p$ <cit.>.
The ECLh is organized in the form
_ECLh = _2 + _4 + ...
where the terms of order $p^{\hat{d}}$ have the generic form <cit.>
_d̂ ∼ ∑_k,n_F
c_(d̂) p^d (χv)^k
(ψ ψv^2)^n_F/2 ,
with $\hat{d}= d + n_F/2$, $v=(\sqrt{2} G_F)^{-1/2}=246$ GeV,
$\chi$ ($\psi$) representing any bosonic (fermionic) fields in the ECLh, and
being $p$ any infrared scale appropriately acting on the fields (derivatives,
masses of the particles in the EFT, etc.).
In the lowest order case $\hat{d}=2$ one has couplings $c^{(2)}\sim v^2$.
The counting can be established more precisely
by further classifying what we mean by $p$ in our operators
(explicit derivatives, fermion masses, etc.) <cit.>.
Beyond naive dimensional analysis (NDA),
one usually makes further assumptions on the scaling of the coupling of the composite sectors
and the elementary fermions.
Typically one assumes them to be weakly coupled in order to support the phenomenological observation
$m_\psi\ll 4\pi v\approx 3$ TeV and the moderate size of the Yukawa couplings measured
so far at LHC. [
Based on pure NDA the Yukawa operators in the ECLh would be $\cO(p^1)$,
spoiling the chiral power expansion. In order to avoid this, one needs to make the phenomenologically supoorted
assumption that the constants
$\lambda_\psi$ that parametrize the couplings between the elementary fermions and the composite scalars
$h$ and $\omega^a$ are further suppressed, scaling at least like $\lambda_\psi \sim \cO(p)$ or higher in the
chiral counting <cit.>.
The LO Lagrangian is given by <cit.>,
_2 = v^24 _C ⟨u|_μu^μ|+⟩12 (∂_μh)^2 -
v^2 V
+ _YM + i ψ̅ D ψ - v^2 ⟨J|_S| ⟩ ,
where $\bra ...\ket$
stands for the trace of $2×2$ EW tensors, $_YM$ is the
Yang--Mills Lagrangian for the gauge fields,
$D_μ$ is the gauge covariant derivative acting on the fermions,
$V[h/v]$ is the Higgs potential and $J_S$
denotes the Yukawa operators that couple the SM fermions to $h$ and $ω^a$.
The factors of $v$ in the normalization of some terms are introduced for later convenience.
$_C, V$ and $J_S$ are functionals of $x=h/v$, and
have Taylor expansions,
$_C[x] = 1 + 2 a x + b x^2 +...$,
$J_S[x]=∑_n J_S^(n) x^n/n!$ and
$V[x]= m_h^2 ( 1/2 x^2+12 d_3 x^3 + 18 d_4 x^4 + ... )$,
given in terms of the constants
$a,b,m_h$, etc.~\cite{EW-chiral-counting,ECLh-Gavela,ECLh-other,SILH}.
In the non-linear realization of the spontaneous EW symmetry breaking,
the Goldstones are parameterized through the coordinates $(u_L,u_R)$ of the
$SU(2)_L×SU(2)_R/SU(2)_L+R$ coset space~\cite{CCWZ}, with the $SU(2)$ matrices $u_L,R$
being functions of the Goldstone fields $ω^a$ which enter through the building blocks
\bear
&&u^{}_\mu =
i u_R^\dagger (\partial^{}_\mu-i r^{}_\mu) u^{}_R - iu_L^\dagger (\partial^{}_\mu-i \ell^{}_\mu) u^{}_L\, ,
\quad
\Gamma^{}_\mu = \frac{1}{2} u_R^\dagger (\partial^{}_\mu-i r^{}_\mu) u^{}_R
+\frac{1}{2}u_L^\dagger (\partial^{}_\mu-i \ell^{}_\mu) u^{}_L\, ,
\nn\\
&&\nabla_\mu\, \cdot = \partial_\mu \,\cdot \, +\, [\Gamma_\mu , \,\cdot\, ] \, ,
\quad f_\pm^{\mu\nu}
= u_L^\dagger \ell^{\mu\nu} u_L \pm u_R^\dagger r^{\mu\nu} u^{}_R\, ,
\quad
= \partial_\mu r_\nu -\partial_\nu r_\mu - i [r_\mu, r_\nu]\, ,
\quad (R\leftrightarrow L)\, ,
\nn\\
&& J^{}_{S} = J^{}_{YRL} +J_{YRL}^\dag,
\qquad
J^{}_{P} = i ( J^{}_{YRL} - J_{YRL}^\dag),
\qquad
J^{}_{YRL} = - \Frac{1}{\sqrt{2} v} u_R^\dagger \hat{Y}
\psi^{\alpha}_R \bar{\psi}^{\alpha}_L
u^{}_L ,
\eear
with $ψ_R,L=1/2(1±γ^5)ψ$ and the $SU(2)$ doublet $ψ= (t,b)^T$.
The summation over the Dirac index $α$ in
$ψ^α_R m ψ̅^α_L n
= -ψ̅^α_L n ψ^α_R m $
is assumed and its tensor structure under $$ and indices $m$ and $n$ are left implicit.
The $2×2$ matrix
$Ŷ[h/v]$ is a spurion auxiliary field, functional of $h/v$,
which incorporates the fermionic Yukawa coupling~\cite{EW-chiral-counting,Hirn:2004,flavor-ECLh}.
Other SM fermion doublets and the
flavour symmetry breaking between generations can be incorporated
by adding in $J_YRL$ an additional family index in the fermion fields,
$ψ^A$, and promoting $Ŷ$ to a tensor $Ŷ^AB$
in the generation space~\cite{flavor-ECLh}.
In our analysis, $ℓ_μ, r_μ, Ŷ$ are spurion
auxiliary background fields
that keep the invariance of the ECLh action under $$.
When evaluating physical matrix elements, custodial symmetry is then explicitly broken in the same way as in the SM,
keeping only the gauge invariance under the subgroup
with the auxiliary fields taking the value
ℓ_μ= - g2 W_μ^a σ^a ,
r_μ= - g'2 B_μσ^3 ,
Ŷ[h/v] = ŷ_t[h/v] P_+ + ŷ_b[h/v] P_- $,
with $P_±=(1±σ^3)/2$.
In order to compute the one-loop fluctuations we will consider the coset representatives
$u_L=u_R^†= u$, often expressed in the exponential
parametrization $U=u^2=exp{ i ω^a σ^a/v}$.~\footnote{
Other Goldstone parametrizations are discussed in~\cite{chpt,1loop-AA-scat,1loop-WW-scat-Dobado,1loop-Gavela},
being all of them fully equivalent when describing on-shell matrix elements.
The IR scales in the low-energy theory are
\bear
\partial_\mu, \quad r_\mu, \quad \ell_\mu, \quad m , \quad g^{(')} v , \quad \hat{Y} v
\quad \sim\quad \cO(p)\, ,
\eear
with $m=m_h,W,Z,ψ$.
Accordingly, covariant derivatives scale as the ordinary ones~\cite{chpt} and the Lagrangian is invariant under $$
at every order in the chiral expansion.
Based on this we are going to sort out the building blocks and
operators according to the assignment~\cite{Weinberg:1979,Georgi-Manohar},
\bear
\Frac{\chi}{v} \quad &\sim& \quad \cO(p^0)
\qquad\qquad (\mbox{for the boson fields } \chi=h ,\, \omega^a, \, W_\mu^a,\, B_\mu )\, ,
\nn\\
\Frac{\psi}{v} \quad &\sim& \quad \cO(p^{1/2})
\qquad\quad (\mbox{for the fermion fields } \psi=t ,\, b, \, \mbox{etc})\, .
\eear
Therefore, the chiral order of the various building blocks reads
\bear
\mF_C \,\, \sim \,\, \cO(p^0)\, ,
\qquad u_\mu ,\, \nabla_\mu \,\, \sim \,\, \cO(p^1)\, ,
\qquad r_{\mu\nu},\, \ell_{\mu\nu},\, f_\pm^{\mu\nu},\, J_{YRL},\, J_S ,\, J_P,\, V \,\, \sim \,\, \cO(p^2)\, .
\eear
Hence, the Lagrangian in Eq.~(\ref{eq.L2}) is $(p^2)$ and provides the LO.
%%%In order to make the $\mL_2$
%%%structure more explicit and analogous to the general form in~(\ref{eq.Ld}) we have left the explicit factors $v^2$ in
%%%the Yukawa and Higgs potential terms.
%%%One can also derive the chiral order for the gauge couplings and Yukawas from the scaling of the masses:
%%%g,g',\hat{Y} \quad \sim \quad \cO(p/v)\, .
So far all we did was to sort out the possible terms of the EFT Lagrangian assigning them an order.
The relevance of this classification is that, at low energies,
when one computes the contributions to a given process
the more important ones are given by the Lagrangian operators
with a lower chiral dimension. An arbitrary diagram with vertices from $_d̂$
behaves at low energies like~\cite{EW-chiral-counting,Weinberg:1979,Georgi-Manohar,1loop-AA-scat,Santos:2015}
\bear
\mM &\sim& \Frac{p^2}{v^{E -2} } \, \left(\Frac{p^2}{16\pi^2 v^2}\right)^L
\,\prod_{\hat{d} } \left(\Frac{
%%%f_{\hat{d} }
c_{(\hat{d})} p^{\hat{d}-2}}{v^2}\right)^{N_{\hat{d}}}\, ,
\label{eq.amp-scaling}
\eear
with the IR scales $p$, $L$ the number of loops,
$N_d̂$ the number of subleading vertices from
$_d̂>2$ (with coupling $c_(d̂)$) and $E$ the number
of external legs.
%%%, which can be summarized in the familiar formula for the chiral scaling of
%%%an amplitude~\cite{Weinberg:1979,Georgi-Manohar,Pich-preparation,1loop-AA-scat,EW-chiral-counting}:
%%%\hat{d}_\mM &=& 2\, +\, 2 L \, +\, \sum_{\hat{d} } (\hat{d}-2)\, N_{\hat{d}}\, .
We can have an arbitrary number of $_2$ vertices in the diagram and the amplitude will still have the same scaling
with $p$, provided the number of loops is fixed.
If we add vertices of a higher chiral dimension we will
increase the scaling of the diagram with $p$.
Thus, we have that the one-loop corrections with only $_2$ vertices are $(p^4)$
and their UV divergences are cancelled out by tree-level diagrams
that contain one $_4$ vertex: the renormalization of the effective couplings at $(p^4)$
will render the effective action finite at NLO.
%%%The SM is recovered by setting $\mF_C=(1+h/v)^2$,
%%%$V= \frac{1}{4} \lambda v^2 [(1+h/v)^2-1]^2$
%%%and $\hat{Y}= (1+h/v) \, (y_t P_+ + y_b P_-)$,
%%%defined in terms of the
%%%SM Yukawa coupling constants $y_{q}$. Thus, $\mL^{\rm SM}$ is indeed $\cO(p^2)$ but
%%%its renormalizable structure --which gets manifest in the linear representation of the Higgs doublet--
%%%ensures the precise cancellation of the $\cO(p^4)$ UV divergences from the various contributions one has in
%%%its non-linear representation~\cite{Guo:2015}.
%%%Other SM fermion doublets and the
%%%flavor symmetry breaking between generations can be incorporated
%%%by adding in $J_{YRL}$ an additional family index in the fermion fields,
%%%$\psi^A$, and promoting $\hat{Y}$ to a tensor $\hat{Y}^{AB}$
%%%in the generation space~\cite{flavor-ECLh}.
\section{Fluctuations around a background field }
We are going to consider perturbations $η$ in the fields around their equations of motion (EoM) solutions.
The Lagrangian in the integrand of the generating functional
has also a corresponding expansion in the perturbation,
where each order in $η$ contains relevant information~\cite{heat-kernel}:
\bear
\mL &=& \quad
\underbrace{ \mL^{\cO(\eta^0)} }_{\mbox{Tree-level}} \quad +\quad
\underbrace{ \mL^{\cO(\eta^1)} }_{\mbox{EoM}} \quad +\quad
\underbrace{ \mL^{\cO(\eta^2)} }_{\mbox{1-loop}} \quad +\quad
\underbrace{ \cO(\eta^3) }_{\mbox{Higher loops}} \, .
\eear
The Lagrangian evaluated at the classical solution provides the tree-level contributions to the effective action,
the requirement that the linear term in the $η$ expansion vanishes provides the EoM,
the quadratic fluctuation in $η$ provides the 1-loop contributions to the effective action
and higher loops are encoded in the remaining terms of the $η$ expansion.
In our analysis~\cite{Guo:2015} we were interested in the one-loop UV divergences
at $(p^4)$,
that is those coming from diagrams with $_2$ vertices. We studied
the structures that grow faster with the energy at one-loop, as $(Energy)^4$.
They were given by
the loops of scalars ($h$ and $ω^a$) which are the only ones
that couple derivatively in $_2$.
We made the Goldstone representative choice $u_L=u_R^†$~\cite{Pich:2013}
and performed fluctuations of the scalar fields (Higgs and Goldstones)
around the classical background fields $h̅$ and
\bear
u_{R,L}&=& \bar{u}_{R,L} \, \exp\left\{ \pm i \mF_C^{-1/2}\, \Delta/(2 v) \right\}\, ,
\qquad\qquad
h = \bar{h}\, +\, \epsilon\, ,
\eear
with $Δ=Δ^a σ^a$. Without any loss of generality we
introduced the factor $_C^-1/2$
in the exponent for later convenience, allowing us to write down the second-order fluctuation of the action
in the canonical form~\cite{heat-kernel}.
To obtain the one-loop effective action within the background field method
we then retained the quantum fluctuations $η⃗^T=(Δ^a, ϵ)$
up to quadratic order~\cite{heat-kernel}.
Since we are interested in the loops with only $_2$ vertices, we study the $η$ expansion of the
LO Lagrangian
%%% of the ECLh:
%%%\mL_2 &=& \mL_2^{\cO(\eta^0)} \, +\, \mL_2^{\cO(\eta^1)} \, +\, \mL_2^{\cO(\eta^2)} \, +\, \cO(\eta^3)\, ,
The tree-level effective action is equal to the action evaluated at the classical solution,
$∫d^D x ^(η^0)$.
\subsection{$\cO(\eta^1)$ fluctuations: EoM}
The background field configurations
correspond to the solutions of the classical equations of motion (EoM),
defined by the requirement that the linear term,
\bear
\mL_2^{\cO(\eta^1)} &=& \Frac{v}{2} \bra \Delta \,\,\,
\left( \nabla^\mu(\mF_C u_\mu)\, +\, 2 \mF_C J_P \right) \ket
\,\, +\,\, v\,\epsilon\, \left( \Frac{1}{4} \mF_C' \bra u_\mu u^\mu\ket
- \Frac{\partial^2 h}{v} - V' - \bra J_S'\ket
\right)\, ,
\eear
vanishes for arbitrary $η⃗^T=(Δ^a,ϵ)$. This yields the EoM,
\bear
\nabla^\mu u_\mu &=& -2 \mF_C^{-1} J_{P} - u_\mu \partial^\mu(\ln\mF_C)
\, ,\qquad \qquad
\Frac{\partial^2 h}{v} = \Frac{1}{4} \mF_C' \, \bra u_\mu u^\mu \ket
- V'
-\,\bra J_{S}'\ket \, .
\label{eq.EoM}
\eear
%%%with $J^{}_{P} = i(J^{}_{YRL} -J_{YRL}^\dag)$ and
%%%the covariant derivative $\nabla_\mu \cdot =\partial_\mu + [\Gamma_\mu ,\cdot]$.
Here and in the following, we abuse of the notation by writing the
background fields $u̅_L,R$ and $h̅$ as $u_L,R$ and $h$ for
\subsection{$\cO(\eta^2)$ fluctuations: 1-loop corrections }
The $(η^2)$ term of the expansion of $_2$ reads
\bear
\mL^{\cO(\Delta^2)} &=&
-\, \Frac{1}{4} \bra \Delta \nabla^2 \Delta\ket
\, + \, \Frac{1}{16}\bra [u_\mu,\Delta] \, [u^\mu, \Delta] \ket
\nn\\
\hspace*{-1.7cm}+ \left[ \Frac{ \mF_C^{-\frac{1}{2}} \mK}{8}
\left(\Frac{\partial^2 h}{v}\right)
+ \Frac{\Omega}{16} \left(\Frac{\partial_\mu h}{v}\right)^2 \right]
\bra \Delta^2 \ket
+ \Frac{1}{2\mF_C} \bra \Delta^2 J_S\ket
\nn\\
\mL^{\cO(\epsilon^2)} &= &
-\Frac{1}{2} \epsilon \left[ \partial^2
- \Frac{1}{4} \mF_C'' \bra u_\mu u^\mu \ket
+ V''
+ \bra J_S''\ket
\right]\, \epsilon\, ,
\nn\\
\mL^{\cO(\epsilon\Delta)} &=&
\,- \, \Frac{1}{2} \epsilon \mF_C'
\, \bra u_\mu \nabla^\mu (\mF_C^{-\frac{1}{2}} \Delta) \ket
+ \mF_C^{-\frac{1}{2}} \epsilon \bra \Delta J_P'\ket
\, ,
\eear
in terms of $= _C^-1/2 _C'$ and $Ω=2 _C”/_C - (_C'/_C)^2$.
Through a proper definition of the
differential operator $d_μη⃗=∂_μη⃗ + Y_μη⃗$, one can rewrite $_2^(η^2)$ in the canonical form
\bear
\mL_{2}^{\cO(\eta^2)} &=& - \Frac{1}{2} \vec{\eta}^T\, (d_\mu d^\mu +\Lambda) \vec{\eta}\, ,
\label{eq.Lagr-quad-fluctuation}
\eear
where $d_μ$ and $Λ$ depend on $h$, $u_L,R$
and on the gauge boson and fermion fields (see App.~A in Ref.~\cite{Guo:2015}).
They scale according to the chiral counting as
\bear
d_\mu \,\,\, \sim\,\,\, \cO(p)\,, \qquad\qquad \Lambda\,\,\, \sim\,\,\, \cO(p^2)\, .
\eear
The quadratic form~(\ref{eq.Lagr-quad-fluctuation}) yields a Gaussian integration over $η⃗$ in
the path-integral, which gives the one-loop contribution to the effective action,
\bear
S^{1\ell} &=& \Frac{i}{2} {\rm tr}\, \log\left(d_\mu d^\mu +\Lambda\right)\, .
\eear
where tr stands for the full trace of the operator,
including the trace in the adjoint representation of the flavour space and
that in the coordinate space.
The computation of the full one-loop effective action is in general a difficult task.
However, it is easier
to extract its UV-divergent part.~\footnote{
It is also sometimes possible to compute the effective potential.
See for instance~\cite{Meissner:2015}.
For this, we first transform the trace of the log in configuration space into an integral of an exponential
by means of the Schwinger-DeWitt proper-time representation
embedded in the heat-kernel expansion~\cite{heat-kernel}:
\bear
\bra x|\log\left(d_\mu d^\mu +\Lambda\right)|x\ket &=&
\,-\, \Int_0^\infty \Frac{d\tau}{\tau} \, \underbrace{ \bra x| e^{-\tau(d_\mu d^\mu +\Lambda)} |x\ket }_{\equiv H(x,\tau)}
\,\,\, +\,\,\, C
\nn\\
&&\stackrel{\rm dim-reg}{=}\quad
-\, \Frac{i}{(4\pi)^{D/2} }\, \sum_{n=0}^\infty m^{D-2n} \, \,\Gamma\left(n-\frac{D}{2}\right)\,\, a_n(x)
\,\,\, +\,\,\, C ,
\eear
where we obtain the expansion in term of local operators of increasing
dimension given by the
Seeley-DeWitt coefficients $a_n(x)$ and the potential UV divergences
are contained in the Gamma function
for $D→4$.
Only the terms of the series with $2n≤D$ are divergent and they have their origin
on the short-distance part of the integral, this is, in its lower limit $τ→0$
(the integration variable $τ$ has dimensions of length-square in natural units).
The (infinite) constant $C$ is irrelevant here. In the second line we have used
the Fourier decomposition of the heat-kernel in momentum space,
\bear
H(x,\tau) = \bra x| e^{-\tau (d_\mu d^\mu +\Lambda)} |x\ket
= \Int\Frac{{\rm d^D p}}{(2\pi)^D}
e^{-ipx} e^{-\tau (d_\mu d^\mu +\Lambda)} e^{ipx}
\Frac{i e^{-\tau m^2}}{(4\pi \tau)^{D/2}}\sum_{n=0}^\infty a_n(x)\, \tau^n\, ,
\eear
where the coefficients $a_n(x)$ are extracted by expanding the interaction part of $(d_μd^μ+Λ)$
in the exponential in powers of $τ$
and performing the integral of each corresponding term in dimensional regularization.
In our case,
the residue of the $(D-4)^-1$ pole is given by the trace
The IR regulator $m$ of the heat-kernel integral can be made arbitrary small
and hence the term Tr$\{a_1(x)\}=\, -$Tr$\{\Lambda\}$
does not contribute to the UV divergent part;
%%%when one makes $m\to 0$
note that the particle masses are accounted as a perturbation,
i.e., within $\Lambda(x)$.
\bear
S^{1\ell} %%%_{\cO(p^4)}
&=& \,-\, \divergence \, \Int {\rm d^D x} \,\,\, {\rm Tr}\{ a_2(x)\} \, +\, {\rm finite}
\nn\\
%%%S^{1\ell} %%%_{\cO(p^4)}
&&=\,-\, \divergence \, \Int {\rm d^D x} \,\,\, {\rm Tr}
\left\{ \Frac{1}{12} [d_\mu,d_\nu] \, [d^\mu , d^\nu]
%%%Y_{\mu\nu} Y^{\mu\nu}
\right\} \, +\, {\rm finite}
\nn\\
\,-\, \divergence \, \Int {\rm d^D x} \,\,\, \sum_k \, \Gamma_k \, \mO_k \,
+\, {\rm finite}\, ,
\label{eq.1loop-div}
\eear
where Tr refers to the trace over the $4×4$
operators that acted on
the fluctuation vector $η⃗$ in Eq.~(\ref{eq.Lagr-quad-fluctuation}).
The UV-divergence is determined by the non-derivative quadratic fluctuation $Λ$
and the differential operator $d_μ$ through
$[d_μ, d_ν]=Y_μν =∂_μY_ν-∂_νY_μ+[Y_μ,Y_ν]$,with both $Λ, Y_μν ∼ (p^2)$.
By looking at the second line of~(\ref{eq.1loop-div})
it is then clear that the UV-divergences that
appear from one-loop diagrams with $_2$ vertices are $(p^4)$ and
that they require counterterms of that order
to be cancelled out.
However, as some of this $p$ factors are actually constants (e.g., Higgs masses $m_h$) the structure of
the operators $_k$ resembles that of other operators already present in $_2$.
In general, the one-loop UV-divergences in the effective action will have a local structure and can be written
in terms of the basis of operators of chiral dimension $p^2$, $p^4$,
\bear
S^{1\ell ,\, \infty} &=& \Int{\rm d^D x}\, \bigg(
\mL_2^{1\ell ,\, \infty} \,+\,
\mL_4^{1\ell ,\, \infty} \, +\, ... \bigg)\, .
\eear
Let us summarize the results for the NLO UV-divergences from $h$ and $ω^a$ loops:
\begin{itemize}
\item
{\bf UV divergences with the structure of the $\mL_2$ operators in Eq.~(\ref{eq.L2}):}
\bear
\mL_2^{1\ell,\, \infty} &=&
- \divergence
\bigg\{
\Frac{1}{8}\left[ \Frac{ \mF_C' V'}{\mF_C}(4-\mK^2)
- \mF_C\Omega V''\right] \bra u_\mu u^\mu\ket
\nn\\
&& -\, \frac{3\mF_C' V' \Omega}{8\mF_C}
\left(\Frac{\partial_\mu h}{v}\right)^2
+\left[\Frac{1}{2}\left( V''\right)^2
+ \Frac{3 \mK^2}{8 \mF_C } \left(V' \right)^2 \right]
\nn\\
+ \left( V'' \bra J_{S}'' \ket
- \Frac{3 \mF_C' V' }{2\mF_C} \bra \Gamma_{S} \ket \right)\bigg\}
\, ,
\label{eq.L2-div}
\eear
$\Gamma_{S}= \mF_C^{-1} ( J_{S} - \mF_C' J_{S}' /2)$ is an $\cO(p^2)$ tensor.
These UV divergences are cancelled out through the renormalization of
various parts
of $\mL_2$:
the couplings in the $\mF_C$ term (1st line);
the Higgs kinetic term (1st term in 2nd line),
which requires a NLO Higgs field redefinition;
the coefficients of the Higgs potential, e.g. the
Higgs mass (2nd bracket in 2nd line); the Yukawa term couplings in $\hat{Y}$ (3rd line).
\item
{\bf UV divergences with the structure of the $\mL_4$ operators:}
the $\mL_4^{1\ell,\infty}$ terms are further classified here into two types,
according to whether they include fermion
fields or not.
\begin{enumerate}
\item
{\bf Fermionic operators $\mL_4^{1\ell,\infty}|_{\rm Fer}$: }
\bear
&& \mL_4^{\rm 1\ell,\, \infty}|_{\rm Fer} =
- \divergence
\bigg\{
\bra \left(\Frac{\mK^2}{4}-1\right) \Gamma_{S}
- \Frac{\mF_C \Omega }{8} J_{S}''\ket \, \bra u^\mu u_\mu \ket
\nn\\
+ \Frac{3}{4} \Omega \bra \Gamma_{S}\ket \, \left(\Frac{\partial_\mu
+ \Frac{1}{2} \Omega \bra \Gamma_{P} u^\mu \ket \,
\left(\Frac{\partial_\mu h}{v}\right)
\nn
\\
+ \Frac{1}{2} \bra {J_{S}''}\ket^2
+ \Frac{3}{2} \bra \Gamma_S\ket^2
+ \frac1{\mF_C} \left( 2 \bra\Gamma_{P}^{\,2} \ket
- \bra \Gamma_{P}\ket^2\right)
\,\bigg\} ,
\label{eq.fermion-div}
\eear
with $\Gamma_P= J_{P}' - \mF_C^{-1} \mF_C' J_{P} /2$ being an $\cO(p^2)$ tensor.
\item
{\bf Purely bosonic $\cO(p^4)$ divergences $\mL_4^{1\ell,\infty}|_{\rm Bos}$:}
This is actually the main result of our computation as these $\cO(p^4)$ operators of the effective action
can be only produced from the derivative interactions in $\mL_2$.
They spoil the renormalizability of the SM Lagrangian and lead to the appearance of
``true'' $\cO(p^4)$ UV-divergences, this is, operators with 4 covariant derivatives.~\footnote{
In the case that the covariant derivatives act on $h$ they just reduce to partial derivatives.
Note that the field-strength tensors can be always realized as the commutator of two covariant derivatives.
The outcome is summarized in Table~\ref{tab.div}.
\end{enumerate}
\end{itemize}
%%%Any other operator not listed here is found to be UV-finite.
\begin{table*}[!t]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|c|}
\hline
$c_k$ &
Operator ${\cal O}_k$ &
$\Gamma_k$ & $\Gamma_{k}[0]$ \\
\hline
\hline
$c_1$ & %%%$a_1$ &
$\Frac{1}{4}\bra {f}_+^{\mu\nu} {f}_{+\, \mu\nu}
- {f}_-^{\mu\nu} {f}_{-\, \mu\nu}\ket$
& $\Frac{1}{24}(\mK^2-4)$
& $-\Frac{1}{6}(1-a^2)$
\\
\hline
$(c_2 -c_3)$ & %%%$(a_2-a_3)$ &
$\frac{i}{2} \bra {f}_+^{\mu\nu} [u_\mu, u_\nu] \ket$
& $\Frac{1}{24}(\mK^2-4)$
& $-\Frac{1}{6}(1-a^2)$
\\
\hline
$c_4$ & %%%$a_4$ &
$\bra u_\mu u_\nu\ket \, \bra u^\mu u^\nu\ket $
& $\Frac{1}{96}(\mK^2-4)^2$
& $\Frac{1}{6}(1-a^2)^2$
\\
\hline
$c_5$ & %%%$a_5$ &
$ \bra u_\mu u^\mu\ket^2$
& $\Frac{1}{192} (\mK^2-4)^2 + \Frac{1}{128} \mF_C^2 \Omega^2$
& $\Frac{1}{8}(a^2-b)^2 + \Frac{1}{12} (1-a^2)^2$
\\
\hline
$c_6$ & %%%$\mF_{D7}$ &
$\Frac{1}{v^2}(\partial_\mu h)(\partial^\mu h)\,\bra u_\nu u^\nu \ket$
& $\Frac{1}{16}\Omega (\mK^2-4) - \Frac{1}{96} \mF_C \Omega^2 $
& $-\Frac{1}{6}(a^2-b)(7a^2 -b-6)$
\\
\hline
$c_7$ & %%%$\mF_{D8}$ &
$\Frac{1}{v^2}(\partial_\mu h)(\partial_\nu h) \,\bra u^\mu u^\nu \ket$
& $\Frac{1}{24}\mF_C \Omega^2 $
& $\Frac{2}{3}(a^2-b)^2$
\\
\hline
$c_8$ & %%% $\mF_{D11}$ &
$\Frac{1}{v^4}(\partial_\mu h)(\partial^\mu h)(\partial_\nu h)(\partial^\nu h)$
& $\Frac{3}{32}\Omega^2$
& $\Frac{3}{2}(a^2-b)^2$
\\
\hline
$c_9$ & %%% $b_1$ &
$\Frac{(\partial_\mu h)}{v}\,\bra f_-^{\mu\nu}u_\nu \ket$
& $\Frac{1}{24} \mF_C' \Omega$
& $-\Frac{1}{3}a (a^2-b)$
\\
\hline
$c_{10}$ & %%% $H_1$ &
$ \Frac{1}{2} \bra {f}_+^{\mu\nu} {f}_{+\, \mu\nu}
+ {f}_-^{\mu\nu} {f}_{-\, \mu\nu}\ket$
& $-\Frac{1}{48}(\mK^2+4)$
& $-\Frac{1}{12}(1+a^2)$
\\
\hline
\end{tabular}
\end{center}
\caption{{\small
Purely bosonic operators needed for the renormalization of
the NLO effective Lagrangian $\mL_4$~\cite{Guo:2015}. In the last column, we provide
the first term $\Gamma_{k}[0]$ in the expansion of the $\Gamma_k$ in powers of
$(h/v)$ by using $\mF_C=1+2 a h/v + b h^2/v^2 +\cO(h^3)$.
The first five operators $\mO_{i}$
have the structure of the respective $a_i$ Longhitano operator~\cite{Longhitano:1980iz,Morales:94}
(with $i=1...5$). In addition, $c_6=\mF_{D7}$, $c_7=\mF_{D8}$ and $c_8=\mF_{D11}$ in the notation
of Ref.~\cite{EW-chiral-counting}. The last operator of the list,
$\mO_{10}=2\bra r_{\mu\nu} r^{\mu\nu}+\ell_{\mu\nu}\ell^{\mu\nu}\ket$, only
depends on the EW field strength tensors and its coefficient is labeled as
$c_{10}=H_1$ in the notation of Ref.~\cite{chpt}.
In the notation of~\cite{Santos:2015} $c_{10}=\mF_2$, $c_2-c_3=\mF_3$ and $c_k=\mF_k$ for $k\neq 2,3$.
\label{tab.div}
\end{table*}
One can observe that the non-linearity of the $\mL_2$ Lagrangian
(where, in general, $h$ is not introduced via a complex double $\Phi$)
is the origin of these higher-dimension divergences.
For $\mF_C[h/v] = 1+ 2 a h/v + b h^2/v^2 +...$, the combinations $\mK$ and $\Omega$ that rule the structure of
the divergences are given by
$\left( \mK^2 -4\right)\,=\, 4(a-1) \, +\, \cO(h/v)$,
$\Omega \,=\, 4(b-a^2)\,+\, \cO(h/v)$.
In particular in the linear limit $\mF_C=(1+h/v)^2$ and $\hat{Y}[h/v] = \, Y\, (1+h/v)$,
all the $\cO(p^4)$ divergences from $h$ and $\omega^a$ loops disappear,
\vspace*{-0.35cm}
\bear
\left(\mK^2-4\right)\,=\, \Omega \,=\, 0\, ,
\qquad\qquad
J_S''\,=\, \Gamma_S\,=\, \Gamma_P\,=\, 0\, ,
\eear
where the cancellation in the first identity relies only on the form of $\mF_C$, and
the second one --related to four-fermion operators in $\mL_4^{1\ell,\infty}$--
requires also the linear structure in $\hat{Y}$.
\section{Renormalization at NLO in the ECLh}
In order to have a finite 1-loop effective action
the divergences in Eq.~(\ref{eq.1loop-div}) are canceled by
the counterterms
\bear
\mL^{\rm ct} %%%_{\cO(p^4)}
&=& \sum_k \, c_k \, \mO_k\, ,
\eear
\mbox{such that }
\mL^{\rm ct}\, +\, \mL^{1\ell,\infty} \,=\, {\rm finite}\, ,
where the $\mO_k$ is the previous basis of EFT operators,
translating into the renormalization conditions
\bear
c_k = c_k^r + \divergence \, \Gamma_k\, .
\eear
%%%The $\Gamma_k$'s and $c_k$'s must be understood as functionals of $h/v$,
%%%with a Taylor expansion of the form
%%%\Gamma_k[h/v]=\sum_n \Frac{ \Gamma_{k,n}}{n!} \left(\Frac{h}{v}\right)^n \, ,
%%%\qquad \qquad c_k[h/v]=\sum_n \Frac{ c_{k,n} }{n!} \left(\Frac{h}{v}\right)^n \, .
This leads to the renormalization group equations (RGE)
for the $\cO(p^4)$ coupling constants,
\bear
\Frac{d c_{k,n}^r}{d\ln\mu } &=&
- \, \Frac{\Gamma_{k,n}}{16\pi^2}\, ,
\qquad
\mbox{with } \quad
\Gamma_k[h/v]=\sum_n \Frac{ \Gamma_{k,n}}{n!} \left(\Frac{h}{v}\right)^n \, ,
\quad c_k[h/v]=\sum_n \Frac{ c_{k,n} }{n!} \left(\Frac{h}{v}\right)^n \, .
\eear
Physically, this means that the NLO effective couplings
will appear in the amplitudes in combinations with logarithms of IR scales $p$.
%%%The structure of the amplitudes up to NLO in the chiral counting have the generic form we saw
%%%in~Eq.~(\ref{eq.amp-scaling}) (e.g., for $2\to 2$ processes):
%%%\mM &\sim & \underbrace{ \Frac{p^2}{v^2} }_{\mbox{LO (tree)}}
%%%\, + \, \bigg(
%%%\, \underbrace{ c_{k,n}^r }_{\mbox{ NLO (tree) }} \quad -\quad
%%%\underbrace{ \Frac{ \Gamma_{k,n} }{16\pi^2}\ln\Frac{p}{\mu} \quad +\quad ... }_{
%%%\mbox{NLO (1-loop)} }
%%%\bigg) \,\Frac{p^4}{v^4}\, ,
%%%\mM_{\cO(p^4)} &\sim & \left( c_{k,n}^r(\mu) - \Frac{\Gamma_{k,n}}{16\pi^2}
%%% \ln\Frac{p}{\mu}
%%%\right)\, p^4\, .
%%%where the dots in the NLO part stand for other finite pieces from the loop
%%%(varying from one observable to another~\cite{Gomez-Ambrosio:2015,Azatov:2013,1loop-WW-scat,1loop-AA-scat}).
\section{Conclusions}
Modifying the LO Lagrangian of our EFT action by allowing a non-linear structure
for the Higgs field
($\mF_C\neq (1+h/v)^2$) has important implications not only at lowest order but also at NLO.
Any misalignment between Higgs $h$ and Goldstone fields $\omega^a$ that does not allow
us to combine them into a complex doublet
$\Phi$ produces a whole new set of divergences absent in linear theory.
Nevertheless, it is possible to find combinations of couplings that are
renormalization group invariant (RGI).
%%%They are independent of the non-linearity of the theory and therefore
%%%should also apply to linear theories.
Some examples derived in~\cite{Guo:2015}
are the couplings that determine $\gamma\gamma\to ZZ, \, Z\gamma,\, \gamma\gamma$,
$\gamma\gamma,\, Z\gamma\to h, \, hh,\, hhh...$
%%%and $gg\to h,\, hh,\, hhh...$,
Some of the latter RGI relations were known from previous works
($\gamma\gamma\to ZZ$~\cite{1loop-AA-scat},
$\gamma\gamma,\,Z \gamma\to h$~\cite{1loop-AA-scat,Azatov:2013}).
In addition our result also reproduces the running found
in $WW,\, ZZ$ and $hh$ scattering~\cite{1loop-WW-scat-Dobado,1loop-WW-scat-Espriu}.
Our computation in~\cite{Guo:2015} did not address the following two issues:
on the one hand, the analysis of the additional contributions
to the Higgs potential and their phenomenological implications;
on the other hand, deviations from the linear-Higgs scenario leads to the appearance of UV-divergences and logs related to
four-fermion operators, e.g. $\bra J_{S,P}^2\ket$, which could be strongly constrained by flavour tests.
The study of the latter, together with the full NLO computation including gauge boson and
fermion loops, is postponed for future work.
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\end{document}
|
1511.00581
|
These authors contributed equally to this work.
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
These authors contributed equally to this work.
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
These authors contributed equally to this work.
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
Department of Mathematics & Statistics, University of
Guelph, Guelph, Ontario, Canada
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
State Key Laboratory of Computer Science, Institute of
Software, Chinese Academy of Sciences, Beijing, China
Joint Center for Quantum Information and Computer Science,
University of Maryland, College Park, Maryland, USA
State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science
and Technology of China, Hefei, Anhui 230036, China
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
Department of Mathematics & Statistics, University of
Guelph, Guelph, Ontario, Canada
Canadian Institute for Advanced Research, Toronto,
Ontario, Canada
Institute for Quantum Computing,
University of Waterloo, Waterloo N2L 3G1, Ontario, Canada
Canadian Institute for Advanced Research, Toronto,
Ontario, Canada
Perimeter Institute for Theoretical Physics, Waterloo N2L 2Y5, Ontario,
Entanglement, one of the central mysteries of quantum mechanics,
plays an essential role in numerous applications of quantum
information theory. A natural question of both theoretical and
experimental importance is whether universal entanglement detection
is possible without full state tomography. In this work, we prove a
no-go theorem that rules out this possibility for any non-adaptive
schemes that employ single-copy measurements only. We also examine
in detail a previously implemented experiment, which claimed to
detect entanglement of two-qubit states via adaptive single-copy
measurements without full state tomography. By performing the
experiment and analyzing the data, we demonstrate that the
information gathered is indeed sufficient to reconstruct the state.
These results reveal a fundamental limit for single-copy
measurements in entanglement detection, and provides a general
framework to study the detection of other interesting properties of
quantum states, such as the positivity of partial transpose and the
$k$-symmetric extendibility.
§ INTRODUCTION
Entanglement is one of the central mysteries of quantum mechanics—two or more
parties can be correlated in some way that is much stronger than they
can be in any classical way. Famous thought experiments questioning
the essence of quantum entanglement include the EPR
paradox <cit.> and the Schrodinger's
cat <cit.>, which ask the fundamental question
whether quantum mechanics is incomplete and there are hidden variables
not described in the theory. These debates about the weirdness of
quantum mechanics were later put into a theorem by
Bell <cit.>, which draws a clear line between
predictions of quantum mechanics and those of local hidden variable
theories. Bell's theorem was tested extensively in
experiments <cit.>
and quantum mechanics stands still to date.
More concretely, a bipartite quantum state $\rho_{AB}$ of systems $A$
and $B$ is separable if it can be written as a mixture of product
states $\rho_{AB} = \sum_i p_i \rho_A^i \otimes \rho_B^i$ with
$p_i\geq 0$ and $\sum_i p_i=1$, for some states $\rho_A^i$ of system
$A$ and $\rho_B^i$ of system $B$; otherwise, $\rho_{AB}$ is
entangled <cit.>. However, not every entangled state
$\rho_{AB}$ violates Bell inequalities—some entangled states
do allow local hidden variable descriptions <cit.>.
In practice, entanglement may also be detected by measuring the
`entanglement witnesses', physical observables with certain values
that prove the existence of quantum entanglement in a given state
$\rho_{AB}$ <cit.>. However, none of these entanglement
witnesses could be universal. That is, the value of an entanglement
witness cannot tell with certainty whether an arbitrary state is
entangled or not. On the other hand, the `entanglement' measures do
play such a universal role. By commonly accepted axioms, the quantum
state $\rho_{AB}$ is entangled if and only if it has a nonzero value
of any entanglement measure <cit.>. Unfortunately,
entanglement measures are not physical observables.
These commonly-known restrictions on Bell inequalities, entanglement
witnesses and entanglement measures raise a fundamental question: how
do we universally detect entanglement through physical observables?
The traditional approach to this problem is to completely characterize
the quantum state by means of state tomography <cit.>,
a method that provides
complete information of the state including, of course, entanglement
measures of the state. However, performing quantum state tomography
requires a large number of measurements, a daunting task for growing
system sizes.
A natural idea is to find a way to obtain the value of an entanglement
measure without FST. In fact, there have been
a lot of efforts along this line over the past
decade <cit.>.
However, common techniques to achieve this purpose rely heavily on
collective measurements on many identical copies of the state
$\rho_{AB}$. That is, joint measurement on more than one copy of the
state ($\rho_{AB}^{\otimes r}$ for some integer $r>1$) is needed. This
is bad news for experimentalists, as collective measurements are
usually much more difficult to implement than measuring single-copy
observables. It is then highly desirable to find a method that detects
entanglement without FST by measuring only
single-copy observables. The seeking of such a method has
been pursued in recent years with both theoretical simulations and
experimental realizations, leading to positive signs of
realizing such an appealing
task <cit.>.
In this work, we examine the possibility of detecting entanglement
without FST by measuring only single-copy
observables. Surprisingly, despite the previous signs, we find that
this appealing task is unfortunately impossible, if only single-copy
observables are measured. That is, there is no way to
determine with certainty of any entanglement measure, or even to
determine whether the value is zero or not, without FST.
To be more precise, this means that for any set of
informationally-incomplete measurements, there always exists two
different states, an entangled $\rho_{AB}$ and a separable
$\sigma_{AB}$, giving the same measurement results under this
measurement. This sounds very counter-intuitive at the first sight, as
entanglement is just a single value, while quantum state tomography
requires measuring a set of observables that are
informationally-complete, scaling as the squared dimension
of the Hilbert space of the system.
Our observation is that universal detection of any property
without FST enforces strong geometrical structural conditions on the
set of states having that property. The set of separable states does
not satisfy such conditions due to its nonlinear nature and,
therefore, universal detection of entanglement without FST using
single-copy measurements is not possible. There is a nice geometric
picture of this fact: unless the shape of the separable states is
`cylinder-like', it is not possible to find a projection of the state
space to a lower dimensional hyperplane with non-overlapping image for
the set of separable states and entangled states.
If one allows adaptive measurements (the observable to be measured can
depend on previous measurement results), a protocol was implemented
in <cit.>, claiming to have detected entanglement
of a two-qubit state $\rho_{AB}$ via single-copy measurements without
FST. The protocol involves local filters that require repeated
tomography on each single qubit, which leads to a bound on the
entanglement measure concurrence <cit.> of $\rho_{AB}$,
in case the single-qubit reduced density matrices $\rho_A$ and
$\rho_B$ are not maximally mixed.
We design an experiment to implement this adaptive protocol as proposed
in <cit.>, and show that for
certain $\rho_{AB}$, given the experimental
data collected, the state $\rho_{AB}$ is already completely determined. In
other words, once the concurrence of $\rho_{AB}$ is
determined, the protocol already leads to a FST of $\rho_{AB}$, i.e.
the protocol does not lead to the universal detection of entanglement
without FST.
This supplements
our no-go result with non-adaptive measurements.
Additionally, it is worthy emphasizing that to our best knowledge this is the first experimental realization of quantum filters (or equivalently, the amplitude-damping channel) via the ancilla-assisted approach. Compared to the optical platform which does not demand extra ancilla qubits to realize an amplitude damping channel <cit.>, our approach is more general and can be extended to other systems straightforwardly.
We further show that, however, if one
allows joint measurements on $r$-copies (i.e. $\rho_{AB}^{\otimes r}$) even for $r=2$,
one can indeed find protocols that detects the entanglement of $\rho_{AB}$ without
FST. Therefore our no-go result
reveals a fundamental limit for single-copy measurements, and provides
a general framework to study the detection of other interesting
quantities for a bipartite quantum state, such as the positivity of
partial transpose <cit.> and $k$-symmetric
extendibility <cit.>.
§ RESULTS
We discuss a no-go result stating that it is impossible to determine universally
whether a state is entangled or not without FST, with only
single-copy measurements. We first
prove a no-go theorem for non-adaptive measurements, and then
examine the protocol with adaptive measurements as proposed
in <cit.> in detail. We design
an experiment to implement this adaptive protocol, and
demonstrate that the information gathered is indeed sufficient
to reconstruct the state.
Non-adaptive measurement. For any given bipartite state $\rho_{AB}$,
one is only allowed to measure
physical observables on one copy of this given state. That is, we can
only measure Hermitian operators $S_k$ that are acting on
$\mathcal{H}_A \otimes \mathcal{H}_B$. For simplicity, we consider the
case where both $A$, $B$ are qubits. Our method
naturally extends to the general case of any bipartite systems (see
the Supplementary Information for details).
Now we consider a two-qubit state $\rho_{AB}$. In order to obtain some
information about $\rho$, we measure a set $\mathcal{S}$ of physical
observables $\mathcal{S}=\{S_1,S_2,\cdots,S_k\}$. An
informationally-complete set of observables contains $k=15$
linearly independent $S_i$'s. A simple choice of $\mathcal{S}$ is the
set of all two-qubit Pauli matrices other than the identity, i.e.
$\mathcal{S}=\{\sigma_i\otimes\sigma_j\}$ with $i,j=0,1,2,3$, where
$\sigma_0=I, \sigma_1=X, \sigma_2=Y, \sigma_3=Z$ and $(i,j)\ne (0,0)$.
Assume that we can decide universally whether an arbitrary $\rho_{AB}$
is entangled or not, without measuring an
informationally-complete set of observables. That is, there
exists a set $\mathcal{S}$ of at most $k=14$ physical observables such
that, by measuring $\mathcal{S}$, we can tell for sure whether
$\rho_{AB}$ is entangled or not. For our purpose, it suffices to
assume $k=14$.
The set of all two-qubit state $\rho_{AB}$, denoted as $\mathcal{A}$,
is characterized by $15$ real parameters, forming a convex set in
$\mathbb{R}^{15}$. The separable two-qubit states $\mathcal{S}$ form a
convex subset of $\mathcal{A}$. It is well-known that $\mathcal{S}$
has a non-vanishing volume <cit.>. Denote the set of
entangled states by $\mathcal{E}$, i.e.,
$\mathcal{E} = \mathcal{A} \setminus \mathcal{S}$.
The set of measurements $\mathcal{S}$ with $k=14$ can be visualized as
the definition of projections of $\mathcal{A}$ (hence also
$\mathcal{S}$) onto a $14$-dimensional hyperplane. If the measurement
of observables in $\mathcal{S}$ can tell for sure whether $\rho_{AB}$
is entangled or not, the images on the hyperplane of the separable
states $\mathcal{S}$ and the entangled states $\mathcal{E}$ must have
no overlap. We illustrate this geometric idea in
Fig. <ref>.
Geometry of separable and entangled states. The top pink oval represents
the set of all states, denoted by $\mathcal{A}$. Figure (b) shows a set (indicated by blue) that
is an intersection of a generalized cylinder with $\mathcal{A}$ (i.e.
The projection onto the plane that is orthogonal to the boundary
lines of the cylinder separates this set with the rest of the states.
Figure (a) has a set (indicated by blue) inside $\mathcal{A}$ which is not
`cylinder-like'. Hence in fact no projection onto any plane exists
that can separable the set with the rest states. The bottom
ovals are the images of the top sets onto a plane, which clear
show a separation of the images of the blue set from the pink set in Figure (b),
but an overlap of images in Figure (a).
In fact, the only possibility to separate any set from the rest of the
states without FST is that the set is an intersection
of the set of all states (i.e. set $\mathcal{A}$ as in Fig. <ref>)
with a generalized cylinder (i.e. a set of the form $\Omega\times (-\infty,+\infty)$,
where $\Omega$ is a convex set of dimension $14$), In this sense,
we call these sets `cylinder-like', where the corresponding states can be
separated from the rest of states from some $14$ (or lower) dimensional projection.
Hence, to show that entanglement detection
without tomography is impossible, it suffices to prove that
$\mathcal{S}$ is not `cylinder-like' (in $\mathbb{R}^{15}$). To do
this, we show that for any projection onto a $14$-dimensional
hyperplane with normal direction $R$, there always exists
a two-qubit state $\rho$ that is on the boundary of the set
$\mathcal{S}$, such that $\rho+tR$ is entangled for some $t$ (see
Supplementary Information for details). That is, $\rho$ and $\rho+tR$ have the same
image on the $14$-dimensional hyperplane.
This geometric picture leads to a general framework to study the
detection of other interesting quantities for a bipartite quantum
state with single-copy measurements. Indeed, our proof also showed
that the sets of states with positive partial transpose (PPT) is not
`cylinder-like', hence cannot be universally detected by single-copy
measurements without full state tomography. With a similar method, we can
show that the sets of states allowing $k$-symmetric extension are also
not `cylinder like', even for two-qubit system. This reveals a
fundament limit of single-copy measurements, that is, full state
tomography is essentially needed to universally detect many
non-trivial properties of quantum states (e.g., separability, PPT,
$k$-symmetric extendability, see
Supplementary Information for details).
Adaptive measurement.
In case of adaptive protocols,
the observable to be measured in each step can depend on previous
measurement results. This kind of measurement protocol can be
formulated as follows. First an observable $H_1$ is chosen, and
$\tr(H_1\rho)$ is measured. Suppose the measurement result is
$\alpha_1$. Based on $\alpha_1$, observable $H_{2,\alpha_1}$ is
chosen, and $\tr(H_{2,\alpha_1}\rho)$ is measured. Suppose the
measurement result is $\alpha_2$. Based on $\alpha_1,\alpha_2$,
observable $H_{3,\alpha_1,\alpha_2}$ is chosen,
$\tr(H_{3,\alpha_1,\alpha_2}\rho)$ is measured and so on.
The protocol in <cit.> to determine the
concurrence <cit.> of a two-qubit state without
FST falls into the category of adaptive measurements. We
implement this protocol and show that given the
experimental data collected for certain state $\rho_{AB}$, this
protocol in fact leads to FST of $\rho_{AB}$. That is,
this protocol does not
lead to universal detection of entanglement without FST.
First let us briefly introduce the idea of entanglement distillation
via an iteratively filtering procedure <cit.>
depicted in Fig. <ref>a. For an unknown two-qubit state
$\rho_{AB}^{0}$, we measure the local reduced density matrices
$\rho_{A}^{0}=\tr_B(\rho_{AB}^{0})$ and
$\rho_{B}^{0}=\tr_A(\rho_{AB}^{0})$ for both qubits. In case
$\rho_{A}^{0}$ and $\rho_{B}^{0}$ are not fully mixed, we
design the first filter $\mathcal{F}_A^0 = 1/\sqrt{2\rho_{A}^{0}}$
based on the information of $\rho_{A}^{0}$, and evolve $\rho_{AB}^{0}$
to $\rho_{AB}^{1}$ by applying $\mathcal{F}_A^0$. Similarly, the same
procedure is repeated for qubit B. The iterative applications of
filters are kept on going, and at step $k$,
the reduced density
matrices of the qubits will be $\rho_{A}^{k}$ and $\rho_{B}^{k}$.
In case both $\rho_A^{0}$
and $\rho_B^{0}$ are not identity, the iterative procedure described
above leads to a `distillation' of the density matrices
$\rho_{A}^{k}$ and $\rho_{B}^{k}$ and it is guaranteed
that they both converge to identity eventually <cit.>.
All of the reduced density matrices $\rho_{A}^{i}$ and $\rho_{B}^{i}$ ($i=0,1,\ldots,k$) are recorded
during the iterative procedure. At step $k$, when $\rho_{A}^{k}$ and $\rho_{B}^{k}$
are sufficient close to identity,
they can be used to reconstruct
a bound on the value of entanglement in $\rho_{AB}^{0}$
through the optimal witness $W(\rho_{AB}^{0})$ that is
only dependent on $\rho_{A}^{i}$ and $\rho_{B}^{i}$ ($i=0,1,\ldots,k$) (up to
local unitary transformations),
whose value hence tells whether $\rho_{AB}^{0}$
is entangled or not <cit.>.
At the first sight, the above procedure seems feasible to determine
the value of entanglement without FST, since only single-qubit density matrices
$\rho_{A}^{i}$ and $\rho_{B}^{i}$ ($i=0,1,\ldots,k$) are repeatedly measured
and only local unitary transformations are used in constructing the optimal witness.
That is, it seems that the two-qubit correlations in $\rho_{AB}^{0}$ are never
measured, which hence not lead to FST.
However, a detailed look shows it is not the case.
The key observation here is that, `local filters' are in fact
`weak' measurements that do record the correlations in $\rho_{AB}^{0}$.
This is because that the filters cannot be implemented with probability one,
so the correlation in $\rho_{AB}^{0}$
is `encoded' in the information that all the filters are implemented successfully. In other
words, what these local filters and local tomography on each single qubit
does, is in fact an FST of $\rho_{AB}^{0}$.
In order to demonstrate the relationship between
the local filters and FST, we simulate the local filter procedure
by choosing different number of applied filters as depicted in
Fig. <ref>a. It turns out, in many case
$k=4$ (five filters) is enough to uniquely determine
$\rho_{AB}^{0}$ based on the data of
$\rho_{A}^{i}$ and $\rho_{B}^{i}$ ($i=0,1,\ldots,k$). Thus,
the information of $\rho_{A}^{i}$ and $\rho_{B}^{i}$ lead
to an FST of $\rho_{AB}^{0}$.
As an example, we illustrate the simulation with the input state chosen as equation (<ref>) with $\lambda=0.2$, and the result is shown in
Fig. <ref>b.
Initially, we have $15$ real parameters (i.e. degrees of freedom, DOF for short) to determine $\rho_{AB}^{0}$
(ignoring the identity part due to the normalization condition). When more and more
filters are applied, DOF is decreasing eventually since we are acquiring more and more knowledge about the original input state. For example, the initial local reduced density matrices $\rho_{A}^{0}$ and $\rho_{B}^{0}$ before applying any filter can already reduce DOF to $9$; $\rho_{B}^{1}$ after the first filter provides three more constraints so DOF lowers to $6$, and so on. It is found that with $5$ filters, the input
state $\rho_{AB}^{0}$ can be uniquely determined via the collected
information of the reduced density matrices. And this
procedure works similarly for many other two-qubit state $\rho_{AB}^{0}$,
where $5$ filters are found to be enough to reconstruct $\rho_{AB}^{0}$,
as we will show in our experiment results.
Circuit and simulation results of the theoretical protocol. (a) Schematic circuit for
implementing the filter-based entanglement distillation proposal
for an unknown two-qubit state $\rho_{AB}^{0}$.
$\mathcal{F}_{A,B}^i=1/\sqrt{2\rho_{A,B}^{i}}$ ($i\geq 0$) is the
$i$th local filter applied on A or B, where
$\rho_{A}^{i}=\text{tr}_B(\rho_{AB}^{i})$ and
$\rho_{B}^{i}=\text{tr}_A(\rho_{AB}^{i})$ are the local reduced
density matrices of the current two-qubit state $\rho_{AB}^{i}$. The
gray dots mean a single-qubit tomography is implemented at that
place. (b) Simulated variation of concurrence and fidelity as the
increase number $1\leq m \leq 5$ of applied filters. The simulated
state is chosen as equation (<ref>) with $\lambda=0.2$. For
any given $m$, we collected all the available reduced density
matrices at this stage and reconstructed 100 possible input state.
When $m \leq 4$, the reconstructed state is not unique due to the
lack of constraints, so both the concurrence and fidelity have
some distributions. When $m=5$, the input state can be uniquely
determined, and the concurrence and fidelity converges to a single
point. The dashed blue and red curves show the envelopes of the
variations of concurrence and fidelity along with $m$,
Experimental protocol in NMR setup. To experimentally implement the protocol as presented in Fig. <ref>a,
we first discuss how to realize the local filters in NMR system.
Without loss of generality, we can consider a local filter
$\mathcal{F}_A$ applied on qubit A as an example. For any
$\mathcal{F}_A$, it can always be decomposed into the form of
$U_A\Lambda_A V_A$ via singular value decomposition, where $U_A$ and
$V_A$ are single-qubit unitaries and $\Lambda_A$ is a diagonal Kraus
\begin{equation}
\Lambda_A =
\left(
\begin{array}{cc}
1 & 0 \\0 & \sqrt{ 1-\gamma_A}\\
\end{array}
\right).
\end{equation}
$\gamma_A\in [0,1]$ relies on $\mathcal{F}_A$ and indicates the
probability that the excited state $\ket{1}$ decays to the ground
state $\ket{0}$ when a system undergoes $\Lambda_A$. Although
non-unitary, $\Lambda_A$ can be expanded to a two-qubit unitary with
the aid of an ancilla qubit 1. Basically, if a two-qubit unitary can
\begin{equation}
\label{2qubittransform}
\begin{split}
\ket{0}_1\ket{0}_A &\rightarrow \ket{0}_1\ket{0}_A,\\
\ket{0}_1\ket{1}_A &\rightarrow \sqrt{1-\gamma_A}\ket{0}_1\ket{1}_A+\sqrt{\gamma_A}\ket{1}_1\ket{1}_A,
\end{split}
\end{equation}
the quantum channel on the system qubit A would be $\Lambda_A$ by
post-selecting the subspace in which the ancilla qubit 1 is $\ket{0}$.
One possible unitary transformation that satisfies
equation (<ref>) is
\begin{equation}
\mathcal{U}_{1A}=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & \sqrt{1-\gamma_A} & 0 & \sqrt{\gamma_A}\\
0 & 0 & 1 & 0 \\
0 & -\sqrt{\gamma_A} & 0 & \sqrt{1-\gamma_A} \\
\end{array}
\right).\label{U1A}
\end{equation}
The operation $\mathcal{U}_{1A}$ is thus a controlled rotation: when
the system qubit $A$ is $\ket{0}$, the ancilla remains invariant; when
$A$ is $\ket{1}$, the ancilla undergoes a rotation
$R_{-y}(\theta_A)=e^{i\theta_A\sigma_y/2}$ where
$\theta_A=2\text{arccos}\sqrt{1-\gamma_A}$. Therefore, in an
ancilla-assisted system with the ancilla initialized to $\ket{0}$,
the local filter $\mathcal{F}_A$ can be accomplished through a
two-qubit unitary gate $(I\otimes U_A)\mathcal{U}_{1A}(I\otimes V_A)$
followed by post-selecting the subspace in which ancilla is $\ket{0}$.
Molecular structure and Hamiltonian parameters of $^{13}$C-labeled trans-crotonic acid. C$_1$, C$_2$, C$_3$ and C$_4$ are used as four qubits in the experiment, and M, H$_1$ and H$_2$ are decoupled throughout the experiment. In the table, the chemical shifts with respect to the Larmor frequency 176.05MHz and J-coupling constants (in Hz) are listed by the diagonal and off-diagonal numbers, respectively. The relaxation time scales T$_{1}$ and T$_{2}$ (in Seconds) are shown at bottom.
NMR implementation. To implement the aforementioned filter-based entanglement distillation protocol as presented in Fig. <ref>a in NMR, we need a 4-qubit quantum processor consisting of two system qubits A and B, and two ancilla qubits 1 and 2. Our 4-qubit sample is $^{13}$C-labeled trans-crotonic acid dissolved in d6-acetone. The structure of the molecule is shown in Fig. <ref>, where C$_1$ to C$_4$ denote the four qubits. The methyl group M, H$_1$ and H$_2$ were decoupled throughout all experiments. The internal Hamiltonian of this system can be described as
\begin{align}\label{Hamiltonian}
\mathcal{H}_{int}=\sum\limits_{j=1}^4 {\pi \nu _j } \sigma_z^j + \sum\limits_{j < k,=1}^4 {\frac{\pi}{2}} J_{jk} \sigma_z^j \sigma_z^k,
\end{align}
where $\nu_j$ is the chemical shift of the jth spin and $\emph{J}_{jk}$ is the J-coupling strength between spins j and k. We assigned C$_3$ and C$_2$ as system qubits A and B, and C$_4$ and C$_1$ as ancilla qubits 1 and 2 to assist in mimicking the filters, respectively. All experiments were conducted on a Bruker DRX 700MHz spectrometer at room temperature.
NMR sequence to realize the filter-based proposal of entanglement distillation. In particular, this sequence displays how to realize the first two filters $\mathcal{F}_{A}^0$ and $\mathcal{F}_{B}^1$ in terms of NMR pulses. All other sequences can be obtained analogously. A (marked by C$_3$) and B (marked by C$_2$) are system qubits to implement the proposal, while qubit 1 (marked by C$_4$) and 2 (marked by C$_1$) are ancilla qubits to assist in mimicking the filters. First the 4-qubit system is prepared to the PPS by spatial average technique which is shown before the initialization step. Then the system qubits are initialized to $\ket{\phi_{\mathcal{B}}}$, $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$ via three independent experiments illustrated in the lower-right inset, respectively. The part after the initialization step is the sequence for realizing filters. $V_A$, $V_B$, $U_A$, $U_B$, $\theta_1$ and $\theta_2$ all depend on the measured results of reduced density matrices. Refer to the main text for detailed information of the parameters.
Our target input state was chosen as a mixed state involving one Bell-state portion and two product-state portions, with the weight of Bell-state portion tunable. The state is written as
ρ_AB^0 = λ|ϕ_ℬ⟩⟨ϕ_ℬ|+(1-λ)( |ϕ_1⟩⟨ϕ_1| + |ϕ_2⟩⟨ϕ_2|)/2,
\begin{align}
\ket{\phi_{\mathcal{B}}} &= \left ( \ket{00} +\ket{11}\right )/\sqrt{2}, \\ \nonumber
\ket{\phi_{1}} &= \left ( \ket{0} -i\ket{1}\right )\left ( \ket{0} +\ket{1}\right )/2,\\ \nonumber
\ket{\phi_{2}} &= \left ( \ket{0} +\ket{1}\right )\left ( \ket{0} -2i\ket{1}\right )/\sqrt{10}
\end{align}
have concurrences $1$, $0$ and $0$, respectively. The parameter
$\lambda$ in $[0, 1]$ is thus proportional to the value of entanglement of
$\rho_{AB}^{0} $. In experiment, we varied $\lambda$ from 0.2 to 0.7
with step size 0.1 for every point, and implemented the proposal
correspondingly. Considering the two ancilla, the overall input state
for our 4-qubit system is thus
$\ket{0}\bra{0}\otimes\rho_{AB}^{0} \otimes\ket{0}\bra{0}$. As shown
in Fig. <ref>, we prepared a pseudo-pure state from the
thermal equilibrium via spatial average technique <cit.> and then created
three components $\ket{\phi_{\mathcal{B}}}$, $\ket{\phi_{1}}$ and
$\ket{\phi_{2}}$ on the system qubits, respectively (see Methods for detailed descriptions). Subsequently, each component undergoes the whole
filtering and single-qubit readout stage, with the final result
obtained by summarizing over all three experiments.
A two-qubit state tomography was implemented on the system qubits after creating $\rho_{AB}^{0}$. $\rho_0^{e}$ was reconstructed in experiment and its fidelity compared with the expected $\rho_{AB}^{0}$ is over 98.2% for any $\lambda$ (Supplementary Table S1). This two-qubit state tomography is not required in the original proposal <cit.> in which only single-qubit measurements are necessary. However, since we claim that the filter-based proposal has already provided sufficient information to reconstruct the initial two-qubit state $\rho_0^{e}$, we need to compare it with $\rho_{f}^{e}$ which is reconstructed after running the entire proposal. To support our viewpoint, we have to show that $\rho_0^{e}$ and $\rho_{f}^{e}$ are the same up to minor experimental errors. This comparison is the only purpose of doing a two-qubit state tomography here.
Now we show how to realize local filtering operations in NMR. By measuring the local reduced density matrix $\rho_{A}^{0}$ of the input state $\rho_{AB}^{0}$, the first filter in Fig. <ref>a was calculated via $\mathcal{F}_A^0 = 1/\sqrt{2\rho_{A}^{0}}$ and decomposed into $U_A^0\Lambda_A^0 V_A^0$. Since $U_A^0$ and $V_A^0$ are merely local unitaries on qubit A, they can be realized by local radio-frequency (RF) pulses straightforwardly. $\Lambda_A^0$, which can be expanded to a 2-qubit controlled rotation $\mathcal{U}_{1A}$ (see equation (<ref>)) in a larger Hilbert space, was performed by a combination of local RF pulses and J-coupling evolutions<cit.>
where $U(\theta_A/2\pi J_{1A})$ represents the J-coupling evolution
$e^{-i\theta_A\sigma_z^1\sigma_z^A/4}$ between qubit 1 and A, and
$\theta_A=2\text{arccos}\sqrt{1-\gamma_A}$ depends on $\Lambda_A^0$.
After this filter, the system evolved to $\rho_{AB}^{1}$ and a
single-qubit tomography on qubit B was implemented, as shown by the
gray dots in Fig. <ref>a. The same procedure was repeated for
qubit B to realize the second filter $\mathcal{F}_B^1 =
1/\sqrt{2\rho_{B}^{1}}$. In experiment, these two filters
$\mathcal{F}_A^0$ and $\mathcal{F}_B^1$ were carried out
simultaneously using the partial decoupling technique as shown in Fig.
<ref>, with additional Z rotations in the tail to compensate
the unwanted phases induced by the chemical shift evolutions. In Fig.
<ref>, $\theta_1$ and $\theta_2$ pulses are used to realize
$R_{-y}^1(\theta_A/2)$ and $R_{-y}^2(\theta_B/2)$, respectively, and
the free evolution time $\tau_1$ and $\tau_2$ are defined as
τ_1 = θ_1/4πJ_1A + θ_2/4πJ_B2
τ_2 = θ_1/2πJ_1A - θ_2/2πJ_B2.
Here we have assumed that $\tau_2>0$ ($\theta_1/2\pi J_{1A} > \theta_2/2\pi J_{B2}$). When $\tau_2<0$, the circuit just needs to be modified slightly by adjusting the positions of refocusing $\pi$ pulses. All the other filters have analogical structures with the one shown in Fig. <ref>, and they were always carried out on qubit A and B simultaneously since they commute.
Every time after performing one local filter, we implemented a single-qubit tomography on the other qubit rather than the working qubit on which the filter was applied. The reason is that the working qubit has evolved to identity due to the properties of the filter. The tomographic result was used to design the next filter on the other qubit. In principle, before applying any filters, it is necessary to reset the two ancilla qubits to $\ket{00}$. As it is difficult to refresh the spins in NMR, an alternative way was adopted in our experiments. For example, to realize $\mathcal{F}_A^2$, we packed it together with $\mathcal{F}_A^0$ and generated a new operator. It can be regarded as a 2-in-1 filter and implemented in the same way. Hence, we avoided the reset operations throughout the experiments and for any individual experiment we just started from the original two-qubit state $\rho_{AB}^{0}$. This feedback-based filtering operations continued to be executed till five filters accomplished and seven 1-qubit tomographies carried out, as shown in Fig. <ref>a.
From the above discussions, we have shown that the NMR experiments only contain free J-coupling evolutions and single-qubit unitaries. See the circuit in Fig. <ref>. For the J-coupling evolutions, we drove the system to undergo the free Hamiltonian in equation (<ref>) for some time. For local unitaries, we utilized GRadient Ascent Pulse Engineering (GRAPE) techniques <cit.> to optimize them. The GRAPE approach provided 1 ms pulse width and over 99.8% fidelity for every local unitary, and furthermore all pulses were rectified via a feedback-control setup in NMR spectrometer to minimize the discrepancies between the ideal and implemented pulses <cit.>.
Experimental results and error analysis. We prepared six input states by varying $\lambda$ from 0.2 to 0.7 with 0.1 step size in the form of equation (<ref>). After the preparations, we performed two-qubit full state tomography on each state, and reconstructed them as $\rho_0^e$ where the superscript $e$ means experiment. The fidelity between the theoretical state $\rho_{AB}^{0}$ and measured state $\rho_0^e$ is over 98.2% for each of the six input states. The infidelity can be attributed to the imperfections of PPS, GRAPE pulses and minor decoherence effect. Nevertheless, this infidelity is merely used to evaluate the precision of our input state preparation. For the latter experiments, we only compared the experimental results with $\rho_0^e$, as $\rho_0^e$ was the actual state from which we started the filter-based experiment.
After initial state preparation and each filter, we obtained the reduced density matrix of qubit A and/or B by single-qubit tomography in the subspace where the ancilla qubits are $\ket{00}$ (see Methods). Refer to Fig. <ref>a to see the seven gray dots where single-qubit tomography occurred. The average fidelity between the measured single-qubit state and the expected state computed by $\rho_0^e$ is about 99.5% (Supplementary Table S1), which demonstrates that our filtering operations and single-qubit tomographies are accurate.
Fidelities between $\rho_0^e$ and $\rho_f^e$ for different $\lambda$'s. $\rho_0^e$ is obtained from two-qubit state tomography right after the creation of the input state $\rho_{AB}^{0}$, and $\rho_f^e$ from the maximum likelihood reproduction of $\rho_{AB}^{0}$ based on the own seven single-qubit states. The error bar comes from the fitting uncertainty when extracting the NMR spectra into quantum states. All fidelities are over 92.0%, which means the initial two-qubit state is able to be well-reconstructed merely by the seven single-qubit states.
With the seven single-qubit states in hand, we could reproduce the initially prepared two-qubit state $\rho_0^e$. The maximum likelihood method was adopted here and $\rho_f^e$ was found to be closest to the experimental raw data. Not surprisingly, $\rho_f^e$ is very similar to $\rho_0^e$, and the fidelity between them for every $\lambda$ is over 92.0% as illustrated in Fig. <ref>. Moreover, the real parts of the density matrices $\rho_0^e$ and $\rho_f^e$ are shown in Fig. <ref>. The experimental results clearly reveal that the information of the seven single-qubit states collected during the filter-based entanglement distillation procedure already enables the reproduction of the initial two-qubit state. In other words, this filter-based proposal to universally detect and distill entanglement is equivalent compared to doing a two-qubit state tomography.
Afterwards, we computed the concurrence for each case with different input two-qubit state. Concurrence is an entanglement monotone defined for a mixed state $\rho$ of two qubits
𝒞(ρ) = max( 0, λ_1-λ_2-λ_3-λ_4),
where $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are the eigenvalues of
R = √(√(ρ)( σ_y⊗σ_y ) ρ^* ( σ_y⊗σ_y ) √(ρ))
in decreasing order. Apparently, the concurrence is proportional to $\lambda$ since $\lambda$ is the weight of Bell-state which is the only term contributing to entanglement. In Fig. <ref>, the brown curve displays the value of concurrence as a monotonically increasing function of Bell-state weight $\lambda$. The blue squares represent the concurrence of $\rho_0^e$, the state obtained from two-qubit state tomography on the experimentally prepared state. Recall that the preparation fidelity is always over 98.2% so the blue squares do not deviate much from the brown curve. The red circles represent the concurrence of $\rho_f^e$, which ideally should be the same as blue squares if there are no experimental errors. However, in experiment we have inevitable errors from many factors such as the imprecision of the single-qubit readout stage, the imperfect application of filters and the relaxation, and we need to take them into account.
For convenience, we assume the errors originate from three primary aspects and they are additive. One error is caused by the imprecision of the single-qubit tomography procedure. As we used a least-square fitting algorithm to analyze the outcome spectra and converted the data into quantum states, the fitting induced about 3.00% uncertainty to the single-qubit readout result. The second is the error from applying imperfect filters in experiment. It mainly comes from the errors of accumulating GRAPE pulses, which is about 1.59% for each filter operation. The third error, to the lesser extent, is about 1.20% caused by decoherence. Therefore, in total we estimated at most 5.79% error might occur in the entire process. We dealt with it as an artificial noise and embedded it into the theoretical input state $\rho_{AB}^{0}$. In simulation, we first discretized $\lambda$ to 200 values from $\lambda=0.1$ to $\lambda=0.8$. For a given $\lambda$, 2500 states were randomly sampled deviated from $\rho_{AB}^{0}$ within 5.79% noise range. For every sampled state, the concurrence was calculated and projected onto one point in Fig. <ref>. Hence, a colored band-region was generated considering the density of points. All of our experimental results have fallen into this region, which is consistent with the simulation model.
Concurrence of $\rho_{AB}^{0}$, $\rho_0^e$, and $\rho_f^e$ as a function of the Bell-state weight $\lambda$. The brown curve shows the concurrence computed by the theoretical state $\rho_{AB}^{0}$, and exhibits the value of concurrence as a monotonically increasing function of $\lambda$. The blue squares represent the concurrence of $\rho_0^e$, the state obtained from two-qubit state tomography right after the input state preparation. Generally we can roughly assume this state is the truly prepared state, and the following filtering operations are always applied this state as long as we neglect the measurement error of reconstructing $\rho_0^e$. The red circles represent the concurrence of $\rho_f^e$, the state reproduced from the seven single-qubit states. Ideally $\rho_0^e$ and $\rho_f^e$ should be the same if there are no experimental errors. The colored band-region accounts for an artificial noise of the strength 5.79%, which is roughly estimated from the fitting error 3.00%, GRAPE imperfection error 1.59%, and decoherence error 1.20%. We added this noise on the theoretical state $\rho_{AB}^{0}$, and randomly sampled 2500 states within the noise range for every $\lambda$ (200 values in [0.1, 0.8]). The colored band-region is thus plotted based on the density of projected points out of 2500.
Density matrices between $\rho_0^e$ and $\rho_f^e$ for different $\lambda$'s. Only the real parts are displayed. The upper row shows the density matrices of $\rho_0^e$, which are obtained directly after the input state preparation via two-qubit state tomography. The lower row shows the density matrices of $\rho_f^e$, which are reconstructed through the single-qubit information after implementing the entire circuit in Fig. <ref>a. For any $\lambda$, the fidelity between $\rho_0^e$ and $\rho_f^e$ is always above 92%.
§ DISCUSSION
We proved a no-go theorem that there is no way to detect entanglement for an arbitrary bipartite state $\rho_{AB}$ without FST, if only single-copy non-adaptive measurements are allowed. Our observation is due to a nice geometric picture: unless the shape of the
separable states is `cylinder-like', it is not possible to find a projection of the state space to a lower dimensional hyperplane with non-overlapping image for the set of separable states and entangled states. Our method provides a general framework to study the detection of other interesting quantities for a bipartite quantum state, such as positive partial transpose and $k$-symmetric extendibility.
We also have investigated the case of adaptive measurements. It is proposed in <cit.> that
the entanglement measure concurrence for two-qubit states can be determined without FST, via only single-copy measurements.
To implement this protocol, we developed an ancilla-assisted approach to realize the filters. Practically, our technique can be extend to other quantum systems other than optics to implement an amplitude damping channel, which is of great importance in quantum information. By
implementing this protocol, we show that given the experimental data collected for certain state $\rho_{AB}$, this protocol in fact leads to FST of $\rho_{AB}$.
Therefore, this protocol does not lead to universal detection of entanglement of $\rho_{AB}$ without FST.
Our study thus reveals a fundamental relationship between entanglement detection and quantum state tomography. That is, universal detection of entanglement without FST is impossible with only single-copy measurements.
A natural question is what if joint measurements on $r$ copies of the state $\rho_{AB}$ (i.e. $\rho_{AB}^{\otimes r}$) for $r>1$ are allowed. In this case, one indeed can detect entanglement universally for any $\rho_{AB}$ without reconstructing the state, and one example for determining the concurrence of a two-qubit $\rho_{AB}$ is given in <cit.>. However, the protocol of <cit.> involves joint measurements on $4$ copies of $\rho_{AB}$ (i.e. $\rho_{AB}^{\otimes 4}$), which makes the protocol hard to be implemented in practice. It will be interesting to find a smaller $r$ such that joint measurements on c are enough to universally detect the entanglement in $\rho_{AB}$ without full state tomography.
In fact, there are cases that this is possible even for $r=2$. For instance we have found such a scheme that detects the entanglement of an arbitrary two-qubit state $\rho_{AB}$ without FST, if we allow joint measurements on $2$-copies. The idea is that $\rho_{AB}$ is entangled if and
only if <cit.>
\text{Det}(\rho_{AB}^{T_A})<0,
where $\rho_{AB}^{T_A}$ is the partial transpose of $\rho_{AB}$ on system $A$. So
we only need to design a scheme with measurements on $\rho_{AB}^{\otimes r}$ which
can give the value of $\text{Det}(\rho_{AB}^{T_A})$. This can indeed be done
without FST (see supplemental materials for details).
Furthermore, if only single-copy measurements are allowed, one cannot determine
the value of $\text{Det}(\rho_{AB}^{T_A})$, even with adaptive measurements. Assume such adaptive measurements exist. Now, we let the input state is maximal mixed state $I/4$, after the measurement, one can compute the determinant. Notice that there exists at least one nonzero traceless $R$ not measured, which means that this measurements can not distinguish between $I/4$ and $I/4+tR$. Therefore, $\text{Det}(I/4+tR^{\Gamma})=\text{Det}(I/4)$ for sufficient small $t$. This then leads to $R=0$.
This strongly supports our no-go results, which indicates that even with adaptive measurements,
universal detection of entanglement with single-copy measurements is impossible without FST.
§ METHODS
Initialization in NMR. We first create the PPS from the thermal equilibrium state, which is a highly mixed state and not yet ready for quantum computation tasks. Since our sample consisting of four $^{13}$C's is a homonuclear system, we simply set the gyromagnetic ratio of $^{13}$C to 1 and write the thermal equilibrium state as
\begin{align}\label{thermal}
\rho_{\text{thermal}}=\frac{{I}}{2^N}+\epsilon\sum _{i=1}^N \sigma_z^i,
\end{align}
where $N=4$ is the number of qubits, ${I}$ is the $2^N\times2^N$ unity matrix, and $\epsilon\approx 10^{-5}$ represents the polarization at room temperature. This initialization step was realized by the spatial average technique <cit.>, and the related pulse sequence is depicted in Fig. <ref>. In particular, the gradient pulses represented by Gz crush all coherence in the instantaneous state. The final state after the entire PPS preparation sequence is
\begin{align}\label{pps}
\rho_{0000}=\frac{1-\epsilon}{16}{I}+\epsilon\ket{0000}\bra{0000}.
\end{align}
It is worthy stressing that the large identity does not evolve under any unitary propagator, and it cannot be observed in NMR. Thus we only need to focus on the deviation part $\ket{0000}$ as the entire system behaves exactly the same as it does.
Our aim input state is $\rho_{AB}^0$ in equation (<ref>). This mixed state consists of three components: $\ket{\phi_{\mathcal{B}}}$, $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$ with a weight for each. Typically we repeated every experiment by three times, and created one component in each round as the input. The sequences to prepare all three components are shown in the lower-right inset of Fig. <ref>, with all gates applied only on system qubits A and B. (i) For $\ket{\phi_{\mathcal{B}}}$, we applied a Hadamard gate on qubit A, and then a controlled-NOT between A and B; (ii) for $\ket{\phi_{1}}$, we applied $R_x^A(\pi/2)$ and $R_y^B(\pi/2)$; (iii) for $\ket{\phi_{2}}$, we applied $R_y^A(\pi/2)$ and $R_x^B(0.7\pi)$. Subsequently, each component undergoes the whole filtering and measurement procedure respectively, with the final result obtained by summarizing over all three experiments.
Single-qubit tomography after each filter. The entanglement distillation procedure described in Ref. <cit.> involves iterative local filter operations, meaning that every filter depends on the single qubit measurement result before. In experiment we performed single-qubit tomography on system qubit C$_2$ or C$_3$ correspondingly. It requires the measurement of the expectation values of $\sigma_x$, $\sigma_y$ and $\sigma_z$, respectively. In our 4-qubit system, this single-qubit tomography (assuming the measurement of C$_3$) is equivalent to measuring $\ket{0}\bra{0}\otimes \sigma_{x,y,z}\otimes I \otimes \ket{0}\bra{0}$, since we only need to focus in the subspace where the ancilla qubits are $\ket{00}$. To get the expectation values of the observables, a spectrum fitting procedure was utilized to extract these results from NMR spectra. Supplementary Table S1 summarizes all of the single-qubit state fidelities and two-qubit state fidelities for every $\lambda$, and as an example Supplementary Fig. S1 shows the NMR spectra after each filter to measure <$\sigma_x$> and <$\sigma_y$> of the current single qubit (C$_2$ or C$_3$) when $\lambda =0.5$.
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Notes. After finishing this paper, we noticed a related recent work <cit.>, where a similar idea
of showing the set of separable states is not `cylinder-like' is developed.
Author Contributions. D.L. and T.X. designed and carried out the NMR experiments and simulations; N.Y., Z.J., J.C. and B.Z. made the theoretical proposal and contributed to the analysis of results. X.P. and R.L. supervised the experiment. All authors contributed to the writing of the paper and discussed the experimental procedures and results.
Acknowledgments. We thank Y. Zhang for bringing the recent work <cit.> into our attention. We thank X. Ma for insightful discussions, and are grateful to the following funding sources: NSERC (D.L., N.Y., J.B., B.Z. and R.L.); Industry Canada (R.L.); CIFAR (B.Z. and R.L.); National Natural Science Foundation of China under Grants No. 11175094 and No. 91221205 (T.X. and G.L.), No. 11425523 and No. 11375167 (X.P.); National Basic Research Program of China under Grant No. 2015CB921002 (T.X. and G.L.).
Supplementary information: Tomography is necessary for universal entanglement detection
with single-copy observables
§ A GENERAL METHOD TO SHOW THAT THE SET OF SEPARABLE STATES IS
NOT `CYLINDER-LIKE'.
We use
two-qubit states as an example, however our method generalized naturally to
the case of any bipartite systems (see supplemental materials for details).
Without loss of generality, we assume there is a set of $S_i$, where $1\leq i\leq 14$, such that there is a function
$g(\tr(S_1\rho_{AB}),\tr(S_2\rho_{AB}),\cdots,\tr(S_{14}\rho_{AB}))=1$ for entangled $\rho_{AB}$, and $0$ otherwise. One can hence view $g$ as an analogue of an entanglement measure which is nonzero if and only if $\rho_{AB}$ is entangled.
Then one can find another observable $R$ such that $\tr R=0$ and $\tr(R^{\dag}S_i)=0$ for any $0\leq i\leq 14$.
Our key observation is that for any non zero traceless $R$ there exists some $\rho_{AB}$ and real $t$ such that $\rho_{AB}+tR$ is non-negative (hence is a quantum state), and $\rho_{AB}$ is separable, $\rho_{AB}+tR$ is entangled. That is, $g(\rho_{AB})$ cannot exist.
To show this, we consider some state on the boundary of separable states, $i.e.$, the two-qubit isotropic states $\rho^{iso}(\alpha)=(1-\alpha)I/4+\alpha\op{\Phi}{\Phi}$ with $\ket{\Phi}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ being the Bell state. It is known that the isotropic state $\rho^{iso}(\alpha)$ is separable if and only if $\alpha\geq 1/3$ <cit.>.
Now we let $\rho_{AB}=\rho^{iso}(\frac{1}{3})=\frac{1}{6}I+\frac{1}{3}\op{\Phi}{\Phi}$,
so $\rho_{AB}$ is separable.
For any $R$, for sufficient small $t$, $\rho_{AB}+tR$ is non-negative since $\rho_{AB}$ is positive(full rank).
Choose $t>0$ such that $\rho_{AB}+tR$ and $\rho_{AB}-tR$ are both non-negative. Therefore, $\rho_{AB}+tR$ and $\rho_{AB}-tR$ are separable.
Notice that $\tr(\sigma\op{\Phi}{\Phi})\leq 1/2$ holds for any separable state $\sigma$ <cit.>, we have
\begin{eqnarray*}
1/2\geq \tr[(\rho_{AB}+tR)\op{\Phi}{\Phi}]\Longrightarrow 0\geq \tr[R\op{\Phi}{\Phi}],\\
1/2\geq \tr[(\rho_{AB}-tR)\op{\Phi}{\Phi}]\Longrightarrow 0\leq \tr[R\op{\Phi}{\Phi}].
\end{eqnarray*}
Thus, $\tr[R\op{\Psi}{\Psi}]=0$ holds for arbitrary maximally entangled state $\ket{\Psi}=(U\otimes I)\ket{\Phi}$. In other words,
\begin{eqnarray*}
R=I\otimes M+N\otimes I,
\end{eqnarray*}
with $\tr(M)=\tr(N)=0$.
Notice that for any non-singular matrix $S_A$ such that $S_ANS_A^{\dag}$ is traceless, $R'=(S_A\otimes I)R(S_A\otimes I)^{\dag}$ also satisfies the property that $\rho_{AB}+tR'$ is separable if and only if $\rho_{AB}$ is separable and $\rho_{AB}+tR'$ is non-negative. According to the previous arguments, we know that $R'$ can be written as $I\otimes M'+N'\otimes I$. Directly, one can conclude that $M=0$. By choosing $S_B$ such that $S_BNS_B^{\dag}$ being traceless, we can obtain that $N=0$. Therefore, $R$ must be $0$. In other words, tomography is required for detecting two-qubit entanglement by using the one copy non-adaptive measurement.
§ $K$-SYMMETRIC EXTENSION
In this section, we will show that the sets of states allowing $k$-symmetric extension are not
‘cylinder like’ for $k\geq 2$, even for two-qubit system, where a state $\rho_{AB}$ is called $k$-symmetric extendable if and only if there exists $\sigma_{AB_1B_2\cdots B_k}$ such that $\sigma_{AB_i}=\rho_{AB}$ for all $1\leq i\leq k$.
We first recall that: A two-qudit Werner state is a state invariant
under the $U\otimes U$ operator for all unitary $U$ and
has the following form
\begin{equation*}
\rho_W(\psi^{-}) = \frac{1+\psi^{-}}{2} \rho^{+} +
\frac{1-\psi^{-}}{2} \rho^{-},
\end{equation*}
where $\psi^{-} \in [-1,1]$ is the parameter, $\rho^{+}$ and
$\rho^{-}$ are the states proportional to the projection of the
symmetric subspace and anti-symmetric subspace
$\rho_W(\psi^{-})$ is $k$-symmetric extendable iff
$\psi^{-} \ge -(d-1)/k$ proved in <cit.>.
Assume that the set of states allowing $k$-symmetric extension is cylinder like. In other words there exists some traceless Hermitian operator $R$ such that $\rho+tR$ is $k$-symmetric extendable if $\rho$ is $k$-symmetric extendable and $\rho+tR$ is positive semidefinite.
Choose $\rho_0=\rho_W(-(d-1)/k)$, then $\rho_0>0$ and for any $R$, there exists small $t$ such that $\rho_0+tR$ and $\rho_0-tR$ are all positive semidefinite. Therefore, they are both $k$-symmetric extendable. As a direct consequence, we have Werner states $\sigma_1$ and $\sigma_2$ are both $k$-symmetric extendable,
\begin{eqnarray*}
\sigma_1&=&\int_U (U\otimes U)(\rho_0+tR)(U\otimes U)^{\dag} dU\\
&=& \rho_0+t \tr(RP^{+})(\rho^{+}-\rho^{-}),\\
\sigma_2&=&\int_U (U\otimes U)(\rho_0-tR)(U\otimes U)^{\dag} dU,\\
&=& \rho_0-t \tr(RP^{+})(\rho^{+}-\rho^{-}),\\
\end{eqnarray*}
where $P^{+}$ and $P^{-}$ are the projection onto the symmetric subspace and anti-symmetric subspace
By using condition of <cit.>, one can conclude that
\begin{eqnarray*}
\frac{1-(d-1)/k}{2}+t \tr(RP^{+})\geq \frac{1-(d-1)/k}{2},\\
\frac{1-(d-1)/k}{2}-t \tr(RP^{+})\geq \frac{1-(d-1)/k}{2}.
\end{eqnarray*}
\begin{equation*}
\tr(RP^{+})=\tr(R\rho^{+})=\tr(RP^{-})=0.
\end{equation*}
Similar technique can be applied for $(V\otimes I)\rho (V\otimes I)^{\dag}$ with unitary $V$. That leads us to
\begin{equation*}
\tr(R(V\otimes I)\rho^{+} (V\otimes I)^{\dag})=0.
\end{equation*}
That is, $R^{\Gamma}$, the partial transpose of $R$, is orthogonal to all maximally entangled state $(V\otimes I)\ket{\Phi}$,
\begin{eqnarray*}
&&\tr(R^{\Gamma}(V\otimes I)\op{\Phi}{\Phi}(V\otimes I)^{\dag})\\
&=&\tr(R(V\otimes I)(P^{+}-P^{-})(V\otimes I)^{\dag})\\
\end{eqnarray*}
Now we write $R$ as follows,
\begin{eqnarray*}
R^{\Gamma}_{AB}=I\otimes M+N\otimes I+X_{AB},
\end{eqnarray*}
with $\tr(M)=\tr(N)=0$, and $X_{A}=X_{B}=0$. This can be done by simply choosing $M=\frac{(R^{\Gamma}_{AB})_B}{d}$, $N=\frac{(R^{\Gamma}_{AB})_A}{d}$ and $X_{AB}=R^{\Gamma}_{AB}-I\otimes \frac{(R^{\Gamma}_{AB})_B}{d}-\frac{(R^{\Gamma}_{AB})_A}{d}\otimes I$.
For sufficient small $s$, $I_{AB}/d+sX_{AB}$ is a choi matrix of some unital quantum channel.
According to Theorem 1 of <cit.>, we know that $I_{AB}/d+sX_{AB}$ is a linear combination of the density matrix of maximally entangled states, so is $X_{AB}$. Thus,
\begin{eqnarray*}
\end{eqnarray*}
where we use the fact that $I\otimes M+N\otimes I$ is orthogonal to all maximally entangled states.
Thus, $X_{AB}=0$ and $R$ can be written as
\begin{eqnarray*}
R^{\Gamma}_{AB}=I\otimes M+N\otimes I.
\end{eqnarray*}
Now, notice that for any non-singular matrix $S_A$ such that $S_ANS_A^{\dag}$ is traceless, $R'=(S_A\otimes I)R(S_A\otimes I)^{\dag}$ also satisfies that $\rho_{AB}+tR'$ is $k$-symmetric extendable if and only if $\rho_{AB}$ is $k$-symmetric extendable and $\rho_{AB}+tR'$ is non-negative. Then we know that $R'$ can be written as $I\otimes M'+N'\otimes I$, too. Directly, one can conclude that $M=0$.
Thus, we only need to deal with
\begin{eqnarray*}
R=N^{\Gamma}\otimes I.
\end{eqnarray*}
In the following, we deal with the two-qubit case. Note that there exists local unitary which transforms $R$ into diagonal version $R=Z\otimes I=\diag\{1,1,-1,-1\}$.
We only construct some state $\rho$ which is not 2-symmetric extendable <cit.> and $\rho+R$ is separable. Then such $\rho+R$ is $k$-symmetric extendable for all $k$ while $\rho$ is not 2-symmetric extendable.
\begin{eqnarray}
\rho:&=&\left(\begin{array}{cccc}
x & 0 & 0 & \sqrt{xw}\\
0 & y & \sqrt{yz} & 0\\
0 & \sqrt{yz} & z & 0\\
\sqrt{xw} & 0 & 0 & w
\end{array}\right),\\
\rho+R:&=&\left(\begin{array}{cccc}
x+1 & 0 & 0 & \sqrt{xw}\\
0 & y+1 & \sqrt{yz} & 0\\
0 & \sqrt{yz} & z-1 & 0\\
\sqrt{xw} & 0 & 0 & w-1
\end{array}\right)
\end{eqnarray}
By using the condition of <cit.>, $\rho$ is not 2-symmetric extendable iff
\begin{eqnarray*}
(x+z)^2+(y+w)^2&<& x^2+y^2+z^2+w^2+2xw+2yz,\\
\Leftrightarrow (x-y)(w-z)&>&0.
\end{eqnarray*}
$\rho+R$ is separable iff
\begin{eqnarray*}
&&(x+1)(w-1)\geq xw,yz,\\
&& (y+1)(z-1)\geq xw,yz.
\end{eqnarray*}
It is direct to see that one can choose some $w>z>x>y>0$ such that
\begin{eqnarray*}
&&(x+1)(w-1)\geq xw\geq yz,\\
&& (y+1)(z-1)\geq xw \geq yz.
\end{eqnarray*}
Actually, we can choose $\epsilon$ to be sufficient small, and
\begin{eqnarray*}
\end{eqnarray*}
Therefore, for all nonzero $R$, one can always find $\rho$ and $\rho+R$ such that $\rho+R$ is separable and $\rho$ is not 2-symmetric extendable. This shows that the set of states allowing $k$-symmetric extension is also not `cylinder-like', even for two-qubit system.
§ THE CASE OF JOINT MEASUREMENTS
We discuss the case of joint measurements on $r$ copies of $\rho_{AB}$
(i.e. $\rho_{AB}^{\otimes r}$) with $r>1$. We take the two-qubit case as
an example. In this case, it is known that $\rho_{AB}$ is entangled if and
only if <cit.>
\begin{equation}
\text{Det}(\rho_{AB}^{T_A})<0,
\end{equation}
where $\rho_{AB}^{T_A}$ is the partial transpose of $\rho_{AB}$ on system $A$.
Notice that the determinant of $\rho_{AB}^{T_A}$ is a polynomial of degree $4$
in terms of the matrix entries of $\rho_{AB}$, so it can be detected by measuring
only a single observable on $4$ copies of $\rho_{AB}$ (i.e. $\rho_{AB}^{\otimes 4}$) <cit.>.
For the case of joint measurements with $r=2$, however, one cannot measure
only a single observable to get $\text{Det}(\rho_{AB}^{T_A})$. Nevertheless, we
show that the value of $\text{Det}(\rho_{AB}^{T_A})$ can be get without full state tomography with only a single joint measurement on $\rho_{AB}^{\otimes 2}$.
To see how this works, we rewrite $\rho_{AB}^{T_A}$ as
\[
\rho_{AB}^{T_A}=\begin{pmatrix}
R & S\\
S^{\dag} & T
\end{pmatrix},
\]
with $R$ is a $3\times 3$ matrix, $S$ is $3\times 1$,
and $T$ is $1\times 1$.
We can first determine
$R$ by single-copy measurements on $\rho_{AB}^{T_A}$,
where $9$ independent observables need to be measured.
After that, $T$ is known by the normalization condition $\tr\rho_{AB}^{T_A}=1$
This does not lead to a full state tomography on $\rho_{AB}$,
since $S$ is undetermined, with $6$ free real parameters.
However, after knowing $R$ and $T$, $\text{Det}(\rho_{AB}^{T_A})$
is a polynomial of degree $2$
in terms of the matrix entries of $\rho_{AB}$, so it can be detected by measuring
only a single observable on $2$ copies of $\rho_{AB}$ (i.e. $\rho_{AB}^{\otimes 2}$).
Together with the measurements on $R$ and $S$, we have total $10$ measurement
outcomes that determine universally whether $\rho_{AB}$ is entangled or not. And
this does not lead to full state tomography of $\rho_{AB}$, since $\rho_{AB}$ needs
$15$ real parameters to determine.
In this way, we determine whether $\rho_{AB}$ is entangled or not without full state tomography, by measuring a single observable on $\rho_{AB}^{\otimes 2}$ together
with single copy measurements. This indicates that our no-go results for single copy measurements fail in the case if joint measurements are allowed, even for $r=2$.
§ EXPERIMENTAL RESULTS OF SINGLE-QUBIT TOMOGRAPHY AFTER EACH FILTER
Table <ref> summarizes all of the single-qubit state fidelities and two-qubit state fidelities for every $\lambda$, and as an example Fig. <ref> shows the NMR spectra after each filter to measure <$\sigma_x$> and <$\sigma_y$> of the current single qubit (C$_2$ or C$_3$) when $\lambda =0.5$.
Fidelities of the experimental results compared with the theoretical ones for every $\lambda$. $F(\rho_{AB}^0, \rho_0^e)$ is the fidelity between the theoretical 2-qubit state $\rho_{AB}^0$ and $\rho_0^e$ which is the truly prepared 2-qubit state. Meanwhile, the fidelities of the five single-qubit states after each filter are also shown.
NMR spectra of measuring <$\sigma_x$> and <$\sigma_y$> of single qubit after each filter for $\lambda =0.5$. (a-e) are produced after filter 1-5, respectively. (a, c, e) show spectra of C$_2$, and (b, d) show C$_3$. The red curve is the simulation result assuming the input state is $\ket{0}\bra{0}\otimes \rho_0^e \otimes \ket{0}\bra{0}$, and the blue curve is the experimental result which can be used to extract <$\sigma_x$> and <$\sigma_y$> of the current qubit. The black curve shows the fitting spectrum to obtain <$\sigma_x$> and <$\sigma_y$>. To measure <$\sigma_z$>, we rotated it to $\sigma_x$ with a $\pi/2$ pulse around $y$-axis and then measured. It can be seen that the fitting matches extremely well with the experimental result, which means our readout values are very accurate.
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|
1511.00478
|
On the number of representations of $n$ as a
linear combination of four triangular
numbers II
Min Wang$^1$ and Zhi-Hong Sun$^2$
$\ ^1$School of Mathematical Sciences, Soochow
Suzhou, Jiangsu 215006,
P.R. China
Email: [email protected] $\
^2$School of Mathematical Sciences, Huaiyin Normal University,
Huaian, Jiangsu 223001, P.R. China Email:
[email protected] Homepage:
Let $\Bbb Z$ and $\Bbb N$ be the set of integers
and the set of positive integers, respectively. For
$a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of
representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and
let $t(a,b,c,d;n)$ be the number of
representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2
+dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper
we reveal the connections
between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in\Bbb N$
and $2\nmid a$. We show that
for $(a,b,c,d)=
(a,a,2a,8m),\ (a,3a,8k+2,8m+6),\ (a,3a,8m+4,8m+4)\ (n\equiv
m+\frac{a-1}2 \pmod 2)$ and $(a,3a,16k+4,16m+4)\ (n\equiv
\frac{a-1}2\pmod 2)$. We also obtain explicit formulas for
$t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,1,2,8),\
(1,1,2,16),(1,2,3,6),\ (1,3,4,12),\ (1,1,$ $3,4),\ (1,1,5,5),\
(1,5,5,5),\ (1,3,3,12),\ (1,1,1,12),\ (1,1,3,12)$ and $(1,3,3,4)$.
Keywords: representation; triangular number
Mathematics Subject Classification 2010: Primary 11D85,
Secondary 11E25
The second author
is supported by the National Natural Science Foundation of China
(grant No. 11371163).
§ 1. INTRODUCTION
Let $\Bbb Z$ and $\Bbb N$ be the set of integers
and the set of positive integers, respectively.
For $n \in \Bbb N$ let
$$\sigma(n)=\sum_{d \mid n,d\in\Bbb N}d.$$
For convenience
we define $\sigma(n)=0$ for $n\notin \Bbb N$.
Let $\Bbb Z^4=\Bbb Z\times \Bbb Z\times \Bbb
Z\times \Bbb Z$. For $a,b,c,d\in\Bbb N$ and $n\in\Bbb N \cup \{0\}$
$$N(a,b,c,d;n)=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ n=ax^2+by^2+cz^2+dw^2
\}\big|$$
$$t(a,b,c,d;n)=\Big|\Big\{(x,y,z,w)\in \Bbb Z^4\ |\ n\
=a\f{x(x-1)}2+ b\f{y(y-1)}2+c\f{z(z-1)}2+d\f{w(w-1)}2\Big\}\Big|.$$
The numbers $\f{x(x-1)}2\ (x\in\Bbb Z)$ are called triangular
In 1828 Jacobi showed that
$$N(1,1,1,1;n)=8\sum_{d\mid n,4\nmid d}d.\tag 1.1$$
In 1847 Eisenstein (see [D]) gave formulas for $N(1,1,1,3;n)$ and
$N(1,1,1,5;n)$. From 1859 to 1866 Liouville made about 90
conjectures on $N(a,b,c,d;n)$ in a series of papers. Most
conjectures of Liouville have been proved. See [A1, A2, AALW1-AALW5,
AAW], Cooper's survey paper [C], Dickson's historical comments [D]
and Williams' book [W2].
$$t'(a,b,c,d;n)=\Big|\Big\{(x,y,z,w)\in \Bbb N^4\ |\ n=a\f{x(x-1)}2+
As $\f12
x(x-1)=\f12(-x+1)(-x)$ we have
In [L] Legendre stated that
$$t'(1,1,1,1;n)=\sigma(2n+1).\tag 1.2$$
In 2003, Williams [W1] showed that
$$t'(1,1,2,2;n)=\f 14\sum_{d\mid 4n+3}\big(d-(-1)^{\f{d-1}2}\big).$$
For $a,b,c,d\in\Bbb N$ with $5\le a+b+c+d\le 8$ let
where $i_j$ is the number of elements in $\{a,b,c,d\}$ which are
equal to $j$. When $5\le a+b+c+d\le 7$, in 2005 Adiga, Cooper and
Han [ACH] showed that
$$C(a,b,c,d)t'(a,b,c,d;n)=N(a,b,c,d;8n+a+b+c+d).\tag 1.3$$
$a+b+c+d=8$, in 2008 Baruah, Cooper and Han [BCH] proved that
$$C(a,b,c,d)t'(a,b,c,d;n)=N(a,b,c,d;8n+8)-N(a,b,c,d;2n+2).\tag 1.4$$
In 2009,
Cooper [C] determined $t'(a,b,c,d;n)$ for $(a,b,c,d)=(1,1,1,3),\
(1,3,3,3),$ $(1,2,2,3),\ (1,3,6,6),\ (1,3,4,4),\ (1,1,2,6)$ and
In a previous paper [WS], the authors obtained
formulas for $t(a,b,c,d;n)$ in the cases
$(a,b,c,d)=(1,2,2,4),\ (1,2,4,4),\ (1,1,4,4),\ (1,4,4,4)$,
$(1,1,9,9),\ (1,9,9,$ $9)$, $(1,1,1,9)$, $(1,3,9,9)$ and $(1,1,3,9).$
Ramanujan's theta functions $\varphi(q)$ and $\psi(q)$ (see [Be])
are defined by
q^{n^2}\qtq{and} \psi(q)=\sum_{n=0}^{\infty}q^{n(n+1)/2}\
(|q|<1).\tag 1.5$$
It is evident that for $|q|<1$,
\varphi(q^b)\varphi(q^c)\varphi(q^d),$$
\psi(q^c)\psi(q^d).$$
From [BCH, Lemma 4.1] we know that for
$$\align &\varphi(q)=\varphi(q^4)+2q\psi(q^8),\tag 1.6
\\&\psi(q)\psi(q^3)=\varphi(q^6)\psi(q^4)+q\psi(q^{12})
\varphi(q^2),\tag
\\&\psi(q)^2=\varphi(q)\psi(q^2).\tag
By (1.6), for $k\in\Bbb N$,
=\varphi(q^{16k})+2q^{4k}\psi(q^{32k})+2q^k\psi(q^{8k}).\tag 1.9$$
In this paper, by using Ramanujan's theta functions we reveal
some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose
$a,n\in\Bbb N$ and $2\nmid a$. We show that
for $(a,b,c,d)=
(a,a,2a,8m),\ (a,3a,8k+2,8m+6),\ (a,3a,8m+4,8m+4)\ (n\e m+\f{a-1}2
\mod 2)$ and $(a,3a,16k+4,16m+4)\ (n\e \f{a-1}2\mod 2)$. Using the
formulas for $N(a,b,c,d;n)$ in [AALW1-AALW5] and [AAW] we also
obtain explicit formulas for $t(a,b,c,d;n)$ in the cases
$(a,b,c,d)=(1,1,2,8),\ (1,1,2,16),(1,2,3,6),\ (1,3,4,$ $12)$,
$(1,1,3,4)$, $\ (1,1,5,5),\ (1,5,5,5),\ (1,3,3,12),\ (1,1,1,12),\
(1,1,3,12)$ and $(1,3,3,4)$.
Throughout this paper $\sls am$ is the
Legendre-Jacobi-Kronecker symbol. For $n\in\Bbb N$, $a(n)$ is given by
a(n)q^n\ (|q|<1).$$
§ 2. FOUR RELATIONS BETWEEN $T(A,B,C,D;N)$ AND
Theorem 2.1 Let $m,n\in\Bbb N$ and $a\in\{1,3,5,\ldots\}$.
$$t(a,a,2a,8m;n)=\f 23N(a,a,2a,8m;8n+8m+4a)-2N(a,a,2a,8m;2n+2m+a).$$
Proof. Suppose $|q|<1$ and $a=2s+1$. By (1.9),
\\&=\varphi(q^{2s+1})^2\varphi(q^{4s+2})\varphi(q^{8m})
\\&=\big(\varphi(q^{32s+16})+2q^{8s+4}\psi(q^{64s+32})+2q^{2s+1}\psi(q^{16s+8})\big)^2
\\&\q\times\big(\varphi(q^{16s+8})+2q^{4s+2}\psi(q^{32s+16})\big)\cdot\big(\varphi(q^{32m})+2q^{8m}\psi(q^{64m})\big)
\\&=
\big(\varphi(q^{32s+16})^2+4q^{16s+8}\psi(q^{64s+32})^2
\\&\q+4q^{8s+4}\varphi(q^{32s+16})\psi(q^{64s+32})
\\&\q+8q^{10s+5}\psi(q^{64s+32})\psi(q^{16s+8})\big)
\\&\q\times
\big(\varphi(q^{16s+8})\varphi(q^{32m})+2q^{8m}\varphi(q^{16s+8})
\psi(q^{64m})+2q^{4s+2}\psi(q^{32s+16})\varphi(q^{32m})
\\&\q+4q^{8m+4s+2}\psi(q^{32s+16})\psi(q^{64m})\big).
\endaligned\tag 2.1$$
Note that $\varphi(q^{8k_1})^{m_1}\psi(q^{8k_2})^{m_2}
=\sum_{n=0}^{\infty}b_nq^{8n}$ for any nonnegative integers
$k_1,k_2,m_1$ and $m_2$. From (2.1) we deduce that
\\&=4q^{4s+2}\psi(q^{16s+8})^2\cdot2q^{4s+2}\psi(q^{32s+16})\varphi(q^{32m})
\\&\q+4q^{4s+2}\psi(q^{16s+8})^2\cdot4q^{8m+4s+2}\psi(q^{32s+16})\psi(q^{64m})
\\&\q+4q^{8s+4}\varphi(q^{32s+16})\psi(q^{64s+32})\cdot\varphi(q^{16s+8})\varphi(q^{32m})
\\&\q+4q^{8s+4}\varphi(q^{32s+16})\psi(q^{64s+32})\cdot2q^{8m}\varphi(q^{16s+8})
\psi(q^{64m})
\endalign$$
and so
\\&=8q^{8s}\psi(q^{16s+8})^2\psi(q^{32s+16})\varphi(q^{32m})
\\&\q+4q^{8s}\varphi(q^{32s+16})\psi(q^{64s+32})\varphi(q^{16s+8})\varphi(q^{32m})
\\&\q+8q^{8s+8m}\varphi(q^{32s+16})\psi(q^{64s+32})\varphi(q^{16s+8})
\psi(q^{64m}).
\endalign$$
Replacing $q$ with $q^{1/8}$ in the above formula we obtain
\\&=8q^{s}\psi(q^{2s+1})^2\psi(q^{4s+2})\varphi(q^{4m})
\\&\q+4q^{s}\varphi(q^{4s+2})\psi(q^{8s+4})\varphi(q^{2s+1})\varphi(q^{4m})
\psi(q^{8m}).\endalign$$
By (1.8),
\f{\psi(q^{4s+2})^2}{\psi(q^{8s+4})}\cdot \psi(q^{8s+4})
\\&=12q^{s}\psi(q^{2s+1})^2\psi(q^{4s+2})\varphi(q^{4m})
+24q^{s+m}\psi(q^{2s+1})^2\psi(q^{4s+2}) \psi(q^{8m}).\endalign$$
the other hand, from (2.1) we have
\\&=4q^{2s+1}\varphi(q^{8s+4})\psi(q^{16s+8})\varphi(q^{4s+2})\varphi(q^{8m}).
\endalign$$
Replacing $q$ with $q^{1/2}$ in the above formula we obtain
$$\align &\sum_{n=0}^{\infty}N(2s+1,2s+1,4s+2,8m;2n+1)q^{n}
\\&=4q^{s}\varphi(q^{4s+2})\psi(q^{8s+4})\varphi(q^{2s+1})
\varphi(q^{4m})
\\&-3\sum_{n=0}^{\infty}N(2s+1,2s+1,4s+2,8m;2n+1)q^{n}
\\&=24q^{s+m}\psi(q^{2s+1})^2\psi(q^{4s+2})
\psi(q^{8m})
\\&=24q^{s+m}\sum_{n=0}^{\infty}t'(2s+1,2s+1,4s+2,8m;n)q^n
\\&=\f32q^{m+s}
\sum_{n=0}^{\infty}t(2s+1,2s+1,4s+2,8m;n)q^n.\endalign$$
the coefficients of $q^{n+m+s}$ on both sides we obtain the result.
Theorem 2.2 Let $a\in\{1,3,5,\ldots\}$,
$k,m\in\{0,1,2,\ldots\}$ and $n\in\Bbb N$. Then
$$\align t(a,3a,8k+2,8m+6;n)
Proof. Suppose $|q|<1$ and $a=2s+1$. By (1.9),
$$\aligned &\sum_{n=0}^{\infty}N(2s+1,6s+3,8k+2,8m+6;n)q^{n}
\\&=\varphi(q^{2s+1})\varphi(q^{6s+3})\varphi(q^{8k+2})
\varphi(q^{8m+6})
\\&=\big(\varphi(q^{32s+16})+2q^{8s+4}\psi(q^{64s+32})+2q^{2s+1}\psi(q^{16s+8})\big)
\\&\q\times\big(\varphi(q^{96s+48})+2q^{24s+12}\psi(q^{192s+96})+2q^{6s+3}\psi(q^{48s+24})\big)
\\&\q\times\big(\varphi(q^{32k+8})+2q^{8k+2}\psi(q^{64k+16})\big)
\cdot\big(\varphi(q^{32m+24})+2q^{8m+6}\psi(q^{64m+48})\big)
\\&=
\big(\varphi(q^{32s+16})\varphi(q^{96s+48})+2q^{24s+12}\varphi(q^{32s+16})\psi(q^{192s+96})
\\&\q +2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})+4q^{32s+16}\psi(q^{64s+32})\psi(q^{192s+96})
\\&\q+4q^{14s+7}\psi(q^{64s+32})\psi(q^{48s+24})
\\&\q+4q^{26s+13}\psi(q^{16s+8})\psi(q^{192s+96})+4q^{8s+4}
\psi(q^{16s+8})\psi(q^{48s+24}) \big)
\\&\q\times \big(\varphi(q^{32k+8})\varphi(q^{32m+24})+
\\&\q +4q^{8m+8k+8}\psi(q^{64k+16})\psi(q^{64m+48})\big).
\endaligned\tag 2.2$$
Note that $\varphi(q^{8k_1})^{m_1}\psi(q^{8k_2})^{m_2}
=\sum_{n=0}^{\infty}b_nq^{8n}$ for any nonnegative integers
$k_1,k_2,m_1$ and $m_2$. From (2.2) we deduce that
\\&=2q^{24s+12}\varphi(q^{32s+16})\psi(q^{192s+96})
\cdot\varphi(q^{32k+8})\varphi(q^{32m+24})
\\&\q+2q^{24s+12}\varphi(q^{32s+16})\psi(q^{192s+96})
\cdot4q^{8m+8k+8}\psi(q^{64k+16})\psi(q^{64m+48})
\\&\q+2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})
\cdot\varphi(q^{32k+8})\varphi(q^{32m+24})
\\&\q+2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})
\cdot4q^{8m+8k+8}\psi(q^{64k+16})\psi(q^{64m+48})
\\&\q+4q^{8s+4}\psi(q^{16s+8})\psi(q^{48s+24})
\cdot\varphi(q^{32k+8})\varphi(q^{32m+24})
\\&\q+4q^{8s+4}\psi(q^{16s+8})\psi(q^{48s+24})
\cdot4q^{8m+8k+8}\psi(q^{64k+16})\psi(q^{64m+48})
\endalign$$
and so
\\&=2q^{24s+8}\varphi(q^{32s+16})\psi(q^{192s+96})\varphi(q^{32k+8})\varphi(q^{32m+24})
\\&\q+8q^{24s+8m+8k+16}\varphi(q^{32s+16})\psi(q^{192s+96})\psi(q^{64k+16})\psi(q^{64m+48})
\\&\q+2q^{8s}\psi(q^{64s+32})\varphi(q^{96s+48})\varphi(q^{32k+8})\varphi(q^{32m+24})
\\&\q+8q^{8m+8k+8s+8}\psi(q^{64s+32})\varphi(q^{96s+48})\psi(q^{64k+16})\psi(q^{64m+48})
\\&\q+4q^{8s}\psi(q^{16s+8})\psi(q^{48s+24})\varphi(q^{32k+8})\varphi(q^{32m+24})
\\&\q+16q^{8m+8k+8s+8}\psi(q^{16s+8})\psi(q^{48s+24})\psi(q^{64k+16})\psi(q^{64m+48}).
\endalign$$
Replacing $q$ with
$q^{1/8}$ in the above we obtain
\\&=2q^{3s+1}\varphi(q^{4s+2})\psi(q^{24s+12})\varphi(q^{4k+1})\varphi(q^{4m+3})
\\&\q+2q^{s}\psi(q^{8s+4})\varphi(q^{12s+6})\varphi(q^{4k+1})\varphi(q^{4m+3})
\\&\q+8q^{3s+m+k+2}\varphi(q^{4s+2})\psi(q^{24s+12})\psi(q^{8k+2})\psi(q^{8m+6})
\\&\q+8q^{m+k+s+1}\psi(q^{8s+4})\varphi(q^{12s+6})\psi(q^{8k+2})\psi(q^{8m+6})
\\&\q+4q^{s}\psi(q^{2s+1})\psi(q^{6s+3})\varphi(q^{4k+1})\varphi(q^{4m+3})
\\&\q+16q^{m+k+s+1}\psi(q^{2s+1})\psi(q^{6s+3})\psi(q^{8k+2})\psi(q^{8m+6}).
\endalign$$
Applying (1.7) we get
\\&=6q^s\psi(q^{2s+1})\psi(q^{6s+3})\varphi(q^{4k+1})\varphi(q^{4m+3})
\endaligned$$
By (1.6),
\\&=\varphi(q^{2s+1})\varphi(q^{6s+3})\varphi(q^{8k+2})\varphi(q^{8m+6})
\\&=\big(\varphi(q^{8s+4})+2q^{2s+1}\psi(q^{16s+8})\big)\big(\varphi(q^{24s+12})+2q^{6s+3}
\psi(q^{48s+24})\big)\varphi(q^{8k+2})\varphi(q^{8m+6})
\endalign$$
and so
\\&=2q^{6s+3}\varphi(q^{8s+4})\psi(q^{48s+24})\varphi(q^{8k+2})\varphi(q^{8m+6})
\\&\q+2q^{2s+1}\psi(q^{16s+8})\varphi(q^{24s+12})\varphi(q^{8k+2})\varphi(q^{8m+6}).
\endalign$$
Replacing $q$ with $q^{1/2}$ in the above formula we obtain
\\&=2q^{3s+1}\varphi(q^{4s+2})\psi(q^{24s+12})\varphi(q^{4k+1})\varphi(q^{4m+3})
\endalign$$
Now applying (1.7) we get
\\&\q-3\sum_{n=0}^{\infty}N(2s+1,6s+3,8k+2,8m+6;2n+1)q^{n}
\\&=24q^{m+k+s+1}\psi(q^{2s+1})\psi(q^{6s+3})\psi(q^{8k+2})\psi(q^{8m+6})
\\&=24q^{m+k+s+1}\sum_{n=0}^{\infty}t'(2s+1,6s+3,8k+2,8m+6;n)q^{n}
\\&=\f32q^{m+k+s+1}\sum_{n=0}^{\infty}
Comparing the
coefficients of $q^{m+n+k+s+1}$ on both sides we obtain the result.
Theorem 2.3 Let $a\in\{1,3,5,\ldots\}$, $m\in\{0,1,2,\ldots\}$
and $n\in\Bbb N$. If $n\e m+\f{a-1}2\mod 2$, then
$$\align t(a,3a,8m+4,8m+4;n)
Proof. Suppose $|q|<1$ and $a=2s+1$. Using (1.9) we see that
\\&=\varphi(q^{2s+1})\varphi(q^{6s+3})\varphi(q^{8m+4})^2
\\&=\big(\varphi(q^{32s+16})+2q^{8s+4}\psi(q^{64s+32})
\\&\q\times\big(\varphi(q^{96s+48})+2q^{24s+12}
\psi(q^{192s+96})
\\&\q\times\big(\varphi(q^{32m+16})+2q^{8m+4}
\psi(q^{64m+32})\big)^2
\\&=
\big(\varphi(q^{32s+16})\varphi(q^{96s+48})+2q^{24s+12}
\varphi(q^{32s+16})\psi(q^{192s+96})
\\&\q +2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})+4q^{32s+16}\psi(q^{64s+32})\psi(q^{192s+96})
\\&\q+4q^{14s+7}\psi(q^{64s+32})\psi(q^{48s+24})
\\&\q+4q^{26s+13}\psi(q^{16s+8})\psi(q^{192s+96})+4q^{8s+4}
\psi(q^{16s+8})\psi(q^{48s+24}) \big)
\\&\q\times \big(\varphi(q^{32m+16})^2+
+4q^{16m+8}\psi(q^{64m+32})^2 \big).
\endaligned\tag 2.3$$
Note that $\varphi(q^{8k_1})^{m_1}\psi(q^{8k_2})^{m_2}
=\sum_{n=0}^{\infty}b_nq^{8n}$ for any nonnegative integers
$k_1,k_2,m_1$ and $m_2$. From (2.3) we deduce that
\\&=\varphi(q^{32s+16})\varphi(q^{96s+48})
\cdot4q^{8m+4}\varphi(q^{32m+16})\psi(q^{64m+32})
\\&\q+2q^{24s+12}\varphi(q^{32s+16})\psi(q^{192s+96})
(\varphi(q^{32m+16})^2 +4q^{16m+8}\psi(q^{64m+32})^2)
\\&\q+2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})
(\varphi(q^{32m+16})^2 +4q^{16m+8}\psi(q^{64m+32})^2)
\\&\q+4q^{32s+16}\psi(q^{64s+32})\psi(q^{192s+96})
\cdot4q^{8m+4}\varphi(q^{32m+16})\psi(q^{64m+32})
\\&\q+4q^{8s+4}\psi(q^{16s+8})\psi(q^{48s+24})
+4q^{16m+8}\psi(q^{64m+32})^2) \endalign$$
\\&=4q^{8m}\varphi(q^{32s+16})\varphi(q^{96s+48})
\varphi(q^{32m+16})\psi(q^{64m+32})
\\&\q+2q^{24s+8}\varphi(q^{32s+16})\psi(q^{192s+96})
\varphi(q^{32m+16})^2
\\&\q+8q^{24s+16m+16}\varphi(q^{32s+16})\psi(q^{192s+96})
\psi(q^{64m+32})^2
\\&\q+2q^{8s}\psi(q^{64s+32})\varphi(q^{96s+48})
\varphi(q^{32m+16})^2
\\&\q+8q^{8s+16m+8}\psi(q^{64s+32})\varphi(q^{96s+48})
\psi(q^{64m+32})^2
\\&\q+16q^{32s+8m+16}\psi(q^{64s+32})\psi(q^{192s+96})
\varphi(q^{32m+16})\psi(q^{64m+32})
\\&\q+4q^{8s}\psi(q^{16s+8})\psi(q^{48s+24})
\varphi(q^{32m+16})^2
\\&\q+16q^{8s+16m+8}\psi(q^{16s+8})\psi(q^{48s+24})
\psi(q^{64m+32})^2 .\endalign$$
Replacing $q$ with
$q^{1/8}$ we then obtain
\\&=4q^{m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+2q^{3s+1}\varphi(q^{4s+2})\psi(q^{24s+12}) \varphi(q^{4m+2})^2
+2q^{s}\psi(q^{8s+4})\varphi(q^{12s+6}) \varphi(q^{4m+2})^2
\\&\q+8q^{3s+2m+2}\varphi(q^{4s+2})\psi(q^{24s+12}) \psi(q^{8m+4})^2
+8q^{s+2m+1}\psi(q^{8s+4})\varphi(q^{12s+6}) \psi(q^{8m+4})^2
\\&\q+16q^{4s+m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+4q^{s}\psi(q^{2s+1})\psi(q^{6s+3})
\varphi(q^{4m+2})^2 +16q^{s+2m+1}\psi(q^{2s+1})\psi(q^{6s+3})
\psi(q^{8m+4})^2 .\endalign$$
By (1.7),
\psi(q^{24s+12})=\psi(q^{2s+1})\psi(q^{6s+3}).\tag 2.4$$
\\&=4q^{m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+16q^{4s+m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+6q^{s}\psi(q^{2s+1})\psi(q^{6s+3})
\varphi(q^{4m+2})^2 +24q^{s+2m+1}\psi(q^{2s+1})\psi(q^{6s+3})
\psi(q^{8m+4})^2 .\endaligned\tag 2.5$$
On the other hand, using
(1.9) we see that
\\&=\varphi(q^{2s+1})\varphi(q^{6s+3})\varphi(q^{8m+4})^2
\\&=\big(\varphi(q^{8s+4})+2q^{2s+1}\psi(q^{16s+8})\big)
\big(\varphi(q^{24s+12})
\endalign$$
and so
\\&=\big(2q^{6s+3}\varphi(q^{8s+4})\psi(q^{48s+24})
\endalign$$
Replacing $q$ with $q^{1/2}$ we then obtain
\\&=\big(2q^{3s+1}\varphi(q^{4s+2})\psi(q^{24s+12})
\endalign$$
Now applying (2.4) we get
=2q^{s}\psi(q^{2s+1})\psi(q^{6s+3})\varphi(q^{4m+2})^2.\tag 2.6$$
From (2.5) and (2.6) we deduce that
\\&-3\sum_{n=0}^{\infty}N(2s+1,6s+3,8m+4,8m+4;2n+1)q^{n}
\\&=4q^{m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+16q^{4s+m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q +24q^{s+2m+1}\psi(q^{2s+1})\psi(q^{6s+3})
\psi(q^{8m+4})^2
\\&=4q^{m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+16q^{4s+m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{4m+2})\psi(q^{8m+4})
\\&\q+24q^{s+2m+1}\sum_{n=0}^{\infty}t'(2s+1,6s+3,8m+4,8m+4;n)q^n.
\endalign$$
Suppose $n\e m+s\mod 2$. Then $s+2m+1+n\e m+1\mod 2$. Comparing the coefficients
of $q^{s+2m+1+n}$ in the above expansion we obtain
\\&-3N(2s+1,6s+3,8m+4,8m+4;2(2m+n+s+1)+1)
\\&=24t'(2s+1,6s+3,8m+4,8m+4;n)=\f32t(2s+1,6s+3,8m+4,8m+4;n).\endalign$$
This completes the
Theorem 2.4 Let $a\in\{1,3,5,\ldots\}$,
$k,m\in\{0,1,2,\ldots\}$ and $n\in\Bbb N$. If $n\e \f{a-1}2\mod 2$,
$$\align t(a,3a,16k+4,16m+4;n)
Proof. Suppose $|q|<1$ and $a=2s+1$. Using (1.9) we see that
\\&=\varphi(q^{2s+1})\varphi(q^{6s+3})\varphi(q^{16k+4})
\varphi(q^{16m+4})
\\&=\big(\varphi(q^{32s+16})+2q^{8s+4}\psi(q^{64s+32})+2q^{2s+1}\psi(q^{16s+8})\big)
\\&\q\times \big(\varphi(q^{96s+48})+2q^{24s+12}\psi(q^{192s+96})+2q^{6s+3}\psi(q^{48s+24})\big)
\\&\q\times
\big(\varphi(q^{64k+16})+2q^{16k+4}\psi(q^{128k+32})\big)\cdot
\big(\varphi(q^{64m+16})+2q^{16m+4}\psi(q^{128m+32})\big)
\\&=
\big(\varphi(q^{32s+16})\varphi(q^{96s+48})+2q^{24s+12}
\varphi(q^{32s+16})\psi(q^{192s+96})
\\&\q +2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})+4q^{32s+16}\psi(q^{64s+32})\psi(q^{192s+96})
\\&\q+4q^{14s+7}\psi(q^{64s+32})\psi(q^{48s+24})
\\&\q+4q^{26s+13}\psi(q^{16s+8})\psi(q^{192s+96})+4q^{8s+4}
\psi(q^{16s+8})\psi(q^{48s+24}) \big)
\\&\q\times \big(\varphi(q^{64k+16})\varphi(q^{64m+16})+
\\&\q+2q^{16k+4}\psi(q^{128k+32})\varphi(q^{64m+16}) +
\endaligned\tag 2.7$$
Note that $\varphi(q^{8k_1})^{m_1}\psi(q^{8k_2})^{m_2}
=\sum_{n=0}^{\infty}b_nq^{8n}$ for any nonnegative integers
$k_1,k_2,m_1$ and $m_2$. From (2.7) we deduce that
\\&=\varphi(q^{32s+16})\varphi(q^{96s+48})
\big(2q^{16m+4}\varphi(q^{64k+16})\psi(q^{128m+32}) +
\\&\q+2q^{24s+12}\varphi(q^{32s+16})\psi(q^{192s+96})
\big(\varphi(q^{64k+16})\varphi(q^{64m+16})
\\&\qq+4q^{16k+16m+8}\psi(q^{128k+32})\psi(q^{128m+32})\big)
\\&\q+2q^{8s+4}\psi(q^{64s+32})\varphi(q^{96s+48})
\big(\varphi(q^{64k+16})\varphi(q^{64m+16})
\\&\qq+4q^{16k+16m+8}\psi(q^{128k+32})\psi(q^{128m+32})\big)
\\&\q+4q^{32s+16}\psi(q^{64s+32})\psi(q^{192s+96})
\big(2q^{16m+4}\varphi(q^{64k+16})\psi(q^{128m+32})
\\&\qq+2q^{16k+4}\psi(q^{128k+32})\varphi(q^{64m+16})\big)
\\&\q+4q^{8s+4}\psi(q^{16s+8})\psi(q^{48s+24})
\big(\varphi(q^{64k+16})\varphi(q^{64m+16})
\\&\qq+4q^{16k+16m+8}\psi(q^{128k+32})\psi(q^{128m+32})\big)\endalign$$
and so
\\&=2q^{16m}\varphi(q^{32s+16})\varphi(q^{96s+48})
\varphi(q^{64k+16})\psi(q^{128m+32})
\\&\q+2q^{16k}\varphi(q^{32s+16})\varphi(q^{96s+48})
\psi(q^{128k+32})\varphi(q^{64m+16})
\\&\q+2q^{24s+8}\varphi(q^{32s+16})\psi(q^{192s+96})
\varphi(q^{64k+16})\varphi(q^{64m+16})
\\&\q+8q^{24s+16k+16m+16}\varphi(q^{32s+16})\psi(q^{192s+96})
\psi(q^{128k+32})\psi(q^{128m+32})
\\&\q+2q^{8s}\psi(q^{64s+32})\varphi(q^{96s+48})
\varphi(q^{64k+16})\varphi(q^{64m+16})
\\&\q+8q^{8s+16k+16m+8}\psi(q^{64s+32})\varphi(q^{96s+48})
\psi(q^{128k+32})\psi(q^{128m+32})
\\&\q+8q^{32s+16m+16}\psi(q^{64s+32})\psi(q^{192s+96})
\varphi(q^{64k+16})\psi(q^{128m+32})
\\&\q+8q^{32s+16k+16}\psi(q^{64s+32})\psi(q^{192s+96})
\psi(q^{128k+32})\varphi(q^{64m+16})
\\&\q+4q^{8s}\psi(q^{16s+8})\psi(q^{48s+24})
\varphi(q^{64k+16})\varphi(q^{64m+16})
\\&\q+16q^{8s+16k+16m+8}\psi(q^{16s+8})\psi(q^{48s+24})
\psi(q^{128k+32})\psi(q^{128m+32}).\endalign$$
Replacing $q$ with
$q^{1/8}$ in the above we obtain
\\&=2q^{2m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{8k+2})\psi(q^{16m+4})
\\&\q+2q^{2k}\varphi(q^{4s+2})\varphi(q^{12s+6})
\psi(q^{16k+4})\varphi(q^{8m+2})
\\&\q+2q^{3s+1}\varphi(q^{4s+2})\psi(q^{24s+12})
\varphi(q^{8k+2})\varphi(q^{8m+2})
\\&\q+2q^{s}\psi(q^{8s+4})\varphi(q^{12s+6})
\varphi(q^{8k+2})\varphi(q^{8m+2})
\\&\q+8q^{3s+2k+2m+2}\varphi(q^{4s+2})\psi(q^{24s+12})
\psi(q^{16k+4})\psi(q^{16m+4})
\\&\q+8q^{s+2k+2m+1}\psi(q^{8s+4})\varphi(q^{12s+6})
\psi(q^{16k+4})\psi(q^{16m+4})
\\&\q+8q^{4s+2m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{8k+2})\psi(q^{16m+4})
\\&\q+8q^{4s+2k+2}\psi(q^{8s+4})\psi(q^{24s+12})
\psi(q^{16k+4})\varphi(q^{8m+2})
\\&\q+4q^{s}\psi(q^{2s+1})\psi(q^{6s+3})
\varphi(q^{8k+2})\varphi(q^{8m+2})
\\&\q+16q^{s+2k+2m+1}\psi(q^{2s+1})\psi(q^{6s+3})
\psi(q^{16k+4})\psi(q^{16m+4}).\endalign$$
By (1.7),
\psi(q^{24s+12})=\psi(q^{2s+1})\psi(q^{6s+3}).$$
\\&=2q^{2m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{8k+2})\psi(q^{16m+4})
\\&\q+2q^{2k}\varphi(q^{4s+2})\varphi(q^{12s+6})
\psi(q^{16k+4})\varphi(q^{8m+2})
\\&\q+8q^{4s+2m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{8k+2})\psi(q^{16m+4})
\\&\q+8q^{4s+2k+2}\psi(q^{8s+4})\psi(q^{24s+12})
\psi(q^{16k+4})\varphi(q^{8m+2})
\\&\q+6q^{s}\psi(q^{2s+1})\psi(q^{6s+3})
\varphi(q^{8k+2})\varphi(q^{8m+2})
\\&\q+24q^{s+2k+2m+1}\psi(q^{2s+1})\psi(q^{6s+3})
\psi(q^{16k+4})\psi(q^{16m+4}).\endaligned\tag 2.8$$
On the other
hand, using (1.9) we see that
\\&=\varphi(q^{2s+1})\varphi(q^{6s+3})\varphi(q^{16k+4})\varphi(q^{16m+4})
\\&=\Big(\varphi(q^{8s+4})+2q^{2s+1}\psi(q^{16s+8})\Big)\Big(\varphi(q^{24s+12})
\endalign$$
and so
\\&=2q^{6s+3}\varphi(q^{8s+4})\psi(q^{48s+24})\varphi(q^{16k+4})
\varphi(q^{16m+4})
\\&\q+2q^{2s+1}\psi(q^{16s+8})\varphi(q^{24s+12})\varphi(q^{16k+4})
\varphi(q^{16m+4}).
\endalign$$
Replacing $q$ with $q^{1/2}$ in the above formula we obtain
\\&=2q^{3s+1}\varphi(q^{4s+2})\psi(q^{24s+12})\varphi(q^{8k+2})
\varphi(q^{8m+2})
\varphi(q^{8m+2}).
\endalign$$
Now applying (1.7) we get
\\&=2q^{s}\psi(q^{2s+1})\psi(q^{6s+3})\varphi(q^{12s+6})
\varphi(q^{8k+2}).\endaligned\tag 2.9$$
From (2.8) and (2.9) we
deduce that
\\&\q-3\sum_{n=0}^{\infty}N(2s+1,6s+3,16k+4,16m+4;2n+1)q^{n}
\\&=2q^{2m}\varphi(q^{4s+2})\varphi(q^{12s+6})
\varphi(q^{8k+2})\psi(q^{16m+4})
\\&\q+2q^{2k}\varphi(q^{4s+2})\varphi(q^{12s+6})
\psi(q^{16k+4})\varphi(q^{8m+2})
\\&\q+8q^{4s+2m+2}\psi(q^{8s+4})\psi(q^{24s+12})
\varphi(q^{8k+2})\psi(q^{16m+4})
\\&\q+8q^{4s+2k+2}\psi(q^{8s+4})\psi(q^{24s+12})
\psi(q^{16k+4})\varphi(q^{8m+2})
\\&\q+24q^{s+2k+2m+1}\psi(q^{2s+1})\psi(q^{6s+3})
\psi(q^{16k+4})\psi(q^{16m+4}).\endaligned$$
Suppose $n\e s\mod 2$.
Then $n+s+2k+2m+1\e 1\mod 2$. Comparing the coefficients of
$q^{n+s+2k+2m+1}$ in the above expansion we obtain
\\&-3N(2s+1,6s+3,16k+4,16m+4;2(n+s+2k+2m+1)+1)
\\&=24t'(2s+1,6s+3,16k+4,16m+4;n)=\f32t(2s+1,6s+3,16k+4,16m+4;n).\endalign$$
This completes the
§ 3. FORMULAS FOR $T(1,1,2,8;N)$, $T(1,1,2,16;N)$, $T(1,2,3,6;N)$
AND $T(1,3,4,12;N)$
Lemma 3.1 ([AALW2, Theorem 4.3]) Suppose
$n\in\Bbb N$, $n=2^{\a}n_1$ and $2\nmid n_1$. Then
$$\aligned N(1,1,2,8;n)=\cases 2\sigma(n_1)+2\sls 2{n_1}
\sum\limits\Sb (r,s)\in\Bbb Z^2, 4\mid r-1\\n_1=r^2+4s^2\endSb r
&\t{if $n\e1\mod 4$,}
\\2\sigma(n_1)&\t{if $n\e3\mod 4$,}
\\12\sigma(n_1)&\t{if $n\e 4\mod 8$.}
\endcases\endaligned$$
Theorem 3.1 Let $n\in\Bbb N$. Then
$$\aligned t(1,1,2,8;n)=\cases 4\sigma(2n+3)
&\t{if $2\mid n$,} \\4\sigma(2n+3)+4(-1)^{\f {n-1}2}\sum\limits\Sb
(r,s)\in\Bbb Z^2, 4\mid r-1\\2n+3=r^2+4s^2\endSb r &\t{if $2\nmid
Proof. Taking $a=m=1$ in
Theorem 2.1 we see that
$$t(1,1,2,8;n)=\f 23N(1,1,2,8;8n+12)-2N(1,1,2,8;2n+3).$$
Now applying Lemma 3.1 we deduce the result.
Lemma 3.2 ([AALW2, Theorem 4.15]) Let $n\in\Bbb N$ and
$n=2^{\a}n_1$ with $2\nmid n_1$. Then
$$\aligned N(1,1,2,16;n)=\cases 2\sum_{d\mid n_1} \f {n_1}d\Ls 2d+2\sum\limits\Sb
(r,s)\in\Bbb Z^2, 4\mid r-1\\n_1=r^2+2s^2\endSb r &\t{if $n\e 1\mod
2$,}\\12\sum_{d\mid n_1} \f {n_1}d\Ls 2d&\t{if $n\e 4\mod
Theorem 3.2 Suppose $n\in\Bbb N$. Then
$$t(1,1,2,16;n)=4\sum_{d\mid {2n+5}} \f {2n+5}d\Ls 2d-4\sum\limits\Sb
(r,s)\in\Bbb Z^2, 4\mid r-1\\2n+5=r^2+2s^2\endSb r .$$
Proof. Taking $a=1$ and $m=2$ in
Theorem 2.1 we see that
$$t(1,1,2,16;n)=\f 23N(1,1,2,16;8n+20)-2N(1,1,2,16;2n+5).$$
Now applying Lemma 3.2 we deduce the result.
Lemma 3.3 ([AALW1, Theorem 1.15]) Let $n\in\Bbb N$,
$n=2^{\a}3^{\beta}n_1$ and $\t gcd (n_1,6)=1.$ Then
$$\aligned N(1,2,3,6;n)=\cases (3^{\beta+1}-2)\sigma(n_1)+a(n)
&\t{if $n\e1\mod 2$,}
\\6(3^{\beta+1}-2)\sigma(n_1)&\t{if $n\e0\mod 4$.}
\endcases\endaligned$$
Theorem 3.3 Suppose $n\in\Bbb N$ and $2n+3=3^{\beta}n_1$ with
$n_1\in\Bbb N$ and $3\nmid n_1$. Then
Proof. Taking $a=1$ and $k=m=0$ in
Theorem 2.2 we see that
$$t(1,2,3,6;n)=\f 23N(1,2,3,6;8n+12)-2N(1,2,3,6;2n+3).$$
Now applying Lemma 3.3 we deduce the result.
Lemma 3.4 ([AALW1, Theorem 1.17]) Let
$n\in\Bbb N$ and $n=2^{\a}3^{\beta}n_1$ with $gcd(n_1,6)=1$. Then
$$\aligned N(1,3,4,12;n)=\cases 8\sigma(n_1)
&\t{if $n\e4\mod8$,}
\\\sigma(n_1)+a(n)&\t{if $n\e1\mod4$,}
\\\sigma(n_1)-a(n)&\t{if $n\e3\mod4$.}\endcases\endaligned$$
Theorem 3.4 Suppose $n\in\Bbb N$ and $2n+5=3^{\beta}n_1$ with
$3\nmid n_1$. Then
$$t(1,3,4,12;n)= 2(\sigma(n_1)-(-1)^na(2n+5)).$$
Proof. Suppose $|q|<1$.
Then clearly
By (1.9),
\\&=\big(\varphi(q^{16})+2q^4\psi(q^{32})+2q\psi(q^{8})\big)
\\&\q\times\big(\varphi(q^{48})+2q^{12}\psi(q^{96})+2q^3\psi(q^{24})\big)
\\&\q\times\big(\varphi(q^{16})+2q^4\psi(q^{32})\big)\cdot
\\&=\big(\varphi(q^{16})\varphi(q^{48})+2q^{12}\varphi(q^{16})\psi(q^{96})
\\&\q+2q^4\psi(q^{32})\varphi(q^{48})+4q^{16}\psi(q^{32})\psi(q^{96})
\\&\q+2q\psi(q^{8})\varphi(q^{48})+4q^{13}\psi(q^{8})\psi(q^{96})
\\&\q\times
\big(\varphi(q^{16})\varphi(q^{48})+2q^{12}\varphi(q^{16})
\psi(q^{96})+2q^{4}\psi(q^{32})\varphi(q^{48})
\\&\q+4q^{16}\psi(q^{96})\psi(q^{32})\big).
\endalign$$
Note that $\varphi(q^{8k_1})^{m_1}\psi(q^{8k_2})^{m_2}
=\sum_{n=0}^{\infty}b_nq^{8n}$ for $|q|<1$ and any nonnegative
integers $k_1,k_2,m_1$ and $m_2$. From the above we deduce that
\\&=\varphi(q^{16})\varphi(q^{48})\cdot2q^{12}\varphi(q^{16})\psi(q^{96})
\\&\q+2q^{12}\varphi(q^{16})\psi(q^{96})\cdot\varphi(q^{16})\varphi(q^{48})
\\&\q+2q^4\psi(q^{32})\varphi(q^{48})\cdot\varphi(q^{16})\varphi(q^{48})
\\&\q+4q^{16}\psi(q^{32})\psi(q^{96})\cdot2q^{12}\varphi(q^{16})\psi(q^{96})
\\&\q+4q^4\psi(q^{8})\psi(q^{24})\cdot\varphi(q^{16})\varphi(q^{48})
\endalign$$
and so
\\&=2q^{8}\varphi(q^{16})\varphi(q^{48})\varphi(q^{16})\psi(q^{96})
\\&\q+2q^{8}\varphi(q^{16})\psi(q^{96})\varphi(q^{16})\varphi(q^{48})
\\&\q+2\psi(q^{32})\varphi(q^{48})\varphi(q^{16})\varphi(q^{48})
\\&\q+8q^{24}\psi(q^{32})\psi(q^{96})\varphi(q^{16})\psi(q^{96})
\\&\q+4\psi(q^{8})\psi(q^{24})\varphi(q^{16})\varphi(q^{48})
\endalign$$
Replacing $q$ with $q^{1/8}$ in the above we obtain
\\&=2q\varphi(q^{2})\varphi(q^{6})\varphi(q^{2})\psi(q^{12})
\\&\q+2q\varphi(q^{2})\psi(q^{12})\varphi(q^{2})\varphi(q^{6})
\\&\q+2\psi(q^{4})\varphi(q^{6})\varphi(q^{2})\varphi(q^{6})
\\&\q+8q^{3}\psi(q^{4})\psi(q^{12})\varphi(q^{2})\psi(q^{12})
\\&\q+4\psi(q)\psi(q^{3})\varphi(q^{2})\varphi(q^{6})
\\&=4q\varphi(q^{2})\varphi(q^{6})\varphi(q^{2})\psi(q^{12})
\\&\q+4\psi(q)\psi(q^{3})\varphi(q^{2})\varphi(q^{6})
\\&\q+16q^{2}\psi(q^{4})\psi(q^{12})\psi(q^{4})\varphi(q^{6})
\endalign$$
Now applying (1.7) we get
\\&=8\psi(q)\psi(q^{3})\varphi(q^{2})\varphi(q^{6})
\endaligned\tag 3.1$$
On the other hand, using (1.9) we see that
\\&=\big(\varphi(q^{4})+2q\psi(q^8)\big)\big(\varphi(q^{12})
\endalign$$
and so
\\&=2q\psi(q^8)\varphi(q^{12})^2\varphi(q^{4})
\endalign$$
Replacing $q$ with $q^{1/2}$ we then obtain
\\&=2\psi(q^4)\varphi(q^{6})^2\varphi(q^{2})
\endalign$$
Now applying (1.7) we get
=2\psi(q)\psi(q^3)\varphi(q^{6})\varphi(q^{2}).\tag 3.2$$
From (3.1)
and (3.2) we deduce that
\\&=32q^{2}\psi(q)\psi(q^{3})\psi(q^{12})\psi(q^{4})
\\&=2q^{2}\sum_{n=0}^{\infty}t(1,3,4,12;n)q^n
Comparing the coefficients of $q^{n+2}$ on both sides
we obtain
$$t(1,3,4,12;n)=\f 12N(1,3,4,12;8n+20)-2N(1,3,4,12;2n+5).\tag 3.3$$
Now applying Lemma 3.4 we deduce the result.
§ 4. FORMULAS FOR $T(1,1,3,4;N)$, $T(1,1,5,5;N)$,
$T(1,3,3,12;N)$, $T(1,1,1,12;N)$, $T(1,1,3,12;N)$ AND
For $a,b,c,d,n\in\Bbb N$ let
$$N_0(a,b,c,d;n)=\big|\big\{(x,y,z,w)\in\Bbb Z^4\bigm|
n=ax^2+by^2+cz^2+dw^2,\ 2\nmid xyzw\big\}\big|.$$
From [WS, (4.1)]
we know that
$$t(a,b,c,d;n)=N_0(a,b,c,d;8n+a+b+c+d).\tag 4.1$$
For $n\in \Bbb N$ following [AALW4] we define
$$\aligned &A(n)=\sum_{d\mid n}d\Ls{12}{n/d},
\q B(n)=\sum_{d\mid n}d\Ls{-3}{d}\Ls{-4}{n/d},
\\&C(n)=\sum_{d\mid n}d\Ls{-3}{n/d}\Ls{-4}{d},
\q D(n)=\sum_{d\mid n}d\Ls{12}{d},
\\&E(n)=\sum\Sb (i,j)\in\Bbb N\times \Bbb N\\i,j\ odd\\4n=i^2+3j^2\endSb
(-1)^\f{i-1}2 i\qtq {and} F(n)=\sum\Sb (i,j)\in\Bbb N\times\Bbb
N\\i,j\ odd\\4n=i^2+3j^2\endSb (-1)^\f{j-1}2 j.\endaligned$$
that $n=2^{\a}3^{\beta}n_1$, where $\a$ and $\beta$ are non-negative
integers, $n_1\in\Bbb N$ and
$\t{gcd}(n_1,6)=1$. From [AALW4, Theorem 3.1] we know that
&A(n)=2^{\a}3^{\beta}A(n_1), \q
\\&C(n)=(-1)^{\a+\beta+\f{n_1-1}2}3^{\beta}A(n_1)
\qtq{and} D(n)=\Ls 3{n_1}A(n_1).\endaligned\tag 4.2$$
Lemma 4.1 ([AALW3, Theorem 7.2]) Let $n\in\Bbb N$ with $n\e1\mod2$. Then
$$N(1,1,3,4;n)=3A(n)-B(n)+\f 32C(n)-\f 12D(n)+E(n).$$
Lemma 4.2 ([AALW3, Theorem 7.2]) Let $n\in\Bbb N$ with
$n\e1\mod2$. Then
$$N(1,1,4,12;n)=\f 32A(n)-\f 12B(n)+\f 32C(n)-\f 12D(n)+
\f 12E(n)+\f 32F(n).$$
Theorem 4.1 Suppose $n\in\Bbb N$ and $8n+9=3^{\beta}n_1$ with
$3\nmid n_1$. Then
$$ t(1,1,3,4;n)=\f 12\Big(3^{\beta+1}\Ls 3{n_1}-1\Big)
\sum_{d\mid n_1}d\Ls 3d -\sum\Sb a,b\in\Bbb N,\ 2\nmid
a\\4(8n+9)=a^2+3b^2\endSb (-1)^{\f{a-1}2}a.$$
Proof. Since
$$\align &N(1,1,3,4;8n+9)\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+9=x^2+y^2+3z^2+4w^2
\}\big|\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+9=x^2+y^2+3(2z)^2+4w^2 \}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb Z^4\
|\ 8n+9=x^2+y^2+3z^2+4w^2,2\nmid z \}\big|
\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+9=x^2+y^2+12z^2+4w^2
\}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+9=x^2+y^2+3z^2+4w^2,2\nmid {xyzw} \}\big|
\\&=N(1,1,4,12;8n+9)+N_0(1,1,3,4;8n+9)
\\&=N(1,1,4,12;8n+9)+t(1,1,3,4;n),\endalign$$
we have $t(1,1,3,4;n)=N(1,1,3,4;8n+9)-N(1,1,4,12;8n+9).$ Now
applying Lemmas 4.1, 4.2 and (4.2) we deduce the result.
Remark 4.1 Theorem 4.1 was conjectured by the authors in
Lemma 4.3 ([AAW, Theorem 7.1]) Let $n\in\Bbb N$ and
$n=2^{\a}5^{\beta}n_1$ with $n_1\in\Bbb N$ and $\t{gcd}(n_1,10)=1$.
$$\aligned N(1,1,5,5;n)=\cases 2(5^{\beta+1}-3)\sigma(n_1)
&\t{if $2\mid n$,}
\\\f 23(5^{\beta+1}-3)\sigma(n_1)+\f 83c(n)&\t{if $2\nmid n$.}
\endcases\endaligned$$
where $c(n)$ is given by
Theorem 4.2 Suppose $n\in\Bbb N$ and $2n+3=5^{\beta}n_1$ with
$5\nmid n_1$. Then
$$t(1,1,5,5;n)=\f 43(5^{\beta+1}-3)\sigma(n_1)-\f83c(2n+3).$$
Proof. Note that
$$\align &N(1,1,5,5;8n+12)\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+12
=x^2+y^2+5z^2+5w^2 \}\big|\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+12=(2x)^2+(2y)^2+5(2z)^2+5(2w)^2 \}\big|\\&\q+\big|\{(x,y,z,w)\in
\Bbb Z^4\ |\ 8n+12=x^2+y^2+5z^2+5w^2,2\nmid {xyzw} \}\big|
\\&=N(1,1,5,5;2n+3)+N_0(1,1,5,5;8n+12)
\\&=N(1,1,5,5;2n+3)+t(1,1,5,5;n),\endalign$$
applying Lemma 4.3 we deduce the result.
Lemma 4.4 ([AAW, Theorem 6.1]) Let $n\in\Bbb N$. Then
$$ N(1,5,5,5;n)=\sum_{d\mid n}(-1)^{n+d}\Big(\Ls 5d+\Ls 5{n/d}\Big)d.$$
Theorem 4.3 Let $n\in\Bbb N$. Then
$$\align t(1,5,5,5;n)&=\sum_{d\mid {8n+16}}(-1)^{d}\Big(\Ls 5d +\Ls 5{{(8n+16)}/d}
\Big)d\\&\q-\sum_{d\mid {2n+4}}(-1)^{d}\Big(\Ls 5d+\Ls
5{{(2n+4)}/d}\Big) d\endalign$$
Proof. Observe that
$$\align &N(1,5,5,5;8n+16)\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |
\ 8n+16=x^2+5y^2+5z^2+5w^2 \}\big|\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\
|\ 8n+16=(2x)^2+5(2y)^2+5(2z)^2+5(2w)^2
\}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+16=x^2+5y^2+5z^2+5w^2,2\nmid {xyzw} \}\big|
\\&=N(1,5,5,5;2n+4)+N_0(1,5,5,5;8n+16)
\\&=N(1,5,5,5;2n+4)+t(1,5,5,5;n),\endalign$$
applying Lemma 4.4 we deduce the result.
Lemma 4.5 ([AALW3, Theorem
7.2]) Let $n\in\Bbb N$ with $2\nmid n$. Then
$$ N(1,3,3,12;n)=A(n)+B(n)-\f12C(n)-\f12D(n)+F(n)$$
$$ N(3,3,4,12;n)=\f12(A(n)+B(n)-C(n)-D(n)-E(n)+F(n)).$$
Theorem 4.4 Let $n\in\Bbb N$ and $8n+19=3^{\beta}n_1$ with
$n_1\in\Bbb N$ and $3\nmid n_1$. Then
$$\aligned &t(1,3,3,12;n)\\&=\f 12\Big(3^{\beta}\Ls 3{n_1}-1\Big)
\sum_{d\mid n_1}d\Ls 3d+\f12\sum\Sb a,b\in\Bbb N,\ a\e b \e 1\mod
Observe that
$$\align &N(1,3,3,12;8n+19)\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+19=x^2+3y^2+3z^2+12w^2
\}\big|\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+19=(2x)^2+3y^2+3z^2+12w^2 \}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb
Z^4\ |\ 8n+19=x^2+3y^2+3z^2+12w^2,2\nmid x \}\big|
\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+19=4x^2+3y^2+3z^2+12w^2
\}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+19=x^2+3y^2+3z^2+12w^2,2\nmid {xyzw} \}\big|
\\&=N(4,3,3,12;8n+19)+N_0(1,3,3,12;8n+19)
\\&=N(3,3,4,12;8n+19)+t(1,3,3,12;n),\endalign$$
applying Lemma 4.5 and (4.2) we obtain the result.
Lemma 4.6 ([AALW3,
Theorem 7.2]) Let $n\in\Bbb N$ and $2\nmid n$. Then
$$ N(1,1,1,12;n)=3A(n)-B(n)+\f32C(n)-\f12D(n)+3F(n).$$
Theorem 4.5 Let $n\in\Bbb N$ and $8n+15=3^{\beta}n_1$ with
$n_1\in\Bbb N$ and $3\nmid n_1$. Then
$$ t(1,1,1,12;n)=\f 12\Big(3^{\beta+1}\Ls 3{n_1}+1\Big)
\sum_{d\mid n_1}d\Ls 3d+3\sum\Sb a,b\in\Bbb N,\ 2\nmid
a\\4(8n+15)=a^2+3b^2\endSb (-1)^{\f{b-1}2}b.$$
Proof. Since $8n+15=x^2+y^2+z^2+12w^2$ for $x,y,z,w\in\Bbb Z$
implies that $2\nmid xyzw$, we see that
Now the result follows from Lemma 4.6 and (4.2).
Lemma 4.7 ([AALW1, Theorems 1.10 and 1.13]) Let
$n\in\Bbb N$ and $n=3^{\beta}n_1$ with $3\nmid n_1$. For $n\e1\mod4$
we have
Theorem 4.6 Suppose $n\in\Bbb N$ and $8n+17=3^{\beta}n_1$ with
$3\nmid n_1$. Then
$$ t(1,1,3,12;n)=\sigma(n_1)-a(8n+17).$$
Proof. Since
$$\align &N(1,1,3,12;8n+17)\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+17=x^2+y^2+3z^2+12w^2
\}\big|\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+17=x^2+y^2+3(2z)^2+12w^2 \}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb
Z^4\ |\ 8n+17=x^2+y^2+3z^2+12w^2,2\nmid z \}\big|
\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+17=x^2+y^2+12z^2+12w^2
\}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+17=x^2+y^2+3z^2+12w^2,2\nmid {xyzw} \}\big|
\\&=N(1,1,12,12;8n+17)+N_0(1,1,3,12;8n+17)
\\&=N(1,1,12,12;8n+17)+t(1,1,3,12;n),\endalign$$
Lemma 4.7 we deduce the result.
Lemma 4.8 ([AALW1,
Theorems 1.16 and 1.23]) Let $n\in\Bbb N$ and $n=3^{\beta}n_1$ with
$3\nmid n_1$. For
$n\e3\mod4$ we have
Theorem 4.7 Suppose $n\in\Bbb N$ and $8n+11=3^{\beta}n_1$ with
$3\nmid n_1$. Then
$$ t(1,3,3,4;n)=\sigma(n_1)+a(8n+11).$$
Proof. Since
$$\align &N(1,3,3,4;8n+11)\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+11=x^2+3y^2+3z^2+4w^2
\}\big|\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+11=(2x)^2+3y^2+3z^2+4w^2 \}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb
Z^4\ |\ 8n+11=x^2+3y^2+3z^2+4w^2,2\nmid x \}\big|
\\&=\big|\{(x,y,z,w)\in \Bbb Z^4\ |\ 8n+11=4x^2+3y^2+3z^2+4w^2
\}\big|\\&\q+\big|\{(x,y,z,w)\in \Bbb Z^4\ |\
8n+11=x^2+3y^2+3z^2+4w^2,2\nmid {xyzw} \}\big|
\\&=N(3,3,4,4;8n+11)+N_0(1,3,3,4;8n+11)
\\&=N(3,3,4,4;8n+11)+t(1,3,3,4;n),\endalign$$
applying Lemma 4.8 we derive the result.
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|
1511.00250
|
It is known that the Maxwell-Klein-Gordon equations in $\mathbb{R}^{3+1}$ admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of $r$-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.
§ INTRODUCTION
In this paper, we study the asymptotic behavior of solutions to the Maxwell-Klein-Gordon equations on $\mathbb{R}^{3+1}$ with large Cauchy data. To define the equations, let $A=A_\mu dx^\mu$ be a $1$-form. The covariant derivative associated to this 1-form is
\begin{equation*}
D_\mu =\pa_\mu+\sqrt{-1}A_\mu,
\end{equation*}
which can be viewed as a $U(1)$ connection on the complex line bundle over $\mathbb{R}^{3+1}$ with the standard flat metric $m_{\mu\nu}$. Then the curvature $2$-form $F$ is given by
\begin{equation*}
F_{\mu\nu}=-\sqrt{-1}[D_{\mu}, D_{\nu}]=\pa_\mu A_\nu-\pa_\nu A_\mu=(dA)_{\mu\nu}.
\end{equation*}
This is a closed $2$-form, that is, $F$ satisfies the Bianchi identity
\begin{equation}
\label{bianchi}
\pa_\ga F_{\mu\nu}+\pa_\mu F_{\nu\ga}+\pa_\nu F_{\ga\mu}=0.
\end{equation}
The Maxwell-Klein-Gordon equations (MKG) is a system for the connection field $A$ and the complex scalar field $\phi$:
\begin{equation}
\label{EQMKG}\tag{MKG}
\begin{cases}
\pa^\nu F_{\mu\nu}=\Im(\phi \cdot\overline{D_\mu\phi})=J_\mu;\\
D^\mu D_\mu\phi=\Box_A\phi=0.
\end{cases}
\end{equation}
These are Euler-Lagrange equations of the functional
\[
L[A, \phi]=\iint_{\mathbb{R}^{3+1}}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\phi\overline{D^{\mu}\phi}dxdt.
\]
A basic feature of this system is that it is invariant under the following gauge transformation:
\[
\phi\mapsto e^{i\chi}\phi; \quad A\mapsto A-d\chi.
\]
More precisely, if $(A, \phi)$ solves (<ref>), then $(A-d\chi, e^{i\chi}\phi)$ is also a solution for any potential function $\chi$. Note that $U(1)$ is abelian. The Maxwell field $F$ is invariant under the above gauge transformation and (<ref>) is said to be an abelian gauge theory. For the more general theory when $U(1)$ is replaced by a general compact Lie group, the corresponding equations are referred to as Yang-Mills-Higgs equations.
In this paper, we consider the Cauchy problem to (<ref>). The initial data set $(E, H, \phi_0, \phi_1)$ consists of the initial electric field $E$, the magnetic field $H$, together with initial data $(\phi_0, \phi_1)$ for the scalar field. In terms of the solution $(F, \phi)$, on the initial hypersurface, these are:
\begin{equation*}
% \label{IDset}
F_{0i}=E_i,\quad \leftidx{^*}F_{0i}=H_i,\quad \phi(0, x)=\phi_0,\quad D_t\phi(0, x)=\phi_1,
\end{equation*}
where $\leftidx{^*}F$ is the Hodge dual of the 2-form $F$. In local coordinates $(t, x)$,
\[
(H_1, H_2, H_3)=(F_{23}, F_{31}, F_{12}).
\]
The data set is said to be admissible if it satisfies the compatibility condition
\begin{equation*}
div(E)=\Im(\phi_0\cdot \overline{\phi_1}),\quad div (H)=0,
\end{equation*}
where the divergence is taken on the initial hypersurface $\mathbb{R}^3$. For solutions of (<ref>), the energy
\[
E[F, \phi](t):=\int_{\mathbb{R}^3}|E|^2+|H|^2+|D\phi|^2dx
\]
is conserved. Another important conserved quantity is the total charge
\begin{equation}
\label{defcharge}
q_0=\frac{1}{4\pi}\int_{\mathbb{R}^3}\Im(\phi\cdot \overline{D_t\phi})dx=\frac{1}{4\pi}\int_{\mathbb{R}^3}div (E)dx,
\end{equation}
which can be defined at any fixed time $t$. The existence of nonzero charge plays a crucial role in the asymptotic behavior of solutions of (<ref>). It makes the analysis more complicated and subtle. This is obvious from the above definition as the electric field $E_i=F_{0i}$ has a tail $q_0r^{-3}x_i$ at any fixed time $t$.
The Cauchy problem to (<ref>) has been studied extensively. One of the most remarkable results is due to Eardley-Moncrief in <cit.>, <cit.>, in which it was shown that there is always a global solution to the general Yang-Mills-Higgs equations for sufficiently smooth initial data.
This result has later been greatly improved for MKG equations by Klainerman-Machedon in their celebrated work <cit.>. We state their result here as it guarantees a global solution to study in this paper.
There is a unique (up to gauge transformations) global solution of (<ref>) if the initial data are bounded in the energy space, that is, $E[F, \phi](0)$ is finite.
Same statement holds for the non-abelian case of Yang-Mills equations, see e.g. <cit.>, <cit.>,
<cit.>. Since then there has been an extensive literature on generalizations and extensions of the above classical result, aiming at improving the regularity of the initial data in order to construct a global solution. For more details, we refer to <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and reference therein. A common feature of all these works is to
construct a local solution with rough data. Then the global well-posedness follows by establishing a priori bound for some appropriate norms of the solution. In the work of Eardley-Moncrief, based on the conservation law of energy, they showed that the $L^\infty$ norm of the solution never blows up even though it may grow in time $t$. As a consequence the solution can be extended to all time; however the decay property of the solution is unknown. For Theorem <ref>, the difficulty is to construct a solution with finite energy data on a short time interval so that the length of the time interval depends only on the energy norm. Then the solution exists globally in time since the energy is conserved. It is not clear how (if it is possible) this construction of global solution is able to give any asymptotics of the solution. In view of this, although the solution of (<ref>) exists
globally with rough initial data, very little is known about the decay properties.
On the other hand, asymptotic behavior and decay estimates are only understood for linear fields (see e.g.<cit.>) and nonlinear fields with sufficiently small initial data (see e.g.<cit.>, <cit.>). These mentioned results rely on the conformal symmetry of the system, either by conformally compactifying the Minkowski space or by using the conformal killing vector field $(t^2+r^2)\pa_t+2tr\pa_r$ as multiplier. Nevertheless the use of the conformal symmetry requires strong decay of the initial data and thus in general does not allow the presence of nonzero charge except when the initial data are essentially compactly supported. For the case with nonzero charge but still with small initial data, the first related work regarding the asymptotic properties was due to W. Shu in <cit.>. However, that work only considered the case when the solution is trivial outside
a fixed forward light cone. Details for general case were not carried out. A complete proof towards this program was later contributed by Lindblad-Sterbenz in <cit.>, also see a more recent work <cit.>.
The presence of nonzero charge has a long range effect on the asymptotic behavior of the solutions, at least in a neighbourhood of the spatial infinity. This can be seen from the conservation law
of the total charge as the electric field $E$ decays at most $ r^{-2}$ as $r\rightarrow\infty$ at any fixed time. This weak decay rate makes the analysis more complicated even for small initial data. To deal with this difficulty, Lindblad-Sterbenz constructed a global chargeless field and made use of the fractional Morawetz estimates obtained by using $u^p\pa_u+v^p\pa_v$ as multipliers. The latter work <cit.> relied on the observation that the angular derivative of the Maxwell field has zero charge. The Maxwell field then can be estimated by using Poincaré inequality. However, the smallness
of the initial data, in particular the smallness of the charges, still allows one to use perturbation method to obtain the decay estimates for the solutions.
From the above discussion, we see that on one hand the solution of (<ref>) exists globally with data merely in energy space and on the other hand, the asymptotic behavior is only clear for small data. It is then natural to ask whether the global nonlinear charged-scalar-fields (solutions of MKG) exhibit any form of decay properties.
The aim of the present paper is to affirmatively answer this question. We show strong quantitative decay estimates for solutions of (<ref>) with data merely bounded in some weighted energy space.
To be more precise, assume that the initial data $(E, H, \phi_0, \phi_1)$ belong to the weighted energy space with weights $(1+r)^{1+\ga_0}$ for some positive constant $\ga_0$. In particular, the energy is finite. According to Theorem <ref>, there is a global solution $(F, \phi)$ of (<ref>). Let $\tilde{F}$ be the chargeless field[ Compared to the construction in <cit.>, we avoid the use of a smooth cut off function. The reason is that we will carry out estimates respectively in the exterior region $\{t+R<r\}$ and interior region $\{t+R>r\}$, where the field $\tilde{F}$ is smooth.]:
\[
\tilde{F}=F-q_0 r^{-2}\chi_{\{r\geq t+R\}}dt\wedge dr,\quad R>1,
\]
where $q_0$ is the total charge and $\chi_{K}$ is the characteristic function on the set $K$. We can show that the energy flux through the outgoing light cone $\Si_{\tau}$ decays pointwise in terms of $\tau$:
\[
E[\tilde{F}, \phi](\Si_{\tau})\leq C(1+|\tau|)^{-1-\ga}
\]
for all $0<\ga<\ga_0$ and some constant $C$ depending on $\ga_0$, $\ga$ and the initial weighted energy. Moreover, we can establish an integrated local energy decay and a class of $r$-weighted energy decay. These decay estimates are sufficiently strong to capture the asymptotic properties of solutions of (<ref>). In fact, the above energy flux decay immediately leads to the pointwise decay for the spherical average of the scalar field. In our future works, these decay estimates are the first crucial step toward the stronger pointwise decay for large nonlinear charged-scalar-fields.
Furthermore, since the data are assumed to be merely bounded in some weighted energy space, the total charge $q_0$ is generic and can be arbitrarily large. Note that $\tilde{F}=F$ inside the forward light cone $\{t+R\geq r\}$. Our result in particular shows that the charge can only affect the asymptotic behavior of the solution outside this fixed light cone. This phenomenon was conjectured
by W. Shu in <cit.> and has been confirmed for sufficiently smooth and small initial data in <cit.>, <cit.>. We thus give an affirmative answer to the conjecture for all data in the weighted energy space described above. In other words, the quantitative decay estimates together with the precise description of the effect of the charge completely characterize the asymptotic behavior of solutions of (<ref>).
To make the statement of the main result precise, we define some necessary notations.
We use the standard polar local coordinate system $(t, r,
\om)$ of Minkowski space as well as the null coordinates $u=\frac{t-r}{2}$, $v=\frac{t+r}{2}$.
Without loss of generality we only prove estimates in the future, i.e., $t\geq 0$. In our argument, we estimate the decay of the solution with respect
to the foliation $\Si_{\tau}$
depicted in the above Penrose diagram.
Here $\tau<0$ corresponds to the foliation in the exterior region $\{t+R\leq r\}$ and $\tau\geq 0$ foliates the interior region.
In coordinates the leaves $\Si_{\tau}$ can be defined rigorously as follows:
\begin{align*}
&S_\tau:=\{u=u_\tau=\f12(\tau-R), v\geq v_\tau=\f12(|\tau|+R)\},\quad \tau\in \mathbb{R};\\
&\Si_\tau:=\{t=\tau, r\leq R\}\cap\{t\geq 0\}\cup S_\tau,\quad \tau\in\mathbb{R}
\end{align*}
for some constant $R>1$.
Next we introduce a null frame $\{L, \Lb, e_1, e_2\}$, where
\[
L=\pa_v=\pa_t+\pa_r,\quad \Lb=\pa_u=\pa_t-\pa_r
\]
and $\{e_1, e_2\}$ is an orthonormal basis of the sphere with
constant radius $r$. We use $\D$ to denote the covariant derivative associated to the connection field $A$ on the sphere with radius $r$. For any 2-form $F$, denote the null decomposition under the above null frame by
\begin{equation}
\a_i=F_{Le_i},\quad\underline{\a}_i=F_{\Lb e_i},\quad \rho=\f12 F_{\Lb L}, \quad \si=F_{e_1 e_2},\quad i\in{1, 2}.
\end{equation}
Let $E[\phi, F](\Si)$ be the energy flux of the 2-form $F$ and the complex scalar field $\phi$ through the hypersurface $\Si$ in Minkowski space. For the outgoing null hypersurface $S_{\tau}$, one can compute that
\begin{align*}
%E[\phi, F](\mathbb{R}^3)&=\int_{\mathbb{R}^3}|F|^2+|D\phi|^2dx,\quad |F|^2=\rho^2+|\si|^2+\frac{1}{2}(|\a|^2+|\underline{\a}|^2),\\
E[\phi, F](S_\tau)&=\int_{S_\tau}(|D_L\phi|^2+|\D\phi|^2+\rho^2+\si^2+|\a|^2) r^2dvd\om.
\end{align*}
For the admissible initial data set $(E, H, \phi_0, \phi_1)$ for (<ref>), let $q_0$ be the total charge. Define the chargeless electric field $\tilde{E}$ on the initial hypersurface:
\[
\tilde{E}=E-q_0 r^{-2}\chi_{\{R\leq r\}} \om.
\]
The constant $R>1$ was used to define the foliation $\Si_{\tau}$. Our assumption on the initial data is that for some positive constant $0<\ga_0\leq 1$ the following weighted energy
\begin{equation}
\label{IDMKG}
E_0= \int_{\mathbb{R}^3}(1+r)^{1+\ga_0}(|D\phi_0|^2+|\phi_1|^2+|\tilde{E}|^2+|H|^2)dx
\end{equation}
is finite. We now can state our main theorem:
Consider the Cauchy problem to (<ref>). Assume the initial data set $(E, H, \phi_0, \phi_1)$ with charge $q_0$ is admissible
and the weighted energy $E_0$ is finite for some positive constant $\ga_0\leq 1$. Then for all $\ep>0$, $0\leq \ga<\ga_0$, $R>1$ the global solution $(F, \phi)$ obeys the following decay estimates:
1. The energy flux decay and the integrated local energy decay:
\[
E[\phi, \tilde{F}](\Si_{\tau})+\int_{\frac{s}{\tau}\geq 1}\int_{\Si_s }\frac{|\tilde{F}|^2+|D\phi|^2}{(1+r)^{1+\ep}}dxds\leq CE_0(1+|\tau|)^{-1-\ga},\quad \forall\tau\in\mathbb{R};
\]
2. A hierarchy of $r$-weighted energy estimates for all $0\leq p\leq 1+\ga$:
\begin{align*}
&\int_{\frac{s}{\tau}\geq 1}\int_{S_s }r^{p-1}(|D_{L}(r\phi)|^2+|\D(r\phi)|^2+|r\tilde{\rho}|^2+|r\a|^2+|r\si|^2)dvd\om ds\\
+&\int_{S_{\tau}}r^{p}(|D_{L}(r\phi)|^2+|r\a|^2)dvd\om \leq CE_0(1+|\tau|)^{p-1-\ga},\quad \forall\tau\in\mathbb{R};
\end{align*}
for some constant $C$ depending only on $\ga$, $\ga_0$, $\ep$, $R$, $p$ and the charge $q_0$. Here $\tilde{F}$ is the chargeless field $\tilde{F}=F-q_0 r^{-2}\chi_{\{r\geq t+R\}}dt\wedge dr$.
We make several remarks:
The weighted energy space defined in line (<ref>) with $\ga_0=0$ is scale invariant. Thus our assumptions on the initial data can be almost scale invariant. We also note that the charge part
$q_0 r^{-2}$ does not belong to any such weighted energy space with $\ga_0\geq 0$. This in particular implies that the existence of nonzero charge plays an important role in the asymptotics of
the solution at least in the neighborhood of the spatial infinity.
Our approach in this paper can be adapted to obtain similar decay estimates for solutions of (<ref>) in higher dimensions as long as the solution exists globally. The problem in higher dimensions is that
global regularity in energy space is critical in $4+1$-dimension and super-critical in higher dimensions. The global regularity in energy space is known in the critical case with small
energy <cit.>. Recently the smallness assumption has been removed by Oh-Tataru <cit.> and Krieger-Luhrmann<cit.>.
We can relax the assumptions on the components of the initial data $D_{\Lb}\phi$, $\underline{\a}$ to be merely
bounded in the energy space instead of belonging to the weighted energy space. This is because the decay estimates in the interior region rely only on the energy flux through $\Si_0$
and a $r$-weighted energy flux (see definition in Section <ref>) through the forward light cone $\{t=r+R\}$. The latter one is independent of the components $D_{\Lb}\phi$, $\underline{\a}$.
The regularity in the exterior region is propagated. If initially the components $D_{\Lb}\phi$, $\underline{\a}$ are merely bounded in the energy space, then we can only obtain the
boundedness of the energy in the exterior region instead of quantitative decay as in the main theorem. If the initial weighted energy $E_0$ defined in line (<ref>) is finite for some $\ga_0>1$, then we can show that the
integrated local energy and the energy flux through $\Si_{\tau}$ decay with the maximal rates $(1+\tau)^{-2}$ in the interior region $\{t+R\geq r\}$ while in the exterior region the decay estimates are the same
as in the main theorem with all $\ga<\ga_0$. This propagation of regularity in the exterior region has been discussed in <cit.>, <cit.> for the small data case.
For the non-abelian case of Yang-Mills equations, we can obtain the similar decay estimates when the charges are vanishing. For the general case, the definitions of charges that we are aware of are not gauge
invariant. In particular, we may need to work under a fixed gauge condition if we expect similar decay estimates as in the main theorem. We will address this issue in our future works.
As far as we know, Theorem <ref> is the first result that gives the strong decay estimates for the global large solutions of (<ref>). The quantitative decay estimates rely only the weighted energy norm of the initial data. Based on our experience on nonlinear wave equations, these decay estimates, more precisely, the energy flux decay through $\Si_{\tau}$, the integrated local energy estimate together with the hierarchy of $r$-weighted energy estimates, are sufficiently strong to derive the pointwise decay of the solutions if we also have the decay estimates for the derivatives of the solutions, see e.g. <cit.>, <cit.>. However, the MKG equation is nonlinear. Commuting with derivatives will introduce nonlinear terms. The quantitative decay estimates in Theorem <ref> are fundamental to control those nonlinear terms without assuming any smallness. This will be addressed in our future works.
As we have explained previously, the conformal compactification method may not be able to derive the decay estimates in Theorem <ref> due to the existence of nonzero charge and the weak decay of the initial data. The perturbation method works only for small data case, at least some form of smallness is necessary, see e.g. <cit.> in which the incoming energy flux was assumed to be small. In this paper, we use a new approach to study the large data problem to (<ref>). This new approach was originally introduced by Dafermos-Rodnianski in <cit.> for the study of decay of linear waves on black hole spacetimes and has been successfully applied to several linear and nonlinear problems, e.g. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. However these applications were carried out in a linear setting as either the problem itself is linear or the data are small so that perturbation method was applied. Theorem <ref> is the first result showing that this robust new approach is also very useful for some nonlinear large data problems.
In the abstract framework proposed by Dafermos-Rodnianski in <cit.>, the new method for proving the decay estimates as in Theorem <ref> relies on three kinds of basic ingredients and estimates: a uniform energy bound, an integrated local energy decay estimate and a hierarchy of $r$-weighted energy estimates in a neighbourhood of the null infinity. The uniform energy bound usually follows from the energy conservation law or it can be derived by using the vector field $\pa_t$ as multiplier. The integrated local energy decay estimate has been well studied. This type of estimates was first proven by C. Morawetz in <cit.>, <cit.>. In the past decades, the integrated local energy decay estimate has been obtained for variant linear waves on various kinds of backgrounds, see e.g. <cit.>, <cit.>, <cit.>, <cit.> and references therein. The method may vary for different problems. For solutions of (<ref>), we can use the vector field $f(r)\pa_r$ as multiplier to derive the integrated local energy decay estimate.
The very important new ingredient of the new method is the $r$-weighted energy estimates in a neighborhood of the null infinity. This kind of estimates can be obtained by using the vector fields $r^p(\pa_t+\pa_r)$ as multipliers for all $0\leq p\leq 2$. Combining these robust new estimates with the integrated local energy decay estimates discussed above, a pigeon hold argument then leads to the quantitative decay of the energy flux through $\Si_{\tau}$. In other words, to prove the decay estimates as in Theorem <ref>, these three kinds of estimates have to be considered together.
The MKG equations are nonlinear while the new method discussed above is under a linear setting. One of the key observations that allows us to use the new method to obtain the decay estimates in Theorem <ref> is that the total energy momentum tensor (see definitions in Section <ref>) for the full solution $(F, \phi)$ of (<ref>) is divergence free. Based on this fact, we can use the vector fields $\pa_t$, $f(r)\pa_r$, $r^p(\pa_t+\pa_r)$ as multipliers to derive those three kinds of basic estimates for the new approach.
However, due to the presence of nonzero charge, we are not able to get useful $r$-weighted energy estimates for any $p\geq 1$ in the exterior region. This is because the charge part $q_0 r^{-2}$ does not belong to any weighted energy space with weights $r^{p}$, $p\geq 1$. On the other hand, it seems that to close those basic estimates, at lease some $r$-weighted energy estimate with $p>1$ is necessary. To overcome this difficulty caused by the nonzero charge, we define the charge 2-form $\bar F$
\[
\bar F=q_0 r^{-2}\chi_{\{t+R\leq r\}}dt\wedge dr
\]
in the exterior region, which has the same charge $q_0$ as the full solution $F$. By our assumption, the chargeless field $F-\bar F$ is bounded in the weighted energy space with weights $r^{1+\ga_0}$ initially. We thus can apply the vector field method to the energy momentum tensor for the chargeless part of the solution $(F-\bar F, \phi)$ instead of the full solution $(F, \phi)$. However the energy momentum tensor for $(F-\bar F, \phi)$ is not divergence free. It will introduce the following error term:
\[
q_0 \Im(\overline{D\phi}\cdot\phi)
\]
arising from the interaction between the charge and the scalar field $\phi$. Since the charge $q_0$ is large, this error term can not be absorbed in general. The key to control this error term is to make use
of the better decay of the scalar field $\phi$ initially and then use a type of Gronwall's inequality, for details see Section <ref>. This is the reason that we loss a little bit of decay rate ($\ga<\ga_0$) in the main theorem.
Once we have the decay estimates in the exterior region, that is, the estimates in Theorem <ref> for $\tau<0$, we in particular can conclude that the energy flux and the $r$-weighted energy flux through the boundary $\Si_{0}$ of the interior region is finite. The key point that
the charge does not affect the asymptotic behavior of the solution in the interior region is that the $r$-weighted energy flux through the boundary $S_0$ depends only on $F-\bar F$. However the energy flux through $S_{0}$ does rely on the full solution $F$ but the charge part $\bar F$ has finite energy flux through $S_0$. Therefore the energy flux and the $r$-weighted energy flux through $\Si_0$ of the full solution $(F, \phi)$ are finite. This enables us to use the new approach to derive the decay estimates of Theorem <ref> in the interior region.
The paper is organized as follows: we derive the energy identities obtained by using the vector fields $f(r)\pa_r$, $\pa_t$, $r^p\pa_v$ as multipliers in Section <ref>; we then use these identities to derive
decay estimates for the chargeless part of the solution $(F-\bar F, \phi)$ in the exterior region as well as the energy flux through the boundary $S_0$ in Section <ref>; once we have the energy flux
bound through $\Si_0$ and the $r$-weighted energy flux bound through $S_0$, we can obtain the decay estimates for the solution in the interior region and conclude the main theorem in the last Section.
The author would like to thank Mihalis Dafermos and Pin Yu for helpful discussions. He is indebted to Igor Rodnianski for plenty of invaluable comments and suggestions.
§ PRELIMINARIES AND ENERGY IDENTITIES
We define some additional notations used in the sequel. In the exterior region $\{r\geq R+t\}$, for $R\leq r_1< r_2$,
we use $S_{r_1, r_2}$ to denote the following outgoing null hypersurface emanating from the sphere with radius $r_1$
\[
S_{r_1, r_2}:=\{u=-\frac{r_1}{2},\quad r_1\leq r\leq r_2\}.
\]
We note that $S_{r_1, r_2}$ is part of the outgoing null hypersurface $S_{\frac{R-r_1}{2}}$ defined before the main theorem. We use $\bar C_{r_1, r_2}$ to denote the following incoming null hypersurface emanating from the sphere with radius $r_2$
\[
\bar C_{r_1, r_2}:=\{v=\frac{r_2}{2},\quad r_1\leq r\leq r_2\}.
\]
For $0\leq \tau_1<\tau_2$, $\bar C_{\tau_1, \tau_2}$ will be used to denote the null infinity between $\Si_{\tau_1}$ and $\Si_{\tau_2}$. Throughout this paper, $\tau\geq0$ will be the parameter
in the interior region $\{r\leq R+t\}$ and $R\leq r_1\leq r_2$ will be used as the parameters in the exterior region. On the initial hypersurface $\mathbb{R}^3$, the annulus with radii $0\leq r_1<r_2$ is denoted as
\[
B_{r_1, r_2}:=\{t=0,\quad r_1\leq r\leq r_2\}.
\]
We use $\mathcal{D}_{r_1, r_2}$ to denote the region bounded by $S_{r_1, r_2}$, $B_{r_1, r_2}$, $\bar C_{r_1, r_2}$ and
$\mathcal{D}_{\tau_1, \tau_2}$ to denote the region bounded by $\Si_{\tau_1}$, $\Si_{\tau_2}$ for $0\leq \tau_1<\tau_2$. For the complex scalar field $\phi$ and the 2-form $F$, we denote
\[
\bar D\phi:=(D\phi, (1+r)^{-1}\phi),\quad |F|^2:=\rho^2+\si^2+\f12(|\a|^2+|\underline{\a}|^2).
\]
We now review the energy method for the MKG equations. Denote $d\vol$ the
volume form in the Minkowski space. In the local coordinate system $(t, x)$, we have
Here we have chosen $t$ to be the time orientation. For any two forms $\tilde{F}$ satisfying the Bianchi identity (<ref>) and any complex scalar field $\phi$, we define the associated energy momentum tensor
\begin{equation*}
\begin{split}
T[\phi, \tilde{F}]_{\a\b}&=\tilde{F}_{\a\mu}\tilde{F}_\b^{\;\mu}-\frac{1}{4}m_{\a\b}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}+\Re\left(\overline{D_\a\phi}D_\b\phi\right)-\f12 m_{\a\b}\overline{D^\mu\phi}D_\mu\phi.
\end{split}
\end{equation*}
Given a vector field $X$, we have the following identity
\[
\pa^\mu(T[\phi,\tilde{F}]_{\mu\nu}X^\nu) = \Re(\Box_A \phi X^\nu \overline{D_\nu\phi})+X^\nu F_{\nu\a}J^\a+\pa^\mu \tilde{F}_{\mu\a}\tilde{F}_{\nu}^{\;\a}X^{\nu}+T[\phi,\tilde{F}]^{\mu\nu}\pi^X_{\mu\nu},
\]
where $\pi_{\mu\nu}^X=\f12 \mathcal{L}_X m_{\mu\nu}$ is the deformation tensor of the vector field $X$ in Minkowski space and $J_\mu=\Im(\phi\cdot \overline{D_\mu\phi})$. Through out this paper, we raise and lower indices with respect to this flat metric $m_{\mu\nu}$.
Take any function $\chi$. We have the following equality
\begin{align*}
\f12\pa^{\mu}\left(\chi \pa_\mu|\phi|^2-\pa_\mu\chi|\phi|^2\right)= \chi \overline{D_\mu\phi}D^\mu\phi -\f12\Box\chi\cdot|\phi|^2+\chi \Re(\Box_A\phi\cdot \overline\phi).
\end{align*}
We now define the vector field $\tilde{J}^X[\phi, \tilde{F}]$ with components
\begin{equation}
\label{mcurent} \tilde{J}^X_\mu[\phi,\tilde{F}]=T[\phi,\tilde{F}]_{\mu\nu}X^\nu -
\f12\pa_{\mu}\chi \cdot|\phi|^2 + \f12 \chi\pa_{\mu}|\phi|^2+Y_\mu
\end{equation}
for some vector field $Y$ which may also depend on the scalar field $\phi$. We then have the equality
\begin{align*}
\pa^\mu \tilde{J}^X_\mu[\phi,\tilde{F}] =&\Re(\Box_A \phi(\overline{D_X\phi}+\chi\overline \phi))+div(Y)+X^\nu F_{\nu\mu}J^\mu+\pa^\mu \tilde{F}_{\mu\ga}\tilde{F}_{\nu}^{\;\ga}X^{\nu}
\\&+T[\phi,\tilde{F}]^{\mu\nu}\pi^X_{\mu\nu}+
\chi \overline{D_\mu\phi}D^\mu\phi -\f12\Box\chi\cdot|\phi|^2.
\end{align*}
Here the operator $\Box$ is the wave operator in Minkowski space and the divergence of the vector field $Y$ is also taken in the Minkowski space.
Now take any region $\mathcal{D}$ in $\mathbb{R}^{3+1}$. Assume on this region the scalar field $\phi$ and the 2-form $\tilde{F}$ satisfies the following equation
\[
\pa^\nu \tilde{F}_{\mu\nu}=J_{\mu},\quad \Box_A\phi=0.
\]
Here we note that the covariant operator $\Box_A$ is associated to the solution $F$ of (<ref>). Then using Stokes' formula, the above calculation leads to the following energy identity
\begin{align}
\notag &\iint_{\mathcal{D}} div(Y)+X^\nu (F_{\nu\mu}-\tilde{F}_{\nu\mu})J^\mu+T[\phi,\tilde{F}]^{\mu\nu}\pi^X_{\mu\nu}+
\chi \overline{D_\mu\phi}D^\mu\phi -\f12\Box\chi\cdot|\phi|^2d\vol\\
&=\iint_{\mathcal{D}}\pa^\mu \tilde{J}^X_\mu[\phi,\tilde{F}]d\vol=\int_{\pa \mathcal{D}}i_{\tilde{J}^X[\phi,\tilde{F}]}d\vol,
\label{energyeq}
\end{align}
where $\pa\mathcal{D}$ denotes the boundary of the domain $\mathcal{D}$ and $i_Z d\vol$ denotes the contraction of the volume form $d\vol$
with the vector field $Z$ which gives the surface measure of the
boundary. For example, for any basis $\{e_1, e_2,
\ldots, e_n\}$, we have $i_{e_1}( de_1\wedge de_2\wedge\ldots
de_k)=de_2\wedge de_3\wedge\ldots\wedge de_k$.
In this paper, the domain $\mathcal{D}$ will be regular regions bounded by the $t$-constant slices, the outgoing null hypersurfaces $u=constant$ or the incoming null
hypersurfaces $v=constant$. We now compute $i_{\tilde{J}^{X}[\phi,\tilde F]}d\vol$ on these three kinds of hypersurfaces. On
$t=constant$ slice, the surface measure is a function times $dx$. Recall the volume form
\[
\]
Here note that $dx$ is a $3$-form. We thus can show that
\begin{equation}
\label{curlessR}
\begin{split}
D_X\phi)-\f12 X^0\overline{D^\ga\phi}D_\ga\phi-\f12 \pa^t\chi \cdot
\end{split}
\end{equation}
On the outgoing null hypersurface $\{u=constant\}$, we can write the volume form
\[
d\vol=dxdt=r^2drdt d\om=2r^2dvdud\om=-2dudvd\om.
\]
Here $u=\frac{t-r}{2}$, $v=\frac{t+r}{2}$ are the null coordinates and $d\om$ is the standard surface measure on the unit sphere.
Notice that $\Lb=\pa_u$. We can compute
\begin{equation}
\label{curStau}
\begin{split}
i_{\tilde{J}^{X}[\phi, \tilde F]}d\vol=&-2(\Re(\overline{D^{\Lb}\phi}
D_X\phi)-\f12 X^{\Lb}\overline{D^\ga\phi}D_\ga\phi-\f12
\pa^{\Lb}\chi|\phi|^2+\f12\chi \pa^{\Lb}|\phi|^2+Y^{\Lb}\\
\end{split}
\end{equation}
Similarly, on the $v$-constant incoming null hypersurfaces $\{v=constant\}$, we
\begin{equation}
\label{curnullinfy}
\begin{split}
D_X\phi)-\f12 X^{L}\overline{D^\ga\phi}D_\ga\phi-\f12
\pa^{L}\chi|\phi|^2+\f12\chi \pa^{L}|\phi|^2+Y^{L}\\
\end{split}
\end{equation}
We remark here that the above formulae hold for any vector fields $X$, $Y$ and any function $\chi$.
§.§ The $r$-weighted energy estimates using the multiplier $r^pL$
In this section, we establish the $r$-weighted energy identities for solutions of the MKG equations. We obtain the $r$-weighted energy estimate either in the exterior region $\{r\geq R+t\}$ for the domain $\mathcal{D}_{r_1, r_2}$ for $R\leq r_1\leq r_2$ which is bounded by the outgoing null
hypersurface $S_{\frac{R-r_1}{2}}$ emanating from the sphere with radius $r_1$,
the incoming null hypersurface emanating from the sphere with radius $r_2$ and the initial hypersurface $\mathbb{R}^3$ or in the interior region for domain $\tilde{\mathcal{D}}_{\tau_1, \tau_2}$ for
$0\leq \tau_1<\tau_2$ which is bounded by the outgoing null hypersurfaces $S_{\tau_1}$, $S_{\tau_2}$ and the cylinder $\{r=R\}$. Recall here that $R>1$ is a constant determined by the initial data and is
used to define the foliation. In the energy identity (<ref>), we choose the vector fields $X$, $Y$ and the function $\chi$ as follows:
\[
X=r^{p}L, \quad Y=\frac{p}{2}r^{p-2}|\phi|^2L,\quad \chi=r^{p-1}.
\]
Note that we have the equality
\begin{align*}
r^2|D_L\phi|^2=|D_L\psi|^2-L(r|\phi|^2)&,\quad r^2|\D\phi|^2=|\D\psi|^2,\\
r^2|D_{\Lb}\phi|^2=|D_{\Lb}\psi|^2+\Lb(r|\phi|^2)&,\quad \psi=r\phi.
\end{align*}
Throughout this paper, we always use $\psi$ to denote the weighted scalar field $r\phi$.
We then can compute
\begin{align*}
& div(Y)+T[\phi, \tilde{F}]^{\mu\nu}\pi_{\mu\nu}^X+\chi \overline{D^{\mu}\phi}D_\mu\phi-\f12\Box\chi |\phi|^2\\
&\quad -\f12 p(p-1)r^{p-3}|\phi|^2+X^\nu (F_{\nu\a}-\tilde{F}_{\nu\a})J^\a\\
&=\f12 r^{p-3}\left(p(|D_L\psi|^2+|\tilde\a|^2)+(2-p)(|\D\psi|^2+\tilde{\rho}^2+\tilde\si^2)\right)+r^p (F_{L\mu}-\tilde{F}_{L\mu})J^\mu.
\end{align*}
We next compute the boundary terms according to the formulae (<ref>) to (<ref>). We have
\begin{align*}
\int_{S_{r_1, r_2}}i_{\tilde{J}^X[\phi, \tilde{F}]}d\vol&=\int_{S_{r_1, r_2}}r^{p}(|D_L\psi|^2+r^2|\tilde\a|^2)-\f12 L(r^{p+1}\phi) \quad dvd\om,\\
\int_{\bar C_{r_1, r_2}}i_{\tilde{J}^X[\phi, \tilde{F}]}d\vol&=-\int_{\bar C_{r_1, r_2}}r^{p}(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2|\tilde\si|^2)+\f12\Lb(r^{p+1}|\phi|^2) dud\om,\\
\int_{B_{r_1, r_2}}i_{\tilde{J}^X[\phi,\tilde{F}]}d\vol&=\f12\int_{B_{r_1, r_2}}r^p(|D_L\psi|^2+|\D\psi|^2)+r^{p+2}(|\tilde\a|^2+|\tilde{\rho}|^2+\tilde\si^2)\\
&\quad\quad-\pa_r(r^{p+1}|\phi|^2) d\om dr,\\
\int_{r=R, \tau_1\leq t\leq \tau_2 }i_{\tilde{J}^X[\phi, \tilde{F}]}d\vol&=\f12\int_{\tau_1}^{\tau_2}\int_{\om}r^p(|D_L\psi|^2-|\D\psi|^2+r^2(|\tilde \a|^2-|\tilde\rho|^2-|\tilde\si|^2))\\
&\quad\quad-\pa_t(r^{p+1}|\phi|^2) d\om dt.
\end{align*}
The formula on $S_{\tau}$ is the same as that on $S_{r_1, r_2}$. Now notice that on the domain $D_{r_1, r_2}$ in the exterior region we have the identity
\begin{align*}
&\int_{S_{r_1, r_2}}L(r^{p+1}|\phi|^2)dvd\om-\int_{\bar C_{r_1, r_2}}\Lb(r^{p+1}|\phi|^2)dud\om-\int_{B_{r_1, r_2}}\pa_r(r^{p+1}|\phi|^2)d\om dr=0
\end{align*}
and on the domain $\tilde{D}_{\tau_1, \tau_2}$ in the interior region we have
\begin{align*}
&-\int_{S_{\tau_1}}L(r^{p+1}|\phi|^2)dvd\om-\int_{\bar C_{\tau_1, \tau_2}}\Lb(r^{p+1}|\phi|^2)dud\om\\
&+\int_{S_{\tau_2}}L(r^{p+1}|\phi|^2)dvd\om+\int_{\tau_1}^{\tau_2}\int_{\om}\pa_t(r^{p+1}|\phi|^2)d\om dt=0.
\end{align*}
The above calculations then lead to the following $r$-weighted energy identity in the exterior region
\begin{align}
\notag
&\iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}\left(p(|D_L\psi|^2+r^2|\tilde\a|^2)+(2-p)(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2\tilde\si^2)\right)dvd\om du\\
\label{pWescaout}
&+\int_{S_{r_1,r_2}}r^p(|D_L\psi|^2+r^2|\tilde\a|^2)dvd\om+\int_{\bar C_{r_1, r_2}}r^p(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2\tilde\si^2)dud\om\\
\notag
=&\f12\int_{B_{r_1,r_2}}r^p(|D_L\psi|^2+|\D\psi|^2)+r^{p+2}(|\tilde\a|^2+|\tilde{\rho}|^2+\tilde\si^2)drd\om-\iint_{\mathcal{D}_{r_1, r_2}}r^{p}(F_{L\mu}-\tilde{F}_{L\mu})J^\mu dxdt
\end{align}
and the corresponding one in the interior region
\begin{equation}
\label{pWescain}
\begin{split}
&\int_{\tau_1}^{\tau_2}\int_{S_\tau}r^{p-1}\left(p(|D_L\psi|^2+r^2|\tilde\a|^2)+(2-p)(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2\tilde\si^2)\right)dvd\om du\\
&+\int_{S_{\tau_1}}r^p(|D_L\psi|^2+r^2|\tilde\a|^2)dvd\om+\int_{\bar C_{\tau_1, \tau_2}}r^p(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2\tilde\si^2)dud\om\\
=&\int_{S_{\tau_1}}r^p|D_L\psi|^2+r^{p+2}|\tilde{\a}|^2dvd\om-\f12\int_{\tau_1}^{\tau_2}\int_{\om}r^p(|D_L\psi|^2-|\D\psi|^2+r^2(|\tilde{\a}|^2-|\tilde\rho|^2-\tilde\si^2)) d\om dt\\
&-\int_{\tau_1}^{\tau_2}\int_{S_\tau}r^{p}(F_{L\mu}-\tilde{F}_{L\mu})J^\mu dxdt.
\end{split}
\end{equation}
Here we recall that
\[
J_\mu=\Im(\overline{D_\mu\phi}\cdot \phi)=r^{-2}\Im(\overline{D_\mu\psi}\cdot \psi),\quad \psi=r\phi.
\]
In the exterior region, we consider estimates for the chargeless part $\tilde{F}$ while in the interior region, we take $\tilde{F}$ as the full solution $F$. The above $r$-weighted energy identity
explains why the charge can only affect the asymptotic behaviour of the solution in the exterior region. In fact, we see later that the charge only affect the asymptotics for the curvature components
$\rho$. The solution on the interior region only depends on the data on $\Si_0$. More precisely, the energy flux through it and the $r$-weighted energy flux. However, the $r$-weighted energy flux depends only on
the curvature component $\a$ but not $\rho$. This shows that the charge can only affect the asymptotics for the solution on the exterior region. This phenomenon was original conjectured by W. Shu in <cit.>.
§.§ The integrated local energy estimates using the multiplier $f(r)\pa_r$
We consider the integrated local energy estimates either on the exterior region for the domain $\mathcal{D}_{r_1, r_2}$, $R\leq r_1\leq r_2$ or on the interior region for the domain
$\mathcal{D}_{\tau_1, \tau_2}$, $0\leq \tau_1<\tau_2$ which is bounded by the hypersurfaces $\Si_{\tau_1}$ and $\Si_{\tau_2}$. In the energy identity (<ref>), we choose the vector fields
$X$, $Y$ and the function $\chi$ as follows
\[
X=f(r)\pa_r, \quad \chi=r^{-1}f, \quad Y=0
\]
for some function $f$ of $r$. We then can compute
\begin{align*}
&T[\phi, \tilde{F}]^{\mu\nu}\pi^X_{\mu\nu}+ \chi \overline{D_\mu\phi}D^\mu\phi-\f12\Box \chi |\phi|^2\\
=&\f12 f'(|D_t\phi|^2+|D_r\phi|^2+\f12 |\tilde\a|^2+\f12 |\underline{\tilde{\a}}|^2)+(r^{-1}f-\f12 f')(|\D\phi|^2+\tilde\rho^2+\tilde\si^2)-\f12 r^{-1}\pa_r f'|\phi|^2.
\end{align*}
We will construct the function $f$ explicitly later as the choice of the function depends on the region we consider. The basic idea is to choose function $f$
such that $f'$, $r^{-1}f-\f12 f'$, $-\pa_r f'$ are all positive. We now compute the boundary terms according to the formulae (<ref>) to (<ref>). We can show that
\begin{align*}
\int_{S_{r_1, r_2}}i_{\tilde{J}^X[\phi, \tilde{F}]}d\vol&=\f12\int_{S_{r_1, r_2}}f(|D_L\phi|^2-|\D\phi|^2-\tilde{\rho}^2+|\tilde\a|^2-|\tilde\si|^2)-\chi'|\phi|^2\\
&\quad \quad+2\chi \Re(D_L\phi \cdot \overline\phi) \quad r^2dvd\om,\\
\int_{\bar C_{r_1, r_2}}i_{\tilde{J}^X[\phi, \tilde{F}]}d\vol&=\f12\int_{\bar C_{r_1, r_2}}f(|D_{\Lb}\phi|^2-|\D\phi|^2-|\tilde\rho|^2+|\underline{\tilde{\a}}|^2-|\tilde\si|^2)
&\quad \quad-2\chi \Re(D_{\Lb}\phi\cdot \overline{\phi}) \quad r^2dud\om,\\
\int_{B_{r_1, r_2}}i_{\tilde{J}^X[\phi,\tilde{F}]}d\vol&=\int_{B_{r_1, r_2}}f(\Re(\overline{D_t\phi}(D_r\phi+r^{-1}\phi))+\frac{1}{4}f(|\tilde\a|^2-|\tilde{\underline{\a}}|^2) dx.
\end{align*}
Here $\chi=r^{-1}f$. The idea is that we use the energy flux through the corresponding surfaces to bound these boundary terms. In fact, we can compute the energy flux explicitly
\begin{align}
\notag
E[\phi, \tilde{F}](S_{r_1, r_2}):=&2\int_{S_{r_1, r_2}}i_{\tilde{J}^{\pa_t}[\phi, \tilde{F}]}d\vol=\int_{S_{r_1, r_2}}(|D_L\phi|^2+|\D\phi|^2+\tilde{\rho}^2+|\tilde\a|^2+|\tilde\si|^2) r^2dvd\om,\\
\notag
E[\phi, \tilde{F}](\bar C_{r_1, r_2}):=&2\int_{\bar C_{r_1, r_2}}i_{\tilde{J}^{\pa_t}[\phi, \tilde{F}]}d\vol=\int_{\bar C_{r_1, r_2}}(|D_{\Lb}\phi|^2+|\D\phi|^2+|\tilde\rho|^2+|\underline{\tilde{\a}}|^2+|\tilde\si|^2) r^2dud\om,\\
\label{enfluxc}
E[\phi, \tilde{F}](B_{r_1, r_2}):=&2\int_{B_{r_1, r_2}}i_{\tilde{J}^{\pa_t}[\phi,\tilde{F}]}d\vol=\int_{B_{r_1, r_2}}|D\phi|^2+|\tilde\rho|^2+|\tilde\si|^2+\f12(|\tilde\a|^2+|\tilde{\underline{\a}}|^2) dx.
\end{align}
Therefore to control the boundary terms for the multiplier $f(r)\pa_r$, it suffices to control the integral of $|\phi|^2$ in terms of the corresponding energy flux.
We will choose the function $f$ to be bounded. Since $\chi=r^{-1}f$, we have $\chi'\sim r^{-2}$. We thus can use a type of Hardy's inequality adapted to the connection $D$ to bound the integral of $\chi'|\phi|^2$.
Assume the complex scalar field $\phi$ vanishes at null infinity, that is,
\[
\lim_{v\rightarrow\infty}\phi(v, u, \om)=0
\]
for all $u$, $\om$. Then we have
\begin{align*}
\int_{\Si_{\tau}}\left|\frac{\phi}{1+r}\right|^2d\si\leq 12 E[\phi](\Si_{\tau}),\quad \int_{S_{r_1}}\left|\frac{\phi}{1+r}\right|^2r^2dvd\om\leq 12 E[\phi](S_{r_1}),\quad r_1>R,\quad \tau\geq 0.
\end{align*}
Here $d\si=dx$ when $r\leq R$ and $d\si=r^2dvd\om $ on $S_{\tau}$ and $E[\phi](\Si)$ denotes the energy flux through the surface $\Si$.
It suffices to notice that the covariant derivative $D$ is compatible with the inner product $<,>$ on the complex plane. Then the proof goes the same as the case when the connection field $A$ is trivial,
see e.g. <cit.>, <cit.>. Another quick way to reduce the proof of the Lemma to the case with trivial connection field $A$ is to choose a particular gauge such that the scalar field
$\phi$ is real. We can do this is due to the fact that all the quantities we considered here are gauge invariant.
The above lemma implies that the boundary terms arising from the multiplier $f(r)\pa_r$ can be controlled by the corresponding energy flux up to a constant. We now choose $f$ explicitly to establish the
integrated local energy estimates. On the interior region $\mathcal{D}_{\tau_1, \tau_2}$ we choose
$$f=2\ep^{-1}-\frac{2\ep^{-1}}{(1+r)^{\ep}},\quad \chi=r^{-1}f$$
and on the exterior region $\mathcal{D}_{r_1, r_2}$ we take
\[
f=2\ep^{-1}(r_1^{-\ep}-(1+r)^{-\ep}),\quad r_1\geq R>1
\]
for some small positive constant $\ep<\frac{1}{4}$. In any case, $f$ is bounded and we have
\begin{align*}
\f12 f'=\frac{1}{(1+r)^{1+\ep}},\quad -\f12 \Box \chi&=\frac{1+\ep}{r(1+r)^{2+\ep}},\quad \chi- \f12
f'\geq \frac{2\ep^{-1}}{r} - \frac{1+2\ep^{-1}}{r(1+r)^\ep}\geq \frac{1}{r}.
\end{align*}
The last inequality holds for $r>1$.
We still need to estimate the error term $X^\nu(F_{\nu\mu}-\tilde{F}_{\nu\mu})J^{\mu}$. Recall that
\[
\]
Using Cauchy-Schwarz's inequality, we have
\[
2|X^\nu(F_{\nu\mu}-\tilde{F}_{\nu\mu})J^{\mu}|\leq f(\ep_1^{-1}(1+r)^{1+\ep}|F_{r\mu}-\tilde{F}_{r\mu}|^2|\phi|^2+\ep_1(1+r)^{-1-\ep}|D_\mu\phi|^2),\quad \forall \ep_1>0.
\]
The idea to control this error term is that we choose $\ep_1$ sufficiently small so that the second term on the right hand side can be absorbed.
To avoid too many constants, in the rest of this paper, we make a convention that $A\les B$ means $A\leq C B$ for some constant $C$ depends only on
the constants $\ep$, $R$, the charge $q_0$ and $\ga$, $\ga_0$ in the main Theorem <ref>.
Based on the above calculations, we can derive the following integrated local energy estimates in the
interior region for the domain $\mathcal{D}_{\tau_1, \tau_2}$, $0\leq \tau_1<\tau_2$
\begin{align}
\label{ILEMKGin}
&\int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}}\frac{|\bar D\phi|^2+|\tilde{F}|^2}{(1+r)^{1+\ep}}+\frac{|\D\phi|^2+\tilde\rho^2+\tilde\si^2}{1+r}dxdt\\
\notag
\les & E[\phi, \tilde{F}](\Si_{\tau_1})+E[\phi, \tilde F](\Si_{\tau_2})
+E[\phi, \tilde{F}](\bar C_{\tau_1, \tau_2})\\
\notag
\end{align}
and the integrated local energy estimates in the exterior region for the domain $\mathcal{D}_{r_1, r_2}$, $R\leq r_1<r_2$
\begin{align}
\label{ILEMKGout}
&r_1^\ep\iint_{\mathcal{D}_{r_1, r_2}}\frac{|\bar D\phi|^2+|\tilde{F}|^2}{(1+r)^{1+\ep}}+\frac{|\D\phi|^2+\tilde\rho^2+\tilde\si^2}{1+r}dxdt\\
\notag
\les & E[\phi, \tilde{F}](B_{r_1, \infty})+E[\phi, \tilde F](S_{r_1, r_2})
+E[\phi, \tilde{F}](\bar C_{r_1, r_2})\\
\notag
&\quad+r_1^{-2\ep}\iint_{\mathcal{D}_{r_1, r_2}}(1+r)^{1+\ep}|F_{r\nu}-\tilde{F}_{r\nu}|^2|\phi|^2dxdt.
\end{align}
For the estimate in the exterior region, we gain the decay $r_1^\ep$ is due to the fact that the function $f$ has upper bound $2\ep^{-1}r_1^{-\ep}$ while in the interior region $|f|\les 1$. In addition, since
we need to use the Hardy's inequality to control the integral of $|\phi|^2$, we rely on the fact that initially $\phi$ vanishes at the spatial infinity. This is the reason that we use
$E[\phi, \tilde{F}](B_{r_1, \infty})$ instead of $E[\phi, \tilde{F}](B_{r_1, r_2})$ in the estimate.
§.§ Energy estimates using the multiplier $\pa_t$
Our new approach is based on the observation that all the energy should go through the null infinity. Hence the energy flux through the hypersurface $\Si_{\tau}$ which is far away from the light cone has to decay to
zero. The first step is to show that, by using the classical energy method, the energy flux is monotonic.
In the energy identity (<ref>), take $X=\pa_t$, $Y=0$, $\chi=0$. Since $\pa_t$ is killing, based on the calculations (<ref>) in the previous section, we obtain the energy estimate
in the interior region
\begin{equation}
\label{esin}
E[\phi, \tilde{F}](\Si_{\tau_2})+E[\phi, \tilde{F}](\bar C_{\tau_1, \tau_2})\leq E[\phi, \tilde{F}](\tau_1)+2\int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}}|F_{0\mu}-\tilde{F}_{0\mu}||J^\mu| d\vol
\end{equation}
and the energy estimate on the exterior region
\begin{equation}
\label{esout}
E[\phi, \tilde{F}](S_{r_1, r_2})+E[\phi, \tilde{F}](\bar C_{r_1, r_2})\leq E[\phi, \tilde{F}](B_{r_1, r_2})+2\iint_{\mathcal{D}_{r_1, r_2}}|F_{0\mu}-\tilde{F}_{0\mu}||J^\mu| d\vol
\end{equation}
for all $0\leq \tau_1<\tau_2$ and $R\leq r_1<r_2$.
When coupled to the integrated local energy estimates derived in the previous section, we can estimate the last terms containing $J^\mu$ in the above energy estimates by using Cauchy Schwarz's inequality so that
on the right hand side of the inequality there is only integral of $|\phi|^2$ as in the integrated local energy estimates (<ref>), (<ref>).
§ ENERGY ESTIMATES IN THE EXTERIOR REGION
Due to the presence of non-zero charge $q_0$, the component of the curvature $\rho$ has a tail $q_0 r^{-2}$. In general, a useful integrated local energy decay estimate for the
full solution $(F, \phi)$ of (<ref>) at least in the exterior region is not expected as $\rho$ dost not decay in time $t$. A natural way to circumvent this problem is to remove the
charge part of the field and consider estimates for the remained part which is chargeless. As in the work of Lindblad-Sterbenz in <cit.>, we define the
charged 2-form on the exterior region $\{r\geq R+t\}$
\begin{equation*}
\bar F=q_0 d(r^{-1}dt)=q_0 r^{-2}dt\wedge dr.
\end{equation*}
The corresponding null decomposition under the null frame $\{L, \Lb, e_1, e_2\}$ is as follows:
\begin{equation}
\label{charF}
\bar \rho=q_0 r^{-2},\quad \bar \a=\underline{\bar \a}=0,\quad \bar \si=0.
\end{equation}
It can be checked that this charged 2-form $\bar F$ satisfies the linear Maxwell equation
\[
\pa^\mu \bar F_{\mu\nu}=0
\]
in the exterior region $r\geq R+t$. Here we recall that the constant $R$ is a fixed constant determined by the initial data and is used to define the foliation $\Si_{\tau}$. For solution $(F, \phi)$
of (<ref>), we then define the chargeless 2-form
\begin{equation*}
\tilde{F}=F-\bar F.
\end{equation*}
In this section, we do the estimates for the chargeless part $\tilde{F}$ instead of the full solution $F$.
We remark here that since we
do the estimates separately on the exterior region $\{r\geq R+t\}$ and the interior region $\{r\leq R+t\}$, we do not have to take a smooth cut off function as in <cit.>.
Notice that in the integrated local energy estimates (<ref>)
\[
(1+r)^{1+\ep}|F_{r\nu}-\tilde{F}_{r\nu}|^2=(1+r)^{1+\ep}|\bar F_{r\nu}|^2\sim q_0^2 r^{-3+\ep}.
\]
The decay rate is not sufficient to make that term be absorbed. Moreover, since the charge $q_0$ is large, we are even lack of the smallness needed. This means that we need to use other estimates in order to
control the error term arising from the charge part. This forces us to consider the $r$-weighted energy estimate on the exterior region $\{r\geq R+t\}$ first.
Take $\tilde{F}=F-\bar F$ in the $r$-weighted energy
estimate (<ref>) in the exterior region for the charged 2-form $\bar F$ defined above. According to the relation (<ref>), we obtain the following $r$-weighted energy estimate for the chargeless
part of the solution
\begin{equation}
\label{pWeMKGout}
\begin{split}
&\iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}\left(p(|D_L\psi|^2+r^2|\a|^2)+(2-p)(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2\si^2)\right)dvd\om du\\
&+\int_{S_{r_1,r_2}}r^p(|D_L\psi|^2+r^2|\a|^2)dvd\om+\int_{\bar C_{r_1, r_2}}r^p(|\D\psi|^2+r^2|\tilde{\rho}|^2+r^2\si^2)dud\om\\
=&\f12\int_{B_{r_1,r_2}}r^p(|D_L\psi|^2+|\D\psi|^2)+r^{p+2}(|\a|^2+|\tilde{\rho}|^2+\si^2)drd\om-q_0\iint_{\mathcal{D}_{r_1, r_2}}r^{p-2}J_L dxdt.
\end{split}
\end{equation}
Here we recall that
\[
J_L=\Im(\overline{D_L\phi}\cdot \phi)=r^{-2}\Im(\overline{D_L\psi}\cdot \psi),\quad \psi=r\phi.
\]
The integral on the initial hypersurface $B_{r_1, r_2}$ is finite for all $p\leq 1+\ga_0$ by the assumption (<ref>). We will get a useful $r$-weighted energy inequality once we can control
the error term arising from the charge part which involves estimates for the above $J_L$ depending only on the scalar field. In general $r^{-1}\psi$ has the same size as $D_L\psi$. So if the charge $q_0$
is sufficiently small, independent of the initial data, we then can absorb the error term which has no positive sign in the above inequality (<ref>). However, in our setting, the charge $q_0$
is arbitrarily large. Then a possible approach to control this term is to use Gronwall's inequality. The problem is that as having explained the error term has the same decay rate with those terms on the left
hand side of the above energy equality. In other words, we will get a logarithmic growth of the error term instead of boundedness when using Gronwall's inequality.
In order to overcome this potential logarithmic growth, we make use of the better decay of the initial data. More precisely, we are not going to pursue a $r$-weighted energy estimates with the largest possible $p$
value which is $1+\ga_0$ by the initial condition. We instead consider the $r$-weighted energy estimates with the greatest $p$ which is slightly less than $1+\ga_0$. Let
\[
\ep_0=\f12(\ga_0-\ga),
\]
where $\ga<\ga_0$ is the positive constant in the main Theorem <ref>. First using Cauchy-Schwarz's inequality, we can bound
\begin{equation}
\label{csjl}
|q_0\iint_{\mathcal{D}_{r_1, r_2}}r^{p-2}J_L dxdt|\leq \frac{p}{2}\iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}|D_L\psi|^2dudvd\om+\frac{8q_0^2}{p}\iint_{\mathcal{D}_{r_1, r_2}}r^{p-3}|\psi|^2dudvd\om.
\end{equation}
The first term on the right hand side can be absorbed in the previous estimate (<ref>). To estimate the weighted integral of $|\psi|^2$, we first note
\begin{equation}
\label{eqfact}
\pa_L|\psi|\leq |D_L\psi|.
\end{equation}
This is because these are gauge invariant. We can simply assume $\psi$ is real. Then on the fixed outgoing null hypersurface $S_{\tau}$, we can show that
\begin{equation}
\label{Sobpsi}
\begin{split}
\int_{\om}|\psi|^2(u, v, \om)d\om&\les \int_{\om}|\psi|^2(u, v_1, \om)d\om+\int_{\om}(\int_{v_1}^v|D_L\psi|dv)^2d\om\\
&\les\int_{\om}|\psi|^2(u, v_1, \om)d\om+r_1^{-\ga}\int_{v_1}^v\int_{\om}r^{1+\ga}|D_L\psi|^2dvd\om \cdot
\end{split}
\end{equation}
Here $r=\frac{v-u}{2}$, $\tau=\frac{R-r}{2}$ and $v_1<v$. The implicit constant depends only on $\ga$. For $p\leq 1+\ga$, multiply the above inequality by $r^{p-3}$ and then
integrate it from $v_1$ to $v$ on $S_\tau$. We obtain the following estimate:
\begin{equation}
\label{SovpsiI}
\begin{split}
&\int_{v_1}^v\int_{\om}r^{p-3}|\psi|^2(u, v, \om)d\om dv\\
&\les r_1^{p-2}\int_{\om}|\psi|^2(u, v_1, \om)d\om+r_1^{-\ga+p-2}\int_{v_1}^v\int_{\om}r^{1+\ga}|D_L\psi|^2dvd\om \cdot
\end{split}
\end{equation}
The improved estimate for the integral of $|\psi|^2$ comes from the first term on the right hand side of the above estimate as we can choose $v_1=-u$ which means that the point $(u, v_1, \om)$ is on
the initial hypersurface. Denote
\begin{equation*}
E_{r_1, r_2}^p=\f12\int_{B_{r_1,r_2}}r^p(|D_L\psi|^2+|\D\psi|^2+r^{-2}|\psi|^2)+r^{p+2}(|\a|^2+|\tilde{\rho}|^2+\si^2)drd\om.
\end{equation*}
Our assumptions on the initial data implies that $E_{r_1, r_2}^{1+\ga_0}$ is finite. Let $p=1+\ga$ and $v_1=-u$ in (<ref>). From the energy identity (<ref>), the Cauchy-Schwarz's inequality
(<ref>) and the above Sobolev embedding (<ref>), we can conclude that
\begin{equation}
\label{psigron0}
\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om\leq C_1 E_{r_1, r_2}^{1+\ga}+C_1\int_{r_1}^{r_2}s^{-1}\int_{S_{s, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om ds
\end{equation}
for some constant $C_1$ depending only on $\ga$ and the charge $q_0$. Without loss of generality, we can assume $C_1>1$.
Since $s^{-1}$ is not integrable, Gronwall's inequality will lead to a logarithmic growth. We fix $r_2$ and let
$r_1$ be the variable. Denote
\[
G(r_1)=\int_{r_1}^{r_2}s^{-1}\int_{S_{s, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om ds.
\]
Then we have
\[
C_1 E_{r_1, r_2}^{1+\ga}+C_1 G(r_1)+r_1 G'(r_1)\geq 0, \quad G(r_2)=0,\quad G'(r_2)=0,\quad r_1\leq r_2.
\]
Multiply the above inequality by $r_1^{C_1-1}$ and then integrate it from $r_1$ to $r_2$. We obtain
\begin{equation}
\label{psigron1}
G(r_1)\leq C_1 r_1^{-C_1}\int_{r_1}^{r_2}s^{C_1-1}E_{s, r_2}^\ga ds\leq r_1^{-C_1} E_{r_1, r_2}^{C_1+1+\ga}\leq r_1^{-C_1}r_2^{C_1}E_{r_1, r_2}^{1+\ga}\leq E_0 r_1^{-C_1-2\ep_0}r_2^{C_1}.
\end{equation}
Here recall that $E_0=E_{R,\infty}^{1+\ga_0}$ and $2\ep_0=\ga_0-\ga$. Then from the previous two estimates, we conclude that
\begin{equation}
\label{lowerscal}
\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om\leq 2E_0 r_1^{-C_1-2\ep_0}r_1^{C_1+\ep_0}=2E_0 r_1^{-\ep_0},\quad \forall r_1\leq r_2\leq r_1^*= r_1^{1+\frac{\ep_0}{C_1}}.
\end{equation}
Here by our definition, $C_1$ is a constant depending only on the charge $q_0$ and $\ga$. The above estimate implies that due to the better decay of the initial data, we can improve the $r$-weighted energy
estimate with the largest $p$ value for smaller $r_2$. For those points when $r_2$ is large, we rely on the negative $r$ weights in (<ref>).
Fix $u$ and consider the outgoing null hypersurface $S_{r_1, r_2}$ for
$r_2>r_1^*=r_1^{1+\frac{\ep_0}{C_1}}$. For $v\geq u+r_1^*$, let $v_1=u+r_1^*$ in the Sobolev embedding (<ref>). We then can estimate that
\begin{align*}
\int_{S_{r_1, r_2}}r^{\ga-2}|\psi|^2dvd\om&\les \int_{S_{r_1, r_1*}}r^{\ga-2} |\psi|^2dvd\om+(r_1^*)^{\ga-1}\int_{\om}|\psi|^2(u, -u, \om)dvd\om\\
&+(r_1^*)^{\ga-1}r_1^{-\ga}\int_{S_{r_1, r_1^*}}r^{1+\ga}|D_L\psi|^2dvd\om+(r_1^*)^{-1}\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om\\
&\les r_1^{\ga-1}\int_{\om}|\psi|^2(u, -u, \om)dvd\om+r_1^{-1}\int_{S_{r_1, r_1^*}}r^{1+\ga}|D_L\psi|^2dvd\om\\
&\quad+r_1^{-1-\frac{\ep_0}{C_1}}\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om\\
&\les r_1^{\ga-1}\int_{\om}|\psi|^2(u, -u, \om)dvd\om+E_0r_1^{-1-\ep_0}\\
&\quad+r_1^{-1-\frac{\ep_0}{C_1}}\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om.
\end{align*}
Here note that $r_1^*\geq r_1$ and $\ga\leq 1$. In the last line of the previous estimate, the first term can be bounded by the assumption on the initial data. The second term is integrable in terms of $r_1$. The
improved decay rate in the last term allows us to use Gronwall's inequality. We comment here that the above estimate holds for all $r_2\geq r_1$ as when $r_2\leq r_1^*$ the integral on the
left hand side can be controlled by the first two terms on the right hand side. Now back to the $r$-weighted energy identity (<ref>), for $r_2\geq r_1*$ and $p=1+\ga$, the above calculations imply that
\begin{align*}
\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om\leq C_2 E_0 r_1^{-\ep_0}+C_2 \int_{r_1}^{r_2}s^{-1-\frac{\ep_0}{C_1}}\int_{S_{s, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om ds
\end{align*}
for a constant $C_2$ depending only on the charge $q_0$ and $\ga$. The above estimate holds for all $R\leq r_1\leq r_2$. Gronwall's inequality then implies that
\begin{equation}
\label{pWe1ga}
\int_{S_{r_1, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om+\int_{r_1}^{r_2}s^{-1-\frac{\ep_0}{C_1}}\int_{S_{s, r_2}}r^{1+\ga}|D_L\psi|^2dvd\om ds\leq C_3 E_0 r_1^{-\ep_0}
\end{equation}
for some constant $C_3$ depending only on $q_0$, $\ga$. This estimate is sufficient to bound the error term arising from the charge. From (<ref>), we have
\begin{align*}
|q_0\iint_{\mathcal{D}_{r_1, r_2}}r^{p-2}J_L dxdt|&\leq \frac{p}{2}\iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}|D_L\psi|^2dudvd\om+\frac{8q_0^2}{p}\iint_{\mathcal{D}_{r_1, r_2}}r^{p-3}|\psi|^2dudvd\om\\
&\leq \frac{p}{2}\iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}|D_L\psi|^2dudvd\om+C_4 E_{r_1, r_2}^{p}+C_4 E_0 r_1^{-\ga+p-1-\ep_0}\\
& \leq\frac{p}{2}\iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}|D_L\psi|^2dudvd\om+2C_4 E_0 r_1^{-\ga_0+p-1},\quad p\leq 1+\ga
\end{align*}
for some constant $C_4$ depending only on $q_0$, $\ga$. Then from the $r$-weighted energy identity (<ref>), we derive the final $r$-weighted energy estimate we need:
\begin{equation}
\label{pWeMKGout1}
\begin{split}
& \iint_{\mathcal{D}_{r_1, r_2}}r^{p-1}(|D_L\psi|^2+|\D\psi|^2)+r^{p+1}(|\a|^2+|\tilde{\rho}|^2+\si^2)dvd\om du\\
&+\int_{S_{r_1,r_2}}r^p(|D_L\psi|^2+r^2|\a|^2)dvd\om+\iint_{\mathcal{D}_{r_1, r_2}}r^{p-3}|\psi|^2dudvd\om
\les E_0 r_1^{p-1-\ga_0}
\end{split}
\end{equation}
for all $R\leq r_1\leq r_2$, $0\leq p\leq 1+\ga<1+\ga_0$. Here based on our notation, the implicit constant depends only on $q_0$, $\ga$, $\ga_0$. The estimate for the integral of $|\psi|^2$ is
derived from the previous estimate.
Once we have the $r$-weighted energy estimate as well as the control of the integral of $|\phi|^2$, we can improve the integrated local energy estimates and the energy estimates. Take $\tilde{F}=F-\bar F$
in the integrated local energy estimate (<ref>) and the energy estimate (<ref>). We have
\[
|\tilde{F}-F|=|\bar F|\leq |q_0|r^{-2}.
\]
Take $p=\ep$ in the $r$-weighted energy estimate (<ref>). We get
\begin{equation*}
\begin{split}
\iint_{\mathcal{D}_{r_1, r_2}}(1+r)^{1+\ep}|F_{r\nu}-\tilde{F}_{r\nu}|^2|\phi|^2dxdt&\les \iint_{\mathcal{D}_{r_1, r_2}}(1+r)^{\ep-3}|\psi|^2dudvd\om\\
&\les E_0 r_1^{\ep-1-\ga_0}.
\end{split}
\end{equation*}
Here recall that $\psi=r\phi$. Then from the integrated local energy estimate (<ref>) and the energy estimate (<ref>), we can show that
\begin{align*}
&r_1^\ep\iint_{\mathcal{D}_{r_1, r_2}}\frac{|\bar D\phi|^2+|\tilde{F}|^2}{(1+r)^{1+\ep}}+\frac{|\D\phi|^2+\tilde\rho^2+\si^2}{1+r}dxdt\\
&\les E[\phi, \tilde{F}](B_{r_1, \infty})+E_0r_1^{-\ep-1-\ga_0}+\iint_{\mathcal{D}_{r_1, r_2}}|F_{0\nu}-\tilde{F}_{0\nu}||J^\nu|dxdt\\
&\les E_0r_1^{-\ep-1-\ga_0}+\iint_{\mathcal{D}_{r_1, r_2}}\ep_1 r_1^\ep\frac{|\bar D\phi|^2}{(1+r)^{1+\ep}}+\ep_1^{-1}r_1^{-\ep}(1+r)^{\ep-3}|\phi|^2dxdt\\
&\les E_0r_1^{-1-\ga_0}(1+\ep_1^{-1})+\ep_1 r_1^\ep\iint_{\mathcal{D}_{r_1, r_2}}\frac{|\bar D\phi|^2}{(1+r)^{1+\ep}}dxdt
\end{align*}
for all $\ep_1>0$. Here notice that
\[
E[\phi, \tilde{F}](B_{r_1, \infty})\leq E_0 r_1^{-1-\ga_0}.
\]
We also remark here that the regularity of the initial data is propagated in the exterior region as $E[\phi, \tilde{F}](B_{r_1, \infty})$ is the dominant term in the above estimate.
Take $\ep_1$ to be sufficiently small depending only on $q_0$, $\ep$, $\ga$, $\ga_0$ so that the last term in the last line can be absorbed. We derive the integrated local energy
estimate in the exterior region
\begin{align*}
&r_1^\ep\iint_{\mathcal{D}_{r_1, r_2}}\frac{|\bar D\phi|^2+|\tilde{F}|^2}{(1+r)^{1+\ep}}+\frac{|\D\phi|^2+\tilde\rho^2+\si^2}{1+r}dxdt\les E_0r_1^{-1-\ga_0}.
\end{align*}
The previous estimate also gives control of the integral of $|F_{0\mu}-\tilde{F}_{0\mu}||J^\mu|$. Hence from the energy estimate (<ref>), we obtain the energy control of the solution in the exterior region
\begin{equation}
\label{esMKGout1}
E[\phi, \tilde{F}](S_{r_1, r_2})+E[\phi, \tilde{F}](\bar C_{r_1, r_2})\les E_0r_1^{-1-\ga_0}
\end{equation}
for all $R\leq r_1<r_2$. Let $r_2\rightarrow \infty$. We obtain the decay estimates for the chargeless part of the solution in the exterior region.
§ ENERGY ESTIMATES IN THE INTERIOR REGION
In the previous section, we obtained the energy estimates in the exterior region. In particular, we have estimates for the energy flux as well as the $r$-weighted energy
flux through $S_{R, \infty}$ or $S_0$ which is the outgoing null hypersurface starting from the sphere with radius $R$ on the initial hypersurface. These boundary information allows us to obtain
energy estimates for the solutions of the MKG equations in the interior region with foliation $\Si_{\tau}$.
From the $r$-weighted energy estimate (<ref>) and the energy estimate (<ref>) in the exterior region, we in particular have
\begin{equation*}
\begin{split}
&\int_{S_{R, r_2}}r^{1+\ga}(|D_L\psi|^2+r^2|\a|^2)dvd\om\les E_0 R^{\ga-\ga_0}\les E_0, \\
& E[\phi, F](S_{R, r_2})\les E[\phi, \tilde{F}](S_{R, r_2})+\int_{S_{R, r_2}}q_0^2 r^{-4}r^2dvd\om\les E_0
\end{split}
\end{equation*}
for all $r_2> R$. Since the implicit constant is independent of $r_2$, take $r_2\rightarrow \infty$. We obtain
\begin{equation}
\label{bdcin}
\int_{S_{0}}r^{1+\ga}(|D_L\psi|^2+r^2|\a|^2)dvd\om+ E[\phi, F](\Si_{0})\les E_0.
\end{equation}
In the interior region we take $\tilde{F}=F$. From the integrated local energy estimate (<ref>) and the energy estimate (<ref>) in the interior region, we obtain
\begin{equation}
\label{ILEMKGin1}
\begin{split}
& E[\phi, F](\Si_{\tau_2})+E[\phi, F](\bar C_{\tau_1, \tau_2})
+\int_{\tau_1}^{\tau_2}\int_{\Si_{\tau}}\frac{|\bar D\phi|^2+|F|^2}{(1+r)^{1+\ep}}+\frac{|\D\phi|^2+\rho^2+\si^2}{1+r}dxdt\\
&\les E[\phi, F](\Si_{\tau_1})
\end{split}
\end{equation}
for all $0\leq \tau_1<\tau_2$. The improved integrated local energy estimate for $\D\phi$, $\rho$, $\si$ will be used to control the
boundary terms in the $r$-weighted energy estimates. Since $\tilde{F}=F$, the $r$-weighted energy identity (<ref>) implies that
\begin{align}
\notag
&\int_{S_{\tau_2}}r^p|D_L\psi|^2+r^{p+2}|\a|^2dvd\om+\int_{\bar C_{\tau_1, \tau_2}}r^p(|\D\psi|^2+r^2\rho^2+r^2\si^2)dud\om\\
\label{pWeMKGin}
&+\int_{\tau_1}^{\tau_2}\int_{S_\tau}r^{p-1}\left(p(|D_L\psi|^2+r^2|\a|^2)+(2-p)(|\D\psi|^2+r^2\rho^2+r^2\si^2)\right)dvd\om d\tau\\
\notag
=&\int_{S_{\tau_1}}r^p|D_L\psi|^2+r^{p+2}|\a|^2dvd\om-\f12 R^p\int_{\tau_1}^{\tau_2}\int_{\om}|D_L\psi|^2-|\D\psi|^2+R^2(|\a|^2-\rho^2-\si^2) d\om dt.
\end{align}
For $p\in[0, 2]$, every term in the above identity has a positive sign except the last term which is the integral on the boundary of the cylinder with radius $R$. However, we observe that
that term is a constant multiple of $R^p$ which means that we can simply take $p=0$ in the above $r$-weighted energy identity in order to estimate the boundary term. From the above energy estimate (<ref>)
and the bound (<ref>), we conclude that the energy flux through the null infinity $\bar C_{\tau_1, \tau_2}$ is finite. Since the scalar field $\phi$ vanishes at the spatial infinity initially,
we have that the scalar field must also vanish at null infinity. Then using Sobolev embedding and the relation (<ref>), on $S_{\tau}$, $\tau\geq0$, we have
\[
r\int_{\om}|\phi|^2d\om \leq \int_{S_\tau}|D_L\phi|^2r^2dvd\om\leq E[\phi, F](\Si_{\tau}).
\]
Here similar to Lemma <ref> of Hardy's inequality, we can choose a particular gauge so that $\phi$ is real. Therefore the above estimate holds for all connection $D$. Note that
\[
|D_L\psi|^2=r^2|D_L\phi|^2+L(r|\phi|^2),\quad \psi=r\phi.
\]
The previous estimate then implies that
\begin{align*}
\int_{S_\tau}|D_L\psi|^2dvd\om \leq \int_{S_{\tau}}|D_L\phi|^2r^2dvd\om+\int_{\om}r|\phi|^2d\om\leq 2E[\phi, F](\Si_{\tau}).
\end{align*}
Note that $\D\psi=r\D\phi$. Let $p=0$ in the above $r$-weighted energy identity (<ref>). From the energy estimate (<ref>), we can estimate the boundary terms as follows:
\begin{align*}
& \left|\f12 R^p\int_{\tau_1}^{\tau_2}\int_{\om}|D_L\psi|^2-|\D\psi|^2+R^2(|\a|^2-\rho^2-\si^2) d\om dt\right|\\
&\leq 2E[\phi, F](\Si_{\tau_1})+2E[\phi, F](\Si_{\tau_2})+E[\phi, F](\bar C_{\tau_1, \tau_2})+2\int_{\tau_1}^{\tau_2}\int_{S_\tau}\frac{|\D\phi|^2+\rho^2+\si^2}{r}dxdt\\
&\les E[\phi, F](\Si_{\tau_1}).
\end{align*}
Therefore for $0<p<2$, the $r$-weighted energy identity (<ref>) implies that
\begin{align}
\notag
&\int_{S_{\tau_2}}r^p(|D_L\psi|^2+r^{2}|\a|^2)dvd\om+\int_{\tau_1}^{\tau_2}\int_{S_\tau}r^{p-1}(|D_L\psi|^2+|\D\psi|^2+r^2(|\a|^2+\rho^2+\si^2))dvd\om d\tau\\
\label{pWeMKGineq}
\les&\int_{S_{\tau_1}}r^p|D_L\psi|^2+r^{p+2}|\a|^2dvd\om+ E[F, \phi](\Si_{\tau_1}).
\end{align}
Here the implicit constant depends also on $R$ and $p$. Take $p=1+\ga$ in the above $r$-weighted energy inequality. The boundary condition (<ref>) shows that
\begin{align}
\label{pWeMKGinga}
\notag
&+\int_{\tau_1}^{\tau_2}\int_{S_\tau}r^{\ga}(|D_L\psi|^2+|\D\psi|^2)+r^{2+\ga}(|\a|^2+\rho^2+\si^2)dvd\om d\tau\les E_0.
\end{align}
We want to retrieve the integral of the energy flux through the whole hypersurface $\Si_{\tau}$ from the $r$-weighted energy estimate with $p=1$. We thus can make use of the integrated local energy
estimate (<ref>) restricted to the region $r\leq R$. First we note that when $r\leq R$, using Sobolev embedding, we have
\[
\int_{\om}|\phi|^2d\om\les \int_{r\leq R}|D\phi|^2+|\phi|^2dx.
\]
Then from the $r$-weighted energy estimate (<ref>) with $p=1$ we can show that
\begin{align}
\notag
\int_{\tau_1}^{\tau_2}E[\phi, F](\Si_{\tau})d\tau&=\int_{\tau_1}^{\tau_2}\int_{r\leq R}|D\phi|^2+|F|^2dxdt+\int_{\tau_1}^{\tau_2}\int_{\om}|\phi|^2(\tau, R, \om)d\om d\tau\\
\label{pWeMKGin1}
&\quad +\int_{\tau_1}^{\tau_2}\int_{S_\tau}|D_L\psi|^2+|\D\psi|^2+r^2(|\rho|^2+|\a|^2+|\si|^2) dvd\om d\tau\\
\notag
&\les \int_{S_{\tau_1}}r|D_L\psi|^2+r^{3}|\a|^2dvd\om+ E[\phi, F](\Si_{\tau_1})\\
\notag
&\quad+\int_{\tau_1}^{\tau_2}\int_{r\leq R}|\bar D\phi|^2+|F|^2dxdt\\
\notag
&\les \int_{S_{\tau_1}}r|D_L\psi|^2+r^{3}|\a|^2dvd\om+ E[\phi, F](\Si_{\tau_1}).
\end{align}
The $r$-weighted energy
inequality (<ref>) implies that there is a dyadic sequence $\{\tau_{n}\}$ such that
\[
\int_{S_{\tau_{n}}}r^{\ga}|D_L\psi|^2+r^{\ga+2}|\a|^2dvd\om\les E_0 (1+\tau_n)^{-1},\quad \forall n.
\]
Since (<ref>) also implies that
\[
\int_{S_{\tau}}r^{1+\ga}|D_L\psi|^2+r^{\ga+3}|\a|^2dvd\om\les E_0,\quad \forall \tau\geq 0,
\]
interpolation implies that
\[
\int_{S_{\tau_{n}}}r|D_L\psi|^2+r^{3}|\a|^2dvd\om\les E_0 (1+\tau)^{-\ga}, \quad \forall n.
\]
The energy estimate (<ref>) implies that the energy flux $E[\phi, F](\Si_\tau)$ is nonincreasing with respect to $\tau$. Then from the $r$-weighted energy estimate (<ref>) for $p=1$, we
can show that
\begin{align*}
(\tau-\tau_n)E[\phi, F](\Si_{\tau})\les E_0 (1+\tau_n)^{-\ga}+E[\phi, F](\Si_{\tau_n}),\quad \forall \tau\geq \tau_n.
\end{align*}
In particular, for $n=0$, we obtain
\[
E[\phi, F](\Si_{\tau})\les E_0(1+\tau)^{-1},\quad \forall \tau\geq 0.
\]
Since the sequence $\{\tau_n\}$ is dyadic, for $\tau\in [\tau_{n+1}, \tau_{n+2}]$, we have
\[
E[\phi, F](\Si_{\tau})\les (\tau-\tau_n)^{-1}(E_0(1+\tau_n)^{-\ga}+E_0(1+\tau_n)^{-1})\les E_0(1+\tau)^{-1-\ga}.
\]
From the integrated local energy estimate (<ref>), the above decay of the energy flux then implies the decay of the integrated local energy in the interior region. This finishes the proof of the
main Theorem.
DPMMS, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge, UK CB3 0WA
Email address: [email protected]
|
1511.00244
|
Holographic Mutual Information for Singular Surfaces
M. Reza Mohammadi Mozaffar$\, ^{\dag}$, Ali Mollabashi$\, ^{\dag}$ and Farzad Omidi$\, ^{\ddag}$
$^\dag$ School of Physics, $^\ddag$ School of Astronomy,
Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
Emails: m$_{-}$mohammadi, mollabashi, and [email protected]
We study corner contributions to holographic mutual information for entangling regions composed of a set of disjoint
sectors of a single infinite circle in 3-dimensional conformal field theories. In spite of the UV divergence of holographic mutual information, it exhibits a first order phase transition.
We show that tripartite information is also divergent for disjoint sectors, which is in contrast with the well-known feature of tripartite information being finite even when entangling regions share boundaries.
We also verify the locality of corner effects by studying mutual information between regions separated by a sharp annular region.
Possible extensions to higher dimensions and hyperscaling violating geometries is also considered for disjoint sectors.
§ INTRODUCTION
Quantum entanglement is one of the important features of quantum mechanics that emerges in several areas of physics including condensed matter, quantum information and black hole physics.
In the context of quantum field theories (QFTs), the notion of entanglement can reflect various features of the theory, depending on how the Hilbert space of the theory is decomposed. This leads to different types of entanglement in QFTs including those which are denoted in the literature by spatial (or geometric) entanglement <cit.>, momentum space entanglement <cit.>, and field space entanglement <cit.>.
Here we focus on geometric entanglement, which has been widely studied during recent years, in order to extend some recent developments in the context of corner contributions to different entanglement measures.
Entanglement entropy is a well-known measure to quantify quantum entanglement. In the context of geometrical entanglement, in order to define entanglement entropy (EE) we consider a spatial region $V$ on a constant time slice of a $d$ dimensional field theory. Assuming that the total Hilbert space has a decomposition as $\mathcal{H}_{\rm{tot.}}=\mathcal{H}_{V}\otimes \mathcal{H}_{\bar{V}}$, one can define a reduced density matrix corresponding to region $V$ by integrating out the degrees of freedom living in its geometrical complement $\bar{V}$, i.e., $\rho_V=\Tr_{\bar{V}}\rho_{\rm{tot.}}$. Entanglement entropy is then defined as the von Nuemann entropy for this reduced density matrix which is given by $S_{EE}=-\Tr \rho_V \log \rho_V$.
Entanglement entropy in QFTs is a UV-divergent quantity which its leading divergence is proportional to the area of the entangling region $V$, due to the leading contribution of the near-boundary local degrees of freedom i.e. <cit.>
\begin{align}\label{area}
S_{EE}= c^{(d-2)}\frac{\mathcal{A}_V}{\epsilon^{d-2}}+\cdots+s_{\rm univ.}^{(d)}+\mathcal{O}\left(\frac{\epsilon}{\ell}\right),\;\;\;\;\;s_{\rm univ.}^{(d)}=\begin{cases}
c_e^{(d)} \log \frac{\ell}{\epsilon}&~~ d:\rm{even}\\
c_o^{(d)} &~~ d:\rm{odd}
\end{cases}.
\end{align}
In the above expression $1/\epsilon$ is the UV cut-off, $\mathcal{A}_V$ is the area of the entangling surface, $\ell$ is the characteristic length of $V$, and $s_{\rm univ.}^{(d)}$ encodes some universal information about the QFT. Note that by varying $\epsilon$ the two constants $c_e^{(d)}$ and $c_o^{(d)}$ does not change and they are called universal in this sense. The well-known example of this universal information occurs in two dimensional conformal field theories where $c_e^{(2)}$ is proportional to the central charge of the theory <cit.>.
Although the computation of EE even in the simplest case of two dimensional conformal field theories is not an easy task, for the case of field theories supporting a gravity dual in the context of AdS/CFT correspondence <cit.>, one can apply the RT prescription <cit.> to calculate the holographic entanglement entropy (HEE). This set up is based on finding a minimal area surface in the bulk of an asymptotically AdS geometry. This surface anchors on the boundary of $V$ on the conformal boundary of the bulk and the corresponding HEE is given by
\begin{align}
\end{align}
Using this prescription Eq.(<ref>) is reproduced for strongly coupled conformal field theories supporting a gravitational dual <cit.>.
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[blue!60!black,thick] (7.95,0) node $\mathbf{\Omega}$;
Left: The blue plane represents a constant time slice of a $d=3$ CFT with a kink ($k$) entangling region on it. Right: A crease ($k\times R^m$) entangling region as a direct generalization of the kink in higher dimensions.
The structure of the universal terms appearing in the EE depends on the geometry of the entangling region. If the entangling region contains a singularity, it has been shown that extra divergent terms appear in the entanglement entropy <cit.>. Two simple types of singular entangling regions which are known as kink (in $d=3$) and crease (for $d\ge 4$) are shown in Fig.<ref>. The simplest example which is a kink in three dimensions with opening angle $\Omega$ and length $H$ has an extra logarithmic divergent term that modifies Eq.<ref> as
\begin{align}
S_{EE}=c^{(1)} \frac{\mathcal{A}_V}{\epsilon}-a(\Omega)\log \frac{H}{\epsilon}+c_0+\mathcal{O}\left(\frac{\epsilon}{H}\right),
\end{align}
where the function $a(\Omega)$ encodes some universal information about the theory and for a pure state it has a symmetric property as $a(\Omega)=a(2\pi-\Omega)$. The strong subadditivity property and Lorentz invariance of EE impose more constraints on $a(\Omega)$. In the large angle (smooth) limit and small angle (sharp) limit of the opening angle $\Omega$ one finds
\begin{align}
a(\Omega\rightarrow 0)=\frac{\kappa}{\Omega}+\cdots\;\;\;\;,\;\;\;\;a(\Omega\rightarrow \pi)=\sigma(\pi-\Omega)^2+\cdots,
\end{align}
where $\kappa$ and $\sigma$ correspond to some characteristics of the underlying CFT. In particular recently it was shown that the constant $\sigma$ is proportional to the central charge appearing in the two point function of the energy-momentum tensor as $\sigma=\frac{\pi^2}{24}C_T$, where this relation has some universal properties <cit.>. The existence and further generalizations of this universal ratio in more general cases is studied in several directions in
An important feature of the new divergence appearing in such entangling regions is the extensive contribution of its coefficient, $a(\Omega)$, to entanglement entropy which could be easily understood from the locality of the field theory.[Here locality means a one-to-one correspondence between Hilbert space decomposition and factorization of the spatial manifold of the QFT.] Since the UV contributions to entanglement entropy from separate points are supposed not to see each other, if there are more than one corner in the entangling region one may expect an extensive contribution to entanglement entropy from each corner (see <cit.>). We will come back to this point in this section and also in the body of this paper again, specifically when we study mutual information between sectors of an infinite circle.
As mentioned above, the EE for a single line segment in two dimensional CFTs is completely fixed by the central charge. In order to gain some information about the field content of the corresponding CFT, one needs to probe the theory by means of more powerful entanglement measures. In particular mutual information is such a measure which is defined for two disjoint entangling regions $A_1$ and $A_2$ as follows
\begin{align}\label{mutual}
I(A_1, A_2)=S_{A_1}+S_{A_2}-S_{A_1\cup A_2},
\end{align}
where $S_{A_1\cup A_2}$ is the entanglement entropy for the union of two entangling regions.
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[] (-1,0.1) node[left] $S_{\text{dis.}}:$;
[] (9,0.1) node[left] $S_{\text{con.}}:$;
[] (1,0) node[below] $A_1$;
[] (3.5,0) node[below] $A_2$;
[] (11,0) node[below] $A_1$;
[] (13.5,0) node[below] $A_2$;
Two different configurations for computing $S(A_1\cup A_2)$ using RT prescription.
Mutual information is a finite and positive quantity for specific entangling regions which measures the amount of entanglement shared between $A_1$ and $A_2$ regions. It is important to mention that mutual information diverges when the separation between disjoint regions vanishes and they share a boundary. In reference <cit.> it was shown that MI is not only a function of the central charge of the corresponding two dimensional CFT rather it depends on the full operator content of the theory.
On the other hand in the context of holographic CFTs, using the RT prescription it has been shown that holographic mutual information (HMI) exhibits a first order phase transition due to a discontinuity in its first derivative <cit.>. It is believed that this phase transition is a reminiscent of the large central charge limit of the CFT and it disappears if one considers quantum corrections <cit.>. The gravity picture of this phase transition is simply due to a jump between two different configurations candidate for the minimal surface of the entanglement entropy of the union region $S_{A_1\cup A_2}$ (see Fig.<ref>).
This figure demonstrates two possible configurations corresponding to the HEE for the union of two entangling regions.
HMI either vanishes or takes a finite value depending on the value of the ratio between the length of the entangling regions and their separation. It vanishes on one side of a critical value for this ratio and takes a finite value on the other side.
This can be understood as follows: when the disjoint regions are close enough together there is a finite correlation between them, but as they get far apart, the mutual correlation decreases and finally vanishes.
In this paper we are mainly considering three dimensional field theories which we expect the mutual information between infinite strips and disks to be UV-finite. For the case of singular entangling regions since an extra UV-divergent term appears in the entanglement entropy one should consider the relevant divergent terms precisely. For a configuration like that in the right panel of Fig.<ref>, since the singular points are far away from each other one may expect mutual information to still be a UV-finite quantity because of the extensivity of corner contributions to entanglement entropy in local field theories <cit.> which was discussed before. This is not the case when the distance between these points vanishes like the left panel of Fig.<ref>. In such a case the singular points share a local region and thus the extensivity of the corner divergent terms breaks down.[We thank Matthew Headrick for referring and specially thank Horacio Casini for an insightful discussion about this point.] The main part of this paper is an example of this latter case in section <ref>. We also investigate an explicit example of the former case in section <ref> to explicitly show the finiteness of mutual information even between singular regions.
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[densely dashed,blue!20!black] (-1mm,0mm) circle(.1mm);
[densely dashed,blue!20!black] (-5.3mm,0mm) circle(.1mm);
Right: Two singular entangling regions which their singularities are far away from each other. The dashed circle denotes the local region responsible for the main UV-divergent part of the singularity. As long as these points are far from each other, which means their distance $d$ is much larger than the UV cut-off $1/\epsilon$, they contribute to the entanglement entropy (between the union of these regions and its complement) extensively. Left: As these two singular regions get closer to each other (and finally touch), their local regions responsible for the new UV-divergent term intersect with each other and thus their contribution to entanglement entropy is no more extensive.
Beside mutual information, similar quantities are also defined to deal with disjoint regions. Specifically for a system which is composed of at least three disjoint subsystems, another quantity which is called tripartite information is defined as <cit.>
\begin{align}\label{tripartite0}
I^{[3]}(A_1,A_2,A_3)=S_{A_1}+S_{A_2}+S_{A_3}-S_{A_1\cup A_2}-S_{A_1\cup A_3}-S_{A_2\cup A_3}+S_{A_1\cup A_2\cup A_3},
\end{align}
where the $A_i$'s refer to disjoint entangling regions. Tripartite information for smooth subregions is known to be a UV-finite quantity. Remember that while we are dealing with two smooth subregions, if these regions share a boundary the mutual information between them is no longer a UV-finite quantity. In contrast with mutual information, tripartite information is known to be UV-finite even when the smooth regions share boundaries <cit.>. Again we show in section <ref> that this is not the case for singular surfaces when the shared boundary is the same as the singular point. This could be understood as a straightforward generalization of what was explained in Fig.<ref>.
Another important property of tripartite information is the sign of this quantity. In contrast with mutual information, tripartite information in general may be either negative, positive or zero. In the holographic context it has been shown that holographic tripartite information has a definite sign and it is always negative. This property is also known as the monogamy property of HMI <cit.>.
The main goal of this paper is to investigate the holographic mutual and tripartite information and their possible phase transitions in the presence of a kink or a crease singularity in the entangling region. Indeed finding the minimal area surface corresponding to the union of entangling regions is a subtle task. This problem is already only solved analytically for parallel strips or concentric circles[These concentric circles can be mapped to two disjoint disks using a conformal transformation <cit.>.] as entangling regions <cit.>. In particular the authors of <cit.> have introduced an elegant numerical method using Surface Evolver to overcome this difficulty. By employing this numerical technique they could find HMI for various more complicated entangling regions[See <cit.> where the generalization of this method to more general backgrounds is also considered.]. Of course here we do not employ this numerical method. We show that a simple observation can help us to find the HMI for sectors of a single infinite circle using the result of HEE for a kink entangling region in three dimensions. We also show that for more general configurations of this type one can calculate other entanglement measures e.g. holographic tripartite and $n$-partite information.
The remainder of this paper is organized as follows: in section <ref> we briefly review the related literature on holographic entanglement entropy for singular surfaces. In section <ref> we introduce our geometric set-up and study the holographic mutual, tripartite and $n$-partite information where we mainly focus on possible phase transitions of holographic mutual information. In section <ref> we study the possible generalizations of our set-up in section <ref> from three dimensional CFTs to higher dimensions. In section <ref> we study mutual information between singular regions without a shared boundary as an explicit example were the new divergent term is extensive, in contrast with what was studied in <ref> and <ref> where we only study singular surfaces with a common singular point. In the last section beside a summary and some concluding remarks we also discuss about constructing a finite quantity from mutual information for singular surfaces and also discuss about the first law of entanglement for such configurations. Appendix <ref> is devoted to a comparison between singular surfaces in the sharp limit with the well-known results for strip entangling regions. In appendix <ref> we give some results for the entanglement entropy of a kink in holographic theories with a hyperscaling violating geometry as their gravity dual.
§ HOLOGRAPHIC ENTANGLEMENT ENTROPY (REVISITED)
In this section we shortly review the holographic entanglement entropy for singular surfaces which has been previously studied in <cit.>. As mentioned in the previous section in a three dimensional theory only one type of singularity is possible which we refer to it by kink singularity (see the left panel of Fig.<ref>). Most of the analysis of this paper is devoted to this type of singularity. In higher dimensions various types of singularities are possible where two specific ones known as crease (see the right panel of Fig.<ref>) and cone have been studied previously. We will study crease entangling regions in section <ref>.[We do not consider conical entangling regions in this paper. The interested reader may refer to <cit.>.]
In order to clarify the definition of these surfaces consider a $d$ dimensional flat space time, i.e., $R^{1,d-1}$, as follows
\begin{align}\label{metric1}
ds^2=-dt^2+d\rho^2+\rho^2\left(d\theta^2+\sin^2\theta\; d\Omega_n^2\right)+\sum_{i=1}^{m}dx_i^2
\end{align}
where $d=n+m+3$ and $d\Omega_n$ corresponds to the metric of unit $n$-sphere. Following the terminology of <cit.> a kink is defined for $d=3$ and $m=n=0$ which is given by $k=\{t=0, 0<\rho<\infty, -\frac{\Omega}{2}\leq\theta \leq\frac{\Omega}{2}\}$. A crease ($k\times R^{m}$) is the extension of kink to higher dimensions with $n=0$ and $d=3+m$. In this section we only consider kinks in three dimensions, i.e., $m=0$ so the bulk geometry is given by an AdS$_4$ space-time with the following metric
\begin{align}\label{metric}
ds^2=\frac{L^2}{z^2}\left(dz^2-dt^2+d\rho^2+\rho^2 d\theta^2\right),
\end{align}
where the spatial part of the boundary metric is considered in polar coordinates with $\rho$ and $\theta$ as the radial and azimuthal angle respectively. Also $z$ is the radial coordinate of the bulk geometry.
The kink entangling region in three dimensions is defined as
\begin{equation}\label{entangling}
t=\mathrm{const.}\;\;\;,\;\;\;0<\rho< H\;\;\;,\;\;\;-\frac{\Omega}{2}\leq\theta \leq\frac{\Omega}{2},
\end{equation}
where $\Omega$ is the opening angle and $H$ is an IR cut-off on the radial coordinate. Due to the scaling symmetry on the bulk radial coordinate and the boundary coordinates, it was shown in <cit.> that the RT surface in the bulk can be parametrized as $z(\rho,\theta)=\rho\;h(\theta)$ such that $h(\pm\frac{\Omega}{2})=0$. See Fig.<ref> for a schematic plot of the minimal surface. In this case the HEE functional becomes
\begin{align}\label{heefunc}
S=\frac{L^2}{2G_N}\int_{\frac{\epsilon}{h_*}}^H \frac{d\rho}{\rho}\int_0^{\frac{\Omega}{2}-\delta}d\theta\frac{\sqrt{1+h^2+h'^2}}{h^2},
\end{align}
where $\epsilon$ is the inverse UV cut-off defined by $\epsilon=\rho\;h(\frac{\Omega}{2}-\delta)$, and $h_*=h(0)$ is the turning point of the RT surface in the bulk which is defined where $h'(0)=0$ (prime denotes the derivative with respect to $\theta$).
Schematic representation of RT surface for a kink entangling region with opening angle $\Omega$ in a three dimensional CFT.
Applying the standard variational principle leads to the surface which minimizes this functional. According to Eq.(<ref>) where the integrand does not depend explicitly on $\theta$, the corresponding Hamiltonian is a conserved quantity
\begin{align}\label{thetamom}
\mathcal{H}\equiv \frac{1+h^2}{h^2\sqrt{1+h^2+h'^2}}=\frac{\sqrt{1+h_*^2}}{h_*^2}.
\end{align}
Using this relation it is an easy task to find the profile of the minimal surface and hence the HEE. The final result for the HEE is
\begin{align}\label{HEE}
S(\Omega)=\frac{L^2}{2G_N}\frac{H}{\epsilon}-a(\Omega)\log \frac{H}{\epsilon}-\left(\frac{\pi L^2}{4G_N h_*}+a(\Omega)\log h_*\right)+\mathcal{O}\left(\frac{\epsilon}{H}\right),
\end{align}
where the function $a(\Omega)$ is defined as
\begin{align}\label{aomega}
a(\Omega)=\frac{L^2}{2G_N}\int_0^\infty dy\left[1-\sqrt{\frac{1+h_*^2(1+y^2)}{2+h_*^2(1+y^2)}}\right],
\end{align}
and the boundary data (the opening angle $\Omega$) is given in terms of the bulk turning point $h_*$ as
\begin{align}\label{omehah0}
\Omega =\int_0^{h_*} dh\frac{2h^2\sqrt{1+h_*^2}}{\sqrt{1+h^2}\sqrt{h_*^4(1+h^2)-h^4(1+h_*^2)}}.
\end{align}
Left: $\Omega/\pi$ as a function of the turning point $h_*$. Right: $a$ as a function of the opening angle $\Omega$. In both plots the red and violet curves correspond to the small opening angle and smooth limit, respectively.
It is interesting to focus on two specific limits where the kink is extremely sharp $(\Omega \rightarrow 0)$ or it is a smooth surface which is slightly folded $(\Omega \rightarrow \pi)$. In these two limits one can work out the HEE analytically as follows <cit.>:
1) Sharp limit $(\Omega \rightarrow 0)$
\begin{align}\label{smallregion}
\begin{split}
\Omega &=\frac{2\sqrt{\pi}\Gamma(3/4)}{\Gamma(1/4)}h_*-\frac{[3\Gamma^2(3/4)-\Gamma(1/4)\Gamma(5/4)]}{6\sqrt{2\pi}}h_*^3+\cdots,\\
a(\Omega)&=\frac{\kappa}{\Omega}-\frac{L^2}{G_N}\frac{\pi \Gamma(1/4)}{48\sqrt{2}\Gamma^3(3/4)}\Omega+\cdots,\\
S(\Omega)&=\frac{L^2}{2G_N}\frac{H}{\epsilon}-\left[\frac{L^2}{2G_N}\frac{\pi^{3/2}\Gamma(3/4)}{\Gamma(1/4)}+\kappa \left(\log \frac{H}{\epsilon}+\log \Omega+\log\frac{2\Gamma(5/4)}{\sqrt{\pi}\Gamma(3/4)}\right)\right]\frac{1}{\Omega}+\cdots\\
&\sim \frac{L^2}{2G_N}\frac{H}{\epsilon}-\frac{\kappa}{\Omega}\left(\log \frac{H}{\epsilon}+\log \Omega\right)+\cdots,
\end{split}
\end{align}
where $\kappa=\frac{L^2}{2\pi G_N}\Gamma^4(3/4)$. Note that assuming $\Omega\ll\frac{H}{\epsilon}$ we will neglect the last term in the above expression for the HEE in the following discussion.
2) Smooth limit $(\Omega \rightarrow \pi)$
\begin{align}\label{largeregion}
\begin{split}
\Omega &=\pi-\frac{\pi}{h_*}+\cdots\\
S(\Omega)&=\frac{L^2}{2G_N}\frac{H}{\epsilon}-\frac{L^2}{2G_N}\frac{\pi-\Omega}{2}-\sigma \left(\log \frac{H}{\epsilon}-\log \left(1-\frac{\Omega}{\pi}\right)\right)(\pi-\Omega)^2+\cdots,
\end{split}
\end{align}
where $\sigma=\frac{L^2}{8\pi G_N}$.
Fig.<ref> demonstrates the behavior of $\Omega(h_*)$ and $a(\Omega)$. In this figure we also included the above asymptotic results which coincide with the exact results in a wide range of opening angles.
Note that in this figure we have set $\frac{L^2}{2G_N}=1$.
§ HOLOGRAPHIC ENTANGLEMENT MEASURES ($D=3$)
In this section we aim to study holographic entanglement measures with our main focus on holographic mutual information between kink entangling regions in three dimensions. As mentioned in the introduction section the main subtlety to compute quantities such as mutual information is how to compute the last term in Eq.(<ref>). The core of our idea is to consider the set of kinks in interest as sectors of an infinite circle (see Fig.<ref> and Fig.<ref>). In such a configuration the holographic entanglement entropy of a union of two kink entangling regions can be expressed in terms of minimal surfaces anchoring to certain kinks on the boundary. See Fig.<ref> and Fig.<ref> for schematic graphical visualizations.
§.§ Holographic Mutual Information
In this subsection we study the holographic mutual information for the specific configuration mentioned above. The entangling regions are sectors of a single infinite circle with opening angles $\Omega_1$ and $\Omega_2$. These entangling regions touch each other at a single point which is the center of the infinite circle. We denote the angular separation between these regions with $\omega$ (see Fig. <ref>). Note that for a bipartite entangling region there are always two sectors in-between $\Omega_1$ and $\Omega_2$ sectors which we define $\omega$ to be the minimum value of the opening angle between these two sectors. By this definition the value of $\omega$ is restricted to $0\le \omega \le \pi-\left(\Omega_1+\Omega_2\right)/2$. We will consider regions with no angular overlap thus we will always assume $\Omega_1+\Omega_2+\omega < 2\pi$.
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The configuration of a bipartite entangling region. $\Omega_i$'s are the opening angles and $\omega$ is the angular separation and the radial coordinate which runs over $0\le \rho<\infty$.
Schematic representation of the RT surfaces corresponding to $S_{\Omega_1 \cup \Omega_2}$ for disconnected (left) and connected (right) configurations.
Considering such entangling regions, from Eq.(<ref>) one finds the mutual information between these sectors as
\begin{align}\label{mutual1}
I(\Omega_1, \Omega_2)=S_{\Omega_1}+S_{\Omega_2}-S_{\Omega_1 \cup \Omega_2}.
\end{align}
The first two terms in the above equation are easy to compute and the main challenge is to find the suitable expression for the last one, i.e., $S_{\Omega_1 \cup \Omega_2}$. Indeed there are two different configurations which are competing for the value of this term. These two configurations which we denote them by “connected" and “disconnected" configurations are shown schematically in Fig.<ref>. According to this figure the value of $S_{\Omega_1 \cup \Omega_2}$ is given by one of the following expressions
\begin{align}\label{SAUB}
S_{\Omega_1 \cup \Omega_2}=
\begin{cases}
S_{\Omega_1+\Omega_2+\omega}+S_{\omega}\left(\equiv S_{\text{con.}}\right) & ~~ \omega\ll \{\Omega_1,\Omega_2\}\\
S_{\Omega_1}+S_{\Omega_2}\left(\equiv S_{\text{dis.}}\right) & ~~ \omega\gg \{\Omega_1,\Omega_2\}
\end{cases}
\end{align}
depending on the value of three parameters $\Omega_1$, $\Omega_2$, and $\omega$. Note that the contribution of the disconnected configuration is independent of the value of $\omega$. As we will see our numerical results show that there is always a transition between connected and disconnected configurations varying the values of these parameters.
In order to simplify our calculations, we restrict the following discussion to the case of equal opening angles, i.e., $\Omega_1=\Omega_2=\Omega$. In this case we can find a simple explicit expression for HMI which is given by
\begin{align}\label{equalregionsMI}
I&=2\tilde{S}_\Omega-\min\{2\tilde{S}_\Omega, \tilde{S}_{2\Omega+\omega}+\tilde{S}_\omega\},\nonumber\\
\tilde{S}_\Omega &=-a(\Omega) \log \frac{H}{\epsilon}-\left(\frac{\pi L^2}{4G_N h_*(\Omega)}+a(\Omega)\log h_*(\Omega)\right),\nonumber\\
\tilde{S}_\omega &=-a(\omega) \log \frac{H}{\epsilon}-\left(\frac{\pi L^2}{4G_N h_*(\omega)}+a(\omega)\log h_*(\omega)\right),\nonumber\\ \tilde{S}_{2\Omega+\omega}&=-a(2\Omega+\omega)\log \frac{H}{\epsilon}-\left(\frac{\pi L^2}{4G_N h_*(2\Omega+\omega)}+a(2\Omega+\omega)\log h_*(2\Omega+\omega)\right).
\end{align}
Left: Comparing the contributions of connected (solid curve) and disconnected (dashed curve) configurations to the universal part of HEE with $\Omega=\frac{\pi}{4}$. Middle: The coefficient of the universal part of HMI as a function of $\omega$ for $\Omega=\frac{\pi}{4}$. Right: The coefficient of the universal part of HMI as a function of $\Omega$ for $\omega=\frac{\pi}{4}$.
Density plot of the universal part of HMI for regions with equal opening angles. The dashed green curve which is simply a guide to the eye is the transition curve and the dashed red boundary is $2\Omega+\omega <2\pi$ constraint. The white region below this boundary corresponds to the divergence of HMI in the small separation limit.
In Fig.<ref> we have summarized the behavior of the universal part of HMI in this simple set-up. In the left plot the contributions of connected and disconnected configurations to the universal part of HEE for a fixed opening angle are compared. These quantities are defined as follows
\begin{align}
a_{\rm dis.}=2a_{\Omega},\;\;\;a_{\rm con.}=a_{2\Omega+\omega}+a_{\omega}.
\end{align}
In the middle and right plots we have demonstrated the universal part of HMI which is defined as
\begin{align}\label{aI}
a_{I}=-a_{\rm dis.}-\min \{-a_{\rm dis.},-a_{\rm con.}\}
\end{align}
as a function of $\omega$ and $\Omega$ respectively. According to these plots by fixing the opening angles and increasing the separation between the regions the HMI vanishes. Also Fig.<ref> is a density plot which shows the behavior and transition points of the universal part of HMI. It is not hard to show that the area divergent contributions to HMI cancel out and the remaining $\epsilon$-dependence is logarithmic.
Now we can further investigate the transition curve of the HMI in the parameter space. Such a curve is supposed to satisfy
\begin{align}\label{eq:a1}
\end{align}
which depends on $\epsilon$. One can easily check that as $\epsilon\to 0$, solutions of Eq.(<ref>) merge to a cut-off independent curve which we call the universal transition curve for HMI. This universal curve is the solution of
\begin{align}\label{eq:a2}
\end{align}
where the logarithmic UV divergence of HMI cancels out. The universal transition curve is shown by a solid red curve in Fig.<ref>, where the dotted orange line corresponds to the boundary of the $\omega+\Omega\le \pi$ constraint which was mentioned previously.
The solid red curve shows the universal transition curve for HMI. HMI is non zero (positive) below the universal transition curve which is bounded from the other side by the boundary of the geometrical constraint $\omega+\Omega\le\pi$ shown by the dashed orange curve.
In order to further explore the HMI for such configurations, we focus on the sharp kink and the slightly folded limits where we are able to give analytical expressions for HEE and thus HMI.
(i) Sharp limit $\{\omega,\;\Omega\}\ll 1$:
In this case combining Eq.(<ref>) and Eq.(<ref>) leads to
\begin{align}\begin{split}\label{Ismall}
I&=-\frac{2\kappa}{\Omega}\log\frac{H}{\epsilon}+\min\left\{\frac{2}{\Omega},\frac{1}{2\Omega+\omega}+\frac{1}{\omega}\right\}\kappa \log\frac{H}{\epsilon}\\
\begin{cases}
\left(-\frac{2}{\Omega}+\frac{1}{2\Omega+\omega}+\frac{1}{\omega}\right)\kappa \log\frac{H}{\epsilon}\sim\frac{\kappa}{\omega} \log\frac{H}{\epsilon} & ~~ \omega\ll \Omega \ll 1\\
0 & ~~ \Omega\ll \omega \ll 1.
\end{cases}
%&=\Bigg\{ \begin{array}{rcl}
%&\left(-\frac{2}{\Omega}+\frac{1}{2\Omega+\omega}+\frac{1}{\omega}\right)\kappa \log\frac{H}{\epsilon}\sim\frac{\kappa}{\omega} \log\frac{H}{\epsilon}&\,\,\,\,\,\omega\ll \Omega \ll 1,\\
%&0&\,\,\,\,\,\Omega\ll \omega \ll 1.
\end{split}
\end{align}
Actually this final result can be understood considering a conformal map relating the corner geometry to a strip (see appendix A of <cit.>).
This conformal map is given by
\begin{equation}\label{cmap}
t=L e^{\frac{Y}{L}}\cos \xi\;\;\;,\;\;\;\rho=L e^{\frac{Y}{L}}\sin \xi,
\end{equation}
and the entangling region (<ref>) in the new geometry becomes
\begin{equation}
\xi=\frac{\pi}{2}\;\;\;,\;\;\;Y_-<Y<Y_+\;\;\;,\;\;\;-\frac{\Omega}{2}\leq\theta \leq\frac{\Omega}{2},
\end{equation}
where $Y_+=L\log \frac{H}{L}$ and $Y_-=L\log \frac{\epsilon}{L}$. Now by making use of the HEE for a strip the mutual information for two parallel strips with width $\ell$ and separation $x$ is given by <cit.>
\begin{align}
\begin{cases}
\kappa(Y_+-Y_-)\left(-\frac{1}{\ell}+\frac{1}{2(2\ell+x)}+\frac{1}{2x}\right) & ~~ x\ll \ell \\
0 & ~~ x \gg \ell
\end{cases},
%\Bigg\{ \begin{array}{rcl}
%&\kappa(Y_+-Y_-)\left(-\frac{1}{\ell}+\frac{1}{2(2\ell+x)}+\frac{1}{2x}\right)&\,\,\,\,\,x\ll \ell ,\\
%&0&\,\,\,\,\,x \gg \ell.
\end{align}
where $Y_{\pm}$ are the regulator scales for the length of the strip. In the small opening angle limit , i.e., $\{\Omega, \omega\} \ll 1$, one finds $\ell\equiv \Omega L \ll L$ and $x\equiv \omega L \ll L$ and the above expression reduces to Eq.(<ref>). Also note that in $x\ll \ell$ limit only the last term contributes.
Density plot of the universal part of HMI for $\omega=\frac{\pi}{12}$ (left) and $\omega=\frac{\pi}{6}$ (right). The dashed red curve is the boundary of the $\Omega_1+\Omega_2+2\omega< 2\pi$ constraint.
(ii) Smooth limit $\omega \sim 0,\; \Omega \sim \pi,\;\;\;\Omega+\omega=\pi$:
In this case the fact that we always consider a pure state requires that $S_{2\Omega+\omega}=S_{2\pi-\omega}=S_{\omega}$ so Eq.(<ref>) becomes
\begin{align}
I=2S_\Omega-2\min\{S_\Omega, S_\omega\}.
\end{align}
Using Eq.(<ref>) and Eq.(<ref>) for $S_\Omega$ and $S_\omega$ respectively the HMI at the leading order reduces to
\begin{align}\label{Ilarge}
I&=-\frac{L^2}{2G_N}\omega-\min\left\{-\frac{L^2}{2G_N}\omega,-\frac{2\kappa}{\omega}\log\frac{H}{\epsilon}\right\}\sim \frac{2\kappa}{\omega}\log\frac{H}{\epsilon}.
\end{align}
In order to investigate the behavior of HMI in this singular configuration more generally, we consider the case where the opening angles of the entangling regions are not equal, i.e., $\Omega_1\neq \Omega_2$. Fig.<ref> demonstrates the density plot of the universal part of HMI for different values of $\omega$ as a function of entangling opening angles. These plots show the transition points of this quantity and also regions with non-vanishing HMI.
§.§ Holographic Tripartite Information
In this section we study the holographic tripartite information between kink entangling regions which are sectors of a single infinite circle. Tripartite information is defined as follows
\begin{align}\label{tripartite}
I^{[3]}(A_1,A_2,A_3)=S_{A_1}+S_{A_2}+S_{A_3}-S_{A_1\cup A_2}-S_{A_1\cup A_3}-S_{A_2\cup A_3}+S_{A_1\cup A_2\cup A_3},
\end{align}
where the $A_i$'s are entangling regions, which in our case are sectors of a single infinite circle with opening angles $\Omega_i,\;i=1, 2, 3$ and angular separation $\omega_j,\;j=1, 2$. Again we have excluded the maximum opening angle between three different choices to define $\omega_j$'s (see Fig.<ref>). In order to avoid the overlap between different regions we assume $\sum_{ij} (\Omega_i+\omega_j)< 2\pi$. Again as in the case of mutual information we reduce the parameter space from five independent parameters to two by considering the simplest case with $\Omega_i=\Omega, \omega_i=\omega$.
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The configuration of a tripartite entangling region for computing tripartite information, $\Omega_i$'s are the opening angles and $\omega_i$'s are the two smaller angular separations between them. The radial coordinate runs over $0\le \rho<\infty$.
Here the main challenge to calculate holographic tripartite information is finding the minimal area for the union of subsystems. Similar to what we did in the previous subsection, here we can find the relevant configurations for the entanglement entropies appearing in Eq.(<ref>) as minimal surfaces of kink entangling regions with certain opening angles. These configurations are schematically plotted in Fig.<ref>. In the extreme cases the minimal configurations are
\begin{align}
%S_{\Omega_2\cup \Omega_i}=\Bigg\{ \begin{array}{rcl}
%&S_{2\Omega+\omega}+S_{\omega}\equiv S^{(1)}_{\text{dis.}}&\,\,\,\,\,\omega \ll \Omega,\\
%&2S_{\Omega}\equiv S^{(2)}_{\text{dis.}}&\,\,\,\,\,\omega\gg \Omega,
%\end{array}\,\,\;\;\;\;\;i=1 \;\text{or}\; 3,
S_{\Omega_2\cup \Omega_i}=
\begin{cases}\label{union1}
S_{2\Omega+\omega}+S_{\omega}\left(\equiv S^{(1)}_{\text{dis.}}\right) & ~~ \omega \ll \Omega\\
2S_{\Omega}\left(\equiv S^{(2)}_{\text{dis.}}\right) & ~~ \omega\gg \Omega
\end{cases}
\end{align}
for $i=1,3$ and
\begin{align}\label{union2}
%S_{\Omega_1\cup \Omega_3}=\Bigg\{ \begin{array}{rcl}
%&2S_{\Omega}\equiv S^{(2)}_{\text{dis.}}&\,\,\,\,\,\omega\ll \Omega,\\
%&S_{3\Omega+2\omega}+S_{\Omega+2\omega}\equiv S^{(3)}_{\text{dis.}}&\,\,\,\,\,\omega\gg \Omega,
S_{\Omega_1\cup \Omega_3}=
\begin{cases}
2S_{\Omega}\left(\equiv S^{(2)}_{\text{dis.}}\right) & ~~ \omega\ll \Omega\\
S_{3\Omega+2\omega}+S_{\Omega+2\omega}\left(\equiv S^{(3)}_{\text{dis.}}\right) & ~~ \omega\gg \Omega
\end{cases}
\end{align}
and for the case of union of three subsystems
\begin{align}\label{union3}
%S_{\Omega_1\cup \Omega_2\cup \Omega_3}=\Bigg\{ \begin{array}{rcl}
%&S_{3\Omega+2\omega}+2S_{\omega}\equiv S_{\text{con.}}&\,\,\,\,\,\omega\ll \Omega,\\
%&S_{2\Omega+\omega}+S_{\Omega}+S_{\omega}\equiv S^{(4)}_{\text{dis.}}&\,\,\,\,\,\\
%&3S_{\Omega}\equiv S^{(5)}_{\text{dis.}}&\,\,\,\,\,\omega\gg \Omega,\\
%&S_{3\Omega+2\omega}+S_{\Omega+2\omega}+S_{\Omega}\equiv S^{(6)}_{\text{dis.}}&\,\,\,\,\,
S_{\Omega_1\cup \Omega_2\cup \Omega_3}=
\begin{cases}
S_{3\Omega+2\omega}+2S_{\omega}\equiv S_{\text{con.}} & ~~ \omega\ll \Omega\\
S_{2\Omega+\omega}+S_{\Omega}+S_{\omega}\equiv S^{(4)}_{\text{dis.}} & ~~ \omega\gg \Omega\\
3S_{\Omega}\equiv S^{(5)}_{\text{dis.}} & ~~ \omega\gg \Omega\\
S_{3\Omega+2\omega}+S_{\Omega+2\omega}+S_{\Omega}\equiv S^{(6)}_{\text{dis.}} & ~~ \omega\gg \Omega
\end{cases}.
\end{align}
One can calculate the tripartite information by making use of Eq.(<ref>) together with Eq.'s(<ref>), (<ref>), (<ref>) and (<ref>). It is easy to check that the area law divergences cancel out in tripartite information but the logarithmic divergences does not cancel out and the resultant tripartite information is UV-divergent. This could be understood as a straightforward generalization of what was explained in the introduction section below Fig.<ref> and where we referred to it. As an explicit example we have plotted the universal part of this quantity i.e. $a_{I^{[3]}}$, in Fig.<ref> (This universal part is defined similar to Eq.(<ref>) but using the definition of tripartite information). Fig.<ref> also shows the density plot of this quantity. According to these plots the holographic tripartite information in this singular set-up is negative as expected.
Schematic representation of RT surfaces corresponding to $S_{\Omega_1\cup \Omega_2\cup \Omega_3}$ for computing the holographic tripartite information.
Using the small opening angle expansion $\{\Omega,\omega\}\rightarrow 0$ one can also study the behavior of tripartite information semi-analytically. In this case one finds
\begin{align}
I^{[3]}(\Omega, \omega)=\begin{cases}
-\frac{\kappa}{3\Omega}\log\frac{H}{\epsilon}& ~~ \omega\ll \Omega\\
0 & ~~ \omega\gg \Omega\\
\end{cases},
\end{align}
which the final result still depends on the inverse UV cut-off $\epsilon$. In order to derive this result we use the similar analysis as in deriving Eq.(<ref>).
This behavior is in contrast with the well-known feature of tripartite information in the literature, which is thought to be finite even when the regions share boundaries <cit.>. See appendix <ref> for some details about the finiteness of tripartite information for smooth entangling regions in various dimensions.
Universal part of the holographic tripartite information for $\Omega=\frac{\pi}{6}$ (left) and $\omega=\frac{\pi}{4}$ (right) with $H=1$.
§.§ Holographic $n$-partite Information
In this section we generalise our previous study to the case of holographic $n$-partite information. This quantity is a simple generalization of mutual and tripartite information to systems consisting of $n$ disjoint subsystems. The definition of $n$-partite information is as follows <cit.>
\begin{align}\label{eq:npartite}
I^{[n]}(A_{\{i\}})=\sum_{i=1}^nS_{A_i}-\sum_{i<j}^n S_{A_i\cup A_j}+\sum_{i<j<k}^n S_{A_i\cup A_j\cup A_k}
-\cdots\cdots -(-1)^n S_{A_1\cup A_2\cup\cdots\cup A_n},
\end{align}
where for $n=2$ and $n=3$ it reduces to the definition of mutual and tripartite information.[Actually the generalisation to generic $n$ is not unique, but we choose this definition to reproduce the tripartite information for $n=3$. Other definition e.g., multipartite-information does not satisfy this constraint<cit.>.] This quantity is finite and in a general quantum system it can be either negative, positive or zero. Holographic computations show that in certain limits it has a definite sign, positive (negative) for even (odd) $n$<cit.>.
In order to simplify the computations we only consider the case where all the opening angles and separations between them are equal, i.e., $\Omega_1=\Omega_2=\cdots=\Omega_n\equiv\Omega$ and $\omega_1=\omega_2=\cdots=\omega_{n-1}\equiv\omega$. For such a choice similar to the case of mutual information and tripartite information we have a geometric constraint $\Omega+\omega<2\pi/n$. Actually the analysis for finding the $n$-partite information in this singular set-up is very similar to the case of strip entangling regions <cit.> so we just demonstrate the final results (similar analysis for computing HEE for multiple strips has been done in <cit.>). The simplest example is $n=4$ which corresponds to 4-partite information. Fig.<ref> shows the behavior of the universal part of holographic 4-partite information as a function of $\Omega$ and $\omega$. Also in the small opening angle limit $\{\Omega,\omega\}\rightarrow 0$ one finds
\begin{align}
I^{[4]}(\Omega, \omega)=\begin{cases}
\frac{\kappa}{12\Omega}\log\frac{H}{\epsilon}& ~~ \omega\ll \Omega\\
0 & ~~ \omega\gg \Omega\\
\end{cases}.
\end{align}
One can also find the small angle expression for the holographic $n$-partite information for generic $n$ as follows [Note that similar to previous cases, in order to avoid the overlap between different regions we assume $\sum_{ij}\Omega_i+\omega_j< 2\pi$ where $i=1,\cdots, n$ and $j=1,\cdots, n-1$.]
\begin{align}
I^{[n]}(\Omega \sim 0, \omega\sim 0)=\begin{cases}
(-1)^n\frac{2\kappa}{n(n-1)(n-2)\Omega}\log\frac{H}{\epsilon}& ~~ \omega\ll \Omega\\
0 & ~~ \omega\gg \Omega\\
\end{cases}.
\end{align}
Density plot of universal part of the holographic tripartite information for regions with equal opening angles. The dashed red boundary corresponds to $\Omega+\omega<\frac{2\pi}{3}$ constraint.
The above result shows that in this specific construction, considering small separation angles, the holographic $n$-partite information has a definite sign, i.e., it is positive (negative) for even (odd) $n$. Note that using the conformal map which is given by Eq.(<ref>) and the result of $n$-partite information for a set of strips with equal width $\ell$ and separation $x$, considering the $\ell\gg x$ limit (see <cit.>) one can reproduce the above result.
§ HOLOGRAPHIC ENTANGLEMENT MEASURES IN HIGHER DIMENSIONS
In this section we generalise our studies to higher dimensional cases in a specific direction where the entangling region is a crease, i.e., $k\times R^m$ (see the right panel of Fig.<ref>). To do so we first review the computation of HEE for crease entangling regions and continue with computing the holographic mutual and tripartite information for this generalization of the specific configurations we considered in the previous section. The HEE of such entangling regions has been studied in <cit.>.
Universal part of the holographic 4-partite information for $\Omega=\frac{\pi}{6}$ (left) and $\omega=\frac{\pi}{6}$ (right) with $H=1$.
The dual geometry we consider here is an AdS$_{d+1}$ space-time with the following metric in cylindrical coordinates which is the specific case of $n=0$ and $m=d-3$ of Eq.(<ref>)
\begin{align}
ds^2=\frac{L^2}{z^2}\left(dz^2-dt^2+d\rho^2+\rho^2 d\theta^2+\sum_{i=1}^{d-3}\;dx_i^2\right).
\end{align}
We are interested in the following entangling region
\begin{equation}
t=\mathrm{const.}\;\;\;,\;\;\;0<\rho< H\;\;\;,\;\;\;-\frac{\Omega}{2}\leq\theta \leq\frac{\Omega}{2},\;\;\;0<x_i<\tilde{H},
\end{equation}
where both $H$ and $\tilde{H}$ are IR regulators. Due to the symmetries we assume $z=z(\rho, \theta)$, thus the HEE functional becomes
\begin{align}\label{higherdim}
S=\frac{L^{d-1}\tilde{H}^{d-3}}{4G_N}\int d\rho d\theta \frac{\sqrt{\dot{z}^2+\rho^2+\rho^2{z'}^2}}{z^{d-1}},
\end{align}
where $z'=\partial_\rho z$ and $\dot{z}=\partial_\theta z$. Similar to the three dimensional case, using the scaling symmetry, here we consider $z(\rho, \theta)=\rho\;h(\theta)$ and find a conserved quantity as follows <cit.>
\begin{align}\label{Hd}
\mathcal{H}_d\equiv \frac{(1+h^2)^{\frac{d-1}{2}}}{h^{d-1}\sqrt{1+h^2+{h'}^2}}=\frac{(1+h_*^2)^{\frac{d-2}{2}}}{h_*^{d-1}}.
\end{align}
Using this equation the opening angle $\Omega$ is related to the turning point as
\begin{align}
\Omega=\int_0^{h_*}dh \frac{2\mathcal{H}_dh^{d-1}}{\sqrt{1+h^2}\sqrt{(1+h^2)^{d-2}-\mathcal{H}_d^2h^{2(d-1)}}}.
\end{align}
Left: $\Omega/\pi$ as a function of the turning point $h_*$ in different dimensions. Right: $j$ as a function of the opening angle $\Omega$ in different dimensions.
The HEE could be found as
\begin{align}\label{EEhighd}
\end{align}
\begin{align}
j(\Omega)=\int_{h_*}^0 dh\;h^{d-3}J(h)-\frac{1}{h_*}\;\;\;\;,\;\;\;\;J(h)=\frac{\sqrt{1+h^2+{h'}^2}}{h'h^{d-1}}+\frac{1}{h^{d-1}},
\end{align}
noting that $h'$ is determined in terms of $h$ and $h_*$ via Eq.(<ref>). Eq.(<ref>) shows that the logarithmic divergent term in the previous section was a reminiscent of three dimensional field theory and in higher dimensions a new power law divergent term appears. It is important to note that this new power law divergence, i.e., $\epsilon^{-(d-3)}$ does not appear if we consider smooth entangling regions. It is known from reference <cit.> that this behavior is due to adding a flat locus, i.e., $R^m$ to the kink $k$.
Fig.<ref> demonstrates the behavior of $\Omega(h_*)$ and $j(\Omega)$ in higher dimensions. From this plot one can deduce that $j(\Omega)$ vanishes in the smooth limit as expected.
Having the expression for the HEE, in principle we can compute the mutual information Eq.(<ref>) between higher dimensional creases holographically. The subtlety of finding the minimal surface for union of the creases arises again in this case. The difficulty is similar to what happens while dealing with multi-strip entangling regions in higher dimensions. Remember that for a line segment entangling region in a two dimensional theory, the corresponding RT surface is a semi-circle ($s^1$) in an AdS$_3$ geometry. While we are dealing with an infinite strip in AdS$_{d+1}$, the corresponding RT surface is $\tilde{s}^1\times R^{d-2}$ where $\tilde{s}$ refers to a modified semi-circle <cit.>. Using this simple generalization, the authors of <cit.> have found the HMI for strip entangling regions in higher dimensions. However it is not possible to imagine the embedding of the corresponding RT surface for $S_{A_1\cup A_2}$ in the bulk for $d>3$. In the case of our study, where we generalize a kink $k$ in $d=3$ to a crease $k\times R^{d-3}$ in higher dimensions, again there is enough symmetry leading to a conserved quantity (see Eq.(<ref>)). So we expect that the corresponding RT surfaces for computing the HEE for union of crease regions are just simple generalisation of three dimensional case. Note that for disk or spherical entangling regions this procedure does not work any more <cit.>. This makes us to conclude that if we had considered a cone (instead of a crease) as a generalization of a kink in three dimensions, it would be impossible to construct the corresponding RT surfaces analytically.
Normalized holographic mutual (left) and tripartite (right) information in various spatial dimensions for $\Omega=\frac{\pi}{4}$.
Now using the above argument one can find the HMI in our set-up when the opening angles and separation angle are given by $\Omega_1,\;\Omega_2$ and $\omega$ respectively. Doing so the area law divergences trivially cancel out in the expression of HMI and we are left with
\begin{align}
I(\Omega_1, \Omega_2)=\frac{L^{d-1}\tilde{H}^{d-3}}{2G_N}\left[j(\Omega_1)+j(\Omega_2)-\min \left\{j(\Omega_1)+j(\Omega_2),j(\Omega_1+\Omega_2+\omega)+j(\omega)\right\}\right]\frac{1}{(d-3)\epsilon^{d-3}}.
\end{align}
In order to simplify the computations we only consider the equal opening angle case, where the resultant HMI is plotted in Fig.<ref> for various spatial dimensions. In this figure we have plotted a normalized holographic mutual information defined as $\tilde{I}=(d-3)\epsilon^{d-3}I$. This figure shows that the transition always exists, however it depends on UV cut-off similar to the three dimensional case. Similar analysis can be done in the case of holographic tripartite information, see Fig.<ref>. Also note that in order to plot this figure we have set $\frac{L^{d-1}\tilde{H}^{d-3}}{2G_N}=1$.
§ MUTUAL INFORMATION BETWEEN SHARP CONCENTRIC CIRCLES
In the previous sections we studied some entanglement measures including mutual information for configurations which the entangling regions had a common point which was the same as their singular point. We argued in the introduction section (see the caption of Fig.<ref> and where we referred to it) that for such configurations we do not expect mutual information and even tripartite information to be UV-finite quantities. What if we study e.g. mutual information between singular surfaces which do not share a boundary? In such a case we expect the result to be again a UV-finite quantity.
In this section we are going to explore another explicit example for mutual information between singular surfaces in holographic three dimensional conformal field theories. We aim to provide evidence for UV cut-off independence of the mutual information in three dimensions between two singular regions without a common boundary.
Consider two entangling regions $A$ and $B$ separated by an annular region again in the vacuum state of a three dimensional CFT. Region $A$ is a disk with radius $R_-$ and $B$ is the outer region (complement) of a larger disk with radius $R_+$ (see left panel of Fig.<ref>). Mutual information for such a configuration has been studied in details for certain field theories and also for holographic field theories in <cit.>. It would be interesting to consider a slight deformation on the annular region in-between these regions such that it brings in two corners for each entangling region with the same opening angle (see the right panel of Fig.<ref>). The question is whether the mutual information between $A$ and $B$ is still cut-off dependent as a result of the corners appeared perturbatively or not?
[blue!40!black,dashed] (0cm,0cm) circle(2mm);
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[blue!40!black] (-1.5cm,-1mm) node $A$;
[blue!40!black] (-20mm,0mm) node $B$;
[blue!40!black] (-16mm,0mm) node $R_-$;
[blue!40!black] (-12.8mm,2mm) node $R_+$;
[color=blue!40!black,thick,domain=0:3.14,samples=50,smooth] plot (canvas polar
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cs:angle=r,radius= 11.4*1+11.4*.2*sin(r));
[red] (.17cm,.2cm) node $\delta$;
[red] (.17cm,-.2cm) node $\delta$;
[red] (.36cm,.28cm) node $\delta$;
[red] (.36cm,-.28cm) node $\delta$;
Left: Regions $A$ and $B$ which are separated by an annular region. Right: Deformation of the annular region with two singular points at the poles.
We again start with an AdS$_4$ space-time in the Poincare patch as the holographic dual of the three dimensional CFT
and consider the boundary of the entangling regions defined as
Following the method developed to consider generic perturbations on a spherical entangling region in <cit.> and also <cit.> where the entanglement entropy of a particular choice of such deformations is analysed, we consider the following perturbations on the boundaries of the entangling region
1-δ sin(n ϕ) 0≤ϕ<π
1+δ sin(n ϕ) π≤ϕ<2π
where $n$ is a positive integer number controlling the geometry of the perturbed entangling region and $\delta$ is a small parameter controlling the perturbation. Such a generic perturbation makes it possible to study various types of entangling regions but since we are interested in the specific deformation illustrated in the right panel of Fig.<ref>, from now on we restrict our analysis to the case $n=1$.
In order to study the entanglement entropy holographically for such deformed regions and thus compute the mutual information between them, we consider a minimal surface in the bulk parametrized as $\rho(z,\phi)$. The induced metric on such an extended surface in the bulk geometry is
𝒜=∫dz dϕ1/z^2√(ρ^2(1+ρ'^2)+ρ̇^2)
where prime and dot indicate differentiation with respect to $z$ and $\phi$ respectively. The above functional could be minimized up to $\mathcal{O}\left(\delta^2\right)$ via
where $\rho_0(z)$, $\rho_1(z)$ and $\rho_{20}(z)$ must satisfy the following equations
\begin{align}\label{eq:deq}
\begin{split}
\end{split}
\end{align}
Perturbed minimal surfaces versus unperturbed surfaces for $R_-=0.8$, $R_+=1$, $\phi=\pi/2$. The blue dashed curve represents the connected minimal surface and the dashed green ones represent the disconnected minimal surface for $\delta=0$. The red and orange curves represent the connected and disconnected minimal surfaces for $\delta=0.05$.
The solution for the first equation is known to be $\rho_0(z)=\pm\sqrt{R^2-z^2}$ for two symmetric branches. Note that we have ignored $\rho_{22}(z)$ function since it is shown in <cit.> that it does not contribute to the first non-trivial correction in entanglement entropy.
Left: The mutual information for the same parameters as a function of $\epsilon$. The dashed green line represents the mean value for mutual information. Right: Contribution of the sharpness to mutual information which is normalized by the mean value of mutual information as a function of $\epsilon$. This shows that the sharpness contribution is independent of the UV cut-off for about $95\%$. We have set $R_-=0.8$, $R_+=1$ and $\delta=0.05$.
To calculate the mutual information between $A$ and $B$ we have to solve the above equations to find $S_{A}$, $S_{B}$ and $S_{A\cup B}$. To compute $S_{A\cup B}$ we must consider both connected and disconnected configurations which may contribute. By disconnected configurations we mean two minimal surfaces which end on the deformed disks $\rho_{\pm}(\phi)$ which is already considered while we have found $S_{A}$ and $S_{B}$. On the other hand the connected configuration is the minimal surface starting from $\rho_{+}(\phi)$ and ending on $\rho_{-}(\phi)$. Note that the unperturbed connected configuration is also found analytically in <cit.>. We have solved Eq.(<ref>) numerically up to $\mathcal{O}\left(\delta^2\right)$ for functions $\rho_0(z)$, $\rho_1(z)$ and $\rho_{20}(z)$ which contribute to the correction of entanglement entropy at first non-trivial order of the sharpness.
In our numerical analysis we have considered $R_-=0.8$, $R_+=1$, $\delta=0.05$ and we have briefly reported the numerical results in Fig.<ref> and Fig.<ref>. In Fig.<ref> we have plotted minimal surface profiles of such a configuration in a constant-$\phi$ slice. The profiles are found by the perturbation theory which is organized such that the turning point of the minimal surfaces are always unchanged ($z_*=.8$ for $R_-$ and $z_*=1$ for $R_+$). In this figure we have compared the profiles for $\delta=0$ and $\delta=0.05$ at $\phi=\pi/2$ slice which has the maximum separation between the perturbed and unperturbed profiles. Note that from the right panel of Fig.<ref> it is obvious that the maximum separation between the deformed and undeformed configurations happens at $\phi=\pi/2$ and $\phi=3\pi/2$ and they coincide at $\phi=0,\pi$. We have computed the mutual information for such configurations and studied its dependence on the inverse UV cut-off $\epsilon$. As we expected our numerical results show that mutual information is independent of the cut-off for about $95\%$. In the left panel of Fig.<ref> we have plotted the mutual information for various values of inverse UV cut-off $\epsilon$. In the left panel we have plotted $\Delta I\equiv I_\delta-I_0$ normalized by the numerical value of $I_\delta$ which shows deviation between $0-5\%$.
In summary in this section we have considered two singular regions which their singularities are apart from each other. We have numerically analysed a specific configuration which the opening angles are equal to each other (see the right panel of Fig.<ref>). It is shown that the mutual information is independent of the UV cut-off to a great precision. As we have explained in the introduction section, because of the extensivity of the coefficient of the logarithmic divergent term, we expect such an independency even for configurations which the opening angle of the singularities are not the same but the singularities are still apart from each other.
§ CONCLUSIONS AND DISCUSSIONS
In this paper we have mainly studied corner contributions to certain holographic entanglement measures e.g. mutual and tripartite information for three dimensional CFTs. We have considered two different kinds of singular geometries i.e. a set of disjoint sectors of a single infinite circle (cake slices) which have a contact point and also two sharp concentric circles which are completely disjoint. In particular one of our goals was to explore the role of a singular contact point on the behavior of the corresponding holographic mutual information.
In the case of cake slice entangling regions using the previous results for holographic entanglement entropy for a kink we studied holographic mutual information. We have shown that the corresponding HMI is divergent which is due to the common local region shared among these regions near the contact point. Although the resultant HMI was divergent, our analysis revealed that it exhibits a first order phase transition. In this set-up we also studied the holographic tripartite information which is divergent in contrast with the well-known feature of tripartite information, which is finite even when the subsystems share boundaries. Generalizing these results to higher dimensional holographic CFTs we have explored similar behaviors for mutual and tripartite informations.
In the case of two sharp and completely disjoint concentric circles, by performing a numerical study we have found a finite holographic mutual information. Our numerical results show that mutual information is independent of the UV cut-off with a great accuracy. Comparing this result with the HMI for cake slice entangling regions we conclude that a divergent HMI is a reminiscent of the contact point (infinite local correlations) and it does not have anything to do with the geometric singularity.
In the following of this section we discuss about some aspects of other entanglement measures associated to a singular surface. We will focus in the first set-up, i.e., cake slice entangling regions which may help us to gain more insights into certain singular surfaces having a contact point.
§.§ A Universal Measure
In reference <cit.>, the authors have introduced a c-function in three dimensional CFTs by means of the mutual information between concentric disks as
\begin{align}
C(R)=\lim_{a\rightarrow 0}\frac{1}{4\pi}\left(R \frac{\partial I}{\partial R}-I\right),
\end{align}
where $a$ is the difference between the radii of the disks (see left panel of Fig.<ref>). In the $a\rightarrow 0$ limit these two disks coincide and they have a common boundary (which leads to infinite MI). As we have shown in section <ref>, the HMI between sectors of an infinite circle with a contact point at the center of the circle is UV cut-off dependent and thus it is not universal. An important question about such configurations is whether there is any universal measure which we can use for example instead of mutual information and obtain the same physical information?
[thick,blue!40!black] (-1.5cm,0cm) circle(3.2mm);
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[blue!40!black] (-17mm,.2mm) node $R_-$;
[blue!40!black] (-13mm,1.2mm) node $R_+$;
[red] (-1.5cm,3.5mm) node $a$;
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[blue!40!black] (0,4mm) node $\Omega$;
[blue!40!black] (0,-4mm) node $\Omega$;
[red] (4mm,0) node $\omega$;
[red] (-4mm,0) node $\omega$;
Left: The configuration for the mutual information between the disk of Radius $R_-$ and the complement of a disk with radius $R_+$ which is used in <cit.> to define a c-function in three dimensions in the $a\to0$ limit. Right: A similar configuration to concentric disks for singular surfaces in the $\omega\to0$ limit for contacting kinks.
One may consider a similar set-up to what is done for co-centric disks in the vanishing limit of their separation for our configuration. See the right panel of Fig.<ref> for a visualization. It is important to note that in the limit where the entangling regions are slightly bended, $\Omega \rightarrow \pi$ and $\omega \rightarrow 0$, our entangling regions share boundary which has a similar structure as in <cit.>. We define a new quantity from mutual information as
\begin{align}
\mathcal{I}=\lim_{\omega\rightarrow 0}\frac{\omega}{2}\left(\omega \frac{\partial I}{\partial \omega}+I\right),
\end{align}
which captures a universal phase transition. Using the expressions for sharp Eq.(<ref>) and smooth Eq.(<ref>) limits one can easily show that at the leading order $\mathcal{I}\approx\kappa$. It would be interesting to further investigate such a quantity and even try to check whether it satisfies the requirements of a c-function.
§.§ First Law of HEE and Modular Hamiltonian
According to the first law of entanglement entropy the variation of the entanglement entropy at leading order is equal to the variation of the modular Hamiltonian, i.e., $\Delta S=\Delta \langle H_{\rm mod.}\rangle$ <cit.>[See also <cit.>.]. The modular Hamiltonian is a non-local quantity which is defined in terms of the reduced density matrix as $H_{\rm mod.}\sim -\log \rho_{\rm red.}$. In general the computation of this quantity is not an easy task due to its non-locality. However, there are special cases where the modular Hamiltonian becomes local e.g. for spherical entangling regions <cit.>. In this case one can find an exact expression for $H$ in terms of an integral over $T_{00}$ (the time-time component of the stress tensor). This shows that using the first law for EE one may extract some perturbative information about the structure of modular Hamiltonian for more general entangling regions. For example the authors of <cit.> have used the first law to show that in addition to $T_{00}$ other spatial components of stress tensor e.g. $T_{ii}$ also contribute to the modular Hamiltonian for strip entangling regions.
In order to find the leading variation of the HEE for a kink entangling region in a three-dimensional holographic CFT we consider an excited state dual to a black-brane back-ground with the following metric
\begin{align}\label{BHmetric}
ds^2=\frac{L^2}{z^2}\left(\frac{dz^2}{f(z)}-f(z)dt^2+d\rho^2+\rho^2 d\theta^2\right),\;\;\;f(z)=1-m z^3.
\end{align}
In this case one may solve the resultant equations for the minimal surface for small entangling regions, i.e., $m h_*^3\ll1$. In Fig.<ref> we have presented the $\Omega$-dependence of $\Delta S$ in small opening angle limit.
Variation of HEE between the ground state (pure AdS) and thermal state (AdS black-brane) in the small opening angle limit for $H=1$ and $m=0.1$.
We expect that this result may help to more investigate corner contributions to modular Hamiltonian which is a nonlocal quantity.
§ ACKNOWLEDGEMENTS
The authors would like to thank Pablo Bueno, Piermarco Fonda, Matthew Headrick and specially Horacio Casini for correspondence and valuable comments on an early draft of the manuscript. We are grateful to Mohsen Alishahiha for his valuable comments and careful reading of the manuscript. We also thank Erik Tonni for his correspondence. We would like to thank Pablo Bueno and William Witczak-Krempa for sharing unpublished results on corner and cone entanglement beyond the smooth limit.
FO is very grateful to school of Physics of IPM for its warm hospitality during this project.
FO would also like to thank the organizers of the PhD school “Holography: Entangled, Applied, and Generalized" at Niels Bohr Institute in Copenhagen where the final stages of this work took place.
MRMM and AM are supported by Iran National Science Foundation (INSF).
§ HOLOGRAPHIC TRIPARTITE INFORMATION BETWEEN STRIPS
In this section we aim to show that the holographic prescription of entanglement entropy leads to finite tripartite information even for the case when the disjoint regions merge together and share boundaries. Here we consider an entangling region composed of three disjoint infinite strips (three disjoint segments in a 2-dim field theory) with lengths $\ell_i$ where $i=1,2,3$. These are separated with $h_1$ and $h_2$ as in Fig.<ref>. The tripartite information in this case for a two dimensional field theory by definition is given by
\begin{align}
\begin{split}
&+\min\Bigg\{\log\frac{\ell_1\ell_2\ell_3}{\epsilon^3},\log\frac{\ell_3(\ell_1+\ell_2+h_1)h_1}{\epsilon^3},\log\frac{h_1 h_2(\ell_1+\ell_2+\ell_3+h_1+h_2)}{\epsilon^3},\\
\end{split}
\end{align}
There are several cases where this configuration of entanglig regions may share boundaries. By shared boundary we mean that the separation between disjoint subregions `vanishes'. Here `vanishes' precisely means it tends to the value of $\epsilon$. If we consider the entangling regions on a periodic spatial direction, then we denote the spacing between the first and third segments by $h_3$. Now there may be several configurations among $h_i$'s where one, two or even all of them `vanish'. In the following we consider two of these configurations. In the first one $h_1$ `vanishes' while $h_2$ is finite. The case where $h_2$ `vanishes' and $h_1$ is finite could be obtained by a permutation in the indices. In the second case we consider both $h_1$ and $h_2$ `vanish'.
[blue,thick] (0,0)–(1.3,0);
[densely dashed] (1.3,0)–(2,0);
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Entangling region with three disjoint parts.
Case 1:
Consider $\ell_1$ and $\ell_2$ regions to share a boundary, in other words $h_1\to\epsilon$. In this case together with assuming $h_2,\,\ell_i\gg\epsilon$, one can easily check that the above expression for two dimensional field theory gives
\begin{align}
\begin{split}
\end{split}
\end{align}
which is independent of the inverse UV cut-off $\epsilon$ as expected. Note that this is in contrast with what happens to mutual information of two disjoint regions in the limit where their distance `vanishes' which is UV cut-off dependent as
\begin{align}
\begin{split}
\end{split}
\end{align}
A direct generalization of this case to $d>2$ is straightforward where we have to consider infinite strips instead of a line segment. For $d>2$ holographic field theories one can again work out the value of $I^{(3)}$ in the above configuration analytically. For simplicity we consider $\ell_1=\ell_2=\ell_3\equiv\ell$ which leads to
1+3^2-d-2^3-d h_2^d-2(2^d-1-1)<(2ℓ)^d-2
(ℓ/h_2)^d-2-1+3^2-d-2^2-d h_2^d-2(2^d-1-1)≥(2ℓ)^d-2
Again note that the mutual information of two infinite strips when their separation `vanishes' is again a UV cut-off dependent quantity as
Case 2:
One can easily find the result where both $h_1$ and $h_2$ `vanish' simultaneously from the results of case 1 for holographic field theories. For the case of $d=2$ one can easily check that the tripartite information is again a UV cut-off independent quantity as
\begin{align}
\begin{split}
\end{split}
\end{align}
and for the case of $d>2$ it becomes
§ HYPERSCALING-VIOLATING GEOMETRIES
In this appendix we generalise our studies to theories with a hyperscaling-violating geometry as a gravity dual. The HEE for a singular entangling region in this geometry has been previously studied in <cit.>. Here we explore holographic mutual information in specific examples of this kind of backgrounds. As in section <ref> we only consider crease entangling regions and set the dynamical critical exponent $z=1$. The metric for a hyperscaling-violating geometry in four dimensions is given by <cit.>
\begin{align}\label{metrichyper}
ds^2=\frac{1}{z^{2-\theta}}\left(dz^2-dt^2+d\rho^2+\rho^2 d\phi^2\right),
\end{align}
where $\theta$ is the hyperscaling violating exponent and due to the null energy condition it must satisfy either $\theta\geq 2$ or $\theta\leq 0$ <cit.>. Also in order to avoid gravitational instabilities we only consider $\theta\leq 0$ in the following discussion <cit.>. Considering the entangling region as in Eq.(<ref>) and rewriting
$z(\rho,\phi)=\rho\;h(\phi)$ such that $h(\pm\frac{\Omega}{2})=0$, the HEE functional becomes
\begin{align}\label{heefunc-hyper}
S=\frac{L^2}{2G_N}\int_{\frac{\epsilon}{h_*}}^H \frac{d\rho}{\rho^{1-\theta}}\int_0^{\frac{\Omega}{2}-\delta}d\phi\frac{\sqrt{1+h^2+h'^2}}{h^{2-\theta}},
\end{align}
where the notations are similar to the previous sections. Since the integrand does not depend on $\phi$ explicitly we can define a conserved quantity such that
\begin{align}\label{thetamom-hyper}
\mathcal{H}_\theta\equiv \frac{(1+h^2)^{\frac{2-\theta}{2}}}{h^{2-\theta}\sqrt{1+h^2+h'^2}}=\frac{(1+h_*^2)^{\frac{1-\theta}{2}}}{h_*^{2-\theta}}.
\end{align}
Using this expression one can find the relation between the opening angle $\Omega$ and the turning point in the bulk as follows
\begin{align}\label{omehah0-hyper}
\Omega =2\int_0^{h_*} \frac{dh}{\sqrt{1+h^2}\sqrt{(\frac{h_*}{h})^{2(2-\theta)}(\frac{1+h^2}{1+h_*^2})^{1-\theta}-1}}
\end{align}
and if we use the change of variable $y=\sqrt{\frac{1}{h^2}-\frac{1}{h_*^2}}$ the HEE becomes
\begin{align}\label{HEE-hyper}
S(\Omega)=\frac{L^2}{2G_N}\int_{\frac{\epsilon}{h_*}}^H \frac{d\rho}{\rho^{1-\theta}}\int_0^{\sqrt{\frac{\rho^2}{\epsilon^2}-\frac{1}{h_*^2}}}dy \frac{h_* y \left(\frac{1}{h_*^2}+y^2\right)^{-\frac{\theta }{2}}}{\sqrt{1+h_*^2 y^2-\left(\frac{h_*^2+1}{h_*^2 \left(y^2+1\right)+1}\right)^{1-\theta }}}.
\end{align}
The behaviour of the integrand near the boundary, i.e., $y\rightarrow \infty$ is given by
\begin{align}
\frac{h_* y \left(\frac{1}{h_*^2}+y^2\right)^{-\frac{\theta }{2}}}{\sqrt{1+h_*^2 y^2-\left(\frac{h_*^2+1}{h_*^2 \left(y^2+1\right)+1}\right)^{1-\theta }}}\sim
\begin{cases}
y^{-\theta}+\frac{\#}{y^{\theta+2}}+\mathcal{O}(\frac{1}{y^{\theta+4}})&\theta\leq -2,
\end{cases}
\end{align}
which shows that in order to isolate the divergent part one needs to apply a $\theta$-dependent regularization. For example when $\theta =-1$ one finds
\begin{align}\label{HEEhyperfinal}
\end{align}
where the function $a_{-1}(\Omega)$ is defined as
\begin{align}
a_{-1}(\Omega)=\frac{L^2}{2G_N}\int_0^\infty dy\;y\left[\frac{ \left(1+h_*^2y^2\right)^{\frac{1}{2}}}{\sqrt{1+h_*^2 y^2-\left(\frac{h_*^2+1}{h_*^2 \left(y^2+1\right)+1}\right)^{2}}}-1\right].
\end{align}
A similar procedure leads to the divergent parts for other values of $\theta$. Fig.<ref> demonstrates the behavior of $\Omega(h_*)$ and $a_{\theta}(\Omega)$ for $\theta=-1$. In this figure we have also included the case of $\theta=0$ for comparison. Also note that the function $a_{\theta}(\Omega)$ vanishes in $\Omega\rightarrow \pi$ limit which corresponds to the smooth limit as expected. According to Eq.(<ref>) even in the smooth limit in addition to the area law term there exists an extra divergent term. Explicit computations show that when one computes the HMI this divergent term does not cancel out and the resultant HMI becomes divergent. This situation is similar to the case of a spherical entangling region in a $d$-dimensional field theory with $d>3$.
Left: $\Omega/\pi$ as a function of the turning point $h_*$. Right: $a_{\theta}$ as a function of the opening angle $\Omega$.
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|
1511.00247
|
Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics,
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
We study quantum corrections to the classical Bianchi I and Bianchi IX
universes. The modified dynamics is well-motivated from the asymptotic
safety program where the short-distance behavior of gravity is governed by
a non-trivial renormalization group fixed point. The correction terms
induce a phase transition in the dynamics of the model, changing the
classical, chaotic Kasner oscillations into a uniform approach to a point
singularity. The resulting implications for the microscopic structure of
spacetime are discussed.
One of the most fascinating features of general relativity (GR) is
the appearance of spacetime singularities where the theory essentially predicts its own breakdown.
Following the theorems by Hawking and Penrose,
such singularities occur under quite general conditions <cit.>,
e.g. in the form of the (space-like) big bang singularity. This leads to rather
tantalizing questions about a universal behavior of the dynamics close to a singularity,
especially once quantum effects are taken into account.
In the classical treatment of the problem,
the general dynamics near a space-like singularity has been analyzed in
a series of seminal works by Belinski, Khalatnikov and Lifshitz (BKL)
<cit.>, building
on the idea that sufficiently close to the singularity
time derivatives overwhelm the spatial ones.
This results from an asymptotic
ultralocal behavior: at each spacetime point the light cones
collapse into time-like lines, which effectively decouple <cit.>.
Thus a finite region close to the singularity can be
well approximated
by a homogeneous spacetime.
BKL studied a representative class of homogeneous and anisotropic models,
showing that the asymptotic dynamics consists of an endless,
chaotic succession
of phases
where spacetime appears flat and highly anisotropic.
On the other hand, on general grounds one expects that close to a singularity
quantum gravitational corrections should play an important role in the description of the dynamics.
Inquiring how these corrections may modify the BKL scenario
constitutes a crucial step
in understanding the asymptotic approach to a classically singular point.
This has been explored in recent years in the context of Loop Quantum Cosmology <cit.>
and String Theory <cit.>.
In this work we will study for the first time the quantum corrected BKL behavior
in the context of Weinberg's Asymptotic Safety scenario
In this scenario, the
ultraviolet completion of gravity is provided by a scale-invariant
high-energy phase. This phase is realized through a non-trivial fixed
point of the renormalization group (RG) flow which renders
the theory non-perturbatively renormalizable.
We will consider the effect of quantum corrections in two relevant anisotropic models,
namely the (flat) Bianchi I and the (closed) Bianchi IX. Our analysis will show that
in both cases the scale-invariance of the theory at high energies is capable of
changing the qualitative features of the dynamics in novel and unexpected ways.
*Classical BKL dynamics and singularities.
In the BKL analysis
the evolution of spacetime near a singularity
is modeled by a homogeneous and anisotropic Bianchi IX metric, with line element
ds^2 = -dt^2 + (a^2 l_a l_b + b^2 m_a m_b + c^2 n_a n_b ) dx^a dx^b .
Here $a(t), b(t), c(t)$ are scale factors depending on the cosmological time $t$ and the $x$-dependent three-vectors $l_a, m_a$ and $n_a$ define the directions along which spatial distances vary with the corresponding scale-factor.
BKL revealed that the classical dynamics of the scale factors,
governed by eq. (<ref>) with $\lambda_*=0$,
follows a complex pattern of oscillations between so-called Kasner phases, where terms including spatial derivatives of the three-vectors are negligible (see Fig. 2, upper panel). In a Kasner phase the scale factors exhibit a distinctive power-law behavior
a(t) = t^p_1 , b(t) = t^p_2 , c(t) = t^p_3 ,
with the Kasner exponents $p_i$ satisfying
∑_i=1^3 p_i = 1 , ∑_i=1^3 ( p_i )^2 = 1 .
Ordering the Kasner exponents according to $p_1 \ge p_2 \ge p_3$ the solution to the eqs. (<ref>) can be given in terms of a single parameter $u$, see eq. (<ref>) for $r = 1$ below, and is displayed in the second diagram of Fig. <ref>. Thus classically $p_1, p_2 > 0$ and $p_3 < 0$, with $p_1 = 1$ and $p_2 = p_3 = 0$ appearing as a special case.
Each oscillation to a new Kasner phase in the Bianchi IX model changes the value of the Kasner exponents according to
a well-defined alternation rule <cit.>.
The power-law behavior (<ref>) thus characterizes the approach of the model to the initial space-like singularity located at $t=0$. Extending the classification <cit.>, the possible asymptotic behaviors are summarized in Table <ref>.
The classical Bianchi I model, where the Kasner exponents satisfy (<ref>) exhibits either a Pancake or a Cigar-type singularity.
This picture is drastically modified by the quantum effects studied below.
singularity asymptotic Kasner exponents
Point (PT) $p_1, p_2, p_3 > 0$
Barrel (B) $p_1, p_2 > 0$; $p_3 = 0$
Pancake (PC) $p_1 > 0$; $p_2 = p_3 = 0$
Cigar (C) $p_1, p_2 > 0$; $p_3 < 0$
Brick (BR) $p_1 > 0$; $p_2 = 0$; $p_3 < 0$
Plane (PL) $p_1 > 0$; $p_2, p_3 < 0$
Classification of the singular behavior of the scale factors (<ref>). The first four types of spatial singularities follow the classification of <cit.> while the Brick and Plane-type singularities are new features appearing in the quantum-improved model.
*Quantum effects via RG improvement.
At an RG fixed point the theory becomes scale-invariant. This entails in particular that the dimensionful Newton's constant $G$ and the cosmological constant $\Lambda$ acquire a specific energy dependence,
namely, for $k \to \infty$, $G(k) \to \tilde g_* \, k^{-2}$ and $\Lambda(k) \to \tilde \lambda_* \, k^2$,
where $k$ is the RG scale.
The analysis of the gravitational beta
function establishes that $ \tilde g_* > 0$ and $ \tilde \lambda_* > 0$ are numerical coefficients of order unity,
given by the position of the fixed point <cit.>.
The scale-dependence exhibited
by $G$ and $\Lambda$
has profound consequences for the microscopic structure of spacetime at distances
below the Planck scale. In order to study these features based on first principles,
one has to compute the loop corrections capturing
quantum effects at the relevant energy scale. Alternatively, one may exploit the fact that loop corrections are in general minimized
by choosing the RG scale $k$ to be of the order of the characteristic scale of the process one wants to study.
This procedure is known to reproduce the correct Uehling potential
for a static particle <cit.>.
Similar techniques have been applied
to black holes <cit.>
and cosmology <cit.>.
Following this strategy, we apply the technique of “improved equations of motion” in order to study the effect of the fixed point scaling
for the BKL scenario. The starting point are the classical Einstein equations including a cosmological constant,
$G_{\mu\nu} = - \Lambda \, g_{\mu\nu}$,
with $G_{\mu\nu}$ the classical Einstein's tensor.
The RG improvement promotes $\Lambda$ to a scale-dependent coupling constant. Close to the singularity this scale-dependence follows the fixed-point scaling, so that $G_{\mu\nu} = - \lambda_* \, k^2 \, g_{\mu\nu}$.
The RG scale is then identified with the cosmological time $t$, $k = \zeta t^{-1}$, with $\zeta = {\cal O}(1)$, since this sets the typical time-scale of the dynamics. The RG-improved equations of motion [Since we are performing the RG improvement at the level of the classical equations of motion, the contributions of higher-derivative operators presumably present in the fixed point action do not enter into the formalism.] have the form
G_μν = - λ_* t^-2 g_μν
with $ \lambda_* \equiv \tilde \lambda_* \, \zeta^2$ being of order unity.
Notably, the improved equation (<ref>) also arises from a covariant cutoff-identification. Asymptotically, $k^2 \propto \sqrt{C_{\mu\nu\rho\sigma} \, C^{\mu\nu\rho\sigma}} \simeq t^{-2} + {\rm subleading}$, with the Weyl tensor constructed from the classical solution. Thus the implemented cutoff identification agrees with the intuitive expectation that the curvature sets the effective scale where coupling constants are evaluated <cit.>.
Moreover, matter contributions to the field equations, $G_N T_{\mu\nu} \mapsto \tilde g_* \, \zeta^{-2} \, t^2 \, T_{\mu\nu}$, receive an additional suppression factor from the scale identification, and play no role for the dynamics close to $t \to 0$. Thus the RG improved vacuum equations (<ref>) constitute a self-consistent description of the BKL scenario taking into account the quantum corrections expected from a non-trivial fixed point of the RG flow.
Substituting the metric (<ref>) into (<ref>) and subsequently specializing the spatial derivative terms to the Bianchi IX case <cit.> results in
2 α_,ττ - (b^2 - c^2)^2 + a^4 =
2 a^2 b^2 c^2 λ_*/t(τ)^2 ,
2 β_,ττ - (a^2 - c^2)^2 + b^4 =
2 a^2 b^2 c^2 λ_*/t(τ)^2 ,
2 γ_,ττ - (a^2 - b^2)^2 + c^4 =
2 a^2 b^2 c^2 λ_*/t(τ)^2 ,
a = e^α , b = e^β , c = e^γ dt ≡abc dτ .
For $\lambda_*=0$, eqs. (<ref>) are
the classical equations of motion for the Bianchi IX universe with the potential terms originating from the curvature of the spatial slices. The contributions resulting from the RG improvement appear on the
right hand side. Since these terms grow as $t^{-2}$ we expect that they will modify the dynamics close to the spatial singularity situated at $t=0$.
*Quantum improved Kasner solutions.
We shall now show that the extra term indeed alters the singularity structure of the model and induces a phase transition in the classical Bianchi IX oscillations.
We start by studying the dynamics of the system (<ref>) for the case where the potential terms on the left hand side vanish. This corresponds to the Bianchi I universe where the spatial slices are flat. In this case, the scale factors follow the power-law behavior (<ref>)
Admissible Kasner exponents of the quantum improved Kasner system for the illustrative examples (from top to bottom) $\lambda_* = -1/12$, $\lambda_* = 0$, $\lambda_* = 1$, and $\lambda_* = 4$. The labels inserted in the diagram refer to the singularity classification of Table <ref> and indicate what type of spatial singularity is encountered for the corresponding value of $u$.
with the Kasner exponents satisfying
∑_i=1^3 p_i = r , ∑_i=1^3 ( p_i )^2 = r + λ_* ,
r ≡1/2 ( 1 + √(1 + 12 λ_*) ) .
The quantum improved Kasner system admits power-law solutions if $\lambda_* \ge -1/12$ and agrees with the classical system (<ref>) for $\lambda_* = 0$. Following <cit.>, the solutions of the system (<ref>) can be parameterized by
p_1(u) = 1/3 ( r - √(r) ) - √(r) u/1 + u + u^2 ,
p_2(u) = 1/3 ( r - √(r) ) + √(r) u (1+u)/1 + u + u^2 ,
p_3(u) = 1/3 ( r - √(r) ) + √(r) (1+u)/1 + u + u^2 .
Since this parameterization is invariant under the transformations
$p_1(1/u) = p_1(u)$, $p_2(1/u) = p_3(u)$, $p_3(1/u) = p_2(u)$,
all possible asymptotic behaviors are covered by taking $u \in [0,1]$.
Based on the parameterization (<ref>) one can distinguish the following phases, governed by the value of $\lambda_*$:
: - 1/12 ≤λ_* < 0
: λ_* = 0
: 0 < λ_* < 4
: λ_* ≥4
The characteristic features of each phase are depicted in Fig. <ref>, illustrating, from top to bottom,
Phase O, the classical Kasner solution, Phase I, and Phase II.
The classical solutions ($\lambda_* = 0$) in Phase C generically admit two positive and one negative Kasner exponent.
The parameter $\lambda_*$ essentially shifts the system of Kasner exponents to more negative ($\lambda_* < 0$) or more positive ($\lambda_* > 0$) values. Depending on the value of $u$ the
space in Phase O admits classical Kasner solutions as well as a new class of
where two Kasner exponents are negative and one is positive. Phase I supports both classical Kasner behavior and a new set of solutions where all three Kasner exponents are positive. In Phase II, $\lambda_* \ge 4$, the value of $\lambda_*$ is sufficiently positive that the classical Kasner behavior does not occur anymore and the Kasner exponents are positive semidefinite for all values $u$.
Note that Asymptotic Safety predicts $\lambda_* > 0$ so that the quantum improved system is either in Phase I or Phase II.
*Modified asymptotics of the quantum BKL system.
The full Bianchi IX system (<ref>) can be solved numerically.
Comparison between the classical (Phase C) and quantum improved (Phase I) evolution of $\alpha$, $\beta$, $\gamma$ in the Bianchi IX model using identical initial conditions. Including the quantum dressing the system exhibits a dynamical phase transition from Kasner oscillations to a non-chaotic approach to a point singularity as $\tau$ decreases.
Typical solutions for the cases $\lambda_* =0$ (Phase C) and $\lambda_* =1$ (Phase I) are shown in Fig. <ref>.
Phase C exhibits the endless chaotic oscillations discovered by BKL.
In Phase I this chaotic behavior is modified by the appearance of the PT solutions of the improved Bianchi I model.
For sufficiently large negative $\tau$ the
oscillatory Kasner behavior ends and all scale factors vanish for $t \rightarrow 0$, resulting in a point singularity.
This change in the dynamics can be understood as follows.
In the classical case, bounces in the Bianchi IX model are
triggered by one of the terms in the classical potential becoming large, i.e.,
when the scale-factor with negative Kasner exponent grows as $t \rightarrow 0$. As a result of the bounce
the Kasner exponents are swapped and the system enters into the next Kasner phase.
This mechanism stops operating if the system bounces into a phase where all three Kasner exponents are positive.
In this case all scale factors vanish simultaneously as $t \rightarrow 0$ and the contribution of the classical potential
is subleading compared to the quantum corrections. Thus, the asymptotic dynamics is governed by the
quantum improved Kasner system restricted to the parameter range $u$ where all Kasner exponents are positive.
We have shown how the asymptotic dynamics of a Bianchi IX model near a spatial singularity
changes with respect to the classical BKL result once quantum corrections motivated by Weinberg's Asymptotic Safety scenario are taken into account.
The system exhibits a new type of asymptotic behavior where all Kasner exponents become positive and a pointlike singularity is approached. The transition between the chaotic Kasner oscillations and the non-chaotic approach to the singularity is triggered by quantum corrections dominating over the classical regime when curvature becomes large.
While this phase transition is of broad interest in general relativity and quantum gravity, the modification of the classical Kasner behavior is particularly relevant for characterizing the short distance structure of the asymptotically safe quantum spacetime <cit.>. Our results are in complete agreement with observation <cit.> that the cosmological constant crucially influences the short distance behavior of quantum spacetime. Notably, the renormalization group improvement studied in this letter constitutes a characterization of the quantum spacetime which is complementary to the one provided by the spectral dimension $d_s$, which asymptotes to $d_s = 2$ in the fixed point regime <cit.>. The result that the quantum improved anisotropic spacetimes generically develop a point singularity where the scale factors of all three spatial dimensions vanish simultaneously supports the picture that the reduction of the spectral dimension does not necessarily imply the dimensional reduction of position space to the same value. This may serve as an illustrative example that in quantum gravity the dimensionality of effective position and momentum spaces do not necessarily need to agree. Naturally, it would be very interesting to initiate a similar position-space study in other quantum gravity programs to investigate whether a similar mechanism is operative and to compare the resulting refined picture of microscopic quantum spacetimes.
We thank T. Ottenbros for collaboration at the intermediate stage of this project and M. Reuter for helpful discussions. The research of F. S. and G. D. is supported by the Netherlands Organisation for Scientific
Research (NWO) within the Foundation for Fundamental Research on Matter (FOM) grants
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|
1511.00359
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In [Acta Math. Univ. Comenianae Vol. LXXX, 1 (2011), pp. 63–70], Yang, Chen and Shi examined the system of difference equations
\[
x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots,
\]
where $q$ is a positive integer with $p < q$, $p \nmid q$, $p \geq 3$ is an odd number, both $a$ and $b$ are nonzero real constants, and
the initial values $x_{-q+1},x_{-q+2},\ldots,$ $x_0,y_{-q+1},y_{-q+2},\ldots,y_0$ are nonzero real numbers.
At the end of their note, they posted a question regarding the behaviour of solutions of the given system when $p$ is even.
More precisely, they asked what the solutions of the system may look like if $p$ is even.
In this note we answer this question raised by the authors.
Particularly, we show that the system may or may not admit a periodic solution depending on the coprimality of the parameters $p$ and $q$ and on the parity of the integer $p/\gcd(p,q)$.
§ INTRODUCTION
In a recent paper, Yang, Chen and Shi investigated the system of difference equation
\begin{equation}
\label{problem}
x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots,
\end{equation}
where $q$ is a positive integer with $p < q$, $p \nmid q$, $p \geq 3$ is an odd number, both $a$ and $b$ are nonzero real constants, and
the initial values $x_{-q+1},x_{-q+2},\ldots, x_0,$
$y_{-q+1},y_{-q+2},\ldots,y_0$ are nonzero real numbers.
They have showed in <cit.> that all real solutions of the system are eventually periodic with period $2pq$ (resp. $4pq$) when $(a/b)^q = 1$ (resp.
$(a/b)^q = -1$). Further, the authors <cit.> characterized the asymptotic behavior of the solutions of (<ref>) when $a \neq b$.
Their work is actually a generalization of a result seen in <cit.>, wherein Özban investigated the behavior of the positive solutions of the system of rational difference equations
\begin{equation}
\label{oz3}
x_n=\frac{a}{y_{n-3}},\qquad y_n=\frac{by_{n-3}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots,
\end{equation}
where $q > 3$ is a positive integer with $3\nmid q$, $a$ and $b$ are positive constants.
Özban particularly showed in <cit.> that for the case $b = a$, $p = 3$, $q > 3$, and $p$ not dividing $q$,
all positive solutions of the system (<ref>) are eventually periodic with period $6q$.
Meanwhile, Özban's work <cit.> was inspired by a result found by Yang, Liu and Bai in <cit.>.
In their four page note, Yang et al. <cit.> investigated the system (<ref>) for the case where $p$ and $q$ are positive integers with $p\leq q$, and $a$ and $b$ are positive constants.
They have showed that if $b = a$, and $q$ is divisible by $p$, then every positive solution of (<ref>) is periodic with period $2q$.
In <cit.>, an intriguing result regarding the behavior of positive solutions of the higher-order difference equation
\begin{equation}
\label{bratislav}
x_n=\frac{cx_{n-p}x_{n-p-q}}{x_{n-q}},\qquad n=0,1,\ldots,
\end{equation}
where $p,q \in \mathbb{N}$ and $c>0$, was obtained by Iričanin and Liu through an elegant and short way.
One particular results they presented in (<ref>) reads as follows: if $c=1$ in (<ref>) and $\gcd(p,q)=1$, and $p$ is odd, then all positive solutions of (<ref>) are eventually periodic with period $2pq$.
Their result was in fact inspired by earlier results presented in <cit.> and <cit.>.
Similar nonlinear systems of rational difference equations were also investigated, see, e.g., Cinar <cit.>, Cinar and Yalçinkaya <cit.>, Cinar et al. <cit.>,
and Özban <cit.>.
Now, our aim in this work is to answer a question raised by Yang et al. at the end of their paper <cit.>.
Particularly, we shall described here the behavior of solutions of (<ref>) in the case when $p$ is even.
We mention that Yang et al. <cit.> already made a preliminary observation on the behavior of solutions of (<ref>) when $p$ is even.
More specifically, they have observed, after some numerical experimentations, that (<ref>) is non-periodic when $p$ is even.
Here we show, through an analytical approach, that this observation is in fact true whenever $\gcd(p,q)=1$, i.e. $q$ is odd
and that, in addition, every positive solution of (<ref>) when $b=a$ has an exponentially growing/decaying subsequence irrespective of the choice of positive initial values $x_{-q+1},x_{-q+2},\ldots, x_0,y_{-q+1},y_{-q+2},\ldots,y_0$ on this case.
Every solution, however, will be periodic of period $m$, where $m$ denotes the least common multiple of $p$ and $2q$, if $p/\gcd(p,q)$ is odd.
On the other hand, if $p/\gcd(p,q)$ is even, then every solution of (<ref>) behaves in a similar fashion as in the case when $\gcd(p,q)=1$.
That is, there is some subsequence of the solution $\{x_n\}$ (resp. $\{y_n\}$) which tends to infinity (resp. converges to zero) (cf. Theorem <ref>).
We emphasize that the problem raised by Yang et al. in <cit.> remains open since Iričanin and Liu <cit.> only deal with the case when $p$ is odd in (<ref>), which is not of interest here.
By an eventually periodic solution $\{(x_n,y_n)\}:=\{(x_n,y_n)\}_{n=-(q-1)}^\infty$ of (<ref>), we mean that
there exist an integer $n_0 \geq -q+1$ and a positive integer $\pi$ such that
\[
(x_{n+n_0+\pi},y_{n+n_0+\pi})=(x_{n+n_0},y_{n+n_0}), \quad n=1,2,\ldots,
\]
and $\pi$ is called a period (cf. <cit.>).
If, regardless of the choice of initial values $x_{-q+1}$, $x_{-q+2}, \ldots, x_0, y_{-q+1},y_{-q+2},\ldots,y_0$, no such values of $n_0$ and $\pi$ exist, then every solution of (<ref>) is not and can never be periodic.
§ MAIN RESULTS
In this section we show that the system (<ref>) can never have a periodic or eventually periodic solution when $p$ is even and $q$ is odd.
If, however, $q$ is even, then the existence of periodic solution of (<ref>) depends on the parity of $p/\gcd(p,q)$.
Our approach parallels that seen in <cit.>.
We only consider the case when $b=a$ in (<ref>) with all of its initial values taken from the set of positive real numbers.
The same inductive lines, however, can be followed to show a similar result for the case $b=-a$ and even for the more general case given by system (<ref>).
To show that (<ref>) has no periodic solution, it suffices to prove that every solution of (<ref>) has an increasing subsequence (or, perhaps, a decreasing subsequence) regardless of the choice of initial values $x_{-q+1},x_{-q+2},\ldots, x_0$, $y_{-q+1},y_{-q+2},\ldots,y_0$.
With this idea in mind, we now proceed as follows.
First, we transform the first equation in (<ref>) to
\[
x_nx_{n-q}=x_{n-p}x_{n-p-q}, \qquad n=0,1,\ldots.
\]
Since $x_n>0$ for all $n\geq 0$, then taking the natural logarithm of both sides of the above equation and making the subtitution $a_n:=\ln x_n$, we get
\[
\]
Using the ansatz $a_n =\lambda^n \in \mathbb{R}$, we obtain the polynomial equation
\[
\]
From here on, we consider two possibilities: (i) $\gcd(p,q)=1$; and (ii) $\gcd(p,q)>1$.
CASE 1: Suppose that $\gcd(p,q)=1$ or equivalently, $q$ is odd.
Then, it is evident that $\lambda=-1$ is a root of $P(\lambda)=0$ of order two.
Denote this repeated root by $\lambda_1$ and $\lambda_{p+1}$, i.e., let $\lambda_1=\lambda_{p+1}=-1$.
Then, the explicit formula for the sequence $\{a_n\}$ is of the form
\[
a_n=c_1\lambda_1^n+c_{p+1}n \lambda_{p+1}^n+\sum_{\substack{{i=2}\\{i\neq p+1}}}^{p+q} c_i \lambda_i^n,
\]
for some real numbers $c_1,c_2,\ldots,c_{p+q}$.
Note that $P(\lambda)=0$ has all of its roots on the unit disk $|\zeta|\leq 1$.
Denote these roots by $\{\lambda_i\}_{i=1}^{p+q}$ where $\{\lambda_i\}_{i=1}^p$ are the corresponding roots of $\lambda^p-1=0$ and $\{\lambda_i\}_{i=p+1}^{p+q}$ are the roots of $\lambda^q+1=0$.
Clearly, $\lambda_i^p=1$ for all $1\leq i\leq p$ and $\lambda_i^{2q}=1$ for all $p+1\leq i\leq p+q$.
Since $p \nmid q$, then $\lambda_i^{2pq}=1$ for all $1\leq i\leq p+q$.
\[
a_{2pqn}=c_1+2c_{p+1}pqn+\sum_{\substack{{i=2}\\{i\neq p+1}}}^{p+q} c_i.
\]
Suppose $c_{p+1}>0$, then for sufficiently large $N \in \mathbb{N}$, $a_{2pqn}$ will eventually be increasing for $n\geq N$, in fact we'll have
\[
a_{2pqn} \longrightarrow \infty \quad \text{as}\quad n \rightarrow \infty.
\]
Going back to the relation $a_n=\ln x_n$, we see that
\[
x_{2pqn}=\exp\left\{ a_{2pqn}\right\} \longrightarrow \infty \quad \text{as}\quad n \longrightarrow \infty.
\]
Moreover, we have
\[
y_{2pqn}\longrightarrow 0\quad \text{as}\quad n \longrightarrow \infty.
\]
Therefore, the subsequence $\{x_{2pqn}\}$ (resp. $\{y_{2pqn}\}$) will tend to infinity (resp. converges to zero) exponentially.
Thus, every solution of (<ref>) can never be periodic when $\gcd(p,q)=1$, or equivalently, when $q$ is odd.
CASE 2: Now, the case $\gcd(p,q)>1$ needs a little more work.
Suppose $\gcd(p,q)=2^ru$ for some odd integer $u\geq1$ and integer $r>0$.
Then, the roots of $P(\lambda)=0$ can be expressed as
\begin{equation}
\label{roots}
\begin{cases}
\ \exp\left\{ \dfrac{(2k+1)\pi i}{q}\right\}, & k=0,1,\ldots, q-1,\\[1em]
\ \exp\left\{ \dfrac{2l \pi i}{p}\right\}, & l=0,1,\ldots, p-1.
\end{cases}
\end{equation}
Let $p=2^rus$ and $q =2^rut$ where $s<t$ and $s\nmid t$.
The roots of $P(\lambda)=0$ are simple if and only if
\[
\frac{2k+1}{2^r u t} \neq \frac{2l}{2^r u s}, \qquad \text{for each}\ k,l \in \mathbb{N}_0,
\]
or equivalently
\begin{equation}
\label{condition}
(2k+1)s\neq 2l t, \qquad \text{for each}\ k,l \in \mathbb{N}_0.
\end{equation}
We consider two separate subcases, namely: (a) $p/\gcd(p,q)$ is odd; and (b) $p/\gcd(p,q)$ is even.
Subcase 2.1: Clearly, if $s$ is odd (i.e., $p/\gcd(p,q)$ is odd), then the inequality (<ref>) always holds.
Hence, the roots of (<ref>) are distinct.
Moreover, since $\lambda^{2q}=1$, then the explicit formula for $a_{mn}$ takes the form
\[
a_{mn+t}=\sum_{i=1}^{p+q} c_i\lambda_i^{mn+t}=\sum_{i=1}^{p+q} c_i\lambda_i^{t}=a_t, \qquad \forall n=0,1,\ldots,
\]
for each $t=\{0,1,\ldots,m-1\}$, where $m:={\rm lcm}(p,2q)$ denotes the least common multiple (lcm) of $p$ and $2q$.
Therefore, $a_n$ is eventually periodic with period $m$.
Since $a_n=\ln x_n$ for all $n=0,1,\ldots$, then $x_n$, as well as $y_n$, are also eventually periodic.
Subcase 2.2: Now, if $p/\gcd(p,q)$ is even (i.e., $s=2^{r_0}s_0<t$ for some integers $r_0,s_0>0$ with $s_0$ being odd), then $t$ must be odd, otherwise $\gcd(p,q) > 2^ru$.
Evidently, (<ref>) does not hold since the equality $(2k+1)2^{r_0-1}s_0= l t$ may hold true by choosing appropriate values for $l,k \in \mathbb{N}_0$.
For instance, if $r_0=1$, then we can choose $l=s_0$ and $k=(t-1)/2$.
In general, we can take $l=2^{r_0-1}s_0$ and $k=(t-1)/2$.
Since the inequality was not satisfied, then $P(\lambda)=0$ has at least one repeated root.
Without loss of generality, let $\lambda_j$ be a root of $P(\lambda)=0$ of order two and $m$ be the least common multiple of $p$ and $2q$.
By arguing as in the first case, we see that the subsequence
\begin{align*}
a_{mn}&=c_jm n\lambda_j^{mn}+\sum_{i=1}^{p+q} c_i\lambda_i^{mn}=c_j m n+\sum_{i=1}^{p+q} c_i\\
&\longrightarrow \infty\qquad \text{as}\quad n\longrightarrow \infty.
\end{align*}
Again, going back to the relation $a_n=\ln x_n$, the above result leads us to conclude that when $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is even,
a subsequence of the solution to (<ref>) grows/decays exponentially.
Thus, every solution in this case is non-periodic.
We remark that every solution to (<ref>) when $b=-a$ has the same behavior as those in the case when $b=a$.
The only difference is that every real solution of (<ref>) for $b=-a$ oscillates at $0$.
This can be seen easily from the relation $x_nx_{n-q}=-x_{n-p}x_{n-p-q}$.
In fact, if $\{(x_n,y_n)\}$ is a solution to (<ref>) with positive initial values, then there is some subsequence $\{|x_{mn+t}|\}$ (resp. $\{|y_{mn+t}|\}$)
which tends to infinity (resp. converges to zero) exponentially.
We summarize our discussion in the following theorem for the case $b=a$.
A similar conclusion can be established for $b=-a$.
Let $\{(x_n,y_n)\}$ be a solution to (<ref>).
Then, the the following statements are true:
(i) If $\gcd(p,q)=1$, then the solution $\{(x_n,y_n)\}$ of system (<ref>) has a subsequence $\{x_{2pqn}\}$ (resp. $\{y_{2pqn}\}$) that tends to infinity (resp. converges to zero) exponentially, and vice versa.
(ii) If $\gcd(p,q)>1$, and $p/\gcd(p,q)$ is odd, then the solution $\{(x_n,y_n)\}$ of system (<ref>) is eventually periodic with period $m$, where $m$ denotes the least common multiple of $p$ and $2q$.
(iii) If $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is even, then the solution $\{(x_n,y_n)\}$ behaves in a similar fashion as in (i).
Some illustrations which shows different behaviors of various solutions of system (<ref>) for the case $b=a$ with random positive initial values are illustrated in Figures (1)–(5).
It is noticeable from the above illustration that the system of difference equation
\[
x_n=\frac{1}{y_{n-2}}, \quad y_n=\frac{y_{n-2}}{x_{n-3}y_{n-3}},\qquad n=0,1,\ldots
\]
has a solution $\{x_n\}$ (resp. $\{y_n\}$) with a subsequence $\{x_{6n+s}\}$ (resp. $\{y_{6n+t}\}$) ($0\leq s,t<6$) that tends to infinity.
This result, moreover, agrees with Theorem <ref>–(i)
The above figure illustrates the behavior of another solution of the system
\[
x_n=\frac{1}{y_{n-2}}, \quad y_n=\frac{y_{n-2}}{x_{n-3}y_{n-3}},\qquad n=0,1,\ldots.
\]
Observe that after $N=25$, there is some subsequences $\{x_{6n+s}\}$ and $\{y_{6n+t}\}$ which both tends to infinity for $n \geq N$
The above figure shows an interesting behavior of solutions of the system
\[
x_n=\frac{1}{y_{n-6}}, \quad y_n=\frac{y_{n-6}}{x_{n-10}y_{n-10}},\qquad n=0,1,\ldots.
\]
Clearly, as the above system satisfies the conditions in Theorem <ref>–(ii), we then have a periodic solution of period ${\rm lcm}(6,2\times 10)=60$
Another interesting illustration for Theorem <ref>–(ii) is shown above.
In this example, the system
\[
x_n=\frac{1}{y_{n-60}}, \quad y_n=\frac{y_{n-60}}{x_{n-84}y_{n-84}},\qquad n=0,1,\ldots,
\]
has been considered. A simple computation for the period $m$ of the solution gives us $m={\rm lcm}(60,2\times 84)=840$
The above figure illustrates the behavior of a particular solution of the system
\[
x_n=\frac{1}{y_{n-4}}, \quad y_n=\frac{y_{n-4}}{x_{n-6}y_{n-6}},\qquad n=0,1,\ldots.
\]
Evidently, that particular solution of the given system behaves accordingly to Theorem <ref>–(iii)
We remark that a similar system has been studied by Özban in <cit.> wherein he investigated the behavior of positive solutions of the system
\[
x_{n+1}=\frac{1}{y_{n-k}},\qquad y_{n+1}=\frac{y_{n-k}}{x_{n-m}y_{n-m-k}},\qquad n=0,1,\ldots,
\]
where $k$ is a nonnegative integer and $m$ is a positive integer.
His main result states that all solutions of the above system of difference equations are periodic with period $2m+ 2k +2$.
In particular, when $b=a$ and $k=0$, all solutions of above equation is periodic with period $2q+2$.
§ CONCLUSION
In this short note, we have investigated the behavior of positive solutions of the system
\[
x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots,
\]
for $b=a$, where $q$ is a positive integer with $p < q$, $p \nmid q$, and $p$ is an even number.
We have found that every solution of the above system when $b=a$, with $p>0$, is non-periodic and has a subsequence that grows/decays exponentially whenever $q$ is odd.
However, a periodic solution of the given system occurs when $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is odd.
In this case, the period of the solution appears to be equal to the least common multiple of $p$ and $2q$.
On the other hand, a similar behavior as for the case when $q$ is odd was observed when $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is even.
Consequently, our result settled the question raised by Yang et al. in <cit.> about the behavior of solution of the given equation on the case when $p$ is even and $q>p$ in the given system.
§ APPENDIX
From the polynomial equation $P(\lambda)=(\lambda^p-1)(\lambda^q+1)=0$, it is clear that $\lambda=1$ is a simple root.
Hence, a particular solution to the non-homogeneous equation
\begin{equation}
\label{receq}
a_n+a_{n-q}-a_{n-p}-a_{n-p-q}=\ln c, \qquad c:=a/b,
\end{equation}
has the form
\[
\]
from which, by simple computation, leads to $A=\ln (c/2p)$.
Thus, if $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is odd, then the general solution of equation (<ref>) takes the form
\begin{align*}
x_n = e^{a_n}= c^{n/2p} \exp &\left\{ \sum_{l=0}^{p-1} \left( \alpha_{l,1} \cos \frac{2l\pi n}{p} + \alpha_{l,2} \sin \frac{2l\pi n}{p}\right)\right.\nonumber\\
&\qquad+ \left. \sum_{k=0}^{q-1} \left( \beta_{k,1} \cos \frac{(2k+1)\pi n}{q} + \beta_{k,2} \sin \frac{(2k+1)\pi n}{q}\right)\right\}.\label{sol}
\end{align*}
We can write $x_n$ as $x_n=c^{n/2p} \hat{x}_n$, where $\hat{x}_n$ denotes the positive solution of equation (<ref>) with $c=1$.
From above discussion, together with Theorem <ref>–(ii), we get the following results.
Assume that $c \in(0,1)$, $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is odd and let $m=\gcd(p,2q)$, then every positive solution of (<ref>) converges geometrically to zero.
Moreover, for each $ t \in \{0, 1, . . . , m - 1\}$, the subsequence $\{x_{mn+t}\}_{n\in\mathbb{N}_0}$ converges monotonically to zero as $n$ tends to infinity.
Assume that $c >1$, $\gcd(p,q)>1$ and $p/\gcd(p,q)$ is odd and let $m=\gcd(p,2q)$, then every positive solution of (<ref>) tends to infinity.
Moreover, for each $ t \in \{0, 1, . . . , m - 1\}$, the subsequence $\{x_{mn+t}\}_{n\in\mathbb{N}_0}$ grows exponentially as $n$ tends to infinity.
Cinar, C.,
On the positive solutions of the difference equation system $x_{n+1}=\frac{1}{y_n}, y_{n+1}=\frac{y_n}{x_{n-1}y_{n-1}}$,
Appl. Math. Comp.,
158 (2004), pp. 303–305.
Cinar, C., Yalçinkaya, I.,
On the positive solutions of the difference equation system $x_{n+1}=\frac{1}{z_n}, y_{n+1}=\frac{1}{x_{n-1}y_{n-1}}, z_{n+1}=\frac{1}{x_{n-1}}$,
Int. Math. J.,
5 (2004), pp. 517–519.
Cinar, C., Yalçinkaya, I.,
On the positive solutions of the difference equation system $x_{n+1}=\frac{1}{z_n}, y_{n+1}=\frac{x_n}{x_{n-1}}, z_{n+1}=\frac{1}{x_{n-1}}$,
Int. Math. J.,
5 (2004), pp. 525–527.
Cinar, C., Yalçinkaya, I.,
On the positive solutions of the difference equation system $x_{n+1}=\frac{1}{z_n}, y_{n+1}=\frac{y_n}{x_{n-1}y_{n-1}}, z_{n+1}=\frac{1}{x_n}$,
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18 (2005), pp. 91–93.
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1511.00560
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Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
Ghent University, Department of Physics and Astronomy, Krijgslaan, 281-S9, 9000 Gent, Belgium
Ghent University, Department of Physics and Astronomy, Krijgslaan, 281-S9, 9000 Gent, Belgium
ITEP, B. Cheremushkinskaya 25, Moscow, 117218 Russia
Moscow Inst Phys & Technol, Dolgoprudny, Moscow Region, 141700 Russia
School of Biomedicine, Far Eastern Federal University, Sukhanova str 8, Vladivostok 690950 Russia
We revisit non-interacting string partition functions in Rindler space by summing over fields in the spectrum. In field theory, the total partition function splits in a natural way in a piece that does not contain surface terms and a piece consisting of solely the so-called edge states. For open strings, we illustrate that surface contributions to the higher spin fields correspond to open strings piercing the Rindler origin, unifying the higher spin surface contributions in string language. For closed strings, we demonstrate that the string partition function is not quite the same as the sum over the partition functions of the fields in the spectrum: an infinite overcounting is present for the latter. Next we study the partition functions obtained by excluding the surface terms. Using recent results of <cit.>, this construction, first done by Emparan <cit.>, can be put on much firmer ground. We generalize to type II and heterotic superstrings and demonstrate modular invariance. All of these exhibit an IR divergence that can be interpreted as a maximal acceleration close to the black hole horizon. Ultimately, since these partition functions are only part of the full story, divergences here should not be viewed as a failure of string theory: maximal acceleration is a feature of a faulty treatment of the higher spin fields in the string spectrum. We comment on the relevance of this to Solodukhin's recent proposal in <cit.>. A possible link with the firewall paradox is apparent.
§ INTRODUCTION
Black hole horizons continue to baffle physicists at the quantum level, especially in light of the recent firewall paradox (see e.g. <cit.><cit.><cit.><cit.><cit.><cit.>).
Partly motivated by this, there has been a recent renewed interest in better understanding black hole horizons within quantum gravity <cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.>. Also within string theory specifically, multiple ideas have been explored. In <cit.><cit.><cit.><cit.>, a new massless mode was studied that lives on the Euclidean geometry close to the tip of the cigar. This mode is absent in field theory. In <cit.><cit.><cit.><cit.>, we interpreted this mode as the thermal scalar field representing the dominant contribution to the thermal string gas surrounding the black hole horizon.[The physical significance of this thermal scalar field was further analyzed in <cit.><cit.><cit.>.] The cigar theory was further studied recently in <cit.><cit.><cit.><cit.>. In <cit.><cit.><cit.>, real-time aspects of strings near horizons were discussed (non-adiabatic string production and elongation of strings falling near a black hole horizon). Long strings were also argued to be very important in between both horizons of rotating black holes in <cit.><cit.>. All of these works have in common that they focus on aspects of strings near black holes that differ from the naive field theory extrapolation.
In this paper, we will highlight yet another one of these aspects. We aim at describing manifestly non-interacting strings in Euclidean Rindler space and its conical orbifolds. On conical spaces, this description is obscured even in field theory. It has been known for quite some time that computing the one-loop vacuum amplitude for free fields on a conical space can yield negative contributions to the entropy, associated to a surface term on the black hole horizon <cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.>. This peculiarity arises only for spins $s \geq 1$ and obscures a thermodynamic interpretation. It is closely related to the difficulty in defining entanglement entropy in lattice gauge theories, a topic that is attracting a lot of attention recently, see e.g. <cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.>.
In either of these contexts, one encounters a surface contribution that represents the entanglement of the so-called edge states attached to the entangling surface.[The negativity of this contribution in continuum field theory was recently clarified in <cit.><cit.>.] Hence, within this context, the total partition function splits in a natural way in a piece that does not contain the surface terms and a piece consisting of solely these terms.
Susskind and Uglum looked at the analogous question for closed strings and illustrated that negative contributions can arise from worldsheets that intersect the origin in Euclidean Rindler space <cit.>. These were interpreted in real-time as emission and reabsorption processes with emergent open strings whose endpoints are stuck at the horizon. The QFT and string case were argued to be manifestations of the same phenomenon, for which some evidence was gathered <cit.>. However, the full equality of these features has not been demonstrated.
Very recently, the authors of <cit.> provided a new powerful technique to compute an arbitrary higher spin partition function on a conical space. Moreover, they demonstrate that the open string partition function is of precisely this same form, upon summing over the string spectrum. This allows a more direct comparison between the surface contributions of fields, and the analogous origin-intersecting worldsheet contributions of strings. Also the larger question on if and how string theory can be looked at as simply a sum over field theories can be addressed.
Secondly, the method also allows a clear distillation of the surface contributions from the “normal" part. Extracting only the latter, and then summing over the string spectrum, one obtains candidate string partition functions that are manifestly related to the sum over their field content. We will explore this road throughout this work.
Our final goal will be to get indications on whether this procedure done at the level of the states in the string spectrum, can really be identified with string partition functions that exclude worldsheets that intersect the origin (figure <ref>).
Open and closed string diagrams at one loop in flat space (or its conical cousins) where the origin of the 2d plane is excised. Worldsheets are then automatically forced not to intersect the origin. Drawn here are the winding one graphs.
Throughout this work we attempt to better understand if and how the contact terms are encoded within string theory as worldsheets that intersect the black hole horizon. If this interpretation is correct, several consistency checks can be done for the remainder of the partition functions.
Firstly, once horizon-intersecting worldsheets are excluded, the remainder should have a conventional thermodynamic interpretation (e.g. no peculiar negative contributions to the entropy).
Secondly, winding number and discrete momentum around the origin should be good quantum numbers, since one can now define wrapping numbers of the worldsheet in a clean fashion.
Thirdly, if good string theory models can be built in this way, the open and closed string sectors should be related by worldsheet duality.
Finally, for closed strings one needs modular invariance in order to have a consistent torus interpretation.
Our main effort throughout this work will be to try to establish these results and hence demonstrate that mathematically one can make sense of the theory that excludes horizon-intersecting worldsheets.
On a related note, it is interesting to think about the proposal made by Solodukhin in <cit.> concerning the equality between black hole entropy and entanglement entropy. He proposed that the tree-level contribution matches the sum of the contact terms at one loop, up to a sign. This would cause the black hole entropy (up to one loop) to be fully equal to the entanglement entropy of the fields and be manifestly positive. When applying this proposal in string theory, these are precisely the partition functions that we will construct here, and we will argue that within perturbative string theory such a mechanism does not seem viable.
The partition functions we will construct in this way by dropping contact terms, experience a thermal divergence that can be associated to a maximal acceleration of string theory. Maximal acceleration was argued to be a general feature of any consistent theory of quantum gravity in the past <cit.>. The arguments are however based on local considerations which makes applying them to string theory questionable.
A naive argument to do expect this in string theory is as follows. The local Unruh temperature around a black hole increases all the way to infinity. When it crosses the Hagedorn temperature, the supposedly ultimate temperature in nature, the strings cannot be in equilibrium anymore and they fall into the black hole. A slightly more refined argument was given in <cit.>, where the authors argue that a detector emersed in the Hagedorn heat bath at string length from the horizon, would be seen by inertial observers as emitting long strings. This would provide a mechanism for energy loss and would prevent a further acceleration.
A much more elaborate indication of this effect, was given in <cit.><cit.><cit.><cit.><cit.><cit.>, where explicit computations were carried out. The strategy was to compute a property of a massive scalar in Rindler space (the propagator or the stress tensor vev), and then sum over the complete string spectrum to obtain the string theory result, while ignoring possible spin-dependence. A divergence was found in each case, that can be interpreted as a maximal acceleration.[Additional classical string considerations also lead to this concept in <cit.><cit.>. A boundary state construction with this property is given in <cit.>.]
Of course, the strategy has been to sum over the complete string spectrum while assuming higher spin fields just behave as several copies of massive scalars. This has been a general strategy in string theory, ever since its conception.
To put these considerations in another perspective, consider the celebrated computation by Polchinski in <cit.>, in which he showed that the free energy of a string gas in flat space can be viewed as the sum of the free energies of the particles in the string spectrum. A hidden assumption here is that for each higher spin field, one can utilize the same expression for the free energy. For flat space, of course, this works out nicely. This strategy is doomed to fail for a black hole, since the surface interactions with the horizon ensure a non-trivial difference between the fields of different spin: one would need to know the free energy of every field in the string spectrum explicitly, before attempting the summation over the spectrum.
On a larger level, despite the fact that we expect quantum horizons to be only fully understood in a non-perturbative treatment, we believe it is worthwhile to further investigate the perturbative story as well, since many features of perturbative string theory in a black hole geometry are still ill-understood and a detailed study of them has already lead to some surprising conclusions in the past.
With our interest on thermodynamics in mind, it is useful to first set the stage. All expressions for partition functions $Z$ we will write down are for single particle loops (or string loops) on the thermal manifold. As is well known, a simple exponentiation gives the field theory vacuum amplitude where an arbitrary number of vacuum loops are included. This single particle partition function has an expansion into a vacuum part and a thermal part. The vacuum part follows formally by taking the $T\to0$ limit of the partition function. Our focus in this work, is on the remaining thermal part. Within closed string theory, the full partition function and the vacuum part are both separately modular invariant. This requires the thermal part to be modular invariant on its own. Even more so, the thermodynamic entropy $S = -\left(\beta\partial_{\beta}-1\right)Z$ where $Z=-\beta F$ also needs to be modular invariant. One of our main goals is to explicitly verify this modular invariance of our candidate non-interacting thermal partition functions. We will write down most of our results for general $N$. If one is interested in Rindler thermodynamics, one should take $N\to1$ in the end.
A gas of non-interacting bosonic matter in any space at temperature $1/\beta$ has for its free energy:
\begin{equation}
\label{fre}
\beta F = \sum_n \rho_B(E_n)\ln\left(1-e^{-\beta E_n}\right) = -\sum_{m=1}^{\infty}\sum_n \rho_B(E_n) \frac{e^{-m\beta E_n}}{m}.
\end{equation}
For spacetime fermions, one has instead
\begin{equation}
\beta F = -\sum_n \rho_F(E_n)\ln\left(1+e^{-\beta E_n}\right) = -\sum_{m=1}^{\infty}\sum_n \rho_{F}(E_n) \frac{(-)^{m-1}e^{-m\beta E_n}}{m}.
\end{equation}
If the theory is spacetime supersymmetric, $\rho_B= \rho_F = \rho$ and the total free energy becomes:
\begin{equation}
\label{freesusy}
\beta F = -\sum_{m=1}^{\infty}\sum_n \rho(E_n) \frac{e^{-m\beta E_n}}{m}(1 - (-)^{m}),
\end{equation}
which implies sectors with $m$ even are absent. Hence the characteristic factor of $1-(-)^m$ is one of the signatures of spacetime supersymmetry in thermodynamical quantities. The modular invariants we will construct for type II and heterotic strings will indeed include such a factor.
A further immediate property (well-known in flat space) is that $F<0$ and $S>0$ for non-interacting matter.
This paper is structured as follows. In section <ref> we provide an intuitive argument to show that open strings stuck at the horizon are important for black holes, even in the Lorentzian case. While we do not build upon this argument in the remainder, it is instructive to keep it in mind when tempted to dismiss the exotic open-closed interactions as a non-physical feature only occuring on the thermal manifold. Section <ref> recapitulates the work of <cit.> with a particular emphasis on the field content of the open string partition functions. We provide a detailed comparison and an interpretation of the higher spin surface contributions in string language. Starting with section <ref>, we take a look at closed strings. Firstly, we will prove that surface terms of the higher spin fields in the spectrum cannot be the end of the story in that case. Section <ref> provides a first attempt at non-interacting closed strings, by utilizing worldsheet open-closed duality to make some statement about closed strings. We also extend this computation to type II superstrings and we find similar conclusions. The main part of this work, section <ref>, discusses the construction of non-interacting closed string partition functions for bosonic, type II and heterotic strings. A particular emphasis is placed on modular invariance of these partition functions and the relation with earlier work by Emparan <cit.>. Section <ref> contains a detailed analysis on the interpretation of the IR divergence that arises in the partition functions for each string type. We end with a conclusion and outlook in <ref>. Some technical details are included in the appendix, as well as several formulas on the fixed winding heat kernels on flat cones.
§ RELEVANCE OF OPEN STRINGS FOR LORENTZIAN BLACK HOLES
Before delving into the computations, we would like to give a qualitative argument showing that the exotic open-closed interactions on the Euclidean black hole horizon envisaged by Susskind and Uglum, actually are very important in real time as well. To that end, it is instructive to recapitulate a basic physical argument in favor of the Unruh effect <cit.>. Suppose we coordinatize our flat metric as
\begin{equation}
ds^2 = - dT^2 + dX^2 = -\rho^2d\omega^2 + d\rho^2.
\end{equation}
The coordinate frame is shown below in figure <ref>. Rindler space covers only a quarter of 2d Minkowski space. Constant Rindler time slices (i.e. constant $\omega$) are semi-infinite lines originating at the origin. The infinite past ($\omega = - \infty$) and infinite future ($\omega = + \infty$) in Rindler time are the two diagonal lines drawn in the figure.
As is well known, an accelerating observer in flat space experiences the Minkowski vacuum as being thermally populated. An intuitive account for this effect can be given by explaining how the accelerating observer describes the vacuum fluctuations <cit.>. The heat bath seen by the Rindler observer arises because of eternal vacuum fluctuations as shown in figure <ref> (a).
(a) Vacuum fluctuations in QFT. The leftmost fluctuation is invisible to the Rindler observer. The rightmost loop is also seen as a vacuum fluctuation by the Rindler observer. The middle fluctuation is the important one: it is long-lived according to the Rindler observer. (b) The same diagrams within string theory, with the same interpretations.
The vacuum loop that encircles the origin is the relevant one to describe the Unruh heat bath. It is viewed by a fiducial observer as being eternal: the vacuum fluctuations close to the Rindler origin are no longer virtual. The analogous torus diagrams in string theory have been drawn in figure <ref> (b).
In string theory however, a second set of embeddings is possible, leading to an open string gas with fixed endpoints on the horizon as shown in figure <ref>.
String vacuum fluctuation that crosses the Rindler origin. This is seen by the Rindler observer as an open string with ending points fixed (and immobile) on the horizon.
As mentioned above, this intuitive account of the Unruh effect actually contains much of the basic physical principles at work here. And it clearly demonstrates the relevance of open strings when considering the heat bath created by the closed string Minkowski vacuum. Usually, open string theory requires closed string theory to make sense of its interactions. Now it is apparent that a black hole horizon also requires the opposite to be true: closed string theory requires open strings.[An alternative argument for the necessity of open strings is to try to define entanglement entropy for closed strings. The Hilbert space does not factorize since strings can pierce through the entangling surface. These strings are viewed from either side of the surface, as open strings attached to the entangling surface. It is up to the reader which argument is preferred.]
The remainder of the results of this work focus on the thermal (Euclidean) theory, but it is interesting to keep the intuition developed in this section in mind when contemplating the physical relevance of these open strings.
§ OPEN STRING PARTITION FUNCTION AND ITS FIELD CONTENT
A general way of studying string theory on a flat cone is to consider orbifolding the plane using a $\mathbb{Z}_N$ subgroup of $SO(2)$. For string theory, such discrete cones are apparently the only ones where a consistent modular invariant partition function is known. Afterwards, to study thermodynamics, one performs a continuation in the variable $N$ to a real number.
Within QFT, one can also perform this orbifolding procedure, but one is also free to simply study the field theory on a generic cone directly and avoid the artificial orbifolding. This for instance allows a description directly in terms of a wrapping number of particle paths around the conical singularity.
For the special $\mathbb{Z}_N$ cones, both descriptions should agree of course.
The logic throughout this work will be to start with string theory on the $\mathbb{Z}_N$ cones, and link it to its description in terms of the fields in the spectrum. Each higher spin field in the spectrum contains a “normal" part and a surface part. We will remove the surface part and study the remainder.[We study string theory as a sum of scalars and spin $1/2$ fermions with the appropriate high level degeneracy of states. We remark that for the entropy it has been pointed out (see e.g. <cit.>) that fermions do not contain a contact term: only bosonic fields do.] The latter partition function allows us to take $N$ a real number immediately. This construction will be shown to yield modular invariant partition functions.
In this section, we will start by taking a closer look at the open string partition function as obtained through summing all fields in the spectrum.
§.§ Bosonic fields
In <cit.>, He et al. wrote down the (single-particle) partition function of a generic higher spin bosonic field of mass $m$ on the flat cone $\mathbb{C}/\mathbb{Z}_N \times \mathbb{R}^{D-2}$ as
\begin{equation}
Z = \int_{0}^{+\infty}\frac{ds}{2s}\frac{V_{D-2}}{(2\pi )^{D-2}}\int d^{D}k \frac{1}{N}\sum_{j=1}^{N-1}\sum_{a=1}^{N_a}\frac{e^{\frac{2\pi i j s_a}{N}}}{4\sin^2\left(\frac{\pi j}{N}\right)}\delta(k_0)\delta(k_1)e^{-s(k^2+m^2)},
\end{equation}
upon subtracting the $j=0$ contribution (which is independent of the conical angle). The number $s_a$ denotes the spin of the $SO(2)$ subgroup of $SO(D)$ in the 2d plane of the cone.
Or upon integrating over $k$:
\begin{equation}
\label{ftspf}
Z = \int_{0}^{+\infty}\frac{ds}{2s}\frac{V_{D-2}}{(4\pi s)^{(D-2)/2}}\frac{1}{N}\sum_{j=1}^{N-1}\sum_{a=1}^{N_a}\frac{e^{\frac{2\pi i j s_a}{N}}}{4\sin^2\left(\frac{\pi j}{N}\right)}e^{-s m^2}.
\end{equation}
The thermodynamic entropy can be readily found as $ S= \partial_N\left(NZ\right)$.
For instance, for a spin-$0$ field, one obtains
\begin{equation}
\label{spin0Z}
Z = \frac{N}{12}\int_{0}^{+\infty}\frac{ds}{2s}\frac{V_{D-2}}{(4\pi s)^{(D-2)/2}}e^{-s m^2},
\end{equation}
after subtracting the non-thermal part and making use of the sum:
\begin{equation}
\label{bossum}
\sum_{j=1}^{N-1}\frac{1}{\sin^2\left(\frac{\pi j}{N}\right)} = \frac{N^2-1}{3}.
\end{equation}
For open bosonic strings, the partition function on $\mathbb{C}/\mathbb{Z}_N \times \mathbb{R}^{D-2}$ can be written down as well as
\begin{align}
\label{obspf}
Z &= V_{D-2} \int_{0}^{+\infty}\frac{dt}{2t}(8\pi^2\alpha' t)^{-12} \frac{1}{N}\sum_{j=1}^{N-1}\frac{\eta(it)^{-21}}{\sin\left(\frac{\pi j}{N}\right)\vartheta_1\left(\frac{j}{N},it\right)} \nonumber \\
&= V_{D-2} \int_{0}^{+\infty}\frac{ds}{2s}(4\pi s)^{-12} \frac{1}{N}\frac{1}{\eta\left(\frac{is}{2\pi\alpha'}\right)^{21}}\frac{1}{2}\frac{e^{\frac{s}{8\alpha'}}}{\prod_{n=1}^{+\infty}(1-q^n)} \nonumber \\
&\times \sum_{j=1}^{N-1}\frac{1}{\sin^2\left(\frac{\pi j}{N}\right)}\prod_{n=1}^{+\infty}\sum_{p_n,q_n=0}^{+\infty}e^{\frac{2\pi i j}{N} (p_n-q_n)}e^{- \frac{n s}{\alpha'}(p_n+q_n)},
\end{align}
where in the second line we have set $s=2 \pi \alpha' t$.
The string expression (<ref>) can be directly compared to the field expression (<ref>). Upon expanding the Dedekind $\eta$ functions in a power series as well, this identifies $\alpha' m^2 = N_o - 1$, where $N_o$ is the open string oscillator number of the 24 oscillators (of which two are in the conical plane). The $SO(2)$ spin of each state can then be identified as
\begin{equation}
s_a = \sum_{n}(p_n-q_n).
\end{equation}
Of course, we still need to check that the combinatorics work out, i.e. that every state we construct is really represented in the above string construction. As a simple example of this point, consider the case where the exponent $\sum_n n(p_n+q_n)$ equals $2$. The number of such terms is the number of partitions of $2$. There are several options for creating this: $p_2=1$, $q_2=1$, $p_1=q_1=1$, $p_1=2$ or $q_2=2$. In terms of states, we have: $\alpha_2^x$, $\alpha_2^y$, $\alpha_1^x \alpha_1^y$, $\alpha_1^x \alpha_1^x$ and $\alpha_1^y \alpha_1^y$.[Here $x$ and $y$ denote the 2d plane with the conical singularity.] The first two options have spin $1$. The final three states carry spin $2$ or spin $0$, where only 1 linear combination carries spin $0$: $\alpha_1^x \alpha_1^x + \alpha_1^y \alpha_1^y$. This is indeed also the case for $\sum_{n}(p_n-q_n)$.
It turns out that we should interpret $p_n$ as counting $\alpha_n^x + i \alpha_n^y$ and $q_n$ as counting $\alpha_n^x - i \alpha_n^y$, and indeed, this is how string partition functions on cones are typically computed in the first place: by combining the fields as $Z = X+iY$ and $\bar{Z} = X-iY$ <cit.><cit.>.
In appendix <ref>, we illustrate this for level $3$ and then demonstrate that it is generally true at any level. Hence summing the particle partition function over the full string spectrum gives precisely the above string partition function.
The particle partition functions however contain surface terms at higher spin. The open string partition function on the other hand contains exotic configurations where an open string pierces the horizon, interpreted as an open string emitting and reabsorbing another open string (figure <ref>).
An open string piercing the Rindler origin, performing a loop around the thermal direction and forming a cylindrical worldsheet. Fixed timeslices are interpreted as an emission and reabsorption of an open string.
The conventional thermodynamic contribution on the other hand consists of open strings that do not intersect the origin, such as those displayed in figure <ref>.
An open string performing a loop around the thermal direction and forming a cylindrical worldsheet. Fixed timeslices are interpreted as a non-interacting open string.
The reason that this configuration is manifestly equal to the non-interacting thermal trace $Z= \text{Tr}e^{-\beta H}$, is that it can simply be described as an open string moving a distance $\beta$ in Euclidean time and then reidentifying the configuration. The only unknown here is the open string Rindler Hamiltonian $H$, which performs a time translation of an open string. But one does not need to know it explicitly to ensure this interpretation; its existence is sufficient.
Now, due to the equality between the sum-over-fields approach and the full string result, we conclude that surface contributions to the higher spin fields can be identified in string language as the exotic open string interaction diagrams where the cylinder worldsheet pierces the horizon. This is the interpretation as suggested quite some time ago by Kabat <cit.>, but the above results make this much more explicit for open strings.[A curious fact about these emergent open strings attached to the origin is that their interactions with the real (open or closed) string gas cannot be turned off in the $g_s \to 0$ limit. A puzzle that arises then is that, since the Lorentzian computation (tracing over fixed energy states in a canonical ensemble) should always match the Euclidean computation, how do these fixed open strings influence the $g_s\to0$ behavior of the Lorentzian string gas. Apparently, their influence must also be felt in that language. And indeed, the argument presented in the previous section, although qualitative, shows that also in that language these open strings are important.]
This also demonstrates that the correct density of states (the same as in flat space) has been utilized here: for every single field on (Lorentzian) Rindler space, we compute the heat kernel. Finally summing these with the flat space density of states agrees precisely with the conical partition function of open string theory.
Now, let us turn to the manifestly non-interacting partition function, where we drop these strange surface interactions.[In the remainder of this work, we will call the resulting partition functions the non-interacting partition functions.] From the particle heat kernels, it is clear that these contributions arise from the spin-dependent parts. One can hence turn off these contributions by simply removing the spin-dependent exponential in the heat kernels (<ref>). Note that this procedure makes the partition function larger: the surface contributions are of negative sign in $Z$. They hence contribute positively to the free energy ($Z=-\beta F$). The same negativity of the surface interactions is true for the entropy $S$ for any higher spin field as can be seen in the work of <cit.>. Within string theory, this procedure translates into the removal of the same factor. For open bosonic strings, one obtains for instance:
\begin{align}
\label{nonintpf}
Z &= V_{D-2} \int_{0}^{+\infty}\frac{ds}{2s}(4\pi s)^{-12} \frac{1}{N}\frac{1}{\eta\left(\frac{is}{2\pi\alpha'}\right)^{21}}\frac{1}{2}\frac{e^{\frac{s}{8\alpha'}}}{\prod_{n=1}^{+\infty}(1-q^n)} \nonumber \\
&\times \sum_{j=1}^{N-1}\frac{1}{\sin^2\left(\frac{\pi j}{N}\right)}\prod_{n=1}^{+\infty}\sum_{p_n,q_n=0}^{+\infty}e^{- \frac{n s}{\alpha'}(p_n+q_n)}.
\end{align}
§.§ Fermionic fields
The extension to fermionic fields was also given in <cit.>. The authors wrote down a formula combining both bosonic and fermionic fields as:
\begin{equation}
\label{ftspff}
Z = (-)^F \int_{0}^{+\infty}\frac{ds}{2s}\frac{V_{D-2}}{(4\pi s)^{(D-2)/2}}\frac{1}{N}\sum_{j=1}^{N-1}\sum_{a=1}^{N_a}\frac{e^{\frac{2\pi i j 2s_a}{N}}}{4\sin^2\left(\frac{2\pi j}{N}\right)}e^{-s m^2}.
\end{equation}
Only odd $N$ are allowed in this formula, since fermionic fields only have an orbifold interpretation in this case. For an elementary spin-$\frac{1}{2}$ field for instance, one finds
\begin{equation}
\label{spin1/2Z}
Z = \sum_{a=1}^{2}\frac{N}{24} \int_{0}^{+\infty}\frac{ds}{2s}\frac{V_{D-2}}{(4\pi s)^{(D-2)/2}}e^{-s m^2},
\end{equation}
upon subtracting the non-thermal part. The relevant sum that was performed here is given by
\begin{equation}
\label{fermsum}
\sum_{j=1}^{N-1}\frac{e^{\frac{2\pi j}{N}}}{\sin^2\left(\frac{2\pi j}{N}\right)} = \frac{-N^2+1}{6}.
\end{equation}
Hence this recovers the old result that each Majorana component of a 2d spinor contributes half as much as a real scalar (<ref>). Kabat indeed proved that a spin-$\frac{1}{2}$ 2-component spinor has the same thermal entropy as a real scalar in Rindler space <cit.>.
Also the open superstring partition function can be written down and compared to its particle content. This was done in <cit.> and we will refrain at this point from making any more detailed checks (we come back to this partition function further on).
Just as for bosonic higher spin fields, where we drop all spin-dependent parts and effectively reduce them to scalars, we will do the same with higher spin fermionic fields and treat them as spin-$\frac{1}{2}$ fermions, which have no surface interactions.
§ CLOSED STRING THEORY AS A SUM OVER FIELDS?
It is by now clear that the open string entropy can be viewed as a sum of the field theory entropies of all the states in the spectrum. Our goal is to do the same analysis for closed strings. However, the string partition functions themselves that can be constructed as $\mathbb{C}/\mathbb{Z}_N$ orbifolds do not lend themselves to an analogous comparison <cit.>. The reason is the second quantum number (next to $j$) that is summed over, for which a direct thermal interpretation is more difficult to make. In fact, the partition functions turn out to be different as we now illustrate.[Related to this is the fact that the perspective of open strings as a sum-over-fields has always been more direct than in the closed string case. Examples of this are for instance that the one-loop cosmological constant for closed strings is different than what one would obtain if one sums over all fields in the spectrum. Or that it is apparently much simpler to construct an open string field theory action than it is to construct a closed one.]
The bosonic closed string partition function on the $\mathbb{Z}_N$ orbifold can be written down as
\begin{align}
\label{fundpf}
Z &= V_{D-2} \int_{\mathcal{F}}\frac{d\tau^2}{4\tau_2}(4\pi^2\alpha' \tau_2)^{-12} \frac{1}{N}\sum_{m,w=0, (m,w) \neq(0,0)}^{N-1}\frac{\left|\eta(\tau)\right|^{-42}e^{2\pi\tau_2\frac{w^2}{N^2}}}{\left|\vartheta_1\left(\frac{m}{N} + \frac{w}{N}\tau,\tau\right)\right|^2}.
\end{align}
On the other hand, the sum of fields approach yields
\begin{align}
\label{stripf}
Z = V_{D-2} \int_{\mathcal{E}}\frac{d\tau^2}{4\tau_2}(4\pi^2\alpha' \tau_2)^{-12} \frac{1}{N}\sum_{j=1}^{N-1}\frac{\left|\eta(\tau)\right|^{-42}}{\left|\vartheta_1\left(\frac{j}{N} ,\tau\right)\right|^2},
\end{align}
which is basically simply the square of the open string partition function (<ref>). The difference is the second sum over $w$ and the difference in integration region over either the fundamental domain or the entire strip. The big question is now whether the expressions (<ref>) and (<ref>) could be equal. One can readily prove here that this is impossible for any finite $N$. If the range of all of the summations were infinite, then one could immediately use the standard unfolding theorem. The finite range obscures the question at hand, and makes it indeed improbable for something similar to succeed.
The proof proceeds by trying to apply the McClain-Roth-O'Brien-Tan theorem <cit.><cit.> as much as possible. We hence start in the modular fundamental domain and try to build up the strip domain by applying suitable modular transformation to the $w\neq0$ sectors. This can be formulated in a mathematical language as a build-up of not the entire strip, but instead of (parts of) the fundamental domain for the Hecke congruence subgroups $\Gamma_0(N)$ of the modular group. This was studied previously in <cit.><cit.><cit.>. This is done explicitly for several low values of $N$ in appendix <ref>. These domains are never equal to the full strip (except for $N\to\infty$) and hence the above partition functions (<ref>) and (<ref>) cannot be equal.
The difference however, can be interpreted as coming from the small $\tau_2$ UV region, and much like the cosmological constant in closed string theory, it appears that string theory also handles the UV in a different fashion for conical entropies.
The detailed comparison done in appendix <ref> also shows that the sum-over-fields approach actually hugely overcounts the stringy result. Every torus configuration is counted an infinite number of times. The way this happens is quite analogous to the vacuum energy in flat space closed string theory: in that case the sum-over-fields approach gives a modular invariant but integrated over the strip domain. This domain can then be folded into the fundamental domain, but an overall infinity is included in the process (basically counting the number of images of the fundamental domain that lie within the strip) due to the immense overcounting of the tori within field theory. Whether one still calls this partition function modular invariant is only a matter of taste; it is pathological and infinitely large when trying to interpret it as a modular invariant integrated over the fundamental domain.
We would like to point out the difference between this situation and the flat space closed string case. For flat space, the free energy (and entropy) of the string gas is nicely given by the sum of the free energies (or entropies) of the fields in the spectrum; the discrepancy between string and field theory only occurs for the non-thermal vacuum energy part. For the conical manifolds (and hence the approach to Rindler entropy), it also seems to happen for the thermal part.
§ WORLDSHEET DUALITY AS A ROAD TO NON-INTERACTING CLOSED STRINGS
It would be interesting to have a deeper understanding of the above feature, but we will take a more pragmatic approach in the remainder of this work.
An indirect way of saying something about closed strings is through open-closed duality. Using this, one can at least obtain information on the conformal weights of closed strings on the same space. This is the approach we will utilize in this section.
§.§ Bosonic strings
Let us first look at the full open bosonic string partition function (<ref>). Worldsheet (open-closed) duality can be used in the standard fashion: one transforms $t\to 1/t$ and then analyzes the large $t$ limit. One finds an expansion where all closed string states appear propagating along the cylinder. The theta-function has the property
\begin{equation}
\vartheta_1\left(\nu,\frac{i}{t}\right) = -i\sqrt{t}e^{-\pi\nu^2 t}\vartheta_1\left(\nu i t, it\right),
\end{equation}
and one finds for the most dominant closed string state propagating in the closed twist $j$ channel:[We do not keep track of the polynomial prefactors.]
\begin{equation}
Z_j \sim \int^{+\infty}\frac{dt}{t}e^{\pi t \left(2 - \frac{j}{N} + \frac{j^2}{N^2}\right)},
\end{equation}
which is indeed the most dominant closed string tachyon of twist $j$ <cit.><cit.>. As we expect, worldsheet duality tells us something about the conformal weights of the states.
Next we try to do the same thing for the partition function for which the surface contributions have been deleted. Dropping the spin-dependent exponent, one recovers the pure thermal (non-interacting) contribution. It is worthwile to rewrite expression (<ref>) a bit. For the open string partition function, one can arrive there by replacing
\begin{equation}
\vartheta_1\left(\frac{j}{N}|\tau\right) \to \lim_{\nu \to 0}\frac{\vartheta_1(\nu|\tau)}{\sin(\pi \nu)} \sin\left(\frac{\pi j}{N}\right),
\end{equation}
in the first line of equation (<ref>), which basically eliminates the spin-dependent exponentials in the expansion written above.
One can write down an expression for the resulting non-interacting partition function by defining $\tilde{\vartheta}_1$ as the $\vartheta_1$-function with the sine factor removed:
\begin{equation}
\label{deftilde}
\tilde{\vartheta}_1\left(\nu|\tau\right) =\frac{\vartheta_1( \nu|\tau)}{\sin(\pi \nu)}.
\end{equation}
Then we can write:
\begin{align}
Z &= V_{D-2} \int_{0}^{+\infty}\frac{dt}{2t}(8\pi^2\alpha' t)^{-12} \frac{1}{N}\sum_{j=1}^{N-1}\frac{\eta(it)^{-21}}{\sin^2\left(\frac{\pi j}{N}\right)\tilde{\vartheta}_1\left(0,it\right)} \nonumber \\
&= V_{D-2} \frac{1}{N} \int_{0}^{+\infty}\frac{dt}{2t}(8\pi^2\alpha' t)^{-12} \frac{N^2-1}{3}\frac{\eta(it)^{-21}}{\tilde{\vartheta}_1\left(0,it\right)}.
\end{align}
Note that there is a thermal contribution $Z\sim N$ and a non-thermal one $Z\sim1/N$.
Again doing the worldsheet duality, one instead finds
\begin{equation}
Z_j \sim \int^{+\infty}\frac{dt}{t} e^{2\pi t},
\end{equation}
for any $j$, showing that it is the closed string tachyon that propagates most dominantly, even in the twisted channels. The non-interacting partition function hence diverges for bosonic strings. The divergence is independent of the twisted sector $j$. This derivation of the most dominant closed string state evades having to contemplate the density of states for closed strings (presumably the same as for flat space).
§.§ Extension to superstrings
The extension to superstrings is readily made. In Green-Schwarz language, the open superstring partition function equals
\begin{equation}
V_{D-2}\int_{0}^{+\infty} \frac{dt}{2t}(8\pi^2\alpha't)^{-4}\sum_{j=1}^{N-1}\frac{\vartheta_1\left(\frac{j}{N},it\right)^4}{N\sin\left(\frac{2\pi j}{N}\right)\vartheta_1\left(\frac{2j}{N},it\right)\eta(it)^9}.
\end{equation}
Performing worldsheet duality again, one finds for large $t$:
\begin{align}
\vartheta_1\left(\frac{j}{N},\frac{i}{t}\right)^4 &\sim e^{-\pi \frac{j^2}{N^2}4t}e^{\pi\frac{j}{N}4t}e^{-\pi t}, \\
\vartheta_1\left(\frac{2j}{N},\frac{i}{t}\right)^{-1} &\sim e^{\pi \frac{4j^2}{N^2}t}e^{-\pi\frac{2j}{N}t}e^{\pi t/4} \underbrace{e^{-2\pi t\left(\frac{2j}{N}-1\right)}}_{\text{if }j>N/2}, \\
\eta^{-9}\left(\frac{i}{t}\right) &\sim e^{2\pi t \frac{9}{24}}.
\end{align}
Hence one arrives at
\begin{align}
Z_j &\sim \int^{+\infty}\frac{dt}{t} e^{2\pi t\frac{j}{N}}, \quad j < \frac{N}{2}, \\
Z_j &\sim \int^{+\infty}\frac{dt}{t} e^{2\pi t\left(1-\frac{j}{N}\right)}, \quad j > \frac{N}{2}.
\end{align}
In this Green-Schwarz language, the sectors with $j>N/2$ will correspond to the oddly twisted ones in the RNS language, whereas the sectors with $j<N/2$ are evenly twisted. Using the Riemann identity and a shifting of $w$ and $m$ in the closed string partition function, these indeed get redistributed into even and odd sectors in the RNS superstring language <cit.><cit.>.
This dominant closed string propagating state is again well-known and is the most dominant state in the twisted sectors.
The non-interacting partition function for open superstrings can be found quite analogously as above. One relies heavily on formulas derived in <cit.> to obtain this. We include the relevant formulas for reference in appendix <ref>. After performing the sum of the twisted sectors $j$, one obtains
\begin{align}
\label{nonintfermpf}
Z &= V_{D-2}\int_{0}^{+\infty} \frac{dt}{2t}(8\pi^2\alpha't)^{-4} \frac{\frac{N^2-1}{3}\vartheta_3\left(0,it\right)^4 - \frac{N^2-1}{3}\vartheta_4\left(0,it\right)^4 + \frac{N^2-1}{6}\vartheta_2\left(0,it\right)^4}{N \tilde{\vartheta}_1\left(0,it\right)\eta(it)^9} \nonumber \\
&= V_{D-2}\frac{N}{2}\int_{0}^{+\infty} \frac{dt}{2t}(8\pi^2\alpha't)^{-4} \frac{\vartheta_2\left(0,it\right)^4}{\tilde{\vartheta}_1\left(0,it\right)\eta(it)^9} + (\text{Temp-independent}).
\end{align}
In fact, $\tilde{\vartheta}_1(0,it) \sim \eta(it)^3$. Hence
\begin{equation}
Z \sim \int^{+\infty}\frac{dt}{t} \frac{\vartheta_2^4(it)}{\eta(it)^{12}},
\end{equation}
which will be consistent with the non-interacting closed string partition function we will construct below.
Since for large $t$,
\begin{align}
\vartheta_2\left(0,i/t\right) &\sim 1, \\
\tilde{\vartheta}_1(0,i/t) &\sim e^{-\pi t/4},
\end{align}
we get for the most dominant contribution of type II closed superstring propagation:
\begin{equation}
Z_j \sim \int^{+\infty}\frac{dt}{t} e^{\pi t},
\end{equation}
which is again the closed string tachyon. Hence, even closed type II superstrings have a divergent non-interacting partition function in the twisted sectors. Note that tachyons in the twisted sectors can be interpreted as thermal tachyons, which are relevant for thermodynamics.
§.§ All windings must be present in the non-interacting partition function due to open-closed duality
As a further application of the open-closed duality, we here explain that this leads to the conclusion that all winding numbers must be present in the closed string spectrum of the non-interacting partition function.
We focus here on the $N=1$ limit, although a generic conical deficit does not alter any of our conclusions. We have in mind here the decomposition of the partition function into the different wrapping numbers around the polar origin; the link of the orbifolding procedure and its twisted sectors (labeled by $j=1...N$) with the actual wrapping numbers is a bit more difficult to make and we do not focus on this here. We elaborate on this link further on.
The polar origin represents a topological defect around which string worldsheets can wrap. Imagine wrapping an open string worldsheet around the origin. The worldsheets are labeled by an integer, the wrapping number. Performing an open-closed duality, one views these worldsheets as closed strings moving parallel to the polar axis. These closed strings now have a definite winding number, corresponding to the wrapping number of the original open string.[Indefinite wrapped open string worldsheets (where the worldsheet intersects the polar axis) correspond to indefinite winding number of the closed strings. These are excluded here, since we are interested in dropping all of the surface interactions.] The situation for wrapping number two is sketched in figure <ref>.
An open string worldsheet of wrapping number two can be equivalently seen as a twice wound closed string moving from the green to the red curve. All winding numbers of closed strings must be present simply because all open string wrapping numbers must be present in the non-interacting open string partition function in such a topologically non-trivial situation.
Wrapping numbers of such configurations are of course Poisson dual to discrete momentum in flat space, just as closed string winding is Poisson dual to open string wrapping. So we expect every winding number in the closed string theory to be present in the theory.[One can readily prove this intuition analytically for the simpler $\mathbb{R}^{25}\times S_1$ manifold. For open strings propagating on this space, the partition function includes the discrete momentum contribution
\begin{equation}
\sum_{n\in\mathbb{Z}}e^{-2\pi t \left(\frac{\alpha'n^2}{R^2}\right)}.
\end{equation}
Upon Poisson resummation, one obtains contributions to the open string path integral with a definite wrapping number along the compactified circle:
\begin{equation}
\label{poissonresu}
\sum_{n\in\mathbb{Z}}e^{-2\pi t \left(\frac{\alpha'n^2}{R^2}\right)} \propto \sum_{m\in\mathbb{Z}}e^{- \frac{\pi}{2t} \left(\frac{m^2R^2}{\alpha'}\right)}.
\end{equation}
If one is interested in using open-closed duality on the other hand, one focuses on the small $t$ limit of this expression. In this limit, all terms in the above sum contribute equally, and one again needs to perform a Poisson resummation. In the small $t$ expansion, the contribution from the open string oscillators $\eta(it)^{-24}$ starts with a term $\sim e^{\frac{2\pi}{t}}$. The final result for the partition function is a series expansion in closed string states of ever increasing mass upon identifying $\tau_2 = \frac{1}{2t}$, and indeed, equation (<ref>) turns into the standard winding contribution of the closed string partition function. Note that discrete momentum of the closed strings is completely missed in this approach. ]
We have hence demonstrated that all winding numbers in closed string theory should be present in the non-interacting partition function, simply because the open string partition function can be wrapped any number of times around the polar origin.
A further expectation that we can illustrate here is that the most dominant contribution in each winding sector will be divergent in precisely the same way as the closed string tachyon. One naive way of anticipating this, is when we take the mass formula for closed bosonic strings in $\mathbb{R}^{25}\times S_1$, it is of the form:
\begin{equation}
m^2 = \frac{w^2R^2}{\alpha'^2} - \frac{4}{\alpha'}.
\end{equation}
Taking the limit as $R\to0$, one indeed finds all $w$ states to experience the same closed string tachyon divergence. This result will be borne out in the detailed computations to be described further on.
§ CLOSED STRING MODULAR INVARIANTS
In the previous section, we have obtained information on the non-interacting closed string partition function through worldsheet duality. In this section, we explicitly sum over the fields in the closed string spectrum and form non-interacting partition functions. In this language, these partition functions are integrated over the modular strip and hence modular invariance (necessary for a torus interpretation) is not manifest. The purpose of this section is to demonstrate that these partition functions can be rewritten as modular invariants integrated over the fundamental domain, and this for all types of strings (bosonic, type II and heterotic). These partition functions will exhibit a divergence (as demonstrated above already from the open string perspective), that can be interpreted in terms of maximal acceleration. Moreover, we will demonstrate that for type II and heterotic strings, these modular invariants precisely encode the thermal sign factors that we expect <cit.>.
§.§ Bosonic string
For the closed bosonic string, one can readily write down the non-interacting partition function as well, simply obtained by dropping the spin-dependent exponential in equation (<ref>) and summing over the closed string spectrum:
\begin{equation}
\label{bospf}
Z = \frac{V_{D-2}}{N} \sum_{j=1}^{N-1}\int_{\mathcal{E}} \frac{d\tau d\bar{\tau}}{\tau_2} \frac{1}{(4\pi^2\alpha' \tau_2)^{12}}\frac{\left|\eta\right|^{-48}}{4\sin^2\left(\frac{j\pi}{N}\right)}.
\end{equation}
As usual, the integral over $\tau_1$ is generated by enforcing the level-matching condition on the excited string states. The modular integration region $\mathcal{E}$ is the modular strip. The sum over $j$ can be readily done. Analytically continuing as $N \to \frac{2\pi}{\beta}$, one can rewrite this in the form
\begin{equation}
\label{emparan}
Z = \int_{-1/2}^{1/2}d\tau_1\int_{0}^{\infty}\frac{ds}{2s}\zeta(s)\left|\eta\right|^{-48},
\end{equation}
with $\zeta(s)$ the heat kernel of the Laplacian operator on the Euclidean manifold. The heat kernels are given by <cit.><cit.>
\begin{align}
\zeta_{\text{flat}}(s) &= \frac{L}{\sqrt{4\pi s}}, \\
\zeta_{\text{cone}}(s) &= \frac{\beta}{2\pi}\frac{A}{4\pi s}+\frac{1}{12}\left(\frac{2\pi}{\beta}-\frac{\beta}{2\pi}\right).
\end{align}
Indeed, setting $s=\pi\alpha'\tau_2$ and dropping the temperature-independent parts, one finds agreement with (<ref>). Equation (<ref>) was written down by Emparan 20 years ago <cit.>. The main difference is that in the case at hand, the computation is much better motivated and it is clear that Emparan computed the non-interacting free energy, obtained by neglecting all spin-dependent parts of the heat kernels and treating every higher spin field as a scalar. Two remarks are in order.
Firstly, the above partition function diverges for any $N$ as $\sim e^{4\pi/\tau_2}$. This means the thermal entropy of the non-interacting bosonic string gas diverges as $N\to1$.
Secondly, the above heat kernel on the cone makes explicit the spatial dependence as the variable $\rho$ is the radial polar coordinate in this coordinate system. This will lead to a spacetime interpretation of the divergence, as discussed by Emparan as well <cit.>.
The above partition function is modular invariant <cit.>. Using the methods of <cit.><cit.>, we can write this in a modular invariant way. For a modular invariant function $f$, one can unfold the fundamental domain in this case as
\begin{equation}
\int_{\mathcal{E}} \frac{d\tau d\bar{\tau}}{\tau_2^2} f(\tau) \tau_2 = \int_{\mathcal{F}} \frac{d\tau d\bar{\tau}}{\tau_2^2} f(\tau) \frac{3}{\pi^2}\sum_{m,w\in\mathbb{Z}}'\frac{\tau_2}{\left|m+w\tau\right|^2},
\end{equation}
where $\mathcal{F}$ is the modular fundamental domain. The prime in the summations indicates that the $m=w=0$ term has been excluded. The proof of this formula follows <cit.><cit.> precisely. An alternative way of appreciating this result is by integrating the flat space identity
\begin{equation}
\int_{\mathcal{F}}\frac{d^2 \tau}{4\pi\tau_2^{2}}f(\tau)\sum_{m,w \in \mathbb{Z}}'e^{-\frac{\beta^2}{4\pi \alpha'\tau_2}\left|m+w\tau\right|^2} = \int_{\mathcal{E}}\frac{d^2 \tau}{4\pi\tau_2^{2}}f(\tau)\sum_{m \in \mathbb{Z}}'e^{-\frac{\beta^2}{4\pi\alpha'\tau_2}\left|m\right|^2},
\end{equation}
with respect to $\beta^2$ and then setting $\beta \to 0$:
\begin{equation}
\label{modident}
\int_{\mathcal{F}}\frac{d^2 \tau}{4\pi\tau_2^{2}}f(\tau)\sum_{m,w \in \mathbb{Z}}'\frac{\tau_2}{\left|m+w\tau\right|^2} = \frac{\pi^2}{3}\int_{\mathcal{E}}\frac{d^2 \tau}{4\pi\tau_2^{2}}f(\tau)\tau_2.
\end{equation}
This alternative is more of a mnemonic than a proof, but it is this perspective that will allow the most transparent generalization to type II and heterotic strings further on.
We hence interpret $w$ and $m$ as the winding numbers of the torus worldsheet in the fundamental domain along the two torus cycles, and the single $m$ quantum number in the strip as the thermodynamic expansion parameter as in equation (<ref>). As a check that these interpretations appear to be correct, we note that the partition function in the strip domain is proportional to
\begin{equation}
Z \sim \sum_{m=1}^{+\infty}\frac{1}{m^2\beta^2}\beta,
\end{equation}
which is a sum of a monotonically decreasing function of $m\beta$ multiplied with $\beta$. This is precisely the same sort of functional dependence we expect from a non-interacting partition function (<ref>).
The actual proof that these interpretations are correct is provided in appendix <ref>. We prove there that the heat kernel for a bosonic massless particle on a flat cone[For a massive boson of mass $M$, one simply multiplies this with $\exp(-sM^2)$.]
\begin{align}
\zeta_{\text{cone}}(s) = \frac{\beta}{2\pi}\frac{A}{4\pi s}+\frac{1}{12}\left(\frac{2\pi}{\beta}-\frac{\beta}{2\pi}\right),
\end{align}
can be decomposed into a wrapping-zero contribution
\begin{equation}
G^{(0)}(s) = \left(\frac{A}{4\pi s}-\frac{1}{12}\right)\frac{\beta}{2\pi},
\end{equation}
and the remaining part that can be seen as the sum of
\begin{equation}
G^{(m)}(s) = \frac{1}{2\pi \beta m^2},
\end{equation}
for $m\neq0$, the wrapping number of the particle path. The latter indeed sums into $\frac{1}{12}\frac{2\pi}{\beta}$. These expressions are explicitly proven using the fixed winding heat kernel expressions known in the literature <cit.><cit.><cit.>.[It is interesting at this point to compare the orbifolding procedure in field theory with the actual decomposition in wrapping numbers a bit in more detail. We have for $\beta=2\pi/N$:
\begin{align}
\zeta_{\text{cone}}(s) &= \frac{1}{N}\frac{A}{4\pi s}+\frac{1}{12}\left(N-\frac{1}{N}\right) \nonumber\\
&= \frac{\beta}{2\pi}\left(\frac{A}{4\pi s}-\frac{1}{12}\right)+\frac{1}{12}\frac{2\pi}{\beta}.
\end{align}
The first line shows the orbifold decomposition of the heat kernel, where the first term is the projected untwisted sector and the other terms represent the sum of the twisted sectors. Each twisted sector is weighted by $1/\text{sin}^2\left(\frac{\pi j}{N}\right)$ as reviewed earlier. This last part contains both a thermodynamical part and an additive contribution to the free energy. The second line on the other hand shows the decomposition into wrapping zero (the first term) and the sum over all non-zero wrappings. Clearly, the remaining terms are fully thermodynamical (and do not contain an additive shift).
In general, the second description is much more physical, but the orbifold construction is much simpler to compute.] Hence, the strip quantum number $m$, which is implicitly introduced in equation (<ref>) somewhat arbitrary, has the correct meaning in terms of wrapping number of the heat kernel.
One hence rewrites (<ref>) as
\begin{equation}
Z = \frac{V_{D-2}}{N}\frac{3}{4\pi^2}\int_{\mathcal{F}} \frac{d\tau d\bar{\tau}}{\tau_2} \frac{1}{(4\pi^2\alpha' \tau_2)^{12}}\left|\eta\right|^{-48}\frac{N^2-1}{3}\sum_{m,w\in\mathbb{Z}}'\frac{1}{\left|m+w\tau\right|^2}.
\end{equation}
For arbitrary conical deficits, one obtains upon dropping a $\beta$-independent part:
\begin{equation}
Z = V_{D-2}\frac{1}{2\pi \beta} \int_{\mathcal{F}} \frac{d\tau d\bar{\tau}}{\tau_2} \frac{1}{(4\pi^2\alpha' \tau_2)^{12}}\left|\eta\right|^{-48}\sum_{m,w\in\mathbb{Z}}'\frac{1}{\left|m+w\tau\right|^2}.
\end{equation}
An important feature of this partition function, is that it does not have an overall infinity present (which the sum over all fields including surface contributions does have as discussed in section <ref>). This partition function is well-behaved as a modular invariant.[Barring of course the exponential $\tau_2\to\infty$ divergence present, but this one is not fully pathological. It signals a physical feature of these partition functions that we will discuss a lot more in what follows.]
§.§ Type II superstrings
For type II superstrings, one can generalize this. The fermions add with the same sign to the non-interacting entropy as the bosons. One finds the following modular combination in the integral:
\begin{equation}
\frac{\frac{N^2-1}{3}\left|\vartheta_3^4-\vartheta_4^4\right|^2 + \frac{N^2-1}{3}\left|\vartheta_2^4\right|^2 + \frac{N^2-1}{6}\left(\vartheta_3^4-\vartheta_4^4\right)\bar{\vartheta_2}^4 + \frac{N^2-1}{6}\left(\bar{\vartheta_3}^4-\bar{\vartheta_4}^4\right)\vartheta_2^4}{\left|\eta\right|^{24}}.
\end{equation}
Just like in the open superstring case, upon dropping the non-thermal $N$-independent part, our expression reduces to
\begin{equation}
\frac{N^2}{3}\frac{\left|\vartheta_3^4-\vartheta_4^4\right|^2 + \left|\vartheta_2^4\right|^2 + \frac{1}{2}\left(\vartheta_3^4-\vartheta_4^4\right)\bar{\vartheta_2}^4 + \frac{1}{2}\left(\bar{\vartheta_3}^4-\bar{\vartheta_4}^4\right)\vartheta_2^4}{\left|\eta\right|^{24}} = \frac{N^2}{3}\frac{3\left|\vartheta_2^4\right|^2}{\left|\eta\right|^{24}}.
\end{equation}
The relative factors of $1/2$ are again caused by the fact that a Majorana fermion component (NS-R and R-NS) has half the contribution to the entropy of a real scalar (NS-NS and R-R). This leads to
\begin{align}
Z &= \frac{V_{D-2}}{N} \frac{N^2}{12}\int_{\mathcal{E}} \frac{d\tau d\bar{\tau}}{\tau_2} \frac{1}{(4\pi^2\alpha' \tau_2)^{4}}\frac{3\left|\vartheta_2^4\right|^2}{\left|\eta\right|^{24}}.
\end{align}
Alternatively, and more rudimentary, one can find this as well using the fact that the oscillators for type II superstrings contribute as
\begin{equation}
\left|\frac{\prod_{n}(1+q^n)}{\prod_n(1-q^n)}\right|^{16},
\end{equation}
both for spacetime bosons and fermions. Both hence contribute with the same sign. Incidentally,
\begin{equation}
\frac{\left|\vartheta_2\right|^8}{\left|\eta\right|^{24}} \sim \left|\frac{\prod_{n}(1+q^n)}{\prod_n(1-q^n)}\right|^{16}.
\end{equation}
As $\tau_2\to\infty$, one finds the typical GSO projection (no tachyon). However, as $\tau_2\to0$, one finds a thermal divergence. This means the non-interacting thermal entropy is divergent as well for type II superstrings $\sim e^{2\pi/\tau_2}$.
Next we rewrite this in the modular fundamental domain. For type II superstrings, an analogous identity as before holds:
\begin{align}
\int_{\mathcal{F}}\frac{d^2\tau}{\tau_2^6}\frac{1}{\left|\eta\right|^{24}}&\tau_2 \sum_{m,w}'\frac{1}{\left|m+w\tau\right|^2}\left[\vartheta_2^4\bar{\vartheta}_2^4 + \vartheta_3^4\bar{\vartheta}_3^4 + \vartheta_4^4\bar{\vartheta}_4^4\right. \nonumber \\
&\left.+ (-)^{w+m}\left(\vartheta_2^4\bar{\vartheta}_4^4 + \vartheta_4^4\bar{\vartheta}_2^4\right) - (-)^m\left(\vartheta_2^4\bar{\vartheta}_3^4+\vartheta_3^4\bar{\vartheta}_2^4\right) - (-)^w \left(\vartheta_3^4\bar{\vartheta}_4^4 + \vartheta_4^4\bar{\vartheta}_3^4\right)\right] \nonumber \\
&= \int_{\mathcal{E}} \frac{d^2\tau}{\tau_2^6}\frac{1}{\left|\eta\right|^{24}}\tau_2 \underbrace{\sum_{m}'\frac{1-(-)^m}{m^2}}_{\pi^2/2}2\left|\vartheta_2\right|^8.
\end{align}
This can be found by integrating the flat space unfolding theorem in $\beta^2$ and then letting $\beta \to 0$. This manifestly preserves modular invariance throughout the process. Hence also for type II superstrings, modular invariance is present for the non-interacting $N$-dependent part of the partition function. This is required since the $N$-independent part that is dropped is modular invariant on its own. Modular invariance is a necessary condition for the partition function to be interpreted as a torus path integral.
Note the natural appearance of a factor $1-(-)^m$ in this process. A related fact is that
\begin{equation}
\sum_{m}'\frac{1}{m^2} = -2\sum_{m}'\frac{(-)^m}{m^2},
\end{equation}
showing that the alleged bosonic contribution is indeed twice the fermionic contribution, and providing faith to our interpretation as this $m$ as the wrapping number of the particle paths around the origin.
In the end, one finds
\begin{align}
Z &= \frac{V_{D-2}}{N} \frac{N^2}{12}\int_{\mathcal{E}} \frac{d\tau d\bar{\tau}}{\tau_2} \frac{1}{(4\pi^2\alpha' \tau_2)^{4}}\frac{3\left|\vartheta_2^4\right|^2}{\left|\eta\right|^{24}} \nonumber \\
&= V_{D-2} \frac{N}{4\pi^2}\int_{\mathcal{F}}\frac{d^2\tau}{\tau_2}\frac{1}{(4\pi^2\alpha'\tau_2)^4}\frac{1}{\left|\eta\right|^{24}} \sum_{m,w}'\frac{1}{\left|m+w\tau\right|^2}\left[\vartheta_2^4\bar{\vartheta}_2^4 + \vartheta_3^4\bar{\vartheta}_3^4 + \vartheta_4^4\bar{\vartheta}_4^4\right. \nonumber \\
&\left.+ (-)^{w+m}\left(\vartheta_2^4\bar{\vartheta}_4^4 + \vartheta_4^4\bar{\vartheta}_2^4\right) - (-)^m\left(\vartheta_2^4\bar{\vartheta}_3^4+\vartheta_3^4\bar{\vartheta}_2^4\right) - (-)^w \left(\vartheta_3^4\bar{\vartheta}_4^4 + \vartheta_4^4\bar{\vartheta}_3^4\right)\right].
\end{align}
We have proven elsewhere <cit.> that the non-interacting torus path integral leads to a modular invariant result, consistent with the above description.
Similarly to the bosonic case, the divergence is the same for any odd $n$ and every $m$ and is hence independent of the winding number of the string. The restriction to odd $n$ here is the thermal GSO projection. This is in accord with the early results of <cit.> where an intuitive argument was also laid forward in favor of this.
§.§ Heterotic string
For heterotic string theory, the same story applies. The oscillators yield the contribution
\begin{equation}
\sim \frac{\vartheta_2^4}{\eta^{12}}\bar{\eta}^{-8} \bar{\eta}^{-16}\bar{\Gamma}_{int}.
\end{equation}
The flat space heterotic modular invariant is given by
\begin{equation}
\sum_{n,m}\int_{\mathcal{F}}\frac{d^2\tau}{\tau_2^6}\frac{1}{\eta^{12}\bar{\eta}^{24}}e^{-\frac{\beta^2\left|m+n\tau\right|^2}{4\pi\alpha'\tau_2}}(-)^{nm}\left[\vartheta_3^4-(-)^n\vartheta_4^4-(-)^m\vartheta_2^4\right]\bar{\Gamma}_{int},
\end{equation}
\begin{align}
\bar{\Gamma}_{int} &= \frac{1}{2}\left(\bar{\vartheta_3}^{16}+\bar{\vartheta_2}^{16}+\bar{\vartheta_4}^{16}\right), \quad SO(32), \\
\bar{\Gamma}_{int} &= \left(\frac{1}{2}\left(\bar{\vartheta_3}^{8}+\bar{\vartheta_2}^{8}+\bar{\vartheta_4}^{8}\right)\right)^2, \quad E_8 \times E_8,
\end{align}
the internal lattice modular combination.
The unfolding procedure puts this equal to
\begin{equation}
\sum_{m}\int_{\mathcal{E}}\frac{d^2\tau}{\tau_2^6}\frac{1}{\eta^{12}\bar{\eta}^{24}}e^{-\frac{\beta^2m^2}{4\pi\alpha'\tau_2}}\left[\vartheta_3^4-\vartheta_4^4-(-)^m\vartheta_2^4\right]\bar{\Gamma}_{int}.
\end{equation}
Again integrating both formulas in $\beta^2$ and then setting $\beta\to 0$, one finds an equality very similar to the one above.[Just like above, the $(-)^m$ can be extracted from the theta functions by using a global $1-(-1)^m$ which only selects odd $m$.] For odd $m$, the sum is again the same series as for the type II string, and Jacobi's identity allows us to rewrite the theta functions all in terms of $\vartheta_2$. So the technical details are all the same as for the type II superstring and a modular invariant partition function is constructed.
From this, it is clear that also the resulting non-interacting partition function diverges for any $N$.
§.§ Some comments
Several important comments are in order.
* The fact that the non-interacting thermal entropy diverges for bosonic, type II and heterotic superstrings resonates with the fact that the non-interacting sum-over-states quantities are expected to experience maximal acceleration phenomena: it is impossible to have an arbitrarily high acceleration for a single particle or string <cit.><cit.>. We will come back to this in the next section.
* The modular invariants make clear that all windings contribute equally to the divergence, a property which has been discovered by Parentani and Potting several years ago <cit.>.
* The $1-(-1)^m$ is indicative that this quantum number $m$ is indeed the correct one, since spacetime supersymmetric partition functions should contain this. The reader might be puzzled at this point, since we are discussing conical manifolds which manifestly break spacetime SUSY. However, the Lorentzian spectrum on Rindler space is spacetime supersymmetric and hence we expect the free energy to take the form of equation (<ref>) where this factor is indeed present.
* Unlike the $\mathbb{C}/\mathbb{Z}_N$ orbifold models, the modular invariants we obtained here are valid for any real $N$. Modular invariance is not broken. This corresponds to the fact that any Lorentzian particle state has a meaningful heat kernel on a cone with arbitrary conical deficit.
§ SPACETIME INTERPRETATION OF DIVERGENCES
As discussed previously, using the explicit heat kernel on the cone, it is possible to give a spacetime interpretation to the divergences arising in the constructed partition functions. We first show how this works for the different types of string theory, and then we explain its relation to the beautiful physical picture by Parentani and Potting <cit.>.
§.§ Local Divergences
§.§.§ Bosonic strings
The partition function can be written as
\begin{equation}
\label{divpf}
Z = \int_{-1/2}^{1/2}d\tau_1\int_{0}^{\infty}\frac{ds}{2s}\zeta(s)\left|\eta\right|^{-48}.
\end{equation}
The conical heat kernel can be written as <cit.><cit.>:
\begin{align}
\zeta_{\text{cone}}(s) &= \frac{\beta}{2\pi}\frac{A}{4\pi s} - \frac{1}{4\pi s}\int_{0}^{+\infty}d\rho \rho \int_{-\infty}^{+\infty}dw e^{-\frac{\rho^2\cosh(w/2)^2}{s}}\cot\left(\frac{\pi}{\beta}(\pi+iw)\right) \nonumber \\
&= \frac{\beta}{2\pi}\frac{A}{4\pi s}+\frac{1}{12}\left(\frac{2\pi}{\beta}-\frac{\beta}{2\pi}\right).
\end{align}
Here the variable $w$ is an additional dummy variable that has no direct physical interpretation. $\rho$ on the other hand is the radial coordinate in the 2d plane under consideration.
We will first review this spacetime interpretation as was given by Emparan in <cit.>. The idea is to analyze a possible divergence in the integral over $s$ as a function of $\rho$. So we swap the integral over $s$ with the spatial integral over $\rho$ and the dummy integral over $w$. Of course, there is the possibly hazardous power divergence from the prefactor of $1/s$. We know this is absent in string theory and we ignore it. A much more physical divergence arises if there is some exponential divergence if $s\to0$. Since we are considering the modular strip domain, this is to be interpreted as an IR thermal divergence that is indeed relevant.
Using $\left|\eta(\tau)\right|^{-48} \propto e^{4\pi^2\alpha'/s}$, in the limit $s \to 0$, the integral over $s$ converges if
\begin{equation}
\rho^2\cosh^2(w/2) > 4\pi^2\alpha',
\end{equation}
for all $w$. It is therefore sufficient for it to hold if $w=0$, so $\rho > \rho_{\text{crit}} = 2\pi\sqrt{\alpha'}$. This gives $T_{\text{crit}}= \frac{1}{2\pi \rho_{\text{crit}}} = T_{H}/\pi$.
This perspective is however not without reservation: we naively swapped the $w$- and $s$-integrals. Is this allowed? Also, the different wrapping numbers are already combined into a closed expression, and hence it is not obvious that the divergence arises from all of them in the same way.
To answer these questions, we will perform the same computation again, but instead using a different formula for the heat kernel. This will also clearly demonstrate that the divergence indeed arises from all winding numbers (as we have demonstrated several times already).
The idea is to use formula (<ref>) for a fixed winding number $n$ and insert this expression into the above expression (<ref>). The small $s$ behavior of these fixed winding heat kernels was analyzed in equation (<ref>) for the zero-winding contribution and in equation (<ref>) for the non-zero winding contribution. The zero-winding contribution is to be dropped when considering thermodynamic quantities. The non-zero winding part on the other hand, has for its small $s$ asymptotics:
\begin{equation}
Z \sim \int_{0}\frac{ds}{s}\frac{1}{12} e^{\frac{4\pi^2\alpha'}{s}}e^{-\frac{\rho^2}{s}},
\end{equation}
which has no exponential (thermal) divergence only when $\rho > 2\pi\sqrt{\alpha'}$, the same result as above. This computation was done independently of the wrapping number $m$ and we hence see from this perspective as well that all windings contribute in the same way to the thermal divergence.
§.§.§ Type II strings
For type II superstrings, the only difference is the oscillator contribution, which this time yields $\sim e^{2\pi^2\alpha '/s}$, in the end also giving the same formula
\begin{equation}
T_{\text{crit}} = \frac{T_H}{\pi},
\end{equation}
but this time with the type II flat Hagedorn temperature used for $T_H$.
§.§.§ Heterotic strings
For heterotic strings, the situation requires a bit more care. To analyze the divergence in the strip domain, the approach reviewed in <cit.> is ideally suited. We first expand the modular functions as
\begin{align}
\frac{\vartheta_3^4 - \vartheta_4^4 + \vartheta_2^4}{4\eta^{12}} &= \sum_{K=0}^{+\infty}S_Kq^K, \\
\frac{\bar{\Gamma}_{int}}{\bar{\eta}^{24}} &= \sum_{L=-1}^{+\infty}T_L \bar{q}^L.
\end{align}
The partition function to be analyzed can then be expanded as
\begin{align}
Z \sim \frac{V_7}{(2\pi)^{10}}\int_{0}^{+\infty}\frac{d\tau_2}{\tau_2^6} &\int_{-1/2}^{+1/2}d\tau_1 \sum_{K,L}S_K T_L q^K\bar{q}^L \nonumber \\
&\times \int_{0}^{+\infty}d\rho \rho \int_{-\infty}^{+\infty}dw e^{-\frac{\rho^2\cosh(w/2)^2}{\pi\alpha'\tau_2}}\cot\left(\frac{\pi}{\beta}(\pi+iw)\right),
\end{align}
where the relevant part of the heat kernel has already been filled in. The integral over $\tau_1$ enforces $K=L$. The integral over $\tau_2$ can be done in terms of a modified Bessel function $K_5$, the details can be found in <cit.>. The requirement is then finally that the sum over $K$ converges. For this, the exponential behavior needs to be damped. Just like in the flat case, the $S$ and $T$ coefficients scale as
\begin{align}
S_K &\sim \exp(\sqrt{2}\pi 2 \sqrt{K}), \\
T_K &\sim \exp(2\pi 2 \sqrt{K}),
\end{align}
for large $K$. The modified Bessel function is of the form
\begin{equation}
K_{5}\left(\frac{4\rho\cosh(w/2)\sqrt{K}}{\sqrt{\alpha'}}\right) \sim \exp\left(-\frac{4\rho\cosh(w/2)\sqrt{K}}{\sqrt{\alpha'}}\right).
\end{equation}
Hence convergence of the sum over $K$ requires (for $w=0$)
\begin{equation}
(2+\sqrt{2})\pi < \frac{2 \rho }{\sqrt{\alpha'}},
\end{equation}
which leads again to the critical temperature
\begin{equation}
T_{\text{crit}} = \frac{T_H}{\pi},
\end{equation}
but this time with the heterotic Hagedorn temperature filled in.
§.§ Physical interpretations
A very beautiful interpretation of these divergences was made in <cit.> in a propagator context. The idea can be readily adapted to our case for the partition function $Z$ and goes as follows. For all types of string theory, the divergence comes from a region close to the black hole horizon, which can be written suggestively as
\begin{equation}
2\rho < \frac{1}{T_H},
\end{equation}
where one plugs in the correct flat space Hagedorn temperature for the type of string (bosonic, type II or heterotic) one is considering.
Usually, the Hagedorn temperature is determined when the circumference of the singly wound string around the thermal circle becomes too small. In formulas
\begin{equation}
\text{circumference} < \frac{1}{T_H},
\end{equation}
where in a thermal theory with topologically supported thermal circle, the circumference is always the inverse temperature $\beta$.
For the case at hand, there is no physical topologically supported circle. However, considering the coincident fixed winding heat kernel at a distance $\rho$, there is a minimal circumference for any trajectory to have: twice the radial distance $\rho$. This is illustrated in figure <ref>.
Strings with non-zero winding that go through a fixed point at a radial distance $\rho$, have a minimal length of $2\rho$. This minimal length is independent of the winding number, as long as it is non-zero.
Afterwards, we integrate over the coordinate $\rho$ to obtain the traced heat kernel, where all locations $\rho$ that are too close to the origin (black hole horizon) lead to a divergence, in precisely the same way as when the circumference is below the Hagedorn scale in a toroidally compactified model.
We note that the associated critical temperature $T_{\text{crit}} = \frac{T_H}{\pi}$ is local, and relies on an extrapolation of QFT in curved backgrounds into the stringy regime. Hence one should not attach too much value to it. The main conclusion is that the non-interacting partition functions for all string types exhibit a divergence, coming from the near-horizon region $\rho< \frac{1}{2T_H}$.
As a further characterization of this divergence, we may look at it field by field. The Minkowski vacuum which we assume to be used for each field, has a vanishing stress tensor (by definition). The Rindler observer explains this as due to a cancellation of the Casimir contribution with the thermal contribution. When constructing the non-interacting partition functions, we are making modifications on the thermal part of each field. The Casimir part was left alone. This implies that for each field, there is no longer a perfect cancellation and the stress tensor vev is non-zero in the constructed vacuum. These build up as one sums over the spectrum and ultimately lead to a divergence in the full string theory. This vev implies structure is present near the black hole horizon and an infalling observer would no longer be able to pass safely through the horizon. The presence of a divergence is hence observer-independent.
§ DISCUSSION AND OUTLOOK
In this paper, we have studied the partition functions of open and closed strings on $\mathbb{Z}_N$ cones.
For open strings, we have demonstrated that the one-loop open string partition function contains exotic interactions with the horizon that are to be interpreted as the sum of the surface contributions of all the higher spin fields in the string spectrum.
Whereas for open strings, the situation is more or less completely clear now, this is not so for closed strings. We demonstrated that there is a difference between the sum-over-fields approach and the full stringy result for the conical partition functions. This conclusion would not have been possible to make without the explicit expressions found in <cit.> for the higher spin conical partition functions.
To gain a better understanding of the surface interactions and the interpretation in terms of worldsheets either including or excluding the origin, we considered the partition functions obtained by deleting the surface interactions by hand within the higher spin contributions in the string spectrum. As discussed in the introduction, if this is a good operation within string theory, several consistency conditions are expected to be fulfilled.
The main part of this work has been to establish these consistency requirements, providing faith in the identification of horizon-intersecting worldsheets as the surface interactions.
The constructions done here are identical to those Emparan considered for bosonic strings over 20 years ago <cit.>. Inspired by some recent results on entanglement entropy in Rindler space <cit.>, we have been able to provide more rigor to this construction and extend Emparan's idea to type II and heterotic strings. The main technical results are partition functions for type II and heterotic strings, that show explicit spacetime supersymmetry and modular invariance for any conical deficit. These partition functions also have a thermal divergence for any conical opening angle that is independent of the winding and momentum around the conical singularity.
The upshot for closed strings is that we have a priori three candidate partition functions. The first is the full stringy result which is of course modular invariant and contains worldsheets intersecting the conical singularity. The second is the sum-over-fields approach, by including the full contribution from each higher spin field. This however is not the same as the string result, as an infinite overcounting of tori configurations is done, manifestly leading to an overall divergence. This candidate partition function hence seems invalid. The third construction is to sum over all fields and exclude all surface interactions. The result is modular invariant and is related by worldsheet duality to the same procedure for open strings. String worldsheets are not allowed to intersect the conical singularity in this case. A thermal divergence is present for all $\beta$ that can be related to maximal acceleration.
In this respect, we have succeeded in deepening our understanding of the link between string theory and the field theories of the states in its spectrum, within these conical backgrounds.
One of the lessons to be learned from our endeavors here is that the surface interactions first discovered by Kabat for gauge fields, are an important and integral part of perturbative string theory. Within string language, one can associate a clear geometric picture to these.
Recent work by Wall and Donnelly <cit.> has taught us that for gauge fields, one can view the surface term as representing the edge modes present on the entangling surface, the negativity of these arising as a regularization artifact in the continuum limit. More precisely, in Kabat's original computation <cit.>, they arise by utilizing heat kernel regularization. String theory however has an innate preference for heat kernel regularization (as the Schwinger parameter is directly related to the torus modulus $\tau_2$). The negative contributions are intrinsic to the perturbative worldsheet formulation of string theory, and one should be wary of dismissing them simply as a regularization feature in this case. It will be interesting to investigate this further.
Let's now come back to the proposal by Solodukhin in <cit.> as discussed in the Introduction, where he envisioned an equality between the tree-level contribution to the Bekenstein-Hawking entropy and the sum of the contact terms at one loop, up to the sign. Thus the black hole entropy up to one loop would be equal to just the sum of the “normal” contributions of the fields in the spectrum. However, in our string theory context, this is precisely the partition function we developed throughout this work. This would imply a divergent black hole entropy due to the maximal acceleration, something we deem impossible.
An interesting extension would be to look into related orbifolds with fixed points, such as $\mathbb{C}^2/\mathbb{Z}_{N(k)}$, and obtain the analogous non-interacting partition function. Again modular invariance should be checked explicitly.
The best check of our results here would be to simply compute the first quantized torus string path integral with fixed winding number in the strip modular domain and check whether it agrees with expression (<ref>). This is the analogous computation of that which was done for scalar particles in the past <cit.><cit.><cit.>. Unfortunately, the computation seems untractable to perform.
Our original motivation for this work was to play devil's advocate and provide more detail on this sum-over-fields approach to black hole thermodynamics utilized in the older string literature, to hopefully ultimately show that this route is shaky. However, we appear to reach the opposite conclusion: good modular invariants can be constructed by simply summing the (spin-independent parts of the) Lorentzian fields in the string spectrum. The resulting partition functions show properties that we expect them to have such as spacetime supersymmetry.
Of course, these non-interacting partition functions are not the ultimate goal when one is interested in thermodynamics in Rindler space, as one cannot approximate the full thermodynamic quantities by these in any way. The reason is that sending $g_s$ to zero (the only way to really achieve the supposedly non-interacting theory), actually retains the open-closed interactions on the horizon. Our entire endeavor in this paper has been to understand the difference between these two types of partition functions (that either include or exclude the surface interactions) to ultimately gain a better understanding of the surface interactions themselves.
The fact that these partition function exhibit a divergence associated to maximal acceleration should not be taken as a failure of string theory, since these do not correspond to any physical observables. The entropy computed using the full partition function (i.e. reincluding the surface interactions) is finite for type II superstring theory <cit.><cit.>.
Hence the maximal acceleration phenomenon is fiction when treating the complete string theory. These partition functions, while mathematically consistent (modular invariant for a torus interpretation and spacetime SUSY is apparent), cannot be reached in a physical scheme when considering string thermodynamics.
It is tantalizing to suspect that these non-interacting partition functions and their maximal acceleration divergence are related to the recent firewall paradox. Especially since we have demonstrated that a crucial role in eliminating the divergence is played by spin, the role of which in formulating firewall paradoxes has to the best of our knowledge not been studied thoroughly. Closely related, the Hilbert space does not cleanly factorize here due to the surface contributions, whereas this factorization is assumed in formulating the firewall paradox. So we would suggest that the sum of the surface contributions cancels the maximal acceleration “firewall” originating from approximating all fields as scalars or spin $1/2$ fermions. A more detailed comparison is beyond the scope of this work and is left to future work.
A basic question does remain at this point: how can one describe these divergences within a field theory action of wound strings? We saw in earlier work <cit.> that the field theory of the thermal scalar nicely agrees with properties of the string partition functions on $\mathbb{Z}_N$ cones. How does this work in this case? It seems that all winding modes are not aware of the intrinsic conical feature in the space, but instead simply behave the same as the $R\to0$ limit of a circular dimension with radius $R$. In some sense, this is as we would expect since we excised the conical singularity itself from the space by forcing all string worldsheets not to intersect it. A more precise investigation of this point will be very interesting and must unfortunately be left to future work.
It is our hope that the results reported in this paper will help unveil the true nature of quantum black hole horizons within string theory.
§ ACKNOWLEDGEMENTS
The authors thank T. Takayanagi for e-mail correspondence. TM gratefully acknowledges financial support from the UGent Special Research Fund, Princeton University, the Fulbright program and a Fellowship of the Belgian American Educational Foundation. The work of VIZ was partially supported by the RFBR grant 14-02-01185.
§ DETAILED COMPARISON BETWEEN THE STRING RESULT AND THE SUM OVER FIELDS
Here we show that one can indeed interpret the quantity $\sum_n(p_n-q_n)$ as the $SO(2)$ spin. First we demonstrate this for level $3$, after which the generalization will be obvious.
At level $3$, simple counting shows that 10 states exist of the form
\begin{align}
&\alpha_1^x\alpha_1^x\alpha_1^x, \quad \alpha_1^x\alpha_1^x\alpha_1^y, \quad \alpha_1^x\alpha_1^y\alpha_1^y, \quad \alpha_1^y\alpha_1^y\alpha_1^y, \\
&\alpha_2^x\alpha_1^x, \quad \alpha_2^x\alpha_1^y, \quad \alpha_2^y\alpha_1^x, \quad \alpha_2^y\alpha_1^y, \\
&\alpha_3^x, \quad \alpha_3^y.
\end{align}
These states can be reorganized into states with definite $SO(2)$ spin as:
\begin{align}
\begin{array}{|c|c|}
\hline
\text{state} & \text{spin}\\
\hline
\left(\alpha_1^x+i\alpha_1^y\right)^3 & 3 \\
\left(\alpha_1^x-i\alpha_1^y\right)^3 & -3 \\
\left(\alpha_1^x+i\alpha_1^y\right)^2\left(\alpha_1^x-i\alpha_1^y\right) & 1 \\
\left(\alpha_1^x+i\alpha_1^y\right)\left(\alpha_1^x-i\alpha_1^y\right)^2 & -1 \\
\alpha_3^x+i\alpha_3^y & 1 \\
\alpha_3^x-i\alpha_3^y & -1 \\
\left(\alpha_1^x+i\alpha_1^y\right)\left(\alpha_2^x-i\alpha_2^y\right) & 0 \\
\left(\alpha_1^x-i\alpha_1^y\right)\left(\alpha_2^x+i\alpha_2^y\right) & 0 \\
\left(\alpha_1^x+i\alpha_1^y\right)\left(\alpha_2^x+i\alpha_2^y\right) & 2 \\
\left(\alpha_1^x-i\alpha_1^y\right)\left(\alpha_2^x-i\alpha_2^y\right) & -2 \\
\hline
\end{array}
\end{align}
On the other hand, from the expansion of the string partition function, one has the following table for the spins computed from the formula above
\begin{align}
\begin{array}{|c|c|}
\hline
\text{state} & s_a = \sum_{n}(p_n-q_n) \\
\hline
p_1 = 3 & 3 \\
q_1 = 3 & -3 \\
p_1=2, q_1=1 & 1 \\
p_1=1, q_1=2 & -1 \\
p_3=1 & 1 \\
q_3=1 & -1 \\
p_1=1, q_2=1 & 0 \\
p_2=1, q_1=1 & 0 \\
p_1=1, p_2=1 & 2 \\
q_1=1, q_2=1 & -2 \\
\hline
\end{array}
\end{align}
showing that the count matches: each state in the expansion is constructed from the elementary oscillator excitations.
The above description shows that both descriptions match in general, if we identify $p_n$ as counting $\alpha_n^x + i \alpha_n^y$ and $q_n$ as counting $\alpha_n^x - i \alpha_n^y$.
Hence this is the interpretation of the oscillator expansion numbers $p_n$ and $q_n$.
§ OPEN SUPERSTRING FORMULAS
As shown in <cit.>, the open superstring partition function can be completely decomposed into its underlying particle partition functions. The string partition function is given by
\begin{equation}
Z = V_{D-2}\int_{0}^{+\infty} \frac{dt}{2t}(8\pi^2\alpha't)^{-4}\sum_{j=1}^{N-1}\frac{\vartheta_1\left(\frac{j}{N},it\right)^4}{N\sin\left(\frac{2\pi j}{N}\right)\vartheta_1\left(\frac{2j}{N},it\right)\eta(it)^9}.
\end{equation}
The $\vartheta_1$ function in the denominator can be series-expanded just like for the bosonic string, and leads to a double series in $p_n$ and $q_n$. The new feature is the $\vartheta_1^4$ in the numerator. One first utilizes the Riemann identity to deconstruct this into the bosonic and fermionic contributions:
\begin{equation}
\vartheta_3^3(\tau)\vartheta_3\left(\frac{2j}{N},\tau\right) - \vartheta_4^3(\tau)\vartheta_4\left(\frac{2j}{N},\tau\right) - \vartheta_2^3(\tau)\vartheta_2\left(\frac{2j}{N},\tau\right) = 2\vartheta_1\left(\frac{j}{N},\tau\right)^4.
\end{equation}
Performing a series expansion on the $j$-dependent theta-functions, one can write the partition function as
\begin{align}
\label{nonintpff}
Z &= V_{D-2} \int_{0}^{+\infty}\frac{ds}{2s}(4\pi s)^{-4} \frac{1}{N}\frac{1}{\eta\left(\frac{is}{2\pi\alpha'}\right)^{9}}\frac{1}{4}\frac{e^{\frac{s}{8\alpha'}}}{\prod_{n=1}^{+\infty}(1-q^n)}\sum_{j=1}^{N-1}\frac{1}{\sin^2\left(\frac{2 \pi j}{N}\right)} \nonumber \\
&\times \left[ \vartheta_3^3\left(0,\frac{is}{2\pi\alpha'}\right) \sum_{m\in\mathbb{Z}}\prod_{n=1}^{+\infty}\sum_{p_n,q_n=0}^{+\infty}e^{\frac{4\pi i j}{N} (p_n-q_n+m)}e^{- \frac{s}{\alpha'}\left(n(p_n+q_n)+m^2/2\right)}\right. \nonumber \\
&\left.\quad - \vartheta_4^3\left(0,\frac{is}{2\pi\alpha'}\right) \sum_{m\in\mathbb{Z}}\prod_{n=1}^{+\infty}\sum_{p_n,q_n=0}^{+\infty}(-)^m e^{\frac{4\pi i j}{N} (p_n-q_n+m)}e^{- \frac{s}{\alpha'}\left(n(p_n+q_n)+m^2/2\right)} \right. \nonumber \\
&\left.\quad - \vartheta_2^3\left(0,\frac{is}{2\pi\alpha'}\right) \sum_{m\in\mathbb{Z}}\prod_{n=1}^{+\infty}\sum_{p_n,q_n=0}^{+\infty}e^{\frac{4\pi i j}{N} \left(p_n-q_n+m-\frac{1}{2}\right)}e^{- \frac{s}{\alpha'}\left(n(p_n+q_n)+(m-1/2)^2/2\right)}\right].
\end{align}
Just as for the bosonic string, the first exponential can be associated to the spacetime spin of each boson or fermion. Dropping this contribution, one finds for the first two terms precisely the same bosonic sum as before (these are the bosons).[One needs to use
\begin{equation}
\label{bossum}
\sum_{j=1}^{N-1}\frac{1}{\sin^2\left(\frac{2\pi j}{N}\right)} = \sum_{j=1}^{N-1}\frac{1}{\sin^2\left(\frac{\pi j}{N}\right)} = \frac{N^2-1}{3},
\end{equation}
where $N$ is odd.] The third sum on the other hand, requires a spin $1/2$ sum (<ref>). Altogether, one retrieves the result shown in equation (<ref>).
§ MODULAR DOMAINS FOR $\MATHBB{Z}_N$ ORBIFOLDS
§.§ Unfolding the fundamental domain
Before starting with the proof, we will check whether the divergence of expression (<ref>) as $\tau_2\to0$ reproduces the winding tachyon divergence of (<ref>) as $\tau_2\to\infty$. This is a necessary condition for a possible equality. After using some theta-identities, one retrieves the behavior (as $\tau_2\to0$)
\begin{equation}
\left|\vartheta\left[
\begin{array}{c}
1/2 \\
1/2 + j/N \end{array}
\right]\right|^{-2} \to e^{\frac{2\pi}{\tau_2}\left(\frac{j^2}{N^2}-\frac{j}{N}+\frac{1}{4}\right)}.
\end{equation}
Hence the small $\tau_2$ behavior of (<ref>) yields indeed
\begin{equation}
\sim e^{\frac{2\pi}{\tau_2}\left(\frac{j^2}{N^2}-\frac{j}{N}+2\right)},
\end{equation}
which is the winding tachyon divergence. Nonetheless, the two partition function are definitely not equal as we now demonstrate.
We try to apply the theorem established in <cit.><cit.> to the extent that is possible. We hence start in the modular fundamental domain and try to build up the strip domain by applying suitable modular transformation to the $w\neq0$ sectors.
The first step is to prove that the quantum numbers $m$ and $w$ transform as a doublet under $SL(2,\mathbb{Z})$ and allow one to undo the modular transformation at hand. This was already proven in the early literature on this model, and we will not repeat it. The summary is the transformation rules
\begin{align}
T&: m \to m+w ,\\
S&: m\to -w, \quad w\to m.
\end{align}
We first reorder the sums over both quantum numbers such that they include both positive and negative entries, for instance
\begin{equation}
w:0\to N-1 \quad \mapsto \quad w: -\frac{N-1}{2} \to \frac{N-1}{2},
\end{equation}
and the same for $m$ and $j$. We henceforth restrict our attention to odd $N$. This makes the discussion more symmetric, and for type II superstrings we are restricted to odd $N$ in any case.
The lowest non-trivial value of $N$ is then $N=3$. There are three possible values of $w$ and three of $m$, yielding 9 states. Removing the $w=m=0$ state, we have 8 states left. The strategy is to take any fixed state ($m, w$) and construct a suitable modular transformation to get to $w=0$ in a transformed domain that is included within the strip. For each such state, the strategy is exactly the same as in flat space. Let us present just the gist of it. The $PSL(2,\mathbb{Z})$ tranformation that we seek acts on $\tau$ as
\begin{equation}
\tau \to \frac{a\tau+b}{c\tau+d},
\end{equation}
where $c$ and $d$ can be fixed by imposing that this transformation sets the new $w$ equal to zero. This entails $cm+dw=0$, which leads to $c=w/r$ and $d=-m/r$, for $r$ the gcd of $m$ and $w$. The remaining two parameters are then fixed by demanding the transformed modulus to be inside the strip modular domain combined with the determinantal condition $ad-bc=1$.
Performing such modular transformations on the fundamental domain, one can reach the result of figure <ref>.
Regions in the modular plane that are reached by unfolding the fundamental domain in the case of $N=3$.
The two states that are left alone are $m=1, w=0$ and $m=-1, w=0$. Upon using the $S$-transformation, one finds the two states $m=0, w=1$ and $m=0, w=-1$. The region is mapped into the central wedge in the figure. One can perform analogous modular transformations to reach the other two regions that contain the contributions from the sectors: $m=1, w=-1$ and $m=-1, w=1$ for the leftmost wedge and $m=w=1$ and $m=w=-1$ for the rightmost wedge. The required modular transformations are respectively,
\begin{equation}
\frac{-1}{\tau+1}, \quad \frac{-1}{\tau-1}.
\end{equation}
All four wedges contain only the strip quantum number $j=\pm1$ as it should be. Hence taking the partition function (<ref>) with $w=0$, but integrated along the union of these four wedges, one finds back the original result of the fundamental domain (<ref>).
But this is not the full modular strip of equation (<ref>)! Hence, at least for $N=3$, the sum-over-fields result (yielding the modular strip) and the stringy result (yielding the fundamental domain) cannot be equal in any way.
This result also allows us to explain why the winding tachyon divergence is present in the strip partition function as $\tau_2\to0$. We simply need to ask where the large $\tau_2$ region for a generic $w$ gets mapped into. One of the sectors that carries the winding tachyon divergence is the ($0,w$) sector. Since we started in the fundamental domain and transformed this divergence into the central lower wedge of this figure; it must hence be present there. The large $\tau_2$ region gets mapped by an $S$-transformation into the origin.
It is irrelevant for this divergence whether the full strip is filled in or not; as long as the small $\tau_2$ zone is present (which it is for any $N$), we are guaranteed to find indeed the same divergence.
Let us press on and look at higher values of $N$. For $N=5$, four additional regions in the strip domain open up, shown in blue in figure <ref>.
Regions in the modular plane that are reached by unfolding the fundamental domain in the case of $N=5$. There are four additional blue regions generated in this case.
These four extra regions however, only contain the $j=\pm1$ part of the strip sum; the $j=\pm2$ terms are completely missed.
The modular transformation to reach the four blue regions, starting with $\tau$ in the modular domain are (from left to right):
\begin{equation}
\frac{-\tau}{2\tau-1}, \quad \frac{-1}{\tau+2},\quad \frac{-1}{\tau-2}, \quad \frac{\tau}{2\tau+1}.
\end{equation}
To be complete, let us mention which original states get mapped into each of the regions. The four red regions each contain 4 states. The top region contains ($\pm1,0$) and ($\pm2,0$). The three lower regions contain (from left to right):[This should be read as having matched signs for $m$ and $w$. E.g. the first entry contains ($1,-1$) and ($-1,1$).] ($\pm1,\mp1$), ($\pm2,\mp2$) and ($0,\pm1$), ($0,\pm2$) and ($\pm1,\pm1$), ($\pm2,\pm2$). Finally, the four blue regions each only contain two states (from left to right): ($\mp1,\mp2$) and ($\pm2,\mp1$) and ($\pm2,\pm1$) and ($\pm1,\mp2$).
As a further example, for $N=7$ eight further additional regions are created, shown in green in figure <ref>.
Regions in the modular plane that are reached by unfolding the fundamental domain in the case of $N=7$. There are eight additional green regions generated in this case.
However, both these and the previous blue regions, only contain the $j=\pm1$ part; the $j=\pm2,\pm3$ terms are not generated. The set of modular transformation to reach the eight green regions, starting with $\tau$ in the modular domain are (from left to right):
\begin{equation}
\frac{-\tau+1}{2\tau-3}, \quad \frac{-\tau-1}{3\tau+2}, \quad \frac{-\tau}{3\tau-1}, \quad \frac{-1}{\tau+3},\quad \frac{-1}{\tau-3}, \quad \frac{\tau}{3\tau+1}, \quad \frac{\tau-1}{3\tau-2}, \quad \frac{\tau+1}{2\tau+3}.
\end{equation}
Just to check whether the count of the number of sectors match, we started with 48 sectors in the fundamental domain. Each of the red regions contain 6 states, each of the blue and green regions contain only 2 states. The total is $4\times 6 + 4 \times 2 + 8 \times 2 = 48$ indeed.
One can see that increasing $N$ further will generate additional regions, but the contributions to the $j$-sum in each region will only fill up slowly, according to the number theoretic properties of the number $N$. This also precludes a bit whether the $N\to\infty$ limit really gives a nice construction here of the modular strip. For $N$ very large but finite, there are still multiple zones that do not even have half of their states included. For $N$ strictly infinite however, one should find agreement with the modular strip, but the limit appears to be ill-defined.
On a more mathematical level, the above displayed regions are fundamental domains for the Hecke congruence subgroups $\Gamma_0(N)$ of the modular group.
§.§ Folding the modular strip
Now let's try to work in the opposite direction. We start with the sum-over-fields expression (<ref>) and try to fold this into the fundamental domain. Following the same theorem, one readily finds that
\begin{align}
Z &= V_{D-2} \int_{\mathcal{F}}\frac{d\tau^2}{4\tau_2}(4\pi^2\alpha' \tau_2)^{-12} \frac{1}{N}\sum_{m,w=0, (m,w) \neq(0,0)}^{gcd(m,w) < N}\frac{\left|\eta(\tau)\right|^{-42}e^{2\pi\tau_2\frac{w^2}{N^2}}}{\left|\vartheta_1\left(\frac{m}{N} + \frac{w}{N}\tau,\tau\right)\right|^2}
\end{align}
agrees with the result (<ref>). The only difference with the full string result (<ref>) is the extension to all sets of integers with $gcd(m,w) < N$. From the periodicity of the terms in the partition function as $m\to m+N$ and $w\to w+N$, one sees that this is an infinite overcounting of the actual stringy result.
This expression is (formally) modular invariant, since the set of all integers $m$ and $w$ restricted to $gcd(m,w)=x$ for any fixed $x$ form an orbit under the full modular group: $PSL(2,\mathbb{Z})$ transformations are unable to change the value of $x$. Since the action of $PSL(2,\mathbb{Z})$ on the doublet ($m$, $w$) is 1:1, modular transformations simply permute the different terms in the sum for every fixed value of the $gcd$.
§ SOME INTERESTING FORMULAS FOR THE FIXED WINDING HEAT KERNELS
The Euclidean Green's propagator (heat kernel) for a massless scalar particle in a 2d flat plane with polar coordinates $\rho$ and $\phi$, whose trajectory is constrained to wrap the origin $m$ times, is given by <cit.><cit.><cit.>:
\begin{equation}
\label{propag}
G^{(m)}(\rho,0;\rho',\phi; s) = \frac{1}{4\pi s}e^{-\frac{\rho^2+\rho'^2}{4s}}\int_{-\infty}^{+\infty}d\nu I_{\left|\nu\right|}\left(\frac{\rho\rho'}{2s}\right)e^{-2\pi i m \nu - \phi i \nu}.
\end{equation}
As a check, one can sum this expression over all $m$. Using the following formulas
\begin{align}
\sum_{m\in\mathbb{Z}}e^{-2\pi i m \nu} &= \sum_{k\in\mathbb{Z}}\delta(k-\nu), \\
\sum_{k\in\mathbb{Z}}I_k(x)t^k &= e^{\frac{x}{2}(t+1/t)},
\end{align}
and the fact that $I_k = I_{-k}$ for integer $k$, we get
\begin{equation}
G(\rho,0;\rho',\phi; s) = \frac{1}{4\pi s}e^{-\frac{\rho^2+\rho'^2-2\rho\rho'\cos\phi}{4s}},
\end{equation}
which is indeed the flat space heat kernel between these two points. Taking the coincident limit, and then integrating over the full 2d area, one simply finds:
\begin{equation}
G(s) = \frac{A}{4\pi s},
\end{equation}
for the 2d area $A$.
It is interesting to try to reverse the order of these operations. First, we look at the integrated coincident heat kernel. Afterwards, we sum over all $m$. There are a few things we can learn already just by staring at formula (<ref>) long enough. Firstly, the positive and negative wrappings sum into a real quantity and the latter is monotonically decreasing as $\left|n\right|$ increases. Secondly, all of these real contributions are strictly positive. We will see in the end that our resulting formulas respect these properties.
To get started, we must regulate the integrals, as one obtains a divergence for each wrapping number and their geometric interpretation is a priori obscured. So we replace
\begin{equation}
e^{-\frac{\rho^2+\rho'^2}{4s}} \to e^{-\frac{(\rho^2+\rho'^2)(1+\epsilon)}{4s}}.
\end{equation}
in equation (<ref>).
The physical interpretation of this regulator $\epsilon$ can be made apparent, by summing over $n$, taking the coincident limit and then finally again integrating over the plane (the order of operations done above). This gives
\begin{equation}
G(s) = \frac{2\pi}{4\pi s} \int_{0}^{+\infty}d\rho \rho e^{-\epsilon \frac{\rho^2}{2s}} = \frac{2\pi s }{4 \pi s\epsilon}= \frac{A}{4\pi s} ,
\end{equation}
so $A = \frac{2\pi s}{\epsilon}$.
Now let's try to reverse the order of the operations.
With this regulator, the integrated coincident heat kernel becomes
\begin{equation}
G^{(m)}(s) = \int_0^{+\infty}d\rho \rho \frac{1}{2 s}e^{-\frac{\rho^2(1+\epsilon)}{2s}}\int_{-\infty}^{+\infty}d\nu I_{\left|\nu\right|}\left(\frac{\rho^2}{2s}\right)e^{-2\pi i m \nu }.
\end{equation}
The integral over $\rho$ can be rewritten as
\begin{equation}
\int_0^{+\infty}d\rho \rho e^{-\frac{\rho^2(1+\epsilon)}{2s}} I_{\left|\nu\right|}\left(\frac{\rho^2}{2s}\right) = s \int_0^{+\infty}dt e^{-t(1+\epsilon)} I_{\left|\nu\right|}\left(t\right),
\end{equation}
which is the Laplace transform of $I_{\left|\nu\right|}(t)$:
\begin{equation}
\mathcal{L}\left(I_{\left|\nu\right|}(t)\right)(p) = \frac{1}{\sqrt{p^2-1}\left(p+\sqrt{p^2-1}\right)^{\left|\nu\right|}},
\end{equation}
where we should take $p=1+\epsilon$.
The above integrated heat kernel becomes
\begin{equation}
\label{halfway}
G^{(m)}(s) = \frac{1}{2}\int_{-\infty}^{+\infty}d\nu e^{-2\pi i m \nu } \frac{1}{\sqrt{2\epsilon+\epsilon^2}\left(1 + \epsilon+\sqrt{2\epsilon+\epsilon^2}\right)^{\left|\nu\right|}}.
\end{equation}
As a check, upon summing this again over $m$, one obtains the replacement $\nu \to k$, an integer. With $\epsilon > 0$, one then recognizes a geometric series:
\begin{equation}
-1 + 2 \sum_{k=0}^{+\infty}\frac{1}{\left(1 + \epsilon + \sqrt{2\epsilon+\epsilon^2}\right)^{k}} \approx \frac{\sqrt{2}}{\sqrt{\epsilon}} + \frac{\sqrt{2\epsilon}}{4}+\mathcal{O}\left(\epsilon^{3/2}\right).
\end{equation}
Combining this with the expansion
\begin{equation}
\frac{1}{\sqrt{2\epsilon+\epsilon^2}} \approx \frac{1}{\sqrt{2\epsilon}} - \frac{\sqrt{2\epsilon}}{8} + \mathcal{O}\left(\epsilon^{3/2}\right),
\end{equation}
this leads again to $G(s) = \frac{A}{4\pi s}$ as it should.
Miraculously, one can continue analytically, since the integral in equation (<ref>) is simply the Fourier transform of $e^{-a\left|t\right|}$:
\begin{equation}
\mathcal{F}\left(e^{-\left|t\right|\ln \alpha}\right)(\omega) = \frac{2\ln \alpha}{\left(\ln \alpha\right)^2+\omega^2},
\end{equation}
where $\alpha= 1+ \epsilon+\sqrt{2\epsilon+\epsilon^2} > 1$. We obtain
\begin{equation}
G^{(m)}(s) = \frac{1}{2\sqrt{2\epsilon+\epsilon^2}}\frac{2\text{ln}\left(1+ \epsilon+\sqrt{2\epsilon+\epsilon^2}\right)}{\left(\text{ln}\left(1+ \epsilon+\sqrt{2\epsilon+\epsilon^2}\right)\right)^2+4\pi^2m^2} .
\end{equation}
For $n=0$, the heat kernel becomes
\begin{equation}
G^{(0)}(s) = \frac{1}{2\epsilon} -\frac{1}{12}= \frac{A}{4\pi s}-\frac{1}{12}.
\end{equation}
The other heat kernels ($m\neq0$) are given by
\begin{equation}
G^{(m)}(s) = \frac{1}{4\pi^2m^2}.
\end{equation}
Summing the latter leads to $+1/12$, again combining into the correct flat space heat kernel.
It is natural that these $m\neq 0$ terms do not diverge as $s\to 0$, as the path always has to be of a macroscopic distance to loop around the origin. They do not scale as the transverse area, which is also expected since the points far from the origin behave completely different than those close to the origin, effectively making the radial direction behave as if it were compact. That it is independent of $s$ is unexpected.[Naively performing the substitution $u=\frac{\rho^2}{2s}$ in equation (<ref>) would suggest that for all $n$ the resulting expression is independent of $s$. We should be careful though, as the integrals can be divergent, in which case an $s$-dependent regulator might be physically required.]
As a check on our analytical computations, we checked numerically that the following statements hold for large $\rho$:
\begin{align}
\label{wind0}
&\int_{-\infty}^{+\infty}d\nu I_{\left|\nu\right|}\left(\frac{\rho^2}{2s}\right) = e^{\frac{\rho^2}{2s}}+ \mathcal{O}\left(e^{-\frac{\rho^2}{2s}}\right), \\
\label{windnon0}
&\int_{-\infty}^{+\infty}d\nu I_{\left|\nu\right|} \left(\frac{\rho^2}{2s}\right) \cos(2\pi \nu m) = \mathcal{O}\left(e^{-\frac{\rho^2}{2s}}\right),
\end{align}
confirming that the $m=0$ sector will have a divergent result due to the large $\rho$ integration, unlike the $m\neq0$ sectors. Thus quite literally, the $m\neq0$ sectors are confined to the origin and behave as if in a potential well, cutting off the large $\rho$ region. The first of these formulas actually shows that for large $\rho$, one has
\begin{equation}
G^{(0)}(\rho,0;\rho,0; s) = \frac{1}{4\pi s} + \mathcal{O}\left(e^{-\frac{\rho^2}{2s}}\right),
\end{equation}
just like in ordinary flat space. This makes sense since the no-winding restriction is not felt at very large distance from the origin.
§.§ Cones
The above expression for the heat kernel is perfectly capable of reproducing known formulas for conical spaces. Suppose we consider a conical geometry with periodicity $\beta$. A moment's thought reveals that the path integral with fixed wrapping number $m$ on the cone can be equivalently seen as the path integral on $\mathbb{C}$ with wrapping number $\left\lfloor \frac{m\beta}{2\pi}\right\rfloor$ and an extra angular difference $\Delta\phi =\frac{m\beta}{2\pi}\ - \left\lfloor \frac{m\beta}{2\pi}\right\rfloor$.
Using the general expression (<ref>) for this case, we readily obtain
\begin{equation}
G^{(m)}_{\beta}(\rho,0;\rho',\phi; s) = \frac{1}{4\pi s}e^{-\frac{\rho^2+\rho'^2}{4s}}\int_{-\infty}^{+\infty}d\nu I_{\left|\nu\right|}\left(\frac{\rho\rho'}{2s}\right)e^{-2\pi i \frac{2\pi}{\beta}m \nu - \phi i \nu}.
\end{equation}
Directly summing this expression over $m$ cannot be done so trivially as before.[One runs into the series
\begin{equation}
\sum_{k\in\mathbb{Z}}I_{kT}(x),
\end{equation}
which proves to be difficult to manipulate further. One can use the Schläfli representation of the Bessel function to rewrite this as a contour integral, but we do not want to go in that direction.]
Continuing instead by first integrating over the coordinates, the procedure above is readily modified, and leads in the end to
\begin{equation}
\boxed{
G^{(0)}(s) = \left(\frac{A}{4\pi s}-\frac{1}{12}\right)\frac{\beta}{2\pi}}.
\end{equation}
The other heat kernels ($m\neq0$) are given by
\begin{equation}
\boxed{
G^{(m)}(s) = \frac{1}{2\pi \beta m^2}}.
\end{equation}
The latter sums into $\frac{1}{12}\frac{2\pi}{\beta}$.
A more heuristic argument can be found in <cit.>.
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1511.00049
|
Department of Mathematics, Texas A&M University, College Station, Texas 77843
We show that the empirical distribution of the eigenvalues of the sample covariance matrix of certain random vectors (not necessarily independent entries) with bounded
marginal $L^{4}$ norms converges weakly to a compound free Poisson distribution.
§ MAIN RESULT
Marchenko and Pastur <cit.> showed that the empirical distribution of the eigenvalues of the sample covariance matrix of a random vector uniformly distributed on
the unit sphere converges weakly to the Marchenko-Pastur law. There has been many generalizations to general random vectors (see <cit.>). The main result of this
paper is
Suppose that $f_{1},\ldots,f_{N}$ are independent random vectors on $\mathbb{C}^{n}$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(f_{j},x)|^{4}\leq\frac{L}{n^{2}}\text{ and }\mathbb{E}\|f_{j}\|^{k}\leq L_{k},\quad j=1,\ldots,N,\;k\geq 1\]
for some $L>0$ and $L_{k}>0$, $k\geq 1$ independent of $n$ and $N$. If $n,N\to\infty$ in such a way that $\frac{n}{N}\to\lambda\in(0,\infty)$ and
\[\left\|\sum_{j=1}^{N}\mathbb{E}\|f_{j}\|^{2(k-1)}f_{j}\otimes f_{j}-a_{k}I\right\|\leq Cn^{-\epsilon_{0}},\quad k\geq 1,\]
for some $a_{k}\in\mathbb{C}$, $k\geq 1$ and $C,\epsilon_{0}>0$ independent of $n$ and $N$, then
\[\mathbb{E}\circ\mathrm{tr}(f_{1}\otimes f_{1}+\ldots+f_{N}\otimes f_{N})^{p}\to\sum_{\pi\in\mathrm{NC}(p)}\prod_{B\in\pi}a_{|B|}.\]
Notation: tr means normalized trace. $\mathrm{NC}(p)$ is the set of all noncrossing partitions on $\{1,\ldots,p\}$.
1. An immediate consequence of Theorem <ref> is that the theorem of Marchenko and Pastur still holds if the random vector is distributed (but not uniformly distributed)
on the unit sphere provided that it has bounded marginal $L^{4}$ norms.
2. The condition $\displaystyle\sup_{x\in S^{n-1}}\mathbb{E}|(f_{i},x)|^{4}\leq\frac{L}{n^{2}}$ cannot be removed from Theorem <ref>. For example, when $N=n$ and each
$f_{i}$ is uniformly distributed on the canonical basis $\{e_{i}\}_{i=1}^{n}$ for $\mathbb{C}^{n}$, we have $a_{k}=1$ and
\[\mathbb{E}\circ\mathrm{tr}(f_{1}\otimes f_{1}+\ldots+f_{n}\otimes f_{n})^{p}\to B_{p},\]
where $B_{p}$ is the Bell number, the number of partitions on $\{1,\ldots,p\}$.
§ A GRAPH INEQUALITY
This section is devoted to proving the following lemma.
Let $S_{1},\ldots,S_{r}$ be subsets of a set $E$ such that every element $e\in E$ is contained in exactly two of the sets $S_{1},\ldots,S_{r}$. Assume that $|S_{1}|\leq
\ldots\leq|S_{r}|$. Let $t\geq 0$. Then
\[\min(t,|S_{1}|)+\min(t,|S_{2}\backslash S_{1}|)+\ldots+\min(t,|S_{r}\backslash(S_{1}\cup\ldots\cup S_{r-1})|)\geq\frac{\min(t,|S_{1}|)}{2}r.\]
Let $S_{1},\ldots,S_{r}$ be subsets of a set $E$ such that every element $x\in E$ is contained in exactly two of the sets $S_{1},\ldots,S_{r}$. Then
\[|E|=\frac{1}{2}\sum_{k=1}^{r}|S_{k}|.\]
By assumption, $\displaystyle\sum_{k=1}^{r}I_{S_{k}}(x)=2$ for all $x\in E$. So
\[\sum_{k=1}^{r}|S_{k}|=\sum_{k=1}^{r}\sum_{x\in E}I_{S_{k}}(x)=\sum_{x\in E}\sum_{k=1}^{r}I_{S_{k}}(x)=\sum_{x\in E}2=2|E|.\]
In Lemma <ref> and <ref> below, $\Lambda^{c}$ is understood as $\{1,\ldots,r\}\backslash\Lambda$. Also when $k=1$, $S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})$ is
understood as $S_{1}$.
Let $S_{1},\ldots,S_{r}$ be subsets of a set $E$ such that every element $x\in E$ is contained in exactly two of the sets $S_{1},\ldots,S_{r}$. If $\Lambda\subset
\{1,\ldots,r\}$ and $1\leq k_{0}\leq r$, then
\[\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\geq\frac{1}{2}\left(\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda}}|S_{k}|-\sum_{\substack{1\leq k
\leq k_{0}-1\\k\in\Lambda^{c}}}|S_{k}|\right).\]
\begin{align*}
&\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\\=&\sum_{k=1}^{r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|-\sum_{k\in\Lambda^{c}}|S_{k}\backslash
(S_{1}\cup\ldots\cup S_{k-1})|\\=&|E|-\sum_{k\in\Lambda^{c}}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\text{ since }E=\bigcup_{k=1}^{r}S_{k}\\=&
\frac{1}{2}\sum_{k=1}^{r}|S_{k}|-\sum_{k\in\Lambda^{c}}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\text{ by Lemma }\ref{22}\\=&
\frac{1}{2}\sum_{k\in\Lambda}|S_{k}|+\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}|-\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|-
\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\\=&
\frac{1}{2}\sum_{k\in\Lambda}|S_{k}|+\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}\backslash(S_{1}
\cup\ldots S_{k-1})|\\=&
\frac{1}{2}\sum_{k\in\Lambda}|S_{k}|+\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-\frac{1}{2}\sum_{\substack{1\leq k\leq k_{0}-1\\ k\in\Lambda^{c
}}}|S_{k}\backslash(S_{1}\cup\ldots S_{k-1})|\\&-\frac{1}{2}\sum_{\substack{k_{0}\leq k\leq n\\k\in\Lambda^{c}}}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\\\geq&
\frac{1}{2}\sum_{k\in\Lambda}|S_{k}|+\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-\frac{1}{2}\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda^{c}
}}|S_{k}|\\&-\frac{1}{2}\sum_{k_{0}\leq k\leq n}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\\=&
\frac{1}{2}\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda}}|S_{k}|+\frac{1}{2}\sum_{\substack{k_{0}\leq k\leq n\\k\in\Lambda}}|S_{k}|+\frac{1}{2}\sum_{k\in\Lambda^{c}}
|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-\frac{1}{2}\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda^{c}}}|S_{k}|
\\&-\frac{1}{2}\sum_{k_{0}\leq k\leq n}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\\=&
\frac{1}{2}\left(\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda}}|S_{k}|-\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda^{c}}}|S_{k}|\right)+\\&
\frac{1}{2}\left(\sum_{\substack{k_{0}\leq k\leq n\\k\in\Lambda}}|S_{k}|+\sum_{k\in\Lambda^{c}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-
\sum_{k_{0}\leq k\leq n}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\right).
\end{align*}
To complete the proof, it suffices to show that
\begin{equation}\label{21e}
\sum_{\substack{k_{0}\leq k\leq r\\k\in\Lambda}}|S_{k}|+\sum_{k\in\Lambda^{c}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-
\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\geq 0.
\end{equation}
To begin,
\begin{align}\label{22e}
&\sum_{\substack{k_{0}\leq k\leq r\\k\in\Lambda}}|S_{k}|+\sum_{k\in\Lambda^{c}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-
\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\nonumber\\\geq&
\sum_{\substack{k_{0}\leq k\leq r\\k\in\Lambda}}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|+\sum_{\substack{k_{0}\leq k\leq r\\k\in\Lambda^{c}}}|S_{k}\cap(S_{1}\cup\ldots\cup
S_{k-1})|\nonumber\\&-\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\nonumber\\=&
\sum_{k_{0}\leq k\leq r}|S_{k}\cap(S_{1}\cup\ldots\cup S_{k-1})|-\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\nonumber\\=&
\sum_{k_{0}\leq j\leq r}|S_{j}\cap(S_{1}\cup\ldots\cup S_{j-1})|-\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|.
\end{align}
By assumption, every element in $V$ is contained in at least two of the sets $S_{1},\ldots,S_{r}$. Therefore, if an element $e$ of $S_{k}$ is not in $S_{1}\cup\ldots\cup S_{k-1}$
then $e$ must be in $S_{k+1}\cup\ldots\cup S_{r}$. Thus,
\[|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\leq|S_{k}\cap(S_{k+1}\cup\ldots\cup S_{r})|\leq\sum_{k+1\leq j\leq r}|S_{k}\cap S_{j}|.\]
\begin{eqnarray}\label{23e}
\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|&\leq&\sum_{k_{0}\leq k\leq r}\sum_{k+1\leq j\leq n}|S_{k}\cap S_{j}|\nonumber\\&=&
\sum_{k_{0}+1\leq j\leq r}\sum_{k_{0}\leq k\leq j-1}|S_{k}\cap S_{j}|\nonumber\\&\leq&
\sum_{k_{0}\leq j\leq r}\sum_{1\leq k\leq j-1}|S_{k}\cap S_{j}|.
\end{eqnarray}
By assumption, every element in $E$ is contained in at most two of the sets $S_{1},\ldots,S_{n}$. So the sets $S_{1}\cap S_{j},\ldots,S_{j-1}\cap S_{j}$ are disjoint.
So $\displaystyle\sum_{1\leq k\leq j-1}|S_{k}\cap S_{j}|=|S_{j}\cap(S_{1}\cup\ldots\cup S_{j-1})|$. Thus, by (<ref>),
\[\sum_{k_{0}\leq k\leq r}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\leq\sum_{k_{0}\leq j\leq r}|S_{j}\cap(S_{1}\cup\ldots\cup S_{j-1})|.\]
Combining this with (<ref>), we obtain (<ref>). This completes the proof.
Let $m\geq 1$. Let $\Lambda_{1}$ and $\Lambda_{2}$ be subsets of $\{1,\ldots,m\}$. If $|[l,m]\cap\Lambda_{1}|\leq|[l,m]\cap\Lambda_{2}|$ for all $l\in\{1,\ldots,m\}$ then
there exists a strictly increasing function $f:\Lambda_{1}\to\Lambda_{2}$ such that $f(k)\geq k$ for all $k\in\Lambda_{1}$.
Since by assumption $|\Lambda_{1}|\leq|\Lambda_{2}|$, the function $f:\Lambda_{1}\to\Lambda_{2}$ defined by sending the $i$th largest element of $\Lambda_{1}$ to the $i$th
largest element of $\Lambda_{2}$ is well defined and strictly increasing. It remains to show that $f(k)\geq k$ for all $k\in\Lambda_{1}$. For each $i=1,\ldots,
|\Lambda_{1}|$, let $k_{i}$ be the $i$th largest element of $\Lambda_{1}$. By assumption, $|[k_{i},m]\cap\Lambda_{1}|\leq|[k_{i},m]\cap\Lambda_{2}|$ for all $i=1,\ldots,
|\Lambda_{1}|$. Note that $[k_{i},m]\cap\Lambda_{1}=\{k_{1},k_{2}\ldots,k_{i}\}$. So $|[k_{i},m]\cap\Lambda_{1}|=i$. Therefore, $|[k_{i},m]\cap\Lambda_{2}|\geq i$ for all $i=1,
\ldots,|\Lambda_{1}|$. So the $i$th largest element of $\Lambda_{2}$ is at least $k_{i}$. So $f(k_{i})\geq k_{i}$ for all $i=1,\ldots,|\Lambda_{1}|$ so $f(k)\geq k$ for all
Let $S_{1},\ldots,S_{r}$ be subsets of a set $E$ such that every element $x\in E$ is contained in exactly two of the sets $S_{1},\ldots,S_{r}$. Assume that $|S_{1}|\leq
\ldots\leq|S_{r}|$. If $\Lambda\subset\{1,\ldots,r\}$ then
\[\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\geq\frac{1}{2}|S_{1}|(|\Lambda|-|\Lambda^{c}|).\]
Case I: For every $1\leq l\leq r$, $|[l,r]\cap\Lambda^{c}|<|[l,r]\cap\Lambda|$.
From the first four lines of the proof of Lemma <ref>, we have
\[\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|=\frac{1}{2}\sum_{k=1}^{r}|S_{k}|-\sum_{k\in\Lambda^{c}}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|.\]
\[\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\geq\frac{1}{2}\sum_{k=1}^{r}|S_{k}|-\sum_{k\in\Lambda^{c}}|S_{k}|=\frac{1}{2}\sum_{k\in\Lambda}|S_{k}|-
\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}|.\]
Taking $m=r$, $\Lambda_{1}=\Lambda^{c}$ and $\Lambda_{2}=\Lambda$ in Lemma <ref>, we obtain an injective function $f:\Lambda^{c}\to\Lambda$ such that $f(k)\geq k$ for
all $k\in\Lambda^{c}$. Therefore,
\begin{eqnarray*}
\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|&=&\frac{1}{2}\sum_{j\in\Lambda}|S_{j}|-\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}|\\&=&
\frac{1}{2}\sum_{j\in f(\Lambda^{c})}|S_{j}|+\frac{1}{2}\sum_{j\in\Lambda\backslash f(\Lambda^{c})}|S_{j}|-\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}|\\&=&
\frac{1}{2}\sum_{k\in \Lambda^{c}}|S_{f(k)}|+\frac{1}{2}\sum_{j\in\Lambda\backslash f(\Lambda^{c})}|S_{j}|-\frac{1}{2}\sum_{k\in\Lambda^{c}}|S_{k}|\\&=&
\frac{1}{2}\sum_{k\in \Lambda^{c}}(|S_{f(k)}|-|S_{k}|)+\frac{1}{2}\sum_{j\in\Lambda\backslash f(\Lambda^{c})}|S_{j}|\\&\geq&
0+\frac{1}{2}|\Lambda\backslash f(\Lambda^{c})||S_{1}|.
\end{eqnarray*}
The last inequality follows from the fact that $f(k)\geq k$ for all $k\in\Lambda^{c}$ and the assumption that $|S_{1}|\leq\ldots\leq|S_{r}|$. Since $|\Lambda\backslash
f(\Lambda^{c})|=|\Lambda|-|f(\Lambda^{c})|=|\Lambda|-|\Lambda^{c}|$, it follows that
\[\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\geq\frac{1}{2}(|\Lambda|-|\Lambda^{c}|)|S_{1}|.\]
Case II: There exists $1\leq k_{0}\leq r$ such that $|[k_{0},r]\cap\Lambda^{c}|\geq |[k_{0},r]\cap\Lambda|$.
We may assume that $k_{0}$ is the smallest one with such property. We may also assume that $k_{0}>1$. Otherwise, the result is trivial. Thus, we have
$|[l,k_{0}-1]\cap\Lambda^{c}|<|[l,k_{0}-1]\cap\Lambda|$ for all $l\in\{1,\ldots,k_{0}-1\}$. Otherwise, an $l$ failing this property would contradict with the minimality of
$k_{0}$. Taking $m=k_{0}-1$, $\Lambda_{1}=[1,k_{0}-1]\cap\Lambda^{c}$ and $\Lambda_{2}=[1,k_{0}-1]\cap\Lambda$ in Lemma <ref>, we obtain an injective function
$f:[1,k_{0}-1]\cap\Lambda^{c}\to[1,k_{0}-1]\cap\Lambda$ satisfying $f(k)\geq k$ for all $k\in[1,k_{0}-1]\cap\Lambda^{c}$.
By Lemma <ref>, we have
\begin{align*}
&\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\\\geq&\frac{1}{2}\left(\sum_{\substack{1\leq k\leq k_{0}-1\\k\in\Lambda}}|S_{k}|-\sum_{\substack{1\leq k
\leq k_{0}-1\\k\in\Lambda^{c}}}|S_{k}|\right)\\=&\frac{1}{2}\left(\sum_{j\in[1,k_{0}-1]\cap\Lambda}|S_{j}|-\sum_{k\in[1,k_{0}-1]\cap\Lambda^{c}}|S_{k}|\right)\\=&
\frac{1}{2}\left(\sum_{j\in\{f(k):k\in[1,k_{0}-1]\cap\Lambda^{c}\}}|S_{j}|+\sum_{j\in[1,k_{0}-1]\cap\Lambda\backslash\{f(k):k\in[1,k_{0}-1]\cap\Lambda^{c}\}}|S_{j}|-\sum_{k\in[1,k_{0}-1]\cap\Lambda^{c}}|S_{k}|\right)\\=&
\frac{1}{2}\left(\sum_{k\in[1,k_{0}-1]\cap\Lambda^{c}}|S_{f(k)}|+\sum_{j\in[1,k_{0}-1]\cap\Lambda\backslash\{f(k):k\in[1,k_{0}-1]\cap\Lambda^{c}\}}|S_{j}|-\sum_{k\in[1,k_{0}-1]\cap\Lambda^{c}}|S_{k}|\right)\\=&
\frac{1}{2}\left(\sum_{k\in[1,k_{0}-1]\cap\Lambda^{c}}(|S_{f(k)}|-|S_{k}|)+\sum_{j\in[1,k_{0}-1]\cap\Lambda\backslash\{f(k):k\in[1,k_{0}-1]\cap\Lambda^{c}\}}|S_{j}|\right)
\\\geq&\frac{1}{2}(0+|[1,k_{0}-1]\cap\Lambda\backslash\{f(k):k\in[1,k_{0}-1]\cap\Lambda^{c}\}||S_{1}|).
\end{align*}
The last equality follows from the fact that $f(k)\geq k$ for all $k\in[1,k_{0}-1]\cap\Lambda^{c}$ and the assumption that $|S_{1}|\leq\ldots\leq|S_{r}|$. Therefore,
\begin{eqnarray*}
\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|&\geq&\frac{1}{2}(|[1,k_{0}-1]\cap\Lambda|-|\{f(k):k\in[1,k_{0}-1]\cap\Lambda^{c}\}|)|S_{1}|\\&=&
\frac{1}{2}(|[1,k_{0}-1]\cap\Lambda|-|[1,k_{0}-1]\cap\Lambda^{c}|)|S_{1}|\\&=&
\frac{1}{2}(|\Lambda|-|[k_{0},r]\cap\Lambda|-|\Lambda^{c}|+|[k_{0},r]\cap\Lambda^{c}|)|S_{1}|\\&\geq&
\frac{1}{2}(|\Lambda|-|\Lambda^{c}|)|S_{1}|.
\end{eqnarray*}
The last inequality follows from Case II assumption.
Let $\Lambda=\{1\leq k\leq r:|S_{k}\backslash(S_{1}\cup\ldots\cup S_{k-1})|\leq t\}$. Then
\begin{align*}
&\min(t,|S_{1}|)+\min(t,|S_{2}\backslash S_{1}|)+\ldots+\min(t,|S_{n}\backslash(S_{1}\cup\ldots\cup S_{n-1})|)\\=&\sum_{k\in\Lambda}|S_{k}\backslash(S_{1}\cup\ldots\cup
\end{align*}
If $|\Lambda^{c}|\geq\frac{r}{2}$ then
\[\min(t,|S_{1}|)+\min(t,|S_{2}\backslash S_{1}|)+\ldots+\min(t,|S_{r}\backslash(S_{1}\cup\ldots\cup S_{r-1})|)\geq\frac{tr}{2}\]
and the result follows. If $|\Lambda|\geq\frac{r}{2}$ then $|\Lambda|-|\Lambda^{c}|\geq 0$ so by Lemma <ref>, it follows that
\begin{align*}
&\min(t,|S_{1}|)+\min(t,|S_{2}\backslash S_{1}|)+\ldots+\min(t,|S_{n}\backslash(S_{1}\cup\ldots\cup S_{n-1})|)\\\geq&\frac{1}{2}|S_{1}|(|\Lambda|-|\Lambda^{c}|)+
\frac{\min(t,|S_{1}|)}{2}r.
\end{align*}
§ PROOF OF THE MAIN RESULT
If $y$ and $z$ are nonnegative random variables then for every $0<\epsilon<1$,
\[\mathbb{E}yz\leq(\mathbb{E}y)^{1-\epsilon}(\mathbb{E}y(z^{\frac{1}{\epsilon}}))^{\epsilon}.\]
By Hölder's inequality, $\mathbb{E}yz=\mathbb{E}y^{1-\epsilon}(y^{\epsilon}z)\leq(\mathbb{E}y)^{1-\epsilon}(\mathbb{E}(y^{\epsilon}z)^{\frac{1}{\epsilon}})^{\epsilon}=(\mathbb{E}y)^{1-\epsilon}(\mathbb{E}y(z^{\frac{1}{\epsilon}}))^{\epsilon}$.
Let $f_{1},\ldots,f_{r}$ be a random vector on $\mathbb{C}^{n}$ such that for every $\delta>0$ there exists $M_{\delta}>0$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(f,x)|^{4}\leq\frac{M_{\delta}}{n^{2(1-\delta)}}\text{ and }\mathbb{E}\|f\|^{k}\leq L_{k},\quad f\in\{f_{1},\ldots,f_{r}\},\;k\geq 1.\]
Then for every $\epsilon>0$ and $x_{1},\ldots,x_{r}\in\mathbb{C}^{n}$ with $\|x_{i}\|\leq 1$,
\[\mathbb{E}|(f_{1},x_{1})|\ldots|(f_{r},x_{r})|\leq\frac{C_{\epsilon}}{n^{\frac{1}{2}\min(r,4)(1-\epsilon)}},\]
where $C_{\epsilon}$ depends on $\epsilon$ and certain $M_{\delta}$ and $L_{k,\delta}$ but not on $n$.
By Hölder's inequality,
\[\mathbb{E}|(f_{1},x_{1})|\ldots|(f_{r},x_{r})|\leq(\mathbb{E}|(f_{1},x_{1})|^{r})^{\frac{1}{r}}\ldots(\mathbb{E}|(f_{r},x_{r})|^{r})^{\frac{1}{r}}\]
so it suffices to prove the lemma when $f_{1}=\ldots=f_{r}=f$ and $x_{1}=\ldots=x_{r}=x$. If $r>4$ then by Lemma <ref>, for every $\epsilon>0$,
\begin{eqnarray*}
\mathbb{E}|(f,x)|^{r}&\leq&\mathbb{E}|(f,x)|^{4}\|f\|^{r-4}\\&\leq&(\mathbb{E}|(f,x)|^{4})^{1-\frac{\epsilon}{2}}(\mathbb{E}|(f,x)|^{4}\|f\|^{\frac{2(r-4)}{\epsilon}})^{\frac{\epsilon}{2}}\\&\leq&(\mathbb{E}|(f,x)|^{4})^{1-\frac{\epsilon}{2}}(\mathbb{E}\|f\|^{4+\frac{2(r-4)}{\epsilon}})^{\frac{\epsilon}{2}}\\&\leq&\left(\frac{M_{\frac{\epsilon}{2}}}{n^{2(1-\frac{\epsilon}{2})}}\right)^{1-\frac{\epsilon}{2}}(L_{4+\frac{2(r-4)}{\epsilon}})^{\frac{\epsilon}{2}}\\&\leq&
\frac{M_{\frac{\epsilon}{2}}^{^{1-\frac{\epsilon}{2}}}}{n^{2(1-\epsilon)}}(L_{4+\frac{2(r-4)}{\epsilon}})^{\frac{\epsilon}{2}}.
\end{eqnarray*}
If $r\leq 4$ then by Hölder's inequality,
\[\mathbb{E}|(f,x)|^{r}\leq(\mathbb{E}|(f,x)|^{4})^{\frac{r}{4}}\leq\frac{M_{\epsilon}^{\frac{r}{4}}}{n^{\frac{r}{2}(1-\epsilon)}}.\]
Let $G=(V,E)$ be a graph with no loops but perhaps with multiple edges. Let $(\mathscr{B}_{v})_{v\in V}$ be independent $\sigma$-subalgebras of a probability space $(\Omega,\mathscr{B},\mathbb{P})$. For each $e\in E$, let $u_{1}(e)$ and $u_{2}(e)$ be the two endpoints of $e$ and let $h_{e}^{(1)}$ and $h_{e}^{(2)}$ be $\mathscr{B}_{u_{1}(e)}$-measurable and $\mathscr{B}_{u_{2}(e)}$-measurable random vectors on $\mathbb{C}^{n}$. Assume that for every $\delta>0$, there exist $M_{\delta}>0$ and $L_{k,\delta}$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(h,x)|^{4}\leq\frac{M_{\delta}}{n^{2(1-\delta)}}\text{ and }\mathbb{E}\|h\|^{k}\leq L_{k,\delta}n^{\delta},\quad h\in\bigcup_{e\in E}\{h_{e}^{(1)},h_{e}^{(2)}\},\;k\geq 1.\]
If every vertex has degree at least $4$, then for every $\epsilon>0$,
\[\mathbb{E}\prod_{e\in E}|\langle h_{e}^{(1)},h_{e}^{(2)}\rangle|\leq\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)}},\]
where $C_{\epsilon}$ depends on $\epsilon$, the graph $G$ and certain $M_{\delta}$ and $L_{k,\delta}$ but not on $n$.
Let $v_{1},\ldots,v_{|V|}$ be an enumeration of $V$ with ascending order according to their degrees, i.e., defining $S_{j}$ to be the set of all edges incident to $v_{j}$, we have $|S_{1}|\leq|S_{2}|\leq\ldots|S_{|V|}|$. For each $j=1,\ldots,|V|$, if $e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})$ then either $u_{1}(e)=v_{j}$ or $u_{2}(e)=v_{j}$ and so by interchanging the values of $u_{1}(e)$ and $u_{2}(e)$ (and accordingly also $h_{e}^{(1)}$ and $h_{e}^{(2)}$), if necessary, we may assume that $u_{1}(e)=v_{j}$. Thus, for every $\eta>0$
\begin{align}\label{First}
&\mathbb{E}\prod_{e\in E}|\langle h_{e}^{(1)},h_{e}^{(2)}\rangle|\nonumber\\=&\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}|\langle h_{e}^{(1)},h_{e}^{(2)}\rangle|\nonumber\\=&
\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|(\|h_{e}^{(2)}\|+\eta)\nonumber\\=&
\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}(\|h_{e}^{(2)}\|+\eta)\nonumber\\=&
\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\prod_{e\in E}(\|h_{e}^{(2)}\|+\eta),
\end{align}
where as before, when $j=1$, $S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})$ is understood as $S_{1}$.
Since $u_{1}(e)=v_{j}$, $h_{e}^{(1)}$ is $\mathscr{B}_{v_{j}}$-measurable. On the other hand, by assumption, $h_{e}^{(2)}$ is $\mathscr{B}_{u_{2}(e)}$-measurable; and since $G$ has no loops, $u_{2}(e)\neq u_{1}(e)=v_{j}$. Therefore, by Lemma <ref>,
\begin{equation}\label{CondExp}
\mathbb{E}_{\mathscr{B}_{v_{j}}}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\leq\frac{C_{\epsilon}}{n^{\frac{1}{2}\min(|S_{j}\backslash(S_{1}\cup\ldots S_{j-1})|,4)(1-\epsilon)}}.
\end{equation}
Note that the right hand side is a constant. We claim that
\begin{equation}\label{Claim}
\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\leq\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)}},
\end{equation}
where $C_{\epsilon}$ denotes any positive number depending on $\epsilon$, the graph $G$ and certain $M_{\delta}$ and $L_{k,\delta}$ but not on $n$.
To prove the claim, we write
\begin{align*}
&\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\\=&\mathbb{E}\left(\prod_{e\in S_{1}}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\right)\left(\prod_{j=2}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\right).
\end{align*}
All the edges $e$ in the first parenthesis are incident to $v_{1}$, whereas all the $e$ in the second parenthesis are not incident to $v_{1}$. Thus, the term in the second parenthesis is independent of $\mathscr{B}_{v_{1}}$ and so
\begin{align*}
&\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\\=&
\mathbb{E}\left(\mathbb{E}_{\mathscr{B}_{v_{1}}}\prod_{e\in S_{1}}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\right)\left(\prod_{j=2}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\right)\\\leq&
\frac{C_{\epsilon}}{n^{\frac{1}{2}\min(|S_{1}|,4)(1-\epsilon)}}\mathbb{E}\left(\prod_{j=2}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\right),
\end{align*}
where the inequality follows from (<ref>). Continuing this procedure, we obtain
\begin{align*}
&\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\\\leq&\frac{C_{\epsilon}}{n^{\frac{1}{2}\min(|S_{1}|,4)(1-\epsilon)}}\frac{C_{\epsilon}}{n^{\frac{1}{2}\min(|S_{2}\backslash S_{1}|,4)(1-\epsilon)}}\ldots\frac{C_{\epsilon}}{n^{\frac{1}{2}\min(|S_{|V|}\backslash(S_{1}\cup\ldots\cup S_{|V|-1})|,4)(1-\epsilon)}}.
\end{align*}
By Lemma <ref>, it follows that
\[\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\\\leq\frac{C_{\epsilon}}{n^{\frac{1}{4}\min(|S_{1}|,4)|V|(1-\epsilon)}},\]
possibly with different $C_{\epsilon}$. Since by assumption, $|S_{1}|\geq 4$, the claim (<ref>) is proved.
Having proved (<ref>), before we apply Lemma <ref>, we estimate
\begin{align*}
&\mathbb{E}\left(\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\right)\left(\prod_{e\in E}(\|h_{e}^{(2)}\|+\eta)\right)^{\frac{1}{\epsilon}}\\\leq&
\mathbb{E}\left(\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\|h_{e}^{(1)}\|\right)\left(\prod_{e\in E}(\|h_{e}^{(2)}\|+\eta)^{\frac{1}{\epsilon}}\right)\\\leq&
\mathbb{E}\left(\prod_{e\in E}\|h_{e}^{(1)}\|\right)\left(\prod_{e\in E}(\|h_{e}^{(2)}\|+\eta)^{\frac{1}{\epsilon}}\right)\\\leq&
\left(\prod_{e\in E}\mathbb{E}\|h_{e}^{(1)}\|^{2|E|}\prod_{e\in E}\mathbb{E}(\|h_{e}^{(2)}\|+\eta)^{\frac{2|E|}{\epsilon}}\right)^{\frac{1}{2|E|}},
\end{align*}
where the last inequality follows from Hölder's inequality. Combining this estimate with (<ref>), (<ref>) and Lemma <ref>, we obtain
\[\mathbb{E}\prod_{e\in E}|\langle h_{e}^{(1)},h_{e}^{(2)}\rangle|\leq\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)^{2}}}\left(\prod_{e\in E}\mathbb{E}\|h_{e}^{(1)}\|^{2|E|}\prod_{e\in E}\mathbb{E}(\|h_{e}^{(2)}\|+\eta)^{\frac{2|E|}{\epsilon}}\right)^{\frac{\epsilon}{2|E|}}.\]
Taking $\eta$ to be arbitarily small, we have
\begin{eqnarray*}
\mathbb{E}\prod_{e\in E}|\langle h_{e}^{(1)},h_{e}^{(2)}\rangle|&\leq&\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)^{2}}}\left(\prod_{e\in E}\mathbb{E}\|h_{e}^{(1)}\|^{2|E|}\prod_{e\in E}\mathbb{E}\|h_{e}^{(2)}\|^{\frac{2|E|}{\epsilon}}\right)^{\frac{\epsilon}{2|E|}}\\&\leq&
\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)^{2}}}\left(\prod_{e\in E}(L_{2|E|,1}n)\prod_{e\in E}(L_{\frac{2|E|}{\epsilon},1}n)\right)^{\frac{\epsilon}{2|E|}}\\&\leq&
\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)^{2}}}\left(\prod_{e\in E}L_{2|E|,1}\prod_{e\in E}L_{\frac{2|E|}{\epsilon},1}\right)^{\frac{\epsilon}{2|E|}}n^{\epsilon},
\end{eqnarray*}
where the second inequality follows from the assumption. This completes the proof with a different $\epsilon$.
Suppose that $(\mathscr{B}_{j})_{j\in J}$ are independent $\sigma$-subalgebras of a probability space $(\Omega,\mathscr{B},\mathbb{P})$. Let $j:\{1,\ldots,p\}\to J$ be such that $\ker j$ is a crossing partition on $\{1,\ldots,p\}$. For each $i=1,\ldots,p$, let $f_{i}^{(1)},f_{i}^{(2)}$ be $\mathscr{B}_{j(i)}$-measurable functions on $\Omega$. Assume that for every $\delta>0$, there exist $M_{\delta}>0$ and $L_{k,\delta}>0$, $k\geq 1$ such that
\begin{equation}\label{31e}
\sup_{x\in S^{n-1}}\mathbb{E}|(f,x)|^{4}\leq\frac{M_{\delta}}{n^{2(1-\delta)}}\text{ and }\mathbb{E}\|f\|^{k}\leq L_{k,\delta}n^{\delta},\quad f\in\{f_{1}^{(1)},f_{1}^{(2)},\ldots,f_{p}^{(1)},f_{p}^{(2)}\},\;k\geq 1
\end{equation}
Then for every $\epsilon>0$,
\[|\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})(f_{2}^{(1)}\otimes f_{2}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})|\leq\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|+1-\epsilon}},\]
where $C_{\epsilon}>0$ depends on $\epsilon,p$ and certain $M_{\delta}$ and $L_{k,\delta}$ but not on $n$.
We may assume that $j(1)\neq j(2)\neq\ldots\neq j(p)\neq j(1)$ and each $j(i)$ appears at least twice in the list $j(1),\ldots,j(p)$. Otherwise, if $j(i)=j(i+1)$ then
\[(f_{i}^{(1)}\otimes f_{i}^{(2)})(f_{i+1}^{(1)}\otimes f_{i+1}^{(2)})=\langle f_{i+1}^{(1)},f_{i}^{(2)}\rangle(f_{i}^{(1)}\otimes f_{i+1}^{(2)})=(\langle f_{i+1}^{(1)},f_{i}^{(2)}\rangle f_{i}^{(1)})\otimes f_{i+1}^{(2)}.\]
Note that $\langle f_{i+1}^{(1)},f_{i}^{(2)}\rangle f_{i}^{(1)}$ and $f_{i+1}^{(2)}$ are $\mathscr{B}_{j(i)}$-measurable since $j(i)=j(i+1)$. Also, by Hölder's inequality, $\langle f_{i+1}^{(1)},f_{i}^{(2)}\rangle f_{i}^{(1)}$ satisfies (<ref>) perhaps with different $M_{\delta}$ and $L_{k,\delta}$. Thus, the result follows by induction hypothesis since the product $(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(f_{p-1}^{(1)}\otimes f_{p}^{(2)})$ of $p$ terms becomes a product of $p-1$ terms. (The $i$th term and the $(i+1)$th term are combined.)
Similar argument works if we have $j(p)\neq j(1)$.
If there is a $j(i)$ that appears only once in the list $j(1),\ldots,j(p)$, then by independence of $(\mathscr{B}_{j})_{j\in J}$,
\begin{align}\label{32e}
&\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(f_{p-1}^{(1)}\otimes f_{p}^{(2)})\nonumber\\=&
\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(f_{i}^{(1)}\otimes f_{i}^{(2)})\ldots(f_{p-1}^{(1)}\otimes f_{p}^{(2)})\nonumber\\=&
\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)})\ldots(f_{p-1}^{(1)}\otimes f_{p}^{(2)})\nonumber\\=&
\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)})(f_{i+1}^{(1)}\otimes f_{i+1}^{(2)})\ldots(f_{p-1}^{(1)}\otimes f_{p}^{(2)})\nonumber\\=&
\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots((\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)})f_{i+1}^{(1)}\otimes f_{i+1}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})\nonumber\\=&
\frac{1}{n}\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(n(\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)})f_{i+1}^{(1)}\otimes f_{i+1}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)}).
\end{align}
Note that $\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)}$ is a deterministic matrix and
\begin{eqnarray*}
|\langle(\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)})x,y\rangle|&=&|\mathbb{E}\langle x,f_{i}^{(2)}\rangle\langle f_{i}^{(1)},y\rangle|\\&\leq&\mathbb{E}|\langle x,f_{i}^{(2)}\rangle||\langle f_{i}^{(1)},y\rangle|\\&\leq&(\mathbb{E}|\langle f_{i}^{(2)},x\rangle|^{2})^{\frac{1}{2}}(\mathbb{E}|\langle f_{i}^{(1)},y\rangle|^{2})^{\frac{1}{2}}\\&\leq&(\mathbb{E}|\langle f_{i}^{(2)},x\rangle|^{4})^{\frac{1}{4}}(\mathbb{E}|\langle f_{i}^{(1)},y\rangle|^{4})^{\frac{1}{4}}\\&\leq&\left(\frac{M_{\delta}}{n^{2(1-\delta)}}\right)^{\frac{1}{4}}\left(\frac{M_{\delta}}{n^{2(1-\delta)}}\right)^{\frac{1}{4}}\\&=&\frac{\sqrt{M_{\delta}}}{n^{1-\delta}},\quad x,y\in S^{n-1}.
\end{eqnarray*}
\[\|n\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)}\|\leq\sqrt{M_{\delta}}n^{\delta}.\]
Hence, $n(\mathbb{E}f_{i}^{(1)}\otimes f_{i}^{(2)})f_{i+1}^{(1)}$ is $\mathscr{B}_{j(i+1)}$ and still satisfies (<ref>) perhaps with different $M_{\delta}$ and $L_{k,\delta}$. Thus, in view of (<ref>), the result follows by induction hypothesis since the product $(f_{1}^{(1)}\otimes f_{1}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})$ of $p$ terms becomes a product of $p-1$ terms. (The $i$th term is absorbed by the $(i+1)$th term.)
Therefore, we may justifiably assume that $j(1)\neq j(2)\neq\ldots\neq j(p)\neq j(1)$ and each $j(i)$ appears at least twice in the list $j(1),\ldots,j(p)$.
\begin{align*}
&|\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})(f_{2}^{(1)}\otimes f_{2}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})|\\=&\frac{1}{n}|\mathbb{E}\langle f_{1}^{(2)},f_{2}^{(1)}\rangle\langle f_{2}^{(2)},f_{3}^{(1)}\rangle\ldots\langle f_{p}^{(2)},f_{1}^{(1)}\rangle|\leq\frac{1}{n}\mathbb{E}|\langle f_{1}^{(2)},f_{2}^{(1)}\rangle||\langle f_{2}^{(2)},f_{3}^{(1)}\rangle|\ldots|\langle f_{p}^{(2)},f_{1}^{(1)}\rangle|.
\end{align*}
For notational convenience, let $j(p+1)=j(1)$ and $f_{p+1}^{(1)}=f_{1}^{(1)}$. Then we have
\begin{equation}\label{FirstEstimate}
|\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})(f_{2}^{(1)}\otimes f_{2}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})|\leq\frac{1}{n}\mathbb{E}\prod_{i=1}^{p}|\langle f_{i}^{(2)},f_{i+1}^{(1)}\rangle|.
\end{equation}
We use Lemma <ref> to estimate this. First, we take the vertex set $V=\{j(1),\ldots,j(p)\}$ and the edge set $E=\{1,\ldots,p\}$, where for each $i\in E$, the two endpoints are $u_{1}(i)=j(i)$ and $u_{2}(i)=j(i+1)$. There are no loops since we assume that $j(i)\neq j(i+1)$ for all $i=1,\ldots,p$. For each $i\in E$, take $h_{i}^{(1)}=f_{i}^{(2)}$ and $h_{i}^{(2)}=f_{i+1}^{(1)}$. To see that every vertex has degree at least $4$, recall that we assume that for every $j\in V=\{j(1),\ldots,j(p)\}$, there exist $i_{1}\neq i_{2}$ in $\{1,\ldots,p\}$ such that $j(i_{1})=j(i_{2})=j$. Since $j(1)\neq j(2)\neq\ldots\neq j(p)\neq j(1)$, $i_{1}$ and $i_{2}$ cannot be consective numbers. Therefore, the vertex $j$ is incident with the four distinct edges $i_{1}-1,i_{1},i_{2}-1,i_{2}$. (When $i_{1}=1$, $i_{1}-1=p$.) Thus, the assumptions of Lemma <ref> are satisfied and so we obtain
\[\mathbb{E}\prod_{i=1}^{p}|\langle f_{i}^{(2)},f_{i+1}^{(1)}\rangle|\leq\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|(1-\epsilon)}}.\]
The result follows by combining this with <ref>.
In Lemma <ref>, the assumption that $\ker j$ is a crossing partition is necessary because it guarantees that repeating the procedure of (1) combining the $i$th term and the $(i+1)$th term when $j(i)=j(i+1)$ and (2) the $i$th term being absorbed by the $(i+1)$th term when $j(i)$ appears only once in the list $j(1),\ldots,j(p)$ does not make reduce $\{1,\ldots,p\}$ to a singleton. Without the crossing assumption, one would have got Lemma <ref> below.
As an immediate consequence of Lemma <ref>, we have
Suppose that $(f_{j})_{j\in J}$ is an independent family of random vectors on $\mathbb{C}^{n}$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(f_{j},x)|^{4}\leq\frac{L}{n^{2}}\text{ and }\mathbb{E}\|f_{j}\|^{k}\leq L_{k},\quad j\in J,\;k\geq 1\]
for some $L>0$ and $L_{k}>0$, $k\geq 1$ independent of $N$. Let $j:\{1,\ldots,p\}\to J$ be such that $\ker j$ is a crossing partition on $\{1,\ldots,p\}$. Then for every $\epsilon>0$,
\[|\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})|\leq\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|+1-\epsilon}},\]
where $C_{\epsilon}>0$ depends on $\epsilon,p,L$ and certain $L_{k}$ but not on $n$.
The following lemma is the analog of Lemma <ref> for noncrossing partition.
Suppose that $(\mathscr{B}_{j})_{j\in J}$ are independent $\sigma$-subalgebras of a probability space $(\Omega,\mathscr{B},\mathbb{P})$. Let $j:\{1,\ldots,p\}\to J$ be such that $\ker j$ is a noncrossing partition on $\{1,\ldots,p\}$. For each $i=1,\ldots,p$, let $f_{i}^{(1)},f_{i}^{(2)}$ be $\mathscr{B}_{j(i)}$-measurable functions on $\Omega$. Assume that for every $\delta>0$, there exist $M_{\delta}>0$ and $L_{k,\delta}>0$, $k\geq 1$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(f,x)|^{4}\leq\frac{M_{\delta}}{n^{2(1-\delta)}}\text{ and }\mathbb{E}\|f\|^{k}\leq L_{k,\delta}n^{\delta},\quad f\in\{f_{1}^{(1)},f_{1}^{(2)},\ldots,f_{p}^{(1)},f_{p}^{(2)}\},\;k\geq 1.\]
Then for every $\epsilon>0$,
\begin{equation}\label{BoundExpe}
\|\mathbb{E}(f_{1}^{(1)}\otimes f_{1}^{(2)})(f_{2}^{(1)}\otimes f_{2}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})\|\leq\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|-\epsilon}},
\end{equation}
where $C_{\epsilon}>0$ depends on $\epsilon,p$ and certain $M_{\delta}$ and $L_{k,\delta}$ but not on $n$.
The only differences are that on the left hand side of (<ref>), one has norm of expectation instead of trace expectation and that on the right hand side of (<ref>), one only has $\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|-\epsilon}}$ instead of $\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|+1-\epsilon}}$ in Lemma <ref>. The proof of Lemma <ref> is exactly the same as the beginning of the proof of Lemma <ref>. One needs the fact that for every noncrossing partition $\pi$ on $\{1,\ldots,p\}$, at least one of the following holds.
* There exists $i\in\{1,\ldots,p-1\}$ such that $i$ and $i+1$ are in the same block of $\pi$.
* $\pi$ has a singleton block.
This is because every noncrossing partition contains an interval block.
As an immediate consequence of Lemma <ref>, we have
Suppose that $(f_{j})_{j\in J}$ is an independent family of random vectors on $\mathbb{C}^{n}$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(f_{j},x)|^{4}\leq\frac{L}{n^{2}}\text{ and }\mathbb{E}\|f_{j}\|^{k}\leq L_{k},\quad j\in J,\;k\geq 1\]
for some $L>0$ and $L_{k}>0$, $k\geq 1$ independent of $N$. Let $j:\{1,\ldots,p\}\to J$ be such that $\ker j$ is a noncrossing partition on $\{1,\ldots,p\}$. Then for every $\epsilon>0$,
\[\|\mathbb{E}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})\|\leq\frac{C_{\epsilon}}{n^{|\{j(1),\ldots,j(p)\}|-\epsilon}},\]
where $C_{\epsilon}>0$ depends on $\epsilon,p,L$ and certain $L_{k}$ but not on $n$.
Suppose that $f_{1},\ldots,f_{N}$ are independent random vectors on $\mathbb{C}^{n}$ such that
\[\sup_{x\in S^{n-1}}\mathbb{E}|(f_{j},x)|^{4}\leq\frac{L}{n^{2}}\text{ and }\mathbb{E}\|f_{j}\|^{k}\leq L_{k},\quad j=1,\ldots,N,\;k\geq 1\]
for some $L>0$ and $L_{k}>0$, $k\geq 1$ independent of $n$ and $N$. If $n,N\to\infty$ in such a way that $\frac{n}{N}\to\lambda\in(0,\infty)$ and
\[n^{\epsilon_{0}}\left\|\sum_{j=1}^{N}\mathbb{E}\|f_{j}\|^{2(k-1)}f_{j}\otimes f_{j}-a_{k}I\right\|\to 0,\quad k\geq 1,\]
for some $a_{k}\in\mathbb{C}$, $k\geq 1$ and $\epsilon_{0}>0$ independent of $n$ and $N$, then for every noncrossing partition $\pi$ on $\{1,\ldots,p\}$,
\[\left|\sum_{\substack{j:\{1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi}}\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})-\prod_{B\in\pi}a_{|B|}\right|\to 0.\]
We prove by induction on $p$. For $p=1$, the result is obvious. For $p\geq 2$, since $\pi$ is a noncrossing partition on $\{1,\ldots,p\}$, there is an interval block $B_{0}\in\pi$. For simplicity, since the trace is cyclic invariant, we may assume that $B_{0}=\{1,\ldots,q\}$ for some $1\leq q\leq p$. Thus, for every $j:\{1,\ldots,p\}\to\{1,\ldots,N\}$ with $\ker j=\pi$, we have
\begin{align*}
&\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\=&
\mathrm{tr}\mathbb{E}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(q)}\otimes f_{j(q)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\=&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)}),
\end{align*}
since $j(1)=\ldots=j(q)$. Note that every $j:\{1,\ldots,p\}\to\{1,\ldots,N\}$ with $\ker j=\pi$ corresponds to $j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}$ with $\ker l=\pi\backslash\{B_{0}\}$ and $j(1)\in\{1,\ldots,N\}\backslash\{j(q+1),\ldots,j(p)\}$. Thus,
\begin{align*}
&\sum_{\substack{j:\{1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi}}\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\=&
\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{1,\ldots,N\}\backslash\{j(q+1),\ldots,j(p)\}}\\&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\=&
\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{1,\ldots,N\}}\\&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\
&-\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{j(q+1),\ldots,j(p)\}}\\&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\
\mathrm{tr}\left(\sum_{j(1)\in\{1,\ldots,N\}}\mathbb{E}\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)}\right)\\&\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\right)\\
&-\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{j(q+1),\ldots,j(p)\}}\\&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\=&
\mathrm{tr}a_{q}I\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\right)\\
&+\mathrm{tr}\left(\sum_{j(1)\in\{1,\ldots,N\}}\mathbb{E}\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)}-a_{q}I\right)\\&\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\right)\\
&-\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{j(q+1),\ldots,j(p)\}}\\&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)}).
\end{align*}
By induction hypothesis, the first term
\[\mathrm{tr}a_{q}I\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\right)\]
converges to $\displaystyle a_{q}\prod_{B\in\pi\backslash\{B_{0}\}}a_{|B|}=\prod_{B\in\pi}a_{|B|}$. For the second term,
\begin{align*}
&\bigg|\mathrm{tr}\left(\sum_{j(1)\in\{1,\ldots,N\}}\mathbb{E}\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)}-a_{q}I\right)\\&\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\right)\bigg|\\\leq&
\left\|\sum_{j(1)\in\{1,\ldots,N\}}\mathbb{E}\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)}-a_{q}I\right\|\\&
\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\|\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\|\right)\\\leq&
\left\|\sum_{j(1)\in\{1,\ldots,N\}}\mathbb{E}\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)}-a_{q}I\right\|\\&
\left(\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\frac{C_{\frac{\epsilon_{0}}{2}}}{n^{|\{j(q+1),\ldots,j(p)\}|-\frac{\epsilon_{0}}{2}}}\right)\\
&\text{ by Lemma }\ref{BoundExp2}\\\leq&
\left\|\sum_{j(1)\in\{1,\ldots,N\}}\mathbb{E}\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)}-a_{q}I\right\|C_{\frac{\epsilon_{0}}{2}}n^{\frac{\epsilon_{0}}{2}}\to 0.
\end{align*}
For the third term,
\begin{align*}
&\bigg|\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{j(q+1),\ldots,j(p)\}}\\&
\mathrm{tr}\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\bigg|\\\leq&
\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{j(q+1),\ldots,j(p)\}}\\&
\|\mathbb{E}(\|f_{j(1)}\|^{2(q-1)}f_{j(1)}\otimes f_{j(1)})\|\|\mathbb{E}(f_{j(q+1)}\otimes f_{j(q+1)})\ldots(f_{j(p)}\otimes f_{j(p)})\|\\&
\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}\sum_{j(1)\in\{j(q+1),\ldots,j(p)\}}
\frac{C_{\frac{1}{4}}}{n^{1-\frac{1}{4}}}\frac{C_{\frac{1}{4}}}{n^{|\{j(q+1),\ldots,j(p)\}|-\frac{1}{4}}}\\
&\text{ by Lemma }\ref{BoundExp2}\text{ with }\epsilon=\frac{1}{4}\\\leq&
\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}
\sum_{\substack{j:\{q+1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j=\pi\backslash\{B\}}}
\frac{C}{n^{|\{j(q+1),\ldots,j(p)\}|+\frac{1}{2}}}\leq\frac{C}{n^{\frac{1}{2}}}\to 0.
\end{align*}
\begin{eqnarray*}
\mathbb{E}\circ\mathrm{tr}(f_{1}\otimes f_{1}+\ldots+f_{N}\otimes f_{N})^{p}&=&\sum_{j:\{1,\ldots,p\}\to\{1,\ldots,N\}}\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\&=&\sum_{\substack{j:\{1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j\text{ noncrossing}}}\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\&&+
\sum_{\substack{j:\{1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j\text{ crossing}}}\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})
\end{eqnarray*}
The first term converges to $\displaystyle\sum_{\pi\in\mathrm{NC}(p)}\prod_{B\in\pi}a_{|B|}$ by Proposition <ref>. For the second term,
\begin{align*}
&\sum_{\substack{j:\{1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j\text{ crossing}}}\mathbb{E}\circ\mathrm{tr}(f_{j(1)}\otimes f_{j(1)})\ldots(f_{j(p)}\otimes f_{j(p)})\\\leq&\sum_{\substack{j:\{1,\ldots,p\}\to\{1,\ldots,N\}\\\ker j\text{ crossing}}}\frac{C_{\frac{1}{2}}}{n^{|\{j(1),\ldots,j(p)\}|+1-\frac{1}{2}}}\text{ by Proposition }\ref{FirstProp}\text{ with }\epsilon=\frac{1}{2}\\\leq&\frac{C}{n^{\frac{1}{2}}}\to 0.
\end{align*}
Adamczak R. Adamczak, On the Marchenko-Pastur and circular laws for some classes of random matrices with dependent entries, Electron. J. Probab. 16 (2011),
Marchenko V. Marchenko, L. Pastur, Distribution of eigenvalues of some sets of random matrices, Math USSR-Sb. 1, (1967), 507-536.
|
1511.00380
|
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India
We study the percolative superconducting transition as the density of randomly placed attractive centers grows in a host metal. Employing the Hubbard-Stratanovich transformation for the interaction and allowing for spatial, thermal fluctuations of the pairing field, we obtain real-space features of the transition from weak to strong coupling. Spectral and transport properties are studied in detail. BCS-BEC crossover is discussed in the context of site dilution of attractive centers.
PACS : 74.81.-g, 74.20.-z, 74.78.-w, 71.10.Fd
§ INTRODUCTION
The interplay between superconductivity (SC) and disorder is a long standing problem <cit.>. For weak disorder the two are not expected to be inimical to each other
since pairing takes place between time-reversed states <cit.> that are present when only potential impurities disorder the system <cit.> . In this limit the superconducting state is not
expected to be much different from the mean field BCS state. In particular, it remains homogeneous
in low disorder regime. However, when the disorder is strong, it dramatically alters the superconducting phase <cit.>. Large phase fluctuations can reduce the
superconducting transition temperature from its mean field value and in a temperature window between $T_c$ and $T_{BCS}$, where $T_c$ is the transition
temperature and $T_{BCS}$ is the mean field value expected from the BCS theory, a pseudogap phase is expected <cit.>. This is a region where
preformed pairs exist, but global coherence is absent. Strong disorder also makes the phase highly inhomogeneous.
Theoretical studies at strong disorder <cit.> reveal the inhomogeneous natures of the SC state, formation of superconducting puddles and presence of a pseudogap phase where
long-range SC order diminishes although the quasiparticle gap remains open. While Josephson effect between puddles can give rise to global SC state<cit.>, strong phase fluctuations
among them may lead to an insulating state<cit.>. Recent experimental studies have probed this behavior <cit.>.
They reveal fragmentation of the superconducting state into islands, pseudogap like features in the normal state, and a change in the normal state
resistivity suggestive of a metal-insulator transition.
Competition between superconductivity and disorder is expected to be more interesting in two dimensions since arbitrarily small disorder is capable of localizing
electrons <cit.> while the superconducting transition itself is of Berezenskii-Kosterlitz-Thouless (BKT) type. Experimental studies show a superconductor-insulator transition
in many two dimensional systems<cit.>, a theoretical understanding of which is still not very satisfactory. Another complication arises as one increases the strength of attractive
interaction since, in absence of disorder, this is expected to lead to a BCS-BEC crossover <cit.>. The latter results from the Bose condensation of local pairs of electrons arising due to enhanced double occupancy for large local attractive interactions. The physics is very different from the BCS limit; there is still a large, local pairing gap that is visible in the
spectral function, but in contrast to the BCS limit, $T_c$ is much reduced and does not scale with the pairing gap, though the zero temperature pairing gap continues to increase with interaction. Phase fluctuations play a dominant role here and is the cause of suppression of SC order even when there are strong local pairing tendencies.. The effect of disorder in this limit has not been explored adequately.
Further, there are various systems in which a SC ground state is arrived at by doping an insulating host. For example, PbTe is a semiconductor,
but when doped with Tl (Pb$_{(1-x)}$Tl$_x$Te) becomes superconducting beyond a critical $x_c \sim 0.3$ <cit.> . $T_c$ increases with $x$ suggesting that Tl induces pairing. It is also known that $T_c$ decreases when a superconducting material is doped with certain atoms. Examples include MgB$_2$ doped with carbon <cit.>
(Mg$_{(1-x)}$C$_x$B$_2$) or aluminium (Mg$_{(1-x)}$Al$_x$B$_2$) <cit.>. A simple way of looking at this problem is to assume that a host system is
doped with inhomogeneous attractive centers which promote local pairing<cit.>. As the number of such attractive centers increases, superconducting islands start to form.
However, onset of superconductivity requires percolation of these puddles, thereby establishing global phase coherence. This problem has several interesting features.
Disorder and superconductivity contribute on an equal footing and the superconducting state is expected to be intrinsically inhomogeneous. In absence of attractive centers,
the host could be metallic or nonconducting, though we study only the former in this paper.
One could also explore the BCS-BEC crossover in the context of dilution of attractive centers if one is able to handle the regime of large attractive interactions.
Theoretical studies based on the above picture have been carried out previously using mean field theory <cit.> or quantum Monte Carlo (QMC)<cit.>. The former does not have the prospect of studying
large interaction strengths. The latter can handle the entire range, but is numerically expensive with obvious system size limitations;
transport is harder to evaluate. Recently, dynamical mean field theory (DMFT) <cit.> was employed in conjunction with coherent potential approximation (CPA) to treat disorder<cit.>.
However, neglect of spatial correlations lead to unphysical results at strong interactions.
To this end, we use a numerically less expensive method that captures the entire parameter regime while retaining thermal fluctuations of the pairing field
and is able to shed light on the physics in real space.
We employ the random attractive Hubbard model <cit.> where the number of attractive centers is determined by the dilution on an average.
We use a real-space Hubbard-Stratanovich (HS) <cit.>
transformation by introducing auxiliary fields in the pairing and charge channels that couple to electrons. For simplicity, we assume the auxiliary fields to be classical;
while we allow spatial fluctuations of amplitude and phase of the pairing field, we neglect their dynamics. This results in studying the self-consistent quantum dynamics of electrons coupled to thermally fluctuating, classical pairng field which is treated numerically using Monte Carlo method<cit.> . The details of the model and numerical procedure are given in Section II.
We discuss the critical temperature of the superconducting transition and its variation with dilution and strength of interaction in
Section III. Afterwards, we present spectral and transport properties of this model. Being a real-space method, this gives us a direct image of the physics in real space,
while allowing to access the BCS-BEC crossover regime. We conclude by pointing out certain limitations of the present approach and possible extensions.
§ MODEL AND THE STATIC AUXILIARY FIELD METHOD
To study the nature of percolative superconductivity due to variation in the density of attractive centers, we employ a minimal model, which is the attractive Hubbard model with site dilution, that captures the essential features of the problem. The Hamiltonian employed is
\begin{equation}
H = -t \sum_{\left< ij \right>, \sigma} c^{\dag}_{i \sigma} c_{j \sigma} - \sum_i U_i n_{i \uparrow} n_{i \downarrow} - \mu \sum_i n_i.
\end{equation}
Here, $t$ is the nearest-neighbor hopping integral (which we take to be unity to set the energy scales), $U_i$ is the strength of attractive interaction that
is site-dependent, $\mu$ is the chemical potential which fixes the mean electron density. In this paper, we fix the electron density to be $n = 0.875$.
However, the physics is not very sensitive to changes in average electron density, except at half filling.
The case of half filling is special, which we discuss in the last section. Site dilution is introduced via site dependent $U_i$
that follows a bimodal probability distribution such that $U_i = U$ with probability $P(U) = \delta $ and $U_i = 0$ with probability
$P(U) = 1-\delta$ <cit.>, where $\delta$ is the average number of sites having attractive centers (e.g., $\delta =1$
when all sites have attractive centers, which is the clean limit). We employ the Hubbard-Stratanovich transformation to reduce the interacting, quartic Hamiltonian
to a quadratic fermionic Hamiltonian coupled to a pairing field $\Delta_i$, which is a complex variable and a real, scalar-valued charge (or, equivalently density) field $\phi_i$.
The resulting Hamiltonian reads :
\begin{eqnarray}
H_{eff} = &-& t \sum_{\left< ij \right>,\sigma} c^{\dag}_{i \sigma} c_{j \sigma} - \mu \sum_i n_i + \sum_i \left( \Delta_i c^{\dag}_{i \uparrow} c^{\dag}_{i \downarrow} + h.c. \right)
\nonumber \\
& +& \sum_i {{{\left| \Delta_i \right| }^2} \over U_i} + \sum_i \phi_i n_i + \sum_i {\phi_i^2 \over U_i},
\end{eqnarray}
where $\Delta_i = \left< c_{i \uparrow} c_{i \downarrow} \right>$ and $n_i = \sum_{\sigma} c^\dag_{i \sigma} c_{i \sigma}$. The partition function can be evaluated in terms of the effective Hamiltonian and is given by
\begin{equation}
{\cal {Z}} = \int {\cal{D}} \Delta {\cal {D} }\Delta^* {\cal {D}} \phi {\cal {D}} \left[ c^\dag, c \right] e^{-\beta H_{eff}},
\end{equation}
so that the probability of occurrence of a particular configuration of $\Delta_i$ at inverse temperature $\beta = 1/\left( k_B T \right)$ is obtained from
\begin{equation}
P \left( \Delta_i \right) = {1 \over {\cal{Z}}} \int {\cal {D}} \phi {\cal {D}} \left[ c^\dag, c \right] e^{-\beta H_{eff}}.
\end{equation}
The saddle-point solutions of the action corresponding to the effective Hamiltonian give Bogoliubov-de Gennes (BdG) equations for the pairing field $\Delta_i$ and the charge field $\phi_i$.
While at this level the action is exact, to make progress, we assume that the pairing fields are static (i.e., we neglect quantum fluctuations),
but their amplitudes and phases are site-dependent and thermally fluctuating <cit.> . Charge field is also assumed to be classical. At finite temperatures,
this necessitates thermally averaging over their most probable configurations, which we carry out using a Monte Carlo (MC) estimation of their weights based on Metropolis algorithm.
This essentially means that for a given electron density and temperature, we start with a random configuration of attractive centers by fixing the amount of site dilution,
and a judicious choice of the pairing and charge fields at every site. This leaves us with a problem of electrons moving in random (classical) fields that requires
an exact diagonalization of the fermion problem. A thermal sampling of the most probable configurations of the auxiliary fields is performed by Monte Carlo updating of the (classical) fields.
Thermodynamic properties of the system as well as spectral features of electrons and transport are obtained by averaging over configurations thus obtained.
This method, which requires exact diagonalization of the electron system at every Monte Carlo step, obviously restricts the system size and to circumvent it
we use a traveling cluster algorithm (TCA)<cit.> . Here, the fermion problem is diagonalized on a smaller cluster around the chosen MC update site,
embedded in a much larger lattice. The cluster moves during every MC update restoring ergodicity. A similar approach was recently used successfully for the case of repulsive Hubbard-Holstein model in two dimensions <cit.>.
Before presenting our results, we review the previous works based on the above model. These include mean field calculations based on BdG equations<cit.> with disorder treated
using CPA, quantum Monte Carlo<cit.> , and the dynamical mean field theory with iterated perturbation theory (IPT)
as an impurity solver in conjunction with CPA to handle disorder<cit.>. In general, a critical concentration of attractive centers, $\delta_c$, is required to get superconducting ground state.
The system undergoes a first order metal-SC transition at $\delta_c$. While $\delta_c$ increases with $U$ in mean field calculations, it is seen to decrease and then saturate with $U$ in QMC.
DMFT studies reveal that $\delta_c$ decreases sharply with increasing $U$.
For all $U \geq$ 2.7, $\delta_c \sim$ 0. This is obviously an artifact of the infinite coordination number employed in DMFT, neglecting spatial correlations. They also find that suppressing dynamic fluctuations leads to $\delta_c =$ 0, suggesting that arbitrary small number of sttractive dopants are needed for onset of superconductivity.
In general, $\delta_c$ displays a strong dependence on $U$, which suggests that the transition from the metallic to SC state cannot be thought of entirely in terms of percolation alone.
In the next section, we provide results on the metal-SC transition in the model based on our study focusing on the variation of structure factor with temperature which determines
§ ORDER PARAMETER AND CRITICAL TEMPERATURE
Once the system reaches equilibrium, we use the thermal averaged structure factor for the pairing field $\Delta_i = \left< c_{i \uparrow} c_{i \downarrow} \right>$, to track the onset of superconductivity as a function of dilution and temperature:
\begin{equation}
S\left( {\bf q} \right) = {1 \over N^2} \sum_{ij} \left< \Delta_i \Delta^*_j \right> e^{i {\bf q} \cdot \left( {\bf r}_i - {\bf r}_j \right)}.
\end{equation}
For a uniform superconducting solution, we look at the wave vector ${\bf q} = \left( 0, 0 \right)$ and the corresponding structure factor $S( {\bf 0})$. For a given dilution, $S( {\bf 0})$ is vanishingly small at high temperatures
and starts picking up at a characteristic temperature, which we identify with the superconducting transition temperature $T_c$.
A typical result is plotted in Fig. 1 for $U=8$. We note that there is a critical concentration of attractive centers, $\delta_c$,
needed to have "global" superconductivity, which for this case happens to be roughly $\delta_c = 0.6$. Further, the saturation value and $T_c$ increases with $\delta$.
As we will discuss later, the onset of superconductivity is brought about by percolation of locally superconducting islands and having larger number of attractive centers helps in enhancing
the superconducting correlations, and hence the transition temperature itself. For most of our discussions, we have used a system size of 32 $\times$ 32
with the size of the traveling cluster being 8 $\times$ 8. There is a marginal decrease in $T_c$ as system size increases which is to be expected in a two-dimensional system.
However, we expect that in a three dimensional system, even with a small hopping between layers (or, in other words a large anisotropy between them),
transition temperatures would stabilize. Of course, we do not take up this task since it is computationally expensive and more importantly, we are interested
in the generic features of the problem, demonstrating the usefulness of the procedure. Results for other values of $U$ show similar behavior.
However, a notable change is the non-monotonic variation of $T_c$ as a function of $U$ which we discuss next.
(Color online) Structure factor $S({\bf 0})$ for the pairing field as a function of temperature for $ U =$ 8 and for different values of dilution $\delta$.
(Color online) Transition temperature $T_c$ as a function of dilution $\delta$ for different values of interaction strength $U$.
Fig. 2 shows the variation of $T_c$ as a function of $\delta$ for different values of $U$. The critical density of attractive centers needed for onset of
"globally phase coherent" superconductivity increases with $U$ and appears to saturate around $U \sim 6$. For small values of $U$, $T_c$ is determined
by the pairing scale at which the amplitude of the Cooper pair become non-zero. This is the BCS limit where phase fluctuations hardly play any role.
However, as $U$ increases, the pair size (or equivalently, the coherence length) comes down and onset of superconductivity is determined
by the phase coherence temperature, instead of the pairing scale. In fact, at large $U$, the pairing amplitude remains almost constant across the transition
at the sites where there is an attractive center. However, the relative phases among the sites fluctuate wildly and global coherence is established
only at a very low temperature (compared to the BCS mean field value) and is determined by phase fluctuation scale which goes as $\sim t^2/U$. This results in a nonmonotonic variation of $T_c$ with $U$ arising due to BCS-BEC crossover and is most clearly seen in Fig. 2 for $\delta=$ 1 (the clean limit). However, such a behavior sets in
roughly at $U =$ 6, establishing this crossover in an intrinsically disordered system. This also signals a clear separation of energy scales. The zero-temperature pairing gap in the quasiparticle spectrum continues to increase with $U$, though it determines the superconducting $T_c$ only in the weak-coupling limit. This also results in a nontrivial behavior of the normal phase, wherein it changes from a Fermi liquid to a gapped phase at large interaction strengths.
There is a smooth crossover
between these two regimes with an intermediate high-temperature normal phase that intervenes in the crossover region with anomalous properties,
which we will discuss in the next section.
Our method incorporates spatial fluctuations of the pairing field, both its amplitude and phase, in an unbiased way and in fact, this is a crucial ingredient
to obtain the BCS-BEC crossover. The latter arises due to site-dependent phase fluctuations, which cannot be captured in the
conventional BCS framework as was discussed earlier. Further, unique to this problem s the local charge fluctuations that can be quite large due to site dilution.
Next, we discuss the role of each of these, the charge and the amplitude and phase of pairing field fluctuations and their role in nucleating/stabilizing superconductivity
as a fundtion of temperature and dilution. In the clean limit with $\delta = 1$, all sites have uniform charge distribution and fluctuations are negligible.
However, as sites are being diluted, there is a strong tendency to have average charge density to be larger near attractive centers and this can be seen most clearly in the lowest rows
of Fig. 4 corresponding to $U =$ 8.
(Color online) Real-space configurations of the distribution of attractive centers and various auxiliary fields at $U =$ 2 at the lowest temperature ($T=$ 0.001 in units of $t$.
The three columns correspond to different dilution : $\delta =$ 0.2 (first), $\delta =$ 0.3 (second) and $\delta =$ 0.7 (third). The first row gives the distribution of attractive centers with blue circles denoting sites with $U_i = U$. The other rows depict the phase $\cos (\arg (\Delta_i))$ (second) and the amplitude $\left| \Delta_i \right|$ (third) of the pairing fields, and the charge field $n_i$ (fourth). The system size is 32 $\times$ 32.
On the contrary, the local amplitude of the pairing field $\left| \Delta_i \right|$, where $\Delta_i = \left| \Delta_i \right| e^{i \theta_i}$, is almost vanishing at every site
for large enough dilution, except in small islands where it is nonzero, grows in size as dilution decreases. This in fact, is the origin of the percolative nature of the transition
as a function of $\delta$ for a fixed $U$. There are puddles where amplitude is nonzero, but there are large intervening regions where it is vanishingly small.
There is phase coherence within a given puddle, but that cannot stabilise a global superconducting state. Beyond a percolation threshold $\delta_c$,
which happens mostly in the BCS-BEC crossover region, there is sufficient pairing amplitude at most sites since the dilution is less.
However, physics in this strong-coupling region is determined entirely by phase fluctuations. To bring out this feature more clearly, we present the same physical quantities in Fig. 4 for
$U =$ 8. The behaviour of $\left< n_i \right>$ and $\left| \Delta_i \right|$ are qualitatively different at larger values of $U$.
The site-to-site charge fluctuations increase enormously, with attractive centers having localised pair of electrons with opposite spins.
As $\delta$ varies from 0.4 to 0.8, charge-rich sites increases in number. The charge density at the charge-rich sites reaches close to 2 for systems with large $U$.
Amplitude of the order takes zero almost everywhere in the lattice for $\delta=0.40$ or smaller than that. It starts to take non-zero values from $\delta=0.50$ onwards.
However, there is a strong fluctuation of these amplitudes compared to $U=2$. Average value of this amplitude on sites with $U_i=U$ increases with $\delta$ for $\delta>0.50$
whereas $|\mathbf{\Delta}|$ does not change much on the sites with $U_i=0$.
Naturally, the picture that emerges is that, as expected, fluctuations of the phase degrees of freedom do not play a major role for small values of $U$. The transition is entirely BCS-like. Variation of $\delta$ affects the percolative nature of the transition since, the otherwise locally phase coherent islands have to overlap to give rise to a globally superconducting state. However, as can be seen from the second rows of Figs. 3 and 4, the phase of the order parameter changes dramatically
as we change $U$. At large $U$, even though the amplitudes have acquired reasonably large values at every site, their phases become uncorrelated due to
thermal fluctuations of these soft degrees of freedom. This reduces $T_c$ as $U$ increases. The nonmonotonic variation of $\delta_c$ and $T_c$ as a function of $U$ suggests that
the transition from a metallic to a superconducting ground state cannot be thought of as entirely due to percolation of amplitude puddles, as noted in an earlier work<cit.> . The contrast between the two extreme ends of weak and strong coupling is striking: There are phase-coherent patches of relatively small amplitudes
at small $U$ and leads to a percolative transition as the number of attractive centers increases; however, strong phase fluctuations suppress the $T_c$
at large $U$ even though the pairing amplitude is quite strong enough at most of the sites.
(Color online) Real-space configurations of the distribution of attractive centers and various auxiliary fields at $U =$ 8 at the lowest temperature ($T=$ 0.001 in units of $t$.
The three columns correspond to different dilution : $\delta =$ 0.4 (first), $\delta =$ 0.6 (second) and $\delta =$ 0.8 (third). The first row gives the distribution of attractive centers with blue circles denoting sites with $U_i = U$. The other rows depict the phase $\cos (\arg (\Delta_i))$ (second) and the amplitude $\left| \Delta_i \right|$ (third) of the pairing fields, and the charge field $n_i$ (fourth). The system size is 32 $\times$ 32.
§ ELECTRONIC SPECTRAL FUNCTIONS
(Color online) Single-particle density of states at the lowest temperature ($T =$ 0.001 in units of $t$) as a function of dilution for (a) $U =$ 2 and (b) $U =$ 8.
In this section we look at the spectral properties of the system, in particular, concentrating on the single particle density of states. This is expected to show a bulk
superconducting gap and coherence peaks when the system turns superconducting. Fig. 5a shows its variation as a function of $\delta$ at the lowest temperature
we have accessed in the weak-coupling regime with $U =$ 2.. There is no gap until about $\delta \sim$ 0.3 and a gap appears as $\delta$ approaches 0.4. A clear spectral gap is seen at
larger values of $\delta$, which increases with increase in $\delta$. Also visible are the coherence peaks on either side of the gap. These results match very well with those obtained from the structure factor in Fig. 1.
In fact, this gap vanishes as we increase the temperature across $T_c$ in this region of parameter space (for small $U$). However, the temperature variation of the spectral gap changes dramatically as we increase $U$. The temperature at which the gap vanishes increases very rapidly with $U$ even though as mentioned in the previous section, the $T_c$ comes down drastically. This shows a clear separation the pairing scale determined from the spectral gap and superconducting scale that determines $T_c$.
We also find that the gap in the density of states is larger at larger values of $U$, as expected. However, the effect of $\delta$ is more subtle.
While the gap decreases with $\delta$, it exists even when superconductivity is not present in the system. We show typical results for a representative value of
$U = 8$ in Fig. 5b. Even when very few sites have attractive centers, when $U$ is large enough, local pairing tendencies are stronger.
Thus centers which have large $U$ become doubly occupied and gain an energy of the order of $-U$.
In the large $U$ limit this states with energy $-U$ will be isolated from the
kinetic energy band. If the average no of particle is nearly half then one has to fill up higher
energy states above the gap. Density at state at the Fermi energy($\omega=0$) will be non-zero; the ground state of the system would be a metal as can be seen for $\delta =$ 0.2 and 0.4.
Fig. 6 shows the behavior of single particle density of states for two different dilution $\delta =$ 0.5 (Fig. 6a) and 0.8 (Fig. 6b) at two representative temperatures. As expected,
sharp coherence peaks are not seen in the spectral function. What is striking is that at larger $\delta$, the spectral gap, even though diminished in size, persists at high temperatures
(Color online) Single-particle density of states for $U =$ 6 with $\delta =$ 0.5 (a) and 0.8 (b) at different temperatures.
§ OPTICAL TRANSPORT
(Color online) The real part of the optical conductivity $\sigma_R(\omega)$ as a function of frequency $\omega$ at the lowest temperature ($T =$ 0.001 in units of $t$) for different values of site dilution and interaction strengths $U =$ 2 (upper panel) and $U =$ 8 (lower panel).
Optical conductivity is expected to be directly correlated to the spectral features discussed previously and we explore it in this section.
In the superconducting state, $\sigma(\omega)$ has two contributions; there is a zero frequency diamagnetic response that is proportional to the superfluid stiffness.
In addition, there is a $\omega$-dependent regular part arising due to various effects such as pair breaking, quasiparticle scattering etc.
We concentrate on the latter in this section.
In the superconducting ground state, in the BCS limit, the latter contributes only at frequencies larger than twice the superconducting gap. At nonzero temperatures,
there is finite contribution even inside this frequency window, but exponentially suppressed.
For small values of $U$, the above features appear to be generic and a representative behavior is shown in Fig. 7a for $U =$2. The gapped spectrum results in the vanishing of the optical conductivity for $\delta \geq$ 0.4. At small $\delta$, the system remains a metal. However, there is intrinsic disorder present due to small number of attractive centers which give rise to enhanced scattering even at low temperatures and results in a non-Drude behavior of $\sigma(\omega)$.
In the large $U$ limit optical conductivity behaves differently as a function $\delta$. Strong optical response at larger $\omega$ arises due to excitations across the SC gap, while the excitations within the kinetic band, separated from the lower band at $-U$, give rise to low frequency non-Drude-like behavior. (See Fig. 7b.)
(Color online) Two representative phase diagrams of the site-diluted attractive Hubbard model as a function of temperature $T$ and strength of the attractive interaction $U$ for (a) $\delta =$ 0.8
and for (b) $\delta =$ 0.5. SC, NM, PG, and G represent superconducting, normal metal, pseudogap phase, and gapped phases.
We summarise these findings in the next two figures. In Fig. 8, we give two representative phase diagrams of the site-diluted attractive Hubbard model as a function of
$T$ and $U$ for two specific values of $\delta =$ 0.8 (Fig. 8a) and $\delta =$ 0.5 (Fig. 8b). At small values of $U$, the system turns from a BCS-like superconducting state to a normal metal above $T_c$, which scales with the pairing gap. However, at larger $U$ even though the pairing gap continues to increase, $T_c$ reduces from the mean field value due to strong phase fluctuations. The normal state above $T_c$ has quasiparticle gap in the spectrum. However, there is an intervening region, where, a hard gap does not appear in the spectrum, but there is significant reduction of spectral weight at low frequencies. We call this the pseudogap phase.
For smaller $\delta$, SC ground state vanishes above a critical value of $U$ since there is a critical dilution $\delta_c$ that increases with $U$. Finally, Fig. 9 shows the three dimensional phase diagram of the model as a function of $T$, $\delta$, and $U$, clearly brining out the variantion of $\delta_c$ with $U$ and the nonmonotonic behavior of $T_c$ as the strength of interaction changes.
(Color online) Phase diagram of the site-diluted attractive Hubbard model as a function of the superconducting transition temperature $T_c$, attractive interaction $U$, and
the average density of attractive centers $\delta$.
§ CONCLUSIONS
In this paper, we studied random local attraction driven metal-superconductor transition in two dimensions using the random attractive Hubbard model. A real space Monte Carlo method was employed after introducing
pairing and density fields via Hubbard-Stratanovich transformation. The method is capable of capturing the physics from weak to strong coupling regimes and gives
a real space picture of the transition, both of which are crucial to the problem at hand. In particular, effect of intrinsic disorder on the BCS-BEC crossover has been studied.
The main results are as follows. As observed in previous studies, we find that there is a critical concentration of attractive centers needed for the onset of globally phase coherent superconductivity. The critical concentration increases with $U$ and appears to saturate above some value.
For small values of $U$ the transition is of percolative nature and the physics is akin to that of a BCS superconductor.
The picture that emerges is that of superconducting puddles, internally phase coherent, percolating at a critical concentration of attractive centers
to give a globally superconducting state. However, the scenario is different at larger strengths of interaction. In this regime, the zero-temperature pairing gap continues to increase with $U$,
local pairing tendencies persist even above the transition temperature, but a dominant role is played by strong spatial phase fluctuations, resulting in a BCS-BEC crossover. Transition temperatures are suppressed even though there is a robust spectral gap due to local pairing and $T_c$ shows a nonmonotonic behavior as $U$ is varied. The high temperature normal state
transforms from metal to a gapped phase as $U$ increases and has pseudogap features in the intermediate region due to the existence of short-range order of the amplitude of the order parameter. Spectral functions and transport properties corroborate these findings.
Next, we comment on a few shortcomings of our study. First, we have allowed for numerically exact treatment of thermal fluctuations of the order parameter (and also, of the charge field)
but neglected their dynamics. Thus quantum fluctuations of the order parameter are not taken into account. This may affect the low temperature properties,
especially near the critical concentration of impurities. Second, while the method works very well away from half filling,
it is not expected to capture the charge density wave (CDW) instability at half filling that competes with superconductivity. One expects that $T_c$ will reduce
to zero at larger values of $U$ and the system will transform to a CDW phase. The present method allows for large charge fluctuations at sites as $U_i$ varies. This calls for caution. In real systems, there will be long-range Coulomb interaction, which would not allow such charge fluctuations as it costs Coulomb energy. We wish to address this problem in future. In real two-dimensional systems there cannot be finite temperature phase transitions, in contrast to what is seen here, since thermal fluctuations prohibit any order at nonzero temperatures.
However, $T_c$ shown here should be thought of as a sort of crossover scale below
which correlation length increases rapidly. If so, even a weak coupling to a third dimension will stabilize the SC phase at nonzero temperatures.
The present work could be extended in many ways, some of which are currently underway.
One could include the dynamics of order parameter field in a semiclassical way after obtaining the equilibrium configurations.
This would allow us to extract some interesting physical properties in the normal phase having preformed pairs at moderate-to-large coupling, reminiscent of the Nernst effect in cuprates,
and its persistence in the BEC regime. One could study the competition between disorder arising due to site dilution with those of alternate origin, for example, the presence of magnetic impurities. The latter is expected to have a detrimental effect on the BCS state. The effect of site dilution in an insulating host is an interesting problem where an insulating gap and SC gap compete with each other <cit.>; the application of the present method shows a further suppression of $T_c$
as one moves towards the clean limit ($\delta \sim$ 1) and a transition to a charge-modulated insulating phase<cit.>. A further extension would be to study the role of site dilution in SC systems with order parameter symmetry different from the $s$-wave considered here as well as in imbalanced lattice fermion system<cit.> that could support breached pair and Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) states.
§ ACKNOWLEDGEMENTS
We acknowledge the use of high performance computing clusters at HRI.
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1511.00229
|
DESY 15-198
arXiv:1511.00229 [hep-ph]
DO–TH 15/15
October 2015
A Kinematic Condition on Intrinsic Charm
Johannes Blümlein
Deutsches Elektronen–Synchrotron, DESY,
Platanenallee 6, D-15738 Zeuthen, Germany
We derive a kinematic condition on the resolution of intrinsic charm and discuss phenomenological consequences.
Intrinsic charm in nucleons has been proposed as a phenomenon, which can be described in the light-cone
wave function formalism <cit.> using old fashioned perturbation theory <cit.>.
It is characterized by a Fock state
\begin{eqnarray}
|q_1\, q_2\, q_3\, Q_4\, Q_5 \rangle,
\label{eq:STATE}
\end{eqnarray}
with massless quarks $q_i$ and heavy quarks $Q_j$ of mass $M_Q$.[One-loop radiative corrections
calculated in <cit.>]. The emergence of this state can be viewed as a definite
quantum fluctuation in front of a general hadronic background, which can be resolved in deep-inelastic
lepton-nucleon scattering.
Extrinsic heavy flavor contributions <cit.>, on the other hand, are due to factorized single
massless parton induced processes, exciting the heavy quark contributions. For neutral current interactions the process
results from vector boson-gluon fusion <cit.> and appears in first order in the strong coupling
$\alpha_s$ at the quantum level.
Both processes are distinct and of very different nature. As has been shown in Ref. <cit.>
the intrinsic charm contributions are situated at larger values of $x$, while major
contributions of the extrinsic charm appear at low values of $x$. While the intrinsic charm
contribution appears in the scaling limit already, extrinsic charm contributes on the quantum level only.
In the following we derive the condition under which intrinsic charm is unambiguously visible in deep-inelastic scattering.
We follow Drell and Yan, Ref. <cit.>, and compare the lifetime, $\tau_{\rm life}$, of the intrinsic
charm state
with the interaction time in the deep-inelastic process, $\tau_{\rm int}$, demanding
\begin{eqnarray}
\frac{\tau_{\rm life}}
{\tau_{\rm int}} \gg 1
\label{eq:COND}
\end{eqnarray}
as a necessary criterion for the observation of the phenomenon. Eq. (<ref>) delivered a clear condition on the applicability
of the (massless) parton model singling out the corresponding ranges in $x$ and $Q^2$. Here the major requests are
that the virtuality $Q^2$ of the process is much larger than any transverse momentum squared in the hadronic wave-function,
$Q^2 \gg k_{\perp,i}^2$, and the Bjorken variable $x$
shall neither get close to 1 nor take too small values, <cit.>.
Usually, in the excluded regions other contributions, like higher twist terms are present and/or there is a need of novel
small-$x$ resummations, which are both of comparable or even of larger size than the terms computed. In the following
we will apply Eq. (<ref>) to the case of the state (<ref>).
In an infinite momentum frame we may express the momentum transfer by the electro-weak boson probing the nucleon, $q$, as
follows <cit.>
\begin{eqnarray}
q = (q_0; q_3, q_\perp),~~q_0 = \frac{2 m_p \nu +q^2}{4 P},~~q_3 = -\frac{2 m_p \nu -q^2}{4 P},
\end{eqnarray}
where $q^2 = -Q^2$, $m_p$ the proton mass, $\nu$ the energy transfer to the nucleon in the proton rest
frame, and $P$ is the large (`infinite') momentum.
The interaction time $\tau_{\rm int}$ is given by
\begin{eqnarray}
\tau_{\rm int} = \frac{1}{q_0} = \frac{4 P}{2 m_p \nu +q^2} = \frac{4 P x}{Q^2(1-x)}~.
\label{eq:INT}
\end{eqnarray}
Here $x$ denotes the momentum fraction of the struck quark. Likewise, we obtain for the lifetime
of the intrinsic charm state
\begin{eqnarray}
\tau_{\rm life} = \frac{1}{\sum_i E_i - E} = \left. \frac{2 P}{\left(\sum_{i=1}^5 \frac{M_i^2
+k_{\perp,i}^2}{x_i}\right) - m_p^2} \right|_{\sum_j x_j =1},
\end{eqnarray}
with $E_i = \sqrt{x_i^2 P^2 + M_i^2 +k_{\perp,i}^2}$ the energies of the partons in the state and $E$ the total
energy, applying the infinite momentum representation, consistently neglecting sub-leading terms $\sim 1/P$
in the large momentum. $M_i$ denotes the mass of the $i$th quark, $k_{\perp,i}$ its transverse
momentum, and $x_i$ its momentum fraction. Deriving $\tau_{\rm life}$ for intrinsic charm, we consider three
massless valence quarks and the heavy quark-antiquark pair in the Fock state. We set the masses of the three
light valence quarks to zero and neglect the effect of transverse momenta, as in the
derivation of Eq. (8) <cit.>, but retain the term $m_p^2/M_Q^2$ here. One obtains
\begin{eqnarray}
\tau_{\rm life}(x) &=& \left. \frac{2 P}{M_Q^2} \int_0^{1-x_5} dx_4 \int_0^{1-x_4-x_5} dx_3
\frac{1-x_3-x_4-x_5}{\frac{1}{x_4} + \frac{1}{x_5} - \frac{m_p^2}{M_Q^2}}\right|_{x_5 = x}
\nonumber\\
&=& \frac{P x}{6 M_Q^2 (1-c x)^4}
\Biggl\{
(1-x) (1-c x)
[ 2 +
x [5 - x - c (1-x) [4 + x (5 - 2 c (1-x))]]
\nonumber\\ &&
+6 x (1 - c x (1-x))^2 \ln
\left[\frac{x}{1- c x (1-x)}\right]\Biggr\},
\label{eq:TLIV}
\end{eqnarray}
with $c = m_p^2/M_Q^2$. The integrals in (<ref>) are the same as used to derive the probability distribution
$P(x)$ in <cit.> and Eq. (<ref>), however, the energy denominator appears in the first power.
[]Left panel: Normalized intrinsic charm distribution with finite $m_p$; Right panel: ratio of the probability distribution
for intrinsic charm including the effect of the proton mass to the case $m_p/M_Q \rightarrow 0$. The parameter $c$ is given
by $c \simeq 0.348$ and $M_Q = 1.59~\GeV$ in the pole mass scheme <cit.>.
Lower values of $m_c \sim
1.3$ GeV
used e.g. in <cit.> at NLO are fully compatible with the NNLO value applied in the present study,
cf. <cit.>.
One may estimate also a lifetime for extrinsic $c\bar{c}$-production, if viewed as Fock state. Due to
the factorized production, one considers the state $|c \bar{c} X\rangle$, with $X$ the hadronic remainder of
momentum fraction $x_1$, which yields
\begin{eqnarray}
\tau_{\rm life}^{\rm ext}(x) &=& \left. \frac{2 P}{M_Q^2} \int_0^1 dx_4 \int_0^1 dx_1 \delta(1 - x_1 - x_4 -x_5)
\frac{1}{\frac{1}{x_4} +\frac{1}{x_5} - c} \right|_{x_5 = x}
\nonumber\\
&=& \frac{2 P x}{M_Q^2 (1-c x)^2} \left\{(1-x)(1 - cx) + x \ln\left[\frac{x}{1- c x(1-x)}\right]\right\}~.
\label{eq:TLIVex}
\end{eqnarray}
The lowest order probability distribution for intrinsic charm, accounting for the nucleon mass effect, is
given by
\begin{eqnarray}
P(x) &=& \left. N(c) \int_0^{1-x_5} dx_4 \int_0^{1-x_4-x_5} dx_3 \frac{1 - x_3 - x_4 -
\frac{1}{x_5} - c\right)^2} \right|_{x_5 = x}
\\
&=& \frac{N(c) x^2}{6 (1-c x)^5}
\Biggl\{
(1-x) (1-c x) [1 + x[10 + x -c (1-x) (x (10-c (1-x))+2)]]
\nonumber\\ &&
+ 6 x [1+ x (1-c (1-x))] [1-c (1-x) x]
[\ln(x)-\ln[1-c (1-x) x]]\Biggr\},
\label{eq:PP}
\end{eqnarray}
with $N(c)$ determined such that $\int_0^1 dx P(x) = N_{\rm IC}$, the integral fraction of
intrinsic charm. Here we retained the effect of the proton mass, which was neglected in
<cit.>, and illustrate the distribution in Figure 1a. One obtains a modification of the
intrinsic charm distribution due to the finite nucleon mass effect of up to 10%, as shown in
see Figure 1b. Setting $c \rightarrow 0$ leads to the previous result <cit.>
\begin{eqnarray}
P(x) &=& 600 N_{\rm IC} x^2 \Biggl[(1-x)(x^2+10 x +1) +6 x (x+1) \ln(x) \Biggr]~.
\end{eqnarray}
The ratio $\rho(x)= \tau_{\rm life}/\tau_{\rm int}$ is Lorentz-invariant and has to be larger than a suitable
bound of $O(5 ... 10)$. The size of this value is fixed using standard requests applied also for the parameter setting
in experimental pulse resolution techniques e.g. in particle detectors. Here $\rho = 5$ would refer to a failure rate of
20% and $\rho = 10$ of 10%[I would like to thank Dr. J. Bernhard from the Compass experiment for a corresponding
Full line: lower boundary in $Q^2/\GeV^2$ for which
$\rho(x) \geq 5$ as a function of
$x$. Long dashed line: lower boundary for extrinsic charm production
resulting from Eq. (<ref>) also for $\rho(x) \geq 5$;
Short dashed line: highest $Q^2$-bin of the EMC experiment <cit.>.
Using this condition we may determine the allowed $Q^2$-range as a function of
$x$ in which the quantum fluctuation leading to intrinsic charm can be unambiguously resolved by
deep-inelastic scattering. Eqs. (<ref>,<ref>) lead to the function
\begin{eqnarray}
\rho(x) \equiv \frac{\tau_{\rm life}(x)}{\tau_{\rm int}(x)} \geq \frac{Q^2}{12 M_Q^2} \simeq
\frac{Q^2}{30.34~\GeV^2}.
\end{eqnarray}
The function $\rho(x)$ rises with growing values of $x$, i.e. a minimal bound of $Q^2 > 151~\GeV^2$ is obtained,
demanding the
ratio to be $\rho(x) \geq 5$. We show the corresponding $x$-dependence in Figure 2. We also show the boundary implied
for extrinsic charm production.
Let us consider the kinematics of the EMC experiment at CERN <cit.>, which probably was the first measuring charm
final states of a larger amount in deep-inelastic scattering
<cit.>. These data have frequently been analyzed also searching for intrinsic charm. The highest $Q^2$ bin is centered
at $Q^2 \simeq 170~\GeV^2$. The kinematic range allowed for a clear intrinsic charm signal demanding $\rho(x) \geq 5$ is
obtained as $x \lsim 0.01$, far below the peak-region at $x \simeq 0.22$ of the predicted distribution. On the other
hand, the bound resulting for extrinsic charm production, cf. Eq. (<ref>), covers a wider range, also of
the kinematic region probed by the EMC experiment. Note that, furthermore, the accessible range in $x$ is
strongly correlated to the
probed region in $Q^2$ in deep-inelastic scattering experiments. This has to be taken into account interpreting low energy data as
those of the EMC experiment in terms of intrinsic charm effects. Several phenomenological analyses have been carried out to
search for intrinsic charm, cf. e.g. Refs. <cit.>. Other analyses came to very similar conclusions of a possible
integral fraction of $N_{\rm IC}$ in the range of up to $O(1..3\%)$. In all these analyses the life-time
constraint (<ref>) has not been considered.
Discoveries need clean conditions. The $Q^2$ bound illustrated by Figure 2 points to a much more fortunate situation to search
for intrinsic charm effects opening up at high energy colliders if compared to fixed target experiments, such as at HERA or
within future projects like the EIC <cit.> and LHeC <cit.>, also operating at high luminosity.
Condition (<ref>) is more easily fulfilled there because of the much wider kinematic range. As a consequence,
intrinsic charm can be searched for in a dedicated way only at high energies.
I would like to thank H. Fritzsch and G. Branco for organizing a nice conference on High Energy
Physics in the beautiful Algarve, where this note has been worked out. Conversations with S. Brodsky
and M. Klein are gratefully acknowledged. This work was supported in part by the European Commission
through contract PITN-GA-2012-316704 (HIGGSTOOLS).
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|
1511.00276
|
Green's function asymptotics]Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds
Kha]Minh Kha
M.K., Department of Mathematics, Texas A&M University,
College Station, TX 77843-3368, USA
[2010]35P99, 35J08, 35J10, 35J15, 58J05, 58J37, 58J50
The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for “generic" periodic second-order, self-adjoint, elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of $\mathbb{R}^n$ only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold.
§ INTRODUCTION
The behavior at infinity of the Green's function of the Laplacian $-\Delta$ on an Euclidean space below and at the boundary of the spectrum is well known. The main term of the asymptotics for the Green's function of any bounded below periodic second-order elliptic operator below and at the bottom of the spectrum was found in <cit.> (see also <cit.> for discrete setting).
For such operators, the band-gap structure of their spectra is known (e.g., <cit.>), and thus, spectral gaps may exist. Thus, it is interesting to derive the behavior of the Green's functions inside and at the edges of these gaps.
Recently, the corresponding results for a “generic" periodic elliptic operator in $\mathbb{R}^n$
were established in <cit.>.
Meanwhile, many classical properties of solutions of periodic Schrödinger operators on Euclidean spaces were generalized successfully to solutions of periodic Schrödinger operators on coverings of compact manifolds (see e.g., <cit.>).
Hence, a question arises of whether one can obtain
analogs of the results of <cit.> as well.
The main theorems <ref> and <ref> of this article provide such results for periodic operators on an abelian covering of a compact Riemannian manifold. The results are in line with Gromov's idea that the large scale geometry of a co-compact normal covering is captured mostly by its deck transformation group (see e.g., <cit.>). For instance, the dimension of the covering manifold does not enter explicitly to the asymptotics. Rather, the torsion-free rank $d$ of the abelian deck transformation group influences these asymptotics significantly.
One can find a similar effect in various
results involving analysis on Riemannian co-compact normal coverings such as
the long time asymptotic behaviors of the heat kernel on a noncompact abelian Riemannian covering <cit.>, and the analogs of Liouville's theorem <cit.> (see also <cit.> for an excellent survey on analysis on co-compact coverings).
We discuss now the main thrust of this paper.
Let $X$ be a noncompact Riemannian manifold that is a normal abelian covering of a compact Riemannian manifold $M$ with the deck transformation group $G$. For any function $u$ on $X$ and any $g \in G$, we denote by $u^g$ the “shifted" function
$$u^g(x)=u(g\cdot x),$$
for any $x \in X$.
Consider a bounded below second-order, symmetric elliptic operator $L$ on the manifold $X$ with smooth coefficients. We assume that $L$ is a periodic operator on $X$, i.e., the following invariance condition holds:
for any $g\in G$ and $u \in C^{\infty}_c(X)$. The operator $L$, with the Sobolev space $H^2(X)$ as its domain, is an unbounded self-adjoint operator in $L^2(X)$.
The following result for such operators is well-known (see e.g., <cit.>):
The spectrum of the above operator $L$ in $L^{2}(X)$ has a band-gap structure, i.e., it is the union of a sequence of closed bounded intervals $[\alpha_j, \beta_j] \subset \mathbb{R}$ $(j=1,2,...)$ (bands or stability zones of the operator $L$):
\begin{equation}
\label{band-gap}
\sigma(L)=\bigcup_{j=1}^{\infty}[\alpha_j, \beta_j],
\end{equation}
such that $\alpha_{j} \leq \alpha_{j+1}$, $\beta_{j} \leq \beta_{j+1}$ and $\lim_{j \rightarrow \infty}\alpha_j=\infty$.
The bands can overlap when the dimension of the covering $X$ is greater than $1$, but they can leave open intervals in between, called spectral gaps.
A finite spectral gap is of the form $(\beta_{j}, \alpha_{j+1})$ for some $j \in \mathbb{N}$ such that $\alpha_{j+1}>\beta_{j}$, and the semifinite spectral gap is the open interval $(-\infty, \alpha_{1})$, which contains all real numbers below the bottom of the spectrum of $L$.
In this text, we study Green's function asymptotics for the operator $L$ at an energy level $\lambda \in \mathbb{R}$,
such that $\lambda$ belongs to the union of all closures of finite spectral gaps[All of the results still hold for the case when $\lambda$ does not exceed the bottom of the spectrum, i.e. for the semi-infinite gap.].
We divide this into two cases:
* Case I: (Spectral gap interior) The level $\lambda$ is in a finite spectral gap $(\beta_j, \alpha_{j+1})$ such that $\lambda$ is close either to the spectral edge $\beta_j$ or to the spectral edge $\alpha_{j+1}$.
* Case II: (Spectral edge case) The level $\lambda$ coincides with one of the spectral edges of some finite spectral gap, i.e., $\lambda=\alpha_{j+1}$ (lower edge) or $\lambda=\beta_{j}$ (upper edge) for some $j \in \mathbb{N}$.
In Case I, the Green's function $G_{\lambda}(x,y)$ is the Schwartz kernel of the resolvent operator $R_{\lambda,L}:=(L-\lambda)^{-1}$, while in Case II, it is the Schwartz kernel of the weak limit of resolvent operators $R_{\lambda,L}:=(L-\lambda \pm \varepsilon)^{-1}$ as $\varepsilon \rightarrow 0$ (the sign $\pm$ depends on whether $\lambda$ is an upper or a lower spectral edge).
Note that in the flat case $X=\mathbb{R}^d$, Green's function asymptotics of periodic elliptic operators were obtained in <cit.> for Case I ($d \geq 2$), and in <cit.> for Case II ($d \geq 3$). As in <cit.>, we will deduce all asymptotics from an assumed “generic" spectral edge behavior of the dispersion relation of the operator $L$, which we will briefly review in Section 2.
The organization of the paper is as follows. In Subsection 2.1, we will review some general notions and results about group actions on abelian coverings. Then in Subsection 2.2, we introduce additive and multiplicative functions defined on an abelian covering, which will be needed for writting down the main formulae of Green's function asymptotics. Subsection 2.3 contains not only a brief introduction to periodic elliptic operators on abelian coverings, but also the necessary notations and assumptions for formulating the asymptotics. The main results of this paper are stated in Section 3. In Section 4, the Floquet-Bloch theory is recalled and the problem is reduced to studying a scalar integral. Some auxiliary statements that appeared in <cit.> are collected in Section 5, and the final proofs of the main results are provided in Section 6. Section 7 provides the proofs of some technical claims that were postponed from previous sections. Section 8 discusses analogous results for Green's functions of nonsymmetric periodic elliptic operators of second-order on abelian coverings below and at the generalized principal eigenvalues, and then describes the corresponding Martin compactifications and the Martin integral representations for such operators.
The last sections contain some concluding remarks and acknowledgements.
§ NOTIONS AND PRELIMINARY RESULTS
§.§ Group actions and abelian coverings
Let $X$ be a noncompact smooth Riemannian manifold of dimension $n$ equipped with an isometric, properly discontinuous, free, and co-compact action of an finitely generated abelian discrete group $G$. The action of an element $g \in G$ on $x \in X$ is denoted by $g\cdot x$. Due to our conditions, the orbit space $M=X/G$ is a compact smooth Riemannian manifold of dimension $n$ when equipped with the metric pushed down from $X$.
We assume that $X$ and $M$ are connected.
Thus, we are dealing with a normal abelian covering of a compact manifold
$$X \xrightarrow{\pi} M(=X/G),$$
where $G$ is the deck group of the covering $\pi$.
Let $d_X(\cdot, \cdot)$ be the distance metric on the Riemannian manifold $X$. It is known that $X$ is a complete Riemannian manifold since it is a Riemannian covering of a compact Riemannian manifold $M$ (see e.g., <cit.>). Thus, for any two points $p$ and $q$ in $X$, $d_X(p,q)$ is the length of a length minimizing geodesic connecting these two points.
Let $S$ be any finite generating set of the deck group $G$. We define the word length $|g|_S$ of $g \in G$ to be the number of generators in the shortest word representing $g$ as a product of elements in $S$:
$$|g|_S=\min\{n \in \mathbb{N} \mid g=s_1 \dots s_n, s_i \in S\cup S^{-1}\}.$$
The word metric $d_S$ on G with respect to S is the metric on $G$ defined by the formula
for any $g, h \in G$.
We introduce a notion in geometric group theory due to Gromov that we will need here (see e.g., <cit.>).
Let $Y, Z$ be metric spaces. A map $f: Y \rightarrow Z$ is called a quasi-isometry, if the following conditions are satisfied:
* There are constants $C_1, C_2>0$ such that
$$C_1^{-1}d_Y(x,y)-C_2 \leq d_{Z}(f(x),f(y)) \leq C_1 d_Y(x,y)+C_2$$
for all $x,y \in Y$.
* The image $f(Y)$ is a net in $Z$, i.e., there is some constant $C>0$ so that if $z\in Z$, then there exists $y \in Y$ such that $d_Z(f(y),z)<C$.
We remark that given any two finite generating sets $S_1$ and $S_2$ of $G$, the two word metrics $d_{S_1}$ and $d_{S_2}$ on $G$ are equivalent (see e.g., <cit.>).
The next result, which directly follows from the Švarc-Milnor lemma (see e.g., <cit.>, <cit.>), establishes a quasi-isometry between the word metric $d_S(\cdot, \cdot)$ of the deck group $G$ and the distance metric $d_X(\cdot, \cdot)$ of the Riemannian covering $X$ of a closed connected Riemannian manifold $M$.
For any $x \in X$, the map
\begin{equation*}
\begin{split}
(G, d_S) &\rightarrow (X, d_X) \\
g &\mapsto g \cdot x
\end{split}
\end{equation*}
given by the action of the deck transformation group $G$ on $X$ is a quasi-isometry.
Since $G$ is a finitely generated abelian group, its torsion free subgroup is a free abelian subgroup $\mathbb{Z}^d$ of finite index. Hence, we obtain a normal $\mathbb{Z}^d$-covering
$$X \rightarrow M'(=X/\mathbb{Z}^d),$$
and a normal covering of $M$ with a finite number of sheets
$$M' \rightarrow M.$$
Then $M'$ is still a compact Riemannian manifold.
By switching to the normal subcovering $X \xrightarrow{\mathbb{Z}^d} M'$, we assume from now on that the deck group $G$ is $\mathbb{Z}^d$ and substitute $M'$ for $M$. This will not reduce generality of our results [The same reduction holds for any finitely generated virtually abelian deck group $G$.].
* Hereafter, we choose the symmetric set $\{-1,1\}^d$ to be the generating set $S$ of $\bZ^d$. Then the function $z=(z_1, \dots, z_d) \mapsto \sum_{j=1}^d |z_j|$ is the word length function $|\cdot|_S$ on $\bZ^d$ associated with $S$.
* For a general Riemannian manifold $Y$, we denote by $\mu_Y$ the Riemannian measure of $Y$. We use the notation $L^2(Y)$ for the Lebesgue function space $L^2(Y, \mu_Y)$. Also, the notation $L^2_{comp}(Y)$ stands for the subspace of $L^2(Y)$ consisting of compactly supported functions.
It is worth mentioning that in our case, the Riemannian measure $\mu_X$ is the lifting of the Riemannian measure $\mu_M$ to $X$. Thus, $\mu_X$ is a $G$-invariant Riemannian measure on $X$.
* We recall that a fundamental domain $F(M)$ for $M$ in $X$ (with respect to the action of $G$) is an open subset of $X$ such that
for any $g \neq e$, $\displaystyle F(M) \cap g\cdot F(M)=\emptyset$ and the subset
$$X\setminus\bigcup_{g \in G} g\cdot F(M)$$
has measure zero. One can refer to <cit.> for constructions of such fundamental domains. Henceforth, we use the notation $F(M)$ to stand for a fixed fundamental domain for $M$ in $X$.
The closure of $F(M)$ contains at least one point in $X$ from every orbit of $G$, i.e.,
\begin{equation}
\label{closure_F(M)}
X=\bigcup_{g \in G} g\cdot \overline{F(M)}.
\end{equation}
Thus, if $F: X \rightarrow \mathbb{R}$ is the lifting of an integrable function $f: M \rightarrow \mathbb{R}$ to $X$, then
\begin{equation}
\label{integralF(M)}
\int_{M} f(x)d\mu_M(x)=\int_{\overline{F(M)}} F(x)d\mu_X(x).
\end{equation}
§.§ Additive and multiplicative functions on abelian coverings
To formulate our main results in Section 3, we need to introduce an analog of exponential type functions on the noncompact covering $X$.
We begin with a notion of additive and multiplicative functions on $X$ (see <cit.>).
* A real smooth function $u$ on $X$ is said to be additive if there is a homomorphism $\alpha: G \rightarrow \mathbb{R}$ such that
\begin{equation*}
u(g\cdot x)=u(x)+\alpha(g), \quad \mbox{for all} \quad (g,x) \in G \times X.
\end{equation*}
* A real smooth function $v$ on $X$ is said to be multiplicative if there is a homomorphism $\beta$ from $G$ to the multiplicative group $\mathbb{R}\setminus \{0\}$ such that
\begin{equation*}
v(g\cdot x)=\beta(g) v(x), \quad \mbox{for all} \quad (g,x) \in G \times X.
\end{equation*}
* Let $m \in \mathbb{N}$. A function $h$ (resp. $H$) that maps $X$ to $\mathbb{R}^m$ is called a vector-valued additive (resp. multiplicative) function on $X$ if every component of $h$ (resp. $H$) is also additive (resp. multiplicative) on $X$.
Following <cit.>, we can define explicitly some additive and multiplicative functions for which the group homomorphisms $\alpha$, $\beta$ appearing in Definition <ref> are trivial.
Let $f$ be a nonnegative function in $C_{c}^{\infty}(X)$ such that $f$ is strictly positive on $\overline{F(M)}$. For any $j=1, \dots, d$, we define the following function
\begin{equation*}
\label{exp_function}
H_j(x)=\sum_{g \in \bZ^d}\exp{(-g_j)}f(g\cdot x).
\end{equation*}
We also put $H(x):=(H_1(x),\dots, H_d(x))$.
Then $H_j$ is a positive function satisfying the multiplicative property $H_j(g\cdot x)=\exp{(g_j)}H_j(x)$, for any $g=(g_1, \dots, g_d) \in \bZ^d$. The multiplicative function $H$ plays a similar role to the one played by the exponential function $e^{x}$ on the Euclidean space $\mathbb{R}^d$.
By taking logarithms, we obtain an additive function on $X$, which leads to the next definition.
We introduce the following smooth $\mathbb{R}^d$-valued function on $X$:
$$h(x):=(\log{H_1(x)}, \cdots, \log{H_d(x)}).$$
Then $h=(h_1, \dots, h_d)$ with $h_j(x)=\log{H_j(x)}$. Thus, $h$ satisfies the following additivity:
\begin{equation}
\label{additivity}
h(g \cdot x)=h(x)+g, \quad \mbox{for all} \quad (g,x) \in G \times X.
\end{equation}
Here we use the natural embedding $G=\bZ^d \subset \mathbb{R}^d$.
Clearly, the definitions of functions $H$ and $h$ depend on the choice of the function $f$ and the fundamental domain $F(M)$. So, there is no canonical choice for constructing additive and multiplicative functions. Nevertheless, a more invariant approach to defining additive and multiplicative functions on Riemannian co-compact coverings can be found in <cit.>.
The following important comparison between the Riemannian metric and the distance from the additive function $h$ in Definition <ref> will be needed later.
There are some positive constants $R_h$ (depending on $h$) and $C>1$ such that
whenever $d_X(x,y)\geq R_h$, we have
$$C^{-1}\cdot d_X(x,y) \leq |h(x)-h(y)| \leq C\cdot d_X(x,y).$$
Here $|\cdot|$ is the Euclidean distance on $\mathbb{R}^d$, and the constant $C$ is independent of the choice of $h$.
As a consequence, the pseudo-distance $d_h(x,y):=|h(x)-h(y)| \rightarrow \infty$ if and only if $d_X(x,y) \rightarrow \infty$.
The proof of this statement is given in Section 7.
For any additive function $h$ satisfying additivity, $\mathcal{A}_h$ is the set consisting of unit vectors $s \in \mathbb{S}^{d-1}$ such that there exist two points $x$ and $y$ satisfying $d_X(x,y)>R_h$ and
The set $\mathcal{A}_h$ is called the admissible set of the additive function $h$, and its elements are admissible directions of $h$.
For the proof of the following proposition, one can see in Section 7.
For any additive function $h$ on $X$, one has
\begin{equation}
\label{rational}
\mathbb{Q}^{d}\cap \mathbb{S}^{d-1}=\{ g/|g| \mid g \in \bZ^d\setminus \{0\}\} \subset \mathcal{A}_h.
\end{equation}
Hence, the admissible set $\mathcal{A}_h$ of $h$ is dense in the sphere $\mathbb{S}^{d-1}$.
In particular, when $d=2$, $\mathcal{A}_h$ is the whole unit circle $\mathbb{S}^1$.
When the dimension $n$ of $X$ is less than $(d-1)/2$ (e.g., if $d>5$ and $X$ is the standard two dimensional jungle gym $JG^2$ in $\mathbb{R}^d$, see <cit.>), the $(d-1)$-dimensional Lebesgue measure of the admissible set $\mathcal{A}_h$ of any additive function $h$ on $X$ is zero. To see this, we first denote by $X_h$ the $2n$-dimensional smooth manifold $\{(x,y) \in X \times X \mid d_X(x,y)>R_h\}$, and then consider the smooth mapping:
\begin{equation*}
\begin{split}
\Psi: X_h &\rightarrow \mathbb{S}^{d-1}
\\ (x,y) &\mapsto \frac{h(x)-h(y)}{|h(x)-h(y)|}.
\end{split}
\end{equation*}
Then $\mathcal{A}_h$ is the range of $\Psi$. Since $\dim{X_h}<\dim{\mathbb{S}^{d-1}}$, every point in the range of $\Psi$ is critical and thus, $\mathcal{A}_h$ has measure zero by Sard's theorem.
* Here is a family of non-trivial examples of additive functions in the flat case, i.e., when the covering space $X$ is $\mathbb{R}^d$ and the base is the $d$-dimensional torus $\mathbb{T}^d$.
Let $d \geq 1$ and $\varphi$ be a real smooth function in $\mathbb{R}^d$ such that $\varphi$ is $\bZ^d$-periodic.
It is shown in <cit.> that there exists a unique map $F_{\varphi}=((F_{\varphi})_1, \dots, (F_{\varphi})_d): \mathbb{R}^d \rightarrow \mathbb{R}^d$ satisfying $F_{\varphi}(0)=0$, the additive condition additivity, i.e.,
$F_{\varphi}(x+n)=F_{\varphi}(x)+n$ for any $(x,n) \in \mathbb{R}^d \times \bZ^d$, and the equation
$$\Delta (F_{\varphi})_i=\nabla \varphi \cdot \nabla (F_{\varphi})_i,$$
for any $1 \leq i \leq d$. Note that $F_{\varphi}$ is just the identity mapping in the trivial case when $\varphi=0$. Moreover, it is also known <cit.> that when $d=2$, $F_{\varphi}$ is a diffeomorphism of $\mathbb{R}^d$ onto itself. In particular, for any $\bZ^2$-periodic function $\varphi$,
$|F_{\varphi}(x)-F_{\varphi}(y)| \geq C_{\varphi}|x-y|$ for any $x,y \in \mathbb{R}^2$ for some $C_{\varphi}>0$.
However, when $d\geq 3$, $F_{\varphi}$ may admit a critical point for some $\bZ^d$-periodic function $\varphi$.
* Let $X \xrightarrow{p} M$, $Y \xrightarrow{q} N$ be normal $\bZ^{d_1}$ and $\bZ^{d_2}$ coverings of compact Riemannian manifolds $M$ and $N$ respectively. Then $X \times Y \xrightarrow{p \times q} M \times N$ is also a normal $\bZ^{d_1+d_2}$ covering of $M \times N$. Consider any $\mathbb{R}^{d_1}$-valued function $h_1$ (resp. $\mathbb{R}^{d_2}$-valued function $h_2$) defined on $X$ (resp. $Y$). Let us denote by $h_1 \oplus h_2$ the following $\mathbb{R}^{d_1+d_2}$-valued function on $X \times Y$:
$$(h_1 \oplus h_2) (x,y)=(h_1(x), h_2(y)), \quad (x,y) \in X \times Y.$$
Then it is clear that $h_1 \oplus h_2$ is additive (resp. multiplicative) on $X \times Y$ if and only if both functions $h_1$ and $h_2$ are additive (resp. multiplicative). Moreover, $\mathcal{A}_{h_1 \oplus h_2} \subseteq \{\left(a_1\cdot\mathcal{A}_{h_1}, a_2 \cdot \mathcal{A}_{h_2}\right) \mid 0<a_1, a_2<1 \hspace{4pt} \mbox{and} \hspace{4pt} a_1^2+a_2^2=1\}$.
§.§ Some notions and assumptions
Let $L$ be a bounded from below and symmetric second-order elliptic[The ellipticity is understood in the sense of the nonvanishing of the principal symbol of the operator $L$ on the cotangent bundle of the underlying manifold (with the zero section removed).] operator on $X$ with smooth[The smoothness condition is assumed for avoiding lengthy technicalities and it can be relaxed.] coefficients such that the operator commutes with the action of $G$.
An operator that commutes with the action of $G$ is called a $G$-periodic (or sometimes periodic) operator for brevity.
Notice that on a Riemannian co-compact covering, any $G$-periodic elliptic operator with smooth coefficients is uniformly elliptic in the sense that
\begin{equation*}
\label{uniform_ell}
|L_0^{-1}(x,\xi)|\leq C|\xi|^{-2}, \quad (x,\xi) \in T^*X, \xi \neq 0.
\end{equation*}
Here $|\xi|$ is the Riemannian length of $(x,\xi)$ and $L_0(x,\xi)$ is the principal symbol of $L$.
The periodic operator $L$ can be pushed down to an elliptic operator $L_M$ on $M$ and thus, $L$ is the lifting of an elliptic operator $L_M$ to $X$. By a slight abuse of notation, we will use the same notation $L$ for both elliptic operators acting on $X$ and $M$.
Under these assumptions on $L$, the symmetric operator $L$ with the domain $C^{\infty}_{c}(X)$ is essentially self-adjoint in $L^2(X)$, i.e., the minimal operator $L_{min}$ coincides with the maximal operator $L_{max}$ (see e.g., <cit.> for notation $L_{min}$ and $L_{max}$).
This fact can be found in <cit.>, for instance [In <cit.>, Atiyah proved for symmetric elliptic operators acting on Hermitian vector bundles over any general co-compact covering manifold (not necessary to be a Riemannian covering). Later, in <cit.>, Brunning and Sunada extended Atiyah's arguments to the case including compact quotient space $X/G$ with singularities.
Hence, there exists a unique self-adjoint extension in the Hilbert space $L^2(X)$ of $L$, which we denote also by $L$. Since $L$ is a uniformly elliptic operator on the manifold $X$ of bounded geometry, its domain is the Sobolev space $H^2(X)$ <cit.>, and henceforward, we always work with this self-adjoint operator $L$.
* The dual (or reciprocal) lattice is $2\pi \mathbb{Z}^d$ and its fundamental domain is the cube $[-\pi,\pi]^{d}$ (Brillouin zone).
* For any $m \in \mathbb{N}$, the $m$-dimensional torus $\mathbb{R}^m/\mathbb{Z}^m$, is denoted by $\mathbb{T}^m$.
From now on, we fix any smooth function $h$ satisfying additivity in Definition <ref>. The following lemma is a preparation for the next definition.
For any $k \in \mathbb{C}^d$, we have
$$e^{-ik\cdot h(x)}L(x,D)e^{ik\cdot h(x)}=L(x,D)+B(k),$$
where $B(k)$ is a smooth differential operator of order $1$ on $X$ that commutes with the action of the deck group $G$.
Thus by pushing down, the differential operators $e^{-ik\cdot h(x)}L(x,D)e^{ik\cdot h(x)}$ and $B(k)$ can be considered also as differential operators on $M$.
Moreover, given any $m \in \mathbb{R}$, the mapping
$$k \mapsto e^{-ik\cdot h(x)}L(x,D)e^{ik\cdot h(x)}$$
is analytic in $k$ as a $B(H^{m+2}(M), H^{m}(M))$-valued function.
It is standard that the commutator $[L, e^{ik\cdot h(x)}]$ is a differential operator of order $1$ on $X$.
Now one can write
\begin{equation*}
\label{B(k)}
B(k)=e^{-ik\cdot h(x)}Le^{ik\cdot h(x)}-L=e^{-ik\cdot h(x)}[L, e^{ik\cdot h(x)}]
\end{equation*}
to see that $B(k)$ is also a smooth differential operator of order $1$. Also, one can check that $B(k)$ commutes with the action of $G$ by using $G$-periodicity of the operator $L$ and additivity of $h$. This proves the first claim of the lemma. From a standard fact (see e.g., <cit.>), the operator $e^{-ik\cdot h(x)}Le^{ik\cdot h(x)}$ defined on $X$ can be written as a sum $\sum_{|\alpha|\leq 2}k^{\alpha}L_{\alpha},$
where $L_{\alpha}$ is a $G$-periodic differential operator on $X$ of order $2-|\alpha|$ which is independent of $k$. By pushing the above sum down to a sum of operators on $M$, the claim about analyticity in $k$ is then obvious.
For any $k \in \mathbb{C}^d$, we denote by $L(k)$ the elliptic operator
\begin{equation*}
\label{conjugatingLk}
e^{-ik\cdot h(x)}L(x,D)e^{ik\cdot h(x)}
\end{equation*}
in $L^2(M)$ with the domain the Sobolev space $H^2(M)$.
In this definition, the vector $k$ is called the quasimomentum [The name comes from solid state physics <cit.>.].
* When dealing with real quasimomentum $k$, it is enough to consider $k$ in any shifted copy of the Brillouin zone $[-\pi, \pi]^d$, since the operators $L(k)$ and $L(k+2\pi \gamma)$ are unitarily equivalent, for any $\gamma \in \mathbb{Z}^d$.
* The operator $L(k)$ is self-adjoint in $L^{2}(M)$ for each $k \in \mathbb{R}^d$, with the domain $H^{2}(M)$. Due to the ellipticity of $L$, each of the operators $L(k)$ ($k \in \mathbb{R}^d$) has discrete real spectrum and thus, we can list its eigenvalues in non-decreasing order:
\begin{equation*}
\label{eigenv}
\lambda_{1}(k) \leq \lambda_{2}(k) \leq ... \quad .
\end{equation*}
Hence, we can single out continuous and piecewise-analytic band functions $\lambda_{j}(k)$ for each $j \in \mathbb{N}$ <cit.>.
* By Lemma <ref>, the operators $L(k)$ are perturbations of the self-adjoint operator $L(0)$ by lower order operators $B(k)$ for each $k \in \mathbb{C}^d$. Consequently, the spectra of the operators $L(k)$ on $M$ are all discrete (see <cit.>).
* We now describe another equivalent model of the operators $L(k)$, which sometimes can be useful (see <cit.>). For any quasimomentum $k \in \mathbb{C}^d$, we denote by $\gamma_k$ the character (i.e., a $1$-dimensional representation) $e^{ik\cdot g}$ of the abelian group $G$ and consider the $1$-dimensional flat vector bundle $E_k$ over $M$ associated with this representation. For any real number $s$, let $H^{s}_k(X)$ be the space of $H^s$-sections of $E_k$. Since $L$ is $G$-periodic, $L$ maps continuously $H^2_k(X)$ into $L^2_k(X)$. This defines an elliptic operator over the space $\mathcal{E}(M, E_k)$ of smooth sections of $E_k$ over the compact manifold $M$. Moreover, this elliptic operator is unitarily equivalent to the operator $L(k)$ in Definition <ref>.
Notice that in the case of abelian coverings, a third equivalent model using differential forms and the Jacobian torus $J(M)$ to define can be found in <cit.>.
Now we can restate the band-gap structure of $\sigma(L)$ presented in Theorem <ref> in more details.
The spectrum of $L$ is the union of all the spectra of $L(k)$ when $k$ runs over the Brillouin zone (or any its shifted copy), i.e.
\begin{equation}
\label{fl_spectrum}
\sigma(L)=\bigcup_{k \in [-\pi, \pi]^d}\sigma(L(k)).
\end{equation}
In other words, the spectrum of $L$ is the range of the multivalued function
\begin{equation*}
\label{sp_function}
k \rightarrow \lambda(k):=\sigma(L(k)), \quad k\in [-\pi, \pi]^d,
\end{equation*}
As a result, the range of the band function $\lambda_j$ (see remark <ref>) constitutes exactly the band $[\alpha_j, \beta_j]$ of the spectrum of $L$ shown in band-gap.
The notions which we will introduce now are important concepts in solid state physics (see e.g., <cit.>) as well as in general theory of periodic elliptic operators (see e.g., <cit.>).
* A Bloch solution with quasimomentum $k$ of the equation $L(x,D)u=0$ is a solution of the form
\begin{equation*}
u(x)=e^{ik\cdot h(x)}\phi (x),
\end{equation*}
where $h$ is any fixed additive function on $X$ and the function $\phi$ is invariant under the action of the deck transformation group $G$.[It is easy to see that this definition is independent of the choice of $h$.]
* The Bloch variety $B_{L}$ of the operator $L$ consists of all pairs $(k,\lambda) \in \mathbb{C}^{d+1}$ such that the equation $Lu=\lambda u$ on $X$ has a non-zero Bloch solution $u$ with quasimomentum $k$.
The Bloch variety $B_{L}$ can be seen as the graph of the multivalued function $\lambda(k)$, which is also called the dispersion relation:
\begin{equation*}
\label{disp_relation}
B_L=\{(k,\lambda): \lambda \in \sigma(L(k))\}.
\end{equation*}
* The Fermi surface $F_{L,\lambda}$ of the operator $L$ at the energy level $\lambda \in \mathbb{C}$ consists of all quasimomenta $k \in \mathbb{C}^{d}$ such that the equation $Lu=\lambda u$ on $X$ has a non-zero Bloch solution $u$ with quasimomentum $k$. We shall write $F_{L}$ instead of $F_{L,0}$ when $\lambda=0$.
Equivalently, Fermi surfaces are level sets of the dispersion relation.
The next statement can be found in <cit.> (see also <cit.>).
There exist entire ($2\pi \mathbb{Z}^d$-periodic in $k$) functions of finite orders on $\mathbb{C}^{d}$ and on $\mathbb{C}^{d+1}$ such that the Fermi and Bloch varieties are the sets of all zeros of these functions respectively.
As a consequence, the band functions $\lambda_j(k)$ are piecewise analytic on $\mathbb{C}^d$.
Note that the piecewise analyticity of the band functions is shown initially in <cit.> for Schrödinger operators in the flat case.
Without loss of generality, it is enough to assume henceforth that $0$ is the spectral edge of interest (by adding a constant into the operator $L$ if neccessary) and there is a spectral gap below this spectral edge $0$. Therefore,
$0$ is the lower spectral edge of some spectral band [The upper spectral edge case is treated similarly.], i.e., $0$ is the minimal value of some band function $\lambda_j(k)$ for some $j \in \mathbb{N}$ over the Brillouin zone.
As in <cit.>, the following analytic assumptions are imposed on the band function $\lambda_j$:
Assumption A
There exists $k_0 \in [-\pi, \pi]^d$ and a band function $\lambda_{j}(k)$ such that:
A1 $\lambda_{j}(k_0)=0$.
A2 $\min_{k \in \mathbb{R}^d, i \neq j}|\lambda_{i}(k)|>0$.
A3 $k_0$ is the only (modulo $2\pi \mathbb{Z}^d$) minimum of $\lambda_{j}$.
A4 The Hessian matrix $H:=\Hess{(\lambda_j)}(k_0)$ of $\lambda_j$ at $k_0$ is positive-definite.
A5 All components of the quasimomentum $k_0$ are equal to either $0$ or $\pi$.
* For the flat case, the main theorem in <cit.> shows that the conditions A1 and A2 are `generically' satisfied, i.e., they can be achieved by small perturbation of the potential of a periodic Schrödinger operator. The same proof in <cit.> still works for periodic Schrödinger operators on a general abelian covering.
* In mathematics and physics literature, the conditions A3 and A4 are commonly believed to be `generically' true (see e.g., <cit.>). In particular, A4 is often assumed to define effective masses of Bloch electrons <cit.>. Additionally, we remark that the condition A3 can be relaxed (see Section 9).
* It is known <cit.> that spectral edges could occur deeply inside the Brillouin zone, however, the condition A5 holds in many practical cases. We shall only use this condition for the spectral gap interior case.
* Due to results of <cit.> (in the flat case) and of <cit.> (in the general case), all these assumptions A1-A5 hold at the bottom of the spectrum for non-magnetic Schrödinger operators.
Here are some notations that will be used thoughout this paper.
* The real parts of a complex vector $z$ and of a complex matrix $A$ are denoted by $\Re(z)$ and $\Re(A)$, respectively.
* For any two functions $f$ and $g$ defined on $X \times X$, if there exist constants $C>0$ and $R>0$ such that $|f(x,y)|\leq C|g(x,y)|$ whenever $d_X(x,y)>R$, we write $f(x,y)=O(g(x,y))$.
We say that a set $W$ in $\mathbb{C}^d$ is symmetric
if for any $z \in W$, we have $\overline{z} \in W$.
The following proposition will play a crucial role in establishing Theorem <ref>.
There exists an $\epsilon_0>0$ and a symmetric open subset $V \subset \mathbb{C}^d$ containing the quasimomentum $k_0$ from Assumption A such that the band function $\lambda_j$ in Assumption A has an analytic continuation into a neighborhood of $\overline{V}$, and the following properties hold for any $z$ in a symmetric neighborhood of $\overline{V}$:
* $\lambda_{j}(z)$ is a simple eigenvalue of $L(z)$.
* $|\lambda_j(z)|<\epsilon_0$ and $\displaystyle \overline{B}(0,\epsilon_0) \cap \sigma(L(z))=\{\lambda_j(z)\}$.
* There is a nonzero $G$-periodic function $\phi_z$ defined on $X$ such that
Moreover, $z \mapsto \phi_z$ can be chosen analytic as a $H^2(M)$-valued function.
* $\displaystyle 2\Re(\Hess{(\lambda_{j})}(z))>\min \sigma(\Hess{(\lambda_{j})}(k_0))\cdot I_{d \times d}$.
* $\displaystyle F(z):=(\phi_{z}(\cdot),\phi_{\overline{z}}(\cdot))_{L^{2}(M)} \neq 0.$
Due to Remark <ref>, for any $z \in \mathbb{C}^d$, the operator $L(z)$ has discrete spectrum and thus, it is a closed operator with nonempty resolvent set. Moreover, the operator domain $H^2(M)$ of $L(z)$ is independent of $z$. Also, by Lemma <ref>, for any $\phi \in H^2(M)$, $L(z)\phi$ is a $L^2(M)$-valued analytic function of $z$. These imply that $\{L(z)\}_{z \in \mathbb{C}^d}$ is an analytic family of type $\mathcal{A}$ (see e.g., <cit.>).
Now (P1)-(P4) would follow easily from analytic perturbation theory <cit.> using conditions A1, A2 and A4, while (P5) is due to (P3) and the inequality $F(k_0)=\|\phi_{k_{0}}\|^{2}_{L^{2}(M)}>0$.
Define $\mathcal{V}:=\{\beta \in \mathbb{R}^d \mid k_0+i\beta \in \overline{V}\}.$
Now we introduce the function
$E(\beta):=\lambda_{j}(k_0+i\beta)$, which is defined on $\mathcal{V}$.
The next lemma (see <cit.>) is the only place in this paper where the condition A5 is used.
Assume that the operator $L$ is real [Namely, $Lu$ is real whenever $u$ is real.] and the condition A5 is satisfied. Then $E$ is a real-valued function. By reducing the neighborhood $V$ in Proposition <ref> if necessary, the function $E$ can be assumed real analytic and strictly concave function from $\mathcal{V}$ to $\mathbb{R}$ such that its Hessian at any point $\beta$ in $\mathcal{V}$ is negative-definite.
For $\lambda \in \mathbb{R}$, we put
$$K_{\lambda}:=\{\beta \in \mathcal{V}: E(\beta)\geq \lambda \}$$
$$\Gamma_{\lambda}:=\{\beta \in \mathcal{V}: E(\beta)=\lambda \}$$
Due to Lemma <ref>, $K_{\lambda}$ is a strictly convex $d$-dimensional compact set in $\mathbb{R}^d$, and its boundary $\Gamma_{\lambda}$ is a compact hypersurface in $\mathbb{R}^d$ whose Gauss-Kronecker curvature is nowhere zero.
Therefore, there exists a diffeomorphism $\beta$ from $\mathbb{S}^{d-1}$ onto $\Gamma_{\lambda}$ such that
\begin{equation*}
\label{E:gradient_E_s}
\nabla E(\beta_{s})=-|\nabla E(\beta_s)|s.
\end{equation*}
In addition,
$$\lim_{|\lambda| \rightarrow 0}\max_{s \in \mathbb{S}^{d-1}}|\beta_s|=0.$$
By letting $|\lambda|$ be sufficiently small,
we will suppose that there is an $r_0>0$ (independent of $s$) such that
\begin{equation}
\label{E:beta_s_in_V}
\{k+it\beta_{s} \mid (t,s) \in [0,1] \times \mathbb{S}^{d-1}, \hspace{3pt} |k-k_0|\leq r_0\} \subset V.
\end{equation}
§ THE MAIN RESULTS
We recall that $h$ is a fixed additive function satisfying additivity in Definition <ref>.
First, we consider the case when $\lambda$ is inside a gap and is near to one of the edges of the gap.
The following result is an analog for abelian coverings of compact Riemannian manifolds of <cit.>.
(Spectral gap interior)
Suppose that $d \geq 2$, $L$ is real, and the conditions A1-A5 are satisfied.
For $\lambda<0$ sufficiently close to $0$ (depending on the dispersion branch $\lambda_j$ and the operator $L$), the Green's function $G_{\lambda}$ of $L$ at $\lambda$ admits the following asymptotics as $d_{X}(x,y) \rightarrow \infty$:
\begin{equation}
\label{main_asymp}
\begin{split}
G_{\lambda}(x,y)&=\frac{e^{(h(x)-h(y))(ik_{0}-\beta_{s})}}{(2\pi|h(x)-h(y)|)^{(d-1)/2}}\cdot\frac{|\nabla E(\beta_s)|^{(d-3)/2}}{\det{(-\mathcal{P}_s \Hess{(E)}(\beta_{s})\mathcal{P}_s)}^{1/2}}\\& \times \frac{\phi_{k_{0}+i\beta_{s}}(x)\overline{\phi_{k_{0}-i\beta_{s}}(y)}}{(\phi_{k_{0}+i\beta_{s}},\phi_{k_{0}-i\beta_{s}})_{L^{2}(M)}}
+e^{(h(y)-h(x))\cdot \beta_{s}}r(x,y).
\end{split}
\end{equation}
$$\displaystyle s=(h(x)-h(y))/|h(x)-h(y)| \in \mathcal{A}_h,$$
and $\mathcal{P}_s$ is the projection from $\mathbb{R}^{d}$ onto the tangent space of the unit sphere $\mathbb{S}^{d-1}$ at the point $s$.
Also, there is a constant $C>0$ (independent of $s$ and of the choice of $h$) such that the remainder term $r$ satisfies
$$|r(x,y)| \leq Cd_X(x,y)^{-d/2},$$
when $d_X(x,y)$ is large enough.
By using rational admissible directions (see rational) in the formula main_asymp, the large scale behaviors of the Green's function along orbits of the $G$-action admit the following nice form in which the additive function $h$ is absent.
Under the same notations and hypotheses of Theorem <ref> and suppose that $\lambda<0$ is close enough to $0$, as $|g| \rightarrow \infty$ ($g \in \bZ^d$), we have
\begin{equation}
\label{main_rational}
\begin{split}
G_{\lambda}(x,g \cdot x)&=\frac{e^{g\cdot (ik_{0}-\beta_{g/|g|})}}{(2\pi|g|)^{(d-1)/2}}\cdot\frac{|\nabla E(\beta_{g/|g|})|^{(d-3)/2}}{\det{(-\mathcal{P}_{g/|g|} \Hess{(E)}(\beta_{g/|g|})\mathcal{P}_{g/|g|})}^{1/2}}\\& \times \frac{\phi_{k_{0}+i\beta_{g/|g|}}(x)\overline{\phi_{k_{0}-i\beta_{g/|g|}}(g \cdot x)}}{(\phi_{k_{0}+i\beta_{g/|g|}},\phi_{k_{0}-i\beta_{g/|g|}})_{L^{2}(M)}}
+e^{g\cdot \beta_{s}}O(|g|^{-d/2}).
\end{split}
\end{equation}
We also give another interpretation of <cit.> in the special case $X=\mathbb{R}^2$ as follows:
Let $\varphi$ be any real, $\bZ^2$-periodic and smooth function on $\mathbb{R}^2$, and we recall the notation $F_{\varphi}$ from Example <ref>. Let $s$ be any unit vector in $\mathbb{R}^2$ and $y \in \mathbb{R}^2$. Then as $|t| \rightarrow \infty$ $(t \in \mathbb{R})$, the Green's function $G_{\lambda}$ of $L$ at $\lambda$ ($\approx 0$) has the following asymptotics
\begin{equation*}
\label{main_flat}
\begin{split}
G_{\lambda}(F_{\varphi}^{-1}(ts+F_{\varphi}(y)),y)&=\frac{e^{ts\cdot(ik_{0}-\beta_{s})}}{(2\pi |\nabla E(\beta_s)|\cdot \det{(-\mathcal{P}_s \Hess{(E)}(\beta_{s})\mathcal{P}_s)\cdot |t|})^{1/2}}\\& \times \frac{\phi_{k_{0}+i\beta_{s}}(F_{\varphi}^{-1}(ts+F_{\varphi}(y)))\overline{\phi_{k_{0}-i\beta_{s}}(y)}}{(\phi_{k_{0}+i\beta_{s}},\phi_{k_{0}-i\beta_{s}})_{L^{2}(\mathbb{T}^2)}}
+e^{ts\cdot \beta_{s}}O(|t|^{-1}).
\end{split}
\end{equation*}
We now switch to the case when $\lambda$ is on the boundary of the spectrum. Recall that we assume the spectral edge $\lambda$ is zero. The following result is a generalization of <cit.>.
(Spectral edge case)
Assume that $d \geq 3$ and the operator $L$ satisfies the assumptions A1-A4. For a small $\varepsilon>0$, we denote by $R_{-\varepsilon}=(L+\varepsilon)^{-1}$ the resolvent of $L$ near the spectral edge $\lambda=0$
(which exists, due to Assumption A). Then:
* For any $\phi, \varphi \in L^2_{comp}(X)$, as $\varepsilon \rightarrow 0$, we have:
$$\langle R_{-\varepsilon}\phi, \varphi \rangle \rightarrow \langle R\phi, \varphi \rangle.$$
for an operator $R: L^2_{comp}(X) \rightarrow L^2_{loc}(X)$.
* The Schwartz kernel $G(x,y)$ of the operator $R$, which we call the Green's function of $L$ (at the spectral edge 0), has the following asymptotics when $d_X(x,y) \rightarrow \infty$:
\begin{equation}
\label{main_asymp_KR}
\begin{split}
G(x,y)&=\frac{\Gamma(\frac{d-2}{2})e^{i(h(x)-h(y))\cdot k_0}}{2\pi^{d/2}\sqrt{\det H}|H^{-1/2}(h(x)-h(y))|^{d-2}}\cdot \frac{\phi_{k_0}(x)\overline{\phi_{k_0}(y)}}{\|\phi_{k_0}\|_{L^2(M)}^2}\\& \times \left(1+O\left(d_X(x,y)^{-1}\right)\right)+O\left(d_{X}(x,y)^{1-d}\right),
\end{split}
\end{equation}
where $H$ is the Hessian matrix of $\lambda_j$ at $k_0$.
Here the notation $\Gamma(z)$ means the Gamma function $\Gamma(z)=\displaystyle\int_{0}^{\infty}x^{z-1}e^{-x}d{x}$.
* An interesting feature in the main results is that the dimension $n$ of the covering manifold $X$ does not explicitly enter into the asymptotics main_asymp and main_asymp_KR (especially, see also main_rational). Anyway, it certainly influences the geometry of the dispersion curves and therefore the asymptotics too. However, as the Riemannian distance between $x$ and $y$ becomes larger, one can see that in the asymptotics, the role of the dimension $n$ is rather limited, while the influence of the rank $d$ of the torsion-free subgroup of the deck group $G$ is stronger.
* Note that for a periodic elliptic operator of second order on $\mathbb{R}^d$, at the bottom of its spectrum, the operator is known to be critical when the dimension $d \leq 2$ (see <cit.>).
This also holds true for the Laplacian on a co-compact Riemannian covering (see <cit.>).
Therefore, the assumption $d \geq 3$ is needed in Theorem <ref>.
* The asymptotics main_asymp and main_asymp_KR can be described in terms of the Albanese map and the Albanese pseudo-distance on the abelian covering $X$ (see these definitions in <cit.>), provided that the additive function $h$ is chosen to be harmonic (see also <cit.>).
Proving Theorem <ref> by generalizing <cit.> is similar to establishing Theorem <ref> by generalizing <cit.>. Thus, after finishing the proof of Theorem <ref>, we will sketch briefly the proof of Theorem <ref> in Section 6.
We outline the general strategy of both the proofs of Theorem <ref> and Theorem <ref>.
As in <cit.>, the idea is to show that only one branch of the dispersion relation $\lambda_j$ appearing in the Assumption A will control the asymptotics.
* Step 1: We use the Floquet transform to reduce the problems of finding asymptotics of Green's functions to the problems of obtaining asymptotics of some integral expressions with respect to the quasimomentum $k$.
* Step 2: We localize these expressions around the quasimomentum $k_0$ and then we cut an “infinite-dimensional" part of the operator to deal only with the multiplication operator by the dispersion branch $\lambda_j$.
* Step 3: The dispersion curve around this part is almost a paraboloid according to the assumption A4, thus,
we can reduce this piece of operator to the normal form in
the free case. In this step, we obtain some scalar integral expressions which are close to the ones arising when dealing with the Green's function of the Laplacian operator at the level $\lambda$. Our remaining task is devoted to computing the asymptotics of these scalar integrals.
§ A FLOQUET-BLOCH REDUCTION OF THE PROBLEM
First, we introduce the following fundamental domain $\mathcal{O}$ (with respect to the dual lattice $2\pi \bZ^d$ and the quasimomentum $k_0$ in Assumption A):
𝒪=k_0+[-π, π]^d.
In another word, $\mathcal{O}$ is just a shifted version of the Brillouin zone so that the quasimomentum $k_0$ is its center of symmetry.
If $k_0$ is a high symmetry point of the reduced Brillouin zone (i.e., $k_0$ satisfies Assumption A5), then $k_0=(\delta_1 \pi, \delta_2 \pi,..., \delta_d \pi)$, where $\delta_{j} \in \{0,1\}$ for $j \in \{1,...,d\}$. Hence, fund-dom becomes:
$$\mathcal{O}=\prod_{j=1}^d [(\delta_{j}-1)\pi, (\delta_j+1)\pi].$$
§.§ The Floquet transforms on abelian coverings
The following transform will play the role of the Fourier transform for the periodic case. Indeed, it is a version of the Fourier transform on the group $\mathbb{Z}^d$ of periods.
The Floquet transform $\mathcal{F}$ (which depends on the choice of $h$)
\begin{equation*}
f(x) \rightarrow \widehat{f}(k,x)
\end{equation*}
maps a compactly supported function $f$ on $X$ into a function $\widehat{f}$ defined on $\mathbb{R}^d \times X$ in the following way:
\begin{equation*}
%\label{eqn:hat f(k)}
\widehat{f}(k,x):=\sum_{\gamma \in \mathbb{Z}^d}f(\gamma \cdot x)e^{-i h(\gamma \cdot x) \cdot k}.
\end{equation*}
From the above definition, one can see that $\widehat{f}$ is $\mathbb{Z}^d$-periodic in the $x$-variable and satisfies a cyclic condition with respect to $k$:
\[ \left\{
\begin{array}
\label{E:periodic}
\widehat{f}(k,\gamma \cdot x)=\widehat{f}(k,x), \quad \forall \gamma \in \mathbb{Z}^d \\
\widehat{f}(k+2\pi \gamma,x)=e^{-2\pi i \gamma \cdot h(x)}\widehat{f}(k,x), \quad \forall \gamma \in \mathbb{Z}^d \\
\end{array}
.\right. \]
Thus, it suffices to consider the Floquet transform $\widehat{f}$ as a function defined on $\mathcal{O} \times M$. Usually, we will regard $\widehat{f}$ as a function $\widehat{f}(k,\cdot)$ in $k$-variable in $\mathcal{O}$ with values in the function space $L^{2}(M)$.
The next lemma lists some well-known results of the Floquet transform. Although the lemma is stated for abelian coverings, its proof does not require any change from the proof for the flat case. We omit the details since these can be found in <cit.>, for instance.
* The transform $\mathcal{F}$ is an isometry of $L^{2}(X)$ onto
\begin{equation*}
\int_{\mathcal{O}}^{\oplus}L^{2}(M)=L^{2}(\mathcal{O},L^{2}(M))
\end{equation*}
and of $H^{2}(X)$ onto
\begin{equation*}
\int_{\mathcal{O}}^{\oplus}H^{2}(M)=L^{2}(\mathcal{O},H^{2}(M)).
\end{equation*}
* The following two equivalent inversion formulae $\mathcal{F}^{-1}$ are given by
\begin{equation}
\label{E:inversion1}
f(x)=(2\pi)^{-d}\int_{\mathcal{O}}e^{ik \cdot h(x)}\widehat{f}(k,x)dk, \quad x \in X.
\end{equation}
\begin{equation}
\label{E:inversion2}
f(x)=(2\pi)^{-d}\int_{\mathcal{O}}e^{ik \cdot h(x)}\widehat{f}(k,\gamma^{-1} \cdot x)dk, \quad x \in \gamma \cdot \overline{F(M)}.
\end{equation}
* The action of any periodic elliptic operator $P$ in $L^{2}(X)$ under the Floquet transform $\mathcal{F}$ is given by
\begin{equation*}
\mathcal{F}P(x,D)\mathcal{F}^{-1}=\int_{\mathcal{O}}^{\oplus}P(k)dk,
\end{equation*}
where $P(k)(x,D)=e^{-ik\cdot h(x)}P(x,D)e^{ik\cdot h(x)}$.
In other words,
\begin{equation*}
\widehat{Pf}(k)=P(k)\widehat{f}(k), \quad \forall f \in H^{2}(X).
\end{equation*}
The direct integral decomposition of $P$ in Lemma <ref> (iii) has an important consequence that the spectrum of any periodic elliptic operator $P$ on $X$ is the union of the spectra of operators $P(k)$ on $M$ over the fundamental domain $\mathcal{O}$.
From now on, we will consider the Green's function $G_{\lambda}(x,y)$ at the level $\lambda$ in Case I (i.e., Spectral gap interior) in the rest of this section.
§.§ A Floquet reduction of the problem
We begin with the following proposition, which says roughly that if one starts moving $k$ from some shifted copy of the Brillouin zone along the direction $i\beta_s$, then $k_0+i\beta_s$ is the first quasimomentum $k$ that belongs to the Fermi surface $F_{L, \lambda}$.
If $|\lambda|$ is small enough (depending on the dispersion branch $\lambda_j$ and $L$), then for any $(t,s) \in [0,1] \times \mathbb{S}^{d-1}$, we have $\lambda \in \sigma(L(k+it\beta_s))$ if and only if $(k,t)=(k_0,1)$.
This statement is proven in <cit.> for the flat case. The case of an abelian covering does not require any change in the proof. The main ingredients in the proof are the upper-semicontinuity of the
spectra of the analytic family $\{L(k)\}_{k \in \mathbb{C}^d}$ and the fact that $E$ is a real function, whose Hessian is negative definite (Lemma <ref>).
We consider the following real, smooth linear elliptic operators on $X$:
\begin{equation*}
\label{E:L_t_s}
L_{t,s}=e^{t\beta_s\cdot h(x)}Le^{-t\beta_s\cdot h(x)}, \quad (t,s) \in [0,1] \times \mathbb{S}^{d-1}.
\end{equation*}
Notice that these operators are $G$-periodic, and when pushing $L_{t,s}$ down to $M$, we get the operator $L(-i\beta_s)$. We also use the notation $L_s$ for $L_{1,s}$.
Due to Remark <ref>, we can apply the identity fl_spectrum to the operator $L_{t,s}$ to obtain
\begin{equation}
\label{E:band_functions_L_t_s}
\sigma(L_{t,s})=\bigcup_{k \in \mathcal{O}} \sigma(L_{t,s}(k))=\bigcup_{k \in \mathcal{O}} \sigma(L(k+it\beta_s)) \supseteq \{\lambda_{j}(k+it\beta_s)\}_{k\in \mathcal{O}}.
\end{equation}
We now fix a real number $\lambda$ such that the statement of Proposition <ref> holds.
By E:band_functions_L_t_s and Proposition <ref>, $\lambda$ is in the resolvent set of $L_{t,s}$ for any $(t,s) \in [0,1) \times \mathbb{S}^{d-1}$.
Let $R_{t,s,\lambda}$ be the resolvent operator $(L_{t,s}-\lambda)^{-1}$.
Using Lemma <ref> (iii), for any $f \in L^{2}_{comp}(X)$, we have
\begin{equation*}
\widehat{R_{t,s,\lambda}f}(k)=(L_{t,s}(k)-\lambda)^{-1}\widehat{f}(k), \quad (t,k) \in [0,1) \times \mathcal{O}.
\end{equation*}
Due to Lemma <ref> (i), the sesquilinear form $(R_{t,s,\lambda}f,\varphi)$ is equal to
\begin{equation*}
(2\pi)^{-d}\int_{\mathcal{O}}\left( (L_{t,s}(k)-\lambda)^{-1}\widehat{f}(k), \widehat{\varphi}(k) \right)dk,
\end{equation*}
where $\varphi \in L^2_{comp}(X)$.
In the next lemma, the weak convergence as $t \nearrow 1$ of the operator $R_{t,s,\lambda}$ in $L^{2}_{comp}(X)$ is proved and thus, we can introduce the limit operator $\displaystyle R_{s,\lambda}:=\lim_{t \rightarrow 1^{-}}R_{t,s,\lambda}$.
Let $d\geq 2$. Under Assumption A, for $f,\varphi$ in $L^{2}_{comp}(X)$, the following equality holds:
\begin{equation}
\label{E:limit}
\lim_{t \rightarrow 1^{-}}(R_{t,s,\lambda}f,\varphi)=(2\pi)^{-d}\int_{\mathcal{O}}\left( L_{s}(k)-\lambda)^{-1}\widehat{f}(k),\widehat{\varphi}(k)\right)dk.
\end{equation}
The integral in the right hand side of E:limit is absolutely convergent.
This lemma is a direct corollary of Lemma <ref>, Proposition <ref> and the Lebesgue Dominated Convergence Theorem as being shown in <cit.>. We skip the proof.
For any $(t,s) \in [0,1) \times \mathbb{S}^{d-1}$, let $G_{t,s, \lambda}$
be the Green's function of $L_{t,s}$ at $\lambda$, which is the kernel of $R_{t,s,\lambda}$.
$$G_{t,s,\lambda}(x,y)=e^{t\beta_s \cdot (h(x)-h(y))}G_{\lambda}(x,y).$$
Taking the limit and applying Lemma <ref>, we conclude that the function
$$G_{s, \lambda}(x,y):=e^{\beta_s \cdot (h(y)-h(x))}G_{\lambda}(x,y)$$
is the integral kernel of the operator $R_{s,\lambda}$ defined as follows:
\begin{equation}
\label{E:R_s}
\widehat{R_{s,\lambda}f}(k)=(L_{s}(k)-\lambda)^{-1}\widehat{f}(k).
\end{equation}
Hence, the problem of finding asymptotics of $G_{\lambda}$ is now equivalent
to obtaining asymptotics of any function $G_{s,\lambda}$,
where $s$ is an admissible direction in $\mathcal{A}_h$.
In addition, by E:inversion1 and E:R_s, the function $G_{s,\lambda}$, which is also the Green's function of the operator $L_{s}$ at $\lambda$, is the integral kernel of the operator $R_{s,\lambda}$ that acts on $L^{2}_{comp}(X)$ in the following way:
\begin{equation}
\label{E:R_s_exp}
R_{s,\lambda}f(x)=(2\pi)^{-d}\int_{\mathcal{O}}e^{ik\cdot h(x)}(L_{s}(k)-\lambda)^{-1}\widehat{f}(k,x)dk, \quad x \in X.
\end{equation}
This accomplishes Step 1 in our strategy of the proof.
§.§ Isolating the leading term in $R_{s,\lambda}$ and a reduced Green's function
The purpose of this part is to complete Step 2, i.e., to localize the part of the integral in E:R_s_exp, that is responsible for the leading term of the Green's function asymptotics.
For any $z \in V$, we denote by $P(z)$ the spectral projector $\chi_{B(0, \varepsilon_0)}(L(z))$, i.e.,
\begin{equation*}
\label{E:Riesz_projector}
P(z)=-\frac{1}{2\pi i}\oint_{|\alpha|=\epsilon_0}(L(z)-\alpha)^{-1}d\alpha.
\end{equation*}
By (P2), $P(z)$ projects $L^{2}(M)$ onto the eigenspace spanned by $\phi_z$. We also put $Q(z):=I-P(z)$ and denote by $R(P(z))$, $R(Q(z))$ the ranges of the projectors $P(z)$, $Q(z)$ correspondingly.
Using (P6) and the fact that $P(k+i\beta_s)^*=P(k-i\beta_s)$,
we can deduce that if $|k-k_0| \leq r_0$ (see E:beta_s_in_V), the following equality holds
\begin{equation}
\label{E:form_P_s}
P(k+i\beta_s)u=\frac{(u, \phi_{k-i\beta_s})_{L^{2}(M)}}{(\phi_{k+i\beta_s}, \phi_{k-i\beta_s})_{L^{2}(M)}}\phi_{k+i\beta_s},
\quad \forall u \in L^{2}(M).
\end{equation}
Let $\eta$ be a cut-off smooth function on $\mathcal{O}$ supported on $\{k \in \mathcal{O} \mid |k-k_0|<r_0\}$ and equal to $1$ around $k_0$.
According to E:R_s_exp, for any $f \in C^{\infty}_{c}(X)$, we want to find $u$ such that
Then the Green's function $G_{s, \lambda}$ satisfies
$$\int_{X}G_{s, \lambda}(x,y)f(y)d\mu_{X}(y)=\mathcal{F}^{-1}\widehat{u}(k,x)=u(x),$$
where $\mathcal{F}$ is the Floquet transform introduced in Definition <ref>.
By Proposition <ref>, the operator $L_s(k)-\lambda$ is invertible for any $k$ such that $k \neq k_0$. Hence, we can decompose $\widehat{u}(k)=\widehat{u_0}(k)+(L_s(k)-\lambda)^{-1}(1-\eta(k))\widehat{f}(k)$, where $\widehat{u_0}$ satisfies the equation
Observe that $R(P(z))$ and $R(Q(z))$ are invariant subspaces for the operator $L(z)$ for any $z \in V$. Thus, if $u_1, u_2$ are functions such that $\widehat{u_1}(k)=P(k+i\beta_s)\widehat{u_0}(k)$ and $\widehat{u_2}(k)=Q(k+i\beta_s)\widehat{u_0}(k)$, we must have
\begin{equation}
\label{u1}
\end{equation}
Due to (P2), when $k$ is close to $k_0$, $\lambda=\lambda_{j}(k_0+i\beta_s)$ must belong to the resolvent of the operator $L_{s}(k)|_{R(Q(k+i\beta_s))}$.
Hence, we can write $\widehat{u_2}(k)=\eta(k)(L_s(k)-\lambda)^{-1}Q(k+i\beta_s)\widehat{f}(k)$.
Therefore, $\widehat{u}(k)$ equals
The next theorem shows that for finding the asymptotics, we can ignore the infinite-dimensional part of the operator $R_{s, \lambda}$, i.e., the second term in the above sum of two operators.
Let $T_{s}$ be the operator acting on $L^2(X)$ as follows:
Then the Schwartz kernel $K_{s}(x,y)$ of the operator $T_{s}$ is continuous away from the diagonal of $X$, and moreover, it is also rapidly decaying in a uniform way with respect to $s \in \mathbb{S}^{d-1}$,
i.e., for any $N>0$,
$$\sup_{s \in \mathbb{S}^{d-1}}|K_{s}(x,y)|=O(d_X(x,y)^{-N}).$$
A proof using microlocal analysis will be mentioned in Section 7.
Now let $V_{s}:=R_{s, \lambda}-T_{s}$. Then
the Schwartz kernel $G_0(x,y)$ of the operator $V_{s}$ satisfies the following relation:
\begin{equation}
\label{G_0}
\int_X G_0(x,y)f(y)d\mu_{X}(y)=\mathcal{F}^{-1}\widehat{u_1}(k,x)=u_1(x).
\end{equation}
In what follows, we will find an integral representation of the kernel $G_0$. We will see that $G_0$ provides the leading term of the asymptotics of the kernel $G_{s, \lambda}$. For this reason, $G_0$ is called a reduced Green's function.
To find $u_1$, we use the equation u1 and apply E:form_P_s to deduce
\begin{equation*}
\end{equation*}
Using $\widehat{u_1}(k)=P(k+i\beta_s)\widehat{u_1}(k)$ and E:form_P_s again, the above identity becomes
\begin{equation*}
\label{E:def_u1}
\widehat{u_1}(k,x):=\frac{\eta(k)\phi_{k+i\beta_s}(x)(\widehat{f}(k),\phi_{k-i\beta_s})_{L^{2}(M)}}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}, \quad k \neq k_0.
\end{equation*}
By the inverse Floquet transform E:inversion1, for any $x \in X$,
\begin{equation*}
u_{1}(x)=(2\pi)^{-d}\int_{\mathcal{O}}e^{ik\cdot h(x)}\frac{\eta(k)\phi_{k+i\beta_s}(x)(\widehat{f}(k),\phi_{k-i\beta_s})_{L^{2}(M)}}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}dk.
\end{equation*}
Now we repeat some calculations in <cit.> to have
\begin{equation*}
\begin{split}
u_1(x)&=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}\int_{M}\frac{e^{ik\cdot h(x)}\eta(k)\widehat{f}(k,y)\overline{\phi_{k-i\beta_s}(y)}\phi_{k+i\beta_s}(x)}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}d\mu_{M}(y)dk\\
&=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}\int_{\overline{F(M)}}\sum_{\gamma \in G}\frac{e^{ik\cdot (h(x)-h(\gamma^{-1}\cdot y))}\eta(k)\overline{\phi_{k-i\beta_s}(y)}\phi_{k+i\beta_s}(x)}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}d\mu_{X}(y)dk\\
&=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}\sum_{\gamma \in G}\int_{\gamma \cdot \overline{F(M)}}f(y)\frac{e^{ik\cdot (h(x)-h(y))}\eta(k)\overline{\phi_{k-i\beta_s}(\gamma^{-1} \cdot y)}\phi_{k+i\beta_s}(x)}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}d\mu_{X}(y)dk\\
&=\frac{1}{(2\pi)^{d}}\int_{X}f(y)\left(\int_{\mathcal{O}}\frac{e^{ik\cdot (h(x)-h(y))}\eta(k)\overline{\phi_{k-i\beta_s}(y)}\phi_{k+i\beta_s}(x)}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}dk\right)d\mu_{X}(y).
\end{split}
\end{equation*}
In the second equality above, we use the identity integralF(M).
Consequently, from G_0, we conclude that our reduced Green's function is
\begin{equation}
\label{E:formula_G0}
G_0(x,y)=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}e^{ik\cdot (h(x)-h(y))}\eta(k)\frac{\phi_{k+i\beta_s}(x)\overline{\phi_{k-i\beta_s}(y)}}{(\phi_{k+i\beta_s},\phi_{k-i\beta_s})_{L^2(M)}(\lambda_j(k+i\beta_s)-\lambda)}dk.
\end{equation}
§ SOME AUXILIARY STATEMENTS
In this part, we provide the analogs of some results from <cit.>, which do not require any significant change in the proofs when dealing with the case of abelian coverings. Instead of repeating the details, we will make some brief comments about the main ingredients of these results.
The first result studies the local smoothness in $(z,x)$ of the eigenfunctions $\phi_z(x)$ of the operator $L(z)$ with the eigenvalue $\lambda_j(z)$.
Suppose that $B \subset \mathbb{R}^d$ is the open ball centered at $k_0$ with radius $r_0$ (see E:beta_s_in_V). Then for each $s \in \mathbb{S}^{d-1}$, the functions $\displaystyle \phi_{k\pm i\beta_s}(x)$ are smooth on a neighborhood of $\overline{B} \times M$ in $\mathbb{R}^d \times M$. In addition, for any multi-index $\alpha$, the functions $\displaystyle D^{\alpha}_{k}\phi_{k\pm i\beta_s}(x)$ are also jointly continuous in $(s,k,x)$. In particular, we have
\begin{equation*}
\sup_{(s,k,x) \in \mathbb{S}^{d-1}\times \overline{B} \times M} |D^{\alpha}_{k}\phi_{k \pm i\beta_s}(x)|<\infty.
\end{equation*}
To obtain Lemma <ref>, one can modify the proof of <cit.> without any significant change.
Indeed, the three main ingredients in the proof are the smoothness in $z$ of the family of operators $\{L(z)\}_{z \in V}$ acting between Sobolev spaces (Lemma <ref>), the property (P3) for bootstraping regularity of eigenfunctions in $k$, and the standard coercive estimates of elliptic operators $L(z)$ on the compact manifold $M$ (see e.g., <cit.>) for bootstraping regularity in $x$.
The next result is the asymptotics of the scalar integral expression obtained from the integral representation E:formula_G0 of the reduced Green's function $G_0$.
Suppose that $d\geq 2$ and $B$ is the open ball defined in Proposition <ref>. Let $\eta(k)$ be a smooth cut off function around the point $k_0$, and $\{\mu_{s}(k,x,y)\}_{s \in \mathbb{S}^{d-1}}$ be a family of smooth $\mathbb{C}^d$-valued functions defined on $\overline{B} \times M \times M$. We also use the same notation $\mu_{s}(k,x,y)$ for its lift to $\overline{B} \times X \times X$.
For each quadruple $(s,a,x,y) \in \mathbb{S}^{d-1} \times \mathbb{R}^d \times X \times X$, we define
$$I(s,a):=\frac{1}{(2\pi)^d}\int_{\mathcal{O}}e^{ik\cdot a}\frac{\eta(k)}{\lambda_j(k+i\beta_s)-\lambda}dk$$
$$J(s,a, x, y):=\frac{1}{(2\pi)^d}\int_{\mathcal{O}}e^{ik\cdot a}\frac{\eta(k)(k-k_0)\cdot \mu_s(k,x,y)}{\lambda_j(k+i\beta_s)-\lambda}dk.$$
Assume that the size of the support of $\eta$ is small enough. Fix a direction $s \in \mathbb{S}^{d-1}$ and consider all vectors $a$ such that $\displaystyle s=\frac{a}{|a|}$. Then when $|a|$ is large enough, we have
\begin{equation}
\label{E:asymp_I}
I(s,a)=\frac{e^{ik_0 \cdot a}|\nabla E(\beta_s)|^{(d-3)/2}}{(2\pi|a|)^{(d-1)/2}\det{(-\mathcal{P}_s \Hess{(E)}(\beta_s)\mathcal{P}_s)}^{1/2}}+O(|a|^{-d/2})
\end{equation}
\begin{equation}
\label{E:asymp_J}
\sup_{(x,y) \in X \times X}|J(s,a,x,y)|=O(|a|^{-d/2}).
\end{equation}
Moreover, if all derivatives of $\mu_s(k,x,y)$ with respect to $k$ are uniformly bounded in $s \in \mathbb{S}^{d-1}$,
then all the terms $O(\cdot)$ in E:asymp_I and E:asymp_J are also uniform in $s \in \mathbb{S}^{d-1}$ when $|a| \rightarrow \infty$.
The proof of Proposition <ref> can be extracted from <cit.>. The main ingredient (see <cit.>) is an application of the Weierstrass Preparation Lemma in several complex variables to have a factorization of the denominator $\lambda_j(k+i\beta_s)-\lambda$ of the integrands of $I, J$ into a form that is close to the normal form in the free case. This trick was used in <cit.> in the discrete setting.
The next result <cit.> will be needed in the proof of Theorem <ref>.
Assume $d \geq 3$. Let $a \in \mathbb{R}^d$. Let $\eta$ be a smooth function satisfying the assumptions of Proposition <ref>, and let $\mu(k,x,y)$ be a smooth $G$-periodic function from a neighborhood of $\overline{B} \times X \times X$ to $\mathbb{C}^d$. Then the following asymptotics hold when $|a| \rightarrow \infty$:
\begin{equation*}
\label{E:asymp_KR}
\begin{split}
&\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}e^{ik\cdot a}\frac{\eta(k)}{\lambda_j(k)}dk=\frac{\Gamma(\frac{d}{2}-1)e^{ik_0 \cdot a}}{2\pi^{d/2}(\det{H})^{1/2}|H^{-1/2}(a)|^{d-2}}(1+O(|a|^{-1}),\\
\mbox{and}&\\
&\sup_{x,y \in X}\left|\int_{\mathcal{O}}e^{ik\cdot a}\frac{\eta(k)(k-k_0)\cdot \mu(k,x,y)}{\lambda_j(k)}dk\right|=O(|a|^{-d+1}).
\end{split}
\end{equation*}
§ PROOFS OF THE MAIN RESULTS
We fix an admissible direction $s$ of the additive function $h$ and consider any $x,y \in X$ such that
$$\displaystyle \frac{h(x)-h(y)}{|h(x)-h(y)|}=s \in \mathcal{A}_h.$$
As we discussed in Section 4, the Green's function $G_{\lambda}$ satisfies
\begin{equation}
\label{G_s}
G_{\lambda}(x,y)=e^{\beta_s \cdot (h(y)-h(x))}G_{s, \lambda}(x,y),
\end{equation}
where $G_{s, \lambda}$ is the Schwartz kernel of the resolvent operator $R_{s}$. Also, $R_{s, \lambda}=V_{s}+T_{s}$. Due to Theorem <ref>, the Schwartz kernel of $T_{s}$ decays rapidly (uniformly in $s$) when $d_X(x,y)$ is large enough. Hence, to find the asymptotics of the kernel of $R_{s, \lambda}$, it suffices to consider the kernel $G_0$ of the operator $V_{s}$.
\begin{equation}
\label{a}
\end{equation}
$$\tilde{\mu}_{\omega}(k,p,q):=\frac{\phi_{k+i\beta_{\omega}}(p)\overline{\phi_{k-i\beta_{\omega}}(q)}}{(\phi_{k+i\beta_{\omega}},\phi_{k-i\beta_{\omega}})_{L^2(M)}}, \quad (\omega,p,q) \in \mathbb{S}^{d-1} \times M \times M.$$
By Lemma <ref>, $\tilde{\mu}_{\omega}$ is a smooth function on $\overline{B} \times M \times M$. By Taylor expanding around $k_0$, $\tilde{\mu}_{\omega}(k,p,q)=\tilde{\mu}_{\omega}(k_0,p,q)+(k-k_0)\cdot \mu_{\omega}(k,p,q)$ for some smooth $\mathbb{C}^d$-valued function $\mu_{\omega}(k,p,q)$ defined on $\overline{B} \times M \times M$. From Lemma <ref> and the definition of $\tilde{\mu_{\omega}}$,
$$\sup_{(\omega,k,x,y) \in \mathbb{S}^{d-1} \times\overline{B} \times M \times M} |D^{\alpha}_{k}\tilde{\mu}_{\omega}(k,x,y)|<\infty,$$
for any multi-index $\alpha$.
Thus, all derivatives of $\mu_{\omega}$ with respect to $k$ are also uniformly bounded in $\omega \in \mathbb{S}^{d-1}$.
We now can rewrite E:formula_G0 as follows:
\begin{equation*}
\label{E:formula_G0_1}
\begin{split}
G_0(x,y)&=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}e^{ik\cdot a}\frac{\eta(k)}{\lambda_j(k+i\beta_s)-\lambda}\left(\tilde{\mu}_s(k_0,x,y)+(k-k_0)\cdot \mu_s(k,x,y)\right)dk\\
&=I(s,a)\frac{\phi_{k_0+i\beta_s}(x)\overline{\phi_{k_0-i\beta_s}(y)}}{(\phi_{k_0+i\beta_s},\phi_{k_0-i\beta_s})_{L^2(M)}}+J(s,a, x, y).
\end{split}
\end{equation*}
Here the integrals $I(s,a)$ and $J(s,a,x,y)$ are defined in Proposition <ref>. By using Proposition <ref>, we obtain the following asymptotics whenever $|a|$ is large enough:
\begin{equation}
\label{E:formula_G0_2}
\begin{split}
G_0(x,y)&=\Big(\frac{e^{ik_0 \cdot a}|\nabla E(\beta_s)|^{(d-3)/2}}{(2\pi|a|)^{(d-1)/2}\det{(-\mathcal{P}_s \Hess{(E)}(\beta_s)\mathcal{P}_s)}^{1/2}}+O(|a|^{-d/2})\Big)\\
&\times \frac{\phi_{k_0+i\beta_s}(x)\overline{\phi_{k_0-i\beta_s}(y)}}{(\phi_{k_0+i\beta_s},\phi_{k_0-i\beta_s})_{L^2(M)}}+O(|a|^{-d/2}),
\end{split}
\end{equation}
where all the terms $O(\cdot)$ are uniform in $s$. Due to a and Proposition <ref>, $O(|a|^{\ell})=O(d_X(x,y)^{\ell})$ for any $\ell \in \bZ$, provided that $d_X(x,y)>R_h$. Hence, by choosing the constant $R_h$ larger if necessary, we can assume that when $d_X(x,y)>R_h$, the asymptotics E:formula_G0_2 would follow.
Finally, we substitute a to the asymptotics E:formula_G0_2 and then use G_s to deduce Theorem <ref>.
We recall that $\lambda=\lambda_j(k_0)=0$ and $R_{-\varepsilon}$ is the resolvent operator $(L+\varepsilon)^{-1}$ when $\varepsilon>0$ is small enough. We will repeat the Floquet reduction approach in Section 4. Given any $f, \varphi \in L^2_{comp}(X)$, the sesquilinear form $\langle R_{-\varepsilon}f, \varphi \rangle$ is
$$(2\pi)^{-d}\int_{\mathcal{O}}\left((L(k)+\varepsilon)^{-1}\widehat{f}(k), \widehat{\varphi}(k)\right)dk.$$
The first conclusion of this theorem is achieved by a smilar argument in <cit.>. Hence the operator $R=\lim_{\varepsilon \rightarrow 0^+}R_{-\varepsilon}$ is defined by the identity $\widehat{Rf}(k)=R(k)\widehat{f}(k)$ and the Green's function $G$ is the Schwartz kernel of the operator $R$. To single out the principal term in $R$, we first choose a neighborhood $V \subset \mathcal{O}$ of $k_0$ such that when $k \in V$, there is a non-zero $G$-periodic eigenfunction $\phi_k(x)$ of the operator $L(k)$ with the corresponding eigenvalue $\lambda_j(k)$ and moreover, the mapping $k \mapsto \phi_k(\cdot)$ is analytic in $k$ as a $H^2(M)$-valued function. This is always possible due to Lemma <ref>. For such $k \in V$, let us denote by $P(k)$ the spectral projector of $L(k)$ that projects $L^2(M)$ onto the eigenspace spanned by $\phi_k$. The notation $R(I-P(k))$ stands for the range of the projector $I-P(k)$.
Then we pick $\eta$ as a smooth cut off function around $k_0$ such that $\supp(\eta) \Subset V$.
Define the operator
As in Theorem <ref>, the Schwartz kernel $K(x,y)$ of $T$ is rapidly decaying as $d_X(x,y) \rightarrow \infty$. Thus, the asymptotics of the Green's function $G$ are the same as the asymptotics of the Schwartz kernel $G_0$ of the operator $R-T$. To find $G_0$, we repeat the arguments in Subsection 4.3 to derive the formula
$$G_0(x,y)=\frac{1}{(2\pi)^{d}}\int_{\mathcal{O}}e^{ik\cdot (h(x)-h(y))}\frac{\eta(k)}{\lambda_j(k)}\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{\|\phi_{k}\|^2_{L^2(M)}}dk, \quad x,y \in X.$$
As in the proof of Theorem <ref>, we set $a:=h(x)-h(y)$ and rewrite the smooth function
$$\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{\|\phi_{k}\|^2_{L^2(M)}}=\frac{\phi_{k_0}(x)\overline{\phi_{k_0}(y)}}{\|\phi_{k_0}\|^2_{L^2(M)}}+(k-k_0)\cdot \mu(k,x,y),$$
for some smooth $G$-periodic function $\mu: \overline{B}\times X \times X \rightarrow \mathbb{C}^d$.
Now by applying Proposition <ref>, the proof is completed.
§ PROOFS OF TECHNICAL STATEMENTS
§.§ Proofs of Proposition <ref> and Proposition <ref>
Fixing a point $x_0 \in X$, we let
$$R:=\max_{x \in K} d_X(x_0,x),$$
$$\widetilde{R_h}:=\max_{(x,y) \in K \times K}|h(x)-h(y)|.$$
Due to Proposition <ref> and the fact that $|\cdot|_S$ is equivalent to $|\cdot|$ on $\bZ^d$, there exist $C_1>1$ and $C_2>0$ such that
$$C_1^{-1}\cdot d_X(g_1 \cdot x_0, g_2 \cdot x_0)-C_2 \leq |g_1-g_2| \leq C_1\cdot d_X(g_1 \cdot x_0,g_2 \cdot x_0)+C_2,$$
for any $g_i \in \bZ^d$, $i=1,2$.
Now we consider any two points $x,y$ in $X$. By closure_F(M), we can select $\tilde{x}$, $\tilde{y}$ in $K$ such that $x=g_1 \cdot \tilde{x}$ and $y=g_2 \cdot \tilde{y}$ for some $g_1, g_2 \in \bZ^d$.
Since $\bZ^d$ acts by isometries, we get
\begin{equation}
\label{E:isometry}
d_X(g_1 \cdot x_0, g_1 \cdot \tilde{x})=d_X(x_0,\tilde{x}) \quad \mbox{and} \quad d_X(g_2 \cdot x_0, g_2 \cdot \tilde{y})=d_X(x_0,\tilde{y}).
\end{equation}
By additivity, we have
Using triangle inequalities and E:isometry, we obtain
\begin{equation*}
\begin{split}
|h(x)-h(y)| &\leq \widetilde{R_h}+|g_1-g_2| \leq C_1\cdot d_X(g_1 \cdot x_0, g_2 \cdot x_0)+\widetilde{R_h}+C_2
\\&\leq C_1 \cdot d_X(x,y)+C_1 \cdot(d_X(x_0, \tilde{x})+d_X(x_0, \tilde{y}))+\widetilde{R_h}+C_2
\\&\leq C_1 \cdot d_X(x,y)+(2C_1 R+\widetilde{R_h}+C_2).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
|h(x)-h(y)| &\geq |g_1-g_2|-\widetilde{R_h} \geq C_1^{-1} \cdot d_X(g_1 \cdot x_0, g_2 \cdot x_0)-(\widetilde{R_h}+C_2)
\\&\geq C_1^{-1} \cdot d_X(x,y)-(C_1^{-1} \cdot(d_X(x_0, \tilde{x})+d_X(x_0, \tilde{y}))+\widetilde{R_h}+C_2)
\\&\geq C_1^{-1} \cdot d_X(x,y)-(2C_1 R+\widetilde{R_h}+C_2).
\end{split}
\end{equation*}
The statement follows if we put $C:=2C_1$ and $R_h:=2C_1 (2C_1 R+\widetilde{R_h}+C_2)$.
By Definition <ref>, any rational point in the unit sphere $\mathbb{S}^{d-1}$ is an admissible direction of the additive function $h$ and thus we have rational. By using the stereographic projection, one can see that the subset $\mathbb{Q}^{d}\cap \mathbb{S}^{d-1}$ is dense in $\mathbb{S}^{d-1}$. Hence, the density of $\mathcal{A}_h$ follows.
Now we consider the case $d=2$.
For any point $x_0 \in X$, we denote by $\mathcal{A}_h(x_0)$ the subset of $\mathcal{A}_h$ consisting of unit vectors $s$ such that there exists a point $x$ in $\{ x \in X \mid d_X(x,x_0)>R_h\}$ satisfying either
It is enough to prove that for any $x_0$, $\mathcal{A}_h(x_0)=\mathbb{S}^1$. Without loss of generality, we suppose that $h(x_0)=0$. Let $Y$ be the range of the continuous function $\displaystyle x \mapsto \frac{h(x)}{|h(x)|}$, which is defined on the connected set $\{ x \in X \mid d_X(x,x_0)>R_h\}$.
Then $Y$ is a connected subset that contains $\mathbb{Q}^2 \cap \mathbb{S}^1$ since $h(n\cdot x_0)=n$ for any $n \in \bZ^d$. Suppose for contradiction, there is a unit vector $s$ such that $s \notin \mathcal{A}_h(x_0)$ and hence, $Y \subseteq \mathbb{S}^1 \setminus \{\pm s\}$. Thus, $Y$ cannot be connected, which is a contradiction.
§.§ Proof of Theorem <ref>
It suffices to prove the following claim:
Let $\phi$ and $\theta$ be two functions in $C^{\infty}_c(X)$ such that the metric distance on $X$ between the supports of these two functions is bigger than $R_h$. Let $K_{s,\phi,\theta}$ be the Schwartz kernel of the operator $\phi T_s \theta$.
Then $K_{s,\phi,\theta}$ is continuous and rapidly decaying (uniformly in $s$) on $X \times X$, i.e., for any $N>0$, we have
$$\sup_{s \in \mathbb{S}^{d-1}}|K_{s,\phi,\theta}(x,y)| \leq C (1+d_X(x,y))^{-N},$$
for some positive constant $C=C(N, \|\phi\|_{\infty}, \|\theta\|_{\infty})$.
Let $K_s(k,x,y)$ be the Schwartz kernel of the operator $T_s(k)$.
The next lemma is an analog for abelian coverings of <cit.>.
Let $\phi$ and $\theta$ be any two compactly supported functions on $X$ such that $\supp(\phi) \cap \supp(\theta) =\emptyset$. Then the following identity holds for any $(x,y) \in X \times X$:
\begin{equation*}
K_{s,\phi,\theta}(x,y)=\frac{1}{(2\pi)^d}\int_{\mathcal{O}} e^{ik\cdot (h(x)-h(y))}\phi(x)K_s(k,\pi(x),\pi(y))\theta(y)dk,
\end{equation*}
where $\pi$ is the covering map $X \rightarrow M$.
Let $\mathcal{P}$ be the subset of $C^{\infty}_c(X)$ consisting of all functions $\psi$ whose support is connected, and if $\gamma \in G$ such that $\displaystyle \supp{\psi^{\gamma}} \cap \supp{\psi} \neq \emptyset$ then $\gamma$ is the identity element of the deck group $G$. Since any compactly supported function on $X$ can be decomposed as a finite sum of functions in $\mathcal{P}$, we can assume that both $\phi$ and $\theta$ belong to $\mathcal{P}$. Then the rest is similar to the proof of <cit.>.
Another key ingredient in proving Theorem <ref> is the following result:
Let $\displaystyle \dim{M}=n$. Then for any multi-index $\alpha$ such that $|\alpha|\geq n$, $D^{\alpha}_k K_s(k,x,y)$ is a continuous function on $M \times M$. Furthermore, we have
\begin{equation*}
\sup_{(s,k,x,y) \in \mathbb{S}^{d-1} \times \mathcal{O} \times M \times M}|D_{k}^{\alpha}K_s(k,x,y)|< \infty.
\end{equation*}
Before providing the proof of Proposition <ref>, let us use it to prove Theorem <ref>.
The exponential function $e^{2\pi i\gamma \cdot h(x)}$ is $G$-periodic for any $\gamma \in G$, and hence, it is also defined on $M$. We use the same notation $e^{2\pi i\gamma \cdot h(x)}$ for the corresponding multiplication operator on $L^2(M)$. Then we can write
$$T_s(k+2\pi \gamma)=e^{-2\pi i\gamma \cdot h(x)}T_s(k)e^{2\pi i\gamma \cdot h(x)}, \quad (k,\gamma) \in \mathcal{O} \times G$$
It follows that for any multi-index $\alpha$,
\begin{equation}
\label{perboundary}
e^{i(k+2\pi \gamma)\cdot(h(x)-h(y))}\nabla_{k}^{\alpha}K_s(k+2\pi \gamma,\pi(x),\pi(y))=e^{ ik\cdot(h(x)-h(y))}\nabla_{k}^{\alpha}K_s(k,\pi(x),\pi(y)).
\end{equation}
Now we apply integration by parts to the identity in Lemma <ref> to obtain
\begin{equation}
\label{int_by_parts}
i^{N}(h(x)-h(y))^{\alpha}K_{s,\phi,\theta}(x,y)=\frac{\phi(x)\theta(y)}{(2\pi)^d}\int_{\mathcal{O}} e^{ik\cdot (h(x)-h(y))}\nabla^{\alpha}_{k} K_s(k,\pi(x), \pi(y))dk.
\end{equation}
Note that due to perboundary, when using integration by parts, we do not have any boundary term.
If $|\alpha|\geq n$, then the above integral is uniformly bounded in $(s,x,y)$ by Proposition <ref>.
When $\phi(x)\theta(y) \neq 0$, we have $d_X(x,y)>R_h$ and so, $h(x) \neq h(y)$ by Proposition <ref>.
Therefore, the kernel $K_{s,\phi, \theta}(x,y)$ is continuous on $X\ \times X$. Now fix $(x,y)$ such that $\phi(x)\theta(y) \neq 0$. Next we choose $\ell_0 \in \{1, \dots, d\}$ such that $|h_{\ell_0}(x)-h_{\ell_0}(y)|=\max_{1 \leq \ell \leq d}|h_{\ell}(x)-h_{\ell}(y)|>0$. Fix any $N \geq n$. Let $\alpha=(\alpha_1,\dots,\alpha_d)=N(\delta_{1,\ell_0},\dots,\delta_{d,\ell_0})$, where $\delta_{\cdot,\cdot}$ is the Kronecker delta. Then $|(h(x)-h(y))^{\alpha}|^{-1}=|h_{\ell_0}(x)-h_{\ell_0}(y)|^{-N} \leq d^{N/2}|h(x)-h(y)|^{-N}$. Consequently, from int_by_parts, we derive a positive constant $C$ (independent of $x,y$) such that
$$\sup_{s \in \mathbb{S}^{d-1}}|K_{s,\phi, \theta}(x,y)| \leq C|\phi(x)\theta(y)| |(h(x)-h(y))^{\alpha}|^{-1}\leq Cd^{N/2}\|\phi\|_{\infty}\|\theta\|_{\infty} |h(x)-h(y)|^{-N}.$$
Using Proposition <ref>, the above estimate becomes
\begin{equation*}
\sup_{(s,x,y) \in \mathbb{S}^{d-1}\times X \times X}(1+d_X(x,y))^{N}|K_{s,\phi, \theta}(x,y)|<\infty,
\end{equation*}
which yields the conclusion.
Back to Proposition <ref>, we first introduce several notions.
Let $\mathcal{S}(M)$ be the space of Schwartz functions on $M$. The first notion is about the order of an operator on the Sobolev scale (see e.g., <cit.>).
A linear operator $A: \mathcal{S}(M) \rightarrow \mathcal{S}(M)$ is said to be of order $\ell \in \mathbb{R}$ on the Sobolev scale $(H^{m}(M))_{m \in \mathbb{R}}$ if for every $m \in \mathbb{R}$ it can be extended to a bounded linear operator $A_{m,m-\ell} \in B(H^{m}(M), H^{m-\ell}(M))$. In this situation, we denote by the same notation $A$ any of the operators $A_{m,m-\ell}$.
A typical example of an operator of order $\ell$ on the Sobolev scale is any pseudodifferential operator of order $\ell$ acting on $M$.
Given $\ell \in \mathbb{R}$. We denote by $\mathcal{S}_{\ell}(M)$ the set consisting of families of operators $\{B_{s}(k)\}_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}}$ acting on $M$ so that the following properties hold:
* For any $(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}$, $B_s(k)$ is of order $\ell$ on the Sobolev scale $(H^{p}(M))_{p \in \mathbb{R}}$.
* For any $ p \in \mathbb{R}$, the operator $B_s(k)$ is smooth in $k$ as a $\displaystyle B(H^{p}(M), H^{p-\ell}(M))$-valued function.
* For any multi-index $\alpha$, $D^{\alpha}_k B_s(k)$ is of order $\ell-|\alpha|$ on the Sobolev scale $(H^{p}(M))_{p \in \mathbb{R}}$ and moreover, for any $p \in \mathbb{R}$, the following uniform condition holds
$$\sup_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}}\|D^{\alpha}_k B_s(k)\|_{B(H^{p}(M), H^{p-\ell+|\alpha|}(M))}<\infty.$$
It is worth giving a separate definition for the class $\displaystyle \mathcal{S}_{-\infty}(M)=\cap_{\ell \in \mathbb{R}}\mathcal{S}_{\ell}(M)$ as follows:
We denote by $\mathcal{S}_{-\infty}(M)$ the set consisting of families of smoothing operators $\{U_{s}(k)\}_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}}$ acting on $M$ so that the following properties hold:
* For any $ m_1, m_2 \in \mathbb{R}$, the operator $U_s(k)$ is smooth in $k$ as a $\displaystyle B(H^{m_1}(M), H^{m_2}(M))$-valued function.
* The following uniform condition holds for any multi-index $\alpha$:
$$\sup_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}}\|D^{\alpha}_k U_s(k)\|_{B(H^{m_1}(M), H^{m_2}(M))}<\infty.$$
We now introduce the class $\tilde{S}^{\ell}(\mathbb{T}^n)$ of parameter-dependent toroidal symbols on the $n$-dimensional torus [Note that for the case $n=d$, the class of parameter-dependent toroidal symbols was introduced in <cit.>. Nevertheless, the techniques and results on parameter-dependent toroidal pseudodifferential operators obtained in <cit.> still work similarly for the general case $n \geq 1$.].
The parameter-dependent class $\tilde{S}^{\ell}(\mathbb{T}^n)$ consists of symbols $\sigma(s,k;x,\xi)$ satisfying the following conditions:
* For each $(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}$, the function $\sigma(s,k;\cdot,\cdot)$ is a symbol of order $\ell$ on the torus $\mathbb{T}^n$ (see e.g., <cit.>).
* Consider any multi-indices $\alpha, \beta, \gamma$ and any $s \in\mathbb{S}^{d-1}$. Then the function $\sigma(s,\cdot;\cdot,\cdot)$ is smooth on $\mathcal{O} \times \mathbb{T}^n \times \mathbb{R}^n$. Furthermore, for some positive constant $C_{\alpha \beta \gamma}$ (independent of $s$,$k$,$x$,$\xi$), we have
\begin{equation*}
\label{E:parameter_class_symbols}
\sup_{s \in \mathbb{S}^{d-1}}|D^{\alpha}_{k}D^{\beta}_{\xi}D^{\gamma}_x \sigma(s,k;x,\xi)| \leq C_{\alpha \beta \gamma} (1+|\xi|)^{m-|\alpha|-|\beta|}.
\end{equation*}
We also define
$$\tilde{S}^{-\infty}(\mathbb{T}^n):=\bigcap_{\ell \in \mathbb{R}}\tilde{S}^{\ell}(\mathbb{T}^n).$$
The class of pseudodifferential operators on the torus $\mathbb{T}^n$ is also provided in the next definition.
* Given a symbol $\sigma(x,\xi)$ of order $\ell$ on the torus $\mathbb{T}^n$, the corresponding periodic pseudodifferential operator $Op(\sigma)$ is defined by
\begin{equation*}
\label{E:def_pseudo}
\left(Op(\sigma)f\right)(x):=\sum_{\xi \in \mathbb{Z}^n}\sigma(x,\xi)\tilde{f}(\xi)e^{2\pi i \xi \cdot x},
\end{equation*}
where $\tilde{f}(\xi)$ is the Fourier coefficient of $f$ at $\xi$.
* For any $\ell \in \mathbb{R} \cup \{-\infty\}$, the set of all families of periodic pseudodifferential operators $\{Op(\sigma(s,k;\cdot,\cdot))\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$, where $\sigma$ runs over the class $\tilde{S}^{\ell}(\mathbb{T}^n)$, is denoted by $Op(\tilde{S}^{\ell}(\mathbb{T}^n))$.
* It is straightforward to check from definitions and the Leibniz rule that for any $\ell_1, \ell_2 \in \mathbb{R} \cup \{-\infty\}$, if $\{A_s(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$, $\{B_s(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$ are two families of operators in the class $\mathcal{S}_{\ell_1}(M)$
and $\mathcal{S}_{\ell_2}(M)$, respectively, then the family of operators $\{A_s(k)B_s(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$ belongs to $\mathcal{S}_{\ell_1+\ell_2}(M)$.
* If the family of operators $\{B_s(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$ belongs to the class $\mathcal{S}_{\ell}(M)$ then by definition, the family of operators $\{D^{\alpha}_k B_s(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$ is in the class $\mathcal{S}_{\ell-|\alpha|}(M)$ for any multi-index $\alpha$.
* $\mathcal{S}_{-\infty}(\mathbb{T}^n)$ is the class $\mathcal{S}$ introduced in <cit.>.
* Given a family of symbols $\{\sigma(s,k;\cdot,\cdot)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}} \in \tilde{S}^{\ell}(\mathbb{T}^n)$, it follows from definitions here and boundedness on Sobolev spaces of periodic pseudodifferential operators (see e.g., <cit.>) that the corresponding family of periodic pseudodifferential operators $\{Op(\sigma(s,k;\cdot,\cdot))\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$ is in the class $\mathcal{S}_{\ell}(\mathbb{T}^n)$. In other words, $Op(\tilde{S}^{\ell}(\mathbb{T}^n)) \subseteq \mathcal{S}_{\ell}(\mathbb{T}^n)$ for any $\ell \in \mathbb{R} \cup \{-\infty\}$.
Roughly speaking, the next lemma says that we can deduce regularity of the Schwartz kernel of an operator provided that it acts “nicely" on Sobolev spaces.
Let $A$ be a bounded operator in $L^2(M)$, where $M$ is a compact $n$-dimensional manifold. Suppose that the range of $A$ is contained in $H^m(M)$, where $m>n/2$ and in addition,
\begin{equation}
\label{sobolev_est}
\|Af\|_{H^m(M)} \leq C\|f\|_{H^{-m}(M)}
\end{equation}
for all $f \in L^2(M)$.
Then $A$ is an integral operator whose kernel $K_A(x,y)$ is a continuous function on $M \times M$. In addition, the kernel of $A$ satisfies the following estimate:
\begin{equation}
\label{E:kernel_estimate_agmon}
|K_A(x,y)| \leq \gamma_0 C,
\end{equation}
where $\gamma_0$ is a constant depending only on $n$ and $m$.
For the Euclidean case, this fact is shown in <cit.>. To prove this on a general compact manifold, we simply choose a finite cover $\mathcal{U}=\{U_p\}$ of $M$ with charts $U_p\cong \mathbb{R}^n$. Then fix a smooth partition of unity $\{\varphi_p\}$ with respect to the cover $\mathcal{U}$, i.e., $\supp{\varphi_p} \Subset U_p$. We decompose $A=\sum_{p,q}\varphi_p A \varphi_{q}$. Given any $f \in L^2(M)$, the estimate sobolev_est will imply the estimate $\|\varphi_p A\varphi_q f\|_{H^m(U_p)} \leq C\|\varphi_q f\|_{H^{-m}(M)}\leq C\|f\|_{H^{-m}(U_q)}$ for any $p,q$. Hence, we obtain the conclusion of the lemma for the kernel of each operator $\varphi_{p}A\varphi_{q}$, and thus for the kernel of $A$ too.
In what follows, we will prove a nice behavior of kernels of families of operators in the class $\mathcal{S}_{\ell}(M)$ following from an application of the previous lemma.
Assume that $\ell \in \mathbb{R} \cup \{-\infty\}$ and $\{A_s(k)\}_{(s,k)}$ is a family of operators in $\mathcal{S}_{\ell}(M)$. Let $K_{A_s}(k,x,y)$ be the Schwartz kernel of the operator $A_s(k)$. Then for any multi-index $\alpha$ satisfying $|\alpha|\geq n+\ell+2$, the kernel $D^{\alpha}_k K_{A_s}(k,x,y)$ is continuous on $M \times M$ and moreover, the following estimate holds:
$$\sup_{s,k,x,y}|D^{\alpha}_k K_{A_s}(k,x,y)|<\infty.$$
For such $|\alpha|\geq n+\ell+2$, we pick some integer $m \in (n/2, (-\ell+|\alpha|)/2]$. Then by Definition <ref>, we have
$$\sup_{s,k}\|D^{\alpha}_k A_s(k)f\|_{H^m(M)} \leq C_{\alpha}\|f\|_{H^{-m}(M)}.$$
Applying Lemma <ref>, the estimates E:kernel_estimate_agmon hold for kernels $D^{\alpha}_k K_{A_s}(k,x,y)$ of the operators $D^{\alpha}_k A_s(k)$ uniformly in $(s,k)$.
The next theorem shows the inversion formula (i.e., the existence of a family of parametrices) in the case of $\mathbb{T}^n$. The proof of this theorem just comes straight from the proof of <cit.>. We omit the details.
Let $r \in \mathbb{N}$. Consider a family of $2r^{th}$ order elliptic operators $\{(\mathcal{Q}_{s}(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$ on the torus $\mathbb{T}^n$. Assume that this family is in $Op(\tilde{S}^{2r}(\mathbb{T}^n))$ and moreover, for each $(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}$, the symbol $\sigma(s,k;x,\xi)$ of the operator $\mathcal{Q}_{s}(k)$ is of the form
\begin{equation*}
\label{form_symbols_1}
\sigma(s,k;x,\xi)=L_0(s,k;x,\xi)+\tilde{\sigma}(s,k;x,\xi),
\end{equation*}
where the families of parameter-dependent symbols $\{L_0(s,k;x,\xi)\}_{(s,k)}$, $\{\tilde{\sigma}(s,k;x,\xi)\}_{(s,k)}$ are in the class $\tilde{S}^{2r}(\mathbb{T}^n)$ and $\tilde{S}^{2r-1}(\mathbb{T}^n)$, respectively. Moreover, suppose that there is some constant $A>0$ such that whenever $|\xi|>A$, we have
$$|L_0(s,k;x,\xi)|\geq 1, \quad (s,k,x) \in \mathbb{S}^{d-1}\times \mathcal{O} \times \mathbb{T}^n.$$
We call $L_0(s,k;x,\xi)$ the “leading part" of the symbol $\sigma(s,k;x,\xi)$.
Then there exists a family of parametrices $\{\mathcal{A}_{s}(k)\}_{(s,k)}$ in $Op(\tilde{S}^{-2r}(\mathbb{T}^n))$ such that
where $\mathcal{R}_{s}(k)$ is some family of smoothing operators in the class $\mathcal{S}_{-\infty}(\mathbb{T}^n)$.
To build a family of parametrices on a compact manifold, we will follow closely the strategy in <cit.> by working on open subsets of the torus first and then gluing together to get the final global result.
There exists a family of operators $\{A_s(k)\}_{(s,k) \in \mathbb{S}^{d-1}\times \mathcal{O}}$ in the class $\mathcal{S}_{-2}(M)$ and a family of operators $\{R_s(k)\}_{(s,k) \in \mathbb{S}^{d-1}\times \mathcal{O}}$ in the class $\mathcal{S}_{-\infty}(M)$ such that
Let $V_p$ ($p=1,\dots,N$) be a finite covering of the compact manifold $M$ by evenly covered coordinate charts. We also choose an open covering $U_p$ ($p=1,\dots,N$) that refine the covering $\{V_p\}$ such that $\overline{U_p} \subset V_p$ for any $p$.
We can assume that each $V_p$ is an open subset of $(0, 2\pi)^n$ in $\mathbb{R}^n$ and hence, we can view each $V_p$ as an open subset of the torus $\mathbb{T}^n$.
To simplify the notation, we will suppress the index $p=1,\dots,N$ which specifies the open sets $V_p$, $U_p$ until the final steps of the proof. Let us denote by $i_U$, $r_U$ the inclusion mapping from $i_U: U \rightarrow \mathbb{T}^n$ and the restriction mapping
$r_U: C^{\infty}(\mathbb{T}^n) \rightarrow C^{\infty}(U)$, correspondingly. We also use the same notation $L_s(k)-\lambda$ for its restrictions to the coordinate charts $V, U$ if no confusion arises. Then $(L_s(k)-\lambda)r_U$ can be considered as an operator on $\mathbb{T}^n$.
Let us first establish the following localized version of the inversion formula
There are families of symbols $\{a(s,k;x,\xi)\}_{(s,k) \in \mathbb{S}^{d-1}\times \mathcal{O}} \in \tilde{S}^{-2}(\mathbb{T}^n)$ and
$\{r(s,k;x,\xi)\}_{(s,k) \in \mathbb{S}^{d-1}\times \mathcal{O}} \in \tilde{S}^{-\infty}(\mathbb{T}^n)$ so that
$$(L_s(k)-\lambda)r_U \mathcal{A}_s(k)=r_U (I-\mathcal{R}_s(k)),$$
where $\mathcal{A}_s(k)=Op(a(s,k;\cdot,\cdot))$, $\mathcal{R}_s(k)=Op(r(s,k;\cdot,\cdot))$.
We denote by $(L_{s}(k)-\lambda)^T$ the transpose operator of $(L_{s}(k)-\lambda)$ on $V$.
Now let $\nu$ be a function in $C^{\infty}_c(V)$ such that $\nu=1$ in a neighborhood of $\overline{U}$ and $0 \leq \nu \leq 1$. Define
Observe that each operator $\mathcal{Q}_s(k)$ is a globally defined $4^{th}$ order differential operator on $\mathbb{T}^n$ with the following principal symbol
Here $\sigma_0(s,k;x,\xi)$ is the non-vanishing symbol of the elliptic operator $L_s(k)-\lambda$.
Thus, each operator $\mathcal{Q}_s(k)$ is an elliptic differential operator on $\mathbb{T}^n$.
In order to apply Theorem <ref> to the family $\{\mathcal{Q}_s(k)\}_{(s,k)}$, we need to study its family of symbols $\{\sigma(s,k;x,\xi)\}_{(s,k)}$.
On the evenly covered chart $V$, we can assume that the operator $L_s(k)-\lambda$ is of the form
$$\sum_{|\alpha| \leq 2}a_{\alpha}(x)(D+(k+i\beta_s)^{T}\cdot\nabla\tilde{h})^{\alpha},$$
for some functions $a_{\alpha} \in C^{\infty}(V)$ and $\tilde{h}$ is a smooth function obtained from the additive function $h$ through some coordinate transformation on the chart $V$.
Similarly, since $(L_{s}(k)-\lambda)^T=L(k-i\beta_s)-\lambda$, one can write the operator $(L_{s}(k)-\lambda)^T$ on $V$ as follows:
$$\sum_{|\alpha| \leq 2}\tilde{a}_{\alpha}(x)(D+(k-i\beta_s)^{T}\cdot\nabla\tilde{h})^{\alpha},$$
for some functions $\tilde{a}_{\alpha} \in C^{\infty}(V)$. Then, on $\mathbb{T}^n$, the operator $\mathcal{Q}_s(k)$ has the following form:
$$\sum_{|\alpha|, |\beta| \leq 2}a_{\alpha}(x)\tilde{a}_{\beta}(x)(D+(k+i\beta_s)^{T}\cdot\nabla\tilde{h})^{\alpha}(D+(k-i\beta_s)^{T}\cdot\nabla\tilde{h})^{\beta}\nu(x)+(1-\nu(x))\Delta^2.$$
$$L_0^{(1)}(s,k;x,\xi):=\sum_{|\alpha|=2} a_{\alpha}(x) (\xi+(k+i\beta_s)^{T}\cdot\nabla \tilde{h})^{\alpha},$$
$$L_0^{(2)}(s,k;x,\xi):=\sum_{|\beta|=2} \tilde{a}_{\beta}(x)(\xi+(k-i\beta_s)^{T}\cdot\nabla \tilde{h})^{\beta}$$
\begin{equation}
\label{form_symbols_2}
\\+(1-\nu(x))|\xi|^4.
\end{equation}
Then the symbol $\sigma(s,k;x,\xi)$ of the operator $\mathcal{Q}_{s}(k)$ can be written as
\begin{equation*}
\end{equation*}
where the family of symbols $\{\tilde{\sigma}(s,k;x,\xi)\}_{(s,k)}$ is in the class $\tilde{S}^{3}(\mathbb{T}^n)$.
Using the boundedness of $\nabla\tilde{h}$ and coefficients $a_{\alpha}$ on the support of $\nu$, we deduce that the family of the symbols of $\{\mathcal{Q}_s(k)\}_{(s,k)}$ is in $\tilde{S}^{4}(\mathbb{T}^n)$.
Thus, our remaining task is to find a constant $A>0$ such that whenever $|\xi|>A$, we obtain $|L_0(s,k;x,\xi)|>1$.
Note that by ellipticity, there are positive constants $\theta_1$, $\theta_2$ such that
$$\sum_{|\alpha|=2} a_{\alpha}(x) \geq \theta_1 |\xi|^2$$
$$\sum_{|\alpha|=2} \tilde{a}_{\alpha}(x) \geq \theta_2 |\xi|^2.$$
We define
$$A_p:=\max_{(s,k,x) \in \mathbb{S}^{d-1} \times \mathcal{O} \times \supp(\nu)}\left(|k^T \cdot \nabla\tilde{h}|^2+\theta_p^{-1}\|a\|_{\infty}|\beta_s^T \cdot \nabla\tilde{h}|^2+\theta_p^{-1}\right), \quad p=1,2.$$
Suppose that $\displaystyle |\xi|^2>2\max_{p=1,2}A_p$, then for any $p=1,2$, we have
\begin{align*}
\sqrt{\nu(x)}|L_0^{(p)}(s,k;x,\xi)| &\geq \Re\left(\sqrt{\nu(x)}L_0^{(p)}(s,k;x,\xi)\right)
\\ & \geq \sqrt{\nu(x)}\left(\theta_p|\xi+k^{T}\cdot\nabla\tilde{h}|^2-\sum_{|\alpha|=2}a_{\alpha}(x)(\beta_s^{T} \cdot \nabla \tilde{h})^{\alpha}\right)
\\ & \geq \sqrt{\nu(x)}\left(\theta_p \left(\frac{|\xi|^2}{2}-|k^T \cdot \nabla\tilde{h}|^2 \right)-\|a\|_{\infty}|\beta_s^T \cdot \nabla\tilde{h}|^2\right)\geq \sqrt{\nu(x)}.
\end{align*}
Thus, due to form_symbols_2, if $|\xi|^2>2\max_{p=1,2}A_p+1$ then $|L_0(s,k;x,\xi)|\geq(\sqrt{\nu(x)})^2+(1-\nu(x))|\xi|^4\geq 1$ as we wish.
Now we are able to apply Theorem <ref> to the family of operators $\{\mathcal{Q}_s(k)\}_{(s,k)}$, i.e., there are families of operators $\{\mathcal{B}_s(k)\}_{(s,k)}\in Op(\tilde{S}^{-4}(\mathbb{T}^n))$ and $\{\mathcal{R}_s(k)\}_{(s,k)}\in \mathcal{S}_{-\infty}(\mathbb{T}^n)$ such that
Let $\mathcal{A}_s(k):=(L_s(k)-\lambda)^T \nu \mathcal{B}_s(k)$. Since $\nu=1$ on a neighborhood of $\overline{U}$, we obtain
\begin{equation*}
\begin{split}
r_U (I-\mathcal{R}_s(k))&=r_U \mathcal{Q}_s(k)\mathcal{B}_s(k)\\
&=r_U \left((L_s(k)-\lambda)(L_s(k)-\lambda)^T\nu\mathcal{B}_s(k)+(1-\nu)\Delta^2 \mathcal{B}_s(k)\right)\\
&=r_U (L_s(k)-\lambda)(L_s(k)-\lambda)^T\nu\mathcal{B}_s(k)\\
&=(L_s(k)-\lambda) r_U (L_s(k)-\lambda)^T\nu\mathcal{B}_s(k)=(L_s(k)-\lambda) r_U \mathcal{A}_s(k).
\end{split}
\end{equation*}
In addition, $\{\mathcal{A}_s(k)\}_{(s,k)}\in Op(\tilde{S}^{-2}(\mathbb{T}^n))$ according to the composition formula <cit.>. Hence, the lemma is proved.
Let $\mu_p \in C^{\infty}_c(U_p)$ ($p=1,\dots,N$) be a partition of unity with respect to the cover $\{U_p\}_{p=1,\dots,N}$ and for any $p=1,\dots,N$, let $\nu_p$ be a function in $C^{\infty}_c(U_p)$ such that it equals $1$ on a neighborhood of $\supp(\mu_p)$. By Lemma <ref>, there are families of operators $\{\mathcal{A}^{(p)}_s(k)\}_{(s,k)} \in Op(\tilde{S}^{-2}(\mathbb{T}^n))$ and $\{\mathcal{R}^{(p)}_s(k)\}_{(s,k)}\in \mathcal{S}_{-\infty}(\mathbb{T}^n)$ such that
\begin{equation}
\label{parametrices_cover}
\end{equation}
Due to pseudolocality, $(1-\nu_p)\mathcal{A}^{(p)}_s(k)\mu_p \in \mathcal{S}_{-\infty}(\mathbb{T}^n)$. This implies that $r_{U_p}\mathcal{A}^{(p)}_s(k)\mu_p-\nu_p\mathcal{A}^{(p)}_s(k)\mu_p \in \mathcal{S}_{-\infty}(\mathbb{T}^n)$, and thus,
$$(L_s(k)-\lambda)r_{U_p}\mathcal{A}^{(p)}_s(k)\mu_p-(L_s(k)-\lambda)\nu_p\mathcal{A}^{(p)}_s(k)\mu_p \in \mathcal{S}_{-\infty}(\mathbb{T}^n).$$
By parametrices_cover, $\mu_pI-(L_s(k)-\lambda)r_{U_p}\mathcal{A}^{(p)}_{s}(k)\mu_p\in \mathcal{S}_{-\infty}(\mathbb{T}^n).$
$$\mu_p I-(L_s(k)-\lambda)\nu_p\mathcal{A}^{(p)}_s(k)\mu_p \in \mathcal{S}_{-\infty}(\mathbb{T}^n).$$
Since both operators $\mu_p I$ and $(L_s(k)-\lambda)\nu_p\mathcal{A}^{(p)}_s(k)\mu_p$ are globally defined on the manifold $M$, it follows that
\begin{equation}
\label{parametrices_cover_2}
\sum_{p} \left(\mu_p I-(L_s(k)-\lambda)\nu_p\mathcal{A}^{(p)}_s(k)\mu_p\right) \in \mathcal{S}_{-\infty}(M).
\end{equation}
Because $Op(\tilde{S}^{-2}(\mathbb{T}^n) \subset \mathcal{S}_{-2}(\mathbb{T}^n)$ (see Remark <ref>), each family of operators $\{\mathcal{A}^{(p)}_s(k)\}_{(s,k)}$ is in the class $\mathcal{S}_{-2}(\mathbb{T}^n)$ for every $p$. Since $\{\nu_p\mathcal{A}^{(p)}_s(k)\mu_p\}_{(s,k)}$ is globally defined on $M$, we also have $\{\nu_p\mathcal{A}^{(p)}_s(k)\mu_p\}_{(s,k)} \in \mathcal{S}_{-2}(M)$ for any $p$.
Now define $A_s(k):=\displaystyle \sum_p \nu_p\mathcal{A}^{(p)}_s(k)\mu_p$ and $R_s(k):=I-(L_s(k)-\lambda)A_s(k)$. Then $\{A_s(k)\}_{(s,k)} \in \mathcal{S}_{-2}(M)$ and moreover, due to parametrices_cover_2, the family of operators $\{R_s(k)\}_{(s,k)}$ is in $\mathcal{S}_{-\infty}(M)$.
The statement of the following lemma is standard.
Let $\mathcal{M}$ be a compact metric space, $\mathcal{D}$ be a domain in $\mathbb{R}^{m}$ ($m \in \mathbb{N}$) and $H_1$, $H_2$ be two infinite-dimensional separable Hilbert spaces. Let $\{T_s\}_{s \in \mathcal{M}}$ be a family of smooth maps from $\mathcal{D}$ to $B(H_1, H_2)$ such that for any multi-index $\alpha$, the map $(s,d) \mapsto D^{\alpha}_dT_s(d)$ is continuous from $\mathcal{M} \times \mathcal{D}$ to $B(H_1, H_2)$.
Suppose that there is a family of maps $\{V_s\}_{s \in \mathcal{M}}$ from $\mathcal{D}$ to $B(H_2, H_1)$
such that $V_s(d)T_s(d)=1_{H_1}$ and $T_s(d)V_s(d)=1_{H_2}$ for any $(s,d) \in \mathcal{M} \times \mathcal{D}$.
Then for each $s \in \mathcal{M}$, the map $d \in \mathcal{D} \mapsto V_s(d)$ is smooth as a $B(H_2, H_1)$-valued function. Furthermore for any multi-index $\alpha$, the map $(s,k) \mapsto D^{\alpha}_d V_s(d)$ is continuous on $\mathcal{M} \times \mathcal{D}$ as a $B(H_2, H_1)$-valued function.
We now go back to the family of operators $\{T_s(k)\}_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}$. The next statement is the main ingredient in establishing Proposition <ref>.
There is a family of operators $\{B_{s}(k)\}_{(s,k)}$ in $\mathcal{S}_{-2}(M)$ such that the family of operators $\{T_s(k)-B_s(k)\}_{(s,k)}$ belongs to $\mathcal{S}_{-\infty}(M)$.
Due to Theorem <ref>, we can find a family $\{A_s(k)\}_{(s,k)} \in \mathcal{S}_{-2}(M)$ and a family $\{R_s(k)\}_{(s,k)} \in \mathcal{S}_{-\infty}(M)$ such that
Also, from the definition of $T_s(k)$, we obtain $T_s(k)(L_s(k)-\lambda)=I-\eta(k)P(k+i\beta_s)$.
Using the above two equalities, we obtain
\begin{equation*}
\label{E:eqn_T(k)}
\end{equation*}
We recall from Section 4 that $P(k+i\beta_s)$ projects $L^2(M)$ onto the eigenspace spanned by the eigenfunction $\phi_{k+i\beta_s}$. Hence, its kernel is the following function
which is smooth due to Lemma <ref>.
Thus, the family of operators $\{\eta(k)P(k+i\beta_s)\}_{(s,k)}$ is in $\mathcal{S}_{-\infty}(M)$. Also, the family of operators $\{\eta(k)Q(k+i\beta_s)\}_{(s,k)}$ belongs to $\mathcal{S}_{0}(M)$.
We put $B_s(k):=A_s(k)-\eta(k)P(k+i\beta_s)A_s(k)$, then $\{B_s(k)\}_{(s,k)} \in \mathcal{S}_{-2}(M)$. Since $T_s(k)-B_s(k)=T_s(k)R_s(k)$, the remaining task is to check that the family of operators $\{T_s(k)R_s(k)\}_{(s,k)}$ belongs to the class $\mathcal{S}_{-\infty}(M)$.
Let us consider any two real numbers $m_1$ and $m_2$. By Lemma <ref>, the operators $L_s(k)-\lambda$ and $L_s(k)Q(k+i\beta_s)-\lambda$ are smooth in $k$ as $B(H^{m_2}(M), H^{m_2-2}(M))$-valued functions such that their derivatives with respect to $k$ are jointly continuous in $(s,k)$.
On the other hand, we can rewrite (see <cit.>):
Hence, by Lemma <ref>,
$T_s(k)$ is smooth in $k$ as a $B(H^{m_2-2}(M), H^{m_2}(M))$-valued function and its derivatives with respect to $k$ are jointly continuous in $(s,k)$. Therefore, for any multi-index $\alpha$, we have
$$\sup_{(s,k) \in \mathbb{S}^{d-1} \times \mathcal{O}}\|D^{\alpha}_k T_s(k)\|_{B(H^{m_2-2}(M), H^{m_2}(M))}<\infty.$$
Moreover since $\{R_s(k)\}_{(s,k)} \in \mathcal{S}_{-\infty}(M)$, $R_s(k)$ is smooth as a $B(H^{m_1}(M), H^{m_2-2}(M))$-valued function and for any multi-index $\alpha$,
$$\sup_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}} \|D^{\alpha}_k R_s(k)\|_{B(H^{m_1}(M), H^{m_2-2}(M))}<\infty.$$
One can deduce from the Leibniz rule that the composition $T_s(k)R_s(k)$ is smooth in $k$ as a $B(H^{m_1}(M), H^{m_2}(M))$-valued function and for any multi-index $\alpha$, the following uniform condition also holds
$$\sup_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}} \|D^{\alpha}_k (T_s(k)R_s(k))\|_{B(H^{m_1}(M), H^{m_2}(M))}<\infty.$$
Consequently, $\{T_s(k)R_s(k)\}_{(s,k)\in \mathbb{S}^{d-1} \times \mathcal{O}}\in\mathcal{S}_{-\infty}(M)$ as we wish.
We now finish this subsection.
Proposition <ref> provides us with the decomposition $T_s(k)=B_s(k)+C_s(k)$, where $\{B_{s}(k)\}_{(s,k)}\in \mathcal{S}_{-2}(M)$ and $\{C_s(k)\}_{(s,k)} \in \mathcal{S}_{-\infty}(M)$. Let $K_{B_s}(k,x,y)$, $K_{C_s}(k,x,y)$ be the Schwartz kernels of $B_s(k)$ and $C_s(k)$, correspondingly. It follows from Corollary <ref> that for any multi-index $\alpha$ satisfying $|\alpha| \geq n$, each kernel $D^{\alpha}_k K_{B_s}(k,x,y)$ is continuous on $M \times M$. Furthermore, we have
$$\sup_{(s,k,x,y) \in \mathbb{S}^{d-1} \times \mathcal{O} \times M \times M}|D^{\alpha}_k K_{B_s}(k,x,y)|<\infty.$$
A similar conclusion also holds for the family of kernels $\{K_{C_s}(k,x,y)\}_{(s,k)}$ and thus, for the family of kernels $\{K_{s}(k,x,y)\}_{(s,k)}$ too. This finishes the argument.
§ ASYMPTOTICS OF GREEN'S FUNCTIONS AND MARTIN COMPACTIFICATIONS FOR NONSYMMETRIC SECOND-ORDER PERIODIC ELLIPTIC OPERATORS
In this section, we discuss briefly analogous results for nonsymmetric $G$-equivariant[Here, without loss of generality, we assume that $G=\mathbb{Z}^d$.] second-order elliptic operators on an abelian covering $X$ below and at the generalized principal eigenvalue, which generalize the main results in <cit.>. Let us consider
now a $G$-periodic linear elliptic operator $A$ of second-order acting on $C^{\infty}(X)$ such that in any coordinate system $(U; x_1, \cdots, x_n)$, $A$ has the form:
$$A=-\sum_{1 \leq i,j \leq n}a_{ij}(x)\partial_{x_i}\partial_{x_j}+\sum_{1 \leq i \leq n}b_i(x)\partial_{x_i}+c(x),$$
where $a_{ij}, b_i, c$ are smooth, real-valued, periodic functions. The matrix $a(x):=(a_{ij}(x))_{1 \leq i,j \leq n}$ is positive definite. The generalized principal eigenvalue of $A$ is defined as follows
$$\Lambda_A=\sup\{\lambda \in \mathbb{R} \mid Au=\lambda u \hspace{4pt} \mbox{for some positive solution} \hspace{4pt} u\}.$$
Let $A^*$ be the formal adjoint operator of $A$. The generalized principal eigenvalues of $A^*$ and $A$ are equal, i.e., $\Lambda_A=\Lambda_{A^*}$. Also, the operators $A-\lambda$ and $A^*-\lambda$ are subcritical [I.e., positive Green's functions exist for these operators.] if $\lambda<\Lambda_{A}$.
Recall that a function $u$ on $X$ is called a $G$-multiplicative function with exponent $k \in \mathbb{R}^d$ if $u(g \cdot x)=e^{k \cdot g}u(x), \forall x \in X, g \in G$ (see Definition <ref>). For any $k \in \mathbb{R}^d$, it is known from <cit.> that there exists a unique real number $\Lambda_A(k)$ so that the equation $Au=\Lambda_A(k)u$ admits a positive $G$-multiplicative solution $u$ with exponent $-k$.
We extract the following results from <cit.>.
* Let $h$ be an additive function on $X$.
Then $\Lambda_A(k)$ is the principal eigenvalue of $A(ik)$ with multiplicity one, where $A(ik)$ is the operator $e^{k\cdot h(x)}Ae^{-k\cdot h(x)}$. Furthermore, $\Lambda_A=\max_{k \in \mathbb{R}^d}\Lambda_A(k)=\Lambda_A(\beta_0)$ for a unique $\beta_0 \in \mathbb{R}^d$.
* The function $\Lambda_A(k)$ is real analytic, strictly concave, bounded from above, and its gradient vanishes at only its maximum point $k=\beta_0$. The Hessian of the function $\Lambda_A(k)$ is negative definite at any $k \in \mathbb{R}^d$.
* For any $\lambda \in \mathbb{R}$, we define
\begin{equation}
\label{levelsets}
\begin{split}
K_{\lambda}&=\{k \in \mathbb{R}^d \mid \Lambda_A(k)\geq \lambda\},\\
\Gamma_{\lambda}&=\partial K_{\lambda}=\{k \in \mathbb{R}^d \mid \Lambda_A(k)=\lambda\}.
\end{split}
\end{equation}
Then $\Gamma_{\lambda}$ (resp. $K_{\lambda}$) is the set consisting of all vectors $k \in \mathbb{R}^d$ such that $Au=\lambda u$ (resp. $Au \geq \lambda u$) for some positive $G$-multiplicative function $u$ with exponent $-k$.
Moreover, if $\lambda=\Lambda_A$, $\Gamma_{\lambda}=K_{\lambda}=\{\beta_0\}$ while if $\lambda<\Lambda_{A}$, $K_{\lambda}$ is a $d$-dimensional strictly convex compact subset in $\mathbb{R}^d$ and its boundary $\Gamma_{\lambda}$ is a compact $d-1$ dimensional analytic submanifold of $\mathbb{R}^d$.
In all cases, $\Gamma_{\lambda}$ is the set of all extreme points of $K_{\lambda}$.
* We define analogous level sets $K^*_{\lambda}$, $\Gamma^*_{\lambda}$ as in levelsets for the formal adjoint operator $A^*$, . Then $K^*_{\lambda}=-K_{\lambda}$ and $\Gamma^*_{\lambda}=-\Gamma_{\lambda}$.
Also, $\Lambda_{A}(k)=\Lambda_{A^*}(-k)$ for any $k \in \mathbb{R}^d$. In particular, if $A=A^*$, $\Lambda_{A}(k)$ is an even function and $K_{\Lambda_A}=\{0\}$ (or $\beta_0=0$).
We are interested in finding the asymptotics at infinity of the Green's function $G_{\lambda}(x,y)$ of the operator $A-\lambda$, where $\lambda \in (-\infty, \Lambda_A]$.
From now on, we fix a point $x_0$ in $X$. Let $\mathcal{K}_{A,\lambda}$ be the set consisting of all positive solutions $u$ of the equation $Au=\lambda u$ such that $u$ is normalized at $x_0$, i.e., $u(x_0)=1$. We denote by $\mathcal{M}_{A, \lambda}$ the subset of $\mathcal{K}_{A,\lambda}$ containing all normalized (at $x_0$) positive $G$-multiplicative solutions with exponents in $\Gamma_{\lambda}$. It was proved in <cit.> that $\mathcal{M}_{A, \lambda}$ coincides with the set of all extreme points of the convex compact set $\mathcal{K}_{A, \lambda}$ [This result also holds for $G$-equivariant elliptic operators of second-order on a Riemannian co-compact nilpotent covering $X$, see e.g., <cit.>.]. As a consequence of the latter fact, it turns out that all such positive $G$-multiplicative solutions are exactly all minimal positive solutions of the equation $(A-\lambda)u=0$.
When $\lambda$ is below the generalized principal eigenvalue $\Lambda_A$,
Theorem <ref> (compare to Proposition <ref> and Lemma <ref>) enables us to define the following notions in a similar manner to the discussion in Subsection <ref> [The role of the function $E$ in Subsection <ref> is now played by the function $\Lambda_A$.].
For each $s \in \mathbb{S}^{d-1}$, let $\beta_s$ be the unique point in $\Gamma_{\lambda}$ such that
$$\frac{\nabla \Lambda_A(\beta_s)}{|\nabla \Lambda_A(\beta_s)|}=-s.$$
For any $k \in \mathbb{R}^d$, let $\phi_k$ and $\phi^*_k$ be periodic, positive, and normalized[Here we mean that $\phi_k(x_0)=\phi^*_k(x_0)=1$.] solutions of the equations $A(ik)u=\Lambda_A(k) u$, $A(ik)^*u^*=\Lambda_{A}(k) u^*$, respectively.
We have:
* Suppose that $d \geq 2$ and $\lambda<\Lambda_A$. Then as $d_{X}(x,y) \rightarrow \infty$, the following asymptotics of the Green's function $G_{\lambda}$ of $A-\lambda$ holds:
\begin{equation*}
\label{main_asymp_nsa}
\begin{split}
G_{\lambda}(x,y)&=\frac{e^{-(h(x)-h(y))\cdot\beta_{s}}}{(2\pi|h(x)-h(y)|)^{(d-1)/2}}\cdot\frac{|\nabla \Lambda_A(\beta_s)|^{(d-3)/2}}{\det{(-\mathcal{P}_s \Hess{(\Lambda_A)}(\beta_{s})\mathcal{P}_s)}^{1/2}} \cdot\frac{\phi_{\beta_{s}}(x)\phi^*_{\beta_{s}}(y)}{(\phi_{\beta_{s}},\phi^*_{\beta_{s}})_{L^{2}(M)}}\\&
+e^{(h(y)-h(x))\cdot \beta_{s}}O(d_X(x,y)^{-d/2}),
\end{split}
\end{equation*}
where $\displaystyle s=(h(x)-h(y))/|h(x)-h(y)| \in \mathcal{A}_h$
and $\mathcal{P}_s$ is the same projection we defined in Theorem <ref>.
* Suppose that $d \geq 3$ and $\Lambda_A=\Lambda_A(\beta_0)$. Then the minimal Green's function $G(x,y)$ of $A-\Lambda_A$ admits the following asymptotics as $d_X(x,y) \rightarrow \infty$:
\begin{equation*}
\label{main_asymp_KR_nsa}
\begin{split}
G(x,y)=\frac{\Gamma(\frac{d-2}{2})e^{-(h(x)-h(y))\cdot \beta_0}}{2\pi^{d/2}\sqrt{\det H}|H^{-1/2}(h(x)-h(y))|^{d-2}}\cdot \frac{\phi_{\beta_0}(x)\phi^*_{\beta_0}(y)}{(\phi_{\beta_0}, \phi^*_{\beta_0})_{L^2(M)}}\cdot\left(1+O\left(d_X(x,y)^{-1}\right)\right),
\end{split}
\end{equation*}
where $H=-\Hess(\Lambda_A)(\beta_0)$.
* By considering the operator $e^{\beta_0 \cdot h(x)}Ae^{-\beta_0 \cdot h(x)}$ instead of $A$, we can assume that $\beta_0=0$.
We follow the proof of Theorem <ref> (see also the outline of the proof at the end of Section <ref>). To apply the Floquet reduction of the problem as in Subsection <ref>, we need to obtain the following analog of Proposition <ref>: for any $t \in [0,1]$ and $k \in \mathbb{R}^d$, then $\lambda$ is in the resolvent set of $A(k+it\beta_s)$ if and only if $k \in 2\pi\mathbb{Z}^d$ and $t=1$. Indeed, if $t=1$, this statement follows directly from <cit.> (see also <cit.>). Otherwise, when $t\in [0,1)$, then by the concavity of $\Lambda_A$ (Theorem <ref>), one has $\Lambda(t\beta_s)\geq t\Lambda(\beta_s)+(1-t)\Lambda(0)>\lambda$. This allows us to apply <cit.> again to conclude that $\lambda$ is not in $\sigma(A(k+it\beta_s))$ for any $k \in \mathbb{R}^d$. By using this fact, for any $s \in \mathcal{A}_h$, the integral kernel $G_{s,\lambda}(x,y)$ of the operator $R_{s,\lambda}$ defined via E:R_s exists (see Lemma <ref>). Now we can repeat the argument [Note that the proof of Theorem <ref> also works in this case.] in Subsection <ref> to see that the asymptotics of $G_{s,\lambda}$ is the same as the asymptotics of the following integral (see E:formula_G0):
$$G_0(x,y)=\frac{1}{(2\pi)^d}\int_{[-\pi, \pi]^d}\frac{e^{ik\cdot (h(x)-h(y))}\eta(k)}{\Lambda_A(\beta_s-ik)-\lambda}\cdot\frac{\phi_{k+i\beta_s}(x)\phi^*_{k+i\beta_s}(y)}{(\phi_{k+i\beta_s}, \phi^*_{k+i\beta_s})_{L^2(M)}}dk,$$
where $\eta$ is a cut-off smooth function on $(-\pi, \pi)^d$ such that $\eta(k)=1$ around $k=0$ and moreover, the function $\Lambda_A(\cdot)$ has an analytic continuation to an open neighborhood of the support of $\eta$. To finish the proof, we just use Proposition <ref> to find the asymptotics of $G_0$ like in the proof of Theorem <ref>.
* This is similar to the proof of Theorem <ref>. We skip the details.
Note that when $A$ is symmetric and $\lambda$ does not exceed the bottom of $\sigma(A)$, the quasimomentum $k_0$ in Assumption A is zero due to Theorem <ref> (d), and thus, the asymptotics in Theorem <ref> and Theorem <ref> are the same one.
As an application of Theorem <ref>, we describe the Martin compactifications, Martin boundaries, and Martin integral representation for such operators (see e.g., <cit.> for some basic background on Martin boundary theory). This also generalizes Theorem 1.5 and Theorem 1.7 in <cit.>.
* Let $d \geq 2$ and $\lambda \in (-\infty, \Lambda_A)$. Then both of the Martin boundary and the minimal Martin boundary of the abelian covering $X$ for the operator $A-\lambda$ are homeomorphic to $\Gamma_{\lambda}$ (or the sphere $\mathbb{S}^{d-1}$). The Martin compactification of $X$ for $A$ is equal to $X \cup \mathbb{S}^{d-1}$, i.e., $X$ is adjoined by the sphere $\mathbb{S}^{d-1}$ at infinity. Furthermore, for any normalized positive solution $u$ of the equation $Au=\lambda u$ (i.e., $u \in \mathcal{K}_{A,\lambda}$), there exists a unique regular Borel probability measure $\mu$ on $\mathbb{S}^{d-1}$ such that
$$u(x)=\int_{\mathbb{S}^{d-1}}e^{-(h(x)-h(x_0))\cdot \beta_s}\phi_{\beta_s}(x)d\mu(s).$$
* Let $d \geq 3$ and $\Lambda_A=\Lambda_A(\beta_0)$. Then the Martin boundary and the minimal Martin boundary coincide with the set $\Gamma_{\Lambda_A}=\{\beta_0\}$. The Martin compactification is the one-point compactification of $X$. Moreover, any positive solution $u$ of the equation $Au=\Lambda_A u$ is a positive scalar multiple of the function $e^{- h(x)\cdot \beta_0}\phi_{\beta_0}(x)$.
* For any $x,y$ in $X$, we define
\begin{equation*}
\label{martin-kernel}
\begin{split}
K_{\lambda}(x,y)&=\frac{G_{\lambda}(x,y)}{G_{\lambda}(x_0,y)}, \quad y \neq x_0,\\
K_{\lambda}(x,y)&=\delta_{x,x_0}, \quad y=x_0.
\end{split}
\end{equation*}
Let us denote by $(\partial_{M}X)_{A-\lambda}$ the Martin boundary for $A-\lambda$ on $X$. Then $(\partial_{M}X)_{A-\lambda}$ consists of all equivalent classes of fundamental sequences
$\{y_m\}_m$ in $X$. Here $\{y_m\}_m$ is called fundamental if it has no accumulation point in $X$ and the sequence $\{K_{\lambda}(\cdot, y_m)\}_m$ converges uniformly on any compact subset of $X$ to a positive solution of the equation $Au=\lambda u$. Two fundamental sequences $\{y_m\}_m$ and $\{z_m\}_m$ are equivalent if on any compact subset of $X$, $\{|K_{\lambda}(\cdot, y_m)-K_{\lambda}(\cdot, z_m)|\}_m$ converges uniformly to zero.
Consider a fundamental sequence $\{y_m\}_m$ in $(\partial_{M}X)_{A-\lambda}$.
Then there exists a subsequence $\{y_{m_k}\}_k$ such that $\{h(y_{m_k})/|h(y_{m_k})|\}_k$ converges to a unit vector $s \in \mathbb{S}^{d-1}$ and $\displaystyle \lim_{k \rightarrow \infty} |h(y_{m_k})|=\infty$.
By Theorem <ref> (a), we have
\begin{equation}
\label{martin-kernel-rhs}
\lim_{k \rightarrow \infty}K_{\lambda}(x,y_{m_k})=e^{-(h(x)-h(x_0))\cdot \beta_s} \frac{\phi_{\beta_s}(x)}{\phi_{\beta_s}(x_0)}=e^{-(h(x)-h(x_0))\cdot \beta_s}\phi_{\beta_s}(x).
\end{equation}
If we denote by $K_{\lambda}(x,s)$ the right-hand side of martin-kernel-rhs, then $K_{\lambda}(\cdot,s)$ is a (minimal) positive solution in $\mathcal{M}_{A,\lambda}$.
Also, $K_{\lambda}(x,y_{m_k}) \rightarrow K_{\lambda}(x,s)$ uniformly on any compact subset of $X$. Since $K(\cdot,s_1) \neq K_{\lambda}(\cdot, s_2)$ if $s_1 \neq s_2$ in $\mathbb{S}^{d-1}$, we must have $\displaystyle \lim_{m \rightarrow \infty} |h(y_{m})|=\infty$ and $\displaystyle\lim_{m \rightarrow \infty}h(y_{m})/|h(y_{m})|=s$.
This implies that
\begin{equation}
\label{martin-boundary}
(\partial_{M}X)_{A-\lambda}=\left\{s \in \mathbb{S}^{d-1} \mid \exists \hspace{3pt} \{y_m\}_m \subset X \hspace{3pt} \mbox{such that} \hspace{3pt} |h(y_m)| \rightarrow \infty, \hspace{3pt} \frac{h(y_m)}{|h(y_m)|} \rightarrow s\right\}.
\end{equation}
The right-hand side of martin-boundary coincides with the closure of $\mathcal{A}_h$, which is $\mathbb{S}^{d-1}$ by Proposition <ref>. This proves the first statement. The latter statement follows from the Martin integral representation theorem (see e.g., <cit.>).
* In this case, by Theorem <ref> (b), the Martin kernel is equal to
$$\lim_{d_X(y,x_0) \rightarrow \infty}\frac{G(x,y)}{G(x_0,y)}=e^{-(h(x)-h(x_0))\cdot \beta_0}\phi_{\beta_0}(x).$$
This proves the statement immediately.
We remark that the integral representation type results stated in Theorem <ref> are special cases of <cit.>, which also holds for periodic elliptic operators of second-order on nilpotent Riemannian co-compact coverings.
§ CONCLUDING REMARKS
* Notice that Theorem <ref> (see also <cit.>) can be applied to operators with periodic magnetic potentials since this result does not require the realness of the operator $L$. On the other hand, the conditions that $L$ has real coefficients and the gap edge occurs at a high symmetry point of the Brillouin zone (i.e., the assumption A5) are assumed in the inside-the-gap situation, mainly because the central symmetry of the relevant dispersion branch $\lambda_j(k)$ (see Lemma <ref>) is needed for the formulation and the proof of Theorem <ref> (see also <cit.>).
* The main results in this paper can be easily carried over to the case when the band edge occurs at finitely many quasimomenta $k_0$ in the Brillouin zone (instead of assuming the condition A3) by summing the asymptotics coming from all these non-degenerate isolated extrema.
It was shown in <cit.> that for a wide class of two dimensional periodic second-order elliptic operators (including the class of operators we consider in this paper and periodic magnetic Schrödinger operators in $2D$), the extrema of any spectral band function (not necessarily spectral edges) are attained on a finite set of values of the quasimomentum in the Brillouin zone.
* The proofs of the main results go through verbatim for periodic elliptic second-order operators acting on vector bundles over the abelian covering $X$.
§ ACKNOWLEDGEMENTS
The work of the author was partly supported by the NSF grant DMS-1517938. The author is grateful to his advisor, Professor P. Kuchment, for insightful discussion and helpful comments. He also thanks the reviewer for giving constructive comments on this manuscript.
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|
1511.00212
|
Communication-avoiding algorithms allow redundant computations to
minimize the number of inter-process communications. In this paper, we
propose to exploit this redundancy for fault-tolerance purpose. We
illustrate this idea with QR factorization of tall and skinny
matrices, and we evaluate the number of failures our algorithm can
tolerate under different semantics.
§ INTRODUCTION
Faut tolerance for high performance distributed applications can be
achieved at system-level or application-level. System-level fault tolerance is transparent for the
application and requires a specific middleware that can restart the
failed processes and ensure coherent state of the application
Application-level fault tolerance requires the application itself to
handle the failures and adapt to them. Of course, it implies that the
middleware that supports the distributed execution must be robust
enough to survive the failures and provide the application with
primitives to handle them <cit.>. Moreover, it requires that
the application uses fault-tolerant algorithms that can deal with
process failures <cit.>.
Recent efforts in the MPI-3 standardization process <cit.>
defined an interface for a mechanism called User-Level Failure
Mitigation (ULFM) <cit.> and Run-Through
Stabilization <cit.>.
This paper deals with the QR factorization of tall and skinny
matrices, and provide three fault-tolerant algorithms in the context
of ULFM. We give the robustness of each algorithm, the semantics of
the fault tolerance and we detail the behavior during failure-free
execution and upon failures.
§ ALGORITHM-BASED FAULT TOLERANCE
FT-MPI <cit.> defined four error-handling semantics
that can be defined on a communicator. SHRINK consists in
reducing the size of the communicator in order to leave no hole in it
after a process of this communicator died. As a consequence, if one
process $p$ which is part of a communicator of size $N$ dies, after
the failure the communicator has $N-1$ processes numbered in
$[0,N-2]$. On the opposite, BLANK leaves a hole in the
communicator: the rank of the dead process is considered as invalid
(communications return that the destination rank is invalid), and
surviving processes keep their original ranks in $[0,N-1]$. While
these two semantics survive failures with a reduced number of
processes, REBUILD spawns a new process to replace the dead
one, giving it the place of the dead process in the communicators it
was part of, including giving it the rank of the dead process. Last,
the ABORT semantics corresponds to the usual behavior of
non-fault-tolerant applications: the surviving processes are
terminated and the application exits.
Using the first three semantics, programmers can integrate
failure-recovery strategies directly as part of the algorithm that
performs the computation. For instance, diskless
checkpointing <cit.> uses the memory of other processes to save
the state of each process. Arithmetic on the state of the processes
can be used to store the checksum of a set of
processes <cit.>. When a process fails, its state can be
recovered from the checkpoint and the states of the surviving
processes. This approach is particularly interesting for iterative
processes. Some matrix operations exhibit some properties on this
checkpoint, such as checkpoint invariant for LU
factorization <cit.>.
A proposal for run-through stabilization introduced new
constructs to handle failures at communicator-level
<cit.>. Other mechanisms, at process-level, have been
integrated as a proposal in the MPI 3.1 standard draft <cit.>. It is called user-level failure
mitigation <cit.>. Failures are detected when an operation
involving a failed process fails and returns an error. As a
consequence, operations that do not involve any failed process can
proceed unknowingly.
§ FAULT-TOLERANT, COMMUNICATION-AVOIDING ALGORITHMS
Communication-avoiding algorithms were introduced on <cit.>. The
idea is to minimize the number of inter-process communications, should
it involve additional computations. These algorithms perform well on
current architectures, ranging from multicore
architectures <cit.> to aggregations of clusters <cit.>,
because of the speed difference between computations and data
As seen in the examples cited in section <ref>, tolerating
failures requires some form of redundancy, such as checkpoints
or checksums stored in additional processes <cit.>.
In this paper, we propose to exploit the redundant computations made
by communication-avoiding algorithms for fault-tolerance purpose. In
this section we illustrate this idea with a communication-avoiding
algorithm for tall and skinny matrices (TSQR). This algorithm can be
used to compute the QR factorization of matrices with a lot of rows
and few columns, or as a panel factorization for QR
factorization <cit.>.
§.§ Computing the R with TSQR
The TSQR relies on successive steps that consist of local QR
factorizations, involving no inter-process communications, and
one inter-process communication. Initially, the matrix is decomposed
in submatrices, each process holding one submatrix. On the first step,
each process performs a QR factorization on its local submatrix. Then
odd-numbered processes send the resulting $\widetilde{R}$ to the previous
even-numbered process: rank 1 sends to rank 0, rank 3 sends to rank
2…. The algorithm itself is given by Algorithm <ref>.
Even-numbered processes concatenate the two $\widetilde{R}$
matrices by creating a new matrix whose upper half is the
$\widetilde{R}$ it has computed and whose bottom half is the
$\widetilde{R}$ it has received. Then the odd-numbered process is done
with its participation to the computation of the $R$. If the $Q$
matrix is computed, it will work again when the moment comes, after
the computation of the $R$ is done.
Even-numbered processes perform a local QR factorization of the
resulting matrix, and produce another $\widetilde{R}$. A similar
communication and concatenation step is performed between processes of
rank $r \pm 2^{step}$, if $r$ denotes the rank of each process. An
illustration of this communication, recombination and local
computation process on four processes is depicted by Figure <ref>.
At each step, half of the participating processes send their
$\widetilde{R}$ and are done. The other half receive an
$\widetilde{R}$, concatenate it with their own $\widetilde{R}$ and
perform a local QR factorization. This communication-computation
progression forms a binary reduction tree <cit.>.
in 0, 1, 2, 3
[draw,thick] ( 0, -4*) rectangle ( 1.8, -4*-3.8 );
at ( -1, -4*-.8 ) $\mathbf{P_\y}$ ;
at ( 1, -4*-.8 ) $A_\y$ ;
in 0, 1, 2, 3
[draw,thick] ( 4, -4*) rectangle ( 5.8, -4*-3.8 );
at ( 5.2, -4*-.6 ) $R_\y$ ;
at ( 4.5, -4*-2.5 ) $V_\y$ ;
[dashed, thick] ( 4, -4*) – ( 5.8, -4*-1.8 );
[ dashed, ->, thick ] (2, -4*-1) – (3.8, -4*-1 );
at ( 3, 1.5 ) QR ;
in 0, 2
[draw,thick] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_\y$ ;
+ 1;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[dashed, thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[dashed, thick] ( 8, -4*) – ( 9.8, -4*-1.8 );
[dashed, thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ dashed, ->, thick ] (6, -4*-5) – (7.8, -4*-2.8 );
[ dashed, ->, thick ] (6, -4*-1) – (7.8, -4*-1 );
at ( 7, 1.5 ) Send/Recv ;
in 0, 2
[draw,thick] ( 12, -4*) rectangle ( 13.8, -4*-3.6 );
at ( 13.2, -4*-.6 ) $R_\y'$ ;
at ( 12.5, -4*-2.5 ) ${V_\y}'$ ;
[dashed, thick] ( 12, -4*) – ( 13.8, -4*-1.8 );
[ dashed, ->, thick ] (10, -4*-1) – (11.8, -4*-1 );
at ( 11, 1.5 ) QR ;
[draw,thick] ( 16, 0 ) rectangle ( 17.8, -3.6 );
[dashed, thick] ( 16,0 ) – ( 17.8, -1.8 );
[dashed, thick] ( 16, -1.8 ) – ( 17.8, -3.6 );
[dashed, thick] ( 16, -1.8 ) – ( 17.8, -1.8 );
at ( 17.2, -.6 ) $R_0'$ ;
at ( 17.2, -2.4 ) $R_2'$ ;
[ dashed, ->, thick ] (14, -1) – (15.8, -1 );
[ dashed, ->, thick ] (14, -9) – (15.8, -3 );
at ( 15, 1.5 ) Send/Recv ;
[draw,thick] ( 20, 0 ) rectangle ( 21.8, -3.6 );
at ( 21.2, -.6 ) $R$ ;
at ( 20.5, -2.5 ) $V$ ;
[ dashed, ->, thick ] (18, -1) – (19.8, -1 );
[dashed, thick] ( 20,0 ) – ( 21.8, -1.8 );
at ( 19, 1.5 ) QR ;
figureComputing the R of a matrix using a
TSQR factorization on 4 processes.
We can see on this example that once it has sent its $\widetilde{R}$,
each process becomes idle. Eventually, process 0 is the only working
process that remains. Half of the processes are idle after the first
step, one quarter are idle after the second step, and so on until only
one process is working at the end.
Submatrix A
Q, R = A
step = 0
step I am a sender
b = step
R, b
I am a receiver
b = step
R', b
A = R, R'
Q, R = A
The root of the tree reaches this point and owns the final $R$
§.§ Redundant TSQR
We have seen in section <ref> that 1) only one process
eventually gets the resulting $R$ and 2) at each step, half of the
working processes get idle. The idea behind Redundant TSQR is
to use these spare processes to produce copies of the intermediate
$\widetilde{R}$ factors, in order to tolerate process failures during
the intermediate steps.
§.§.§ Semantics
With Redundant TSQR, at the end of the computation, all the
processes get the final $R$ matrix. If some processes crash during the
computation but enough processes survive (see
section <ref>), the surviving processes have the
final $R$ factor.
§.§.§ Algorithm
The basic idea is that when two processes communicate with each other,
instead of having one sender and one receiver that assembles the two
$\widetilde{R}$ matrices, the processes exchange their
matrices. Both of them assemble the two matrices and both of them
proceed with the local QR factorization. This algorithm is given by
Algorithm <ref>.
This algorithm is represented on four processes in
Figure <ref>. Plain lines represent the regular TSQR
pattern. During the first communication stage, the redundancies are
represented by dashed lines: $P_1$ and $P_3$ exchange data
with $P_0$ and $P_2$ respectively, and therefore obtain the same
intermediate matrices. Then same data exchange is performed during the
following step, resulting in a first level of redundancy (obtained
from the $P_0 \leftrightarrow P_2$ exchange), represented by loosely
dashed lines, and a secondary level of redundancy (obtained
from the $P_1 \leftrightarrow P_3$ exchange), represented by dashed
in 0, 1, 2, 3
[draw,thick] ( 0, -4*) rectangle ( 1.8, -4*-3.8 );
at ( -1, -4*-.8 ) $\mathbf{P_\y}$ ;
at ( 1, -4*-.8 ) $A_\y$ ;
in 0, 1, 2, 3
[draw,thick] ( 4, -4*) rectangle ( 5.8, -4*-3.8 );
at ( 5.2, -4*-.6 ) $R_\y$ ;
at ( 4.5, -4*-2.5 ) $V_\y$ ;
[ thick] ( 4, -4*) – ( 5.8, -4*-1.8 );
[ ->, thick ] (2, -4*-1) – (3.8, -4*-1 );
at ( 3, 1.5 ) QR ;
in 0, 2
[draw,thick] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_\y$ ;
+ 1;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[ thick] ( 8, -4*) – ( 9.8, -4*-1.8 );
[thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ ->, thick ] (6, -4*-5) – (7.8, -4*-2.8 );
[ ->, thick ] (6, -4*-1) – (7.8, -4*-1 );
in 1, 3
[draw,thick, dashed] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}$ ;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick, dashed] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[ thick, dashed] ( 8, -4*) – ( 9.8, -4*-1.8 );
[thick, dashed] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ ->, thick, dashed ] (6, -4*+3) – (7.8, -4*-.8 );
[ ->, thick, dashed ] (6, -4*-1) – (7.8, -4*-3 );
at ( 7, 1.5 ) Send/Recv ;
in 0, 2
[draw,thick] ( 12, -4*) rectangle ( 13.8, -4*-3.6 );
at ( 13.2, -4*-.6 ) $R_\y'$ ;
at ( 12.5, -4*-2.5 ) ${V_\y}'$ ;
[ thick] ( 12, -4*) – ( 13.8, -4*-1.8 );
[ ->, thick ] (10, -4*-1) – (11.8, -4*-1 );
in 1, 3
[draw,thick, dashed] ( 12, -4*) rectangle ( 13.8, -4*-3.6 );
at ( 13.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}'$ ;
at ( 12.5, -4*-2.5 ) ${V_{\pgfmathprintnumber{\z}}}'$ ;
[ thick, dashed] ( 12, -4*) – ( 13.8, -4*-1.8 );
[ ->, thick, dashed ] (10, -4*-1) – (11.8, -4*-1 );
at ( 11, 1.5 ) QR ;
[draw,thick] ( 16, 0 ) rectangle ( 17.8, -3.6 );
[ thick] ( 16,0 ) – ( 17.8, -1.8 );
[ thick] ( 16, -1.8 ) – ( 17.8, -3.6 );
[thick] ( 16, -1.8 ) – ( 17.8, -1.8 );
at ( 17.2, -.6 ) $R_0'$ ;
at ( 17.2, -2.4 ) $R_2'$ ;
[ ->, thick ] (14, -1) – (15.8, -1 );
[ ->, thick ] (14, -9) – (15.8, -3 );
[draw,thick,loosely dashed] ( 16, -8 ) rectangle ( 17.8, -11.6 );
[ thick, loosely dashed] ( 16,-8 ) – ( 17.8, -9.8 );
[ thick, loosely dashed] ( 16, -9.8 ) – ( 17.8, -11.6 );
[thick, loosely dashed] ( 16, -9.8 ) – ( 17.8, -9.8 );
at ( 17.2, -8.6 ) $R_0'$ ;
at ( 17.2, -10.4 ) $R_2'$ ;
[ ->, thick,loosely dashed ] (14, -1) – (15.8, -9 );
[ ->, thick,loosely dashed ] (14, -9) – (15.8, -11 );
[draw,thick, dashed ] ( 16, -4 ) rectangle ( 17.8, -7.6 );
[ thick, dashed] ( 16,-4 ) – ( 17.8, -5.8 );
[ thick, dashed] ( 16, -5.8 ) – ( 17.8, -7.6 );
[thick, dashed] ( 16, -5.8 ) – ( 17.8, -5.8 );
at ( 17.2, -4.6 ) $R_0'$ ;
at ( 17.2, -6.4 ) $R_2'$ ;
[ ->, thick, dashed ] (14, -5) – (15.8, -5 );
[ ->, thick, dashed ] (14, -13) – (15.8, -7 );
[draw,thick, dashed ] ( 16, -12 ) rectangle ( 17.8, -15.6 );
[ thick, dashed] ( 16,-12 ) – ( 17.8, -13.8 );
[ thick, dashed] ( 16, -13.8 ) – ( 17.8, -15.6 );
[thick, dashed] ( 16, -13.8 ) – ( 17.8, -13.8 );
at ( 17.2, -12.6 ) $R_0'$ ;
at ( 17.2, -14.4 ) $R_2'$ ;
[ ->, thick, dashed ] (14, -5) – (15.8, -13 );
[ ->, thick, dashed ] (14, -13) – (15.8, -15 );
at ( 15, 1.5 ) Send/Recv ;
[draw,thick] ( 20, 0 ) rectangle ( 21.8, -3.6 );
at ( 21.2, -.6 ) $R$ ;
at ( 20.5, -2.5 ) $V$ ;
[ ->, thick ] (18, -1) – (19.8, -1 );
[ thick] ( 20,0 ) – ( 21.8, -1.8 );
[draw,thick, dashed] ( 20, -4 ) rectangle ( 21.8, -7.6 );
at ( 21.2, -4.6 ) $R$ ;
at ( 20.5, -6.5 ) $V$ ;
[ ->, thick, dashed ] (18, -5) – (19.8, -5 );
[ thick, dashed] ( 20, -4 ) – ( 21.8, -5.8 );
[draw,thick, loosely dashed] ( 20, -8 ) rectangle ( 21.8, -11.6 );
at ( 21.2, -8.6 ) $R$ ;
at ( 20.5, -10.5 ) $V$ ;
[ ->, thick, loosely dashed ] (18, -9) – (19.8, -9 );
[ thick, loosely dashed] ( 20, -8 ) – ( 21.8, -9.8 );
[draw,thick, dashed] ( 20, -12 ) rectangle ( 21.8, -15.6 );
at ( 21.2, -12.6 ) $R$ ;
at ( 20.5, -14.5 ) $V$ ;
[ ->, thick, dashed ] (18, -13) – (19.8, -13 );
[ thick, dashed] ( 20, -12 ) – ( 21.8, -13.8 );
at ( 19, 1.5 ) QR ;
figureComputing the R of a matrix using
a TSQR factorization on 4 processes with redundant $\widetilde{R}$ factors.
Submatrix A
Q, R = A
step = 0
b = step
f = R, R', b
$FAIL$ == $f$
A = R, R'
Q, R = A
All the surviving processes reach this point and own the final $R$
Redundant TSQR
§.§.§ Robustness
We can see that at each step, the data exchange creates one extra copy
of each intermediate matrix. As a consequence, the redundancy rate
doubles at each step of the algorithm. Therefore, if $s$ denotes the
step number, the number of existing copies in the system is
$2^s$. Hence, this algorithm can tolerate $2^s - 1$ process failures.
We can see that the number of failures that this algorithm can
tolerate increases as the computation progresses. This fact is a
direct consequence from the fact that the number of redundant copies
of the data is multiplied by 2 at each step. For instance, the
computation can proceed if no more than 1 process have failed by the
end of step 1, no more than 3 processes have failed by the end of step
2, etc. In the meantime, the number of failures in the system increase
with time: the longer a computation lasts, the more processes will
fail <cit.>. Therefore, the robustness of this algorithm
increases with time, which is consistent with the need for robustness.
§.§.§ Behavior upon failures
When a process fails, the other processes proceed with the
execution. Processes that require data from the failed process end
their execution, and those that require data from ended processes end
theirs as well (see line <ref> of Algorithm
in 0, 1, 2, 3
[draw,thick] ( 0, -4*) rectangle ( 1.8, -4*-3.8 );
at ( -1, -4*-.8 ) $\mathbf{P_\y}$ ;
at ( 1, -4*-.8 ) $A_\y$ ;
in 0, 1, 2, 3
[draw,thick] ( 4, -4*) rectangle ( 5.8, -4*-3.8 );
at ( 5.2, -4*-.6 ) $R_\y$ ;
at ( 4.5, -4*-2.5 ) $V_\y$ ;
[ thick] ( 4, -4*) – ( 5.8, -4*-1.8 );
[ ->, thick ] (2, -4*-1) – (3.8, -4*-1 );
at ( 3, 1.5 ) QR ;
in 0, 2
[draw,thick] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_\y$ ;
+ 1;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[ thick] ( 8, -4*) – ( 9.8, -4*-1.8 );
[thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ ->, thick ] (6, -4*-5) – (7.8, -4*-2.8 );
[ ->, thick ] (6, -4*-1) – (7.8, -4*-1 );
in 1, 3
[draw,thick, dashed] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}$ ;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick, dashed] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[ thick, dashed] ( 8, -4*) – ( 9.8, -4*-1.8 );
[thick, dashed] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ ->, thick, dashed ] (6, -4*+3) – (7.8, -4*-.8 );
[ ->, thick, dashed ] (6, -4*-1) – (7.8, -4*-3 );
at ( 7, 1.5 ) Send/Recv ;
[draw,thick] ( 12, 0 ) rectangle ( 13.8, -3.6 );
at ( 13.2, -.6 ) $R_0'$ ;
at ( 12.5, -2.5 ) ${V_0}'$ ;
[ thick] ( 12, 0 ) – ( 13.8, -1.8 );
[ ->, thick ] (10, -1) – (11.8, -1 );
in 1, 3
[draw,thick, dashed] ( 12, -4*) rectangle ( 13.8, -4*-3.6 );
at ( 13.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}'$ ;
at ( 12.5, -4*-2.5 ) ${V_{\pgfmathprintnumber{\z}}}'$ ;
[ thick, dashed] ( 12, -4*) – ( 13.8, -4*-1.8 );
[ ->, thick, dashed ] (10, -4*-1) – (11.8, -4*-1 );
at ( 12.9, -9.2 ) ;
at ( 12.9, -10.3 ) CRASH;
at ( 11, 1.5 ) QR ;
at ( 16.9, -1.3 ) STOP;
[draw,thick, dashed ] ( 16, -4 ) rectangle ( 17.8, -7.6 );
[ thick, dashed] ( 16,-4 ) – ( 17.8, -5.8 );
[ thick, dashed] ( 16, -5.8 ) – ( 17.8, -7.6 );
[thick, dashed] ( 16, -5.8 ) – ( 17.8, -5.8 );
at ( 17.2, -4.6 ) $R_0'$ ;
at ( 17.2, -6.4 ) $R_2'$ ;
[ ->, thick, dashed ] (14, -5) – (15.8, -5 );
[ ->, thick, dashed ] (14, -13) – (15.8, -7 );
[draw,thick, dashed ] ( 16, -12 ) rectangle ( 17.8, -15.6 );
[ thick, dashed] ( 16,-12 ) – ( 17.8, -13.8 );
[ thick, dashed] ( 16, -13.8 ) – ( 17.8, -15.6 );
[thick, dashed] ( 16, -13.8 ) – ( 17.8, -13.8 );
at ( 17.2, -12.6 ) $R_0'$ ;
at ( 17.2, -14.4 ) $R_2'$ ;
[ ->, thick, dashed ] (14, -5) – (15.8, -13 );
[ ->, thick, dashed ] (14, -13) – (15.8, -15 );
at ( 15, 1.5 ) Send/Recv ;
[draw,thick, dashed] ( 20, -4 ) rectangle ( 21.8, -7.6 );
at ( 21.2, -4.6 ) $R$ ;
at ( 20.5, -6.5 ) $V$ ;
[ ->, thick, dashed ] (18, -5) – (19.8, -5 );
[ thick, dashed] ( 20, -4 ) – ( 21.8, -5.8 );
[draw,thick, dashed] ( 20, -12 ) rectangle ( 21.8, -15.6 );
at ( 21.2, -12.6 ) $R$ ;
at ( 20.5, -14.5 ) $V$ ;
[ ->, thick, dashed ] (18, -13) – (19.8, -13 );
[ thick, dashed] ( 20, -12 ) – ( 21.8, -13.8 );
at ( 19, 1.5 ) QR ;
figureRedundant TSQR on 4
processes with one process failure.
For instance, Figure <ref> represents the execution of
Redundant TSQR on four processes. Process $P_2$ crashes at the end of
the first step. The data it contained is also held by process $P_3$,
therefore the execution can proceed. However, process $P_0$ needs data
prom the failed process at the following step. Therefore, process
$P_0$ ends its execution. As a consequence, $P_0$ ends its
execution. At the end of the computation, the final result $R$ has
been computed by processes $P_1$ and $P_3$ and therefore, the final
result is available in spite of the failure.
§.§ Replace TSQR
§.§.§ Semantics
The semantics of Replace TSQR,is similar to the one with
Redundant TSQR: at the end of the computation, all the
processes get the final $R$ matrix. If some processes crash during the
computation but enough processes survive (see section
<ref>), the surviving processes have the final $R$
§.§.§ Algorithm
The fault-free execution of the Replace TSQR algorithm is
exactly the same as with Redundant TSQR (see section
<ref>). The data held by processes along the
reduction tree of the initial TSQR algorithm is replicated on spare
processes that would normally stop their execution.
The difference comes when a failure occurs. In this case, the process
that needs to communicate with another process gets an error when it
tries to communicate with it. Then, it finds a replica of the
process it is trying to communicate with (line
<ref> of Algorithm <ref>) and
exchanges its matrix with it. If no replica can be found alive, it
means that too many processes have failed and no extra copy of this
submatrix exist. Then the process exits. The algorithm is
described by Algorithm <ref>.
Submatrix A
Q, R = A
step = 0;
b = step
f = R, R', b
$FAIL$ == $f$
b = b
$None$ == $b$
f = R, R', b
A = R, R'
Q, R = A
All the surviving processes reach this point and own the
final $R$
Replace TSQR
§.§.§ Robustness
We have seen in section <ref> that this algorithm
can keen progressing as long as there exists at least one copy of each
submatrix. We have seen in section <ref> that at
each step $s$, the number of existing copies in the system is
$2^s$. Hence, this algorithm can tolerate $2^s - 1$ process failures,
just like the Redundant TSQR algorithm (see section
The difference between the Redundant TSQR and the Replace
TSQR is that with the former, the processes that need to
communicate with a failed process exit, whereas with the latter, they
try to find a replica. Therefore, if the root of the tree does not
die, it holds the final result $R$ at the end of the computation.
§.§.§ Behavior upon failures
If a process fails, the processes that try to communicate with it fail
to do so and try to find a replica to communicate with.
in 0, 1, 2, 3
[draw,thick] ( 0, -4*) rectangle ( 1.8, -4*-3.8 );
at ( -1, -4*-.8 ) $\mathbf{P_\y}$ ;
at ( 1, -4*-.8 ) $A_\y$ ;
in 0, 1, 2, 3
[draw,thick] ( 4, -4*) rectangle ( 5.8, -4*-3.8 );
at ( 5.2, -4*-.6 ) $R_\y$ ;
at ( 4.5, -4*-2.5 ) $V_\y$ ;
[ thick] ( 4, -4*) – ( 5.8, -4*-1.8 );
[ ->, thick ] (2, -4*-1) – (3.8, -4*-1 );
at ( 3, 1.5 ) QR ;
in 0, 2
[draw,thick] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_\y$ ;
+ 1;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[ thick] ( 8, -4*) – ( 9.8, -4*-1.8 );
[thick] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ ->, thick ] (6, -4*-5) – (7.8, -4*-2.8 );
[ ->, thick ] (6, -4*-1) – (7.8, -4*-1 );
in 1, 3
[draw,thick, dashed] ( 8, -4*) rectangle ( 9.8, -4*-3.6 );
at ( 9.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}$ ;
at ( 9.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick, dashed] ( 8, -4*-1.8 ) – ( 9.8, -4*-1.8 );
[ thick, dashed] ( 8, -4*) – ( 9.8, -4*-1.8 );
[thick, dashed] ( 8, -4*-1.8 ) – ( 9.8, -4*-3.6 );
[ ->, thick, dashed ] (6, -4*+3) – (7.8, -4*-.8 );
[ ->, thick, dashed ] (6, -4*-1) – (7.8, -4*-3 );
at ( 7, 1.5 ) Send/Recv ;
in 0
[draw,thick] ( 12, -4*) rectangle ( 13.8, -4*-3.6 );
at ( 13.2, -4*-.6 ) $R_\y'$ ;
at ( 12.5, -4*-2.5 ) ${V_\y}'$ ;
[ thick] ( 12, -4*) – ( 13.8, -4*-1.8 );
[ ->, thick ] (10, -4*-1) – (11.8, -4*-1 );
in 1, 3
[draw,thick, dashed] ( 12, -4*) rectangle ( 13.8, -4*-3.6 );
at ( 13.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}'$ ;
at ( 12.5, -4*-2.5 ) ${V_{\pgfmathprintnumber{\z}}}'$ ;
[ thick, dashed] ( 12, -4*) – ( 13.8, -4*-1.8 );
[ ->, thick, dashed ] (10, -4*-1) – (11.8, -4*-1 );
at ( 11, 1.5 ) QR ;
at ( 12.5, -9.2 ) ;
at ( 12.5, -10.3 ) CRASH;
[draw,thick] ( 16, 0 ) rectangle ( 17.8, -3.6 );
[ thick] ( 16,0 ) – ( 17.8, -1.8 );
[ thick] ( 16, -1.8 ) – ( 17.8, -3.6 );
[thick] ( 16, -1.8 ) – ( 17.8, -1.8 );
at ( 17.2, -.6 ) $R_0'$ ;
at ( 17.2, -2.4 ) $R_2'$ ;
[ ->, thick ] (14, -1) – (15.8, -1 );
[ ->, thick ] (14, -13) – (15.8, -3 );
[ ->, thick,loosely dashed ] (14, -1) – (15.8, -13 );
[draw,thick, dashed ] ( 16, -4 ) rectangle ( 17.8, -7.6 );
[ thick, dashed] ( 16,-4 ) – ( 17.8, -5.8 );
[ thick, dashed] ( 16, -5.8 ) – ( 17.8, -7.6 );
[thick, dashed] ( 16, -5.8 ) – ( 17.8, -5.8 );
at ( 17.2, -4.6 ) $R_0'$ ;
at ( 17.2, -6.4 ) $R_2'$ ;
[ ->, thick, dashed ] (14, -5) – (15.8, -5 );
[ ->, thick, dashed ] (14, -13) – (15.8, -7 );
[draw,thick, dashed ] ( 16, -12 ) rectangle ( 17.8, -15.6 );
[ thick, dashed] ( 16,-12 ) – ( 17.8, -13.8 );
[ thick, dashed] ( 16, -13.8 ) – ( 17.8, -15.6 );
[thick, dashed] ( 16, -13.8 ) – ( 17.8, -13.8 );
at ( 17.2, -12.6 ) $R_0'$ ;
at ( 17.2, -14.4 ) $R_2'$ ;
[ ->, thick, dashed ] (14, -13) – (15.8, -15 );
[ ->, thick, dashed ] (14, -5) – (15.8, -13 );
at ( 15, 1.5 ) Send/Recv ;
[draw,thick] ( 20, 0 ) rectangle ( 21.8, -3.6 );
at ( 21.2, -.6 ) $R$ ;
at ( 20.5, -2.5 ) $V$ ;
[ ->, thick ] (18, -1) – (19.8, -1 );
[ thick] ( 20,0 ) – ( 21.8, -1.8 );
[draw,thick, dashed] ( 20, -4 ) rectangle ( 21.8, -7.6 );
at ( 21.2, -4.6 ) $R$ ;
at ( 20.5, -6.5 ) $V$ ;
[ ->, thick, dashed ] (18, -5) – (19.8, -5 );
[ thick, dashed] ( 20, -4 ) – ( 21.8, -5.8 );
[draw,thick, dashed] ( 20, -12 ) rectangle ( 21.8, -15.6 );
at ( 21.2, -12.6 ) $R$ ;
at ( 20.5, -14.5 ) $V$ ;
[ ->, thick, dashed ] (18, -13) – (19.8, -13 );
[ thick, dashed] ( 20, -12 ) – ( 21.8, -13.8 );
at ( 19, 1.5 ) QR ;
figureReplace TSQR on 4
processes with one process failure.
For instance, Figure <ref> represents the execution of
Redundant TSQR on four processes. Process $P_2$ crashes at the end of
the first step. The data it contained is also held by process $P_3$,
therefore the execution can proceed. However, process $P_0$ needs data
prom the failed process at the following step. Therefore, process
$P_0$ ends its execution. As a consequence, $P_0$ fails to communicate
with $P_0$ and finds out that $P_3$ holds the same data as $P_2$. Then
$P_0$ exchanges data with $P_3$.
§.§ Self-Healing TSQR
The previous algorithms described here, Redundant TSQR (section
<ref>) and Replace TSQR (section
<ref>) proceed with the execution without the dead
processes. Here we are describing an algorithm that replaces the dead
process with a new one.
§.§.§ Semantics
With Self-Healing TSQR, at the end of the computation, all the
processes get the final $R$ matrix. If some processes crash during the
computation but enough processes survive at each step (see
section <ref>), the final number of processes is
the same as the initial number and all the processes have the
final $R$ factor.
§.§.§ Algorithm
This algorithm follows the same basic idea as Redundant TSQR
(see section <ref>) in a sense that at each step of the
computation, all the processes send or receive their $\widetilde{R}$
matrices and compute the $R$ of the resulting matrix. As a
consequence, the data required by the computation (the intermediate
submatrices represented in Figure <ref>) are
replicated. This part is described by Algorithm
<ref> with the initialization described by Algorithm
In this algorithm, the failed processes are replaced by newly spawned
ones. We have seen that the data contained by the failed process has
been replicated by the redundant computations. As a consequence, a
failed process can be recovered completely and a newly spawned process
can replace it: see Algorithm <ref>.
The fault-free execution of this algorithm is similar with the
execution represented by Figure <ref>.
Submatrix A
Q, R = A
step = 0
R = R, step
Self-Healing TSQR: initialization
Gets my data from a process that holds the same as me.
t =
R, step = t
Proceed with the computation
R = R, step
At the end of the computation, this process holds the final $R$.
Self-Healing TSQR: process restart
FunctionFunction shtsqr( A, step ):end
Q, R = A
b = step
f = R, R', b
$FAIL$ == $f$
A = R, R'
Q, R = A
All the processes reach this point and own the final $R$
Self-Healing TSQR: computation
§.§.§ Robustness
We have seen in <ref> and
<ref> that at each step $s$, the data necessary for
each process from the original algorithm is replicated $2^s$ times on
other processes. As a consequence, this algorithm can tolerate $2^s -
1$ process failures at each step $s$.
Similarly with Redundant TSQR, the robustness of the algorithm
increases as the need for robustness increases (see section
<ref>). The big difference with Redundant TSQR in
terms of robustness is that Self-Healing TSQR replaces the failed
processes. Therefore, this redundancy rate gives the number of failed
processes that can be accepted at each step. For instance, 1
process can fail at step 1 ; it will be respawned and 3 additional
processes can fail at step 2. As a consequence, the total number of
failures this algorithm can tolerate is $\sum^{p}_{k=1} 2^k$. Besides,
at each step $s$ it can tolerate $2^s - 1$ process failures.
§.§.§ Behavior upon failures
When a process fails, the process that was supposed to communicate
with it detects the failure and spawns a new process. The new process
obtains the redundant data from one of the processes that hold the
same data as the failed process. Then the computation continues
in 0, 1, 2, 3
[draw,thick] ( 0, -4*) rectangle ( 1.8, -4*-3.8 );
at ( -1, -4*-.8 ) $\mathbf{P_\y}$ ;
at ( 1, -4*-.8 ) $A_\y$ ;
in 0, 1, 2, 3
[draw,thick] ( 3.5, -4*) rectangle ( 5.3, -4*-3.8 );
at ( 4.7, -4*-.6 ) $R_\y$ ;
at ( 4, -4*-2.5 ) $V_\y$ ;
[ thick] ( 3.5, -4*) – ( 5.3, -4*-1.8 );
[ ->, thick ] (2, -4*-1) – (3.3, -4*-1 );
at ( 2.7, 1.5 ) QR ;
in 0, 2
[draw,thick] ( 7, -4*) rectangle ( 8.8, -4*-3.6 );
at ( 8.2, -4*-.6 ) $R_\y$ ;
+ 1;
at ( 8.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick] ( 7, -4*-1.8 ) – ( 8.8, -4*-1.8 );
[ thick] ( 7, -4*) – ( 8.8, -4*-1.8 );
[thick] ( 7, -4*-1.8 ) – ( 8.8, -4*-3.6 );
[ ->, thick ] (5.5, -4*-5) – (6.8, -4*-2.8 );
[ ->, thick ] (5.5, -4*-1) – (6.8, -4*-1 );
in 1, 3
[draw,thick, dashed] ( 7, -4*) rectangle ( 8.8, -4*-3.6 );
at ( 8.2, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}$ ;
at ( 8.2, -4*-2.4 ) $R_{\pgfmathprintnumber{\z}}$ ;
[thick, dashed] ( 7, -4*-1.8 ) – ( 8.8, -4*-1.8 );
[ thick, dashed] ( 7, -4*) – ( 8.8, -4*-1.8 );
[thick, dashed] ( 7, -4*-1.8 ) – ( 8.8, -4*-3.6 );
[ ->, thick, dashed ] (5.5, -4*+3) – (6.8, -4*-.8 );
[ ->, thick, dashed ] (5.5, -4*-1) – (6.8, -4*-3 );
at ( 6.2, 1.5 ) S/R ;
at ( 10.1, -9.2 ) ;
at ( 10.1, -10.3 ) CRASH;
[draw,thick] ( 10.5, 0 ) rectangle ( 12.3, -3.6 );
at ( 11.7, -.6 ) $R_0'$ ;
at ( 11, -2.5 ) ${V_0}'$ ;
[ thick] ( 10.5, 0 ) – ( 12.3, -1.8 );
[ ->, thick ] (9, -1) – (10.3, -1 );
in 1, 3
[draw,thick, dashed] ( 10.5, -4*) rectangle ( 12.3, -4*-3.6 );
at ( 11.7, -4*-.6 ) $R_{\pgfmathprintnumber{\z}}'$ ;
at ( 11, -4*-2.5 ) ${V_{\pgfmathprintnumber{\z}}}'$ ;
[ thick, dashed] ( 10.5, -4*) – ( 12.3, -4*-1.8 );
[ ->, thick, dashed ] ( 9, -4*-1) – (10.3, -4*-1 );
at ( 11, 1.5 ) QR ;
[rotate=90] at ( 11.8, -9.8 ) respawn;
[ ->, thick] ( 12.5, -13) – (13, -11.8 );
[rotate = 65] at ( 13.4, -12.8 ) copy;
[draw,thick] ( 12.5, -8 ) rectangle ( 14.3, -11.6 );
at ( 13.7, -8.6 ) $R_2'$ ;
at ( 13, -10.5 ) ${V_2}'$ ;
[ thick] ( 12.5, -8 ) – ( 14.3, -9.8 );
[draw,thick] ( 16.5, 0 ) rectangle ( 18.3, -3.6 );
[ thick] ( 16.5,0 ) – ( 18.3, -1.8 );
[ thick] ( 16.5, -1.8 ) – ( 18.3, -3.6 );
[thick] ( 16.5, -1.8 ) – ( 18.3, -1.8 );
at ( 17.7, -.6 ) $R_0'$ ;
at ( 17.7, -2.4 ) $R_2'$ ;
[ ->, thick ] (13, -1) – (16.3, -1 );
[ ->, thick ] (14.5, -9) – (16.3, -3 );
[draw,thick,loosely dashed] ( 16.5, -8 ) rectangle ( 18.3, -11.6 );
[ thick, loosely dashed] ( 16.5,-8 ) – ( 18.3, -9.8 );
[ thick, loosely dashed] ( 16.5, -9.8 ) – ( 18.3, -11.6 );
[thick, loosely dashed] ( 16.5, -9.8 ) – ( 18.3, -9.8 );
at ( 17.7, -8.6 ) $R_0'$ ;
at ( 17.7, -10.4 ) $R_2'$ ;
[ ->, thick,loosely dashed ] (13, -1) – (16.3, -9 );
[ ->, thick,loosely dashed ] (14.5, -9) – (16.3, -11 );
[draw,thick, dashed ] ( 16.5, -4 ) rectangle ( 18.3, -7.6 );
[ thick, dashed] ( 16.5,-4 ) – ( 18.3, -5.8 );
[ thick, dashed] ( 16.5, -5.8 ) – ( 18.3, -7.6 );
[thick, dashed] ( 16.5, -5.8 ) – ( 18.3, -5.8 );
at ( 17.7, -4.6 ) $R_0'$ ;
at ( 17.7, -6.4 ) $R_2'$ ;
[ ->, thick, dashed ] (13, -5) – (16.3, -5 );
[ ->, thick, dashed ] (14, -13) – (16.3, -7 );
[draw,thick, dashed ] ( 16.5, -12 ) rectangle ( 18.3, -15.6 );
[ thick, dashed] ( 16.5,-12 ) – ( 18.3, -13.8 );
[ thick, dashed] ( 16.5, -13.8 ) – ( 18.3, -15.6 );
[thick, dashed] ( 16.5, -13.8 ) – ( 18.3, -13.8 );
at ( 17.7, -12.6 ) $R_0'$ ;
at ( 17.7, -14.4 ) $R_2'$ ;
[ ->, thick, dashed ] (13, -5) – (16.3, -13 );
[ ->, thick, dashed ] (14, -13) – (16.3, -15 );
at ( 15.5, 1.5 ) S/R ;
[draw,thick] ( 20, 0 ) rectangle ( 21.8, -3.6 );
at ( 21.2, -.6 ) $R$ ;
at ( 20.5, -2.5 ) $V$ ;
[ ->, thick ] (18.5, -1) – (19.8, -1 );
[ thick] ( 20,0 ) – ( 21.8, -1.8 );
[draw,thick, dashed] ( 20, -4 ) rectangle ( 21.8, -7.6 );
at ( 21.2, -4.6 ) $R$ ;
at ( 20.5, -6.5 ) $V$ ;
[ ->, thick, dashed ] (18.5, -5) – (19.8, -5 );
[ thick, dashed] ( 20, -4 ) – ( 21.8, -5.8 );
[draw,thick, loosely dashed] ( 20, -8 ) rectangle ( 21.8, -11.6 );
at ( 21.2, -8.6 ) $R$ ;
at ( 20.5, -10.5 ) $V$ ;
[ ->, thick, loosely dashed ] (18.5, -9) – (19.8, -9 );
[ thick, loosely dashed] ( 20, -8 ) – ( 21.8, -9.8 );
[draw,thick, dashed] ( 20, -12 ) rectangle ( 21.8, -15.6 );
at ( 21.2, -12.6 ) $R$ ;
at ( 20.5, -14.5 ) $V$ ;
[ ->, thick, dashed ] (18.5, -13) – (19.8, -13 );
[ thick, dashed] ( 20, -12 ) – ( 21.8, -13.8 );
at ( 19, 1.5 ) QR ;
figureSelf-Healing TSQR on 4
processes with one process failure.
|
1511.00496
|
Panyushev conjecture]
The reverse operator orbits on $\Delta(1)$ and a conjecture of
Institute of Mathematics, Hunan University,
Changsha 410082, China
We verify conjecture 5.11 of Panyushev [Antichains in weight posets associated with gradings of simple
Lie algebras, Math Z 281(3):1191–1214, 2015].
[2010]Primary 17B20, 05Exx
§ INTRODUCTION
Let $\frg$ be a finite-dimensional simple Lie algebra over $\bbC$. Fix a Cartan subalgebra $\frh$ of $\frg$.
The associated root system is $\Delta=\Delta(\frg, \frh)\subseteq\frh_{\bbR}^*$. Recall that a decomposition
\begin{equation}\label{grading}
\frg=\bigoplus_{i\in \bbZ}\frg(i)
\end{equation}
is a $\bbZ$-grading of $\frg$ if $[\frg(i), \frg(j)]\subseteq \frg(i+j)$ for any $i, j\in\bbZ$.
In particular, in such a case, $\frg(0)$ is a Lie subalgebra of $\frg$. Since each derivation of $\frg$ is inner, there exists $h_0\in\frg(0)$ such that $\frg(i)=\{x\in\frg\mid [h_0, x]=i x\}$. The element $h_0$ is said to be defining for the grading (<ref>). Without loss of generality, one may assume that $h_0\in\frh$. Then $\frh\subseteq\frg(0)$. Let $\Delta(i)$ be the set of roots in $\frg(i)$. Then we can
choose a set of positive roots $\Delta(0)^+$ for $\Delta(0)$ such that
\Delta^+ :=\Delta(0)^+\sqcup \Delta(1)\sqcup \Delta(2)\sqcup \cdots
is a set of positive roots of $\Delta(\frg, \frh)$. Let $\Pi$ be the
corresponding simple roots, and put $\Pi(i)=\Delta(i)\cap \Pi$. Note
that the grading (<ref>) is fully determined by
$\Pi=\bigsqcup_{i\geq 0} \Pi(i)$. We refer the reader to Ch. 3, 3
of <cit.> for generalities on gradings of Lie algebras. Each
$\Delta(i)$, $i\geq 1$, inherits a poset structure from the usual
one of $\Delta^+$. That is, let $\alpha$ and $\beta$ be two roots of
$\Delta(i)$, then $\beta\geq\alpha$ if and only if $\beta-\alpha$ is
a nonnegative integer combination of simple roots.
Recently, Panyushev initiated the study of the rich structure of
$\Delta(1)$ in <cit.>. In particular, he raised five
conjectures concerning the $\mathcal{M}$-polynomial,
$\mathcal{N}$-polynomial and the reverse operator of $\Delta(1)$.
Note that Conjectures 5.1, 5.2 and 5.12 there have been solved by
Weng and the author <cit.>. The current paper aims
to handle conjecture 5.11 of <cit.>. Let us prepare more notation.
Recall that a subset $I$ of a finite poset $(P, \leq)$ is a
lower (resp., upper) ideal if $x\leq y$ in $P$
and $y\in I$ (resp. $x\in I$) implies that $x\in I$ (resp. $y\in
I$). We collect the lower ideals of $P$ as $J(P)$, which is
partially ordered by inclusion. A subset $A$ of $(P, \leq)$ is an
antichain if any two elements in $A$ are non-comparable under
$\leq$. We collect the antichains of $P$ as $\mathrm{An}(P)$. For
any $x\in P$, let $I_{\leq x}=\{y\in P\mid y\leq x\}$. Given an
antichain $A$ of $P$, let $I(A)=\bigcup_{a\in A} I_{\leq a}$. The
reverse operator $\mathfrak{X}$ is defined by
$\mathfrak{X}(A)=\min (P\setminus I(A))$. Since antichains of $P$
are in bijection with lower (resp. upper) ideals of $P$, the reverse
operator acts on lower (resp. upper) ideals of $P$ as well. Note
that the current $\mathfrak{X}$ is inverse to the reverse operator
$\mathfrak{X}^{\prime}$ in Definition 1 of <cit.>, see Lemma
<ref>. Thus replacing $\mathfrak{X}^{\prime}$ by $\mathfrak{X}$ does not affect our
forthcoming discussion on orbits.
We say the $\bbZ$-grading (<ref>) is extra-special if
\begin{equation}\label{extra-special}
\frg=\frg(-2)\oplus \frg(-1) \oplus \frg(0) \oplus \frg(1)
\oplus \frg(2) \mbox{ and }\dim\frg(2)=1,
\end{equation}
Up to conjugation, any simple Lie algebra $\frg$ has a unique extra-special $\bbZ$-grading. Without loss of generality, we assume that $\Delta(2)=\{\theta\}$ , where $\theta$ is the highest root of $\Delta^+$.
Namely, we may assume that the grading (<ref>) is defined by the element $\theta^{\vee}$, the dual root of $\theta$. In such a case, we have
\begin{equation}\label{Delta-one}
\Delta(1)=\{\alpha\in\Delta^+\mid (\alpha, \theta^{\vee})=1\}.
\end{equation}
Let $\mathrm{ht}$ be the height function. Recall that $h:=\mathrm{ht}(\theta)+1$ is the Coxeter number
of $\Delta$. Let $h^*$ be the dual Coxeter number of
$\Delta$. That is, $h^*$ is the height of
$\theta^{\vee}$ in $\Delta^{\vee}$. As noted on p. 1203 of <cit.>,
we have $|\Delta(1)|=2h^*-4$. We call a lower (resp. upper) ideal
$I$ of $\Delta(1)$ Lagrangian if $|I|=h^*-2$. Write
$\Delta_l$ (resp. $\Pi_l$) for the set of all (resp.
simple) long roots. In the simply-laced cases, all
roots are assumed to be both long and short. Note that $\theta$ is
always long, while $\theta^{\vee}$ is always short.
Now Conjecture 5.11 of <cit.> is stated as follows.
Panyushev conjecture. In any extra-special
$\bbZ$-grading of $\frg$, the number of
$\mathfrak{X}_{\Delta(1)}$-orbits equals $|\Pi_l|$, and each orbit
is of size $h-1$. Furthermore, if $h$ is even (which only excludes the case $A_{2k}$ where $h=2k+1$), then each
$\mathfrak{X}_{\Delta(1)}$-orbit contains a unique Lagrangian lower
Originally, the conjecture is stated in terms of upper ideals and
the reverse operator $\mathfrak{X}^{\prime}$. One agrees that we can
equivalently phrase it using lower ideals and $\mathfrak{X}$. The
main result of the current paper is the following.
Panyushev conjecture is true.
After collecting necessary preliminaries in Section 2, the above
theorem will be proven in Section 3. Moreover, we note that by our
calculations in Section 3, one checks easily that for any
extra-special $1$-standard $\bbZ$-grading of $\frg$, all the
statements of Conjecture 5.3 in <cit.> hold.
Notation. Let $\bbN =\{0, 1, 2, \dots\}$, and let
$\mathbb{P}=\{1, 2, \dots\}$. For each $n\in\mathbb{P}$, $[n]$
denotes the poset $(\{1, 2, \dots, n\}, \leq)$.
§ PRELIMINARIES
Let us collect some preliminary results in this section. Firstly,
let us compare the two reverse operators. Let $(P, \leq)$ be any
finite poset. For any $x\in P$, let $I_{\geq x}=\{y\in P\mid y\geq
x\}$. For any antichain $A$ of $P$, put $I_{+}(A)=\bigcup_{a\in A}
I_{\geq a}$. Recall that in Definition 1 of <cit.>, the reverse
operator $\mathfrak{X}^{\prime}$ is given by
$\mathfrak{X}^{\prime}(A)=\max (P\setminus I_{+}(A))$.
The operators $\mathfrak{X}$ and $\mathfrak{X}^{\prime}$ are
inverse to each other.
Take any antichain $A$ of $P$, note that
I(A)))=P\setminus I(A)\mbox{ and } I(\max(P\setminus
I_{+}(A)))=P\setminus I_{+}(A).
Then the lemma follows.
Let $(P_i,\leq), i=1, 2$ be two finite posets. One can define a
poset structure on $P_1\times P_2$ by setting $(u_1, v_1)\leq (u_2,
v_2)$ if and only if $u_1\leq u_2$ in $P_1$ and $v_1\leq v_2$ in
$P_2$. We simply denote the resulting poset by $P_1 \times P_2$. The
following well-known lemma describes the lower ideals of
$[m]\times P$.
Let $P$ be a finite poset. Let $I$ be a subset of $[m]\times P$. For
$1\leq i\leq m$, denote $I_i=\{a\in P\mid (i, a)\in I\}$. Then $I$
is a lower ideal of $[m]\times P$ if and only if each $I_i$ is a
lower ideal of $P$, and $I_m\subseteq I_{m-1}\subseteq \cdots
\subseteq I_{1}$.
In this section, by a finite graded poset we always mean a
finite poset $P$ with a rank function $r$ from $P$ to the positive
integers $\mathbb{P}$ such that all the minimal elements have rank
$1$, and $r(x)=r(y)+1$ if $x$ covers $y$. In such a case, let $P_i$
be the set of elements in $P$ with rank $i$. The sets $P_i$ are said
to be the rank levels of $P$. Suppose that
$P=\bigsqcup_{j=1}^{d} P_j$. Let $P_0$ be the empty set $\emptyset$.
Put $L_i=\bigsqcup_{j=1}^{i} P_j$ for $1\leq j\leq d$, and let $L_0$ be the empty set.
We call those $L_i$ rank level lower ideals.
Let $\mathfrak{X}$ be the reverse operator on $[m]\times P$. In
view of Lemma <ref>, we denote by $(I_1, \cdots,
I_m)$ a general lower ideal of $[m]\times P$, where each $I_i\in
J(P)$ and $I_m\subseteq \cdots \subseteq I_{1}$. We say that the
lower ideal $(I_1, \cdots, I_m)$ is full rank if each $I_i$
is a rank level lower ideal of $P$. Let $\mathcal{O}(I_1, \cdots,
I_m)$ be the $\mathfrak{X}_{[m]\times P}$-orbit of $(I_1, \cdots,
I_m)$. The following lemma will be helpful in determining
$\mathfrak{X}_{[m]\times P}$-orbits consisting of rank level lower
Keep the notation as above. Then
for any $n_0\in \bbN$, $n_i\in\mathbb{P}$ ($1\leq i\leq s$) such that $\sum_{i=0}^{s} n_i =m$, we have
\begin{equation}\label{rank-level}
\mathfrak{X}_{[m]\times P}(L_d^{n_0}, L_{i_1}^{n_1}, \cdots, L_{i_s}^{n_s})=
(L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \cdots, L_{i_s+1}^{n_{s-1}}, L_0^{n_s-1}),
\end{equation}
where $0\leq i_s<\cdots <i_1<d$, $L_d^{n_0}$ denotes $n_0$ copies of $L_d$ and so on.
Note that under the above assumptions, $(L_d^{n_0}, L_{i_1}^{n_1}, \cdots, L_{i_s}^{n_s})$ is a lower ideal of $[m]\times P$ in view of Lemma <ref>. Then analyzing the minimal elements of $([m]\times P)\setminus (L_d^{n_0}, L_{i_1}^{n_1}, \cdots, L_{i_s}^{n_s})$ leads one to (<ref>).
Let $(I_1, \cdots, I_m)$ be an arbitrary lower ideal of $[m]\times P$.
Then $(I_1, \cdots, I_m)$ is full rank if and only if each lower ideal
in the orbit $\mathcal{O}(I_1, \cdots, I_m)$ is full rank.
Use Lemma <ref>.
The above lemma tells us that there are two types of $\mathfrak{X}$-orbits: in the first type each lower ideal is full rank, while in the second type each lower ideal is not. We call them type I and type II, respectively.
For any $n\geq 2$, let $K_{n-1}=[n-1]\oplus([1]\sqcup [1])\oplus [n-1]$ (the ordinal sum, see
p. 246 of <cit.>). We label the elements of $K_{n-1}$ by $1$, $2$, $\cdots$,
$n-1$, $n$, $n^{\prime}$, $n+1$, $\cdots$, $2n-2$, $2n-1$. Figure 1
illustrates the labeling for the Hasse diagram of $K_3$. Note that $L_i$ ($0\leq i\leq 2n-1$) are all the full rank lower ideals. For instance, we have $L_{n}=\{1, 2, \cdots, n, n^{\prime}\}$.
Moreover, we put $I_{n}=\{1, \cdots, n-1,
n\}$ and $I_{n^{\prime}}=\{1, \cdots, n-1, n^{\prime}\}$. The following lemma will be helpful in analyzing the $\mathfrak{X}_{[m]\times K_{n-1}}$-orbits of type II.
Fix $n_0\in \bbN$, $n_i\in\mathbb{P}$ ($1\leq i\leq s$),
$m_j\in\mathbb{P}$ ($0\leq j\leq t$) such that $\sum_{i=0}^{s} n_i +
\sum_{j=0}^{t} m_j=m$. Take any $0\leq j_t< \cdots<j_1<n\leq
i_s<\cdots <i_1<2n-1$, we have
\begin{align*}
\mathfrak{X}_{[m]\times K_{n-1}}&(L_{2n-1}^{n_0}, L_{i_1}^{n_1}, \cdots, L_{i_s}^{n_s}, I_n^{m_0}, L_{j_1}^{m_1}, \cdots, L_{j_t}^{m_t})=\\
( L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \cdots, L_{i_s+1}^{n_{s-1}}, I_{n^{\prime}}^{n_s}, L_{j_1+1}^{m_0},
L_{j_2+1}^{m_1}, \cdots, L_{j_t+1}^{m_{t-1}}, L_0^{m_t-1} ) & \mbox { if } j_1 < n-1;\\
( L_{i_1+1}^{n_0+1}, L_{i_2+1}^{n_1}, \cdots, L_{i_s+1}^{n_{s-1}}, L_{n}^{n_s}, I_n^{m_0},
\, \, \, \, L_{j_2+1}^{m_1}, \cdots, L_{j_t+1}^{m_{t-1}}, L_0^{m_t-1} )& \mbox { if } j_1 = n-1.
\end{cases}
\end{align*}
Analyzing the minimal elements of
$$([m]\times K_{n-1})\setminus (L_{2n-1}^{n_0}, L_{i_1}^{n_1}, \cdots, L_{i_s}^{n_s}, I_n^{m_0}, L_{j_1}^{m_1}, \cdots, L_{j_t}^{m_t})$$
leads one to the desired expression.
The labeled Hasse diagram of $K_3$
§ PANYUSHEV CONJECTURE
This section is devoted to proving Theorem <ref>.
Proof of Theorem <ref>. Note that when
$\frg$ is $A_n$, the extra-special $\Delta(1)\cong [n-1]\sqcup
[n-1]$; when $\frg$ is $C_n$, the extra-special $\Delta(1)\cong
[2n-2]$. One can verify Theorem <ref> for these two cases
without much effort. We omit the details.
For $\frg=B_n$, the extra-special $\Delta(1)= [2]\times [2n-3]$.
Now $|\Pi_{l}|=n-1$, $h-1=2n-1$, and $h^*-2=2n-3$. As in Section 2,
let $L_i$ ($0\leq i\leq 2n-3$) be the rank level lower ideals. For
simplicity, we simply denote $\mathfrak{X}_{[2]\times [2n-3]}$ by
$\mathfrak{X}$. For any $1\leq i\leq n-2$, let us analyze the type I
$\mathfrak{X}$-orbit $\mathcal{O}(L_i, L_i)$ via the aid of Lemma
\begin{align*}
\mathfrak{X}(L_i, L_i)&=(L_{i+1}, L_0),\\
\mathfrak{X}^{2n-4-i}(L_{i+1}, L_0)&=(L_{2n-3}, L_{2n-4-i}),\\
\mathfrak{X}(L_{2n-3}, L_{2n-4-i})&=(L_{2n-3-i}, L_{2n-3-i}),\\
\mathfrak{X}(L_{2n-3-i}, L_{2n-3-i})&=(L_{2n-2-i}, L_{0}),\\
\mathfrak{X}^{i-1}(L_{2n-2-i}, L_{0})&=(L_{2n-3}, L_{i-1}),\\
\mathfrak{X}(L_{2n-3}, L_{i-1})&=(L_{i}, L_{i}).
\end{align*}
Thus $\mathcal{O}(L_i, L_i)$ consists of $2n-1$ elements. Moreover,
in this orbit, $(L_{2n-2-\frac{i+1}{2}}, L_{\frac{i-1}{2}})$ is the
unique ideal with size $2n$ when $i$ is odd, $(L_{n+\frac{i}{2}-1},
L_{n-\frac{i}{2}-2})$ is the unique ideal with size $2n$ when $i$ is
even. Similarly, the orbit $\mathcal{O}(L_0, L_0)$ consists of
$2n-1$ elements and contains a unique ideal with size $2n$:
$(L_{n-1}, L_{n-2})$. Since there are $(n-1)(2n-1)$ lower ideals in
$[2]\times [2n-3]$ by Lemma <ref>, one sees that
all the $\mathfrak{X}$-orbits have been exhausted, and Theorem
<ref> holds for $B_{n}$.
Let us consider $D_{n+2}$, where the extra-special $\Delta(1)\cong
[2]\times K_{n-1}$. We adopt the notation as in Section 2. For simplicity,
we write $\mathfrak{X}_{[2]\times K_{n-1}}$ by $\mathfrak{X}$. We propose
the following.
Claim. $\mathcal{O}(L_i, L_i)$, $0\leq i\leq n-1$,
$\mathcal{O}(I_n, I_n)$, and $\mathcal{O}(I_{n^{\prime}},
I_{n^{\prime}})$ exhausts the orbits of $\mathfrak{X}$ on $[2]\times
K_{n-1}$. Moreover, each orbit has size $2n+1$ and contains a unique
lower ideal with size $2n$.
Indeed, firstly, for any $0\leq i\leq n-1$, observe that by Lemma
<ref>, we have
\begin{align*}
\mathfrak{X}(L_i, L_i)&=(L_{i+1}, I_0),\\
\mathfrak{X}^{2n-i-2}(L_{i+1}, L_0)&=(L_{2n-1}, L_{2n-i-2}),\\
\mathfrak{X}(L_{2n-1}, L_{2n-i-2})&=(L_{2n-i-1}, L_{2n-i-1}),\\
\mathfrak{X}(L_{2n-i-1}, L_{2n-i-1})&=(L_{2n-i}, L_{0}),\\
\mathfrak{X}^{i-1}(L_{2n-i}, L_{0})&=(L_{2n-1}, L_{i-1}),\\
\mathfrak{X}(L_{2n-1}, L_{i-1})&=(L_{i}, L_{i}).
\end{align*}
Thus the type I orbit $\mathcal{O}(L_i, L_i)$ consists of $2n+1$ elements. Moreover,
in this orbit, $(L_{2n-i+\frac{i-1}{2}}, L_{\frac{i-1}{2}})$ is the
unique ideal with size $2n$ when $i$ is odd, $(L_{n+\frac{i}{2}},
L_{n-\frac{i}{2}-1})$ is the unique ideal with size $2n$ when $i>0$
is even, while $(L_{n}, L_{n-1})$ is the unique ideal
with size $2n$ when $i=0$.
Secondly, assume that $n$ is even and let us analyze the orbit
$\mathcal{O}(I_n, I_n)$. Indeed, by Lemma <ref>, we have
\begin{align*}
\mathfrak{X}(I_n, I_n)&=(I_{n^{\prime}}, L_0),\\
\mathfrak{X}^{n-1}(I_{n^{\prime}}, L_0)&=(I_{n}, L_{n-1}),\\
\mathfrak{X}(I_{n}, L_{n-1})&=(L_{n}, I_{n}),\\
\mathfrak{X}^{n-1}(L_{n}, I_{n})&=(L_{2n-1}, I_{n^{\prime}}),\\
\mathfrak{X}(L_{2n-1}, I_{n^{\prime}})&=(I_{n}, I_{n}).
\end{align*}
Thus the type II orbit $\mathcal{O}(I_n, I_n)$ consists of $2n+1$ elements. Moreover,
in this orbit, $(I_n, I_n)$ is the unique ideal with size $2n$. The
analysis of the orbit $\mathcal{O}(I_{n^{\prime}}, I_{n^{\prime}})$
is entirely similar.
Finally, assume that $n$ is odd and let us analyze the orbit
$\mathcal{O}(I_n, I_n)$. Indeed, by Lemma <ref>, we have
\begin{align*}
\mathfrak{X}(I_n, I_n)&=(I_{n^{\prime}}, L_0),\\
\mathfrak{X}^{n-1}(I_{n^{\prime}}, L_0)&=(I_{n^{\prime}}, L_{n-1}),\\
\mathfrak{X}(I_{n^{\prime}}, L_{n-1})&=(L_{n}, I_{n^{\prime}}),\\
\mathfrak{X}^{n-1}(L_{n}, I_{n^{\prime}})&=(L_{2n-1}, I_{n^{\prime}}),\\
\mathfrak{X}(L_{2n-1}, I_{n^{\prime}})&=(I_{n}, I_{n}).
\end{align*}
Thus the type II orbit $\mathcal{O}(I_n, I_n)$ consists of $2n+1$ elements. Moreover,
in this orbit, $(I_n, I_n)$ is the unique ideal with size $2n$. The
analysis of the orbit $\mathcal{O}(I_{n^{\prime}}, I_{n^{\prime}})$
is entirely similar.
To sum up, we have verified the claim since there are $(n+2)(2n+1)$
lower ideals in $[2]\times K_{n-1}$ by Lemma <ref>.
Note that $|\Pi_{l}|=n+2$, $h=h^*=2n+2$ for $\frg=D_{n+2}$, one sees that Theorem
<ref> holds for $D_{n+2}$.
Theorem <ref> has been verified for all exceptional Lie
algebras using . We only present the details for
$E_6$, where $\Delta(1)=[\alpha_2]$, and the Dynkin diagram is as follows.
Note that $|\Pi_l|=6$,
$h-1=11$, $h^*-2=10$. On the other hand, $\mathfrak{X}$ has six
orbits on $\Delta(1)$, each has $11$ elements. Moreover, the size of
the lower ideals in each orbit is distributed as follows:
$\bullet$ $0, 1, 2, 4, 7, \textbf{10}, 13, 16, 18, 19, 20$;
$\bullet$ $3, 4, 5, 6, 9, \textbf{10}, 11, 14, 15, 16, 17$;
$\bullet$ $3, 4, 5, 6, 9, \textbf{10}, 11, 14, 15, 16, 17$;
$\bullet$ $7, 7, 8, 8, 9, \textbf{10}, 11, 12, 12, 13, 13$;
$\bullet$ $5, 6, 6, 8, 9, \textbf{10}, 11, 12, 14, 14, 15$;
$\bullet$ $7, 7, 8, 8, 9, \textbf{10}, 11, 12, 12, 13, 13$.
One sees that each orbit has a unique Lagrangian lower ideal.
This finishes the proof of Theorem <ref>. Acknowledgements The research is supported by
the National Natural Science Foundation of China (grant no.
11571097) and the Fundamental Research Funds for the Central
DW C. -P. Dong, G. Weng, Minuscule representations, the graded poset $\Delta(1)$, and three conjectures of Panyushev, preprint, arXiv:1511.02692v4.
GOV V. V. Gorbatsevich, A. L. Onishchik, E. B. Vinberg, Lie groups and Lie algebras III
(Encyclopaedia Math. Sci., vol. 41), Berlin: Springer (1994).
P D. I. Panyushev, Antichains in weight posets
associated with gradings of simple Lie algebras, Math. Z. 281 (2015), 1191–1214.
St R. P. Stanley, Enumerative combinatorics,
Vol. 1, 2nd Edition, Cambridge University Press, 2012.
|
1511.00597
|
Faculty of Engineering, Izmir University of Economics, Izmir 35330, TURKEY
Department of Physics, North Carolina State University, Raleigh NC 27695-8202, USA
We perform a general computational analysis of possible post-collision mass distributions in high-speed galaxy cluster collisions in the presence of self-interacting dark matter. Using this analysis, we show that astrophysically weakly self-interacting dark matter can impart subtle yet measurable features in the mass distributions of colliding galaxy clusters even without significant disruptions to the dark matter halos of the colliding galaxy clusters themselves. Most profound such evidences are found to reside in the tails of dark matter halos' distributions, in the space between the colliding galaxy clusters.
Such features appear in our simulations as shells of scattered dark matter expanding in alignment with the outgoing original galaxy clusters, contributing significant densities to projected mass distributions at large distances from collision centers and large scattering angles of up to $90^\circ$.
Our simulations indicate that as much as 20% of the total collision's mass may be deposited into such structures without noticeable disruptions to the main galaxy clusters.
Such structures at large scattering angles are forbidden in purely gravitational high-speed galaxy cluster collisions.
Convincing identification of such structures in real colliding galaxy clusters would be a clear indication of the self-interacting nature of dark matter.
Our findings may offer an explanation for the ring-like dark matter feature recently identified in the long-range reconstructions of the mass distribution of the colliding galaxy cluster CL0024+017.
§ INTRODUCTION
Dark matter and dark energy, comprising together 95% of the energy budget in the Universe, remain among the biggest unsolved mysteries of modern physics. Dark matter (DM) has been described conventionally using the Cold Dark Matter (CDM) model, where the primary candidate for DM is an extremely massive ($m_{DM}\approx 10-1000$ GeV) particle interacting exclusively via the weak interaction - the so-called Weakly Interacting Massive Particle (WIMP) <cit.>. In recent years, observations began to suggest that DM can be interacting with the cross-sections large enough to influence the formation of small-scale cosmological structures, the so called Self-Interacting Dark Matter (SIDM) <cit.>. Recent theoretical works put forth various models of such self-interacting DM including mirror DM <cit.>, flavor-oscillating DM <cit.>, SIDM <cit.>
, etc.
Recent observations of colliding galaxy clusters provided a unique opportunity for gaining additional insights about the properties of DM empirically
The observed galaxy clusters may be regarded as natural astrophysical accelerators for high-energy DM particles' collisions.
The observations of two or more galaxy clusters undergoing a high-speed central or near-central passage through each other after a gravitational in-fall
can offer new clues about the microscopic properties of DM
The bullet-type galaxy cluster collisions are such collisions that involve a smaller galaxy cluster, sometimes called the “bullet”, falling onto a much larger cluster. In several cases of such collisions, a bullet-type galaxy cluster collision has been observed shortly after the passage of the bullet through the main cluster <cit.>. Those observations evinced that the galaxy groups in the bullet-type galaxy cluster collisions exhibit a collisionless behavior, namely, passing through each other essentially freely without interactions. On the other hand, the gas component of such colliding galaxy clusters—the intracluster medium (ICM)—exhibits a drastically different behavior with significant ram friction, super-sonic bow-shocks, and strong heating accompanied by X-ray emission as witnessed <cit.>.
One may ask which of these components the DM halos are co-localized with.
Subsequent reconstructions of the mass distribution in some of such bullet-type collisions by means of strong and weak gravitational lensing showed that the DM halos in these collisions are co-localized with the collisionless galaxy groups but not with the collisional ICM gas
<cit.>. This co-localization led to conclusion that the material in the DM halos is collisionless much like the galaxy groups, rather than collisional like the ICM. Arguments such as the preservation of mass-to-light ratios and the coincidence of the centroids of DM halos with that of galaxy groups led to the constraints on the cross-section of possible SIDM particles at approximately $\sigma_{DM}/m_{DM} < 1\ cm^2g^{-1}$
In this work, we perform a computational study of possible post-collision mass distributions that may be realized in high-speed collisions of galaxy clusters in the presence of weakly self-scattering DM.
Several past computational studies focused on simulating the known colliding galaxy clusters and estimating the properties of the dark matter particles from that analysis
Here, we survey rather different possibilities for post-collision mass distributions in high-speed collisions of galaxy clusters under a variety of scenarios
in the presence of self-interacting DM. In that respect, our study does not focus on any specific galaxy cluster collision but aim at an explorative analysis of possible configurations that can be realized in galaxy cluster collisions in the presence of self-interacting DM.
For instance, we do not specifically simulate the classical Bullet cluster 1E 0657-56 although such Bullet cluster observations motivated
our study.
A similar work in spirit has been recently published by Robertson, Massey and Eke <cit.>, where the focus has been on the changes in the shape of DM halos in the presence of DM self-interactions.
In particular, it has been found that shape changes can significantly affect and make unreliable simple analyses of SIDM effects such as the calculation of DM halo centroids' lag.
Our work can be viewed in a complementary light as such investigating the effects of DM self-interactions in the space around the colliding galaxy clusters, in the tails of DM mass distributions, arising due to
astrophysically weak
DM self-interactions and manifesting in the projected mass density maps of galaxy cluster collisions in the regions between and around the colliding galaxy clusters.
We find that while the DM particle interactions with $\sigma_{DM}/m_{DM}\approx 1\ cm^2g^{-1}$ and above cause severe disruption of colliding galaxy clusters, possibly leading to complete destruction and merger of their DM halos over cosmologically short time scales,
a range of weaker DM self-interactions $\sigma_{DM}/m_{DM} \approx 0.1-0.5\ cm^2g^{-1}$ can create weak yet detectable features in DM mass distributions around the colliding galaxy clusters, while not causing major distortions in the main galaxy clusters themselves. One such feature is the shell of scattered DM material that forms due to the scattering of DM particles off each other during the passage of the DM halos of colliding galaxy clusters through each another. Such shells can produce noticeable differences in the projected mass density maps of SIDM galaxy cluster collisions, in the form of extended concentrations of DM at large distances either from the collision center or the outgoing galaxy groups and large scattering angles.
It is interesting that strikingly resembling structures can be spotted in many mass reconstructions of colliding galaxy clusters in the literature
A scattered DM shell may also explain the ring-like DM feature recently observed in a long-range reconstruction of the mass distribution in the galaxy cluster CL0024+17 <cit.>.
Convincing observations of such features in high-speed galaxy cluster collisions can provide a clear evidence of the self-interacting nature of DM.
The remainder of the paper is organized as follows. In Section <ref>, we discuss the methodology of our study. In Section <ref>, we survey different types of post-collision mass distributions with respect to the parameters such as collision's speed, mass, DM self-interaction strength, etc.
In this Section (subsection <ref>), we discuss the conditions necessary for the astrophysical observation of the effects associated with the self-interacting nature of DM in galaxy cluster collisions.
The summary and conclusions follow in Section <ref>.
In Appendix <ref>, we provide the summary of the algorithms used in this work.
In Appendix <ref>, we present the numerical checks related to the convergence and accuracy of our numerical simulations.
§ METHODOLOGY
The bulk of our study focused on carrying out a set of simulations of galaxy cluster collisions using Particle Mesh method and collisional DM particles. In this section, we discuss the details of these simulations' initialization, evaluating DM particle collisions, evaluating gravitational evolution, and selecting the simulation parameters.
§.§ Simulation's initial conditions
We use Plummer profile <cit.> obtained as the result of equilibrating a cloud of collisional DM particles in self-consistent gravitational potential to set up the initial particle distribution of colliding halos. Such Plummer profile reproduces closely the soft-core King model's mass profile, used in some literature in the past as an empirical model of SIDM halos <cit.>, up to the halo's virial radius $r_{200}$ and soft-truncates at $r_{200}$ as is seen Fig. <ref>.
Note that Navarro-Frenk-White (NFW) mass profile <cit.>, popular in CDM literature, is unsuitable for modeling the collisional SIDM halos because NFW profile does not allow for the presence of soft halo cores necessarily present in SIDM halos <cit.>.
Particle-particle scatterings of DM particles in the central regions of SIDM halos are known to reduce the central density of such halos resulting in an isothermal-like behavior of mass density at small radii, differently from the central cusp of NFW profile.
The so called approximate King model's profile, $\rho(r)=\rho_0/(1+r^2/r_c^2)^{3/2}$, had been used to ad-hoc model SIDM halos by introducing into NFW profile a soft core of radius $r_c$.
The SIDM halo profile in this work has been obtained directly by equilibrating self-scattering halo of SIDM particles and can be indeed represented using such King profile very well.
The specific parameters of the initial particle distributions are as follows:
The profiles were created with $N=100,000$ particles with total mass of $2.5\cdot 10^{14}M_\odot$ at per-particle mass of $1.125\cdot 10^{9}M_\odot$. The scale radius parameter of the Plummer density was $r_s=0.24$ Mpc, the core radius was $r_{c}=0.15$ Mpc, and the virial radius was $r_{200}= 0.6$ Mpc, corresponding to the effective concentration parameter of $c=r_{200}/r_c=4$.
The initial mass profile of SIDM halos used in this work, blue diamonds, is the the Plummer density shown with brown dashed line.
Also shown is the approximate King model's profile used in <cit.>, green dash-dotted line.
NFW profile <cit.> (red dashed line) and the isothermal profile <cit.> (black dashed line, $\rho(r)=\rho_0/(1+r^2/r_c^2)$) are also shown for reference. The profile's 1/2-level half-width and the virial radius are indicated with labels $r_{1/2}$ and $r_{200}$, respectively.
§.§ Simulation of SIDM particle-particle scattering
The non-gravitational interactions in SIDM halos, that is, the particle-particle scatterings of SIDM particles, were modeled by scattering simulated particles when they occupied the same cell of the simulation's spatial grid $\mathcal{G}$ (see the next subsection <ref> for the definition of $\mathcal{G}$).
That is, each pair of such particles scattered with the probability
\begin{equation}\label{eq:collalpha}
P=\alpha V_{rel}\Delta t,
\end{equation}
where $V_{rel}=|\vec v_1-\vec v_2 |$ is the relative speed of the two particles, $\Delta t$ is the simulation time step, and $\alpha$ is an effective numerical parameter controlling the intensity of DM self-scattering.
Robertson, Massey and Eke have addressed many of the science goals discussed in the present work by using essentially similar model with different anisotropic DM particle-particle interactions in their recent work <cit.>. In principle, it is conceivable that different particle physics models will predict very different scattering probabilities.
Fundamentally, in quantum field theoretic models, the coupling constant of particle interaction provided by a given model estimates the probability of
particle scattering. The physical coupling constant in the renormalizable quantum field theory may be defined through multi-loop calculations
with self-consistent regularization procedures. The scattering cross section predicted by such models would then be the key to test the validity of the models in comparison with experimental data. Lacking evidence for such sophisticated models at present, we assume in this work the simplest isotropic scattering model parametrized by a single cross-section parameter related to the numerical parameter $\alpha$ via
\begin{equation}\label{eq:alphasigmamass}
\frac{\sigma_{DM}}{m_{DM}}=\alpha \frac{N_{tot}d^3}{M_{tot}},
\end{equation}
where $M_{tot}$ is the total simulation mass, $N_{tot}$ is the total number of simulated particles, and $d$ is the resolution of the simulation grid, $\mathcal{G}$.
Then, the scattering of DM particles was evaluated as follows:
First, the Center-of-Mass velocity, $\vec V_{CM}=(\vec v_1+\vec v_2)/2$, and the relative speed, $V_{rel}=|\vec v_1-\vec v_2 |$, of the scattering particles were computed.
Second, a new direction for the relative velocities, $\vec n$, was selected uniformly at random on a unit sphere, assuming elastic and isotropic scattering.
The velocities of the particles were then updated as
\begin{equation}
\begin{array}{l}
\vec{v}'_{1}=\vec{V}_{CM}+ \frac12V_{rel} \vec n,\\
\vec{v}'_{2}=\vec{V}_{CM}- \frac12V_{rel} \vec n.\\
\end{array}
\end{equation}
Further details of this implementation of the DM particle-particle scattering in our simulations are presented in the algorithm in Appendix <ref>.
§.§ Simulation of SIDM gravitational evolution
Gravitational evolution of SIDM particles was implemented using Particle Mesh algorithm in Matlab.
Namely, a continuous spatial distribution of mass $\rho(\vec x,t)$ was modeled by using a collection of $N$ particles $\vec r_i(t)$, $i=1\dots N$, distributed according to $\rho(\vec x,t)$. As the particles moved in common gravitational potential, $\Phi(\vec x,t)$, self-consistent $\Phi(\vec x,t)$ was calculated by approximating $\rho(\vec x,t)$ on a 3D grid $\mathcal{G}$ by counting the number of the particles in each cell of the grid, $n(\vec x_\mathcal{G},t)$, and numerically solving the Poisson equation
\begin{equation}\label{eqn:poisson}
\nabla^2 \Phi(\vec x,t) = 4\pi G n(\vec x,t),
\end{equation}
where $G$ is the gravitational constant.
The method of Fourier transform was used for the numerical solutions of Eq. (<ref>), where for $\tilde \Phi(\vec k,t)=\int {d\vec x}{(2\pi)^{-3/2}}
e^{-i\vec k\cdot\vec x}\Phi(\vec x,t)$ we got
\begin{equation}\label{eqn:poissonmomentum}
\tilde \Phi(\vec k,t) = -4\pi G \frac{\tilde n(\vec k,t)}{\vec k^2}.
\end{equation}
Thus, $\Phi(\vec x,t)$ was computed by performing two discrete fast Fourier transforms, $n(\vec x,t)\rightarrow \tilde n(\vec k,t)$ and $\tilde \Phi(\vec k,t)\rightarrow \Phi(\vec x,t)$, and making use of Eq. (<ref>) in between them. Once $\Phi(\vec x,t)$ had been computed, the speed and the positions of all the particles in the common gravitational potential were updated according to the regular Newtonian dynamics.
A SIDM self-scattering step had been embedded into this algorithm as described in Section <ref>.
The simulation advanced by using an adaptive time step $\Delta t$ set from the restriction that the maximum change of the speed and the position of the simulated particles was below one grid-cell, and varied between 0.1 My and 10 My.
The detail of the Particle Mesh algorithm is also presented in Appendix <ref>.
The algorithm requires a set of parameters to be set, specifying the total mass of simulated collision, $M_{tot}$, the number of simulated particles in colliding SIDM halos, $N1$ and $N2$, the kinetic collision parameter defined as the square of the galaxy clusters' speed at infinity, $\Delta V2$, and controlling the collision velocity, the initial offsets of colliding halos, $\Delta R$, and the collision impact parameter $\Delta b$. The other algorithm parameters are the number of cells $D$ along each dimension in the cubic spatial grid $\mathcal{G}$ and the cells' dimension $d$, with the total of $D^3$ cells in the grid. These are related to the total linear size of the modeled region of space as $Dd$.
§.§ Selection of simulation parameters
All simulations had been performed using the super-computing facility National Energy Research Scientific Computing Center (NERSC). The majority of simulations used $N=2\cdot 10^5$ total particles, the grid of $D^3=400^3$ cells with cell-resolution $d=15$ kpc and the region of space modeled being a cube of 6 Mpc on the side. Only the dynamics of SIDM halos was simulated and the ICM and the visible matter contributions were not included. All simulations spanned the duration of time $t_{max}$ of 1.5 Gy to 3 Gy chosen so as to cover a single passage of colliding SIDM halos through each other.
We simulated various collision scenarios with regard to the collision speed, collision centrality, symmetricity, and the SIDM self-interaction strength (expressed via the effective strength parameter $\alpha$ or equivalently, $\sigma_{DM}/m_{DM}$ given by Eq. (<ref>)).
The simulations were initialized with all particles divided into two initial halos placed at separation from each other $\Delta R = 2 Mpc$, moving towards each other with initial relative inbound velocity between 500 kmps and 4400 kmps.
In the case of a symmetric galaxy cluster collision, the two halos were initialized with equal number of particles.
For asymmetric collisions, the particles were divided between the two halos in the ratio 5:1.
This choice was motivated by the mass split in the Bullet cluster <cit.>.
In the case of off-central collisions, the center of one of the halos was shifted with respect to the other halo in direction perpendicular to the collision axis by the amount specified by the impact parameter $\Delta b$ chosen equal to the core radius of the larger halo. This choice was motivated by maximizing the effect on non-centrality on the collision, whereas the larger values of the impact parameter resulted in halos missing each others' dense cores and smaller impact parameters resulting in post-collision mass distributions not much different from that of central collisions.
The total mass of simulation here was not varied. This was because the total mass can be reduced from gravitational dynamics by suitably re-scaling the distance and/or time variables in the simulation.
Indeed, consider the Newtonian equations of motion of particles in self-consistent gravitational potential,
\begin{equation}
\begin{array}{l}
\frac{d\vec r_i}{dt}=\vec v_i,\\
\frac{d\vec v_i}{dt}=-G \nabla \int d^3z \frac{\rho(\vec z,t)}{|\vec r_i-\vec z|}.
\end{array}
\end{equation}
If one rescales distances, times, and velocities of all particles according to
\begin{equation}\label{eq:scalegravtransform}
\begin{array}{l}
\vec r= a\vec r\hspace{0.05cm}', \\
\vec v = ab^{-1}\vec v\hspace{0.05cm}', \\
t = bt',
\end{array}
\end{equation}
then the equations of motion become respectively
\begin{equation}
\begin{array}{l}
ab^{-1}\frac{d\vec r\hspace{0.05cm}'_i}{dt'}=ab^{-1}\vec v\hspace{0.05cm}'_i, \\
ab^{-2}\frac{d\vec v\hspace{0.05cm}'_i}{dt'}=-a^{-2}G \nabla' \int d^3z' \frac{\rho'(\vec z\hspace{0.05cm}',t)}{|\vec r\hspace{0.05cm}'_i-\vec z\hspace{0.05cm}'|},
\end{array}
\end{equation}
where we took into account that the element of mass, $dm=d^3r\rho(\vec r,t)$, remained invariant. Simplifying the above we obtain
\begin{equation}\label{eq:scaledgraveqn}
\begin{array}{l}
\frac{d\vec r\hspace{0.05cm}'_i}{dt}=\vec v\hspace{0.05cm}'_i, \\
\frac{d\vec v\hspace{0.05cm}'_i}{dt}=-a^{-3}b^{2}G \nabla \int d^3z' \frac{\rho'(\vec z\hspace{0.05cm}',t)}{|\vec r\hspace{0.05cm}'_i-\vec z\hspace{0.05cm}'|}.
\end{array}
\end{equation}
Therefore, we observe that the total simulation mass $\int d^3r \rho(\vec r,t)$ can be arbitrarily rescaled by scaling up or down either distances, times, or both. For example, $\int d^3r \rho(\vec r,t)$ can be brought to be equal to any (standard) mass $M_{std}$ simply by scaling the distances $\vec r=(M_{tot}/M_{std})^{1/3} \vec r\hspace{0.05cm}'$, where $M_{tot}$ is the total mass in $\rho(\vec r,t)$, or by similarly scaling the time variable, $t=(M_{std}/M_{tot})^{1/2} t'$. Note that the velocities also scale according to Eq. (<ref>).
Thus, the total mass can be reduced from the simulations and all our simulations were performed using a “standard" total collision mass of $5\cdot 10^{14}M_\odot$.
The full list of the parameters used for each class of simulations discussed in this work is given in Table <ref>.
The parameters of the simulations of different galaxy cluster collision scenarios performed in this work.
Type of scenario $M_{tot}$ ($M_\odot$) $D$ (#) $Dd$ (Mpc) $N1$ ($10^5$) $N2$ ($10^5$) $\Delta V2$ (kmps$^2$) $\Delta R1$ (Mpc) $\Delta R2$ (Mpc) $\Delta b$ (Mpc) $\frac{\sigma_{DM}}{m_{DM}}$ ($cm^{-2}g$) $k$ $a$ (%)
fast CDM $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $1300^2$ 1.0 1.0 0.0 0.0 1.6 -
free-fall CDM $5\cdot 10^{14}$ 400 6.0 1.0 1.0 0.0 1.0 1.0 0.0 0.0 1.0 -
slow CDM $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $-(700^2)$ 1.0 1.0 0.0 0.0 0.8 -
symmetric-central-weak $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $1300^2$ 1.0 1.0 0.0 0.45 1.6 10
symmetric-central-strong $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $1300^2$ 1.0 1.0 0.0 1.80 1.6 40
symmetric-noncentral-CDM $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $1300^2$ 1.0 1.0 0.25 0.0 1.6 -
symmetric-noncentral-weak $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $1300^2$ 1.0 1.0 0.25 0.45 1.6 7
symmetric-noncentral-strong $5\cdot 10^{14}$ 400 6.0 1.0 1.0 $1300^2$ 1.0 1.0 0.25 1.80 1.6 31
asymmetric-central-CDM $5\cdot 10^{14}$ 400 6.0 1.5 0.3 $800^2$ 0.0 2.0 0.0 0.0 1.4 -
asymmetric-central-weak $5\cdot 10^{14}$ 400 6.0 1.5 0.3 $800^2$ 0.0 2.0 0.0 0.50 1.4 7
asymmetric-central-strong $5\cdot 10^{14}$ 400 6.0 1.5 0.3 $800^2$ 0.0 2.0 0.0 3.25 1.4 60
asymmetric-noncentral-CDM $5\cdot 10^{14}$ 400 6.0 1.5 0.3 $800^2$ 0.0 2.0 0.25 0.0 1.4 -
asymmetric-noncentral-weak $5\cdot 10^{14}$ 400 6.0 1.5 0.3 $800^2$ 0.0 2.0 0.25 0.80 1.4 9
asymmetric-noncentral-strong $5\cdot 10^{14}$ 400 6.0 1.5 0.3 $800^2$ 0.0 2.0 0.25 4.05 1.4 58
very fast collisions comparison $10^{15}$ 100 6.0 1.0 1.0 $5400^2$ 1.0 1.0 0.0 0.45 8.0 7
very fast collisions comparison $10^{15}$ 100 6.0 1.0 1.0 $3500^2$ 1.0 1.0 0.0 0.45 4.0 8
very fast collisions comparison $10^{15}$ 100 6.0 1.0 1.0 $2000^2$ 1.0 1.0 0.0 0.45 2.0 10
very fast collisions comparison $10^{15}$ 100 6.0 1.0 1.0 $1400^2$ 1.0 1.0 0.0 0.45 1.5 11
§ RESULTS
§.§ Characterization of possible galaxy cluster collision phenomenologies
Our primary goal is to characterize possible effects of astrophysically-weak DM particle-particle self-interactions on the projected mass distributions appearing in collisions of galaxy clusters.
By “weak”, it is meant that DM self-interaction results in the fractions of DM particles scattered during the collisions that are significantly less than one. From the nature of that weak effect, such effects can be expected to appear in the tails of DM halos of the colliding galaxy clusters. To make those effects discernible, we choose a suitable way for visualizing the tails of DM halo distributions using the logarithmic scale for projected mass density maps, adopted for all illustrations in this section.
We point out that the general profile of post-collision mass distributions in colliding galaxy clusters is well characterized by three phenomenological parameters — the collision's kinetic parameter as defined by the ratio of the relative kinetic energy of the colliding galaxy clusters and mutual gravitational energy, $k=|E_K/E_G|$; the fraction of DM halos' non-gravitationally scattered mass, $a$; and the separation of post-collision galaxy clusters in terms of a certain typical radius here chosen as the core radius, $r/r_c$.
We first discuss the kinetic parameter $k=|E_K/E_G|$. This parameter characterizes the degree of the effect that the gravitational effects impart to the post-collision mass distributions. Intuitively, small values of $k$ imply slow collisions in which gravitational interactions have a long time to play dominant role. For large values of $k$, however, the galaxy clusters collide much more rapidly with little to no gravitational distortions.
We specifically define the kinetic parameter $k$ as the ratio of the colliding galaxy clusters' kinetic to mutual gravitational energy at the point of closest approach.
The fraction parameter of the DM halo mass scattered in DM particle-particle collisions, $a$, is another interesting parameter affecting the shape of post-collision galaxy clusters' mass distributions.
The parameter $a$ quantifies the degree of effect that DM self-interactions have on the post-collision mass distribution by setting an upper limit on the relative weight that the non-gravitational scattering features can introduce into post-collision galaxy clusters' mass maps. For instance, for $a=0.1$ at most 10% of the DM halo mass can contribute to self-interaction-related features in the post-collision mass distributions.
The fraction $a$ can be directly related to the effective DM self-interaction parameter in simulations, $\alpha$, and the physical ratio $\sigma_{DM}/m_{DM}$.
For concreteness we define
\begin{equation}\label{eq:colla}
a=2N_{12}/(N_1+N_2)=(\delta M_1+\delta M_2)/M_{tot},
\end{equation}
where $N_{12}$ is the number of DM particle-particle scattering events during the collision and $N_1+N_2$ is the total number of DM particles in the collision.
In terms of real masses of scattered DM, $\delta M_{1,2}$, the same parameter is given as their ratio with the total mass of the collision $M_{tot}$.
The post-collision separation of the galaxy clusters is another phenomenological descriptor of a galaxy cluster collision. We observe that the post-collision mass distributions share significant similarities for the same separations of outgoing galaxy clusters, if expressed in the terms of a typical distance scale associated with the galaxy clusters.
Therefore, such a parameter is advantageous for characterizing the post-collision stage instead of regular time. Of course, post-collision galaxy clusters' separation is also a quantity that can be directly measured from astrophysical data.
We define that parameter as the ratio $r/r_c$ of the distance $r$ between the centers of the outgoing galaxy clusters and the core radius $r_c$ of the larger of the galaxy clusters (that is, the 1/2 half-width of the projected mass distribution of that cluster's DM halo).
§.§ The phenomenology of CDM high-speed galaxy cluster collisions
We first review the galaxy cluster collisions in standard CDM, that is, when the gravity is the only effect present.
We inspect central collisions, in which case the only two free parameters controlling the post-collision mass distribution are $M_{tot}$ and
$k=|E_K/E_G|$, of which the total mass can be excluded by rescaling the distances as discussed above and a single parameter $k$ is left to completely characterize the post-collision mass distribution's properties.
The kinetic collision parameter $k$ has been defined in Section <ref> as the ratio of kinetic and gravitational energy in collision.
By the conservation of energy, $k$ is thus related to the total energy of colliding system, $E$, as $k=1+E/|E_G|$, where $E_G$ is the mutual gravitational energy of colliding galaxy clusters. Three essentially different regimes therefore need to be distinguished:
fast collisions with $E>0$ and $k>1$, “free-fall" collisions with $E=0$ and $k=1$, and slow collisions with $E<0$ and $k<1$.
In fast collisions, the clusters fall towards each other from a finite in-fall velocity at infinity. In “free-fall" collisions, the clusters fall towards each other from zero speed at infinity. The case $k<1$ corresponds to the situation where the clusters fall towards each other from zero speed at a finite distance.
The typical shapes of post-collision mass distributions in CDM for fast collisions regime is explored in Fig. <ref>. For very fast collision with $k\approx 2$ and above, we observe that the colliding galaxy clusters pass through each other without gravitational distortions, essentially maintaining their original shape and velocity after the collision.
For slower collisions, $2>k>1$, the galaxy clusters can substantially interact gravitationally, however, forming high-velocity ejecta in the form of forward conic jets around the clusters' initial velocity vectors. These ejecta give the projected mass density map a notable forward “fan-out” shape, as shown in the left panel of Fig. <ref> in a log-scale.
A narrow, weak “central bridge" of slow trailing material is also observed in this regime in some simulations.
An important observation to make at this point is that such fast ejecta is always in forward and backward cone-directions and is restricted to small scattering angles. This is, in fact, a well-known and expected property of the differential cross-section of gravitational interaction, in which large cross-sections are observed for small scattering angles and very small cross-section are observed towards $90^\circ$ scattering angle. Gravitational interaction, therefore, does not produce orthogonal or “equatorial” ejecta during high-speed collisions.
For a slower case of “free-fall" or $k=1$ collisions, we observe that the colliding galaxy clusters merge in a single passage producing large amount of ejecta in a characteristic “butterfly” shape, as shown in the central panel of Fig. <ref>.
In slow collisions, $k<1$, the clusters rapidly merge with a large amount of close to isotropic ejecta, as shown in the right panel of Fig. <ref>.
Possible phenomenologies of the projected mass density profiles for galaxy cluster collisions in CDM model, for different values of the kinetic parameter $k$.
From left to right shown are the examples of a central symmetric fast collision ($k=1.6$, separation 15$r_c$), a “free-fall" collision ($k=1.0$, maximum separation of approximately $6r_c$), and a slow collision ($k=0.8$, final merger configuration).
The simulation parameters are as defined in Table <ref>.
The projected mass density maps are shown color-coded according to logarithmic scale, with the contour-lines defining the density levels of such maps normalized to the peak projected mass density of one.
The colormap on the right shows the projected mass density in the physical units of $M_\odot/kpc^2$, assuming the total collision mass of $5\cdot10^{14} M_\odot$.
The distance scales shown along x and y axes are in the units of $r_c$.
The colorbar and the distance scale are for this and all similar figures except Fig. <ref> (“ideal" case).
§.§ The phenomenology of SIDM high-speed galaxy cluster collisions
In the interacting DM case, the phenomenology of high-speed galaxy cluster collisions can be substantially different and is governed by two parameters: the kinetic parameter $k$ and the effective DM self-scattering intensity $a$.
With respect to the DM self-scattering strength, we inspect three regimes of strong self-scattering DM, $0.5\leq a$, weak self-scattering, $a\leq 0.2$, and intermediate scattering $0.2\leq a\leq 0.5$. These roughly correspond to $\sigma_{DM}/m_{DM}>2\ cm^{-2}g$ for strong, $\sigma_{DM}/m_{DM}<0.5\ cm^{-2}g$ for weak, and $0.5\ cm^{-2}g<\sigma_{DM}/m_{DM}<2\ cm^{-2}g$ for intermediate DM scattering (see Table <ref>).
The typical shapes of the post-collision mass distributions for all of these regimes are presented in Fig. <ref>. The respective shapes are shown using a table where the columns correspond to different DM self-interaction intensities $a$ and the rows corresponding to different collision scenarios such as symmetric central, asymmetric central, symmetric non-central and asymmetric non-central, as defined in the caption. Note here that the kinetic parameter in symmetric collision scenarios is $k=1.6$ and in asymmetric collision scenarios is $k=1.4$.
In all regimes, we observe that the DM self-interaction can result in additional mass components appearing as diffuse circular mass concentrations centered at the collision center and extending radially out to the distance defined by the outgoing galaxy clusters. The most significant difference is present at $90^\circ$ scattering angles, that is, the equatorial plane perpendicular to the axis of the collision. As have been discussed in the previous section, gravitational interactions cannot produce significant mass ejecta in equatorial plane in fast collisions. DM mass concentrations in that region is a unique consequence of self-interactions of DM particles observed in the simulations.
The relative weight of this additional DM component is defined by the parameter $a$. In the case of strong DM particle-particle scattering, $a > 0.5$ and $\sigma_{DM}/m_{DM} > 2\ cm^{-2}g$, we observe that the mass distribution of colliding galaxy clusters is significantly disrupted in all collision scenarios (see the third column of Fig. <ref>). A very wide approximately spherical hot cloud of DM material forms in that situation around the collision center. In the case of an asymmetric bullet cluster-like collision, the halo of the “bullet" cluster does not survive the passage through the main cluster and is completely dispersed after the first passage, only appearing in the projected mass density maps as a weak extrusion from the main cluster at 1%-level (relative to the peak projected mass density).
We note that this dramatic effect is in contrast to relatively minor effects discussed in the literature for this collision regimes, whereas the effects of such relatively strongly interacting DM in galaxy cluster collisions had been greatly underappreciated in the literature.
In contrast to the previous regime, when $a\leq 0.2$ or $\sigma_{DM}/m_{DM}<0.5\ cm^{-2}g$, the second column of Fig. <ref>, the original DM halos fail to get distorted significantly, consistent with existing observations. However, previously undisclosed features are observed in these simulations contrary to the pure CDM model.
These differences include heavier and substantially wider central regions of projected mass density maps appearing as mass-bridges connecting the halos of outgoing galaxy clusters at 1% to 10% peak density-levels and DM densities appearing at $90^\circ$ scattering angles (see the second and fourth rows of Fig. <ref>).
The latter feature, in particular, presents the greatest interest in that it is completely absent in CDM scenario. Such equatorial mass densities seen in the second column of Fig. <ref> at 1% peak density-levels appear in the projected mass density maps at distances from the center equal to that of the outgoing galaxy clusters.
It is interesting to understand the nature of that latter feature. For that, we can inspect the process of DM halos passing through each other in a galaxy cluster collision. In the weak scattering regime, $a \ll 1$, the mean free path of DM particles is substantially greater than the diameter of DM halos. In that case, the DM particles that scatter leave their respective halos without secondary scattering with large probability. This forms a feature in the shell of scattered DM material expanding radially outwards from the collision center. The conservation of energy and momentum in elastic collisions of DM particles dictates that the speeds of such shell is the same with the original galaxy clusters. As time passes, that shell forms a DM distribution around the collision center at the observed distances.
1 0.251
The phenomenologies of possible post-collision projected mass distributions in collisions of galaxy clusters with self-interacting DM. Different galaxy cluster collision scenarios are shown in different rows:
1st row is central symmetric collisions, 2nd row is central asymmetric collisions, 3rd row is non-central symmetric collisions, and 4th row is non-central asymmetric collisions.
Different DM self-interaction strengths are shown in different columns:
left column - CDM regime, center column - weak DM scattering regime $a=0.1$, right column - strong DM scattering regime $a=0.5$;
The kinetic parameter in symmetric collision scenarios is $k=1.6$ and in asymmetric collision scenarios is $k=1.4$.
The simulation parameters are as defined in Table <ref>.
Note that in that picture, the angular distribution of the material in the DM ejecta-shell should reflect the microscopic properties of the differential cross-section of DM particles.
It is reasonable to expect that such differential cross-section should be isotropic.
Indeed, the low energy spin-averaged cross-sections of all known short-range particle interactions are isotropic.
Under this assumption, the DM ejecta shell will form in a spherically symmetric way forming a narrow isotropic shell expanding radially in alignment with the outgoing galaxy clusters.
To summarize, the new features that we observe SIDM galaxy cluster collisions are the additional mass components introduced into the collision's mass distribution as a spherically symmetric shell radially expanding from the center of the collision in sync with the outgoing galaxy clusters, moving away in-lock with the galaxy clusters and linking them into a spherical-like structure. At larger separations and seen sideways, that feature appears in projected mass density maps as a disk-like diffuse DM concentration lying in between the outgoing galaxy clusters and featuring a thin, ring-like boundary of the width equal to the core-diameters of the collided galaxy clusters.
If seen along the collision axis, the same feature appears as a disk and a ring surrounding the galaxy clusters now placed centrally.
These situations are illustrated “ideally” in Fig. <ref>.
The mass distribution for a self-interacting DM galaxy cluster collision in an “ideal” scenario, where the colliding clusters are very fast and compact. The left panel shows the projected mass density map illustrating the scattered DM shell observed along a direction perpendicular to the collision's axis. The right panel shows the same mass density observed along the collision's axis.
In the former, the DM shell appears as a weak disk-shaped mass distribution with a ring-like rim similar in width to the size of the colliding galaxy clusters, extending outwards from the center of the collision and linking the outgoing galaxy clusters into a ring-like structure. In the latter, the DM shell appears as a weak DM disk and a ring surrounding the centrally placed colliding galaxy clusters, now seen on top of each other.
§.§ Observability conditions of SIDM effects in galaxy cluster collisions
§.§.§ Comparative study of projected mass distributions in CDM and weak SIDM scenarios
1 0.30
The difference in the post-collision mass distributions of fast symmetric galaxy cluster collisions with weakly scattering DM and $k=1.5$. Top row shows the post-collision mass distribution with weakly scattering DM and the bottom row shows the same for standard CDM, for comparison. Three post-collision stages are shown characterized by inter-cluster separation in units of $r_c$: early stage where the galaxy clusters are just barely separated ($\Delta r=2.5r_c$, left), intermediate stage where the galaxy clusters just recently became fully separated ($\Delta r=7.5r_c$, center), and late stage where the galaxy clusters fully separated and moved away to an appreciable distance ($\Delta r=15 r_c$, right).
The simulation parameters are as in Table <ref>.
1 0.30
The difference in the post-collision mass distributions of fast symmetric galaxy cluster collisions with weakly scattering DM and $k=2.0$. Top row shows the post-collision mass distribution with weakly scattering DM and the bottom row shows the same for standard CDM. Left panels show the mass distributions at early separation stage ($\Delta r=2.5r_c$), center panels show the mass distributions at intermediate separation stage ($\Delta r=7.5r_c$), and right panels show the mass distributions at late separation stage ($\Delta r=15 r_c$).
The distance scales are in the units of $r_c$.
The simulation parameters are as in Table <ref>.
1 0.30
The difference in the post-collision mass distributions of fast symmetric galaxy cluster collisions with weakly scattering DM and $k=4.0$. Top row shows the post-collision mass distribution with weakly scattering DM and the bottom row shows the same for standard CDM. Left panels show the mass distributions at early separation stage ($\Delta r=2.5r_c$), center panels show the mass distributions at intermediate separation stage ($\Delta r=7.5r_c$), and right panels show the mass distributions at late separation stage ($\Delta r=15 r_c$).
The distance scales are in the units of $r_c$.
The simulation parameters are as in Table <ref>.
Figs. <ref>-<ref> inspect in details the differences appearing in weakly self-scattering SIDM and CDM for kinetic parameter $k$ between 1.5 and 4.0. In each figure, the projected mass densities for the early separation ($\Delta r=2.5r_c$, where $\Delta r$ is the distance between the centers of the outgoing galaxy clusters), intermediate separation ($\Delta r=7.5r_c$) and late separation stage ($\Delta r=15r_c$) are shown in comparison with the same CDM scenario.
The early separation stage corresponds to the colliding galaxy clusters that just barely emerged out of the collision, with the separation at just about 50% peak-density level.
The intermediate separation stage here corresponds to the colliding galaxy clusters that nearly fully emerged out of the collision but overlap significantly within their $r_{200}$ radii. In late separation stage, the galaxy clusters are well separated by a distance exceeding $r_{200}$.
In the early separation stage, in all scenarios we see quantitative differences in the distribution of mass at central region of colliding galaxy clusters where SIDM mass distributions show substantially heavier and wider central densities, bridging outgoing galaxy clusters at high densities. This difference from CDM scenarios at 10% to 50% is easily noticeable. The origin of this difference is understood physically as the mass contribution coming from DM ejecta shell scattered directly above and below the collision site, along the line of sight, thus remaining near the center of the collision for a far longer times than would be observed in pure CDM.
At intermediate separations, the projection of the DM ejecta shell over the collision's center continues to contribute significantly to central region of the collision forming more prominent bridges linking the outgoing galaxy clusters, even whereas the galaxy clusters in the same CDM scenarios separate at that stage completely - middle panels of Figs. <ref>-<ref>. We believe this effect will be noticeable in astrophysical reconstructions of projected mass densities observed at suitable time epochs.
At late separation stages, scattered DM ejecta shell begins to emerge as a separate component in the mass density field, introducing quantitative and qualitative changes in the mass distributions, as discussed previously. A distinct oval-shaped shell forms at 0.1% to 1% peak-density levels in those scenarios, as seen in Fig. <ref> and Fig. <ref> rightmost panel. These differences are the most pronounced for collisions with high values of kinetic parameter $k=2$ and above.
These differences are dramatically different from CDM but are also quite weak and confined to the tails of DM halos, which may make their detection challenging.
The bridge that we detect could conceivably be mimicked by the collisional intra-cluster gas as real galaxy clusters have significant gas fractions that may be slowed by ram pressure and accumulate at the center of the collision. To that extent, separation of the DM and the baryonic gas in the central mass distribution will require careful analysis including both detailed numerical simulations of the specific collision and experimental estimates of the amounts of hot central gas by using X-ray emissions. On the other hand, while the scattered DM shell manifests itself as added mass at the center of the collision at early post-collision stages, at later stages of galaxy cluster collisions such a shell expands and begins to contribute mass well beyond the central region, where the presence of intra-cluster gas will be substantially less significant.
§.§.§ Quantitative measures of SIDM scattering effects
The projected mass density plotted along the line connecting the centers of the colliding galaxy clusters for the collisions with different values of the kinetic parameter $k$ and different post-collision separation stages. Shown are such mass density plots for early ($\Delta r=2.5r_c$), intermediate ($\Delta r=7.5r_c$) and late ($\Delta r=15r_c$) post-collision separation stages, for weakly interacting DM (solid lines) and CDM (dashed lines), for $k=1.5$ (left), $k=2.0$ (center) and $k=4.0$ (right).
The differences between interacting DM and CDM models focused upon in the main text are shown with the symbol “X".
The CDM halos were first plotted at a fixed separation, and then the SIDM halos were plotted at the same post-collision time, in order to show the offsets between the centroids of the CDM and SIDM halos in galaxy cluster collisions.
While projected mass density maps are telling about the nature of the effects introduced into galaxy cluster collisions by SIDM, specific methods of measuring such effects are advantageous for such effects' detection.
Such measures can be constructed by quantifying the projected mass densities on the line connecting the centers of the outgoing galaxy clusters and measuring angular mass distribution relative to collision center in the collisions' projected density maps.
Fig. <ref> shows such a projected mass distribution plotted against the collision axial line.
At early separation stages, we see that the difference in the projected mass density in central regions in SIDM and CDM can be rather significant. Such differences reach 10% of peak-density values and can be directly measurable.
In late stages, the central region in that scenario features as much as two times more mass than in the same CDM scenarios for collisions with lower $k$. Slower collisions also feature an appreciable lag of DM halo centroids in SIDM as opposed to CDM collision scenarios. This lag is the reflection of the lower outgoing velocity of colliding galaxy clusters with SIDM, and can be seen clearly in the left panel of Fig. <ref> reaching $0.7-0.8r_c$ at later post-collision separations.
However, the differences in “axial” mass density measure decreases with the increase of the kinetic parameter values $k$. In particular, whereas one continues to see a noticeable 10-15% differences in the central mass densities in early separation stages in the middle ($k=2.0$) and right ( $k=4.0$) panels of Fig. <ref>, such differences diminish as the collision progresses. There is also no consistent lag in the outgoing DM halo positions at higher collision velocities. This is related to the change in the outgoing clusters' speed becoming negligible as the speed of the collision increases past $k=2$.
One of the differences that we observed for SIDM simulations is the shell of scattered DM material which can contribute mass at remote locations of the mass density map in directions perpendicular to the collision axis. To quantify this effect, the plot of projected mass in radial sectors vs. the scattering angle can be used. In particular, for SIDM one expects to see a close to uniform DM density in such mass-plot,
whereas in CDM such mass will drop to zero at scattering angle close to $90^\circ$.
Fig. <ref> shows this measure for SIDM simulations and mass measured per $15^\circ$ radial sectors centered on the collision center at different angles away from the equatorial plane. In the case of CDM, we observe this measure dipping to zero at angles close to $0^\circ$ (that is, near the equatorial plane), as expected. In SIDM, we observe the flat segment reminiscent of the scattered isotropic DM material shell. The difference in this measure is seen most prominently for simulations with high $k$ and late post-collision stage. In particular, there is no mass density contributed to sectors around the collision's equator in CDM while as much as 1% of the total mass can be contained in the equatorial radial sector in SIDM scenario.
The plots of the mass contained in projected mass density maps in $15^\circ$ degree sectors centered on the collision's center, as a function of the angle to the collision's equatorial plane, and shown as the percentile of the total density map's mass. Shown are the respective plots for early ($\Delta r=2.5r_c$), intermediate ($\Delta r=7.5r_c$) and late ($\Delta r=15r_c$) post-collision separation stages, for weakly interacting DM (solid lines) and CDM (dashed lines), for $k=1.5$ (left), $k=2.0$ (center) and $k=4.0$ (right). The differences between interacting DM and CDM models focused upon in the main text are shown with the symbol “X".
Yet another quantitative measure of SIDM effects is the presence of DM concentrations in galaxy cluster collisions at large scattering angles and large distances from collision center.
As illustrated in Fig. <ref>,
This feature can be quantified using plots of the mass distribution in projected mass density maps within a narrow $15^\circ$ to $30^\circ$ radial sector built around the equatorial plane of the collision, verses the distance from the collision center.
Fig. <ref> shows such measure in the units of percentile of the total collision mass contained in such a sector $[-30^\circ,30^\circ]$ per $r_c$ distance interval. Clear differences between SIDM and CDM are seen either in intermediate and late post-collision stages. The differences are the most profound in late stages of collisions of very fast galaxy clusters. In CDM in that case, there is practically no contribution in the right panel of Fig.<ref> (dashed line for $\Delta r = 15r_c$), as expected, while for SIDM consistent concentration of DM is observed at the correct distances from the center (that is, the distance of the outgoing galaxy clusters).
Note that the separation $\Delta r$ in Fig. <ref> is that between the centers of the colliding galaxy clusters, so that the distance from the clusters to the collision center is $\Delta r/2$.
The presence of the shell of scattered DM material can be quantitatively detected by plotting the projected mass distribution in a narrow equatorial sector as the function of the distance from the collision's center.
Shown in this illustration is such quantification, using two sectors covering $[-15^\circ,15^\circ]$ (bold dash-dotted line) and $[-30^\circ,30^\circ]$ (bold dotted line) angles around the equatorial plane (thin dashed line).
The plot of the distribution of mass in the projected density maps inside a $[-30^\circ,30^\circ]$ sector around the collision's equatorial plane, as a function of the distance from the collision center. Shown are respective plots for intermediate ($\Delta r=7.5r_c$) and late ($\Delta r=15r_c$) post-collision separations, for weakly interacting DM (solid lines) and CDM (dashed lines), for $k=1.5$ (left), $k=2.0$ (center) and $k=4.0$ (right).
The scattered DM shell in the tail of the radial mass profile of a very high speed galaxy cluster collision seen along its collision axis, observed at very late post-collision stages. The thick black dash-dotted line illustrates the
behavior of the galaxy clusters' mass profile at the boundary of the galaxy clusters, near and at the background density, and, thus, the behavior of the radial mass profile expected in the absence of DM scattering.
Finally, an interesting feature that can appear in astrophysical observations of galaxy cluster collisions due to SIDM is a ring surrounding the colliding galaxy clusters in galaxy cluster collisions observed along the collision axis. Such feature in our simulation can emerge at very late separation stages when the shell of scattered DM moves beyond the galaxy cluster's virial radius. Despite the weak magnitude of that effect, such ring-features may be the most dramatic effect of the self-interacting nature of DM. Fig. <ref> shows an example of such a feature in the tail of the radial mass profile of a simulation of a galaxy cluster collision observed along the axis of the collision.
The size of that feature is just about $0.2\%$ of peak-density, however, its remote location from the collision center may still render such features observable. In fact, we can approximately estimate the relative size of such a feature using the following formula,
\begin{equation}\label{eq:radshellsize}
\frac{\sigma_{shell}}{\sigma_{max}}=\beta a \frac{\log(1+c)-c/(1+c)}{r/r_c},
\end{equation}
where $a$ is the fraction of DM halos scattered into the shell, $c$ is the galaxy cluster's concentration parameter, $r$ is the current radius of the shell, and $\beta\approx 0.5$ is a numerical factor.
In formula (<ref>), we consider that a uniform spherical shell of radius $r$, width (or shell-depth) $h$, and volume-density $\rho$, carrying a total mass $M=4\pi r^2h\rho$, appears in projected density maps as two components: a nearly uniform disk of total mass
$M_{disk}\approx 2\pi r^2h\rho=\frac12M$, and a width $h$ circular rim with the remainder of the mass, that is, having the projected surface density of $\sigma_{shell}\approx\frac12M/(2\pi r h)$.
For a scattered DM shell carrying a fraction $a$ of the total mass of the collided DM halos, $M_{tot}\approx 2\times 4\pi\rho_0 r_c^3 (\log(1+c)-c/(1+c))$, and having a width of $h\approx 2r_c$, we thus obtain $\sigma_{shell}\approx a\rho_0 r_c^2 (\log(1+c)-c/(1+c))/r$. For a more general, nonuniform shell of scattered DM material, we write $\sigma_{shell}= 2\beta a \rho_0 r_c^2 (\log(1+c)-c/(1+c))/r$, where $\beta\approx 0.5$ is a numerical factor defining the fraction of the mass contained in the circular rim for specific radial profile of the shell.
Considering that $\sigma_{max}\approx 2r_c\rho_0$, we then can arrive at Eq. (<ref>).
Eq.(<ref>) allows us to estimate the magnitude of such a scattered DM ring-like feature more generally between 0.1% and 2%.
Interestingly, Ref. <cit.> reports observation of such a ring-like DM structure in the long-range reconstructions of the mass profile of the galaxy cluster CL 0024+17, having compatible magnitude between 1% and 5% .
§.§.§ Optimal conditions for observation of SIDM effects
As discussed, we find that the most profound SIDM effects can be observed in galaxy cluster collisions in the space between the colliding galaxy clusters but not within them. These differences can be well described in terms of a spherical shell of singly-scattered DM particles engulfing the outgoing galaxy clusters and produced due to DM particle-particle scatterings during the central passage of the galaxy clusters. Such shells of scattered DM material may contain 10% to 20% of the entire collision mass without significantly distorting the outgoing galaxy clusters or their DM halos.
For galaxy cluster collisions observed during their early and intermediate post-collision separation stages, the parts of such DM shells scattered directly above and below the center of the collision, along the line of observation, contribute significantly to heavier and wider central regions of the projected mass density maps of such SIDM galaxy clusters, as shown in Figs. <ref>-<ref>.
These differences are very substantial and may account the changes relative to CDM scenarios of up to 10%-20%. Up to 2-fold increase can be observed in such central densities in late separation stages in galaxy cluster collisions with lower collision velocity.
Lag of centroids of DM halos can be also observed for SIDM and slower collisions ($k\approx 1.5$), but not fast collisions ($k\geq 2$).
These are quantitatively measurable by means of projected mass plots on the axial line connecting the centers of the colliding galaxy clusters, as shown in Fig. <ref>.
A qualitative new feature observed for SIDM is the presence of expanding shells of scattered DM material engulfing the outgoing galaxy clusters in such galaxy cluster collisions. Such shell contributes mass densities at 0.1%-5% peak-density levels at very large distances and large scattering angles relative to the collision center. This feature can be quantified using the azimuthal plots of the mass distribution in projected mass density maps relative to the collision center, and the plots of mass distribution within small (for example, $30^\circ$) radial sectors surrounding the equatorial plane of the collision, as shown in Figs. <ref>-<ref>. In such cases, significant deviations from CDM are observed in late post-collision clusters. Whenever the shell of scattered DM materials moves into the region of constant background, a ring-like DM structure can emerge in the mass density observed along the collision axis.
Given these observations, the conditions for best chance of observing such features comprise collisions of galaxy clusters at high speed, during intermediate or late post-collision separation stages.
With respect to the collisions' speed, the higher $k$ above 2.0 may be preferred because they lead to much lesser gravitational disruption of colliding galaxy clusters.
The higher speed of the collision does not affect the DM particle-particle scattering redistributing DM particles azimuthally, as such speed cancels out from the DM particle-particle scattering, but reduces the efficiency of gravitational effects.
Tables <ref> and <ref> list the initial in-fall and the closest-approach speed corresponding to different values of $k$ for different total masses of the collision $M_{tot}$.
As can be seen from these tables, for very heavy galaxy clusters the initial speed required to achieve $k\approx 2-4$ is high but not prohibitively so.
Such speeds can be achieved if the galaxy clusters collide after in-falling towards an element of the large-scale structure or within the context of galaxy clusters' relative motion within a superstructure such as galaxy supercluster.
Examples of galaxy cluster collisions hypothesized to have occurred near a filament are known in the literature <cit.>.
Note that while the closest-approach speeds may seem to be high on first inspection, most of such speed is the speed gained due to gravity during in-fall.
The initial clusters' in-fall speeds in relation to the collision's kinetic parameter $k$ and the collision's total mass $M_{tot}$.
k $10^{15} M_\odot$ $5\cdot10^{14} M_\odot$ $10^{14} M_\odot$ $5\cdot10^{13} M_\odot$ $10^{13} M_\odot$
1 0 kmps 0 kmps 0 kmps 0 kmps 0 kmps
2 2100 kmps 1700 kmps 1000 kmps 800 kmps 470 kmps
4 3600 kmps 2900 kmps 1700 kmps 1400 kmps 780 kmps
8 5600 kmps 4400 kmps 2500 kmps 2100 kmps 1200 kmps
The closest approach relative collision speeds in relation to the the collision's kinetic parameter $k$ and the collision's total mass $M_{tot}$.
k $10^{15} M_\odot$ $5\cdot10^{14} M_\odot$ $10^{14} M_\odot$ $5\cdot10^{13} M_\odot$ $10^{13} M_\odot$
1 4200 kmps 3300 kmps 1900 kmps 1500 kmps 900 kmps
2 5900 kmps 4700 kmps 2800 kmps 2200 kmps 1300 kmps
4 8400 kmps 6600 kmps 3900 kmps 3100 kmps 1800 kmps
8 11900 kmps 9300 kmps 5500 kmps 4400 kmps 2600 kmps
§ SUMMARY AND DISCUSSION
In this work, we performed study of possible configurations of post-collision mass distributions in high-speed galaxy cluster collisions with respect to different hypothetical self-interaction strengths of DM.
All such scenarios can be characterized essentially by two main parameters comprising the ratio of the kinetic and gravitational energy of the collision, $k$, and the fraction of DM halo mass scattered in the collision by DM particle-particle interactions, $a$.
With respect to the kinetic parameter $k$, we observe three main regimes of collisions.
Collisions with very high speed, $k>2$, feature the galaxy clusters passing through each other with little gravitational disturbance. The azimuthal DM particles redistribution effect of weak DM self-scattering is clearly discernible in this setting.
For collisions with $2>k>1$, gravitational effects produce “fan-out” DM ejecta mostly confined to small scattering angles in forward and backward cones of the collision.
In that case, DM self-scattering effects can manifest themselves as discernible new mass concentrations either in central and equatorial regions of the collisions' projected mass density maps, where the efficiency of the gravitational DM scattering is the lowest.
For yet slower collisions, the kinetic energy of the colliding clusters is insufficient to provide for their post-collision separation and rapid mergers are observed with significant and disperse gravitational ejecta of complex shapes and large extents.
With respect to the strength of DM self-interactions, we find
that for strong DM self-scattering in which 50% or more of DM halo mass suffers non-gravitational scattering during the passage, $\sigma_{DM}/m_{DM}>2 cm^{-2}g$, the DM halos are destroyed in the passage. This disruption is severe and results in formation of a single common halo composed of heated DM material. As such, this outcome is far beyond the limited effects such as changes in mass-to-light ratio or a lag of DM halo centroids previously discussed in the literature. Instead, complete and rapid reorganization of the entire DM halo of the colliding galaxy clusters is observed.
For weak DM self-scattering in which 10% to 20% of DM halo particles suffer a non-gravitational scattering, $\sigma_{DM}/m_{DM}<0.5 cm^{-2}g$, the formation of spherical shells of DM particles is observed. This can be understood as the outcome of DM particle-particle scattering during the time of DM halos' central passage through each other, under the conditions of the mean free path of DM particles being significantly greater than the size of the halos, $a\ll 1$.
The shell of such scattered DM material can be observed in the projected mass density maps such as obtained via gravitational lensing under certain conditions.
The said DM structure can be discerned in the projected mass density maps as DM contributions at very large scattering angles and large distances from the collision center or as extended disk and ring-like structures of the size comparable to the post-collision separation of outgoing galaxy clusters.
Such features are forbidden in purely gravitational collisions, as discussed above.
Such DM self-scattering structures can admit up to 20% of the collision's total mass before any significant disruptions begin to be noticeable in the main galaxy clusters' halos.
The remote location of the scattered DM shell either from the outgoing galaxy clusters or central hot ICM may allow such structures to be seen experimentally despite their weak magnitude.
The survey of the literature on weak and strong gravitational lensing reconstructions of mass profiles of colliding galaxy clusters shows that the presence of DM densities at large scattering angles and large distances indeed is a wide-spread feature of such observations. For instance, the reconstructed projected mass density of the colliding galaxy clusters A754 and A520 show both very significant off-axial concentrations of DM at scattering angles close to $90^\circ$ and at the same distance from the collision center as the collided galaxy groups <cit.>. Combined weak and strong lensing reconstructions of the mass density of the Bullet cluster shows large diffuse DM mass concentration in between the outgoing galaxy groups, having disk shape and the size once again similar to the separation of the galaxy groups <cit.>.
Yet more interesting observation has been produced in recent reconstruction of the mass profile of strongly lensing galaxy cluster CL0024+017, now figured as colliding galaxy clusters seen along the axis of the collision, performed to large distances from the center <cit.>. Such reconstruction implicated a weak ring-like DM structure existing around the collided galaxy clusters, as shown in Fig. 7 and Fig. 10 of Ref. <cit.>.
The structure is consistent with the features observed in simulations in this work and has the magnitude of 1-5%.
It is possible to suggest an interpretation of that structure as the remains of a shell of scattered DM material generated in such an ancient galaxy cluster collision.
This work was supported in part by the American Physical Society International Travel Grant Award Program (APS ITGAP) and in part by the US Department of Energy under Contract NO. DE-FG02-03ER41260.
This research also used the resources of the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
YM also would like to acknowledge the support from the Bilim Akademisi—The Science Academy (Istanbul, Turkey) young investigator award under the BAGEP program.
§ DARK MATTER PARTICLE SCATTERING ALGORITHM AND PARTICLE MESH ALGORITHM
DM particle-particle scattering algorithm
$cell \in \mathcal{G}$
particles $i,j \in cell$
Select the pair $(i,j)$ with probability $P=\alpha|\vec{v}_i-\vec{v}_j|\Delta t$
$\vec{V}_{CM}\leftarrow (\vec{v}_i+\vec{v}_j)/2$
$V_{rel}\leftarrow |\vec{v}_i-\vec{v}_j|/2$
Choose $\vec{n}$ uniformly at random on unit sphere
Particle Mesh algorithm for $N$-body gravitational dynamics
simulation parameters $M_{tot},N1,N2,\Delta V2,\Delta R1,\Delta R2,\Delta b$
$\mathcal{G}$ is a cubic 3D grid of size $D^3$
$list \leftarrow \{(\vec{r}_i,\vec{v}_i)\}$ is the list of particles' position and velocity vectors
Form initial particle distributions for the two colliding clusters
$list1\leftarrow N1$ random particles from the standard equilibrium profile (Section <ref>)
$list2\leftarrow N2$ random particles from the standard equilibrium profile (Section <ref>)
$scaling1\leftarrow (({M_{tot}}{N1}/{(N1+N2)})/{M_{std}})^{1/3}$
$scaling2\leftarrow (({M_{tot}}{N2}/{(N1+N2)})/{M_{std}})^{1/3}$
Update all particles in $list1: \vec r_i\leftarrow \vec{r}_i \cdot scaling1,\vec{v}_i\leftarrow\vec{v}_i\cdot scaling1$
Update all particles in $list2: \vec r_i\leftarrow \vec{r}_i \cdot scaling2,\vec{v}_i\leftarrow\vec{v}_i\cdot scaling2$
Update all particles in $list1: (\vec r_i)_x\leftarrow (\vec r_i)_x-\Delta R1$, $(\vec r_i)_y\leftarrow (\vec r_i)_y-\Delta b$
Update all particles in $list2: (\vec r_i)_x\leftarrow (\vec r_i)_x+\Delta R2$
Calculate initial in-fall velocities
$\Delta R\leftarrow \Delta R1+\Delta R2$
$E_G\leftarrow \frac{G(M_{tot})^2}{\Delta R}(\frac{N1}{N1+N2})(\frac{N2}{N1+N2})$
$V2\leftarrow \frac{2E_G}{M_{tot}}+\Delta V2$
Update all particles in $list1: (\vec v_i)_x\leftarrow (\vec v_i)_x+\sqrt{\frac{N2}{N1}V2}$
Update all particles in $list2: (\vec v_i)_x\leftarrow (\vec v_i)_x-\sqrt{\frac{N1}{N2}V2}$
$list\leftarrow \{list1,list2\}$
Main simulation loop
$\Delta t\leftarrow \min(\Delta t_{max},\Delta r_{max}/\max(|\vec{v}_i|))$
Discard the particles that moved out of the bounds of $\mathcal{G}$
Distribute the particles in $list=\{\vec r_i,\vec v_i\}$ into the cells of grid $\mathcal{G}$ based on $\vec{r}_i\in cell$, $n(\vec{x})\leftarrow list\{ \vec{r}_i \}$
$\tilde{n}(\vec{k})\leftarrow FFT(n(\vec{x}))$
$\tilde{\Phi}(\vec{k})\leftarrow -4\pi\tilde{n}(\vec{k})/k^2$ (but set $\tilde{\Phi}(0)\leftarrow 0$)
$\Phi(\vec{x})\leftarrow iFFT(\tilde{\Phi}(\vec{k}))$
For all particles in $list: \vec{a}_i\leftarrow -G \nabla \Phi(\vec{r}_i)$
$\Delta t\leftarrow \min(\Delta t,\Delta v_{max}/\max(|\vec{a}_i|))$
Update all particles in $list: \vec v_i\leftarrow \vec v_i+\vec{a}_i\Delta t$
Evaluate particle-particle scatterings using Algorithm <ref>
Update all particles in $list: \vec r_i\leftarrow \vec r_i+\vec{v}_i\Delta t$
$t\leftarrow t+\Delta t$
§ CONVERGENCE AND ACCURACY OF THE NUMERICAL INTEGRATION METHOD WITH RESPECT TO THE SIDM EFFECTS
The Particle Mesh algorithm used in this work can experience accuracy loss at small scales due to the finite size of the meshgrid used for approximating the dynamical equations. To control for this effect and its impact on the SIDM features elucidated in this work, we simulate the fast CDM and symmetric-central-weak SIDM scenarios from Table <ref> ($k=1.6$) with different sizes of the meshgrid ranging from $D=100^3$ to $D=600^3$ points, and different numbers of particles used in the simulation from $N=100\cdot 10^3$ to $N=400\cdot 10^3$.
The results of these numerical experiments are shown in Figs. <ref> and <ref>.
The red and green curves represent the symmetric-central-weak SIDM results and the fast CDM results, respectively. Here, the distance between the centers of
the outgoing galaxy clusters is given by $\Delta r = 15 r_c$, corresponding to the late separation stage as defined in the main text. In Fig. <ref>, the dotted, dash-dotted, dashed and solid lines represent the different sizes of the meshgrid as $D = 100^3, 200^3, 400^3$ and $600^3$, respectively, while the number of particles is fixed at
$N=200\cdot 10^3$. In Fig. <ref>, the dotted, dash-dotted and dashed lines represent the different numbers of particles used in the simulation as $N = 100\cdot 10^3,
200\cdot 10^3$ and $400\cdot 10^3$, respectively, while the size of the meshgrid is fixed at $D = 400^3$ points.
We observe that in all cases sufficient convergence is achieved by $D=400^3$ and $N=200\cdot 10^3$, which is the choice of the parameers used in the simulations in this work.
The axial (along-the-collision-axis) mass distributions and the sector-azimuthal mass distributions, as introduced in the main text, are not very sensitive to the above simulation parameters, whereas the only significant difference in these quanities is observed for the grid size of $D=100^3$. The radial distribution of DM mass in the equatorial sector of the projected mass density maps is more sensitive, due to the original smallness of that effect. However, even in that case $D=400^3$ and $N=200\cdot 10^3$ suffice to achieve acceptable convergence.
The three SIDM-versus-CDM effects in axial (left), azimuthal (center), and radial (right) distribution of dark matter in a high speed galaxy cluster collision inspected with respect to varying Particle Mesh algorithm's grid size, using $N=200\cdot 10^3$ particles in all simmulations.
Red lines are for SIDM and green lines are for CDM simulations. The convergence of the simulations past the grid sizes of $D=400^3$ is clearly seen.
The three SIDM-versus-CDM effects in axial (left), azimuthal (center), and radial (right) distribution of dark matter in a high speed galaxy cluster collision inspected with respect to varying the number of particles in the simulation.
Red lines are for SIDM and green lines are for CDM simulations.
The convergence of the simulations past $N=200\cdot 10^3$ particles is clearly seen in all cases. $D=400^3$ has been used in all these simulations.
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1511.00152
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We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.
Unsupervised learning, PCA, clustering, K-means, randomized algorithm, big data.
§ INTRODUCTION
Principal component analysis (PCA) is a classical unsupervised analysis method commonly used in all quantitative disciplines <cit.>. Given $n$ observations of $p$-dimensional data
$\left\{ \x_{i}\right\} _{i=1}^n\in\R^p$, standard algorithms to compute PCA require $\order(p^2n)$ flops (we assume throughout the paper that $n\ge p$), which is expensive for large $p$ and/or $n$. The problem is exacerbated in situations with high communication cost, as in distributed data settings such as Hadoop/MapReduce clusters or as in sensor networks. An extreme case is when the data rate is so large that we turn to “streaming” algorithms which examine each new data entry and then discard it, i.e., take at most one pass over the data <cit.>.
Another commonly used unsupervised analytic task is clustering, which refers to identifying groups of similar data samples in a data set. Running a simple clustering method such as the K-means algorithm turns out to be infeasible for distributed and streaming data <cit.>. Therefore, a recent line of research aims at developing scalable learning tools for “big data” and providing fundamental insights and tradeoffs involved in these algorithms.
This paper proposes a specific mechanism to reduce the computational, storage, and communication burden in unsupervised analysis methods such as PCA and K-means clustering, which requires only one pass over the data. The first step is a pre-processing step designed to precondition the data and smooth out large entries, which is based on the well-known fast Johnson-Lindenstrauss result <cit.>. We summarize existing results about this preconditioning and describe how it benefits our approach. The second step is element-wise random sampling: choosing to keep exactly $m$ out of $p$ entries (without replacement) per sample. This step can be viewed as forming a sampling matrix $\RR_i\in\R^{p\times m}$ which contains $m$ distinct canonical basis vectors drawn uniformly at random, and multiplying each preconditioned sample $\mathbf{x}_i$ on the left by $\RR_i\RR_i^T$ (the projection matrix onto the subspace spanned by the columns of $\RR_i$). The two steps of our approach, preconditioning and sampling, can be easily combined into single pass over the data which eliminates the need to revisit past entries of the data.
We then analyze, in a non-Bayesian setting, the performance of our estimators and provide theorems that show as $n$ grows with $p$ fixed, the bounds hold with
high probability
[in the sense of <cit.>] if
$m=\order(\log n/n)$,
cf. Corollary <ref>.
This means that as we collect more samples, the size of the incoming data increases proportional to $n$ but the amount of data we must store only increases like $\log n$, i.e., we compress with ratio $n/\log n$. This is possible because of the careful sampling scheme — if one were to simply keep some of the data points $\xi$, then for a small error in the $\ell_\infty$ norm, one needs to keep a constant fraction of the data.
Similar proposals, based on dimensionality reduction or sampling, have been proposed for analyzing large data sets, e.g., <cit.>. Many of these schemes are based on multiplying the data on the left by a single random matrix $\rand$ and recording $\rand \X$ where $\X=[\x_1,\ldots,\x_n]$ is the $p\times n$ matrix with data samples as columns, $\rand$ is $m\times p$ with $m<p$, and the columns of $\rand \X$ are known as sketches or compressive measurements. The motivation behind sketching is the classical Johnson-Lindenstrauss lemma <cit.> which states that the scaled pairwise distances between low-dimensional sketches are preserved to within a small tolerance for some random matrices $\rand$. Moreover, the more recent field of compressive sensing demonstrates that low-dimensional compressive measurements $\rand \mathbf{X}$ contain enough information to recover the data samples under the assumption that $\mathbf{X}$ is sparse in some known basis representation <cit.>.
However, for PCA, the desired output is the eigendecomposition of $\X\X^T$ or, equivalently, the left singular vectors of $\X$, for which we cannot typically impose structural constraints. Clearly, a single random matrix $\rand^{T}\in\R^{p\times m}$ with $m<p$ only spans at most an $m$-dimensional subspace of $\R^{p}$, and after the transformation $\rand\X$, information about the components of the left singular vectors within the orthogonal complement subspace will be lost. Another technique to recover the left singular vectors is to also record $\X\rand'$ for another random matrix $\rand'$ of size $n \times m'$ with $m'<n$ <cit.>, and with careful implementation, it is possible to record $\rand\X$ and $\X\rand'$ in a one-pass algorithm. However, even in this case, it is not possible to return something as simple as a center estimator in the K-means algorithm without making a second pass over the data.
Our approach circumvents this since each data sample $\x_i$ is multiplied by its own random matrix $\RR_i\RR_i^T$, where $\RR_i\in\R^{p\times m}$ consists of a
randomly chosen subset of $m$ distinct canonical basis vectors. We show that our scheme leads to one-pass algorithms for unsupervised analysis methods such as PCA and K-means clustering, meanwhile containing enough information to recover principal components and cluster centers without imposing additional structural constraints on them.
We expand on comparisons with relevant literature in more detail in Section <ref>, summarize important results about the preconditioning transformation and sub-sampling in Section <ref>, provide theoretical results on the sample mean estimator and covariance estimator in sections <ref> and <ref> respectively, then focus on results relevant for K-means clustering in Section <ref> and finish the paper with numerical experiments in Section <ref> and a conclusion.
§.§ Setup
In this paper, we consider a non-Bayesian data setting where we make no distributional assumptions on the set of data samples $\X=[\x_1,\ldots,\x_n]\in\R^{p\times n}$. For each sample $\x_{i}$, we form a sampling matrix $\RR_i\in\R^{p\times m}$, where the $m$ columns are chosen uniformly at random from the set of canonical basis vectors in $\R^{p}$ without replacement. Thus, we use $m$ for the intrinsic dimensionality of the compressed samples, and write the compression factor as $\gamma=\frac{m}{p}$.
We name column vectors by lower-case bold letters and matrices by upper-case bold letters. Our results will involve bounds on the Euclidean and maximum norm in $\R^p$, denoted $\|\x\|_2$ and $\|\x\|_\infty$ respectively, as well as the spectral and Frobenius norms on the space of $p\times n$ matrices, denoted $\|\X\|_2$ and $\|\X\|_F$ respectively.
Recall that $\|\x\|_\infty \le \|\x\|_2 \le \sqrt{p}\|\x\|_\infty$. We use $\|\X\|_\text{max}$ to denote the maximum absolute value of the entries of a matrix, $\|\X\|_\maxRow$ to be the maximum $\ell_2$ norm of the rows of a matrix, i.e., $\|\X\|_\maxRow = \|\X\|_{2\rightarrow \infty} = \sup_{\y\neq 0} \frac{\|\X\y\|_\infty}{\|\y\|_2}$, and $\|\X\|_\maxCol=\|\X\|_{1\rightarrow 2}$ to be the maximum $\ell_2$ norm of the columns of a matrix.
Let $\mathbf{e}_i$ denote the $i$-th vector of the canonical basis in $\R^{p}$, where entries are all zero except for the $i$-th one which is $1$. In addition, $\diag(\x)$ returns a square diagonal matrix with the entries of vector $\x$ on the main diagonal, and $\diag(\X)$ denotes the matrix formed by zeroing all but the diagonal entries of matrix $\X$. We also represent the entry in the $i$-th row and the $j$-th column of matrix $\X$ as $X_{i,j}$.
§.§ Contributions
In this paper, we introduce an efficient data sparsification framework for large-scale data applications that does not require incoherence and distributional assumptions on the data. Our approach exploits randomized orthonormal systems to find informative low-dimensional representations in a single pass over the data. Thus, our proposed compression technique can be used in streaming applications, where data samples cannot fit into memory <cit.>. We present results on the properties of sampling matrices and randomized orthonormal systems in Theorem <ref> and Section <ref>, respectively.
To demonstrate the effectiveness of our framework, we investigate the application of preconditioned data sparsification in two important unsupervised analysis methods, PCA and K-means clustering. Two unbiased estimators are introduced to recover the sample mean and covariance matrix from the compressed data. We provide strong theoretical guarantees on the closeness of the estimated and true sample mean in Theorem <ref>. Moreover, we employ recent results on exponential concentration inequalities for matrices <cit.> to obtain an exponentially decaying bound on the probability that the estimated covariance matrix
deviates from its mean value in Theorem <ref>. Numerical simulations are provided to
validate the tightness of the concentration bounds derived in this paper. Furthermore, our theoretical results reveal the connections between accuracy and important parameters, e.g., the compression factor and the underlying structure of data.
We also examine the application of our proposed data sparsification technique in the K-means clustering problem. In this case, we first define an optimal objective function based on the Maximum-Likelihood estimation. Then, an algorithm called sparsified K-means is introduced to find assignments as well as cluster centers in one pass over the data. Theoretical guarantees on the clustering structure per iteration are provided in Theorem <ref>. Finally, we present extensive numerical experiments to demonstrate the effectiveness of our sparsified K-means algorithm on large-scale data sets containing up to $10$ million samples.
Beyond just the two unsupervised learning examples, we advocate the general usage of preconditioning followed by sampling. Sampling methods have long been popular due to their simplicity, but unless appropriate weighted sampling distributions are used, the accuracy of these methods is low. The use of the preconditioning obviates the need for weighted distributions, and is easy to analyze and implement.
§ RELATED WORK
Our work has a close connection to the recent line of research on signal processing and information retrieval from compressive measurements that assumes one only has access to low-dimensional random projections of the data <cit.>. For example, Eldar and Gleichman <cit.> studied the problem of learning basis representations (dictionary learning) for the data samples $\X\in\R^{p\times n}$ under the assumption that a single random matrix $\RR\in\R^{p\times m}$, $m<p$, is used for all the samples. It was shown that $\RR^{T}\X$ does not contain enough information to recover the basis representation for the original data samples, unless structural constraints such as sparsity over the set of admissible representations are imposed. This mainly follows from the fact that $\RR$ has a non-trivial null space and, without imposing constraints, we cannot recover the information about the whole space $\R^{p}$. For example, consider a simple case of a rank-one data matrix, $\X=\sigma\mathbf{u}\mathbf{v}^T$, where the goal is to recover the single principal component $\mathbf{u}\in\R^p$ from $\RR^{T}\X$. The singular value decomposition of $\RR^{T}\X$ results in $\RR^{T}\X=\sigma\mathbf{u}'\mathbf{v}^T$, where $\mathbf{u}'=\RR^{T}\mathbf{u}\in\R^m$, so we have retained information on $\mathbf{v}$. However, accurate estimation of $\mathbf{u}$ from $\mathbf{u}'=\RR^{T}\mathbf{u}$ is not possible, unless additional constraints are imposed on $\mathbf{u}$. The line of work <cit.> addressed this issue, in the dictionary learning setting, by observing each data sample $\x_i$ through distinct random matrices.
Another line of work considers covariance estimation and recovery of principal components (PCs) based on compressive measurements <cit.>. In particular, Qi and Hughes <cit.> proposed a method for recovering the mean and principal components of $\X=[\x_1,\ldots,\x_n]$ from compressive measurements $\RR_{i}^{T}\x_i\in\R^{m}$, $i=1,\ldots,n$, where $\RR_i\in\R^{p\times m}$ with $m<p$ is a random matrix with entries drawn i.i.d. from the Gaussian distribution. This method requires computing the projection matrix onto the subspace spanned by the columns of $\RR_i$ which is $\mathbf{P}_i=\RR_i(\RR_i^{T}\RR_i)^{-1}\RR_i^{T}$. Then, each $\mathbf{P}_i\x_i$ can be directly computed from the compressive measurements since $\mathbf{P}_i\x_i=\RR_i(\RR_i^{T}\RR_i)^{-1}\RR_i^{T}\x_i$. It has been shown that the mean and principal components of $\mathbf{P}_i\x_i\in\R^{p}$, $i=1,\ldots,n$, converge to the mean and principal components of the original data (up to a known scaling factor) as the number of data samples $n$ goes to the infinity. However, this method is computationally inefficient because
even if each (pseudo-)random matrix $\RR_i$, $i=1,\ldots,n$, is implicitly stored by a seed, the computational cost of the matrix multiplies and inversions is high.
Their work was extended in <cit.>, where the authors studied the problem of performing PCA on $\RR_i\RR_i^{T}\x_i$ (eliminating the matrix inversion) for a general class of random matrices, where the entries of $\RR_i$ are drawn i.i.d. from a zero-mean distribution with finite moments. Two estimators were proposed to estimate the mean and principal components of the original data with statistical guarantees for a finite number of samples $n$. Moreover, it was shown that one can use very sparse random matrices <cit.> to increase the efficiency in large-scale data sets. However, generating $n$ unstructured random matrices, with $p\times m$ entries in each, can still be memory/computation inefficient for high-dimensional data sets with $p$ large. Another disadvantage of the prior work <cit.> and <cit.> is that the result only holds for data samples drawn from a specific probabilistic generative model known as the spiked covariance model, which may not be informative for real-world data sets.
In the fields of theoretical computer science and numerical linear algebra, there are several alternative lines of work that are related to our proposed approach and we discuss them below in detail.
§.§ Comparison with column sampling approaches
The most natural compression scheme for large-scale data sets would be to directly select a small subset of the original data and then perform data analytic tasks on the reduced subset. The two natural distributions for column sampling are the uniform distribution and a non-uniform data-dependent distribution based on so-called statistical leverage scores <cit.>[Sampling according to column norm is a common third option, but is generally inferior to leverage-score based sampling]. The former method, uniform sampling, is a simple one-pass algorithm but it will perform poorly on many problems if there is any structural non-uniformity in the data <cit.>.
To see this, we recreate the numerical experiment of <cit.> and compare the accuracy of left singular vectors/principal components (PCs) estimated using our proposed approach with those estimated after uniform column sampling. We set the parameters $p=512$ and $n=1024$ and consider $1000$ runs for different values of the compression factor $\gamma=\frac{m}{p}$. In each run, we generate a data matrix $\X\in\R^{p\times n}$ from the multivariate $t$-distribution with $1$ degree of freedom and covariance matrix $\mathbf{C}$ where $C_{ij}=2\times 0.5^{|i-j|}$. In our approach, we precondition the data as described in <ref> and then keep exactly $m$ out of $p$ entries for each data sample to obtain a sparse matrix. To have a fair comparison, we consider randomly selecting $2m$ columns of $\X$ because $n/p=2$ and our sparse matrix has exactly $2mp$ nonzero entries. We measure the accuracy of the estimated PCs based on the explained variance <cit.>: given estimates of $k$ PCs $\widehat{\UU}\in\R^{p\times k}$ (we take $k=10$), the fraction of explained variance is defined as $\trace(\widehat{\UU}^{T}\X\X^{T}\widehat{\UU})/\trace(\X\X^{T})$, and closeness of this fraction to $1$ represents higher accuracy.
Fig. <ref> reports the mean and standard deviation of the explained variance over $1000$ trials. For both approaches, the average explained variance approaches $1$ as $\gamma\rightarrow 1$, as expected, and uniform column sampling is slightly more accurate than our approach.
However, uniform column sampling has an extremely high variance for all values of $\gamma$, e.g., the standard deviations for $\gamma=0.1$, $0.2$, $0.3$ are $0.20$, $0.28$, and $0.31$ respectively.
On the other hand, the standard deviation for our approach is significantly smaller for all values of $\gamma$ (less than $0.04$),
due to the preconditioning.
Thus the worst-case performance of our approach is reasonable, while the worst-case performance of column sampling may be catastrophically bad.
Accuracy of estimated PCs via one-pass methods: uniform column sampling and our proposed precondition+sparsification approach. We plot the mean and standard deviation of the explained variance over $1000$ runs for each value of $\gamma$. The standard deviation for our approach is significantly smaller
compared to the uniform column sampling.
The second common method for column sampling is based on a data-dependent non-uniform distribution and has received great attention in the development of randomized algorithms such as low-rank matrix approximation. These data-dependent sampling techniques are typically computationally expensive and a SVD on the data matrix is required. Recently, there are variants that can be computed more efficiently such as <cit.>. However, these algorithms compute the sampling distribution in at least one pass of all the samples, and a second pass is required to actually sample the columns based on this distribution. Thus at least two passes are required and these algorithms are not suitable for streaming <cit.>.
§.§ Comparison with other element-wise sampling approaches
Analysis of sparsification of matrices for the purpose of fast computation of low-rank approximations goes back to at least Achlioptas and McSherry <cit.>, who propose independently keeping each entry of the matrix in either a uniform fashion or a non-uniform fashion. In the former, each entry of the data matrix is kept with the same probability, whereas in the non-uniform case, each entry is kept with probability proportional to its magnitude squared. Under the latter scheme,
the expected number of nonzero entries can be bounded but
one does not have precise control on the exact number of nonzero entries.
Recently, Achlioptas et al. <cit.> have considered a variant of element-wise sampling where a weight is assigned to each entry of the data matrix according to its absolute value normalized by the $\ell_{1}$ norm of the corresponding row, and then a fixed number of entries is sampled (with replacement) from the matrix. Computing the exact $\ell_{1}$ norms of the rows requires one pass over the data and consequently this method requires two passes over the data. It is shown empirically that disregarding the normalization factor performs quite well in practice, thus yielding a one-pass algorithm. Calculating guarantees on their one-pass variant is an interesting open-problem.
§.§ K-means clustering for big data
Clustering is a commonly used unsupervised learning task that reveals the underlying structure of a data set by splitting the data into groups, or clusters, of similar samples. It has applications ranging from search engines and social network analysis to medical and e-commerce domains. Among clustering algorithms, K-means <cit.> is one of the most popular clustering algorithms <cit.>. K-means is an iterative expectation-maximization type algorithm in which each cluster is associated with a representative vector or cluster center, which is known to be the sample mean of the vectors in that cluster. In each iteration of K-means, data samples are assigned to the nearest cluster centers (usually based on the Euclidean norm) and cluster centers are then updated based on the most recent assignment of the data. Therefore, the goal of K-means is to find a set of cluster centers as well as assignment of the data.
Despite the simplicity of K-means, there are several challenges in dealing with modern large-scale data sets. The high dimensionality and large volumes of data make it infeasible to store the full data in a centralized location or communicate the data in distributed settings <cit.> and these voluminous data sets make K-means computationally inefficient <cit.>.
Recent approaches to K-means clustering for big data have focused on selecting the most informative features of the data set and performing K-means on the reduced set. For example, Feldman et al. <cit.> introduced coresets for K-means, in which one reduces the dimension of the data set from $p$ to $m$ by projecting the data on the top $m$ left singular vectors (principal components) of $\X$, although this requires the SVD of the data matrix and so does not easily apply to streaming scenarios. To reduce the cost of computing exact SVD, the authors in <cit.> proposed using approximate SVD algorithms instead.
Other prior works, such as <cit.>, have used randomized schemes for designing efficient clustering algorithms. In particular, Mahoney et al. <cit.> proposed two provably accurate algorithms to reduce the dimension of the data by selecting a small subset of $m$ rows of the data matrix $\X\in\R^{p\times n}$ (feature selection) or multiplying the data matrix from the left by a random matrix $\rand\in\R^{m\times p}$ (feature extraction). Afterwards, the K-means algorithm is applied on these $n$ data samples in $\R^{m}$ to find the assignment of the original data. In feature selection, the rows of $\X$ are sampled via a non-uniform distribution, which requires two passes over the data. Then, having the distribution, it requires one more pass to actually sample $m$ rows which leads to a three-pass algorithm.
Furthermore, in both feature selection and feature extraction algorithms, K-means is applied on the projected data in $\R^{m}$ so
there is no ready estimate of the cluster centers in the original space $\R^{p}$. One could transform the cluster centers to $\R^p$ with the pseudo-inverse of $\rand$ (recall $\rand$ is low-rank) but this estimate is poor; see Figs. <ref> and <ref> in <ref>.
A better approach to calculate the cluster centers is to use the calculated assignment of vectors in the original space, which requires one additional pass over the data, and consequently neither feature selection nor feature extraction is streaming.
In this paper, we address the need to account for the computational/storage burden and we propose a randomized algorithm for K-means clustering, called sparsified K-means, that returns both the cluster centers and the assignment of the data in single pass over the data. To do this, we take a different approach where a randomized unitary transformation is first applied to the data matrix and we then choose $m$ out of $p$ entries of each preconditioned data uniformly at random to obtain a sparse matrix.
Afterwards, the K-means algorithm is applied on the resulting sparse matrix and, consequently, speedup in processing time and savings in memory are achieved based on the value of compression factor $\gamma=m/p<1$.
We provide theoretical guarantees on the clustering structure of our sparsified K-means algorithm compared to K-means on the original data in each iteration. In fact, this is another advantage of our method over the previous work <cit.>, where guarantees are available only for the final value of the objective function and, thus, it is not possible to directly compare the clustering structure of feature selection and feature extraction algorithms with K-means on the original data.
§ PRELIMINARIES
The randomized orthonormal system (ROS) is a powerful randomization tool that has found uses in fields from machine learning <cit.> to numerical linear algebra <cit.> and compressive sensing <cit.>.
We use the ROS to efficiently precondition the data and smooth out large entries in the
matrix $\X$ before sampling. It is straightforward to combine this preconditioning and sampling operation into a single pass on the data. Furthermore, we can unmix the preconditioned data because the preconditioning transformation is unitary, so by applying its adjoint we undo its effect. Because the operator is unitary, it does not affect our estimates in the Euclidean norm (for vectors) or the spectral and Frobenius norms (for matrices). Moreover, this transformation is stored implicitly and based off fast transforms, so applying to a length $p$ vector takes $\order(p\log(p))$ complexity, and applying it to a matrix is embarrassingly parallel across the columns <cit.>.
Specifically, the ROS preconditioning transformation
uses matrices $\Hadamard$ and $\Diag$ and is defined
\begin{equation}
\x\mapsto\y=\Hadamard\Diag\x\label{eq:ROS}
\end{equation}
where $\Hadamard\in\R^{p \times p}$ is a deterministic orthonormal matrix such as a Hadamard, Fourier, or DCT matrix for which matrix-vector multiplication can be implemented efficiently in $\order(p\log(p))$ complexity without the need to store the matrices explicitly. The matrix $\Diag\in\R^{p \times p}$ is a stochastic diagonal matrix whose entries on the main diagonal are random variables drawn uniformly from $\{\pm 1\}$. The matrix product $\Hadamard\Diag\in\R^{p \times p}$ is an orthonormal matrix and this mapping ensures that, with high probability, the magnitude of any entry of $\Hadamard\Diag\x$ is about $\order(1/\sqrt p)$ for any unit vector $\x$ (cf. Thm. <ref>).
We now present relevant results about the preconditioning transformation $\Hadamard\Diag
$ (<ref>) that will be used in the next sections; more sophisticated results are in <cit.> and Appendix <ref>.
Let $\x\in\R^p$, and $\y=\Hadamard\Diag\x$ a random variable from the ROS applied to $\x$, then for every $j=1,\ldots,p$,
\begin{equation}
\P\left\{ |y_j| \ge (t/\sqrt{p}) \|\x\|_2 \right\} \le 2\exp\left( -\eta t^2/2 \right)
\end{equation}
where $\eta=1$ for $\Hadamard$ a Hadamard matrix and $\eta=1/2$ for $\Hadamard$ a DCT matrix.
As pointed out in <cit.>, the proof follows easily from Hoeffding's inequality (Thm. <ref>). Since $\|\x\|_2=\|\y\|_2$, the theorem implies that no single entry $y_j$ is likely to be far away from the average $\|\y\|_2/\sqrt{p}$.
Using the union bound, we derive the following results:
Let $\X$ be a $p\times n$ matrix with normalized columns, and $\Y=\Hadamard\Diag\X$ a random variable from the ROS applied to $\X$, then for all
\begin{align}
\P\left\{ \|\Y\|_\text{max} \ge \frac{1}{\sqrt{p}}\cdot\sqrt{\frac{2}{\eta}\log\Big(\frac{2np}{\alpha}\Big)} \right\} &\le \alpha \label{eq:maxNorm} \\
\P\left\{ \|\Y\|_\maxRow \ge \sqrt{\frac{n}{p}}\cdot\sqrt{\frac{2}{\eta}\log\Big(\frac{2np}{\alpha}\Big)} \right\} &\le \alpha \label{eq:maxRowNorm}
\end{align}
Taking the union bound over all $np$ entries in the matrix $\Y$ gives (for each column of $\X$, we have $\|\x_i\|_2=1$)
\[
\P\left\{ \|\Y\|_\text{max} \ge t/\sqrt{p} \right\} \le 2np\exp\left( -\eta t^2/2 \right)
\]
which leads to (<ref>) if we choose $t^2=\frac{2}{\eta}\log(2np/\alpha)$.
To derive the second equation, let $\Yi$ denote the $j$-th row of $\Y$. We apply the union bound to get
\[
\P\left\{ \|\Yi\|_\infty \ge t/\sqrt{p} \right\} \le 2n\exp\left( -\eta t^2/2 \right)
\]
and we bound the $\ell_2$ norm of $\Yi\in\R^{1\times n}$ with $\sqrt{n}$ times the $\ell_\infty$ norm,
\begin{equation}\label{eq:RowNormY}
\P\left\{ \|\Yi\|_2 \ge \sqrt{\frac{n}{p}}\cdot t \right\} \le 2n\exp\left( -\eta t^2/2 \right)
\end{equation}
from which (<ref>) follows by taking the union bound over $p$ rows and again choosing $t^2=\frac{2}{\eta}\log(2np/\alpha)$.
Note that for $\Hadamard$ a Hadamard matrix, the result in (<ref>) shows the square of the $\ell_2$ norm of $\Y_{j,:}$ is unlikely to be larger than its mean value. To see this,
note that the $(u,l)$ entry of the Hadamard matrix
$H_{u,l}$ is either $\frac{1}{\sqrt{p}}$ or $-\frac{1}{\sqrt{p}}$, and $\Diag=\diag([D_1,\ldots,D_p])$ where each $D_i$ is $\pm1$ with equal probability. Without loss of generality, we consider the first row of $\Y$, $\Y_{1,:}$, and find the expectation of its $k$-th element squared:
\begin{align}
\E[Y_{1,k}^2] & =\E\Big[\Big(\sum_{l=1}^{p}D_lH_{1,l}X_{l,k}\Big)^2\Big]\nonumber\\
&+\sum_{l_1\neq l_2} \E[D_{l_1}D_{l_2}]H_{1,l_1}H_{1,l_2}X_{l_1,k}X_{l_2,k}\nonumber\\
\end{align}
since $\E[D_l]=0$, $\E[D_l^2]=1$, and $\X$ has normalized columns, i.e., $\sum_{l=1}^{p}X_{l,k}^2=1$. Thus, we get $\E[\|\Y_{1,:}\|_2^2]=\sum_{k=1}^{n}\E[Y_{1,k}^2]=\frac{n}{p}$.
The importance of the preconditioning by the ROS prior to sub-sampling is the reduction of the norms over the worst-case values. As we will see in the subsequent sections, the variance of our sample mean and covariance estimators depends on the quantities such as the maximum absolute value of the entries and the maximum $\ell_2$ norm of the rows of the data matrix. Therefore, the preconditioning step (<ref>) is essential to obtain accurate and reliable estimates. For example, let us consider a data matrix $\X$ with normalized columns, then it is possible for $\|\X\|_\text{max}=1$, which would lead to
weak bounds when inserted into our estimates. The best possible norm is $1/\sqrt{p}$ which occurs if all entries have the same magnitude. Our result in Corollary <ref> states that, under the ROS, with high probability all the entries of $\Y=\Hadamard\Diag\X$ have comparable magnitude and it is unlikely $\|\Y\|_\text{max}$ is larger than $\sqrt{\log(np)}/\sqrt{p}$, which leads to lower variance and improved accuracy of our estimates.
Next, we present a result about the effect of sub-sampling on reduction of the Euclidean norm. Let $\w=\RR\RR^T\x$ denote the sub-sampled version of $\x\in\R^p$. Obviously, $\|\w\|_2^2 \leq \|\x\|_2^2$ and for data sets with a few large entries, $\|\w\|_2^2$ may be very close to $\|\x\|_2^2$ meaning that sub-sampling has not appreciably decreased the Euclidean norm. For example, consider a vector $\x=[1,0.1,0.01,0.001]^T$ where we wish to sample two out of four entries uniformly at random. In this case, $\|\w\|_2^2$ takes extreme values such that it might be very close to either $\|\x\|_2^2$ or zero. However, if we precondition the vector $\x$ prior to sub-sampling based on (<ref>), this almost never happens, and the norm is reduced by nearly $m/p$ as one would hope for:
Let $\x\in\R^p$, and $\y=\Hadamard\Diag\x$ a random variable from the ROS applied to $\x$. Define $\w\in\R^p$ to be a sampled version of $\y$, keeping $m$ of $p$ entries uniformly at random (without replacement). Then with probability greater than $1-\alpha$,
\begin{equation}
\|\w\|_2^2 \le \frac{m}{p} \frac{2}{\eta}\log\Big(\frac{2p}{\alpha}\Big) \|\x\|_2^2
\end{equation}
where $\eta=1$ for $\Hadamard$ a Hadamard matrix and $\eta=1/2$ for $\Hadamard$ a DCT matrix. Moreover, if $\{\w_i\}_{i=1}^{n}\in\R^p$ are sampled versions of the preconditioned data $\Y=\Hadamard\Diag\X$, with probability greater than $1-\alpha$,
\begin{equation}
\|\w_i\|_2^2 \le \frac{m}{p} \frac{2}{\eta}\log\Big(\frac{2np}{\alpha}\Big) \|\x_i\|_2^2,\; i=1,\ldots,n.
\end{equation}
Regardless of how we sample, it holds deterministically that $\|\w\|_2 \le \sqrt{m}\|\y\|_\infty$ and this bound is reasonably sharp since the entries of $\y$ are designed to have approximately the same magnitude. Using Thm. <ref> and the union bound, the probability that $\|\y\|_\infty \ge (t/\sqrt{p}) \|\x\|_2$ is less than $2p e^{-\eta t^2/2}$. Then, we choose $t^2=\frac{2}{\eta}\log(2p/\alpha)$. Finally, we use the union bound when we have $n$ data samples.
As a result of Corollary <ref>, when original data samples $\x_1,\ldots,\x_n$ are first preconditioned and then sub-sampled, by choosing $\alpha=1/100$ we see that $\|\w_i\|_2^2 \leq \frac{m}{p}\frac{2}{\eta}\log(200np) \|\x_i\|_2^2$, $i=1,\ldots,n$, with probability greater than $0.99$.
§ THE SAMPLE MEAN ESTIMATOR
We show that a rescaled version of the sample mean of $\left\{ \mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}\right\} _{i=1}^{n}$, where $m$ out of $p$ entries of $\x_i$ are kept uniformly at random without replacement, is an unbiased estimator for the sample mean of the full data $\left\{ \mathbf{x}_{i}\right\} _{i=1}^{n}$. We will upper bound the error in both $\ell_{\infty}$ and $\ell_{2}$ norms, and show that these bounds are worse when the data set has a few large entries, which motivates our preconditioning.
Let $\overline{\mathbf{x}}_n$ represent the sample mean of $\left\{ \mathbf{x}_{i}\right\} _{i=1}^{n}$, i.e., $\overline{\mathbf{x}}_n=\frac{1}{n}\sum_{i=1}^{n}\mathbf{x}_{i}$.
Construct a rescaled version of the sample mean from $\left\{ \mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}\right\} _{i=1}^{n}$,
where each column of $\mathbf{R}_{i}\in\mathbb{R}^{p\times m}$ is chosen uniformly at random from the set of all canonical basis vectors without replacement:
\begin{equation}
\widehat{\overline{\mathbf{x}}}_{n}=\frac{p}{m}\frac{1}{n}\sum_{i=1}^{n}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}.
\end{equation}
Then, $\widehat{\overline{\mathbf{x}}}_{n}$ is an unbiased estimator
for $\overline{\mathbf{x}}_n$,
i.e., $\mathbb{E}[\widehat{\overline{\mathbf{x}}}_{n}]=\overline{\mathbf{x}}_n$.
Moreover, defining $\tau(m,p)$ as follows:
\begin{equation}\label{eq:tau-m-p}
\tau(m,p) \!:=\!\! \max\!\left\{\!\Big(\frac{p}{m}\!-\!1\Big)\!,\!1\right\} \!=\!
\begin{cases}
(\frac{p}{m}\!-\!1) \!\!\!& \text{if } \frac{m}{p}\!\leq\! 0.5\\
1 \!\!\!& \text{if } \frac{m}{p}\! >\! 0.5
\end{cases}
\end{equation}
then for all $t\geq0$, with probability at least $1-\delta_{1}$,
\begin{equation}
\delta_{1}=2p\exp\left(\frac{-nt^{2}/2}{(\frac{p}{m}-1)\nicefrac{\|\X\|_\maxRow^{2}}{n}+\nicefrac{\tau(m,p)\left\Vert \mathbf{X}\right\Vert_\text{max}t}{3} }\right)\label{eq:fail_prob_sample_mean}
\end{equation}
we have the following $\ell_{\infty}$ result:
\begin{equation}
\Vert \widehat{\overline{\mathbf{x}}}_{n}-\overline{\mathbf{x}}_n\Vert _{\infty}\leq t.\label{eq:error_sample_mean}
\end{equation}
First, we show that $\widehat{\overline{\mathbf{x}}}_{n}$ is an unbiased
\[
\mathbb{E}[\widehat{\overline{\mathbf{x}}}_{n}]=\frac{p}{m}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[\mathbf{R}_{i}\mathbf{R}_{i}^{T}\right]\mathbf{x}_{i}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{x}_{i}=\overline{\mathbf{x}}_n
\]
where we used Thm. <ref>. To upper bound the error, we define:
\begin{equation}
\mathbf{u}:=\widehat{\overline{\mathbf{x}}}_{n}-\overline{\mathbf{x}}_n=\sum_{i=1}^{n}\frac{1}{n}\Big(\frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}-\mathbf{x}_{i}\Big)
\end{equation}
and, thus, the $j$-th entry of $\mathbf{u}=[u_{1},\ldots,u_{p}]^{T}$
can be written as a sum of independent centered random variables:
\begin{equation}
u_{j}=\sum_{i=1}^{n}z_{i},\;\text{where}\; z_{i}=\frac{1}{n}\mathbf{e}_{j}^{T}\Big(\frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}-\mathbf{x}_{i}\Big).
\end{equation}
We now use the Bernstein inequality (Thm. <ref>) to show each entry of $\mathbf{u}$ is concentrated around zero with high probability. To do this, let us define $\tau(m,p):=\max\{(\frac{p}{m}-1),1\}$. We observe that each $z_{i}$ is bounded:
\begin{equation}
\left|z_{i}\right|\leq\frac{1}{n}\left\Vert \frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}-\mathbf{x}_{i}\right\Vert _{\infty}\!\!\!\leq\frac{1}{n}\tau(m,p)\left\Vert \mathbf{X}\right\Vert_\text{max}
\end{equation}
since $\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}$
is a vector with $m$ entries of $\mathbf{x}_{i}$ and the rest equal to zero. Next, we find the variance of $z_i$, $\mathbb{E}[z_{i}^{2}]=(\nicefrac{1}{n^{2}})\mathbf{e}_{j}^{T}\boldsymbol{\Lambda}\mathbf{e}_{j}$, where we use Thm. <ref> to compute $\boldsymbol{\Lambda}$:
\begin{eqnarray*}
\hspace{-7mm}& \boldsymbol{\Lambda}\hspace{-2mm} & =\mathbb{E}\Big[\Big(\frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}-\mathbf{x}_{i}\Big)\Big(\frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}-\mathbf{x}_{i}\Big)^{T}\Big]\\
\hspace{-7mm}& & =\frac{p^{2}}{m^{2}}\mathbb{E}\Big[\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}\mathbf{x}_{i}^{T}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\Big]-\mathbf{x}_{i}\mathbf{x}_{i}^{T}\\
\hspace{-7mm}& & =\frac{-\left(p-m\right)}{m\left(p-1\right)}\mathbf{x}_{i}\mathbf{x}_{i}^{T}+\frac{p\left(p-m\right)}{m\left(p-1\right)}\text{diag}\left(\mathbf{x}_{i}\mathbf{x}_{i}^{T}\right).
\end{eqnarray*}
Thus, $\mathbb{E}[z_{i}^{2}]=(\nicefrac{1}{n^{2}})(\frac{p}{m}-1)(\mathbf{e}_{j}^{T}\mathbf{x}_{i})^{2}$, and $\sigma^{2}=\sum_{i=1}^{n}\mathbb{E}[z_{i}^{2}]\leq(\nicefrac{1}{n^2})(\frac{p}{m}-1)\|\X\|_\maxRow^{2}$.
Using the Bernstein inequality:
\[
\P\left\{ \left|u_{j}\right|\geq t\right\} \leq2\exp\left(\frac{-nt^{2}/2}{(\frac{p}{m}-1)\nicefrac{\|\X\|_\maxRow^{2}}{n}+\nicefrac{\tau(m,p)\left\Vert \mathbf{X}\right\Vert_\text{max}t}{3} }\right).
\]
Finally, we use the union bound over all $p$ entries of $\mathbf{u}$.
The result in Thm. <ref> can be used to upper bound the error of our sample mean estimator $\widehat{\overline{\mathbf{x}}}_{n}$ in $\ell_2$ norm. For all $t\geq0$, with probability at least $1-\delta_1$, where $\delta_1$ defined in (<ref>), we have:
\begin{equation}
\frac{1}{\sqrt{p}}\Vert\widehat{\overline{\mathbf{x}}}_{n}-\overline{\mathbf{x}}_n\Vert _{2}\leq t.
\end{equation}
Solving for $t$ in (<ref>) gives the following expression in terms of failure probability $\delta_1$:
\begin{equation}
t = \frac{1}{n}\left( \frac{\tau(m,p)}{3}\|\X\|_\text{max} \logg
+ \sqrt{ \left( \frac{\tau(m,p)}{3}\|\X\|_\text{max} \logg \right)^2
+ 2\left(\frac{p}{m}-1\right)\logg \|\X\|_\maxRow^{2}
\right)
\end{equation}
We consider a numerical experiment on synthetic data set to show the precision of Thm. <ref>. We set the parameters $p=100$ and compression factor $\gamma=m/p=0.3$ and consider $1000$ runs for different values of $n$. Let $ \mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ denote a multivariate Gaussian distribution parameterized by its mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. In each run, we generate a set of $n$ samples in $\R^{p}$ from the probabilistic generative model $\x_i=\overline{\x}+\epsilon_i$, where $\overline{\x}$ is a fixed vector drawn from the Gaussian distribution and the additive noise $\epsilon_i$ is drawn i.i.d. from $\mathcal{N}(\boldsymbol{0},\mathbf{I}_p)$. We then keep $m=30$ entries from each data sample uniformly at random without replacement to obtain a sparse matrix. Using Thm. <ref>, we find the estimates of the sample mean from the sparse matrix and
compare with the actual
$\ell_\infty$ error between the estimates and true sample means. Fig. <ref> reports the average and maximum of $1000$ runs for each value of $n$ and compares that with the theoretical error bound $t$ in (<ref>) obtained by setting the failure probability $\delta_1=0.001$. The theoretical error bound is quite tight since it is close to the maximum of $1000$ runs.
Verifying the sharpness of Thm. <ref> on the synthetic data. For each $n$
we report the average and maximum of the sample mean estimation error in $1000$ runs compared with the theoretical error bound when $\delta_1=0.001$. The theoretical error bound is tight and decays exponentially as $n$ increases.
In Thm. <ref>, the failure probability $\delta_1$ depends on the properties of the data set such as the maximum absolute value of $\X$.
For a data matrix $\X$ with normalized columns, values of $\|\X\|_\text{max}$ and $\|\X\|_\maxRow^{2}$ can be relatively large due to the existence of large entries in $\X$ and, in the extreme case, we can get $\|\X\|_\text{max}=1$ and $\|\X\|_\maxRow^{2}=n$. Since both $\|\X\|_\text{max}$ and $\|\X\|_\maxRow^{2}$ are in the denominator, large values of these quantities work against the accuracy of the estimator and makes the failure probability $\delta_1$ closer to $1$. This fact motivates the use of preconditioning transformation discussed in Section <ref> to smooth out large entries of $\X$ and reduce the values of $\|\X\|_\text{max}$ and $\|\X\|_\maxRow^{2}$.
In the setting of Thm. <ref>, assume $\X$ is preconditioned by the ROS (<ref>) and $\gamma=m/p\leq0.5$.
By using the results of Corollary <ref>, we can find an upper bound for the failure probability $\delta_1$ that holds with probability greater than $0.99$:
\begin{equation}
\delta_1\leq 2p\exp\left(\frac{-mnt^2}{\nicefrac{4}{\eta}\log(200np)(1+\nicefrac{\sqrt{p}t}{3})}\right),
\end{equation}
and, thus, we can achieve high accuracy, e.g., $\delta_1\leq0.001$, for
\begin{equation}
m\geq\frac{1}{n}\cdot \frac{4}{\eta}\log(200np)\log(2000p)(t^{-2}+\frac{\sqrt{p}t^{-1}}{3})\label{eq:lower-bound-m}
\end{equation}
where $\eta=1$ for $\Hadamard$ a Hadamard matrix and $\eta=1/2$ for $\Hadamard$ a DCT matrix. Recall that $t$ is the upper bound for estimation error in (<ref>). To gain intuition and provide indicative values of $m$, we set $p=512$, $\eta=1$, and $t=0.01$. Then, for example, the values of the lower bound in (<ref>) are $137.2$, $15.1$, and $1.6$ for $n=10^5$, $10^6$, and $10^7$ respectively.
Since $m$ should be a natural number, we need to sample $m=138$, $m=16$, and $m=2$ entries per data sample. This means that as the number of samples $n$ increases, we can sample fewer entries per data sample, which makes our approach applicable to large-scale data sets.
To be formal, as $n$ grows with $p$ and $t$ fixed, if the number of sub-sampled entries per data sample $m$ is proportional to $\order(\log n/n)$, our sample mean estimator is accurate with high probability. Therefore, the amount of data we need to keep increases like $mn\propto \order(\log n)$.
§ THE COVARIANCE ESTIMATOR
In this section, we study the problem of covariance estimation for a set of data samples $\left\{\mathbf{x}_{i}\right\} _{i=1}^{n}$ from $\left\{ \mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}\right\} _{i=1}^{n}$, where $m$ out of $p$ entries of each $\x_i$ are kept uniformly at random without replacement. We propose an unbiased estimator for the covariance matrix of the full data $\Cemp=\nicefrac{1}{n}\sum_{i=1}^{n}\x_i\x_i^T$ and study the closeness of our proposed covariance estimator to $\Cemp$ based on the spectral norm.
we do not impose structural assumptions on the covariance matrix such as $\Cemp$ being low-rank.
To begin, consider a rescaled version of the empirical covariance matrix of the sub-sampled data $\left\{ \mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}\right\} _{i=1}^{n}$:
\begin{equation}
\Cemph:=\frac{p(p-1)}{m(m-1)}\cdot\frac{1}{n}\sum_{i=1}^{n}\RR_i\RR_i^{T}\x_i\x_i^{T}\RR_i\RR_i^{T}.
\end{equation}
Based on Thm. <ref>, we can compute the expectation of $\Cemph$:
\begin{equation}
\E[\Cemph]=\Cemp+\frac{(p-m)}{(m-1)}\diag(\Cemp)
\end{equation}
which consists of two terms, the covariance matrix of the full data $\Cemp$ (desired term) and an additional bias term that contains the elements on the main diagonal of $\Cemp$. However, as in <cit.>, we can easily modify $\Cemph$ to obtain an unbiased estimator:
\begin{equation}
\Cn:=\Cemph-\frac{(p-m)}{(p-1)}\diag(\Cemph)
\end{equation}
where revisiting Thm. <ref> shows that $\Cn$ is an unbiased estimator for $\Cemp$, i.e., $\E[\Cn]=\Cemp$. Next, we present a theorem to show the closeness of our proposed estimator $\Cn$ to the covariance matrix $\Cemp$.
Before stating the result, let us define $\w_i:=\RR_i\RR_i^{T}\x_i$, $i=1,\ldots,n$, which is an $m$-sparse vector containing $m$ entries of $\x_i$. We introduce a parameter $\rho>0$ such that for all $i=1,\ldots,n$ we have $\|\w_i\|_2^2 \leq \rho \|\x_i\|_2^2$. Obviously, $\|\w_i\|_2^2 \leq \|\x_i\|_2^2$ and for data sets with a few large entries, we can have $\|\w_i\|_2^2 = \|\x_i\|_2^2$ meaning that sub-sampling has not decreased the Euclidean norm. Therefore, $\rho\leq1$ and we can always take $\rho=1$. However, if we first apply the preconditioning transformation $\Hadamard\Diag$ (<ref>) to the data, we see that, with high probability, the sub-sampling operation decreases the Euclidean norm by a factor proportional to the compression factor $\gamma=m/p$. In fact, Corollary <ref> with $\alpha=1/100$ shows that $\rho=\frac{m}{p}\frac{2}{\eta}\log(200np)$ with probability greater than $0.99$. As we will see, this is of great importance to decrease the variance of our covariance estimator and achieve high accuracy, which motivates using the preconditioning transformation before sub-sampling.
Let $\Cemp$ represent the covariance matrix of $\left\{ \mathbf{x}_{i}\right\} _{i=1}^{n}$ and construct a rescaled version of the empirical covariance matrix from $\left\{\w_i= \mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{x}_{i}\right\} _{i=1}^{n}$,
where each column of $\mathbf{R}_{i}\in\mathbb{R}^{p\times m}$ is chosen uniformly at random from the set of all canonical basis vectors without replacement:
\begin{equation}
\Cemph=\frac{p(p-1)}{m(m-1)}\cdot\frac{1}{n}\sum_{i=1}^{n}\RR_i\RR_i^{T}\x_i\x_i^{T}\RR_i\RR_i^{T}.
\end{equation}
Let $\rho>0$ be a bound
such that for all $i=1,\ldots,n$, we have $\|\w_i\|_2^2 \leq \rho \|\x_i\|_2^2$
(in particular, we can always take $\rho=1$). Then,
\begin{equation}
\Cn=\Cemph-\frac{(p-m)}{(p-1)}\diag(\Cemph)
\end{equation}
is an unbiased estimator for $\Cemp$, and for all $t\geq 0$,
\begin{equation}
\P \left\{\|\Cn-\Cemp\|_2\leq t \right\}\geq 1-\delta_2,\;\;\delta_2=p\exp\left(\frac{-\nicefrac{t^2}{2}}{\sigma^2+\nicefrac{Lt}{3}}\right)\label{eq:cov-converge-delta2}
\end{equation}
\begin{equation}
L=\hspace{-1mm} \frac{1}{n} \left\{ \left(\frac{p(p-1)}{m(m-1)}\rho+1\right)\|\X\|_\maxCol^2 + \frac{p(p-m)}{m(m-1)}\|\X\|_\text{max}^2\right\}\label{eq:covariance-L}
\end{equation}
and $\sigma^2=\|\E[(\Cn-\Cemp)^2]\|_2$ represents the variance:
\begin{eqnarray}
&\hspace{-9mm}\sigma^2&\hspace{-5mm}\leq \frac{1}{n} \left\{\left(\frac{p(p-1)}{m(m-1)}\rho-1\right)\|\X\|_\maxCol^2\|\Cemp\|_2 \right. \nonumber \\
&&\hspace{-5mm} \left. +\frac{p(p-1)(p-m)}{m(m-1)^2}\rho \|\X\|_\maxCol^2\|\diag(\Cemp)\|_2 \right. \nonumber \\
&&\hspace{-5mm} +\frac{2p(p-1)(p-m)}{m(m-1)^2}\|\X\|_\text{max}^2\frac{\|\X\|_F^2}{n} \nonumber \\
&&\hspace{-5mm} \left. +\frac{p(p-m)^2}{m(m-1)^2}\frac{\max_{j=1,\ldots,p}\sum_{i=1}^{n}X_{j,i}^4}{n} \right\}.\label{eq:covariance_sigma}
\end{eqnarray}
The proof follows from the matrix Bernstein inequality (Thm. <ref>) and delayed to Appendix <ref>.
To gain some intuition and verify the accuracy of Thm. <ref>, we consider a numerical experiment. We set the parameter $p=1000$ and show the accuracy of our proposed covariance estimator $\Cn$ for various numbers of samples $n$ and compression factors $\gamma$. Fig. <ref>(a) shows the closeness of our covariance estimator $\Cn$ to the covariance matrix $\Cemp$, i.e., $\|\Cn-\Cemp\|_2$, over $100$ runs for various values of $n$ when $\gamma=m/p=0.3$ is fixed. In each run, we generate a set of $n$ data samples using the probabilistic generative model $\x_i=\sum_{j=1}^{k}\kappa_{ij}\lambda_{j}\mathbf{u}_j$, where we set $k=5$ and $\mathbf{U}=[\mathbf{u}_1,\ldots,\mathbf{u}_5]$ is a matrix of principal components with orthonormal columns obtained by performing QR decomposition on a $p\times k$ matrix with i.i.d. entries from $\mathcal{N}(0,1)$. The coefficients $\kappa_{ij}$ are drawn i.i.d. from $\mathcal{N}(0,1)$ and the vector $\boldsymbol{\lambda}$ represents the energy of the data in each principal direction and we choose $\boldsymbol{\lambda}=(10,8,6,4,2)$. The empirical value of estimation error $\|\Cn-\Cemp\|_2$ is compared with the theoretical error bound $t$ in (<ref>) when the failure probability is chosen $\delta_2=0.01$, and the resulting error bound is scaled by a factor of $10$.
Verifying the accuracy of Thm. <ref> on synthetic data. We set $p=1000$ and plot the average and maximum of covariance estimation error in $100$ runs for (a) varying $n$ when $\gamma=0.3$ fixed, and (b) varying $\gamma$ when $n=10p$ fixed. The empirical values are compared with the theoretical error bound for $\delta_2=0.01$ and scaled by a factor of $10$.
Our bounds are accurate to within an order of magnitude and they are representative of the empirical behavior of our covariance estimator in terms of $n$ and $\gamma$.
Furthermore, we plot the empirical value of the estimation error vs. compression factor $\gamma$ in Fig. <ref>(b) over $100$ runs when a fixed number of samples $n=10p$ are generated using the same generative model. As before, the empirical value is compared with the theoretical error bound when we choose $\delta_2=0.01$ and our bound is scaled by the same factor of $10$.
Our bounds are accurate to within an order of magnitude, and the theoretical result in Thm. <ref> correctly captures the dependence of the estimation error $\|\Cn-\Cemp\|_2$ in terms of the parameters $n$ and $\gamma$.
For example, as $n$ increases, the estimation error decreases exponentially for a fixed compression factor $\gamma$.
The other important consequence of Thm. <ref> is revealing the connections between accuracy of our covariance estimator $\Cn$ and some properties of the data set $\X$ as well as the compression factor $\gamma$. Note that large values of parameters $L$ and $\sigma^2$ work against the accuracy of $\Cn$ and make the failure probability $\delta_2$ closer to $1$, since they are in the denominator in (<ref>).
Let us assume that $\X$ has normalized columns so that $\|\X\|_\maxCol=1$ and $\|\X\|_F^2=n$.
With this normalization, $\|\Cemp\|_2\le1$, and $\|\diag(\Cemp)\|_2 \le 1$ as well, which follows from exercise 27 in 3.3 <cit.> and $\Cemp \succeq 0$.
In this case, both $L$ and $\sigma^2$ scale as $\frac{1}{n}$ and the estimation error decreases exponentially as $n$ increases for a fixed compression factor. Moreover, for a fixed $n$, $L\propto\order(\frac{p^2}{m^2})$ and $\sigma^2\propto\order(\frac{p^3}{m^3})$.
However, if we precondition the data $\X$ before sub-sampling as discussed in Section <ref>, then $\rho\propto\order(\frac{m}{p})$ and the maximum absolute value of the entries $\|\X\|_\text{max}$ of the preconditioned data is proportional to $\frac{1}{\sqrt{p}}$ ignoring logarithmic factors. Thus, the leading term in $L$ scales as $\order(\frac{p}{m})$ and the leading term in $\sigma^2$ scales as $\order(\frac{p^2}{m^2})$ which means that they are both reduced by a factor of $\frac{m}{p}$ under the preconditioning transformation.
Specifically, simplifying (<ref>) by dropping lower-order terms, assuming $p\gg m\gg 1$, using the normalization of $\X$ discussed above, as well as assuming preconditioning so $\rho=m/p$ and $\|\X\|_\text{max}^2\approx 1/p$, gives
\begin{equation}\label{eq:simplified}
\sigma^2 \lesssim \frac{1}{n} \left\{
\frac{p}{m}\|\Cemp\|_2
+\frac{p^2}{m^2} \|\diag(\Cemp)\|_2
\right\}.
\end{equation}
Using just $\|\Cemp\|$, $\|\diag(\Cemp)\| \le 1$ then gives the bound $\sigma^2 \lesssim\mathcal{O}\left( \frac{1}{n}\frac{p^2}{m^2} \right)$, but this can be improved in special cases.
Based on (<ref>),
we now consider tighter bounds for the few special cases, still assuming the data have been preconditioned so that $\Cemp = \frac{1}{n} \Hadamard\Diag \X\X^T\Diag^T\Hadamard^T$:
* If each $\xi$ is a canonical basis vector chosen uniformly-at-random, then $\Cemp = \diag(\Cemp)= p^{-1} \eye_p$ in the limit as $n\rightarrow\infty$. The third term in (<ref>) dominates, and the bound is $1/n\cdot p^2/m^3$ which is quite strong.
* If each entry of $\X$ is chosen i.i.d. $\mathcal{N}( 0, 1/p )$, then the columns have unit norm in expectation and again $\Cemp \rightarrow p^{-1} \eye_p$ as $n\rightarrow \infty$, so the bound is the same as the previous case (and preconditioning neither helps nor hurts).
* If $\xi = \x$ for all columns $i=1,\ldots,n$ and for some fixed (and normalized) column $\x$, then $\Cemp = \Hadamard\Diag\x\x^T\Diag^T\Hadamard^T$, so $\|\Cemp\|_2 = 1$.
For example, if $\x$ is a canonical basis vector, then $\Cemp$ is the outer-product of a column of the Hadamard matrix, so $\diag(\Cemp) = p^{-1} \eye_p$ and thus $\sigma^2\lesssim 1/n \cdot p/m$, similar to <cit.>. As well as improving $\rho$ and $\|\X\|_\text{max}$, in this case preconditioning has the effect of spreading out the energy in $\Cemp$ away from the diagonal. Without preconditioning, then $\Cemp = \x\x^T$, so if $\x$ is a standard basis vector, $\Cemp=\diag(\Cemp)$. Intuitively, this is a bad case for non-preconditioned sampling, since there is a slim chance of sampling the only non-zero coordinate. Note that this perfectly correlated case is difficult for approaches based on $\rand \X$ since each column is the same so the measurements are redundant, whereas our approach uses a unique sampling matrix for each column $i$ and thus gives more information.
In order to clarify the discussion, we provide a simple numerical experiment on a synthetic data set with a few large entries. We then show the effectiveness of the preconditioning transformation $\Hadamard\Diag$ on the accuracy of our covariance estimator $\Cn$ and, consequently, accuracy of the estimated principal components on this data set.
Effectiveness of preconditioning on the synthetic data set with few large entries. We plot the average of covariance estimation error over $100$ runs for varying $\gamma$ when $p=512$ and $n=1024$ in two cases of sub-sampling $\X$ (without preconditioning) and $\Y=\Hadamard\Diag\X$ (with preconditioning). We compare the empirical values with the theoretical error bound for $\delta_2=0.01$ and scaled by a factor $10$ in these two cases.
The preconditioning transformation $\Hadamard\Diag$ leads to a noticeable reduction of estimation error in both empirical and theoretical results.
In our experiment, we set $p=512$, and generate $n=1024$ data samples from the same probabilistic generative model with $k=10$ and $\mathbf{U}\in\R^{p\times k}$ containing $10$ principal components chosen from the set of all canonical basis vectors, and $\boldsymbol{\lambda}=(10,9,\ldots,1)$. In Fig. <ref>, we plot the average of estimation error $\|\Cn-\Cemp\|_2$ over $100$ runs for various values of the compression factor $\gamma=m/p$.
As described in Section <ref>, we can use a simple unitary transformation $\Hadamard\Diag$ to precondition the data and smooth out large entries. Thus, we consider the case where the data is first preconditioned, i.e., $\Y=\Hadamard\Diag\X$.
In this case the estimation error is $\|\Cn-\Cemp\|_2$, where $\Cemp=\frac{1}{n}\Y\Y^{T}$, and is also plotted in Fig. <ref>. Moreover, we report the theoretical error bounds when we choose $
\delta_2=0.01$ and scale our bounds by the same factor of $10$. The preconditioning decreases the error by almost a factor of $2$, both in experiment and via the theoretical bounds.
To show the importance of this error reduction in covariance estimation, we find the number of recovered principal components obtained from the eigendecomposition of $\Cn$ for both cases. After finding the first $k=10$ eigenvectors of $\Cn$, we compute the inner product magnitude between the recovered and true principal components and we declare that a principal component is “recovered” if the corresponding inner product magnitude is greater than $0.95$. The mean and standard deviation of the number of recovered principal components for varying compression factors $\gamma=m/p$ are reported in Table <ref>. As we see, the preconditioning transformation leads to a significant gain in accuracy of the estimated principal components, especially for small values of the compression factor, which are of great importance for the big data and distributed data settings.
Additionally, the variance in the estimate is much reduced across the whole range of $\gamma$.
Number of recovered principal components for various values of compression factor $\gamma=m/p$ over $100$ runs
2l(without preconditioning) 2l(with preconditioning)
$\gamma$ mean standard deviation mean standard deviation
0.1 0.98 0.99 5.12 0.40
0.2 3.53 1.76 7.01 0.10
0.3 6.85 1.67 8.00 0
0.4 8.18 1.58 8.42 0.49
0.5 9.31 1.03 9.00 0
§ SPARSIFIED K-MEANS CLUSTERING
Clustering is a commonly used unsupervised learning task which refers to identifying clusters of similar data samples in a data set. The K-means algorithm <cit.> is one of the most popular hard clustering
algorithms that has been used in many fields such as data mining and machine learning.
Despite its simplicity, running K-means on large-scale data sets presents new challenges and considerable efforts have been made to introduce memory/computation efficient clustering algorithms. In this paper, we present a variant of the K-means algorithm which allows us to find a set of cluster centers as well as assignment of the data. The idea is to precondition and sample the data in one pass over the data to achieve a sparse matrix, therefore reducing processing time and saving memory, and applicable to streaming and distributed data.
First, we review the standard K-means algorithm. Given a data set $\mathbf{X}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]\in\mathbb{R}^{p\times n}$, the goal of K-means is to partition the data into a known number of $K$ clusters such that $\boldsymbol{\mu}_{k}\in\mathbb{R}^{p}$ is the prototype associated with the $k$-th cluster for $k=1,\ldots,K$. We also introduce a set of binary indicator variables $
c_{ik}\in\{0,1\}$ to represent the assignments, where
$\mathbf{c}_{i}=[c_{i1},\ldots,c_{iK}]^{T}$ is the $k$-th canonical basis vector in $\mathbb{R}^{K}$ if and only if $\mathbf{x}_{i}$ belongs to the $k$-th cluster.
Let us define cluster centers $\boldsymbol{\mu}=\{\boldsymbol{\mu}_{k}\}_{k=1}^{K}$ and the assignments of the data samples $\mathbf{c}=\{\mathbf{c}_{i}\}_{i=1}^{n}$. The K-means algorithm attempts to minimize the sum of the squared Euclidean distances of each data point to its assigned cluster:
\begin{equation}
J(\mathbf{c},\boldsymbol{\mu})=\sum_{i=1}^{n}\sum_{k=1}^{K}c_{ik}\left\Vert \mathbf{x}_{i}-\boldsymbol{\mu}_{k}\right\Vert _{2}^{2}.\label{K-means-obj}
\end{equation}
The objective $J(\mathbf{c},\boldsymbol{\mu})$
is minimized by
an iterative algorithm
that (step one)
updates assignments $\mathbf{c}$ and (step two)
updates $\boldsymbol{\mu}$ as follows:
Step 1: Minimize $J(\mathbf{c},\boldsymbol{\mu})$
over $\mathbf{c}$, keeping $\boldsymbol{\mu}$ fixed:
\begin{equation}
\forall\;
\; i=1,\ldots,n:\; c_{ik}=\begin{cases}
1, & k=\argmin_{j}\left\Vert \mathbf{x}_{i}-\boldsymbol{\mu}_{j}\right\Vert _{2}^{2}\\
0, & \text{otherwise}
\end{cases}
\end{equation}
Step 2: Minimize $J(\mathbf{c},\boldsymbol{\mu})$
over $\boldsymbol{\mu}$, keeping $\mathbf{c}$ fixed:
\begin{equation}
\forall\; k=1,\ldots K:\;\boldsymbol{\mu}_{k}=\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{k}}\mathbf{x}_{i}\label{KMeansStep2}
\end{equation}
where $\mathcal{I}_{k}$ denotes the set of samples assigned to the $k$-th cluster in Step 1 and $n_{k}=\left|\mathcal{I}_{k}\right|$. Therefore, the update formula for cluster center $\boldsymbol{\mu}_{k}$ is the sample mean of the data samples in $\mathcal{I}_{k}$. To initialize K-means, a set of cluster centers can be chosen uniformly at random from the data set $\X$. However, we use the recent K-means++ algorithm <cit.> to choose the initial cluster centers since it improves the performance of K-means over the worst-case random initializations.
Next, we consider a probabilistic mixture model to find an optimal objective function for our sparsified K-means clustering algorithm using Maximum-Likelihood (ML) estimation.
§.§ Optimal Objective Function via ML Estimation
One appealing aspect of the K-means algorithm is that the objective function (<ref>) coincides with the log-likelihood function of a mixture of $K$ Gaussian components (clusters), with mean $\boldsymbol{\mu}_{k}$ and covariance matrix $\boldsymbol{\Sigma}=\lambda\mathbf{I}_{p}$, where $\lambda>0$ is fixed or “known” (its actual value is unimportant), and treating $\mathbf{c}$ and $\boldsymbol{\mu}$ as unknown parameters. We show that under the same assumptions, K-means clustering on sampled preconditioned data enjoys the same ML interpretation.
Let $p(\x_i|\boldsymbol{\mu} ,\mathbf{c}_{i})$ denote the conditional probability distribution of sample $\x_i$ given that a set of centers $\boldsymbol{\mu}$ and a particular value of $\mathbf{c}_{i}$ are known. Under this setup, the conditional distribution of $\x_i$ is Gaussian and it can be written as follows:
\begin{equation}
p\left(\mathbf{x}_{i}|\boldsymbol{\mu} ,\mathbf{c}_{i}\right)=\prod_{k=1}^{K}p\left(\mathbf{x}_{i}|\boldsymbol{\mu}_{k},\boldsymbol{\Sigma}\right)^{c_{ik}}
\end{equation}
\[
p\left(\mathbf{x}_{i}|\boldsymbol{\mu}_{k},\boldsymbol{\Sigma}\right)=\frac{1}{\left(2\pi\lambda\right)^{\frac{p}{2}}}\exp\Big({-}\frac{1}{2\lambda}\left\Vert \mathbf{x}_{i}-\boldsymbol{\mu}_{k}\right\Vert _{2}^{2}\Big).
\]
Given that $\x_i$ belongs to the $k$-th cluster with mean $\boldsymbol{\mu}_k$ and covariance $\boldsymbol{\Sigma}=\lambda\eye_p$, then the preconditioned data $\y_i=\Hadamard\Diag\x_i$ also has a Gaussian distribution with mean $\boldsymbol{\mu}_k':=\E[\y_i]=\Hadamard\Diag\boldsymbol{\mu}_k$ and the same covariance $\boldsymbol{\Sigma}=\lambda\eye_p$ because $\Hadamard$ and $\Diag$ are orthonormal matrices, i.e., $(\Hadamard\Diag)(\Hadamard\Diag)^{T}=\eye_p$. Note that we can also find $\boldsymbol{\mu}_k$ from $\boldsymbol{\mu}_k'$ using the equation:
\begin{equation}
\boldsymbol{\mu}_k=(\Hadamard\Diag)^{T}\boldsymbol{\mu}_k'.\label{eq:mu-mu'}
\end{equation}
We now take $n$ independent realizations of the sampling matrix $\mathbf{R}$, denoted $\mathbf{R}_{1},\ldots,\mathbf{R}_{n}$, each consisting of $m$ canonical basis vectors. Then, given that $\mathbf{y}_{i}$ belongs to the $k$-th cluster with mean $\boldsymbol{\mu}_{k}'$ and covariance matrix $\boldsymbol{\Sigma}=\lambda\mathbf{I}_{p}$, the sub-sampled data $\mathbf{z}_{i}=\mathbf{R}_{i}^{T}\mathbf{y}_{i}$ also has a Gaussian distribution with mean $\mathbb{E}\left[\mathbf{z}_{i}\right]=\mathbf{R}_{i}^{T}\boldsymbol{\mu}_{k}'$ and covariance $\lambda\mathbf{I}_{m}$, since $\RR_i^{T}\RR_i=\eye_m$ based on Thm. <ref>. Hence
\[
p\left(\mathbf{z}_{i}|\boldsymbol{\mu}_{k}',\boldsymbol{\Sigma}\right)=\frac{1}{\left(2\pi\lambda\right)^{\frac{m}{2}}}\exp\Big({-}\frac{1}{2\lambda}\left\Vert \mathbf{z}_{i}-\mathbf{R}_{i}^{T}\boldsymbol{\mu}_{k}'\right\Vert _{2}^{2}\Big)
\]
and, thus, we have the following expression for the conditional distribution of $\mathbf{z}_i$:
\begin{equation}
\end{equation}
Next, we consider ML estimation when we have access to the sampled preconditioned data $\mathbf{Z}=[\mathbf{z}_{1},\ldots,\mathbf{z}_{n}]$:
\[
\]
and taking the logarithm of the likelihood function:
\[
\log p\!\left(\mathbf{Z}|\boldsymbol{\mu}',\mathbf{c}\right)\hspace{-1mm}=\hspace{-1mm}{-}\frac{mn}{2}\!\log\!\left(2\pi\lambda\right)-\frac{1}{2\lambda}\!\sum_{i=1}^{n}\sum_{k=1}^{K}\hspace{-1mm}c_{ik}\left\Vert \mathbf{z}_{i}\hspace{-1mm}-\hspace{-1mm}\mathbf{R}_{i}^{T}\boldsymbol{\mu}_{k}'\right\Vert _{2}^{2}.
\]
Hence, the ML estimate of the unknown parameters $\mathbf{c}$ and $\boldsymbol{\mu}'$ (or equivalently $\boldsymbol{\mu}$) is obtained by minimizing:
\begin{equation}
J'(\mathbf{c},\boldsymbol{\mu}')=\sum_{i=1}^{n}\sum_{k=1}^{K}c_{ik}\left\Vert \mathbf{z}_{i}-\mathbf{R}_{i}^{T}\boldsymbol{\mu}_{k}'\right\Vert _{2}^{2}.\label{eq:CK_means}
\end{equation}
Note that $J'(\mathbf{c},\boldsymbol{\mu}')$ can be written as:
\begin{equation}
J'(\mathbf{c},\boldsymbol{\mu}')=\sum_{i=1}^{n}\sum_{k=1}^{K}c_{ik}\left\Vert \mathbf{R}_{i}^{T}\left(\mathbf{y}_{i}-\boldsymbol{\mu}_{k}'\right)\right\Vert _{2}^{2}\label{eq:CK_means2}
\end{equation}
and for $\mathbf{R}_{i}=\mathbf{I}_{p}$, $i=1,\ldots,n$, it reduces to the objective function of the standard K-means (<ref>) because the preconditioning transformation $\Hadamard\Diag$ is an orthonormal matrix.
§.§ The Sparsified K-means Algorithm
Similar to the K-means algorithm, we minimize the objective function $J'(\mathbf{c},\boldsymbol{\mu}')$ in an iterative procedure that (step one) updates assignments $\mathbf{c}$ and (step two) updates $\boldsymbol{\mu}'$:
Step 1: Minimize $J'(\mathbf{c},\boldsymbol{\mu}')$ holding $\boldsymbol{\mu}'$ fixed:
In (<ref>), the terms involving different $n$ are independent and we assign each sampled preconditioned data $\mathbf{z}_{i}=\RR_i^{T}\y_i\in\R^{m}$ to the closest cluster:
\begin{equation} \label{eq:assignments}
\forall\;
\; i=1,\ldots,n:\; c_{ik}=\begin{cases}
1, & k=\argmin_{j}\left\Vert \mathbf{z}_{i}-\mathbf{R}_{i}^{T}\boldsymbol{\mu}_{j}'\right\Vert _{2}^{2}\\
0, & \text{otherwise}
\end{cases}
\end{equation}
The connection between this step and the first step of K-means is immediate mainly due to the Johnson-Lindenstrauss (JL) lemma. In fact, the accuracy of this step depends on the preservation of the Euclidean norm under selecting $m$ entries of a $p$-dimensional vector. Based on the well-known fast JL transform <cit.>, one needs to first smooth out data samples with a few large entries to ensure the preservation of the Euclidean norm with high probability. In particular, the author in <cit.> showed that selecting $m$ entries of the preconditioned data uniformly at random without replacement preserves the geometry of the data as well as the Euclidean norm. A direct consequence of this result stated in Thm. <ref> shows that pairwise distances between each point and cluster centers are preserved.
Step 2: Minimize $J'(\mathbf{c},\boldsymbol{\mu}')$ holding $\mathbf{c}$ fixed:
Given the assignments $\mathbf{c}$ from Step 1, we can write (<ref>) as:
\begin{equation}
J'(\boldsymbol{\mu}')=\sum_{i\in\mathcal{I}_{k}}\left\Vert \mathbf{R}_{i}^{T}\left(\mathbf{y}_{i}-\boldsymbol{\mu}_{k}'\right)\right\Vert _{2}^{2}
\end{equation}
where $\mathcal{I}_{k}$ represents the set of samples assigned to the $k$-th cluster and recall that $n_{k}=\left|\mathcal{I}_{k}\right|$. The terms involving different $k$ are independent and each term is a quadratic function of $\boldsymbol{\mu}_{k}'$. Thus, each term can be minimized individually by setting its derivative with respect to $\boldsymbol{\mu}_{k}'$ to zero giving:
\begin{equation}
\left(\sum_{i\in\mathcal{I}_{k}}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\right)\boldsymbol{\mu}_{k}'=\sum_{i\in\mathcal{I}_{k}}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{y}_{i}.\label{eq:mu_update_one}
\end{equation}
Note that $\sum_{i\in\mathcal{I}_{k}}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\in \mathbb{R}^{p\times p}$ is a diagonal matrix, where its $j$-th diagonal element counts the number of cases the $j$-th canonical basis vector is chosen in the sampling matrices $\mathbf{R}_{i}$ for all $i\in\mathcal{I}_{k}$, denoted by $n_{k}^{(j)}$. Therefore, for any $j$ with $n_{k}^{(j)}=0$, we cannot estimate the $j$-th entry of $\boldsymbol{\mu}_{k}'$ and the corresponding entry should be removed from (<ref>). Given that $n_{k}^{(j)}>0$ for all $j$, $\boldsymbol{\mu}_{k}'$ is updated as follows:
\begin{equation}
\boldsymbol{\mu}_{k}'=\text{diag}\left(\Big[\frac{1}{n_{k}^{(1)}},\ldots,\frac{1}{n_{k}^{(p)}}\Big]\right)\left(\sum_{i\in\mathcal{I}_{k}}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{y}_{i}\right).\label{eq:mu_update_two}
\end{equation}
Recall that each $\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{y}_{i}$ is the sampled preconditioned data such that $m$ out $p$ entries are kept uniformly at random. Hence, the update formula for $\boldsymbol{\mu}_{k}'$ is the entry-wise sample mean of the sparse data samples in the $k$-th cluster. The sparsified K-means algorithm is summarized in Algorithm <ref>.
Input: Dataset $\mathbf{X}\in\mathbb{R}^{p\times n}$, number
of clusters $K$, compression factor $\gamma=\frac{m}{p}<1$.
Output: Assignments $\mathbf{c}=\{\mathbf{c}_{i}\}_{i=1}^{n}\in\mathbb{R}^{K}$, cluster centers $\boldsymbol{\mu}=\{\boldsymbol{\mu}_k\}_{k=1}^{K}\in\mathbb{R}^{p}$.
Sparsified K-means
Sparsified K-means$\X,K,\gamma$
$\mathbf{Y}\leftarrow\mathbf{H}\mathbf{D}\mathbf{X}$ See Eq. (<ref>)
$\mathbf{w}_{i}=\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{y}_{i}$ $\RR_i\in\R^{p\times m}$:sampling matrix
Find initial cluster centers via K-means++ <cit.>
each iteration
update assignments $\mathbf{c}$ See Eq. (<ref>)
update cluster centers $\boldsymbol{\mu}'$ See Eq. (<ref>)
$\boldsymbol{\mu}=\left(\Hadamard\Diag\right)^{T}\boldsymbol{\mu}'$ See Eq. (<ref>)
Return $\mathbf{c}$ and $\boldsymbol{\mu}$.
Now, we return to equation (<ref>) and study the accuracy of the estimated solution $\boldsymbol{\mu}_{k}'$. To do this, we re-write (<ref>) as:
\begin{equation}
\mathbf{H}_{k}\boldsymbol{\mu}_{k}'=\mathbf{m}_{k}\label{eq:mu_k_update}
\end{equation}
\begin{equation}
\mathbf{H}_{k}=\frac{p}{m}\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{k}}\mathbf{R}_{i}\mathbf{R}_{i}^{T},\;\mathbf{m}_{k}=\frac{p}{m}\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{k}}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\mathbf{y}_{i}.\label{Hkmk}
\end{equation}
Next, we show that $\mathbf{H}_{k}$ converges to the identity matrix $\mathbf{I}_{p}$ as the number of samples in each cluster $n_k$ increases.
Consider $\mathbf{H}_{k}$ defined in (<ref>). Then, for all $t\geq0$:
\begin{equation}
\mathbb{P}\left\{ \left\Vert \mathbf{H}_{k}-\mathbf{I}_{p}\right\Vert _{2}\leq t\right\} \geq 1-\delta_{3}
\end{equation}
where the failure probability,
\begin{equation}
\delta_{3}=p\exp\left(\frac{-n_{k}\nicefrac{t^{2}}{2}}{\left(\frac{p}{m}-1\right)+\left(\frac{p}{m}+1\right)\nicefrac{t}{3}}\right).\label{eq:delta3}
\end{equation}
We can write $\mathbf{S}=\mathbf{H}_{k}-\mathbf{I}_{p}=\sum_{i=1}^{n_{k}}$$\mathbf{Z}_{i}$,
\[
\mathbf{Z}_{i}=\frac{1}{n_{k}}\left(\frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}-\mathbf{I}_{p}\right),\; i=1,\ldots,n_{k}
\]
are independent and symmetric random matrices. Moreover, we have
$\mathbb{E}[\mathbf{Z}_{i}]=\mathbf{0}$ using Thm. <ref>. To apply
the matrix Bernstein inequality given in Appendix <ref>, we should find a uniform bound on the spectral norm of each summand $\|\Z_i\|_2$:
\begin{equation}
\left\Vert \mathbf{Z}_{i}\right\Vert _{2}\leq\frac{1}{{n_{k}}}\left(\left\Vert \frac{p}{m}\mathbf{R}_{i}\mathbf{R}_{i}^{T}\right\Vert _{2}+\left\Vert \mathbf{I}_{p}\right\Vert _{2}\right)=\frac{1}{n_{k}}\left(\frac{p}{m}+1\right)\label{eq:bound_Zi_Hk}
\end{equation}
where it follows from the triangle inequality for the spectral norm and the
fact that $\mathbf{R}_{i}\mathbf{R}_{i}^{T}\in\mathbb{R}^{p\times p}$
is a diagonal matrix with only $m$ ones on the diagonal and the rest
equal to zero.
Next, we find $\mathbb{E}[\mathbf{Z}_{i}^{2}]$ using the results
of Thm. <ref>:
\[
\mathbb{E}\left[\mathbf{Z}_{i}^{2}\right]\hspace{-1mm}=\hspace{-1mm}\frac{1}{n_{k}^{2}}\mathbb{E}\Big[\Big(\frac{p^{2}}{m^{2}}-2\frac{p}{m}\Big)\mathbf{R}_{i}\mathbf{R}_{i}^{T}+\mathbf{I}_{p}\Big]\hspace{-1mm}=\hspace{-1mm}\frac{1}{n_{k}^{2}}\Big(\frac{p}{m}-1\Big)\mathbf{I}_{p}
\]
and thus
\begin{equation}
\sigma^{2}=\Big\Vert \sum_{i=1}^{n_{k}}\mathbb{E}\left[\mathbf{Z}_{i}^{2}\right]\Big\Vert _{2}=\frac{1}{n_{k}}\left(\frac{p}{m}-1\right).\label{eq:Variance_Zi_Hk}
\end{equation}
We now use Theorem <ref> and this completes the proof.
To verify the accuracy of Thm. <ref>, we consider a numerical experiment. We set the parameters $p=100$ and compression factor $\gamma=m/p=0.3$ and show the closeness of $\mathbf{H}_k$ to $\eye_p$ for various values of $n$ over $1000$ runs. For each value of $n$, we generate $n$ sampling matrices $\RR_i$ consisting of $m$ distinct canonical basis vectors uniformly at random. We report the average and maximum of empirical values $\|\mathbf{H}_k-\eye_p\|_2$ over $1000$ runs in Fig. <ref>. We also compare the empirical values with our theoretical error bound $t$ in (<ref>), when the failure probability $\delta_3=0.001$. We see that our error bound is tight and very close to the maximum of $1000$ since $\delta_3=0.001$.
Verifying the accuracy of Thm. <ref>. We set parameters $p=100$ and $\gamma=m/p=0.3$ and plot the average and maximum of $\left\Vert \mathbf{H}_{k}-\mathbf{I}_{p}\right\Vert_2$ over $1000$ runs for varying $n$. We compare the empirical values with our theoretical bound when $\delta_3=0.001$. We see that our bound is tight.
Now, we present a theorem to show the connection between the updated cluster center in our sparsified K-means in (<ref>) and the update formula for the standard K-means algorithm.
Consider the update formula for the $k$-th cluster center in our sparsified K-means algorithm $\boldsymbol{\mu}_k=(\Hadamard\Diag)^{T}\boldsymbol{\mu}_k'$, where $\boldsymbol{\mu}_k'$ is given by the equation $\mathbf{H}_k\boldsymbol{\mu}_k'=\mathbf{m}_k$ in (<ref>). Let $\overline{\x}_k$ denote the sample mean of the data samples in the $k$-th cluster, i.e., $\overline{\x}_k=\frac{1}{n_{k}}\sum_{i\in\mathcal{I}_{k}}\mathbf{x}_{i}$, which is the standard update formula for K-means on $\X$. Then, for all $t\geq0$,
\begin{equation}
\frac{1}{\sqrt{p}}\|\boldsymbol{\mu}_k-\overline{\x}_k\|_2\leq t\left(1+\frac{1}{\sqrt{p}}\|\boldsymbol{\mu}_k\|_2\right)
\end{equation}
with probability greater than $1-\max\{\delta_1,\delta_3\}$, where
\begin{equation}
\delta_{1}=2p\exp\left(\frac{-n_{k}t^{2}/2}{(\frac{p}{m}-1)\nicefrac{\|\Y\|_\maxRow^{2}}{n}+\nicefrac{\tau(m,p)\left\Vert \mathbf{Y}\right\Vert_{max}t}{3} }\right),
\end{equation}
$\tau(m,p)$ is defined in (<ref>) and $\delta_3$ is given in (<ref>). Recall that $\Y=\Hadamard\Diag\X$ is the preconditioned data in our sparsified K-means algorithm.
Based on Thm. <ref> and Thm. <ref>, we re-write equation (<ref>) as:
\[
\left(\eye_p+\mathbf{E}\right)\boldsymbol{\mu}_k'= \Hadamard\Diag\overline{\x}_k + \mathbf{e}
\]
where $\|\mathbf{E}\|_2\leq t$ and $\|\mathbf{e}\|_2 \leq \sqrt{p}t$ with probabilities greater than $1-\delta_3$ and $1-\delta_1$ respectively. Thus, with probability greater than $1-\max\{\delta_1,\delta_3\}$, we have:
\begin{eqnarray}
&&\hspace{-15mm}\|\boldsymbol{\mu}_k'-\Hadamard\Diag\overline{\x}_k\|_2=\|\mathbf{e}-\mathbf{E}\boldsymbol{\mu}_k'\|_2 \nonumber\\
&&\hspace{-15mm}\leq \|\mathbf{e}\|_2 + \|\mathbf{E}\boldsymbol{\mu}_k'\|_2\leq \|\mathbf{e}\|_2+\|\mathbf{E}\|_2\|\boldsymbol{\mu}_k'\|_2 \nonumber\\
&&\hspace{-15mm}\leq t\sqrt{p}\left(1+\frac{\|\boldsymbol{\mu}_k'\|_2}{\sqrt{p}}\right)
\end{eqnarray}
where we used the triangle inequality for the spectral norm. Recall that $\Hadamard\Diag$ is an orthonormal matrix and $\boldsymbol{\mu}_k'=\Hadamard\Diag\boldsymbol{\mu}_k$. Thus, $\|\boldsymbol{\mu}_k'-\Hadamard\Diag\overline{\x}_k\|_2=\|\boldsymbol{\mu}_k-\overline{\x}_k\|_2$ and $\|\boldsymbol{\mu}_k'\|_2=\|\boldsymbol{\mu}_k\|_2$ and this completes the proof.
§ NUMERICAL EXPERIMENTS
We implement the sparsified K-means algorithm in a mixture of Matlab and C, available online[<https://github.com/stephenbeckr/SparsifiedKMeans>]. Since K-means attempts to minimize a non-convex objective, the starting points have a large effect. We use the recent K-means++ algorithm <cit.> for choosing starting points, and re-run the algorithm for different sets of starting points and then choose the results with the smallest objective value. All results except the big-data tests use $20$ different starting trials.
Timing results are from running the algorithm on a desktop computer with two Intel Xeon EF-2650 v3 CPUs at $2.4$–$3.2$ GHz and 8 cores and $20$ MB cache each, and should be interpreted carefully. First, we note that K-means is iterative and so the number of iterations may change slightly depending on the variant.
Furthermore, none of the code was optimized for small problems, so timing results under about $10$ seconds do not scale with $n$ and $p$ as they do at large scale. At the other extreme,
our first series of tests are not on out-of-core data, so the benefits of a single-pass algorithm are not apparent. Our subsequent tests are out-of-core implementations, meaning that they explicitly load data from the hard drive to RAM as few times as possible, and so the number of passes through the data becomes relevant, cf. Table <ref>.
We also caution about interpreting the accuracy results for correct identification of clusters. In our experience, if the accuracy is greater than about $75\%$, then using the result as the initial value for a single-step of standard K-means, thus increasing the number of passes through the data by one, is sufficient to match the accuracy of the standard K-means algorithm.
Low-pass Algorithms for K-means clustering
2cPasses through data...
Algorithm ...to find $\boldsymbol{\mu}$ ...to find $\mathbf{c}$
Sparsified K-means (1-pass) 1 1
Sparsified K-means (2-pass) 2 2
Feature extraction 2 1
Feature selection 4 3
A two-pass sparsified K-means algorithm can be constructed by running the one-pass sparsified K-means in Algorithm <ref> to compute the assignments as well as the cluster centers, then re-computing the cluster center estimates $\boldsymbol{\mu}$ as the average of their assigned data points in the original (non-sampled) domain. Meanwhile, we can re-assign the data samples to the cluster centers in the original domain based on the previous center estimates from one-pass sparsified K-means. The same extra-pass procedure must be applied to feature extraction and feature selection, since their default center estimates $\boldsymbol{\mu}$ are in a compressed domain.
Sparsified K-means, 2-pass
Sparsified K-means 2-pass$\X,K,\gamma$
$(\mathbf{\hat{c}},\boldsymbol{\hat{\mu}})=$Sparsified K-means$\X,K,\gamma$ Alg. <ref>
$\boldsymbol{\mu}_k=\mathbf{0},\; \mathcal{I}_{k}=\emptyset$
Find cluster assignment, i.e., $k$ s.t. $\mathbf{\hat{c}}_i=\mathbf{e}_k$
$\boldsymbol{\mu}_{k} \mathrel{+}= \xi$,
\boldsymbol{\mu}_{k}/|\mathcal{I}_{k}|$
Return $\mathbf{c}$ and $\boldsymbol{\mu}$.
Our tests compare with the feature selection and feature extraction algorithms of <cit.>. In feature selection, one first uses a fast approximate SVD algorithm, e.g., <cit.>, to compute an estimate of the left singular vectors of the data matrix $\X$. Then, the selection of $m$ rows of $\X$ is done with a randomized
sampling approach with probabilities that
are computed from the estimated left singular vectors. This can be written as $\rand\X$, where rows of the sampling matrix $\rand\in\R^{m\times p}$ are chosen from the set of canonical basis vectors in $\R^p$ based on the computed probabilities. In feature extraction, the samples are again $\rand\X$ but the sampling matrix $\rand\in\R^{m\times p}$ is set to be a random sign matrix. Thus, the computational cost of feature extraction is dominated by the matrix-matrix multiplication $\rand\X$, whereas the dominant cost in feature selection is the approximate SVD.
§.§ Sketched K-means for faster computation
Sampling the data leads to both computational time and memory benefits, with computational time benefits becoming more significant for more complicated algorithms, such as the Expectation-Maximization algorithm in Gaussian mixture models that require eigenvalue decompositions. Even for the standard K-means algorithm, sub-sampling leads to faster computation. The most expensive computation in K-means is finding the nearest cluster center for each point, since a naive implementation of this costs $\order(pnK)$ flops per iteration[
We do not consider the variants of K-means based on kd-trees since these have running time exponential in $p$ and are suitable for $p \lesssim 20$
By effectively reducing the dimension from $p$ to $m$, the sparse version sees a speedup of roughly $\gamma^{-1}=p/m$.
Fig. <ref> demonstrates this speedup experimentally on a toy problem. The data of size $p=512$ and $n=10^{5}$ are generated artificially so that each point belongs to one of $K=5$ clusters and is perturbed by a small amount of Gaussian noise. An optimized variant of Matlab's algorithm takes $3448$ seconds to run.
We compare this with random Hadamard mixing followed by $5\%$ sub-sampling, which takes $51$ seconds.
The first two dimensions of the data are shown in Fig. <ref> which makes it clear that there is no appreciable difference in quality, while our sparsified K-means algorithm is approximately $67$ times faster.
Standard K-means and our sparse version of K-means, on synthetic data, $n=10^5$.
§.§ Comparison with dimensionality-reducing approaches on real data
For a realistic clustering application, experiments are performed on the MNIST dataset[<http://yann.lecun.com/exdb/mnist/>] which consists of centered versions of hand-written digits, each digit stored as a $28\times 28$ pixel image. For processing, the images are vectorized so $p=28^2=784$.
The dataset includes both testing and training sets, though for our purposes we combined the two and report in-sample error, since the effect of sampling and dimensionality reduction to out-of-sample error is beyond our scope.
For simplicity of interpreting results, we use data from the samples of three digits (“0”, “3” and “9”), so $K=3$ in the clustering algorithm. There were $6903$, $7141$ and $6958$ examples of each class of image, respectively, so $n=21002$.
The original data provides a ground-truth label, against which we compute accuracy by computing the total number of correctly assigned images, normalized by the total number of images. All the algorithms, except standard K-means, are stochastic, so we re-run the clustering 50 times and record the mean and standard deviation of these 50 trials. Recall that within each run, we choose the best of 20 random starting points.
Accuracy of various K-means algorithms on the MNIST data, 50 trials.
The plot shows the mean and the standard deviation error bars, empirically suggesting that feature-based algorithms show higher variance than the sampling-based algorithm.
For example, at $\gamma=0.1$, the standard deviation for sparsified K-means, sparsified K-means without preconditioning, 2-pass sparsified K-means, feature selection, and feature extraction, is
$0.002$, $0.004$, $0.001$, $0.1151$ and $0.049$,
respectively. Moreover, the accuracy of 2-pass sparsified K-means reaches the accuracy of standard K-means even for small values of the compression factor $\gamma$. For visual clarity, we did not include the standard deviation error bars for $\gamma<0.025$.
Timing and accuracy
Timing of various K-means algorithms on the MNIST data. The three variants of sparsified K-means (with and without preconditioning, and 2-pass) all take approximately the same time on this dataset, so we only show the time for preconditioned sparsified K-means.
The red dashed curve is proportional to $\gamma$, which is the ideal speedup ratio; the constant of proportionality is $5$, chosen to make the curve line up with the sparsified K-means performance. Note that time is in log-scale; at $\gamma=0.3$, sparsified K-means takes about 12 seconds while feature extraction takes 8 seconds.
Clustering accuracy as a function of $\gamma$ is shown in Fig. <ref>, which suggests sparsified K-means is the most accurate of the efficient methods, and that accuracy is further improved in the two-pass version.
Timing results are shown in Fig. <ref>, which also shows the time for our optimized implementation of K-means on the full data.
All the efficient algorithms show a speedup over K-means proportional to $\gamma^{-1}$, as expected, until the sparsity is near $5\%$, at which point various fixed costs in the computation start to dominate; one would expect ideal speedup to continue to lower $\gamma$ if $n$ were larger, cf. Table <ref>.
Sparsified K-means takes longer than standard K-means as $\gamma\rightarrow 1$ since it is inefficient to work with a sparse matrix format when the matrices are not actually sparse.
Center estimation
The estimated cluster centers $\boldsymbol{\mu}$ from several low-pass K-means algorithms are shown in Fig. <ref>, for $\gamma=0.03$. As we see, our sparsified K-means algorithm returns fairly accurate estimates of the true cluster centers in one pass over the data, which represent the three classes of digits in the given unlabeled dataset. However, as described, feature-based algorithms require one more pass over the full dataset after finding assignments to return meaningful estimates of the true cluster centers.
Why does our method give effective 1-pass center estimates, while the other methods do not, even if they have comparable accuracy in estimating assignments?
The reason is that each sample $\mathbf{x}_i$ is sampled with an independent copy of the random sampling operator $\mathbf{R}_{i}\mathbf{R}_{i}^{T}$, and this leads to a consistent estimator. For simplicity, assume assignments have been made correctly for a given cluster $k$, so we know $\mathcal{I}_k$.
Then Thm. <ref> bounds $\|\mathbf{H}_{k}-\mathbf{I}_{p}\|_2$ in terms of $n_k:= |\mathcal{I}_{k}|$, and in particular, we know $\mathbf{H}_{k}$ converges to $\mathbf{I}_{p}$ almost surely as $n_k \rightarrow \infty$ (this follows from the strong law of large numbers).
To be specific, recall that $\boldsymbol{\mu}_k=\frac{1}{n_k}\sum_{i\in \mathcal{I}_{k}}\xi$ (assume $\xi$ are deterministic, though the argument adapts to stochastic $\xi$ under mild assumptions such as finite first two moments), then from the sampled data we can form the center estimate
\[
\widehat{\boldsymbol{\mu}_k} = \frac{1}{n_k} \frac{p}{m} \sum_{i\in \mathcal{I}_{k}} \mathbf{R}_{i}\mathbf{R}_{i}^{T}\x_i
\]
and $\widehat{\boldsymbol{\mu}_k}\rightarrow \boldsymbol{\mu}_k$ almost surely as $n_k\rightarrow \infty$, which follows from the strong law of large numbers and the independence of the $\mathbf{R}_{i}$.
For feature extraction (FE), the collected data are $\{ \rand\x_i \}_{i\in \mathcal{I}_{k}}$, so the obvious center estimate is
\[
\widehat{\boldsymbol{\mu}_k}^\text{FE} = \frac{1}{n_k} \sum_{i\in \mathcal{I}_{k}} \rand^\dagger \rand\x_i
= \frac{1}{n_k} \rand^\dagger \rand \sum_{i\in \mathcal{I}_{k}} \x_i
\]
with $\dagger$ representing the pseudo-inverse.
As $n_k\rightarrow \infty$, this does not converge to $\boldsymbol{\mu}_{k}$ because $\boldsymbol{\Omega}^\dagger \boldsymbol{\Omega} \neq \mathbf{I}_{p}$ (equality is impossible because the term on the left has rank $m<p$). That is, even with more data, the center estimate does not improve because a single copy of the random variable $\boldsymbol{\Omega}$ is used to compress all the data, so it is not consistent. The only solution is to take another pass through the original data $\{\xi\}$ using the estimated cluster assignments to form the sample center estimate.
[true cluster centers]
[K-means, many passes]
[sparsified K-means, 1 pass, no preconditioning]
[sparsified K-means, 2 passes, no preconditioning]
[sparsified K-means, 1 pass, preconditioned]
[sparsified K-means, 2 passes, preconditioned]
[feature extraction, 1 pass]
[feature extraction, 2 passes]
[feature selection, 3 passes]
[feature selection, 4 passes]
Center estimates $\boldsymbol{\mu}$ from
low-pass K-means algorithms.
§.§ Big data tests
We further test on two increasingly large extensions of MNIST.
On the largest extension, we test an out-of-core memory version of our algorithm. Because of the size of the data, we no longer run the K-means algorithm on all the data in order to provided a benchmark, but since the data are generated similarly to the classic MNIST, it would be reasonable to expect that basic K-means would behave similarly and achieve an accuracy close to $92\%$ as in Fig. <ref>.
Since the algorithms take longer to run, we reduce the number of replicates in each trial to $10$, and perform only 10 trials per algorithm. We focus on feature extraction and not feature selection since the smaller-scale tests indicated feature extraction performed better in both speed, accuracy and number of passes through the data.
In-core memory with $n=6\cdot 10^5$
Clustering was performed on data from the first $200,\!000$ samples each of the “0”, “3” and “9” digits from the mnist8M dataset <cit.>, using code from the “Infinite MNIST” project[<http://leon.bottou.org/projects/infimnist>], thus $n=6\cdot 10^5$, with $p=784$ as before. This new dataset artificially creates more training examples by applying pseudo-random deformations and translations to the MNIST images.
Accuracy results are in Fig. <ref>. The preconditioned version of sparsified K-means is much more accurate than the non-preconditioned version, and has better accuracy than feature extraction while also enjoying lower variance and taking only a single pass through the data. If we compute a second pass through the data, accuracy jumps to nearly optimal levels as soon as we sample at least $1\%$ of the data.
Timing results for $\gamma=0.05$ are in Table <ref>. Feature extraction reduces dimension instead of increasing sparsity, and while both algorithms take roughly the same number of flops, feature extraction has roughly a $2\times$ edge in speed since it has simpler data structures which have better data locality and can be exploited by many algorithms. The timing results are broken down into fine detail to show that the majority of time is in the actual K-means iteration on the reduced/sparsified data. We also note that without preconditioning, K-means never converges within 100 iterations in our tests, a sign that it does not capture the structure of the data, and this greatly contributes to the runtime of the algorithm.
Similar to Fig. <ref> but on much larger data, $n=6\cdot 10^5$. The preconditioning helps significantly to increase accuracy, as well as lower variance.
From the same $n=6\cdot 10^5$ simulations as Fig. <ref>, at $\gamma=0.05$. All numbers
are averages over the 10 trials, times in seconds. The K-means algorithm was limited to 100 iterations, and an asterisk denotes the algorithm never converged within this limit (on all trials).
Algorithm Total time Time to sample Time to precondition Iterations of K-means
Sparsified K-means 228.8 s 6.0 s 33.7 s 42.1
Sparsified K-means, 2 pass 237.0 s 6.0 s 33.7 s 42.1
Sparsified K-means, no preconditioning 665.6 s 5.1 s NA 100.0*
Feature extraction 123.1 s 0.4 s NA 41.9
Out-of-core memory with $n=10^7$
We implement out-of-core versions of the sparsified K-means algorithm and the feature extraction algorithm, which efficiently load and compress the data in a batched manner such that the entire matrix is never loaded all at once into the main memory of the computer. The dataset is created using the same “Infinite MNIST” code, so the setup is the same as the previous sections, i.e., $p=784$, but now $n=9,\!631,\!605$, having $3,\!168,\!805$, $3,\!280,\!085$ and $3,\!182,\!715$ examples of the digits “0”, “3” and “9”, respectively. Stored in double-precision using Matlab's default compression, which automatically reduces precision if possible, the matrix is 4.9 GB, and would be 56 GB if loaded into memory in double-precision.
When loading, the matrix is split into 58 chunks, each one (except the last) with dimensions $784 \times 167183$ and approximately $1$ GB in size.
We repeat the experiment three times, using $\gamma \in \{0.01,0.05\}$, and $10$ replicates.
Table <ref> shows the results.
Accuracy is similar to the $n=600,\!000$ simulation, which is not surprising since the data for both experiments were algorithmically generated from the Infinite MNIST code.
As expected, time to load data from disk was significant. For example, in the second pass over data at $\gamma=.05$, loading the data required $125$ seconds while the actual time for computing for the updated mean and assignment, i.e. Alg. <ref> except line 2, was just $28$ seconds.
However, the time to load the data from disk is still not a bottleneck in the overall computation time, since we only need to perform this once (or twice).
In a distributed data setting, where loading the data is even more costly but we would also be able to take $\gamma$ very small, one may prefer the 1-pass variant over the 2-pass variant.
The time to sample is non-negligible since it requires many calls to a pseudo-random number generator and creating sparse arrays. Preconditioning also takes $170$ seconds using the DCT, though we remark that the DCT in Matlab is not well-implemented, and Matlab performs an FFT on the same data in under 1 minute since it directly calls the fftw library. Directly calling fftw's DCT routine would likely be much faster; it would also be possible to accelerate this step using general-purpose graphical processing units (GP-GPU) since applying the DCT to an array is efficiently parallelized.
For a general idea of how long K-means would take without sub-sampling, we run a single iteration of K-means and compare the results in Table <ref>, finding that we reduce the computational time by a factor of almost $40$. The actual time for the full K-means algorithm would not improve by quite this much, since it is not clear if the number of iterations would change, and there is also the fixed cost of loading the data once (for sparsified K-means) or twice (for the 2-pass variant).
We also caution that this was a single run of a single iteration of K-means, so the factor $40$ is of limited precision, but overall, this is better than the result we hope to see since we have $\gamma=0.05$ meaning we have kept $1/20^\text{th}$ of the data. One reason for seeing $40$ times speedup instead of $20$ times speedup may be that Matlab paged memory onto secondary storage for the full version of the algorithm, or that the sparsified data was small enough to fit entirely inside the CPU cache memory (20 MB) rather than RAM.
From the $n=9,\!631,\!605$ simulation.
All numbers are averages over the 3 trials, times in seconds. Accuracy listed as mean $\pm $ standard deviation.
Algorithm Accuracy Iterations of K-means Total to sample to precondition to load data from disk
3*$\gamma=.01\;\Bigg\{$ 1lSparsified K-means $0.745\pm .0008$ 100* 2630s 38s 170s 125s
1lSparsified K-means, 2 pass $0.927\pm .0018$ 100* 2783s 38s 170s 250s
1lFeature extraction $0.680\pm .0610$ 73.8 1123s 18s NA 128s
3*$\gamma=.05\;\Bigg\{$ 1lSparsified K-means $0.887\pm .0002$ 53.5 4380s 137s 159s 126s
1lSparsified K-means, 2 pass $0.933\pm .0001$ 53.5 4538s 137s 159s 254s
1lFeature extraction $0.836\pm .0714$ 70.4 3384s 62s NA 129s
Estimated speedup. From the $n=9,\!631,\!605$ simulation, at $\gamma=0.05$.
2cTime to find assignments 2cTime to update all centers 2cCombined time
(rl)2-3 (rl)4-5 (l)6-7
Algorithm Absolute Speedup Absolute Speedup Absolute Speedup
K-means $130.0$s $1\times$ $150.8$s $1\times$ $280.8$s $1\times$
Sparsified K-means $1.3$s $100\times$ $5.7$s $26.4\times$ $7.0$s $40.1\times$
§.§ Discussion
On the MNIST data, all the fast algorithms show great speedup over standard K-means, and tunable accuracies that can reach the accuracy of standard K-means as $\gamma \rightarrow 1$. In our tests, the one-pass (preconditioned) sparsified K-means algorithm appears to be slightly more accurate than feature extraction, and significantly more accurate than feature selection and the non-preconditioned sparsified K-means.
In addition, our sparsified K-means has significantly lower variance than feature extraction, which means that the output of our method is more reliable and closer to the output of K-means on the full data among different initializations.
Furthermore, based on the MNIST experiments, if one can afford two passes over the data, the accuracy of our two-pass sparsified K-means reaches the accuracy of standard K-means and, at the same time, accurately estimates the cluster centers. For in-core problems, there is negligible cost for the extra pass, so the two-pass variant is the best candidate. For out-of-core problems, the one-pass variant may be preferred.
§ CONCLUSIONS
We have presented a compression scheme for large-scale data sets which leads to both computational time and memory benefits in unsupervised learning tasks such as PCA and K-means clustering. A main feature of our approach is that it requires just one pass over the data thanks to the randomized preconditioning transformation, which makes it applicable to streaming and distributed data settings. In fact, the preconditioning transformation is an essential component of our approach which allows us to achieve accurate and reliable estimates in the data sparsification process and eliminates the need to revisit past entries of the data. A side-benefit of the preconditioning is a reduction in the variance of estimates.
Our sparsified K-means algorithm returns both assignments and cluster centers in a single pass over the data, whereas the state-of-the-art feature-based algorithms require at least two passes. Moreover, our approach leads to per-step guarantees on the clustering structure, as opposed to the guarantees on the overall objective function in feature-based algorithms.
Finally, our compression scheme has the potential to be applicable in many other techniques in signal processing and machine learning, such as subspace learning, K-nearest neighbors, soft K-means, mixture models, and expectation-maximization algorithms. In these settings, the preconditioning and sampling technique could be used to either speed up computation for in-core memory problems, or to create one-pass variants for out-of-core or streaming problems.
§ ACKNOWLEDGMENT
It is a pleasure to thank Michael Wakin and Alex Gittens for informative discussions, and the anonymous referees for constructive comments.
§ EXPONENTIAL CONCENTRATION INEQUALITIES
Below are standard inequalities listed in expedient formats.
Let $z_{1},\ldots,z_{n}$ be independent, centered random variables, and assume that each one is bounded:
\[
\mathbb{E}\left[z_{k}\right]=0\;\text{and}\;\left|z_{k}\right|\leq L_k\;\text{for each}\; k=1,\ldots,n.
\]
Introduce the sum $S=\sum_{k=1}^{n}z_{k}$, and let $\sigma^{2}=\sum_{k=1}^{n}L_k^2$. Then, for all $t\geq0$:
\[
\mathbb{P}\left\{ \left|S\right|\geq t\right\} \leq2\exp\left(\frac{-\nicefrac{t^{2}}{2}}{\sigma^{2}}\right).
\]
Let $z_{1},\ldots,z_{n}$ be independent, centered random variables, and assume that each one is uniformly bounded:
\[
\mathbb{E}\left[z_{k}\right]=0\;\text{and}\;\left|z_{k}\right|\leq L\;\text{for each}\; k=1,\ldots,n.
\]
Introduce the sum $S=\sum_{k=1}^{n}z_{k}$, and let $\sigma^{2}=\sum_{k=1}^{n}\mathbb{E}[z_{k}^{2}]$
denote the variance of the sum. Then, for all $t\geq0$:
\[
\mathbb{P}\left\{ \left|S\right|\geq t\right\} \leq2\exp\left(\frac{-\nicefrac{t^{2}}{2}}{\sigma^{2}+\nicefrac{Lt}{3}}\right).
\]
Let $\mathbf{Z}_{1},\ldots,\mathbf{Z}_{n}$ be independent, symmetric,
centered random matrices with dimension $p$, and assume that each
one is uniformly bounded:
\[
\mathbb{E}\left[\mathbf{Z}_{k}\right]=\mathbf{0}\;\text{and}\;\left\Vert \mathbf{Z}_{k}\right\Vert _{2}\leq L\;\text{for each }k=1,\ldots,n.
\]
Introduce the sum $\mathbf{S}=\sum_{k=1}^{n}\mathbf{Z}_{k}$ , and
let $\sigma^{2}=\left\Vert \sum_{k=1}^{n}\mathbb{E}\left[\mathbf{Z}_{k}^{2}\right]\right\Vert _{2}$ denote the variance. Then, for all $t\geq0$:
\[
\mathbb{P}\left\{ \left\Vert \mathbf{S}\right\Vert _{2}\geq t\right\} \leq p\exp\left(\frac{-\nicefrac{t^{2}}{2}}{\sigma^{2}+\nicefrac{Lt}{3}}\right).
\]
§ PROPERTIES OF THE SAMPLING MATRIX
Consider a sampling matrix $\mathbf{R}=[\mathbf{r}_{1},\ldots,\mathbf{r}_{m}]\in\mathbb{R}^{p\times m}$,
where the $m$ columns are chosen uniformly at random from the set of all
$p$ canonical basis vectors without replacement. Then, these columns form an orthonormal basis, i.e., $\mathbf{R}^{T}\mathbf{R}=\mathbf{I}_{m}$.
Moreover, we have:
\begin{equation}
\mathbb{E}[\mathbf{RR}^{T}]=\frac{m}{p}\mathbf{I}_{p}\label{eq:expectation_RRT}
\end{equation}
and for any fixed vector $\mathbf{x}\in\mathbb{R}^{p}$ and $m\geq2$:
\begin{equation}
\mathbb{E}[\mathbf{RR}^{T}\mathbf{xx}^{T}\mathbf{RR}^{T}]\hspace{-1mm}=\hspace{-1mm}\frac{m(m-1)}{p(p-1)}\mathbf{xx}^{T}\hspace{-1mm}+\hspace{-1mm}\frac{m(p-m)}{p(p-1)}\diag(\mathbf{xx}^{T}).\label{eq:expectation_RRTxxTRRT}
\end{equation}
The columns of $\mathbf{R}$ are distinct canonical basis vectors, thus $\mathbf{R}^{T}\mathbf{R}=\mathbf{I}_{m}$. To prove (<ref>), note that $\mathbb{E}[\mathbf{RR}^{T}]=\sum_{i=1}^{m}\mathbb{E}[\mathbf{r}_{i}\mathbf{r}_{i}^{T}]$, and we will show that
\begin{equation}
\mathbb{E}[\mathbf{r}_{i}\mathbf{r}_{i}^{T}]=\frac{1}{p}\mathbf{I}_{p}\;\text{for}\; i=1,\ldots,m.\label{eq:expectation_ri}
\end{equation}
The main difficulty is that the columns are dependent on each other since the sampling is without replacement.
Let us first consider $i=1$. In this case, the first column $\mathbf{r}_{1}$ is chosen uniformly at random from the set of all canonical basis vectors, i.e., $\mathbb{P}\left\{ \mathbf{r}_{1}=\mathbf{e}_{r_{1}}\right\}=\frac{1}{p}$ for $r_{1}=1,\ldots,p$. Thus
\[
\mathbb{E}\left[\mathbf{r}_{1}\mathbf{r}_{1}^{T}\right]=\sum_{r_{1}=1}^{p}\mathbb{P}\left\{ \mathbf{r}_{1}=\mathbf{e}_{r_{1}}\right\} \mathbf{e}_{r_{1}}\mathbf{e}_{r_{1}}^{T}=\frac{1}{p}\sum_{r_{1}=1}^{p}\mathbf{e}_{r_{1}}\mathbf{e}_{r_{1}}^{T}=\frac{1}{p}\mathbf{I}_{p}.
\]
For $i \in \left\{2,\ldots,m\right\}$, we compute the expectation as follows:
\begin{eqnarray*}
\mathbb{E}\left[\mathbf{r}_{i}\mathbf{r}_{i}^{T}\right] & &\hspace{-5mm} =\mathbb{E}\left[\mathbb{E}\left[\mathbf{r}_{i}\mathbf{r}_{i}^{T}|\mathbf{r}_{1},\ldots,\mathbf{r}_{i-1}\right]\right]\\
& &\hspace{-5mm} =\sum_{\left(r_{1},\ldots,r_{i-1}\right)}\mathbb{P}\left\{ \mathbf{r}_{1}=\mathbf{e}_{r_{1}},\ldots\mathbf{r}_{i-1}=\mathbf{e}_{r_{i-1}}\right\} \\
& &\hspace{-5mm} \times \mathbb{E}\left[\mathbf{r}_{i}\mathbf{r}_{i}^{T}|\mathbf{r}_{1}=\mathbf{e}_{r_{1}},\ldots\mathbf{r}_{i-1}=\mathbf{e}_{r_{i-1}}\right]
\end{eqnarray*}
where the summation is over the set of $(i-1)$ distinct values from $\left\{1,\ldots,p\right\}$, thus $\mathbb{P}\{\mathbf{r}_{1}=\mathbf{e}_{r_{1}},\ldots,\mathbf{r}_{i-1}=\mathbf{e}_{r_{i-1}}\}=\frac{1}{p}\frac{1}{p-1}\ldots\frac{1}{p-(i-2)}$. Also, the expectation does not depend on the permutation of ${r_{1},\ldots,r_{i-1}}$,
so we condense the sum to range over just the set, not permutation, of distinct values, and adjust by multiplying by $\left(i-1\right)!$.
\begin{eqnarray*}
\mathbb{E}\left[\mathbf{r}_{i}\mathbf{r}_{i}^{T}\right]& &\hspace{-5mm}
=\left(i-1\right)!\left(\frac{1}{p}\frac{1}{p-1}\ldots\frac{1}{p-\left(i-2\right)}\right) \\
& & \hspace{-5mm}
\times \sum_{\substack{\{r_{1},\ldots,r_{i-1}\}\\\text{distinct}}}\Big(
\sum_{\substack{r_{i}\;\text{from} \\ \text{remaining values}}}
\mathbb{P}\left\{ \mathbf{r}_{i}=\mathbf{e}_{r_{i}}\right\} \mathbf{e}_{r_{i}}\mathbf{e}_{r_{i}}^{T}\Big)\\
& & \hspace{-5mm}\overset{(a)}{=}\left(i-1\right)!\left(\frac{1}{p}\frac{1}{p-1}\ldots\frac{1}{p-\left(i-2\right)}\frac{1}{p-\left(i-1\right)}\right) \\
& & \hspace{-5mm}
\times \Big(\sum_{\substack{\{r_{1},\ldots,r_{i-1}\}\\\text{distinct}}}
\sum_{\substack{r_{i}\;\text{from} \\ \text{remaining values}}}
\mathbf{e}_{r_{i}}\mathbf{e}_{r_{i}}^{T}\Big)\\
& & \hspace{-5mm}\overset{(b)}{=}\left(i-1\right)!\left(\frac{1}{p}\frac{1}{p-1}\ldots\frac{1}{p-\left(i-2\right)}\frac{1}{p-\left(i-1\right)}\right)\\
& & \hspace{-5mm}
\times \binom{p-1}{i-1}\mathbf{I}_{p}=\frac{1}{p}\mathbf{I}_{p}
\end{eqnarray*}
where (a) follows from $\mathbb{P}\{ \mathbf{r}_{i}=\mathbf{e}_{r_{i}}\}=\frac{1}{p-\left(i-1\right)}$ and (b) is obtained by counting the number of cases where $r_{i}=j$, $1\leq j\leq p$, and this can be easily computed by counting the number of cases that $j$ is not in the set $\{r_{1},\ldots,r_{i-1}\}$ which is $\binom{p-1}{i-1}$.
This completes the proof of (<ref>).
Next, we show that (<ref>) holds. Note that:
\begin{eqnarray*}
& & \hspace{-4mm}\mathbb{E}\left[\mathbf{RR}^{T}\mathbf{xx}^{T}\mathbf{RR}^{T}\right]\!\!=\!\!\!\sum_{\left(r_{1},\ldots,r_{m}\right)}\mathbb{P}\left\{ \mathbf{r}_{1}=\mathbf{e}_{r_{1}},\ldots\mathbf{r}_{m}=\mathbf{e}_{r_{m}}\right\} \\
& & %\hspace{-4mm}
\times \Big(\sum_{i=1}^{m}\mathbf{e}_{r_{i}}\mathbf{e}_{r_{i}}^{T}\Big)\mathbf{xx}^{T}\Big(\sum_{i=1}^{m}\mathbf{e}_{r_{i}}\mathbf{e}_{r_{i}}^{T}\Big)\\
& & \hspace{-4mm}=\Big(\frac{1}{p}\frac{1}{p-1}\ldots\frac{1}{p-\left(m-1\right)}\Big)\left\{ \alpha_{1}\sum_{k=1}^{p}\mathbf{e}_{k}\mathbf{e}_{k}^{T}\mathbf{xx}^{T}\mathbf{e}_{k}\mathbf{e}_{k}^{T}
\vphantom{+\alpha_{2}\sum_{k\neq l}\mathbf{e}_{k}\mathbf{e}_{k}^{T}\mathbf{xx}^{T}\mathbf{e}_{l}\mathbf{e}_{l}^{T}} % for correct size brackets
\right.\\
& & %\hspace{-4mm}
\left.+\alpha_{2}\sum_{k\neq l}\mathbf{e}_{k}\mathbf{e}_{k}^{T}\mathbf{xx}^{T}\mathbf{e}_{l}\mathbf{e}_{l}^{T}\right\}
\end{eqnarray*}
where the summation is over the set of $m$ distinct values from $\left\{1,\ldots,p\right\}$ and we should find the coefficients $\alpha_{1}$ and $\alpha_{2}$. In fact, $\alpha_{1}$ represents the number of cases that each $k$, $1\leq k \leq p$, is among the $m$ numbers chosen from $\{1,\ldots,p\}$ without replacement. Let's fix $r_{1}=1$, we then have $\binom{p-1}{m-1}(m-1)!$ cases. Thus, we see that:
\[
\alpha_{1}=\left[\binom{p-1}{m-1}\left(m-1\right)!\right]m=\frac{m\left(p-1\right)\left(p-2\right)!}{\left(p-m\right)!}.
\]
Similarly, $\alpha_{2}$ represents the number of cases that each pair of $k$ and $l$, $1\leq k,l \leq p$ and $k \neq l$, is among the $m$ numbers chosen from $\{1,\ldots,p\}$. Let's fix $r_{1}=1$ and $r_{2}=2$, we then have $\binom{p-2}{m-2}(m-2)!$ cases. This argument leads to:
\[
\alpha_{2}\!=\!\left[\binom{p-2}{m-2}\left(m-2\right)!\right]m\left(m-1\right)\!=\!\frac{m\left(m-1\right)\left(p-2\right)!}{\left(p-m\right)!}.
\]
Since $\alpha_{2}<\alpha_{1}$, we can write the expectation as follows:
\begin{eqnarray*}
& & \hspace{-5mm}\mathbb{E}\left[\mathbf{RR}^{T}\mathbf{xx}^{T}\mathbf{RR}^{T}\right]=\left(\frac{1}{p}\frac{1}{p-1}\ldots\frac{1}{p-\left(m-1\right)}\right)\\
& & \hspace{-5mm}\times\!\!\left(\!\alpha_{2}\Big(\sum_{k=1}^{p}\mathbf{e}_{k}\mathbf{e}_{k}^{T}\Big)\mathbf{xx}^{T}\Big(\sum_{k=1}^{p}\mathbf{e}_{k}\mathbf{e}_{k}^{T}\Big)\!+\!\left(\alpha_{1}-\alpha_{2}\right)\text{diag}\left(\mathbf{xx}^{T}\right)\!\right)
\end{eqnarray*}
and this completes the proof.
Suppose we sample $m$ entries of $\x\in\R^p$ uniformly at random without replacement. Then, the probability of keeping the $j$-th entry of $\x$ is $\frac{m}{p}$ for all $j=1,\ldots,p$.
The proof follows from the properties of sampling matrices given in Theorem <ref>. For a matrix $\RR\in\R^{p\times m}$ containing $m$ distinct canonical basis vectors, we show that $\E[\RR\RR^T]=\frac{m}{p}\mathbf{I}_p$.
Each component of the sub-sampled data $\w=\RR\RR^T\x$ is a random variable with two possible values: the $j$-th entry of $\w$ takes the same value as the corresponding entry of $\x$ with probability $\pi_j$ and zero otherwise. Since $\E[\w]=\E[\RR\RR^T\x]=\frac{m}{p}\x$, we conclude that $\pi_j=\frac{m}{p}$ for $j=1,\ldots,p$.
§ PROOF OF THEOREM <REF>
We present the proof of Thm. <ref> on the covariance estimator. First, we show that $\Cn$ is an unbiased estimator, i.e., $\E[\Cn]=\Cemp$. Using Thm. <ref>, we compute the following expectations:
\begin{equation}
\E[\Cemph]=\Cemp+\frac{(p-m)}{(m-1)}\diag(\Cemp)\label{eq:expectation-cemph}
\end{equation}
\begin{equation}
\E[\diag(\Cemph)]\hspace{-1mm}=\hspace{-1mm}\diag(\E[\Cemph])\hspace{-1mm}=\hspace{-1mm}\frac{(p-1)}{(m-1)}\diag(\Cemp).\label{eq:expectation-cemph-diag}
\end{equation}
Hence, using (<ref>) and (<ref>), we get $\E[\Cn]=\Cemp$. To find the closeness of $\Cn$ to its expectation $\Cemp$, we use the matrix Bernstein inequality (Thm. <ref>). Note that $(\Cn-\Cemp)$ can be written as a sum of $n$ independent centered random matrices:
\begin{equation}
\Cn-\Cemp=\sum_{i=1}^{n}\Z_i=\sum_{i=1}^{n}\frac{1}{n}(\Z_{i}^{(1)}-\Z_{i}^{(2)}-\Z_{i}^{(3)})
\end{equation}
where $\Z_{i}^{(1)}=\frac{p(p-1)}{m(m-1)}\w_i\w_i^T$, $\Z_{i}^{(2)}=\frac{p(p-m)}{m(m-1)}\diag(\w_i\w_i^T)$, $\Z_{i}^{(3)}=\x_i\x_i^T$, and $\w_i=\RR_i\RR_i^{T}\x_i$ is the sub-sampled data. To apply matrix Bernstein, we should find a uniform bound on the spectral norm of each summand $\|\Z_i\|_2$. We find a uniform bound for the spectral norm of $\Z_{i}^{(1)}$:
\begin{eqnarray}
\hspace{-10mm}&\|\Z_{i}^{(1)}\|_2\hspace{-3mm}&=\frac{p(p-1)}{m(m-1)}\|\w_i\w_i^{T}\|_2=\frac{p(p-1)}{m(m-1)}\|\w_i\|_2^{2} \nonumber\\
\hspace{-10mm}&\hspace{-3mm}&\leq \frac{p(p-1)}{m(m-1)} \rho \|\x_i\|_2^{2}\leq\frac{p(p-1)}{m(m-1)} \rho \|\X\|_\maxCol^2.
\end{eqnarray}
For the second term $\Z_{i}^{(2)}$, it is easy to verify that $\diag(\w_i\w_i^{T})\preccurlyeq\diag(\x_i\x_i^{T})$, where $\mathbf{A}\preccurlyeq\mathbf{B}$ means that $\mathbf{B}-\mathbf{A}$ is positive semidefinite, and thus we get $\|\diag(\w_i\w_i^{T})\|_2\leq\|\diag(\x_i\x_i^{T})\|_2$ which can be used to bound $\|\Z_{i}^{(2)}\|_2$:
\begin{equation}
\|\Z_{i}^{(2)}\|_2\!=\!\frac{p(p-m)}{m(m-1)}\|\diag(\w_i\w_i^{T})\|_2\leq\frac{p(p-m)}{m(m-1)}\|\X\|_\text{max}^2.
\end{equation}
We also find a uniform bound for the spectral norm of $\Z_{i}^{(3)}$:
\begin{equation}
\|\Z_{i}^{(3)}\|_2=\|\x_i\x_i^{T}\|_2=\|\x_i\|_2^{2} \leq \|\X\|_\maxCol^2
\end{equation}
and the triangle inequality for the spectral norm leads to:
\begin{equation}
\|\Z_i\|_2\leq \frac{1}{n}\left(\|\Z_{i}^{(1)}\|_2+\|\Z_{i}^{(2)}\|_2+\|\Z_{i}^{(3)}\|_2\right)\leq L
\end{equation}
where $L$ is given in (<ref>).
Next, we compute the variance of our estimator $\sigma^2$:
\begin{equation}
\sigma^2=\|\E[(\Cn-\Cemp)^2]\|_2=\|\sum_{i=1}^{n}\E[\Z_i^2]\|_2.
\end{equation}
We find the variance $\sigma^2$ by first computing the expectation:
\begin{equation}
\E[\Z_i^2]=\frac{1}{n^2}\E[(\Z_{i}^{(1)}-\Z_{i}^{(2)}-\Z_{i}^{(3)})^2]
\end{equation}
which requires computing the expectation of the product of terms $\Z_{i}^{(1)}$, $\Z_{i}^{(2)}$, and $\Z_{i}^{(3)}$. We begin by computing:
\begin{eqnarray*}
&&\hspace{-8mm}\E[\Z_{i}^{(1)}\Z_{i}^{(1)}]\!\!=\!\! \frac{p^{2}(p-1)^2}{m^{2}(m-1)^2}\E[\w_i\w_i^{T}\w_i\w_i^{T}]\\
&&\hspace{-8mm}=\!\!\frac{p^{2}(p-1)^2}{m^{2}(m-1)^2}\E[\|\w_i\|_2^{2}\w_i\w_i^{T}]\!\preccurlyeq\!\! \frac{p^{2}(p-1)^2}{m^{2}(m-1)^2} \rho \|\x_i\|_2^{2}\E[\w_i\w_i^{T}]\\
\end{eqnarray*}
where the inequality follows the fact that expectation preserves the semidefinite order and we also used Thm. <ref> to compute $\E[\w_i\w_i^{T}]$. Now, we compute $\E[\Z_{i}^{(2)}\Z_{i}^{(2)}]$:
\begin{eqnarray*}
&&\hspace{-8mm}\E[\Z_{i}^{(2)}\Z_{i}^{(2)}]\!=\!\frac{p^{2}(p-m)^2}{m^{2}(m-1)^2}\E[(\diag(\w_i\w_i^{T}))^2] \\
&&\hspace{-8mm}=\!\frac{p^{2}(p-m)^2}{m^{2}(m-1)^2}\frac{m}{p} (\diag(\x_i\x_i^{T}))^2\!=\!\frac{p(p-m)^2}{m(m-1)^2}(\diag(\x_i\x_i^{T}))^2
\end{eqnarray*}
where this follows from $\E[(\diag(\w_i\w_i^T))^2]=\frac{m}{p}(\diag(\x_i\x_i^T))^2$. To see this, let $w_{i,j}$ denote the $j$-th element of $\w_i$ and note that $\diag(\w_i\w_i^T)$ is a diagonal matrix where the $j$-th element is $w_{i,j}^2$. Thus, $(\diag(\w_i\w_i^T))^2$ is also a diagonal matrix where the $j$-th element is equal to $w_{i,j}^4$. Based on Lemma <ref>, the probability of keeping the $j$-th element of $\x_i$ under the uniform sampling without replacement is $\frac{m}{p}$, which means that $\E[w_{i,j}^4]=\frac{m}{p}x_{i,j}^4$.
Next, we can easily find the following expectations:
\begin{eqnarray*}
\end{eqnarray*}
\begin{eqnarray*}
\end{eqnarray*}
\[
\E[\Z_{i}^{(3)}\Z_{i}^{(3)}]=\x_i\x_i^{T}\x_i\x_i^{T}=\|\x_i\|_2^{2}\x_i\x_i^{T}.
\]
Hence, based on the expectations computed above and the triangle inequality, we get:
\begin{eqnarray*}
&&\hspace{-8mm}\sigma^2\!=\!\Big\|\sum_{i=1}^{n}\E[\Z_i^{2}]\Big\|_2\!\leq\! \frac{1}{n}\left(\frac{p(p-1)}{m(m-1)}\rho\!-1\!\!\right)\cdot\Big\|\frac{1}{n}\sum_{i=1}^{n}\|\x_i\|_2^{2}\x_i\x_i^{T} \Big\|_2 \\
&&\hspace{-8mm}+\!\frac{1}{n}\frac{p(p-1)(p-m)}{m(m-1)^2}\rho\Big\|\frac{1}{n}\sum_{i=1}^{n}\|\x_i\|_2^{2}\diag(\x_i\x_i^{T}) \Big\|_2\\
&&\hspace{-8mm}+\!\frac{1}{n^2}\left(\big\|\sum_{i=1}^{n}\E[\Z_i^{(1)}\Z_i^{(2)}]\big\|_2+ \big\|\sum_{i=1}^{n}\E[\Z_i^{(2)}\Z_i^{(1)}]\big\|_2\right).
\end{eqnarray*}
We also have the following two inequalities:
\[
\frac{1}{n}\sum_{i=1}^{n}\|\x_i\|_2^{2}\x_i\x_i^{T}\preccurlyeq\|\X\|_\maxCol^2\cdot\Cemp
\]
\[
\frac{1}{n}\sum_{i=1}^{n}\|\x_i\|_2^{2}\diag(\x_i\x_i^{T})\preccurlyeq\|\X\|_\maxCol^2\cdot\diag(\Cemp).
\]
In the last step, we find an upper bound for the following:
\begin{eqnarray*}
&&\hspace{-8mm}\leq \sum_{i=1}^{n}\E[\|\Z_i^{(1)}\Z_i^{(2)}\|_2]\leq\sum_{i=1}^{n}\E[\|\Z_i^{(1)}\|_2\|\Z_i^{(2)}\|_2]
\end{eqnarray*}
where this follows from the triangle inequality, Jensen's inequality, and the fact that for two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$, we have $\|\mathbf{A}\mathbf{B}\|_2\leq\|\mathbf{A}\|_2\|\mathbf{B}\|_2$. We compute the two terms inside the expectation:
\[
\|\Z_i^{(1)}\|_2=\frac{p(p-1)}{m(m-1)}\|\w_i\|_2^{2}=\frac{p(p-1)}{m(m-1)}\x_i^{T}\RR_i\RR_i^{T}\x_i
\]
\[
\|\Z_i^{(2)}\|_2=\frac{p(p-m)}{m(m-1)}\|\diag(\w_i\w_i^T)\|_2\leq\frac{p(p-m)}{m(m-1)}\|\X\|_\text{max}^{2}.
\]
Hence, using the property $\E[\RR_i\RR_i^{T}]=\frac{m}{p}\eye_p$, we get:
\begin{eqnarray*}
&&\hspace{-8mm}\E[\|\Z_i^{(1)}\|_2\|\Z_i^{(2)}\|_2]\!\leq\! \frac{p^{2}(p-1)(p-m)}{m^{2}(m-1)^{2}}\|\X\|_\text{max}^{2}\E[\x_i^{T}\RR_i\RR_i^{T}\x_i]\\
\end{eqnarray*}
and using $\|\X\|_F^{2}=\sum_{i=1}^{n}\|\x_i\|_2^{2}$, we have:
\[
\Big\|\sum_{i=1}^{n}\E[\Z_i^{(1)}\Z_i^{(2)}]\Big\|_2\leq \frac{p(p-1)(p-m)}{m(m-1)^{2}}\|\X\|_\text{max}^{2}\|\X\|_F^{2}
\]
and this completes the proof.
§ PRESERVATION OF PAIRWISE DISTANCES
Let $\x_1$ and $\x_2$ be two fixed vectors in $\R^{p}$. Consider the structured dimension reduction map consisting of the preconditioning transformation $\Hadamard\Diag$ (<ref>) and the sampling matrix $\RR\in\R^{p\times m}$, where the $m$ columns are chosen uniformly at random from the set of all canonical basis vectors without replacement. Then, with probability at least $1-3\beta^{-1}$,
\begin{equation}
0.40\left\|\x_1-\x_2\right\|_2\leq\left\|\sqrt{\frac{p}{m}}\RR^{T}\Hadamard\Diag(\x_1-\x_2)\right\|_2\leq 1.48\left\|\x_1-\x_2\right\|_2
\end{equation}
given that $4\left[\sqrt{\beta}+\sqrt{8\log(\beta p)}\right]^{2}\log(\beta)\leq m \leq p$.
This result is a straightforward consequence of Theorem 3.1 in <cit.>. Let us denote $\x=\x_1-\x_2$ and represent it as $\x=\VV\cc$, where $\VV\in\R^{p\times \beta}$, $\beta<m$, is an orthonormal matrix and $\cc\in\R^{\beta}$ (e.g. the first column of $\VV$ is $\x/\|\x\|_2$ and the remaining $(m-1)$ columns can be chosen via Gram-Schmidt). We then have the following deterministic lower and upper bounds for $\|\RR^T\Hadamard\Diag\x\|_2=\|\RR^T\Hadamard\Diag\VV\cc\|_2$:
\[
\sigma_\beta(\RR^T\Hadamard\Diag\VV) \|\cc\|_2\leq\|\RR^T\Hadamard\Diag\VV\cc\|_2\leq \sigma_1(\RR^T\Hadamard\Diag\VV) \|\cc\|_2
\]
where $\sigma_1$ and $\sigma_\beta$ denote the largest and smallest singular values. Based on <cit.>, for $m\geq 4[\sqrt{\beta}+\sqrt{8\log(\beta p)}]^{2}\log(\beta)$ and with probability at least $1-3\beta^{-1}$,
\[
0.40\sqrt{m/p}\leq \sigma_\beta(\RR^T\Hadamard\Diag\VV),\;\;
\sigma_1(\RR^T\Hadamard\Diag\VV)\leq1.48\sqrt{m/p}.
\]
Note that $\|\cc\|_2=\|\VV\cc\|_2=\|\x\|_2$ since $\VV$ is an orthonormal matrix and this completes the proof.
|
1511.00564
|
The stochastic logarithmic equation
Viorel Barbu[Octav Mayer Institute of
Mathematics (Romanian Academy) and Al.I. Cuza University and,
700506, Iaşi, Romania. This work was supported by the DFG through
CRC 701 and by CNCS-VEFISCDI (Romania) project PN-II-2012-4-0456.],
Michael Röckner[Fakultät für
Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.
This research was supported by the DFG through CRC 701.],
Deng Zhang[Department of Mathematics,
Shanghai Jiao Tong University, 200240 Shanghai, China. ]
Abstract. In this paper we prove global existence and
uniqueness of solutions to the stochastic logarithmic Schrödinger
equation with linear multiplicative noise. Our approach is mainly
based on the rescaling approach and the method of maximal monotone
operators. In addition, uniform estimates of solutions in
the energy space $H^1(\bbr^d)$ and in an appropriate Orlicz space are also obtained here.
Keywords: Logarithmic Schrödinger equation, maximal
monotonicity, stochastic PDE,
Wiener process.
2000 Mathematics Subject Classification: 60H15; 47H05; 47J05
§ INTRODUCTION AND MAIN RESULT.
The logarithmic Schrödinger equation
\begin{align} \label{deter-equa-x}
i\frac{d u}{dt} + \D u + u \log |u|^2 = 0, \ \ in\ \bbr^+ \times
\bbr^d,
\end{align}
has wide applications in quantum mechanics, quantum optics, nuclear
physics, open quantum systems, Bose-Einstein condensation and so on.
It was first proposed in <cit.> as a model of nonlinear wave
mechanics. As a matter of fact, as shown in <cit.>, the
logarithmic nonlinearity arising in (<ref>) is the
unique nonlinearity for which the separability hypothesis of
noninteracting subsystems of the Schrödinger theory holds. It also
possesses many other attractive features, including the additivity
of the energy for noninteracting subsystems, the validity of the
lower energy bound and Planck's relation for all stationary states.
All these make this equation unique among nonlinear wave equations.
See e.g. <cit.>. We also refer to <cit.> for the derivation of this equation from Nelson's stochastic
quantum mechanics <cit.>.
Motivated by the physical significance above, we are here mainly
concerned with well-posedness of the logarithmic Schrödinger
equation in the stochastic case, that is,
\begin{align} \label{equa-x}
idX&=\Delta Xdt + \lbb X\log|X|^2dt -i\mu Xdt + iXdW,\ t\in(0,T), \nonumber \\
X(0)&=x \in L^2,
\end{align}
Here, $\lbb\in \mathbb{R}$, $W$ is the Wiener process
\begin{align} \label{W}
W(t,\xi) = \sum\limits_{j=1}^n \mu_j e_j(\xi) \beta_j(t),\ t\geq
0,\ \xi\in\mathbb{R}^d,
\end{align}
where $d\geq 1$, $\{\mu_j\}_{j=1}^n$ are complex numbers,
$\{e_j\}_{j=1}^n$ are real-valued functions, and
$\{\beta_j\}_{j=1}^n$ is a family of independent real valued
Brownian motions on a probability space $(\Omega, \mathscr{F},
\mathbb{P})$ with normal (in particular right-continuous) filtration
$(\mathscr{F}_t)_{t\geq 0}$. For simplicity, we assume that $n<\9$.
\begin{align} \label{mu}
\mu(\xi) = \frac12 \sum\limits_{j=1}^n |\mu_j|^2 e_j^2(\xi),\ \
\xi\in\mathbb{R}^d.
\end{align}
The stochastic equation (<ref>) can be derived from
(<ref>) with an additional potential $VX$, where the
random potential $V$ fluctuates rapidly and so can be approximated
by the Gaussian noise $\dot{W}$. Moreover, the linear multiplicative
noise $iXdW$ together with the term $-i\mu X dt$ also plays an
important role in the theory of measurements continuous in time in
open quantum systems. In this case, one main feature is that
$|X(t)|_2^2$ is a continuous martingale. This fact implies the mean
norm square conservation of $X(t)$ and allows to define a new
probability law, the “physical” probability law, which has
important applications to open quantum systems. For more physical
interpretations, we refer to <cit.>, <cit.> and
the references therein.
The stochastic nonlinear Schrödinger equation with the
polynomial nonlinearity $\lbb|X|^{\a-1}X$ was first studied in
<cit.>, based on the mild formulation of the stochastic
equation. The optimal exponents of the nonlinearity for the global
well-posedness were recently achieved in <cit.>,
based on the rescaling transformation (see (<ref>) below) and
the Strichartz estimates established in <cit.> for lower order
perturbations of the Laplacian. However, the contraction mapping
arguments used in the mentioned works are not applicable here, due
to the fact that the function $y\rightarrow y\log|y|^2$ is not
locally Lipschitz.
One of the main features of the logarithmic nonlinearity is the
quasi-monotonicity. Based on this, the global well-posedness of the
deterministic equation (<ref>) was first studied in
<cit.> in the distribution sense for initial data in $L^2$ or
$H^1$. Later, the global well-posedness was also proved in
<cit.> for initial data in $H^1$ and in some convenient Orlicz
space, which is closely related to the logarithmic nonlinearity. We
also refer to <cit.> for the global well-posedness for initial
data in $H^1$ with finite momentum.
Furthermore, stochastic partial differential equations with monotone
coefficients are also extensively studied in the literature. We
refer to <cit.>, <cit.>, <cit.> and the references
therein. Recently, based on the rescaling approach and operatorial
reformulation, the approach of maximal monotone operators was
developed in <cit.> in a general infinite dimensional setting,
which has applications to new existence and uniqueness results of
various stochastic models with linear multiplicative noise.
Inspired by the quasi-monotone feature of the logarithmic
nonlinearity and the works mentioned above, we shall employ the
rescaling transformation and the method of maximal monotone
operators to study the global well-posedness of (<ref>).
However, it should be mentioned that, the results in <cit.> are
not applicable here, since the operator $i\Delta$ in (<ref>)
is not coercive (see <cit.>).
Moreover, another difficulty arises from the passage to the limit in
the approximating equation (see (<ref>) below). Because
even if a space $\mathcal{X}$ is compactly imbedded into another one
$\mathcal{Y}$, we generally do no have the compact imbedding from
$L^p(\Omega; \mathcal{X})$ to $L^p(\Omega; \mathcal{Y})$, $1\leq p
\leq \9$, the classical deterministic method as in
<cit.> to pass to the limit in the nonlinear term
can not directly be applied here.
In order to overcome these difficulties, inspired by <cit.>, we will consider the initial data in the energy space
$H^1(\bbr^d)$ and an appropriate Orlicz space $V$ (see (<ref>)
below). These spaces allow to control the singularity of the
logarithmic nonlinearity at infinity and at the origin respectively.
More importantly, they are also suitable spaces for the maximal
monotonicity of the logarithmic nonlinearity, which makes the
passage to the limit in the approximating equation possible, thereby
yielding the global well-posedness.
To state our results precisely, let us first introduce some
necessary notations. Take $H=L^2(\mathbb{R}^d; \mathbb{C})=:L^2$
with the scalar product defined by $ \<u,v\> = \int_{\mathbb{R}^d}
u\overline{v} d\xi, \ u,v\in H$, and the norm $|u|_2=\<u,u\>^{\frac
12}$. Let $H^1$ denote the classical Sobolev space, i.e. $H^1=\{u\in
L^2: \na u\in L^2\}$ with norm $|u|^2_{H^1} = |u|_2^2 + |\na
u|^2_2$. We also use the standard notation $L^p= L^p(\bbr^d)$,
$1\leq p \leq \9$, for the space of all $p$-integrable complex
functions with the norm $|\cdot|_{L^p}$.
Moreover, as in <cit.>, define the function
\begin{align} \label{def-N}
N(x) = \left\{
\begin{array}{ll}
-x^2 \log x^2, & \hbox{if $0\leq x\leq e^{-3}$;} \\
3x^2 + 4e^{-3}x - e^{-6}, & \hbox{if $e^{-3}\leq x$.}
\end{array}
\right.
\end{align}
$N$ is a positive convex and increasing function, and $N\in
C^1([0,\9)) \cap C^2 ((0,\9))$. The Orlicz space $V$ corresponding
to $N$ is defined by
\begin{align} \label{def-V}
V=\{ u\in L^1_{loc}: N(|u|)\in L^1 \},
\end{align}
equipped with the Luxembourg norm
\begin{align}
\|u\|_V = \inf \{ k>0: \int N(k^{-1}|u(\xi)|)d\xi \leq 1 \}.
\end{align}
Here as usual $L^1_{loc}$ is the space of all locally Lebesgue
integrable functions. It is proved in <cit.> that $N$
is a Young-function which is $\bigtriangleup_2$-regular and $(V,
\|\cdot\|_{V})$ is a separable reflexive Banach space (see also
<cit.> and <cit.>). We also have that (see
<cit.>) for any $u\in V$,
\begin{align} \label{V-N}
\min\{ \|u\|_V, \|u\|^2_V \} \leq \int N(|u(\xi)|) d\xi
\leq \max \{ \|u\|_V, \|u\|^2_V \}
\end{align}
Now, set $U:= H^1 \cap V$. $U$ is a reflexive Banach space equipped
with the norm $\|u\|_{U}= |u|_{H^1} + \|u\|_V$, for any $u \in U$,
and its dual space is $U'=H^{-1}+ V'$ with the norm $\|u\|_{U'}=\inf
\{ |u_1|_{H^{-1}} + \|u_2\|_{V'}: u= u_1+u_2, u_1\in H^{-1}, u_2\in
V' \}$. One advantage for introducing the space $U$ is that the
nonlinear operator $u \mapsto u \log |u|^2$ is continuous from $U$
to $U'$ (see <cit.>).
The precise definition of solutions to (<ref>) is given
A continuous $H$-valued $(\calf_t)$-adapted process $X$ is said to
be a solution to (<ref>) if for any $p\geq 3$, $X\in
L^p(\Omega \times(0,T); U)$, $X\log |X|^2 \in L^{p'}(\Omega \times
(0,T); U')$, and it satisfies $\bbp$-a.s. for all $t\in[0,T]$
\begin{align} \label{equa-x*}
X(t)=& x - \int_0^t \(i\D X(s) ds + \mu X(s)
+ \lbb X(s)\log| X(s)|^2 \) ds
+ \int_0^t X(s)dW(s),
\end{align}
where the stochastic term is taken in Itô's sense.
We also assume that the spatial functions $\{e_j\}_{j=1}^n$ in the
noise $W$ satisfy the hypothesis:
(H) $e_j\in \calb^\9(\bbr^d)$ such that for each $1\leq k \leq
d$, $1\leq m \leq n$,
$$ |\partial_k e_m(\xi)| \leq \lbb(|\xi|),\ \ \xi\in \bbr^d ,$$
where $\calb^\9 = \{ f\in C^\9(\bbr^d),
\partial_\alpha f\in L^\9,\ for\ all\ \alpha\}$, and $\lbb(\cdot) $ is a positive non-increasing function in $C([0,\9]) \cap L^1
The main result of this article is formulated as follows.
Under Hypothesis $(H)$, for any initial datum $x\in U$ and $0<T<\9$,
there exists a unique solution $X$ to (<ref>) in the sense of
Definition <ref>.
Moreover, for any $p\geq 2$,
\begin{align}
\label{integ-l2-x}
&\bbe \|X(t)\|^{p}_{L^\9(0,T;U)} <\9,
\end{align}
\begin{align}
& \bbe \| X(t) \log |X(t)|^2
\|^p_{L^\9(0,T;U')}<\9,
\end{align}
\begin{align}
\bbe \| e^{W(t)} \frac{d}{dt} (e^{-W(t)}X(t))
\|^p_{L^\9(0,T;U')} < \9.
\end{align}
The remainder of this paper is organized as follows. In Section
<ref>, we first apply the rescaling transformation to
reduce the original stochastic equation (<ref>) to a random
equation (see (<ref>)), and then we introduce some
appropriate spaces and prove the maximal monotonicity of the
logarithmic nonlinearity. Section <ref> is mainly concerned
with the approximating equation. We first obtain the $H^1$-global
well-posedness and derive the uniform estimate in the energy space
in Subsection <ref>. Then in Subsection <ref>, in
order to control the singularity of the logarithmic nonlinearity at
the origin, we start with the analysis of the entropy function and
then prove the uniform estimates in the Orlicz space. Section
<ref> is mainly devoted to the proof of the main result.
As mentioned above, the maximal monotonicity will play an important
role in the passage to the limit in the approximating equation. Some
technical details are postponed to the Appendix.
Throughout this paper, $C$ denotes various constants which may
change from line to line.
§ RANDOM EQUATION
Taking into account the quasi-monotone feature of the logarithmic
nonlinearity, we first use the change of variable $X\to e^{-2|\lbb|
t}X$ to reformulate the original equation (<ref>) as
\begin{align} \label{equa-x'}
idX&=\Delta Xdt + \lbb X\log|X|^2dt +(4\lbb|\lbb|t -2i|\lbb| -i\mu) Xdt + iXdW, \nonumber \\
X(0)&=x \in L^2.
\end{align}
Then, applying the rescaling transformation
\begin{align} \label{rescal}
X=e^W y,
\end{align}
which can be seen as a Doss-Sussman transformation generalized to
infinite dimensions, we can reduce the stochastic equation
(<ref>) to a random Schrödinger equation
\begin{align} \label{equa-y}
&\frac{ d y}{d t}(t) = - i e^{-W(t)}\D (e^{W(t)}y(t))
- ( 2|\lbb| + 4i\lbb|\lbb|t + \wh{\mu} ) y(t) \nonumber \\
&\qquad \qquad - \lbb i y(t) \log|e^{W(t)} y(t)|^2,\ \ a.e.\ t\in(0,T), \\
&y(0)=x, \nonumber
\end{align}
where $\wh \mu = \frac 12 \sum\limits_{j=1}^N (\mu_j^2 + |\mu_j|^2)
In order to formulate the definition of solutions to (<ref>),
proceeding as in <cit.>, we consider the Hilbert space
$\mathcal{H}$ of all $H$-valued $(\mathscr{F}_t)_{t\geq 0}$-adapted
processes $y:[0,T] \rightarrow H$ with the scalar product
\begin{align} \label{def-h}
\<y,z\>_{\mathcal{H}} = \mathbb{E} \int_0^T
\<e^{W(t)}y(t),e^{W(t)}z(t)\>dt,
\end{align}
and the norm
\begin{align*}
|y|_{\calh} = \(\mathbb{E}\int_0^T|e^{W(t)}y(t)|_H^2dt\)^{\frac 12}.
\end{align*}
For any $p\geq 3$, consider the space $\calu$ of all
$(\mathscr{F}_t)_{t\geq 0}$-adapted processes $y:= [0,T] \to U$ such
\begin{align} \label{def-calU}
\| y \|_{\calu}^p = \bbe \int_0^T \| e^{W(t)} y(t) \|^{p}_U dt
\end{align}
Let $\calu'$ denote the dual space of $\calu$. In fact, $\calu'$ is
the space of all $(\mathscr{F}_t)_{t\geq 0}$-adapted processes
$y:[0,T] \to \calu'$ such that
\begin{align} \label{def-calU'}
\| y \|_{\calu'}^{p'} = \bbe \int_0^T \| e^{W(t)} y(t) \|^{p'}_{U'}
\end{align}
We have $\calu \subset \calh \subset \calu'$, algebraically and
\begin{align} \label{Def-G}
(\calg y)(t):=& \lbb i y(t) \log|e^{W(t)}y(t)|^2 + 2|\lbb|y(t),\ \ y\in
\end{align}
Analogously to Definition <ref>, the solutions to
(<ref>) is now defined below.
A solution to (<ref>) is a continuous $H$-valued
$(\mathscr{F}_t)$-adapted progress $y$, such that $y\in \calu$,
$y\log|e^Wy|^2 \in \calu'$, and it satisfies $\bbp$-a.s. for all
\begin{align} \label{equa-y*}
y(t) = x - \int_0^t \( i e^{-W(s)}\D (e^{W(s)} y(s))
+ (4i\lbb|\lbb|t + \wh{\mu}) y(s) + \calg(y(s))\) ds.
\end{align}
We refer to <cit.> for a rigorous proof of the
equivalence of solutions to (<ref>) and (<ref>).
Therefore, the proof of Theorem <ref> is now reduce to the
theorem as follows.
Under Hypothesis $(H)$, for any initial datum $x\in U$ and $0<T<\9$,
there exists a unique solution $y$ to (<ref>) in the sense
of Definition <ref>.
Moreover, for all $p\geq 2$,
\begin{align} \label{Bdd-U-y}
&\bbe \|e^{W(t)}y(t)\|^{p}_{L^\9(0,T;U)} <\9,
\end{align}
\begin{align} \label{Bdd-U'-log}
&\bbe \| e^{W(t)}y(t) \log |e^{W(t)}y(t)|^2
\|^{p}_{L^\9(0,T;U')} <\9,
\end{align}
\begin{align} \label{Bdd-U'-dy}
\bbe
\|e^{W(t)}\frac{d}{dt}y(t)\|^p_{L^\9(0,T;U')} <\9.
\end{align}
The remainder of this paper is devoted to the proof of Theorem
<ref>. We will mainly consider the case
$d\geq 3$. The simpler cases $d=1,2$ can be proved similarly.
In the end of this section, let us show the maximal monotonicity of
the operator $ \calg$. Recall that an operator $A:\calx\to\calx'$
(possibly nonlinear) from a Banach space $\calx$ to its dual
$\calx'$ is said to be monotone if
\begin{align*}
Re\ {}_{\calx'}\<Ay_1-Ay_2,y_1-y_2\>_{\calx} \geq 0,\ \forall y_1,y_2\in D(A),
\end{align*}
and maximal monotone if it has no nontrivial monotone extensions in
$\calx \times \calx'$.
For any $p\geq 3$, the operator $\calg$ is maximal monotone from
$\calu$ to $\calu'$.
Proof. In view of <cit.> the
maximality, since the demicontinuity implies the hemicontinuity, it
suffices to prove that $\calg$ is monotone and demicontinous from
$\calu$ to $\calu'$, i.e., if $y_n, y\in \calu$ such that $y_n\to y$
in $\calu$, then
\begin{align} \label{hemi-g}
{}_{\calu'} \<\calg(y_n), z \>_{\calu} \to {}_{\calu'} \<\calg(y),
z\>_{\calu},\ \ z\in \calu.
\end{align}
For this purpose, we first note that by the definition of $\calg$ in
\begin{align*}
&Re\ {}_{\calu'} \< \calg(y_1)-\calg(y_2), y_1-y_2\>_{\calu} \\
=&2|\lbb||y_1-y_2|_{\calh}^2 -2 \lbb Im\ {}_{\calu'} \< y_1 \log|e^{W}y_1| - y_2 \log|e^{W}y_2|, y_1-y_2
\>_{\calu} \geq 0.
\end{align*}
where in the last step we used (<ref>) below with $\ve
=0$, and so the monotonicity of $\calg$ follows.
In order to prove the demicontinuity (<ref>), we will show
\begin{align} \label{bdd-lp'-u'}
\| e^W \calg(y_n)\|_{L^{p'}(\Omega \times (0,T);U')} \leq C < \9,
\end{align}
where $C$ is independent of $n$. Then, for any subsequence of $\{n\}
\to \9$, there exists a further subsequence (still denoted by
$\{n\}$) such that $e^W \calg(y_n) \overset{\omega}{\rightharpoonup}
\eta$, in $L^{p'}(\Omega \times (0,T); U')$, where $
\overset{\omega}{\rightharpoonup}$ stands for weak convergence. But,
since $y_n \to y$ in $\calu$, we have $e^W\calg(y_n) \to e^W
\calg(y)$ in measure $\bbp \otimes dt \otimes d\xi$. Hence, we
conclude that $\eta=e^W \calg(y)$, which implies (<ref>),
since the subsequence was arbitrary.
It remains to prove (<ref>). Set $X_n:= e^W y_n$ and
$L(|X_n|^2):= \log|X_n|^2$. By the definition of $U'$ and $\calg$
we have
\begin{align} \label{split-u'}
&\| e^W \calg(y_n)\|_{L^{p'}(\Omega \times (0,T);U')} \nonumber \\
\leq& 2|\lbb|\ \| X_n \|_{L^{p'}(\Omega \times (0,T);H^{-1})}
+|\lbb| \| I_{\{|X_n| > e^{-3}\}} X_n L(|X_n|^2) \|_{L^{p'}(\Omega \times (0,T);H^{-1})} \nonumber \\
& + |\lbb| \|I_{\{|X_n| \leq e^{-3}\}} X_n L(|X_n|^2) \|_{L^{p'}(\Omega \times (0,T);V')}.
\end{align}
Since for each $\xi \in \{|X_n| > e^{-3} \}$, $
|X_n (\xi) L(|X_n(\xi)|^2)| \leq C_\delta(|X_n(\xi)| + |X_n(\xi)|^{1+\delta})$ with $C_\delta$
independent of $n$. By Sobolev's imbedding theorem with $\delta \leq
\frac{2}{d-2}$,
\begin{align*}
& \| X_n \|_{L^{p'}(\Omega \times (0,T);H^{-1})}
+ \| I_{\{|X_n| > e^{-3}\}} X_n L(|X_n|^2) \|_{L^{p'}(\Omega \times (0,T);H^{-1})} \nonumber \\
\leq& C (\|X_n\|_{L^{p'}(\Omega\times(0,T); L^2)}
+ \|X_n\|^{1+\delta}_{L^{(1+\delta)p'}(\Omega\times(0,T);
L^{2(1+\delta)})}) \\
\leq& C (\|X_n\|_{L^{p'}(\Omega\times(0,T); L^2)}
+ \|X_n\|^{1+\delta}_{L^{(1+\delta)p'}(\Omega\times(0,T);
\end{align*}
Then, taking $\delta$ such that $0<\delta<p-2$, we have
$(1+\delta)p' < p$ and, via the Hölder inequality,
\begin{align} \label{bdd-lp'-h-1}
& \| X_n \|_{L^{p'}(\Omega \times (0,T);H^{-1})}
+\| I_{\{|X_n| > e^{-3}\}} X_n L(|X_n|^2) \|_{L^{p'}(\Omega \times (0,T);H^{-1})}
\nonumber \\
\leq& C_T ( \| X_n \|_{L^{p}(\Omega \times
+ \| X_n \|^{1+\delta}_{L^{p}(\Omega \times
(0,T);H^1)} )
\leq C_T<\9,
\end{align}
where $C_T$ is independent of $n$.
On the other hand, for each $\xi \in \{|X_n| \leq e^{-3} \}$, as in
the proof of <cit.> we have
\begin{align} \label{N'-N}
\wt{N}(|X_n(\xi) L(|X_n(\xi)|^2 )|) \leq 2 N(|X_n(\xi)|),
\end{align}
where $\wt{N}$ is the convex conjugate of $N$.
Then, since $\wt{N}(0)=0$, by (<ref>),
\begin{align} \label{wtN-V}
\int \wt{N}(I_{\{|X_n|\leq e^{-3}\}} |X_n L(|X_n|^2)|) d\xi
=&\int I_{\{|X_n|\leq e^{-3}\}} \wt{N}( |X_n L(|X_n|^2)|) d\xi \nonumber \\
\leq& 2 \int I_{\{|X_n|\leq e^{-3}\}} N(|X_n|) d\xi \nonumber \\
\leq& 2 \max\{\|X_n\|_V, \|X_n\|^2_V\}.
\end{align}
Moreover, similarly to (<ref>), there exist $\kappa, C \in
(2,\9)$ such that
\begin{align} \label{V'-N}
\min \{ \|u\|_{V'},\|u\|^{\kappa}_{V'} \}
\leq C \int \wt{N}(|u|) d\xi.
\end{align}
(See the Appendix for a proof.) Then, (<ref>) and (<ref>)
imply that
\begin{align} \label{bdd-v'}
\| I_{\{|X_n|\leq e^{-3}\}} X_n L(|X_n|^2) \|_{V'}
\leq& C \max \{ \|X_n\|_{V},\|X_n\|^2_{V}, \|X_n\|^{1/\kappa}_V, \|X_n\|^{2/\kappa}_V \} \nonumber \\
\leq& C(\|X_n\|_V^2 + 1),
\end{align}
Hence, since $p\geq 3$, $2p'\leq p$, Hölder's inequality yields
\begin{align} \label{bdd-lp'-v'}
\|I_{\{|X_n| \leq e^{-3}\}} X_n L(|X_n|^2) \|_{L^{p'}(\Omega
\times(0,T);V')}
\leq& C_T(\|X_n\|^2_{L^{2p'}(\Omega\times (0,T); V)}+1)
\nonumber\\
\leq& C_T (\|X_n\|^2_{L^{p}(\Omega\times (0,T); V)}+1) \nonumber\\
\leq& C_T<\9,
\end{align}
where $C_T$ is independent of $n$.
Consequently, (<ref>), (<ref>) and
(<ref>) together yield (<ref>), thereby
completing the proof of Proposition <ref>. $\square$
As in <cit.>, we can also define the operators $\calb, \cala:
\mathcal{U} \to \mathcal{U}'$ by
\begin{align*}
(\cala y)(t)=& ie^{-W(t)}\D (e^{W(t)}y(t)) + 4i\lbb|\lbb|t y(t),\ y\in D(\cala)=\mathcal{U}, \nonumber \\
(\calb y)(t)=&\frac{dy(t)}{dt}+ \wh{\mu}y(t),\ a.e.\ t\in(0,T),\ y\in D(\calb),
\end{align*}
where $
D(\calb) = \{y\in \mathcal{U}:\ y\in AC([0,T];U')\cap C([0,T];H),
\bbp-a.s., \frac{dy}{dt} \in \mathcal{U}', y(0)=x
\}.$
Then, (<ref>) can be reformulated as an operatorial equation
\begin{align*}
\calb y + \cala y + \calg y =0.
\end{align*}
It is clear that $\cala $ is maximal monotone from $\calu$ to
$\calu'$. The same assertion holds also for $\calb$, by similar
arguments as in the proof of <cit.>.
Then, since $D(\cala)=D(\calg)=\calu$, we deduce from <cit.> that $\cala + \calb + \calg$ is also maximal monotone.
However, unlike in <cit.>, we do not have the coercivity (see
<cit.>) in the Schrödinger case, the proof of
<cit.> is not applicable here. In order to
obtain existence of solutions to (<ref>), we shall introduce
and study an associated approximating equation in the next section.
§ APPROXIMATING EQUATION
Consider the approximating equation,
\begin{align} \label{app-equa-y}
& y(t) = x - \int_0^t \(i e^{-W(s)} \D(e^{W(s)}y(s)) +
(4i\lbb|\lbb|t + \wh{\mu} )y(s) + \calg_\ve (y(s))\)ds,\\
&y(0)=x, \nonumber
\end{align}
Here, $t\in (0,T)$, $0\leq \ve \leq 1$,
\begin{align} \label{def-calg}
\calg_{\ve} (y)
:=2 \lbb i y L_{\ve}(e^Wy)
+ 2|\lbb|y,
\end{align}
\begin{align} \label{def-l}
L_{\ve}(u) = \log (\frac{|u| +\ve}{1+\ve |u|}),\ \forall u\in \bbc.
\end{align}
For $\ve =0$, set $L(u):= L_0(u) = \log |u|$, $u\in \bbc$.
We collect some properties of $L_\ve$ in the following lemma, whose
proof is included in the Appendix for completeness.
Let $0<\ve < 1$. Then:
$(i)$ For all $u>0$, $|L_\ve (u)| \leq |\log \ve|$, and $|u L_\ve
(u)| \leq |u L(u)|$.
$(ii)$ For all $u_1, u_2 \in \mathbb{C}$,
\begin{align} \label{Lve-diff}
\left| u_1L_{\ve}(u_1) - u_2L_{\ve}(u_2) \right|
\leq& (1+\log(1/\ve))|u_1-u_2|.
\end{align}
$(iii)$ For all $u_1,u_2\in \mathbb{C}$,
\begin{align} \label{mono-lve-0}
| Im (\ov{u_1} - \ov{u_2}) (u_1 L_\ve(u_1) - u_2 L_\ve(u_2)) |
\leq (1-\ve^2 ) |u_1 - u_2|^2.
\end{align}
The main result in this section is as follows.
Assume $(H)$ and let $0<\ve<1$ be fixed. For any initial datum
$x\in U$ and $0<T<\9$, there exists a unique $U$-valued
$(\mathscr{F}_t)$-adapted process $y_\ve$, such that $y_\ve\in
C([0,T];H^1)$, $\bbp$-a.s., and it satisfies (<ref>) in
the space $U'$ on $[0,T]$, $\bbp$-a.s.
Moreover, for any $p\geq 2$,
\begin{align} \label{esti-u-p}
\bbe \sup\limits_{0\leq t\leq T} \|e^{W(t)}y_\ve(t)\|^p_U
\leq C(T,p)<\9,
\end{align}
\begin{align} \label{esti-u'-p}
\bbe \sup\limits_{0\leq t\leq T} \|e^{W(t)}
\calg_\ve(y_\ve(t))\|^p_{U'}
\leq C(T,p)<\9,
\end{align}
where $C(T,p)$ is independent of $\ve$.
The proof will proceed in two steps. We first prove the global
well-posedness of (<ref>) in the state space $H^1$ in
Subsection <ref>, and then we prove the necessary uniform
estimates in the Orlicz space in Subsection <ref>.
§.§ $H^1$ global well-posedness
Assume $(H)$ and let $0<\ve<1$ be fixed. For each $x\in H^1$ and
$0<T<\9$, there exists a unique $H^1$-valued
$(\mathscr{F}_t)$-adapted process $y_\ve$, such that $y_{\ve} \in
C([0,T]; H^1)$, and it solves (<ref>) in the space
$H^{-1}$ on $[0,T]$, $\bbp$-a.s.
Moreover, for any $p\geq 2$,
\begin{align} \label{esti-bdd-h1-gloabl}
\bbe \sup\limits_{t\in[0,T]} |e^{W(t)}y_{\ve}(t)|_{H^1}^p \leq C(T,p) <\9,
\end{align}
where $C(T,p)$ is independent of $\ve$.
The key observation for the proof lies in the fact that the operator
$y\to y L_{\ve}(e^Wy)$ is Lipschitz on $L^2$ and bounded on $H^1$.
This fact allows to apply a fixed point argument as in
<cit.>. Below, the proof will rely on three lemmas. We first
introduce the evolution operators in Lemma <ref>, and then
we prove the local existence in Lemma <ref>. Finally, in
Lemma <ref> we derive the a priori estimate in
$H^1$-norm, which in turn implies the global well-posedness.
$\mathbb{P}-a.e.$, the operator $y\to -ie^{-W}\Delta (e^Wy) - (
2|\lbb| + 4i\lbb|\lbb|t + \wh{\mu}) y$ generates evolution operators
$U(t,s)=U(t,s,\omega)$ in the space $H^1(\mathbb{R}^d)$, $0\leq
s\leq t\leq T$. For each $x\in H^1(\mathbb{R}^d)$ and $s\in[0,T]$,
the process $[s,T]\ni t \to U(t,s)x$ is continuous and
$(\mathscr{F}_t)$-adapted, hence progressively measurable with
respect to the filtration $(\mathscr{F}_t)_{t\geq s}$.
Moreover, for any $f\in L^1(0,T; H^1)$, then $H^1$-path
\begin{align} \label{equa-U}
y(t) =U(t,0)x + \int_0^t U(t,s) f(s) ds, \ \ 0\leq t\leq T,
\end{align}
satisfies the estimates
\begin{align} \label{Esti-U-0}
\|y\|_{C([0,T]; H)} \leq C_T (|x|_{H} + \|f\|_{L^1(0,T;
\end{align}
\begin{align} \label{Esti-U}
\|y\|_{C([0,T]; H^1)} \leq C_T (|x|_{H^1} + \|f\|_{L^1(0,T;
\end{align}
Here, the process $C_t$, $t\geq 0$, can be taken to be
$(\mathscr{F}_t)$-adapted progressively measurable, increasing and
(For the proof see the Appendix.)
Assume $(H)$ and let $0<\ve<1$ be fixed. For each $x\in H^1$ and
$0<T<\9$, there exists an $H^1$-valued $(\mathscr{F}_t)$-adapted
process $y_{\ve}$ and a stopping time $\tau_{\ve}^*(x) \leq T$, such
that $y_\ve \in C([0,\tau_{\ve}^*(x)); H^1)$, and $y_\ve$ solves the
equation (<ref>) in $H^{-1}$ on $[0,\tau_{\ve}^*(x) )$,
Moreover, $\tau^*_\ve(x) =T$, $\bbp-a.s$, if
\begin{align} \label{yve-blow}
\sup\limits_{t\in[0,\tau_\ve^*(x))} |y_\ve(t)|_{H^1}<\9,\ \ \bbp-a.s.
\end{align}
Proof. Using the evolution operators introduced in Lemma
<ref>, we reformulate the equation (<ref>) in the
mild form
\begin{align} \label{mild-equay-ve}
y= U(t,0)x -2\lbb i \int_0^t U(t,s)\( y(s) L_{\ve} (e^{W(s)} y(s))\)ds,
\end{align}
(Note that, since for $y\in C([0,T];H^1)$, $yL_\ve(e^Wy)\in
L^1(0,T;H^1)$, the equivalence between (<ref>) and
(<ref>) can be proved similarly as in <cit.>.)
Consider the integral operator $F$ defined for any $y \in C([0,T] ;
H^1)$ by
\begin{align*}
F(y)(t) := U(t,0)x -2 \lbb i \int_0^t U(t,s) ( y(s) L_{\ve}
y(s)))ds,\ t\in[0,T].
\end{align*}
We first show that
\begin{align} \label{F-h1-bdd}
F(C([0,T] ; H^1)) \subset C([0,T] ; H^1).
\end{align}
Indeed, by (<ref>),
\begin{align*}
\|F(y)\|_{C([0,T] ; H^1)}
\leq& C_T \(|x|_{H^1}
+ 2|\lbb| \|y L_{\ve}(e^W y)\|_{L^1(0,T;H^1)}
\).
\end{align*}
By Lemma <ref> $(i)$ we have
\begin{align*}
\leq& \sqrt{2} |\log \ve| |y|_{H^1} + |y \na (L_{\ve}(e^Wy))|_2.
\end{align*}
Moreover, straightforward computations show that
\begin{align} \label{cal-na-le}
\na (L_{\ve}(e^Wy))
= \frac{(1-\ve^2) |e^Wy|^{-1} Re(\ov{e^Wy} \na (e^Wy))}{(\ve+ |e^Wy|)(1+ \ve
\end{align}
which implies that
\begin{align} \label{esti-nal}
|\na (L_\ve(e^Wy))| \leq |e^Wy|^{-1} |\na(e^Wy)|.
\end{align}
\begin{align*}
|y\na (L_{\ve}(e^Wy))|_2 \leq |e^{-W} \na (e^Wy)|_2
\leq \sqrt{2}\exp(2|W|_{L^\9})(1+ |\na W|_{L^\9}) |y|_{H^1}.
\end{align*}
\begin{align*}
|y L_{\ve}(e^Wy)|_{H^1}
\leq \sqrt{2}( |\log \ve| + \exp(2|W|_{L^\9})(1+ |\na W|_{L^\9}) ) |y|_{H^1}.
\end{align*}
It follows that
\begin{align} \label{esti-F-bdd}
\|F(y)\|_{C([0,T];H^1)}
\leq C_T |x|_{H^1}
+ C_T D_{1}(T) T \|y\|_{C([0,T];H^1)}
\end{align}
with $D_{1}(T):= 2\sqrt{2}|\lbb|( |\log \ve| + \sup\limits_{t\leq
T}\exp(2|W(t)|_{L^\9})(1+ \sup\limits_{t\leq T}|\na W(t)|_{L^\9}) )
$, thereby yielding (<ref>) as claimed.
Next, we will apply the iteration arguments as in <cit.> to
construct the local solution to (<ref>).
Fix $\oo \in \Omega$. Set $\mathcal {Y}^{\tau_1}_{M_1}:= \{y\in
C([0,\tau_1]; H^1): \|y\|_{C([0,\tau]; H^1)} \leq M_1\}$, where
$\tau_1$ and $M_1$ are random variables to be chosen later.
Similarly to (<ref>), for any $y\in \mathcal
\begin{align} \label{esti-F-bdd-tau}
\|F(y)\|_{C([0,\tau_1];H^1)}
\leq C_{\tau_1} |x|_{H^1}
+ C_{\tau_1} D_{1}(\tau_1) M_1 \tau_1.
\end{align}
Moreover, for any $y, \wt{y} \in \mathcal {Y}^{\tau_1}_{M_1}$, by
\begin{align*}
\|F(y) - F(\wt{y})\|_{C([0,\tau_1];L^2)}
\leq 2|\lbb| C_{\tau_1}
\|yL_{\ve}(e^Wy) - \wt{y}L_{\ve}(e^W\wt{y})\|_{L^1(0,\tau_1;
\end{align*}
which implies by (<ref>) that
\begin{align} \label{esti-F-lip-tau}
\|F(y) - F(\wt{y})\|_{C([0,\tau_1];L^2)}
\leq C_{\tau_1} D_{2}(\tau_1) \tau_1
\|y-\wt{y}\|_{C([0,\tau_1];L^2)},
\end{align}
where $D_{2}(t)= 2|\lbb| (1+ |\log \ve| ) \sup\limits_{s\leq
Then, we define the real-valued continuous,
$(\mathscr{F}_t)$-adapted process $Z(t):=D_1(t)+D_2(t)$, and denote
the $(\mathscr{F}_t)$-stopping time $\tau_1 := \inf\{t\in[0,T]: C_t
Z(t)t\geq \frac 12 \}\wedge T $ and $M:= 2 C_{\tau_1} |x|_{H^1}$.
(<ref>) and (<ref>) imply that
$F(\mathcal{Y}^{\tau_1}_{M_1}) \subset \mathcal{Y}^{\tau_1}_{M_1}$
and $F$ is a contraction in $C([0,\tau_1];L^2)$. Hence, Banach's
fixed point theorem yields a unique $y\in
\mathcal{Y}^{\tau_1}_{M_1}$, such that $y = F(y)$ on $[0,\tau_1]$.
Setting $y_1(t):= y(t\wedge \tau_1)$ and arguing as in the proof of
<cit.> we deduce that
$y_{1}|_{[0,\tau_1]}\in C([0,\tau_1]; H^1)$, $y_1$ is
$(\mathscr{F}_t)$-adapted and it solves (<ref>) on
$[0,\tau_1]$, $\bbp$-a.s.
Applying similar arguments as in <cit.>, we can extend the
solution step by step and construct a sequence $\{(y_m,
\tau_m)\}_{m\geq 1}$, such that for each $m\geq 1$, $\tau_m$ is an
$(\mathscr{F}_t)$-stopping time, $\tau_{m+1} \geq \tau_m$, $y_m$ is
an $H^1$-valued $(\mathscr{F}_t)$-adapted process, such that
$y_m|_{[0,\tau_m]} \in C([0,\tau_m]; H^1)$, $y_m(t) = y_m(t \wedge
\tau_m)$, $t\in [0,T]$, and $y_m$ solves (<ref>) on
$[0,\tau_m]$, $\bbp$-a.s.
More precisely, given the pair $(y_m,\tau_m)$ with such properties
above at the $m$-th step, we set $\mathcal
{Y}^{\sigma_m}_{M_{m+1}}:= \{ z\in C([0,\sigma_m]; H^1):
\|z\|_{C([0,\sigma_m]; H^1)} \leq M_{m+1}\}$, and define for $z\in
\begin{align*}
F_m(z)(t) := & U(\tau_m+t,\tau_m)y_m(\tau_m) \\
& - 2\lbb i \int_0^t U(\tau_m+t,\tau_m+s) ( z(s) L_{\ve}
(e^{W(\tau_m+s)} z(s)))ds.
\end{align*}
Similarly to (<ref>) and (<ref>), for
$z\in \mathcal {Y}^{\sigma_m}_{M_{m+1}}$,
\begin{align*}
\|F_m(z)\|_{C([0,\sigma_m];H^1)}
\leq C_{\tau_m+\sigma_m} (|y_m(\tau_m)|_{H^1}
+ D_{1}(\tau_m+\sigma_m) M_{m+1}\sigma_m).
\end{align*}
and for $z,\wt{z} \in \mathcal {Y}^{\sigma_m}_{M_{m+1}}$,
\begin{align*}
\|F_m(z) - F_m(\wt{z})\|_{C([0,\sigma_m];L^2)}
\leq C_{\tau_m+\sigma_m} D_2(\tau_m+\sigma_m) \sigma_m
\|z-\wt{z}\|_{C([0,\sigma_m];L^2)}.
\end{align*}
Then, define $Z_t^{(m)}:= D_1(\tau_m+t) + D_2(\tau_m+t)$ and
$\sigma_m = \inf \{t\in[0,T-\tau_m]: C_{\tau_m+t} Z_t^{(m)} t>\frac
12 \} \wedge (T-\tau_m)$. It follows that $F_m(\mathcal
{Y}^{\sigma_m}_{M_{m+1}}) \subset \mathcal {Y}^{\sigma_m}_{M_{m+1}}$
and $F_m$ is a contraction in $C([0,\sigma_m];L^2)$. By Banach's
fixed point theorem, we obtain a unique $z_{m+1}\in \mathcal
{Y}^{\sigma_m}_{M_{m+1}}$, such that $z_{m+1} = F_m(z_{m+1})$ on
Therefore, set $\tau_{m+1}: = \tau_m + \sigma_m$ and
\begin{align*}
y_{m+1}(t) :=
\left\{
\begin{array}{ll}
y_m(t), & \hbox{$t\in[0,\tau_m]$;} \\
z_{m+1}((t-\tau_m)\wedge \sigma_m), & \hbox{$t\in(\tau_m,T]$.}
\end{array}
\right.
\end{align*}
Then, we construct a new pair $(y_{m+1},\tau_{m+1})$ with the
properties mentioned above. In particular, $y_{m+1}$ solves
(<ref>) on $[0,\tau_{m+1}]$, $\bbp$-a.s. Iterating this
procedure gives us the desired sequence $\{(y_m,\tau_m)\}_{m\geq
Now, let $\tau_\ve^*(x): =\lim\limits_{m\to \9} \tau_m$ and
$y_{\ve}:= \lim\limits_{m\to \9} y_m I_{[0,\tau_\ve^*(x))}$. It
follows that $\tau_\ve^*(x)$ is an $(\mathscr{F}_t)$-stopping time,
$y_\ve$ is an $H^1$-valued $(\mathscr{F}_t)$-adapted process,
$y_\ve\in C([0,\tau_\ve^*(x)); H^1)$, and it solves the equation
(<ref>) on $[0,\tau_\ve^*(x))$, $\bbp$-a.s.
Finally, by the construction of $\{(y_m,\tau_m)\}_{m\geq 1}$, we use
similar arguments as in <cit.> to obtain the blow-up
alternative, i.e. for $\bbp$-a.e. $\omega$, if $\tau_m(\omega) <
\tau^*_\ve(x)(\omega)$, $\forall m\in \mathbb{N}$, then $
\lim\limits_{t\to \tau_\ve^*(x)(\omega)} |y_\ve(t)(\omega)|_{H^1}
=\9.$ By the construction of $\sigma_m$ above, we consequently
conclude that $\tau_\ve^*(x)=T$ if (<ref>) holds. $\square$
Assume the conditions of Lemma <ref> to hold, and let
$\tau_\ve^*(x)$ and $y_\ve$ be as in Lemma <ref>. Then, for
any $p\geq 2$,
\begin{align} \label{esti-bdd-h1-localye}
\bbe \sup \limits_{t\in[0,\tau_\ve^*(x))}
|e^{W(t)}y_{\ve}(t)|^p_{H^1} \leq C(T,p) < \9,
\end{align}
where $C(T,p)$ is independent of $\ve$.
Proof. Let $X_m:=e^Wy_\ve$, $\phi_j=\mu_je_j$, $1\leq j\leq
m$, and $\{\tau_m\}_{m\geq 1}$ be the sequence of stopping times
constructed in the proof of Lemma <ref>. Since $X_\ve
L_\ve(X_\ve)\in L^2 \subset H^{-1}$, as in the proof of <cit.>, we derive that $\bbp$-a.s., for $t\in[0,\tau_m]$,
\begin{align} \label{ito-app-equa-xa}
& | X_{\ve}(t)|_{H^1}^2 \nonumber \\
=& |x|_{_{H^1}}^2
-4|\lbb| \int_0^t |X_\ve|_{H^1}^2 ds
-2 \int_0^t Re \int \na \ov{X_{\ve}} \na (\mu X_{\ve}) d\xi ds
\nonumber \\
& + \sum\limits_{j=1}^n \int_0^t |\na (X_{\ve}\phi_j)|_{2}^2
ds + 4\lbb \int_0^t Im \int \ \na \ov{ X_{\ve} }\na (X_{\ve}L_{\ve}(X_{\ve})) d\xi
ds\nonumber \\
&+ 2 \sum\limits_{j=1}^n \int_0^t \int |X_\ve|^2 Re \phi_j d\xi d\beta_j(s)
+2 \sum\limits_{j=1}^n \int_0^t Re \int \na \ov{ X_{\ve}}
\na (X_{\ve}\phi_j) d\xi d\beta_j(s).
\end{align}
Then, applying Itô's formula we obtain for any $p\geq 2$,
\begin{align}
&|X_\ve(t)|_{H^1}^p \nonumber \\
=& |x|_{H^1}^p -2p|\lbb| \int_0^t |X_\ve|_{H^1}^p ds \nonumber \\
& -p\int_0^t |X_\ve|_{H^1}^{p-2} Re\int \na\ov{X_\ve} \na(\mu
X_\ve) d\xi ds
+ \frac{p}{2} \sum\limits_{j=1}^n \int_0^t |X_\ve|_{H^1}^{p-2} |\na
(X_\ve\phi_j)|_2^2 ds \nonumber \\
& + \frac{1}{2}p(p-2) \sum\limits_{j=1}^n \int_0^t |X_\ve|_{H^1}^{p-4}
\(\int|X_\ve|^2 Re \phi_j d\xi + Re \int \na\ov{X_\ve}
\na(X_\ve\phi_j)d\xi \)^2 ds \nonumber\\
&+ 2p\lbb \int_0^t |X_\ve|_{H^1}^{p-2} Im \int \na\ov{X_\ve} \na(X_\ve
L_\ve(X_\ve))d\xi ds\nonumber \\
&+ p \sum\limits_{j=1}^n \int_0^t |X_\ve|_{H^1}^{p-2} \int |X_\ve|^2 Re \phi_j d\xi
d\beta_j(s)\nonumber \\
&+ p \sum\limits_{j=1}^n \int_0^t |X_\ve|_{H^1}^{p-2} Re \int \na\ov{X_\ve} \na
(X_\ve\phi_j) d\xi d\beta_j(s)\nonumber
\end{align}
\begin{align}
=:&|x|_{H^1}^p + \sum\limits_{k=1}^7 J_k(t),\ \ t\in[0,\tau_m],\
\bbp-a.s.
\end{align}
Since $e_j\in C_b^{\9}$, $1\leq j\leq n$ and since by
(<ref>) we have
\begin{align*}
\bigg| Im \int \na \ov{X_{\ve}} \na ( X_{\ve}L_{\ve}(X_{\ve})) d\xi \bigg|
= \bigg| Im\ \int \na \ov{ X_\ve} (X_\ve \na L_\ve(X_\ve)) d\xi
\bigg| \leq |\na X_\ve|_2^2,
\end{align*}
it follows that
\begin{align} \label{esti-J-15}
\sum\limits_{k=1}^5 \bbe \sup\limits_{s\leq t} |J_k(s)|
\leq C_p \int_0^t \bbe \sup\limits_{r\leq s}
|X_\ve(r)|^p_{H^1} ds,\ \ t\in[0,\tau_m],
\end{align}
where $C_p$ is independent of $\ve$ and $m$.
As regards the remaining stochastic terms, it follows from the
Burkholder-Davis-Gundy inequality that
\begin{align} \label{esti-J-6}
\bbe \sup\limits_{s\leq t\wedge \tau_m} |J_6(s)|
\leq& C_p \bbe \bigg[ \int_0^{t\wedge \tau_m} \sum\limits_{j=1}^n |X_\ve|^{2p-4}_{H^1} \(\int |X_\ve|^2 Re \phi_j d\xi\)^2 ds \bigg]^{\frac
12} \nonumber \\
\leq& C_p \bbe \( \int_0^{t\wedge \tau_m} |X_\ve|^{2p}_{H^1} ds\)^{\frac 12}
\nonumber \\
\leq& C_p \delta \bbe \sup\limits_{s\leq t\wedge \tau_m} |X_\ve|^p_{H^1}
+ C(p,\delta) \int_0^t \bbe \sup\limits_{r\leq s\wedge \tau_m}
|X_\ve|^p_{H^1} ds,
\end{align}
where we used <cit.> in the last step,
$\delta>0$, and $C_p,C(p,\delta)$ are independent of $\ve$ and $m$.
\begin{align} \label{esti-J-7}
\bbe \sup\limits_{s\leq t\wedge \tau_m} |J_7(s)|
\leq& C_p \bbe \bigg[ \int_0^{t\wedge \tau_m} \sum\limits_{j=1}^n |X_\ve|^{2p-4}_{H^1} \(Re \int \na\ov{X_\ve} \na
(X_\ve\phi_j)d\xi\)^2 ds \bigg]^{\frac 12} \nonumber \\
\leq& C_p\delta \bbe \sup\limits_{s\leq t\wedge \tau_m} |X_\ve|^p_{H^1}
+ C(p,\delta) \int_0^t \bbe \sup\limits_{r\leq s\wedge \tau_m}
|X_\ve|^p_{H^1} ds,
\end{align}
where $C(p,\delta)$ is independent of $\ve$ and $m$.
Therefore, combining (<ref>)-(<ref>), taking
$\delta$ sufficiently small, and applying the Gronwall inequality we
\begin{align*}
\bbe \sup\limits_{t\in[0,\tau_m]} |X_\ve(t)|^p_{H^1} \leq
\end{align*}
where $C(T,p)$ is independent of $\ve$ and $m$. Taking $m\to \9$ and
using Fatou's lemma we consequently
obtain (<ref>). $\square$
Proof of Proposition <ref>. It follows from
(<ref>) that
$\sup\limits_{t\in[0,\tau^*_\ve(x))} |e^{W(t)}y_\ve(t)|_{H^1} <\9$,
$\bbp$-a.s. Then, since $e_j\in C_b^\9$, $1\leq j\leq n$,
$\sup\limits_{t\in[0,\tau^*_\ve(x))} |y_\ve(t)|_{H^1} <\9$,
$\bbp$-a.s., which along with Lemma <ref> implies the
global existence of the solution to (<ref>).
Uniqueness for (<ref>) follows from monotonicity.
Indeed, consider any two solutions $y_1, y_2$ to (<ref>)
with the initial datum $x$, and set $X_i =e^W y_i$, $i=1,2$. Then,
similarly to (<ref>), we derive that
\begin{align} \label{difference}
\bbe |X_1(t) - X_2(t)|_2^2
=& -4 |\lbb| \bbe \int_0^t |X_1 - X_2|_2^2 ds \nonumber \\
&+ 4\lbb \bbe \int_0^t Im \int (\ov{X_1}- \ov{X_2})(X_1 L_\ve(X_1) - X_2 L_\ve(X_2)
) d\xi ds.
\end{align}
By (<ref>),
\begin{align} \label{mono-lve}
&\bigg| \bbe \int_0^t Im \int (\ov{X_1}- \ov{X_2})(X_1 L_\ve(X_1) - X_2 L_\ve(X_2)
) d\xi ds \bigg| \nonumber \\
\leq& (1-\ve^2)\bbe \int_0^t |X_1-X_2|_{2}^2 ds \leq \bbe \int_0^t|X_1-X_2|_2^2ds.
\end{align}
Then, it follows that
\begin{align*}
\bbe |X_1(t) - X_2(t)|_2^2 \leq 0,
\end{align*}
which implies that for each $t\in[0,T]$, $X_1(t)= X_2(t)$,
$\bbp$-a.s. Thus, by the continuity of $y_i$ in $H^1$, $i=1,2$, we
deduce that $X_1(t)= X_2(t)$, $\forall t \in [0,T]$, $\bbp$-a.s.,
thereby obtaining the uniqueness. $\square$
In the next subsection, we shall derive some uniform estimates in
the Orlicz space, which allows to apply the method of maximal
monotone operators to take the limit in the approximating equations.
§.§ Uniform estimates
This subsection is mainly devoted to uniform estimates in the Orlicz
space $V$. Taking into account the definition (<ref>), let us
begin with the estimate of the entropy function below.
Fix $0<\ve \leq 1$. Let $x \in U$, $0<T<\9$, $y_\ve$ be the
approximating solution in Proposition <ref> and
$X_\ve=e^Wy_\ve$. We have for any $p\geq 2$,
\begin{align} \label{esti-log}
\bbe \sup\limits_{t\leq T} \bigg| \int |X_{\ve}(t)|^2 \log |X_{\ve}(t)|^2
d\xi \bigg|^p \leq C(T,p)<\9,
\end{align}
where $C(T,p)$ is independent of $\ve$.
Proof. For $u>0$, set $F_m(u):= \int_0^u (L_{1/m}(\nu) +
1) d\nu$, where $L_{1/m}(\cdot)$ is as defined in (<ref>).
Using the techniques as in <cit.> and <cit.>
we can derive that $\bbp$-a.s., for $t\in [0,T]$,
\begin{align} \label{Ito-Fn}
&\int F_m(|X_\ve(t)|^2) d\xi \nonumber \\
=& \int F_m(|x|^2) d\xi
- 2\int_0^t \int g_m(|X_{\ve}|^2) Im(\ov{X_{\ve}} \na
Re (\ov{X_{\ve}} \na X_{\ve}) d\xi ds \nonumber \\
& -4|\lbb| \int_0^t \int (L_{1/m}(|X_\ve|^2)+1)|X_\ve|^2 d\xi ds
+ \int_0^t \int g_m(|X_{\ve}|^2) (Re \phi_j)^2|X_{\ve}(s)|^4 d\xi ds \nonumber \\
& + 2 \sum\limits_{j=1}^n \int_0^t \int (L_{1/m}(|X_\ve|^2)+1)|X_{\ve}|^2 Re \phi_j d\xi
\end{align}
where $g_m (|X_{\ve}|^2):= 2(1-m^{-2}) (m^{-1}
+|X_{\ve}|^2)^{-1}(1+ m^{-1}|X_{\ve}|^2)^{-1}$, and $\phi_j=\mu_j
e_j$, $1\leq j\leq n$. (See the Appendix for the proof.)
Then, applying Itô's formula we derive that $\bbp$-a.s. for
$t\in [0,T]$,
\begin{align} \label{Ito-Fn-p}
&\(\int F_m(|X_\ve(t)|^2) d\xi \)^p \nonumber \\
=& \(\int F_m(|x|^2) d\xi \)^p \nonumber \\
&-2 p \int_0^t \(\int F_m(|X_\ve|^2) d\xi \)^{p-1}
\(\int g_m(|X_\ve|^2) Im(\ov{X_\ve}\na X_\ve) Re (\ov{X_\ve}\na
X_\ve) d\xi\) ds \nonumber \\
& -4 |\lbb| p \int_0^t \(\int F_m(|X_\ve|^2) d\xi \)^{p-1}
\(\int\(L_{1/m}(|X_\ve|^2)+1\)|X_\ve|^2 d\xi \) \ ds \nonumber \\
&+ p \int_0^t \(\int F_m(|X_\ve|^2) d\xi \)^{p-1}
\(\int g_m(|X_\ve|^2) (Re \phi_j)^2 |X_\ve|^4 d\xi\) ds
\nonumber \\
& +2 p(p-1) \sum\limits_{j=1}^N \int_0^t \(\int F_m(|X_\ve|^2) d\xi \)^{p-2}
\bigg(\int (L_{1/m}(|X_\ve|^2)+1)|X_\ve|^2 Re \phi_j d\xi\bigg)^2ds
\nonumber
\end{align}
\begin{align}
& + 2p \sum\limits_{j=1}^n\int_0^t \(\int F_m(|X_\ve|^2)d\xi \)^{p-1} \(\int |X_\ve|^2 Re
\phi_j d\xi\) d\beta_j(s) \nonumber \\
& + 2p \sum\limits_{j=1}^n \int_0^t \(\int F_m(|X_\ve|^2)d\xi \)^{p-1}
\(\int |X_\ve|^2 L_{1/m}(|X_\ve|^2) Re\phi_j d\xi\)
d\beta_j(s). \nonumber \\
=& \(\int F_m(|x|^2) d\xi \)^p + \sum\limits_{j=1}^6 K_j(t).
\end{align}
Since for $u>0$,
\begin{align} \label{Fm-L}
F_m(u) = uL_{1/m}(u)+ u - (1-m^{-2})\int_0^u
\nu(m^{-1}+\nu)^{-1}(1+m^{-1}\nu)^{-1} d\nu ,
\end{align}
it follows that
\begin{align*}
|F_m(|x|^2)| \leq ||x|^2 L_{1/m}(|x|^2)| + 2|x|^2
\leq ||x|^2 L(|x|^2)| + 2|x|^2 \in L^1(\bbr^d).
\end{align*}
Then, since $F_m(|x|^2) \to |x|^2 L(|x|^2)$, as $m\to \9$, the
dominated convergence theorem yields
\begin{align} \label{conv-Fm}
\int F_m(|x|^2) d\xi \to \int |x|^2 L(|x|^2) d\xi.
\end{align}
In particular,
\begin{align} \label{bdd-Fm}
\sup\limits_{m\geq 1} \bigg|\int F_m(|x|^2) d\xi \bigg| \leq C< \9.
\end{align}
For the other deterministic terms in (<ref>), since $e_j\in
C_b^\9$, $1\leq j\leq n$, and $g_m(|X_\ve|^2) \leq 2|X_\ve|^{-2}$,
using the Young inequality $a^{p-1}b\leq \frac{p-1}{p}\delta a^p +
\frac{1}{p} \delta^{-(p-1)}b^p$, $a^{p-2}b \leq \frac{p-2}{p} \delta
a^p + \frac{2}{p}\delta^{-\frac{p-2}{2}}b^{\frac p2}$, $\delta>0$,
and the boundedness of $H^1$-norm in (<ref>), we
derive that $\bbp$-a.s.
\begin{align} \label{esti-K-15}
&\sum\limits_{j=1}^4 \bbe \sup\limits_{s\leq t} |K_j(s)|
\nonumber \\
\leq& C_p T \delta \bbe \sup\limits_{s\leq t} \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^p \nonumber \\
&+C(p,\delta) \bbe \sup\limits_{s\leq t} \bigg[ \int_0^s |X_\ve|^{2p}_{H^1} ds
+ \int_0^s \(\int \bigg||X_\ve|^2 L_{1/m}(|X_\ve|^2)\bigg|d\xi \)^p
dr \bigg] \nonumber \\
\leq& C(T,p,\delta)
+ C_p T \delta \bbe \sup\limits_{s\leq t} \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^p \nonumber \\
& + C(p,\delta) \bbe \sup\limits_{s\leq t} \int_0^s \(\int \bigg||X_\ve|^2 L_{1/m}(|X_\ve|^2)\bigg|d\xi \)^p
dr, \ t\in[0,T],
\end{align}
where $C_p, C(p,\delta)$ and $C(T,p,\delta)$ are independent of
$\ve$ and $m$.
Moreover, it follows from the Burkholder-Davis-Gundy inequality, the
Young inequality $a^{2p-2}b^2 \leq \frac{p-1}{p} a^{2p} +
\frac{1}{p}b^{2p}$, (<ref>) and Lemma $3.3$ in
<cit.> that
\begin{align} \label{esti-K-6}
&\bbe \sup\limits_{s\leq t} |K_5(s)| \nonumber \\
\leq & C_p \bbe \bigg[ \int_0^t \sum\limits_{j=1}^n \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^{2p-2} \bigg| \int|X_\ve|^2 Re\phi_j d\xi \bigg|^2 ds
\bigg]^{\frac 12} \nonumber \\
\leq& C_p \bbe \bigg[ \int_0^t \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^{2p}\bigg]^{\frac 12}
+C_p \bbe \bigg[ \int_0^t \(\sum\limits_{j=1}^n \bigg| \int|X_\ve|^2 Re\phi_j d\xi \bigg|^2\)^{p} ds
\bigg]^{\frac 12} \nonumber \\
\leq& C(T,p,\delta)+ C_p\delta \bbe \sup\limits_{s\leq t} \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^p
+ C(p,\delta) \int_0^t \bbe \sup\limits_{r\leq s} \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^pds.
\end{align}
\begin{align} \label{esti-K-7}
&\bbe \sup\limits_{s\leq t} |K_6(s)| \nonumber \\
\leq& C_p \bbe \bigg[ \int_0^t \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^{2p}\bigg]^{\frac 12} \nonumber \\
& + C_p \bbe \bigg[ \int_0^t \(\sum\limits_{j=1}^n\bigg| \int |X_\ve|^2 L_{1/m}(|X_\ve|^2) Re \phi_j d\xi \bigg|^2\)^{p} ds \bigg]^{\frac
12} \nonumber \\
\leq& C_p \delta \bbe \sup\limits_{s\leq t} \bigg[ \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^{p} + \(\int \bigg||X_\ve|^2 L_{1/m}(|X_\ve|^2) \bigg|d\xi \)^p
\bigg] \nonumber \\
&+ C(p,\delta) \int_0^t \bbe \sup\limits_{r\leq s} \bigg[ \bigg|\int F_m(|X_\ve|^2) d\xi
\bigg|^{p} + \(\int \bigg||X_\ve|^2 L_{1/m}(|X_\ve|^2) \bigg|d\xi \)^p
\bigg] ds
\end{align}
Thus, it follows from (<ref>)-(<ref>) that
\begin{align} \label{Fm-delta}
&\bbe \sup\limits_{s\leq t} \bigg| \int F_m(|X_\ve(s)|^2) d\xi
\bigg|^p \nonumber \\
\leq& C(T ,p,\delta)
+ C(T,p)\delta \bbe \sup\limits_{s\leq t} \bigg[ \bigg| \int F_m(|X_\ve(s)|^2) d\xi
\bigg|^p
+ \(\int \bigg||X_{\ve}|^2 L_{1/m}(|X_{\ve}|^2) \bigg|d
\xi\)^p \bigg] \nonumber \\
&+ C(T,p,\delta) \int_0^t \bbe \sup\limits_{r\leq s}
\bigg[\bigg| \int F_m(|X_\ve|^2) d\xi \bigg|^p
+ \(
\int
\bigg| |X_{\ve}|^2 L_{1/m}(|X_\ve|^2) \bigg| d\xi
\)^p \bigg] ds.
\end{align}
Since by (<ref>),
\begin{align} \label{Fm-Lm}
\bigg|\ \big| \int F_m(|X_\ve|^2) d\xi \big|
- \big| \int |X_{\ve}|^2 L_{1/m}(|X_\ve|^2) d\xi \big|\ \bigg|
\leq 2 |X_\ve|_2^2,
\end{align}
using (<ref>) we obtain
\begin{align} \label{esti-lm}
&\bbe \sup\limits_{s\leq t} \bigg| \int |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\bigg|^p \nonumber \\
\leq& C(T,p, \delta) + C(T,p)\delta \bbe \sup\limits_{s\leq t}\( \int \bigg||X_{\ve}|^2 L_{1/m}(|X_{\ve}|^2) \bigg| d
\xi\)^p \nonumber \\
&+ C(T,p,\delta) \int_0^t \bbe \sup\limits_{r\leq s} \(
\int
\bigg| |X_{\ve}|^2 L_{1/m}(|X_\ve|^2) \bigg| d\xi
\)^p ds.
\end{align}
Note that $u^2 L_{1/m}(u^2) \leq u^2 \log u^2 \leq C_\delta(u^2 +
u^{2+\delta}) $ for $u>1$. The Sobolev imbedding theorem implies
that for $0<\delta<\frac{4}{d-2}$,
\begin{align} \label{log-x>1}
\int I_{\{|X_\ve| > 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\leq& C_\delta(|X_\ve|_{H^1}^{2} + |X_\ve|_{H^1}^{2+\delta}).
\end{align}
\begin{align*}
&\bigg| \int I_{\{|X_\ve|\leq 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\bigg|^p \\
=& \bigg| \int|X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
- \int I_{\{|X_\ve| > 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\bigg|^p \\
\leq& C_p \bigg| \int|X_\ve|^2 L_{1/m}(|X_\ve|^{2}) d\xi \bigg|^p
+ C(p,\delta) (|X_\ve|_{H^1}^{2p} +
\end{align*}
\begin{align*}
&\( \int \bigg| |X_\ve|^2 L_{1/m}(|X_\ve|^2)\bigg| d\xi \)^p \\
=& \(-\int I_{\{|X_\ve|\leq 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
+ \int I_{\{|X_\ve| > 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\)^p \\
\leq& C_p \bigg| \int I_{\{|X_\ve|\leq 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi \bigg|^p
+ C(p,\delta) (|X_\ve|_{H^1}^{2p} +
\end{align*}
Therefore, inserting the two estimates above into (<ref>)
and then using (<ref>) we get
\begin{align*}
&\bbe \sup \limits_{s\leq t} \bigg| \int I_{\{|X_\ve|\leq 1\}}|X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\bigg|^p \\
\leq& C(T,p,\delta)
+ C(T,p) \delta \bbe \sup \limits_{s\leq t} \bigg| \int I_{\{|X_\ve|\leq 1\}}|X_\ve|^2 L_{1/m}(|X_\ve|^2) d\xi
\bigg|^p \\
& + C(T,p,\delta) \int_0^t \bbe \sup \limits_{r\leq s} \bigg|\int I_{\{|X_\ve|\leq 1\}} |X_\ve|^2 L_{1/m}(|X_\ve|^2)
d\xi \bigg|^p ds.
\end{align*}
Then, taking $\delta$ sufficiently small and applying Gronwall's
inequality we have
\begin{align*}
\bbe \sup\limits_{t\leq T} \bigg| \int I_{\{|X_\ve|\leq 1\}} |X_{\ve}(t)|^2 L_{1/m}(|X_{\ve}(t)|^2) d\xi
\bigg|^p
\leq C(T,p),
\end{align*}
where $C(T,p)$ is independent of $\ve$ and $m$. Hence, by Fatou's
\begin{align} \label{esti-log-x1}
\bbe \sup\limits_{t\leq T} \bigg| \int I_{\{|X_\ve|\leq 1\}} |X_{\ve}(t)|^2 L(|X_{\ve}(t)|^2) d\xi
\bigg|^p
\leq C(T,p),
\end{align}
Consequently, (<ref>) follows immediately from
(<ref>), (<ref>) and (<ref>).
Hence, the proof is
complete. $\square$
Proof of Proposition <ref>. By Proposition
<ref>, there exists a unique $(\mathscr{F}_t)$-adapted
solution $y_\ve$ to (<ref>), and $y_\ve \in C([0,T];
H^1)$, $\bbp$-a.s. Moreover, since $u \mapsto u L_\ve(e^W u)$ is
Lipschitz on $L^2$ and $L^2 \subset U'$, we obtain $\calg_\ve
(y_\ve) \in C([0,T];U')$, $\bbp$-a.s.
It remains to prove (<ref>) and (<ref>). For the
proof of (<ref>), in view of (<ref>), we
only need to prove that for any $p\geq 2$,
\begin{align} \label{esti-v-p}
\bbe \sup\limits_{0\leq t\leq T} \|X_\ve(t)\|^p_V
\leq C(T,p)<\9,
\end{align}
where $X_\ve := e^W y_\ve$, and $V$ is the Orlicz space defined in
To this end, set $B(u):= -u^2 \log u^2 - N(u)$, where $u>0$, and $N$
is as defined in (<ref>). it follows from $(2.6)$ in
<cit.> and the inequality $ab \leq a^2 + b^2$ that
\begin{align*}
\bigg| \int B(|X_\ve(s)|) d\xi \bigg|
\leq C |X_\ve|_2 |X_\ve|_{H^1}^{\frac{d}{d-2}}
\leq C (|X_\ve|^2_2 + |X_\ve|_{H^1}^{\frac{2d}{d-2}}).
\end{align*}
\begin{align*}
\int N(|X_\ve|) d\xi
\leq& \bigg| \int |X_\ve|^2 \log|X_\ve|^2 d\xi \bigg|
+ C (|X_\ve|^2_2 + |X_\ve|_{H^1}^{\frac{2d}{d-2}}).
\end{align*}
Hence by (<ref>) and Lemma <ref>,
\begin{align*}
\bbe \sup\limits_{0\leq t\leq T}
\bigg| \int N(|X_\ve|)d\xi \bigg|^p \leq C(T,p) <\9,
\end{align*}
which along with (<ref>) implies (<ref>), thereby
proving (<ref>).
As regards (<ref>), we note that
\begin{align*}
&\| e^W \calg_\ve(y_\ve)\|_{L^{p}(\Omega;C([0,T]; U'))} \nonumber \\
\leq& 2|\lbb| \|X_\ve\|_{L^{p}(\Omega;C([0,T]; L^2))}
+ 2|\lbb| \| I_{\{|X_\ve| > e^{-3}\}} X_\ve L_\ve(X_\ve)\|_{L^{p}(\Omega;C([0,T]; L^2))} \nonumber\\
&+2|\lbb| \| I_{\{|X_\ve|\leq e^{-3}\}} X_\ve L_\ve(X_\ve) \|_{L^{p}(\Omega;C([0,T]; V'))}
\end{align*}
Since for $\xi\in\{|X_\ve| > e^{-3}\}$,
\begin{align} \label{xlve-leq1}
|X_\ve(\xi) L_\ve(X_\ve (\xi)) |
\leq |X_\ve(\xi)L(X_\ve(\xi)) |
\leq C_\delta (|X_\ve(\xi)| + |X_\ve(\xi)|^{1+\delta}),
\end{align}
by the Sobolev imbedding theorem and (<ref>), it follows
that for $0\leq \delta\leq \frac{2}{d-2}$,
\begin{align*}
&\|X_\ve\|_{L^{p}(\Omega;C([0,T]; L^2))}
+ \| I_{\{|X_\ve|> e^{-3}\}} X_\ve L_\ve(X_\ve)\|_{L^{p}(\Omega;C([0,T];
L^2))} \\
\leq& C( \|X_\ve\|_{L^{p}(\Omega;C([0,T]; L^2))}
+ \|X_\ve\|^{1+\delta}_{L^{(1+\delta)p}(\Omega;C([0,T];
L^{2(1+\delta)}))}) \\
\leq& C( \|X_\ve\|_{L^{p}(\Omega;C([0,T]; L^2))}
+ \|X_\ve\|^{1+\delta}_{L^{(1+\delta)p}(\Omega;C([0,T];
H^1))}) \leq C(T,p)<\9,
\end{align*}
where $C(T,p)$ is independent of $\ve$.
Moreover, since $\wt{N}$ is increasing, by Lemma <ref>
$(i)$, similarly to (<ref>) we have
\begin{align*}
\int \wt{N} (-2I_{\{|X_\ve|\leq e^{-3} \}} X_\ve L_\ve(X_\ve) ) d\xi
\leq & \int I_{\{|X_\ve|\leq e^{-3} \}} \wt{N} (-X_\ve
L(|X_\ve|^2)) d\xi \\
\leq& 2 \max\{\| X_\ve\|_V, \| X_\ve\|^2_V \}.
\end{align*}
Then, as in (<ref>),
\begin{align*}
&\| I_{\{|X_\ve|\leq e^{-3}\}} X_\ve L_\ve(X_\ve) \|_{L^{p}(\Omega;C([0,T];
V'))} \nonumber \\
\leq& C_T(\|X_\ve\|^2_{L^{2p}(\Omega;C([0,T]; V))} + 1)
\leq C(T,p) < \9,
\end{align*}
where the last step is due to (<ref>), thereby proving
(<ref>). The proof of Proposition <ref> is now
complete. $\square$
§ PROOF OF THEOREM <REF>
Let us start with the lemma below.
Let $L_{\ve}$ be defined as in (<ref>) and $p \geq 3$. For any
$X\in L^p(\Omega \times(0,T); U)$,
\begin{align} \label{app-gev-g}
\| XL_\ve(X) - X L(X) \|_{L^{p'}(\Omega \times (0,T); U')} \to 0,\ \ as\ \ve \to
\end{align}
Proof. First note that $X L_\ve(X) \to XL(X)$ a.e., as
$\ve \to 0$, and
\begin{align} \label{u'-split}
&\| X L_\ve(X) - XL(X)
\|_{L^{p'}(\Omega \times (0,T); U')} \nonumber \\
\leq& \| I_{\{|X|\leq e^{-3}\}} (X L_\ve(X) - XL(X))\|_{L^{p'}(\Omega \times (0,T); V')} \nonumber \\
&+ \| I_{\{|X| > e^{-3}\}} (X L_\ve(X) - XL(X))
\|_{L^{p'}(\Omega \times (0,T); H^{-1})}.
\end{align}
Since $|X L_\ve (X)| \leq |X L(X)|$, and as in the proof of
(<ref>), the Sobolev imbedding theorem and the
Hölder inequality imply that for $0<\delta\leq
\min\{\frac{2}{d-2} , p-2\}$,
\begin{align} \label{esti-w'-2}
& \| I_{\{ |X|> e^{-3}\}} X L(X)
\|_{L^{p'}(\Omega \times (0,T);
H^{-1})} \nonumber \\
\leq& C (\|X\|_{L^{p'}(\Omega \times (0,T);
+ \|X\|^{1+\delta}_{L^{(1+\delta)p'}(\Omega \times(0,T);
L^{2(1+\delta)})}) \nonumber \\
\leq& C(T,p) (\|X\|_{L^p(\Omega\times (0,T);H^1)} + \|X\|^{1+\delta}_{L^p(\Omega\times (0,T);H^1)}
\end{align}
the dominated convergence theorem implies that, as $\ve \to 0$,
\begin{align} \label{lve-l-h-1}
&\| I_{\{|X| >e^{-3}\}} (X L_\ve(X) - X L(X))
\|_{L^{p'}(\Omega\times(0,T); H^{-1})} \nonumber \\
\leq& \| I_{\{|X| > e^{-3}\}} (X L_\ve(X) - X L(X))
\|_{L^{p'}(\Omega\times(0,T); L^2)} \to 0.
\end{align}
For the last term in the right hand side of (<ref>), note
that since $\wt{N}$ is increasing, by Lemma <ref> $(i)$ and
\begin{align} \label{nxlve}
\wt{N}(-2|X(\xi)|L_\ve(X(\xi)) )
\leq \wt{N} (-|X(\xi)|L(|X(\xi)|^2)) \leq 2 N(|X(\xi)|).
\end{align}
Moreover, by (<ref>) and Hölder's inequality
\begin{align*}
\| \int N(|X(\xi)|) d\xi \|_{L^{p'}(\Omega\times (0,T))}
\leq& C_T (\|X\|^2_{L^p(\Omega\times(0,T);V)} +1)<\9,
\end{align*}
which implies that $N(|X|) \in L^1(\bbr^d)$, $\bbp \otimes dt$-a.e.
Then, it follows from the dominated convergence theorem that $\bbp
\otimes dt$-a.e.
\begin{align*}
\int \wt{N}( -2X L_\ve(X) I_{\{|X|\leq e^{-3}\}}) d\xi
\to \int \wt{N}(-2X L (X)I_{\{|X|\leq e^{-3}\}} )d\xi,
\end{align*}
which yields by <cit.> that
\begin{align*}
-X L_\ve(X) I_{\{|X|\leq e^{-3}\}}
\to -X L (X)I_{\{|X|\leq e^{-3}\}},\ \ in\ V'.
\end{align*}
Since by Lemma <ref> $(i)$, $\|-XL_\ve(X)I_{\{|X|\leq
e^{-3}\}}\|_{V'} \leq \|-XL(X)I_{\{|X|\leq e^{-3}\}}\|_{V'} $, and
as in (<ref>), we have
\|-X L (X)I_{\{|X|\leq
e^{-3}\}}\|_{V'} \ \in L^{p'}(\Omega \times (0,T))
$. Again, we apply the dominated convergence theorem and get
\begin{align} \label{lve-l-v'}
\| I_{\{|X|\leq e^{-3}\}} (X L_\ve(X) - XL(X))\|_{L^{p'}(\Omega \times (0,T);
V')} \to 0,\ \ \ve\to 0.
\end{align}
Consequently, (<ref>) follows from (<ref>),
(<ref>) and (<ref>). The proof of Lemma
<ref> is thus complete.
Proof of Theorem <ref>. For any $p\geq 2$, by the
uniform estimates (<ref>) and (<ref>), we have
along a subsequence $\{\ve_n\} \to 0$,
\begin{align}
&e^Wy_{\ve_n} \overset{\omega^*}{\rightharpoonup} e^W \wt{y},\ in\ L^p(\Omega; L^\9(0,T;U)),
\label{app-weak.2-C} \\
& e^W\calg_{\ve_n}(y_{\ve_n}) \overset{\omega^*}{\rightharpoonup} e^W\eta,\ in\ L^p(\Omega; L^\9(0,T;U')),
\label{app-weak.1-C}
\end{align}
where $\overset{\omega^*}{\rightharpoonup}$ stands for weak-star
In particular, $e^W\wt{y} \in L^\9(0,T; U)$, and $e^W\eta \in
L^\9(0,T; U')$, $\bbp$-a.s. Since by Hypothesis $(H)$, for any $u\in
U$, we have $\|e^{-W}u \|_{U} \leq c(t) \|u\|_{U}$, where $c(t)=
\sqrt{2}(|e^{-W(t)}|_{L^\9} + |\na e^{-W(t)}|_{L^\9})$. It follows
\begin{align*}
\wt{y} \in L^\9(0,T; U),\ \ \ \eta \in L^\9(0,T; U'),\ \ \bbp-a.s.
\end{align*}
Moreover, for any $p\geq 3$, since $L^p(\Omega; L^\9(0,T;U)) \subset
L^p(\Omega \times (0,T);U))$ and $L^p(\Omega; L^\9(0,T;U')) \subset
L^{p'}(\Omega \times (0,T);U')$, we have (selecting a further
subsequence if necessary)
\begin{align}
&y_{\ve_n} \overset{\omega}{\rightharpoonup} \wt{y},\ in\ \mathcal{U},
\label{app-weak.2} \\
& \calg_{\ve_n}(y_{\ve_n}) \overset{\omega}{\rightharpoonup} \eta,\ in\ \mathcal{U}'.
\label{app-weak.1}
\end{align}
where $\overset{\omega}{\rightharpoonup}$ means weak convergence.
We next take the limit in the approximating equation
(<ref>). Set
$$F_n(y_{\ve_n})(t):= i e^{-W(t)} \Delta (e^{W(t)}y_{\ve_n}(t)) +
(4i\lbb |\lbb|t + \wh{\mu})y_{\ve_n}(t) +
\calg_{\ve_n}(y_{\ve_n}(t)),$$
$$F(\wt{y})(t):=i e^{-W(t)} \Delta (e^{W(t)}\wt{y}(t)) + (4i\lbb |\lbb|t +
\wh{\mu})\wt{y}(t) + \eta(t),$$
where $t\in [0,T]$. Then, from
(<ref>) and (<ref>) it follows that
$F_{n}(y_{\ve_n}) \overset{\omega}{\rightharpoonup} F(\wt{y})$, in
$\calu'$. Thus, for any $u\in U$, $\vf \in L^\9([0,T]\times
\Omega)$, by (<ref>)
\begin{align*}
&\bbe \int_0^T\ {}_{U'}\<e^{W(t)}\wt{y}(t), \vf(t)u\>_{U} dt \\
=& \lim\limits_{n \to \9} \bbe \int_0^T\ {}_{U'}\<e^{W(t)}\wt{y}_{\ve_n}(t), \vf(t)u\>_{U}
=& \bbe \int_0^T\ \<e^{W(t)}x, \vf(t)u\>_H dt
- \lim\limits_{n\to \9} \bbe \int_0^T \int_0^t\ {}_{U'}\<F_n(y_{\ve_n})(s),
\ol{e^{W(t)}}\vf(t)u\>_{U} ds dt.
\end{align*}
Note that, the second term in the right hand side above is equal to
\begin{align*}
&\lim\limits_{n\to \9} \bbe \int_0^T \
\int_s^T\ol{e^{W(t)-W(s)}}\vf(t)u dt\>_{U} ds \\
=& \bbe \int_0^T \ {}_{U'}\<e^{W(s)}F(\wt{y})(s),
\int_s^T\ol{e^{W(t)-W(s)}}\vf(t)u dt\>_{U} ds \\
=& \bbe \int_0^T {}_{U'}\<e^{W(t)}\int_0^t F(\wt{y})(s)ds,
\vf(t)u\>_U dt.
\end{align*}
It follows that for any $u\in U$, $\vf \in L^\9([0,T]\times
\Omega)$,
\begin{align*}
\bbe \int_0^T\ {}_{U'}\<e^{W(t)}\wt{y}(t), \vf(t)u\>_{U} dt
=\bbe \int_0^T\ {}_{U'}\<e^{W(t)}\(x-\int_0^t F(\wt{y})(s)ds\), \vf(t)u\>_{U}
\end{align*}
Thus, $\wt{y} = x- \int_0^{\cdot} F(\wt{y})(s) ds,$ in $\calu'$.
Set $y(t):= x -\int_0^t F(\wt{y})(s)ds$, $t\in[0,T]$. Then, $y\in
AC([0,T]; U')$, $\bbp$-a.s., and $y=\wt{y}$ in $\calu'$, which
implies that $y = \wt{y}$, $\bbp \otimes dt$-a.e. Since for each
$t\in [0,T]$, $\int_0^t F(y)(s)ds = \int_0^t F(\wt{y})(s)ds$,
$\bbp$-a.s., by the continuity of $t \mapsto \int_0^t F(y)(s) -
F(\wt{y})(s) ds$, we thus have that $\bbp$-a.s. for all $t\in
[0,T]$, $\int_0^t F(y)(s)ds = \int_0^t F(\wt{y})(s)ds$, which yields
that $y (t) = \wt{y}(t)$, for all $t\in[0,T]$, $\bbp$-a.s.
\begin{align} \label{equa-y-eta}
y(t)=x - \int_0^t (i e^{-W(s)}\Delta(e^{W(s)}y(s)) + &(4i \lbb|\lbb| s + \wh{\mu})y(s) +
\eta(s))ds,\nonumber\\
&\qquad for\ all\ t\in [0,T],\ \bbp-a.s.
\end{align}
Moreover, taking into account (<ref>),
(<ref>) and (<ref>), we have $y\in
W^{1,p'}(0,T; U') \cap L^p([0,T]; U)$, $\bbp$-a.s., which implies
that $y\in C([0,T]; H)$, $\bbp$-a.s.
In order to prove that $y$ is a solution to (<ref>), we need
to show that
\begin{align} \label{eta-gy}
\eta = \calg(y).
\end{align}
For this purpose, it suffices to prove that
\begin{align} \label{mono-lim}
\limsup\limits_{n\to \9} \int_0^T Re\ {}_{\mathcal{U}_t'} \<\calg_{\ve_n} (y_{\ve_n}),
y_{\ve_n} \>_{\mathcal{U}_t} dt
\leq \int_0^T Re\ {}_{\mathcal{U}_t'} \<\eta,
y\>_{\mathcal{U}_t} dt,
\end{align}
where $\calu_t$ and $\calu'_t$ are defined as in (<ref>)
and (<ref>) respectively, but with $T$ replaced by $t$.
Indeed, by the monotonicity of $\calg_{\ve_n}$, for any positive
function $\varphi \in C([0,T])$,
\begin{align*}
\int_0^T Re\ {}_{\calu'_t} \< \calg_{\ve_n}(y_{\ve_n})-\calg_{\ve_n}(u),
\>_{\calu_t} \vf(t) dt \geq 0,\ \ u\in \calu.
\end{align*}
Then, it follows from (<ref>), (<ref>) and
(<ref>) that
\begin{align*}
\limsup\limits_{n\to\9}
Re\ {}_{\calu'_t} \< \calg_{\ve_n}(y_{\ve_n})-\calg_{\ve_n}(u), y_{\ve_n}-u
\>_{\calu_t}
\leq Re\ {}_{\calu'_t} \< \eta-\calg(u), y-u
\>_{\calu_t}.
\end{align*}
\begin{align*}
&| Re\ {}_{\calu'_t} \< \calg_{\ve_n}(y_{\ve_n})-\calg_{\ve_n}(u), y_{\ve_n}-u
\>_{\calu_t}| \nonumber \\
\leq& \sup\limits_{n\geq1} (\|\calg_{\ve_n}(y_{\ve_n})\|_{\calu'} + \|\calg_{\ve_n}(y)\|_{\calu'})
(\|y_{\ve_n}\|_{\calu'} + \|y\|_{\calu'}) < \9.
\end{align*}
Hence, by Fatou's lemma,
\begin{align*}
0 \leq& \limsup\limits_{n\to\9} \int_0^T Re\ {}_{\calu'_t} \< \calg_{\ve_n}(y_{\ve_n})-\calg_{\ve_n}(u),
\>_{\calu_t} \vf(t) dt \\
\leq& \int_0^T \limsup\limits_{n\to\9} Re\ {}_{\calu'_t} \< \calg_{\ve_n}(y_{\ve_n})-\calg_{\ve_n}(u),
\>_{\calu_t} \vf(t) dt \\
=& \int_0^T Re\ {}_{\calu'_t} \< \eta-\calg(u), y-u
\>_{\calu_t} \vf(t) dt.
\end{align*}
As the integrand is continuous in $t$, and $\vf$ is an arbitrary
positive continuous function, we deduce that,
\begin{align*}
Re\ {}_{\calu'} \< \eta-\calg(u), y-u
\>_{\calu} \geq 0,
\end{align*}
which implies (<ref>) by the maximal monotonicity of $\calg$.
For the proof of (<ref>), we note that by
(<ref>) we have, via Itô's formula,
\begin{align} \label{mono-lim*}
\int_0^T \bbe |e^{W(t)}y_{\ve_n}(t)|_{2}^2 dt
=&|x|_{2}^2 T - 4|\lbb| \int_0^T \bbe \int_0^t |y_{\ve_n}(s)|_2^2 ds dt
\nonumber \\
=& |x|_{2}^2 T - 2 \int_0^T Re\ {}_{\calu'_t}\<
\calg_{\ve_n}(y_{\ve_n}),y_{\ve_n}\>_{\calu_t} dt.
\end{align}
Moreover, as in the proof of <cit.>, applying
Itô's formula to (<ref>) we derive
\begin{align} \label{app.2}
\int_0^T \bbe |e^{W(t)}y(t)|^2_{2} dt
=&|x|^2_{2} T
-2 \int_0^T Re\ {}_{\calu_t'}\<\eta,y\>_{\calu_t} dt.
\end{align}
Thus, by (<ref>), (<ref>) and (<ref>) we
derive that
\begin{align*}
\int_0^T Re\ {}_{\calu_t'}\<\eta, y\>_{\calu_t} dt
=&-\frac 12 \int_0^T \bbe |e^{W(t)} y(t)|_2^2 dt + \frac 12 |x|_2^2 T \\
\geq& \limsup\limits_{n\to\9} \(-\frac 12 \int_0^T \bbe |e^{W(t)}y_{\ve_n}(t)|_2^2 dt + \frac 12
|x|_2^2 T \) \\
=& \limsup\limits_{n\to\9} \int_0^T Re\ {}_{\calu_t'} \<\calg_{\ve_n}(y_{\ve_n}), y_{\ve_n}\>_{\calu_t} dt ,
\end{align*}
which yields (<ref>) as claimed, thereby proving
Therefore, $y$ is a solution to (<ref>) in the sense of
Definition <ref>. Moreover, the estimates
(<ref>)-(<ref>) follow immediately from
(<ref>), (<ref>) and (<ref>).
It is left to prove the uniqueness, which follows from the
monotonicity. In fact, given any two solutions $y_1,y_2$ to
(<ref>), setting $X_i = e^W y_i$, $i=1,2$, by the Ito
formula, we obtain similar formula as in (<ref>) but
with $\ve=0$. Thus, it follows from (<ref>) with $\ve=0$
and similar arguments as those below (<ref>) that
$X_1(t)= X_2(t)$, $\forall t \in[0,T]$, $\bbp$-a.s. The proof of
Theorem <ref> is, therefore, complete. $\square$
§ APPENDIX
Proof of Lemma <ref>. $(i)$. First note that, for
each $0<\ve <1$ fixed,
\begin{align*}
\frac{d}{du} L_\ve(u) = \frac{1-\ve^2}{(\ve + u)(1+ \ve u)} \geq
0,\ u>0,
\end{align*}
which implies that $L_\ve(u)$ is increasing with $u$, and so
$|L_\ve(u)| \leq |\log \ve|$.
Similarly, for each $u>0$ fixed,
\begin{align*}
\frac{d}{d\ve} L_\ve(u) = \frac{1- u^2}{(\ve + u)(1+ \ve u)},\ u>0,
\end{align*}
which yields that $ \ve \mapsto L_\ve(u)$ is increasing with $\ve$
if $u\in[0,1]$, but decreasing if $u\in [1,\9)$. Hence, for $u\in
[0,1]$, we have $\log u \leq L_\ve(u) \leq 0$, and for $u\in
[1,\9)$, $0\leq L_\ve(u) \leq \log u$. Therefore, we obtain for all
$u>0$, $|uL_\ve(u)| \leq |u L(u)|$.
$(ii)$. We may assume $0<|u_2|\leq |u_1|$ without loss of
generality. Note that
\begin{align*}
u_1 L_{\ve}(u_1) - u_2 L_{\ve}(u_2)
= u_2
\(L_{\ve}(u_1) -L_{\ve}(u_2) \)
\end{align*}
Since $|L_{\ve}(u_1)|\leq |\log \ve|$, and
\begin{align} \label{Lve-diff-0}
\left|L_{\ve}(u_1) - L_{\ve}(u_2) \right|
\leq& \frac{1+\ve|u_2|}{|u_2|+\ve}\ \left|\frac{|u_1|+\ve}{1+\ve|u_1|} -
\frac{|u_2|+\ve}{1+\ve|u_2|} \right| \nonumber \\
=& \left|\frac{(1-\ve^2)(|u_1|-|u_2|)}{(|u_2|+\ve)(1+\ve |u_1|)} \right| \nonumber \\
\leq& (1-\ve^2) |u_2|^{-1}|u_1 - u_2|,
\end{align}
we obtain immediately (<ref>).
$(iii)$. We assume $0<|u_2|\leq |u_1|$ without loss of generality.
Note that
\begin{align*}
Im (\ov{u_1} - \ov{u_2}) (u_1 L_\ve(u_1) - u_2 L_\ve(u_2))
= (Im (\ov{u_1} u_2)) (L_\ve(u_1) - L_\ve(u_2)),
\end{align*}
\begin{align*}
|Im(\ov{u_1} u_2)|
= \big|\frac{u_2(\ov{u_1}-\ov{u_2}) + \ov{u_2}(u_2-u_1)}{2i} \big|
\leq |u_2||u_1 - u_2|.
\end{align*}
Thus, taking into account (<ref>) we obtain
(<ref>). $\square$
Proof of Lemma <ref>. This lemma follows
essentially from <cit.>. Using the notations in <cit.>,
we reformulate (<ref>) in form
\begin{align*}
(\partial_t + i\D + \sum\limits_{j=1}^d b^jD_j +
c) y = f,
\end{align*}
meant in the weak sense, where $D_j = - i\partial_j$, $b^j =
-2\partial_j W$, and $c= i \sum\limits_{j=1}^d (\partial_j W)^2 +
i\D W + 2|\lbb| + 4i\lbb|\lbb|t + \wh{\mu} $.
Since for each $1\leq m \leq n$, $e_m \in C_b^\9$ and
$\beta(\cdot)$ is continuous, $\bbp$-a.s., we have $b^j, c \in
C_{\omega}([0,T];\calb^\9)$, where $\calb^\9$ is as defined in
Hypothesis $(H)$, and $C_{\omega}([0,T];\calb^\9) =\{g\in C([0,T];
C^\9), \{g(t,\cdot)\}_{0\leq t\leq T}\ is\ uniformly\ bounded\ in\
\calb^\9. \}$
Moreover, under Hypothesis $(H)$,
\begin{align*}
|Re\ b(t,\xi)| \leq \(2\sum\limits_{m=1}^n |\mu_m|\sup\limits_{t\leq T}
|\beta_m(t)|\) \lbb(|\xi|).
\end{align*}
Hence, the conditions in <cit.> are verified, and
we obtain the existence and uniqueness of the evolution operators
Furthermore, as remarked by the author in <cit.>, the results in
<cit.> holds also for the time-dependent coefficients. Thus,
similarly to <cit.>, we have the estimates
(<ref>) and (<ref>).
Finally, the measurabilities of the processes $U(\cdot, s)x$ and
$C_t$, $t\geq 0$, can be proved similarly as in the proof of Lemma
$3$ and Lemma $4$ in <cit.> (see also <cit.>). The proof is now
complete. $\square$
Proof of (<ref>). Since the nonlinearity $X_\ve
L_\ve(X_\ve) \in L^2 \subset H^{-1}$, we can use similar arguments
as in the proof of <cit.> to
derive that $X_\ve := e^W y_\ve$ satisfies $\bbp$-a.s. for all
\begin{align}
X_\ve(t) =& x -i \int_0^t \Delta X_\ve ds - 2\lbb i \int_0^tX_\ve L_\ve(X_\ve) ds \nonumber \\
&- \int_0^t(4i \lbb |\lbb|s + 2|\lbb| +\mu) X_\ve ds + \sum\limits_{j=1}^n \int_0^t X_\ve \phi_j
d\beta_j(s), \label{equa-x-le}
\end{align}
where the equation is taken in $H^{-1}$.
Proceeding as in <cit.> and <cit.>, we set
$h_\delta = h \ast \psi_\delta$ for any locally integrable function
$h$ mollified by $\psi_\delta$, where $\psi_\delta = \delta^{-d}
\psi(\frac{x}{\delta})$ and $\psi$ is a real-valued, nonnegative,
compactly supported smooth function with unit integral.
Taking convolution of both sides of (<ref>) with the
mollifiers $\psi_\delta$, we have for each $\xi \in \bbr^d$ that
\begin{align}
(X_\ve(t))_\delta(\xi) =& -i \int_0^t \Delta X_{\ve,\delta}(\xi) ds - 2\lbb i \int_0^t (X_\ve L_\ve(X_\ve))_\delta(\xi) ds \nonumber \\
&- \int_0^t (4i \lbb |\lbb|s + 2|\lbb|) X_{\ve,\delta}(\xi) ds
- \int_0^t (\mu X_\ve)_\delta (\xi) ds \nonumber \\
&+ \sum\limits_{j=1}^n\int_0^t (X_\ve \phi_j)_\delta(\xi)
d\beta_j (s),\ \ t\in[0,T], \label{equa-x-le-delta}
\end{align}
where $X_{\ve, \delta}= (X_\ve)_\delta$, and (<ref>)
holds on a set $\Omega_\xi\in \mathscr{F}$ with
Since for any locally integrable function $h$, $h_\delta(\xi)$ is
continuous in $\xi$, using the boundedness of the $H^1$-norm in
(<ref>) and similar arguments as in the proof of
<cit.> and <cit.>, we can prove
the continuity in $\xi$ of all terms in (<ref>).
Thus, (<ref>) holds on a full probability set
$\wt{\Omega}\in \mathscr{F}$, which is independent of $\xi\in
\bbr^d$. For simplicity, below we omit the argument $\xi$ in
Now, applying Itô's formula to the real valued function
$F_m(|X_{\ve,\delta}|^2)$, then integrating over $\bbr^d$,
interchanging the integrals and integrating by parts, we obtain
\begin{align*}
&\int F_m(X_{\ve, \delta}(t)) d\xi \\
=& \int F_m(x) d\xi
- 2\int_0^t \int g_m(|X_{\ve, \delta}|^2)
Re (\ov{X_{\ve,\delta}} \na X_{\ve,\delta})
Im (\ov{X_{\ve,\delta}} \na X_{\ve,\delta}) d\xi ds\\
&+ 4\lbb Im \int_0^t \int ( L_{1/m}(|X_{\ve,\delta}|^2) +1)
\ov{X_{\ve,\delta}} (X_\ve L_\ve(X_\ve))_\delta d\xi ds\\
& - 4|\lbb| \int_0^t \int
(L_{1/m}(|X_{\ve,\delta}|^2)+1)|X_{\ve,\delta}|^2 d\xi ds\\
& -2 \int_0^t \int (L_{1/m}(|X_{\ve,\delta}|^2)+1) Re (\ov{X_{\ve,\delta}} (\mu
X_\ve)_\delta) d\xi ds\\
& + \sum\limits_{j=1}^n \int_0^t \int (L_{1/m}(|X_{\ve,\delta}|^2)+1) |(X_\ve
\phi_j)_\delta|^2 d\xi ds\\
& + \sum\limits_{j=1}^n \int_0^t \int g_m(|X_{\ve, \delta}|^2) (Re (\ov{X_{\ve,
\delta}}(X_{\ve}\phi_j)_\delta))^2 d\xi ds\\
& + 2 \sum\limits_{j=1}^n \int_0^t \int (L_{1/m}(|X_{\ve,\delta}|^2)+1) Re
(\ov{X_{\ve,\delta}} (X_\ve \phi_j)_\delta) d\xi d\beta_j(s),
\end{align*}
where $ g_m(|X_{\ve, \delta}|^2) := 2(1-\frac{1}{m^2})(\frac{1}{m} +
|X_{\ve, \delta}|^2)^{-1} (1+ \frac{1}{m} |X_{\ve,
\delta}|^2)^{-1}$. (Note that, since $|L_{1/m}(|X_{\ve,\delta}|^2)|
\leq \log m$, we can use the (stochastic) Fubini theorem to
interchange the integrals.)
Therefore, since $|L_{1/m}(|X_{\ve,\delta}|^2)| \leq \log m$,
$|g_m(|X_{\ve,\delta}|^2)| \leq 2 |X_{\ve,\delta}|^{-2}$, and
$h_\delta \to h$ in $L^q$, for any $h\in L^q$, $q>1$, using the
boundedness of the $H^1$-norm in (<ref>) and the
generalized dominated convergence theorem, we can take the limit
$\delta \to 0$ above and consequently obtain (<ref>). $\square$
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Cosmol. 16 (2010), no. 4, 288-297.
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1511.00348
|
§ INTRODUCTION
The sign problem is known to be one of the most difficult problems in lattice gauge theory, for example, QCD with the quark chemical potential $\mu$ or including the $\theta$ term where the corresponding Boltzmann weight is complex. So far, a considerable number of research within a framework of Monte Carlo method has been devoted to overcome the sign problem, and there are limited successes, depending on dimensionality or a property of specific model. On the other hand, another possibility to avoid the sign problem is just to leave Monte Carlo method.
The tensor renormalization group (TRG) method is one of such a possibility and has no sign problem, which was originally proposed for studying two-dimensional classical systems by Levin and Nave<cit.>. This method no longer regards the Boltzmann weight as a probability of generating field configurations as in Monte Carlo method.
The TRG method consists of two main steps. The first step is to obtain the tensor network representation of the partition function of a system. In order to obtain the representation, one has to expand the Boltzmann weight using new integers and integrate out the old degrees of freedom. The new integers will become the indices of the tensor. The next step is to reduce the number of the tensors under controlling systematic errors. After the number of the coarse grained tensors decreases, it is possible to calculate the partition function by contracting all indices of the tensors. Although the original TRG was invented for two dimensional system,
the higher order TRG (HOTRG) was introduced by Xie et al.<cit.>
as an extension to higher dimensional systems.
The strong CP problem is one of the interesting topics in QCD;
why the parameter $\theta$ for CP odd operator in the QCD Lagrangian, where such a term is allowed to exist, is so small.
In order to answer the question, understanding of the non-perturbative QCD dynamics including $\theta$-term is indispensable,
but the presence of the this term causes the sign problem.
Instead of dealing with QCD directly, it is reasonable to start to investigate its toy model, CP($N-1$) model, which shares many features with QCD.
A long time ago, Schierholz suggested an interesting scenario to solve the strong CP problem in CP($N-1$) model by analyzing phase diagram in the $\beta - \theta$ plane <cit.>.
Although it is not clear that the solution can be directly applied to QCD,
it is interesting to verify the scenario with another method, namely TRG approach which is absent of the sign problem.
In this report, we apply the HOTRG to CP($N-1$) model in two dimensions, and
present the tensor network representation and numerical results.
Although including $\theta$-term in the tensor network representation is straightforward[An explicit form of tensor in the presence of $\theta$-term shall be given in a separate paper.], we present the tensor at $\theta=0$.
§ TENSOR NETWORK REPRESENTATION OF CP($N-1$) MODEL
The partition function of lattice CP($N-1$) model is given by
\begin{align}
\int \prod_{i} dz_i dz_i^\ast \prod_{<i,j>}dU_{i,j}
{\rm exp}\left\{
\beta N\sum_{i,j}
\Bigl[
z^\ast_i\cdot z_jU_{i,j}+
z^\ast_j \cdot z_i U_{i,j}^\dag
\Bigr]
\right\},
\end{align}
where $z_i$ is $N$-component complex scalar field of unit length, $|z|=1$, and $U_{i,j}$ is link variable described by auxiliary vector field $A_{i,j}$, i.e. $U_{i,j}={\rm exp}\{iA_{i,j}\}$.
In order to obtain a tensor network representation, one has to expand the Boltzmann weight with new integers, and then integrate out the old degrees of freedom (The complex fields $z_i$ and the auxiliary field $A$ in this case). In the end, one can obtain a tensor which has indices of the new integers.
To expand the Boltzmann weight with new integers, we use the characterlike expansion <cit.>,
\begin{align}
{\rm exp}\left\{
\beta N
\Bigl[
z^\ast_i\cdot z_j U_{i,j}+
z_i \cdot z^\ast_j U_{i,j}^\dag
\Bigr]
\right\}
Z_0(\beta)\sum_{l,m=0}^\infty d_{(l;m)}{\rm exp}[i(m-l)A_{i,j}]h_{(l;m)}(\beta)
\label{charactercpn}
\end{align}
where $d_{(l;m)}$ are dimensionalities of characterlike representations, $h_{(l;m)}(\beta)$ are characterlike expansion coefficients, $f_{(l;m)}(z_i,z_j)$ are characterlike expansion characters, and $Z_0(\beta)$ is the normalization factor which makes $h_{(0;0)}(\beta)=1$. The integers $l$ and $m$ will become the indices of the tensor shown below.
The characterlike expansion coefficients $h_{(l;m)}(\beta)$ are expressed by the modified Bessel functions of the first kind
\begin{align}
\end{align}
Since the modified Bessel function of the first kind, $I_n(x)$, decreases rapidly as $n$ increases with a fixed value of $x$, one can safely truncate the sum of $l$ and $m$ in eq. (<ref>) at some order (say $l_{\rm max}$).
We show some explicit form of the dimensionalities of characterlike representations $d_{(l;m)}$ and the characterlike expansion characters $f_{(l;m)}(z_i,z_j)$ with any values of $l$.
For $m=0$,
\begin{align}
\end{align}
\begin{align}
f_{(l;0)}(z_i,z_j)=\sqrt{\frac{(N-1+l)!}{l!(N-1)!}}(z_i \cdot z_j^\ast)^l.
\label{fl0}
\end{align}
For $m=1$,
\begin{align}
\end{align}
\begin{align}
f_{(l;1)}(z_i,z_j)=\sqrt{\frac{(N+l)!(N-1+l)}{l!(N-1)!(N-1)}}\Bigl[(z_i\cdot z_j^\ast)^l(z_i^\ast \cdot z_j)
-\frac{l}{N-1+l}(z_i \cdot z_j^\ast)^{l-1}\Bigr].
\label{fl1}
\end{align}
For $m=2$,
\begin{align}
\end{align}
\begin{align}
\nonumber
&\ \ \ \times \Bigl[(z_i \cdot z_j^\ast)^l(z_i^\ast \cdot z_j)^2-\frac{2l}{N+l}(z_i\cdot z_j^\ast)^{l-1}(z_i^\ast \cdot z_j)+\frac{(l-1)l}{(N-1+l)(N+l)}(z_i\cdot z_j^\ast)^{l-2}\Bigr].
\label{fl2}
\end{align}
Decomposition of characterlike expansion characters $f_{(l;m)}(z_i,z_j)$.
The term, $f_{(l;m)}(z_i,z_j)$, is expressed by the combination of two complex scalar fields, $z_i$ and $z_j$. In order to obtain a tensor network representation, one has to integrate out the complex scalar fields $z$ site by site.
For that purpose, it is convenient to rewrite it as follows,
\begin{align}
\sum_{\{a\}}
\tilde{F}^{a_1,\cdots,a_{l+m}}_{(l;m)}(z_j),
\end{align}
where $\{a\}=a_1,a_2,\cdots, a_{l+m}$, and
$a_n=1,2,\cdots, N$ for
$n=1, 2, \cdots, l+m$.
A pictorial expression of this decomposition is illustrated in Figure <ref>.
The explicit forms of $F$ and $\tilde F$ are as follows.
For $m=0$,
\begin{align}
\left(\frac{(N-1+l)!}{l!(N-1)!}\right)^\frac{1}{4}z_i^{a_1}\cdots z_i^{a_l},
\end{align}
\begin{align}
\tilde{F}^{a_1,\cdots,a_{l}}_{(l;0)}(z_i)
\left(\frac{(N-1+l)!}{l!(N-1)!}\right)^\frac{1}{4}z_i^{\ast a_1}\cdots z_i^{\ast a_l}.
\end{align}
For $m=1$,
\begin{align}
\left(\frac{(N+l)!(N-1+l)}{l!(N-1)!(N-1)}\right)^\frac{1}{4}
\Big[z_i^{a_1} z_i^{\ast a'_1} +\sqrt{\frac{l}{N(N-1+l)}}\delta^{a_1 a'_1}\Big]
z_i^{a_2} \cdots z_i^{a_l},
\end{align}
\begin{align}
\tilde{F}^{a_1,\cdots,a_{l},a'_1}_{(l;1)}(z_i)
\left(\frac{(N+l)!(N-1+l)}{l!(N-1)!(N-1)}\right)^\frac{1}{4}
\Big[z_i^{\ast a_1} z_i^{a'_1} -\sqrt{\frac{l}{N(N-1+l)}}\delta^{a_1 a'_1}\Big]
z_i^{\ast a_2}\cdots z_i^{\ast a_l}.
\end{align}
For $m=2$,
\begin{align}
\nonumber
&= C_{(l;2)}
\Big[z_i^{a_1}z_i^{a_2} z_i^{\ast a'_1} z_i^{\ast a'_2}
-\frac{l(N-1+l)+\sqrt{lN(N-1+l)}}{(N-1+l)(N+l)}\delta^{a_1 a'_1}
z_i^{a_2}z_i^{\ast a'_2}\Big]
z_i^{a_3}\cdots z_i^{a_l},
\end{align}
\begin{align}
\nonumber
&= C_{(l;2)}
\Big[ z_i^{\ast a_1} z_i^{\ast a_2} z_i^{a'_1} z_i^{a'_2}
-\frac{l(N-1+l)-\sqrt{lN(N-1+l)}}{(N-1+l)(N+l)}\delta^{a_2 a'_2}
z_i^{\ast a_1}z_i^{a'_1}\Big]z_i^{\ast a_3}\cdots z_i^{\ast a_l},
\end{align}
\begin{align}
\left(\frac{(N+1+l)!(N-1+l)(N+l)}{2l!N!(N-1)}\right)^\frac{1}{4}.
\end{align}
After the decomposition, the last step is to integrate out the old degrees of freedom, $z$ and $A$.
If we focus on a site $i$, there are two $F$ and two $\tilde{F}$, as illustrated in Figure <ref>.
A tensor expressed in terms of them is given by
\begin{align}
\nonumber
\nonumber
=&\int dz_i dz^\ast_i
\sqrt{d_{(l_s;m_s)}d_{(l_t;m_t)}d_{(l_u;m_u)}d_{(l_v;m_v)}
\times
\tilde{F}^{a_1,\cdots,a_{l_s}}_{(l_s;m_s)}(z_i)
\tilde{F}^{c_1,\cdots,c_{l_u}}_{(l_u;m_u)}(z_i)
\label{cpntensor}
\end{align}
The integration of the complex scalar fields, $z_i$ and $z_i^\ast$, can be done analytically and then the elements of the tensor are fixed.
Integrating out the N-component complex scalar field $z_i$.
At this point, we mention the integration of the link variable.
In contrast to the case of the complex scalar fields, the integration of the link variable is rather simple.
\begin{align}
\int_{-\pi}^{\pi} dA \
{\rm exp}\{{i(m-l)A}\}
\delta_{l, m}.
\end{align}
This just gives a constraint that the integer $l$ is equivalent to the integer $m$.
By using the tensor in eq.(<ref>), we apply the HOTRG and obtain the partition function of CP($N-1$) model.
§ NUMERICAL RESULTS
First, we compare the result of TRG method ($l_{\rm max}=1, 2$) with that of Monte Carlo simulation. Figure <ref> compares the average energy of CP(1) model computed by the two methods. The result of TRG method ($l_{\rm max}=2$) is almost consistent with that of Monte Carlo simulation. The little difference between the two results is considered to the truncation error $l_{\rm max}=2$ of the HOTRG.
It is expected that these two results are consistent at sufficiently large $l_{\rm max}$.
Average energy of CP(1) model computed by HOTRG and Metropolis algorithm. The lattice size is $4 \times 4$. The orange marks indicate the results of HOTRG and the green marks indicate the results of Metropolis algorithm.
Next, Figure <ref> compares the result of the HOTRG with that of the O(3) nonlinear sigma model in two dimensions which is analyzed by the same method. Unmuth-Yockey et al. applied the HOTRG to the O(3) model <cit.>. By following them, we compute the average energy of O(3) model. The energy of the two models is connected to each other in the continuum limit by the relation
Average energy of CP(1) model and O(3) model computed by using HOTRG. The lattice size is $2^{20} \times 2^{20}$. The circle marks indicate the results of CP(1) model and the triangle marks indicate the results of O(3) model.
\begin{align}
\frac{1}{\beta}+E_{\rm O(3)}(\beta)
E_{\rm CP(1)}(\beta)+6.
\end{align}
Using this relation, we mapped the result of the O(3) model into the graph. As $l_{\rm max}$ increases, the systematic errors decrease. In the limit $\beta=\infty$, these two results are expected to be consistent and in fact such a tendency is observed.
§ SUMMARY
In this report, we show a tensor network representation of CP($N-1$) model without the $\theta$-term.
It is confirmed that the numerical results of CP(1) model at $\theta=0$ using the TRG method are consistent with that computed by Monte Carlo simulation and that of O(3) model which is analyzed by the same method in the region $\beta \gg 1$.
For our future work, we shall try to do implementation including the $\theta$-term. In the presence of this term, the integration over link valuables develops the additional terms, and furthermore $l$ no longer equals $m$. In this case, the computational cost of the TRG methods turns out to be very expensive and we may need some techniques to reduce the cost.
Michael Levin and Cody P. Nave, Phys. Rev. Lett. 99, 120601 (2007).
Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang,
Phys. Rev. B 86, 045139 (2012).
G. Schierholz,
Nucl. Phys. Proc. Suppl. 37A, 203 (1994)
J. C. Plefka and S. Samuel,
Phys. Rev. D 55, 3966 (1997)
J. Unmuth-Yockey, Y. Meurice, J. Osborn and H. Zou,
arXiv:1411.4213 [hep-lat].
|
1511.00552
|
Department of Electrical and Computer Engineering,
National University of Singapore, 4 Engineering Drive 3, Singapore
Department of Physics, National University of Singapore,
2 Science Drive 3, Singapore 117551
Department of Electrical and Computer Engineering,
National University of Singapore, 4 Engineering Drive 3, Singapore
Department of Electrical and Computer Engineering,
National University of Singapore, 4 Engineering Drive 3, Singapore
Rayleigh's criterion for resolving two incoherent point sources has
been the most influential measure of optical imaging resolution for
over a century. In the context of statistical image processing,
violation of the criterion is especially detrimental to the
estimation of the separation between the sources, and modern
farfield superresolution techniques rely on suppressing the emission
of close sources to enhance the localization precision. Using
quantum optics, quantum metrology, and statistical analysis, here we
show that, even if two close incoherent sources emit simultaneously,
measurements with linear optics and photon counting can estimate
their separation from the far field almost as precisely as
conventional methods do for isolated sources, rendering Rayleigh's
criterion irrelevant to the problem. Our results demonstrate that
superresolution can be achieved not only for fluorophores but also
for stars.
§ INTRODUCTION
Rayleigh's criterion for resolving two incoherent point sources,
requiring them to be separated at least by a diffraction-limited spot
size on the image plane <cit.>, has been the most
influential measure of optical imaging resolution for over a
century. More recently, insights from quantum optics <cit.> and
statistics <cit.> have led to revolutions in farfield
superresolution techniques <cit.> beyond his
criterion. The techniques proposed in Refs. <cit.>
rely on locating a point source when no other nearby sources are
radiating in the same optical mode. While such techniques have
achieved spectacular success in microscopy, they require sophisticated
control of the emission of special fluorophores and are irrelevant to
astronomy and remote sensing.
For two sources with overlapping radiations on the image plane,
studies have found that signal processing of the imaging data can
still determine their locations, although the precision in the
presence of photon shot noise quickly deteriorates when Rayleigh's
criterion is violated <cit.>. The precision
degradation is mandated by the Cramér-Rao lower error bound
<cit.>, suggesting that the degradation is fundamental to
direct imaging. Given such prior work, conventional wisdom thus
suggests that the positions of two incoherent sources should become
harder to estimate when their radiations overlap, a statistical
phenomenon we call Rayleigh's curse.
Since photon shot noise is now the dominant noise source in
fluorescence microscopy <cit.> as well as stellar imaging
<cit.>, it is timely to inquire
whether a quantum treatment can lead to new insights. Here we attack
the problem from the perpsective of quantum metrology, a branch of
quantum information theory relevant to sensing and imaging
<cit.>. To be specific, we derive the fundamental
quantum limit to the precision of locating two weak thermal optical
point sources in the form of the quantum Cramér-Rao bound (QCRB)
proposed by Helstrom <cit.>. Surprisingly, we find that the
QCRB maintains a fairly constant value for any separation and shows no
sign of Rayleigh's curse. This behavior is in stark contrast to the
QCRB for in-phase coherent sources, in which case Rayleigh's curse is
fundamental <cit.>.
It is known mathematically that there exists a measurement scheme to
attain the QCRB for one parameter asymptotically
<cit.>. For a more concrete experimental
implementation, here we propose the method of SPAtial-mode
DEmultiplexing (SPADE). We show that SPADE can ideally estimate the
separation between the two sources with a quantum-optimal Fisher
information, and we also propose linear optical system designs that
can implement the measurement. Direct imaging is poor at localization
of two close sources precisely because it estimates their separation
poorly, and SPADE is able to overcome this problem and Rayleigh's
curse via further linear optical processing before photon counting.
The subject of quantum imaging has been extensively studied; see
Appendix <ref> for a literature review. Most prior proposals
rely on nonclassical sources or multiphoton coincidence measurements,
however, making them difficult and inefficient to use in
practice. Incoherent sources, such as fluorophores and stars, are of
course much more common, and linear optical methods to enhance the
localization precision for close incoherent sources will be of
monumental interest to both localization microscopy
<cit.> and astrometry
<cit.>. The most relevant prior work remains the
pioneering studies by Helstrom on thermal sources <cit.>, yet
he studied two sources only in the context of binary hypothesis
testing and assumed a given separation in the two-source hypothesis
<cit.>. As the separation is usually unknown and needs to
be estimated in the first place
<cit.>, our
parameter-estimation framework should be more useful.
§ QUANTUM OPTICS FOR WEAK THERMAL SOURCES
To illustrate the essential physics, we follow Lord Rayleigh's lead
<cit.> and assume quasi-monochromatic scalar paraxial waves
and one spatial dimension on the object and image planes. Within each
short coherence time interval for a thermal source at an optical
frequency, it is standard
to assume that the average photon number $\epsilon$ arriving on the
image plane is much smaller than $1$, and useful information is
obtained only after many photons have been measured over many such
intervals. This means that the quantum density operator for the
optical fields on the image plane in each coherence time interval can
be well approximated as
\begin{align}
\rho = (1-\epsilon)\rho_0 +\epsilon \rho_1 + O(\epsilon^2),
\label{rho}
\end{align}
where $\rho_0 = \ket{\textrm{vac}}\bra{\textrm{vac}}$ is the
zero-photon state, $\rho_1$ is a one-photon state, and $O(\epsilon^2)$
denotes terms on the order of $\epsilon^2$; see Appendix <ref>
for a detailed derivation. For the rest of the paper, we neglect the
$O(\epsilon^2)$ terms and use the $\approx$ sign to denote the
first-order approximations. Similar approximations were also used
earlier to study stellar interferometry <cit.>.
A connection with classical statistical optics can be made by
observing that $\rho_1$ is related to the mutual coherence of the
optical fields with respect to the Sudarshan-Glauber distribution. As
shown in Appendix <ref>, the one-photon state for two incoherent
point sources and a diffraction-limited imaging system can be taken as
\begin{align}
\rho_1 &\approx \frac{1}{2}
\bk{\ket{\psi_1}\bra{\psi_1}+\ket{\psi_2}\bra{\psi_2}},
\label{rho1}
\\
\ket{\psi_s} &= \intall dx \psi_s(x)\ket{x}, \quad s = 1,2,
\label{psis}
\end{align}
where $x$ is the image-plane coordinate normalized with respect to the
magnification factor of the imaging system <cit.>,
$\ket{x} = a^\dagger(x)\ket{\textrm{vac}}$ is the photon image-plane position
eigenket defined with respect to annihilation and creation operators
that obey $[a(x),a^\dagger(x')] = \delta(x-x')$
<cit.>, and $\psi_s(x)$ is the image-plane
wavefunction from each source.
We can reproduce the standard Poisson model of direct image-plane
photon counting
<cit.> by
considering the $1-\epsilon$ probability of no photon count and the
$\epsilon \ll 1$ probability of measuring a photon. If a photon is
detected, the probability density of the photon position $x$ is
\begin{align}
\Lambda(x) &= \frac{1}{2}
\bk{\abs{\braket{x|\psi_1}}^2+\abs{\braket{x|\psi_2}}^2}
\nonumber\\
&= \frac{1}{2}\Bk{\abs{\psi_1(x)}^2+\abs{\psi_2(x)}^2}.
\label{intensity}
\end{align}
With $\epsilon \ll 1$, the photon count at each pixel with width $dx$
can be approximated as Poisson with a mean given by
$\epsilon\Lambda(x) dx$. The total photon count over $M$ coherence
time intervals then remains approximately Poisson with a mean
$M\epsilon \Lambda(x) dx = N\Lambda(x) dx$, where $N \equiv M\epsilon$
is the average photon number collected over the $M$ intervals and
$\Lambda(x)$ becomes the mean intensity profile. To illustrate,
Fig. <ref> depicts the wavefunctions and the mean intensity for a
typical imaging system. Note the crucial point that
$\braket{\psi_1|\psi_2} = \intall dx \psi_{1}^*(x) \psi_{2}(x) \neq
0$, and the spatial modes excited by the two sources are in general
not orthogonal, especially when Rayleigh's criterion is violated.
This overlap underlies all the physical and mathematical difficulties
with the resolution problem, as it implies on a fundamental level that
the two modes cannot be separated for independent measurements.
(a) Two photonic wavefunctions on the
image plane, each coming from a point source. $X_1$ and $X_2$ are
the point-source positions, $\theta_1$ is the centroid, $\theta_2$
is the separation, and $\sigma$ is the width of the point-spread
function. (b) If photon counting is performed on the image plane,
the statistics are Poisson with a mean intensity proportional to
$\Lambda(x) = [|\psi_1(x)|^2+|\psi_2(x)|^2]/2$.
§ CLASSICAL AND QUANTUM CRAMÉR-RAO BOUNDS
To investigate the impact of measurement noise on parameter
estimation, suppose that $\rho$ depends on a set of unknown parameters
denoted by $\{\theta_\mu; \mu = 1,2,\dots\}$ <cit.>, and a
quantum measurement is made on the image plane over the $M$ intervals
to estimate $\theta$. Any quantum measurement can be mathematically
described by a positive operator-valued measure (POVM) $E(\mathcal Y)$
<cit.>, such that the probability distribution of measurement
outcome $\mathcal Y$ is
$P(\mathcal Y) = \trace E(\mathcal Y)\rho^{\otimes M}$, with $\trace$
denoting the operator trace and $\rho^{\otimes M}$ denoting a tensor
product of $M$ density operators. Let $\check\theta_\mu(\mathcal Y)$
be an estimator and
\begin{align}
\Sigma_{\mu\nu} \equiv \int d\mathcal Y P(\mathcal Y)
\Bk{\check\theta_\mu(\mathcal Y)-\theta_\mu}
\Bk{\check\theta_\nu(\mathcal Y)-\theta_\nu}
\end{align}
be the error covariance matrix. For any unbiased estimator, the
Cramér-Rao bound is given by
\begin{align}
\Sigma_{\mu\mu} \ge \bk{\mathcal J^{-1}}_{\mu\mu},
\end{align}
\begin{align}
\mathcal J_{\mu\nu} &\equiv \int d\mathcal Y
\frac{1}{P(\mathcal Y)}\parti{P(\mathcal Y)}{\theta_\mu}
\parti{P(\mathcal Y)}{\theta_\nu}
\end{align}
is the Fisher information matrix with respect to $P(\mathcal Y)$
For the Poisson model of direct imaging
\begin{align}
\mathcal J_{\mu\nu}^{(\textrm{direct})} =
N \intall dx \frac{1}{\Lambda(x)}
\parti{\Lambda(x)}{\theta_\mu}\parti{\Lambda(x)}{\theta_\nu}.
\label{Jdirect}
\end{align}
Alternatively, the same result can be derived without the Poisson
approximation by considering the one-photon distribution given by
Eq. (<ref>) and no multiphoton coincidence. As the
Cramér-Rao bound is asymptotically achievable <cit.>, the
Fisher information has become the standard precision measure in modern
fluorescence microscopy <cit.> as well as
astronomy <cit.>.
Direct imaging, though standard, is but one of the infinite
measurement methods permitted by quantum mechanics. The ultimate
performance of any quantum measurement and any unbiased estimator can
be quantified using the quantum Cramér-Rao bound
\begin{align}
\Sigma_{\mu\mu} \ge \bk{\mathcal J^{-1}}_{\mu\mu} \ge
\bk{\mathcal K^{-1}}_{\mu\mu},
\end{align}
where $\mathcal K$ is the quantum Fisher
information matrix in terms of $\rho^{\otimes M}$ <cit.>. To compute
$\mathcal K$ analytically, we assume a spatially invariant imaging
system with $\psi_s(x) =\psi(x-X_s)$, where $\psi(x)$ is the
point-spread function of the imaging system and $X_s$ is the unknown
position of each source <cit.>. Both
$\mathcal J^{(\textrm{direct})}$ and $\mathcal K$ turn out to be
diagonal if we redefine the parameters of interest as the centroid
\begin{align}
\theta_1 &= \frac{X_1+X_2}{2}
\end{align}
and the separation
\begin{align}
\theta_2 = X_2- X_1,
\end{align}
as depicted in Fig. <ref>. We also assume, with little loss of
generality, that the point-spread function has a constant
$x$-independent phase, which can be easily implemented by a two-lens
system <cit.>. The phase is then irrelevant to $\rho_1$ in
Eq. (<ref>) and $\psi(x)$ can be taken as real.
The computation of $\mathcal K$ is described in
Appendix <ref>; the result is
\begin{align}
\mathcal K_{11} &\approx 4N(\Delta k^2-\gamma^2),
\mathcal K_{22} &\approx N\Delta k^2,
\label{K}
\end{align}
with $\mathcal K_{12} = \mathcal K_{21} \approx 0$, where
\begin{align}
\Delta k^2 \equiv \intall dx \Bk{\parti{\psi(x)}{x}}^2
\label{dk2}
\end{align}
is the spatial-frequency variance of the real point-spread function
set by the diffraction limit and
\begin{align}
\gamma \equiv
\intall dx \parti{\psi(x)}{x}\psi(x-\theta_2)
\label{gamma}
\end{align}
is a parameter that depends on $\theta_2$. The prefactor $N$
indicates a shot-noise scaling with respect to the average photon
number, as expected from classical sources
<cit.>. For $\theta_2 \to \infty$,
$\gamma^2 \to 0$, and we recover the standard shot-noise limit to the
localization of isolated sources.
To compare the quantum Fisher information with the classical
information for direct imaging, Fig. <ref> plots the diagonal
elements of $\mathcal J^{(\rm direct)}$ and $\mathcal K$, assuming a
Gaussian point-spread function <cit.>
$\psi(x) = (2\pi\sigma^2)^{-1/4}\exp[-x^2/(4\sigma^2)]$, where
$\sigma = 1/(2\Delta k) = \lambda/(2\pi\textrm{NA})$, $\lambda$ is the
free-space wavelength, and $\textrm{NA}$ is the effective numerical
aperture. The constant $\mathcal K_{22}$ in particular becomes
\begin{align}
\mathcal K_{22} &\approx \frac{N}{4\sigma^2}.
\label{K22_gauss}
\end{align}
The Gaussian case is representative and the same qualitative behaviors
can be observed for other common point-spread functions. For the
centroid, both the classical and quantum information is within a
factor of $2$ of the standard limit $N/\sigma^2$.
$\mathcal J_{11}^{(\rm direct)} \le \mathcal K_{11}$ as it should, but
the small gap between the two means that there is little room for
Plots of Fisher information versus the
separation for a Gaussian point-spread function. $\mathcal K_{11}$
and $\mathcal K_{22}$ are the quantum values for the estimation of
the centroid $\theta_1 = (X_1+X_2)/2$ and the separation
$\theta_2 = X_2- X_1$, respectively, while
$\mathcal J_{11}^{(\rm direct)}$ and
$\mathcal J_{22}^{(\rm direct)}$ are the corresponding classical
values for direct imaging. The horizontal axis is normalized with
respect to the point-spread function width $\sigma$, while the
vertical axis is normalized with respect to $N/(4\sigma^2)$, the
value of $\mathcal K_{22}$.
The difference between the separation information quantities
$\mathcal K_{22}$ and $\mathcal J_{22}^{(\rm direct)}$ in
Fig. <ref> is much more dramatic. Both quantities approach the
same limit $N/(4\sigma^2)$ as $\theta_2 \to\infty$, implying that
direct imaging is quantum-optimal for well-separated sources. For
$\theta_2/\sigma \to 0$, however, the classical information
$\mathcal J_{22}^{(\rm direct)}$ decreases to zero. This means that
direct imaging is progressively worse at estimating the separation for
closer sources, to the point that the information vanishes and the
Cramér-Rao bound diverges at $\theta_2 = 0$. We call this divergent
behavior due to overlapping wavefunctions Rayleigh's curse, as it
implies a severe penalty on the localization precision when the
intensity profiles overlap significantly and Rayleigh's criterion is
violated for a given $N$.
Direct imaging suffers from Rayleigh's curse for any point-spread
function, as $\partial \Lambda(x)/\partial \theta_2$ vanishes at
$\theta_2 = 0$ while $\Lambda(x)$ remains nonzero in regions of $x$
where the derivative vanishes, causing
$\mathcal J_{22}^{(\rm direct)}$ to vanish via Eq. (<ref>).
This is the reason why the Cramér-Rao bounds derived in
Refs. <cit.> for separation estimation all diverge
when Rayleigh's criterion is violated. Remarkably, the quantum
information $\mathcal K_{22}$ in Eq. (<ref>) stays constant
regardless of the separation. If the centroid $\theta_1$ is known,
there exists a POVM with error $\Sigma_{22}$ asymptotically attaining
the single-parameter QCRB <cit.>, viz.,
\begin{align}
\Sigma_{22} \to \frac{1}{\mathcal K_{22}}
\approx \frac{1}{N\Delta k^2},
\quad
N \to \infty.
\end{align}
This means that Rayleigh's curse can be avoided for separation
estimation, and considerable improvements can be obtained, if the
optimal quantum measurement can be implemented. To expound the issue,
Fig. <ref> plots the quantum and classical Cramér-Rao bounds
$1/\mathcal K_{22}$ and $1/\mathcal J_{22}^{(\rm direct)}$,
demonstrating more dramatically the divergent error in the classical
case and the substantial room for improvement offered by quantum
The quantum Cramér-Rao bound
($1/\mathcal K_{22}$) and the classical bound for direct imaging
($1/\mathcal J_{22}^{(\rm direct)}$) on the error of separation
estimation. The bounds are normalized with respect to the quantum
value $4\sigma^2/N$. Rayleigh's curse refers to the divergence of
the classical bound when $\theta_2 \lesssim \sigma$, as discovered
by Refs. <cit.>.
§ SPATIAL-MODE DEMULTIPLEXING (SPADE)
Instead of measuring the position of each photon in the direct imaging
method, we propose a discrimination in terms of the Hermite-Gaussian
spatial modes <cit.> to estimate the separation.
Consider the basis $\{\ket{\phi_q}; q = 0,1,\dots\}$ with
eigenkets given by
\begin{align}
\ket{\phi_q} &= \intall dx \phi_q(x)\ket{x},
\quad
q = 0,1,\dots
\\
\phi_q(x) &= \bk{\frac{1}{2\pi\sigma^2}}^{1/4}\frac{1}{\sqrt{2^q q!}}
\end{align}
where $H_q$ is the Hermite polynomial <cit.>. The POVM for each
coherence time interval can be expressed as projections
\begin{align}
E_0 &= \ket{\textrm{vac}}\bra{\textrm{vac}},
E_1(q) &= \ket{\phi_q}\bra{\phi_q}.
\label{POVM_HG}
\end{align}
Conditioned on a detection event, the probability of detecting the
photon in the $q$th mode becomes
\begin{align}
P_1(q) &\approx \frac{1}{2}
\bk{\abs{\braket{\phi_q|\psi_1}}^2+\abs{\braket{\phi_q|\psi_2}}^2}.
\label{P1q}
\end{align}
Similar to direct imaging, $\epsilon \ll 1$ implies that, over $M$
intervals, the total photon count $m_q$ in each Hermite-Gaussian mode
can be approximated as Poisson with a mean given by $N P_1(q)$.
To proceed further, we assume that the centroid $\theta_1$ is known,
and only $\theta_2$ is to be estimated. Since centroid estimation
using direct imaging is relatively insensitive to the separation, the
assumption of an accurately known centroid is not difficult to
satisfy; see Appendix <ref> for a detailed
discussion. Under this assumption, we can assume $\theta_1 = 0$
without loss of generality, and the wavefunctions become
$\psi_1(x) = \psi\bk{x+\theta_2/2}$, and
$\psi_2(x) = \psi\bk{x-\theta_2/2}$. For simple analytic results, we
further assume that the point-spread function is Gaussian. The
overlap factors in Eq. (<ref>) can then be evaluated by
recognizing that $\ket{\phi_q}$ is mathematically equivalent to an
energy eigenstate of a harmonic oscillator (in the configuration space
of the photon), and $\ket{\psi_1}$ and $\ket{\psi_2}$ are equivalent
to configuration-space coherent states with displacements
$\pm\theta_2/(4\sigma)$. The result is
\begin{align}
P_1(q) &\approx \abs{\braket{\phi_q|\psi_1}}^2 =
\abs{\braket{\phi_q|\psi_2}}^2 =
\exp\bk{-Q}\frac{Q^q}{q!},
\nonumber\\
Q &\equiv \frac{\theta_2^2}{16\sigma^2}.
\label{Q}
\end{align}
This formula is valid even if the two sources have unequal intensities
and $\rho_1$ is any mixture of $\ket{\psi_1}\bra{\psi_1}$ and
$\ket{\psi_2}\bra{\psi_2}$. The classical Fisher information for the
Hermite-Gaussian-basis measurement over $M$ intervals becomes
\begin{align}
\mathcal J_{22}^{(\rm HG)} &\approx
N \sum_{q=0}^\infty P_1(q)\Bk{\parti{}{\theta_2} \ln P_1(q)}^2
\approx \frac{N}{4\sigma^2},
\label{J22_HG}
\end{align}
which is equal to the quantum information given by
Eq. (<ref>) and also free of Rayleigh's curse.
To measure in the Hermite-Gaussian basis, one needs to demultiplex the
image-plane field in terms of the desired spatial modes before
determining the outcome based on the mode in which the photon is
detected. To do so with a high information-extraction efficiency, one
should perform a one-to-one conversion of the Hermite-Gaussian modes
into modes in a more accessible degree of freedom with minimal loss
and measurements that capture as many photons as possible. For
example, we can take advantage of the fact that the Hermite-Gaussian
modes are waveguide modes of a quadratic-index waveguide <cit.>.
Suppose that we couple the image-plane optical field into such a
highly multimode waveguide centered at the centroid position, as shown
in Fig. <ref>. Each mode with index $q$ acquires a different
propagation constant $\beta_q$ along the longitudinal direction $z$.
If a grating coupler <cit.> with spatial frequency $\kappa$
is then used to couple all the modes into free space, each mode will
be coupled to a plane wave with a different spatial frequency
$\beta_q - \kappa$ along the $z$ direction in free space, and a
Fourier-transform lens can be used to focus the different plane waves
onto different spots of a photon-counting array in the far field.
A multimode-waveguide SPADE with a grating
output coupler and farfield photon counting. The photon counter at
the end of the multimode waveguide captures any remaining photon in
the higher-order or leaky modes.
An alternative is to use evanescent coupling with different
single-mode waveguides <cit.>, as depicted in
Fig. <ref>. If each single-mode waveguide is fabricated to have a
propagation constant equal to a different value of $\beta_q$, the
phase-matching condition will cause each mode in the multimode
waveguide to be coupled to a specific fiber.
An alternative design, with evanescent coupling
to single-mode waveguides with different propagation constants for
phase matching. The photon counter at the end of the multimode
waveguide captures any remaining photon in the higher-order or leaky
Given these physical setups, we can now explain the operation of SPADE
in a more intuitive semiclassical optics language: it is based on the
exquisite sensitivity of the mode-coupling efficiencies to the offset
of the wavefunctions from the centroid. The incoherent sources are
literally blinking on the fundamental coherence time scale, causing
each image-plane photon to have a wavefunction given randomly by
$\psi_1(x)$ or $\psi_2(x)$. Either wavefunction can excite the
waveguide modes coherently with the same excitation probabilities,
causing the final photon counts to be as sensitive to the offset for
two sources as it is for one. Put another way, the incoherence between
the two sources implies a random relative phase between the two fields
and enables coupling into the first-order odd mode, which is the main
spatial mode responsible for the high sensitivity to small offsets.
The use of photon counting is essential here to discriminate against
the abundant but uninformative zero-photon events. If homodyne or
heterodyne methods were used instead, they would suffer from excess
vacuum fluctuations when no photon arrives. The poor performance of
heterodyne methods for weak thermal sources is also known in the
context of stellar interferometry <cit.>. The situation
is different from measurements of coherent light, the density operator
of which contains off-diagonal terms with respect to the photon-number
basis and the probabilistic photon picture is less adequate. Our
preliminary calculations <cit.> confirm this
expectation and suggest that heterodyne detection of the spatial modes
still suffers from Rayleigh's curse.
Suppose that a total of $L$ photons are detected over the $M$ trials.
A record of the modes for the $L$ photons $(q_1,\dots,q_L)$ can be
obtained, but in fact a time-resolved record is not necessary, as
$\sum_l q_l$ is a sufficient statistic for estimating $Q$ and
$\theta_2$ <cit.>, meaning that the set of photon numbers
$\{m_q = \sum_l \delta_{q q_l}; q = 0,1,\dots\}$ detected in different
modes are also sufficient. The maximum-likelihood estimator becomes
\begin{align}
\check{Q}_{\rm ML} &= \frac{1}{L} \sum_{q} q m_q,
\check\theta_{2\rm ML} &= 4\sigma\sqrt{\check{Q}_{\rm ML}},
\label{QML}
\end{align}
which is straightforward to implement computationally. For $L = 0$,
one can set $\check\theta_{2}$ to a constant value; the $L = 0$
probability $(1-\epsilon)^M \approx \exp(-N)$ is in any case
negligible for large $N$. Maximum-likelihood estimation can
asymptotically saturate the Cramér-Rao bound
$\Sigma_{22} \ge 1/\mathcal J_{22}^{(\rm HG)}$ for large $M$
<cit.>. With
$\mathcal J_{22}^{(\rm HG)} \approx \mathcal K_{22}$, the QCRB is
asymptotically attainable as well. Appendix <ref> reports
a Monte Carlo analysis of the maximum-likelihood estimator for SPADE,
confirming that the Cramér-Rao bound remains close to the estimation
error for finite photon numbers.
§ BINARY SPADE
Since direct imaging has trouble estimating the separation only when
$\theta_2/\sigma$ is small, and only low-order Hermite-Gaussian modes
in SPADE are excited significantly in that case, we can focus on the
discrimination of low-order modes to simplify the SPADE design. One
such design is depicted in Fig. <ref>, where only the $q = 0$
component is coupled into the single-mode waveguide, while any photon
in the higher-order modes remains in the multimode waveguide for
subsequent detection. An alternative design is depicted in
Fig. <ref>: the $q = 0$ mode is coupled to a single-mode
waveguide, while higher-order modes are necessarily coupled to the
leaky modes of the waveguide, which are also measured.
Binary SPADE with evanescent coupling to only
one single-mode waveguide.
An alternative design of binary SPADE with a
single-mode waveguide and leaky-mode detection.
Conditioned on a detection event, the probability of detecting the
photon in the $q = 0$ mode remains
\begin{align}
P_1(q = 0) &\approx \exp\bk{-Q},
\label{P1q0}
\end{align}
but now the higher-order modes
cannot be discriminated, and the probability of detecting a photon in
any higher-order mode becomes
\begin{align}
P_1(q > 0) &= 1-P_1(q=0) \approx 1-\exp\bk{-Q}.
\label{P1qhigh}
\end{align}
The Fisher information for this scheme is hence
\begin{align}
\mathcal J_{22}^{(\rm b)} &\approx
\frac{N}{4\sigma^2}
\frac{Q\exp(-Q)}{1-\exp(-Q)}.
\label{fi_bspade}
\end{align}
Figure <ref> compares $\mathcal J_{22}^{(\rm b)}$ with the
optimal value $\mathcal J_{22}^{(\rm HG)}\approx \mathcal K_{22}$ as
well as $\mathcal J_{22}^{(\rm direct)}$ for direct imaging. It can be
seen that binary SPADE gives significant information for small
$\theta_2/\sigma$, which happens to be the regime where direct imaging
performs poorly. Binary SPADE actually works less well when the
sources are far apart, and the two methods can complement each other
to enhance the localization precision, as shown in
Appendix <ref>.
Fisher information for separation estimation
versus normalized separation $\theta_2/\sigma$ for a Gaussian
point-spread function. $\mathcal J_{22}^{(\rm HG)}$ is the
information for the ideal Hermite-Gaussian-basis measurement, which
is equal to the quantum value $\mathcal K_{22}$,
$\mathcal J_{22}^{(\rm direct)}$ is for direct imaging, and
$\mathcal J_{22}^{(\rm b)}$ is for binary SPADE. The vertical axis
is normalized with respect to
$\mathcal J_{22}^{(\rm HG)} = \mathcal K_{22} = N/(4\sigma^2)$.
For a total of $L$ detected photons, $L$ and $m_0$, the number of
photons detected in the $q=0$ mode, are sufficient statistics for
estimating $Q$ and $\theta_2$, and $m_0$ follows the binomial
distribution for $L$ trials and success probability $\exp(-Q)$
<cit.>. The maximum-likelihood estimator becomes
\begin{align}
\check Q_{\rm ML}^{(\rm b)} &=-\ln \frac{m_0}{L},
\check\theta_{2\rm ML}^{(\rm b)}
&= 4\sigma \sqrt{\check Q_{\rm ML}^{(\rm b)}}.
\label{ML_bspade}
\end{align}
For $L = 0$ or $m_0 = 0$, one can select finite values for
$\check\theta_2$ to regularize the
estimator. Appendix <ref> reports a Monte Carlo analysis
of the resulting estimation error, confirming that it remains close to
the Cramér-Rao bound for finite photon numbers.
Compared with the large amount of data generated by direct imaging and
the complex algorithms needed to process them, only two photon numbers
are needed by binary SPADE to estimate the separation precisely. The
highly compressed measurement output and computationally simple
estimators, enabled by the coherent optical processing, come as
bonuses with our schemes.
§ OTHER POINT-SPREAD FUNCTIONS
Our analysis of SPADE so far relies on the assumption of a Gaussian
point-spread function. For other point-spread functions, it is
nontrivial to find a suitable basis of spatial modes, although we can
still rely on the mathematical existence of a quantum-optimal
measurement <cit.> to be sure that the QCRB can
be saturated. For a more concrete method, the analysis of the binary
SPADE schemes is fortunately still tractable, if we assume a
single-mode waveguide with a mode profile that matches the
point-spread function $\psi(x)$ centered at the centroid position.
Define $\ket{\psi} = \intall dx \psi(x)\ket{x}$ as the state of one
photon in the waveguide mode. The efficiency of coupling a photon in
state $\ket{\psi_1}$ or $\ket{\psi_2}$ into the waveguide mode becomes
\begin{align}
\abs{\braket{\psi|\psi_1}}^2 &= \abs{\braket{\psi|\psi_2}}^2
=\abs{\intall dx \psi^*(x)\psi\bk{x+\frac{\theta_2}{2}}}^2
\nonumber\\
&= \abs{\intall dk \abs{\Psi(k)}^2\exp\bk{\frac{ik\theta_2}{2}}}^2
\equiv \Upsilon(\theta_2),
\label{f}
\end{align}
where $\Upsilon(\theta_2)$ is the mode overlap factor and
$\Psi(k) \equiv (2\pi)^{-1/2}\intall dx\psi(x)\exp(-ikx)$ is the
optical transfer function of the imaging system before the image plane
<cit.>. For the density operator in Eqs. (<ref>),
(<ref>), and (<ref>), or in fact any mixture of
$\ket{\psi_1}\bra{\psi_1}$ and $\ket{\psi_2}\bra{\psi_2}$, the
probability of finding a photon in the waveguide mode becomes
$P(\psi) \approx \epsilon \Upsilon$, and the probability of finding a
photon in any other mode is
$P(\bar\psi) \approx \epsilon\bk{1-\Upsilon}$. The Fisher information
over $M$ intervals is then
\begin{align}
\mathcal J_{22}^{(\rm b)}
\approx
\frac{N}{\Upsilon(1-\Upsilon)}\bk{\parti{\Upsilon}{\theta_2}}^2.
\end{align}
To study its behavior for small $\theta_2$, expand
$\Upsilon(\theta_2)$ in Eq. (<ref>) as
$\Upsilon(\theta_2) = 1 - \Delta k^2 \theta_2^2/4 + O(\theta_2^4)$
$\Delta k^2 = \intall dk |\Psi(k)|^2 k^2 -[\intall dk |\Psi(k)|^2
k]^2$, giving
\begin{align}
\mathcal J_{22}^{(\rm b)}(\theta_2 = 0) &\approx N \Delta k^2.
\end{align}
$\mathcal J_{22}^{(\rm b)}$ can hence reach the quantum information
$\mathcal K_{22}= N\Delta k^2$ at $\theta_2 = 0$, precisely where
$\mathcal J_{22}^{(\rm direct)}$ vanishes and Rayleigh's curse is at
its worst. For larger $\theta_2$, $\mathcal J_{22}^{(\rm b)}$ is
expected to decrease, as the scheme is unable to discriminate the
higher-order modes that become more likely to be occupied.
Figure <ref> plots $\mathcal K_{22}$, the numerically
computed $\mathcal J_{22}^{(\rm direct)}$, and
$\mathcal J_{22}^{(\rm b)}$ for the sinc point-spread function
<cit.> $\psi(x) = (1/\sqrt{W})\sinc(x/W)$, where
$W = \lambda/(2\textrm{NA})$, $\sinc u \equiv \sin(\pi u)/(\pi u)$ for
$u \neq 0$ and $\sinc(0) \equiv 1$. The information quantities
demonstrate behaviors similar to the Gaussian case.
Fisher information for separation
estimation versus normalized separation $\theta_2/W$ for the sinc
point-spread function. $\mathcal K_{22}$ is the quantum value,
$\mathcal J_{22}^{(\rm direct)}$ is the numerically computed value
for direct imaging, and $\mathcal J_{22}^{(\rm b)}$ is that for
binary SPADE tailored for the sinc function. The vertical axis is
normalized with respect to $\mathcal K_{22} = \pi^2N/(3W^2)$.
§ TWO-DIMENSIONAL IMAGING
The essential physics remains unchanged when we consider
two-dimensional imaging, and we discuss the generalization only
briefly here; the details are given elsewhere <cit.>. The
single-photon ket in Eq. (<ref>) should now be expressed as
$\ket{\psi_s} = \intall dx \intall dy \psi_s(x,y)\ket{x,y}$, where
$\braket{x,y|x',y'} = \delta(x-x')\delta(y-y')$ and $\psi_s(x,y)$ is a
two-dimensional wavefunction <cit.>. In terms
of a point-spread function $\psi(x,y)$ and unknown positions
$(X_1,Y_1)$ and $(X_2,Y_2)$, $\psi_s(x,y) = \psi(x-X_s,y-Y_s)$, and we
can define the four centroid and separation parameters as
$\theta_{1} = (X_1 + X_2)/2$, $\theta_{2} = X_2 - X_1$,
$\theta_{3} = (Y_1 + Y_2)/2$, and $\theta_{4} = Y_2 - Y_1$.
$\mathcal J^{(\rm direct)}$ for the estimation of $\theta_{2}$ and
$\theta_{4}$ decreases to zero when the sources are close, and
Rayleigh's curse still exists for direct imaging
<cit.>. On the other hand, the quantum Fisher information
matrix, to be reported in Ref. <cit.>, again shows no sign of
Rayleigh's curse for two-dimensional separation estimation.
For SPADE, we can use the two-dimensional Hermite-Gaussian basis
<cit.>. Assuming a Gaussian point-spread function and a known
centroid, it is straightforward to show that a measurement of each
photon in the Hermite-Gaussian basis with mode indices $q$ and $p$
obeys a two-variable Poisson distribution, and the classical Fisher
information with respect to $\theta_{2}$ and $\theta_{4}$ remains a
constant and free of Rayleigh's curse, similar to the one-dimensional
case. For other point-spread functions, such as the Airy disk
<cit.>, binary SPADE with a matching mode profile
can estimate the separation without Rayleigh's curse for small
separations in the same way as the one-dimensional case, but
information about the direction of the separation is lost. To obtain
directional information, one needs to discriminate at least some of
the higher-order modes in different directions.
A quadratic-index optical fiber can support two-dimensional
Hermite-Gaussian modes, while a weakly guiding step-index fiber also
has modes closely resembling the Hermite-Gaussian modes
<cit.>. A complication arises for cylindrically symmetric
fibers, as modes with the same total order $q+p$ will have a
degenerate propagation constant, causing multiple modes to satisfy the
same phase-matching conditions in grating or evanescent coupling and
preventing discrimination of modes with the same order. The net
result is that directional information is compromised. One solution is
to turn the point-spread function into an elliptic one with asymmetric
widths and use an elliptic fiber to break the degeneracy.
§ CONCLUSION
We have presented two important results in this paper: the fundamental
quantum limit to locating two incoherent optical point sources and the
SPADE measurement schemes for quantum-optimal separation estimation.
Our quantum bound sets the ultimate limit to localization precision in
accordance with the fundamental laws of quantum mechanics, while SPADE
can extract the full information offered by quantum mechanics
concerning the separation parameter via linear photonics. The proposed
SPADE schemes work well for close sources with significant overlap in
their wavefunctions, avoiding Rayleigh's curse and the divergent error
that plagues direct imaging. The computational simplicity of the
estimators is an additional advantage. Foreseeable applications
include binary-star astrometry <cit.> and
single-molecule imaging <cit.>, either as a replacement of
techniques based on fluorescence resonant energy transfer
<cit.> or as an enhancement of localization
microscopy <cit.> to provide
complementary information about close pairs of fluorophores.
Note added.—Subsequent to the completion of this work (the
first version of this manuscript was submitted to the arXiv preprint
server on Nov 2, 2015 <cit.>), we have developed a semiclassical
but less general theory to explain our results here for pedagogy
<cit.>, discovered an alternative scheme called
Super-Localization via Image-inVERsion interferometry (SLIVER) that
can overcome Rayleigh's curse without the need to tailor the device to
the point-spread function <cit.>, derived the QCRB for
thermal sources without the $\epsilon \ll 1$ approximation using an
alternative approach <cit.>, and shown that variations of
SPADE and SLIVER can attain the bound for arbitrary $\epsilon$
<cit.>, validating the results here. A generalization of
our theory presented here to two dimensions, with similar conclusions,
is described in detail elsewhere <cit.>. Following
Ref. <cit.>, Lupo and Pirandola have derived the ultimate
quantum Fisher information for separation estimation with arbitrary
quantum sources <cit.>, including our independent result on
thermal sources <cit.> as a special case. Experiments
inspired by our theory have been reported in
Refs. <cit.>;
Refs. <cit.> also propose variations of
SPADE that are easier to implement experimentally.
§ AUTHOR CONTRIBUTIONS
M. T. conceived the idea of applying quantum metrology to
two-incoherent-source localization and developed the quantum optics
formalism in Sec. <ref> and Appendix <ref>. R. N.derived the quantum Fisher information in Sec. <ref> and
Appendix <ref> with X.-M. L. and M. T.'s inputs and checks.
M. T. invented and analyzed the SPADE measurement schemes described
in Secs. <ref>–<ref>, while R. N. first recognized
the importance of the first-order Hermite-Gaussian mode to SPADE, as
explained in Sec. <ref>. M. T. supervised the project and
wrote the paper. All authors discussed extensively during the course
of this work.
§ ACKNOWLEDGMENTS
We acknowledge useful discussions with Shan Zheng Ang and Shilin
Ng. This work is supported by the Singapore National
Research Foundation under NRF Grant No. NRF-NRFF2011-07 and the
Singapore Ministry of Education Academic Research Fund Tier 1 Project
§ QUANTUM-IMAGING LITERATURE REVIEW
Helstrom pioneered the application of his quantum estimation and
detection bounds to optical imaging problems
<cit.>, focusing on coherent and
thermal sources. In particular, the now well-known expression for the
shot-noise-limited localization error for one classical source can be
found in Ref. <cit.>; similar expressions in the context of
direct imaging were later reported in
Refs. <cit.>. For more recent studies of
quantum metrology for coherent-state or nonclassical-state imaging,
see, for example, Refs. <cit.>. For studies
on the use of squeezed light for single-object localization, see, for
Refs. <cit.>. None of
these studies considered the problem of locating two close incoherent
The standard quantum model of paraxial imaging and the use of
nonclassical light for that purpose were proposed by Yuen and Shapiro
<cit.>. This topic has been further investigated most
notably by Kolobov and co-workers <cit.>, who
focused on coherent or squeezed light, homodyne detection, and field
fluctuations. Such models are irrelevant to incoherent sources such as
stars and fluorophores, for which the mean field is zero and photon
counting is the more relevant method to minimize vacuum noise; to
quote Helstrom <cit.>,
With such incoherently
illuminated or radiating objects, it is not the field of the light
that is of interest, for that field is best described as a random
process having zero mean value and a most erratic spatiotemporal
variation. Rather it is the mean-square value of the field, averaged
over many cycles of the dominant temporal frequency, that
characterizes the object in the most informative way.
Subsequent work by Kolobov and co-workers
<cit.> considered the
squeezing and measurement of the eigenmodes of an imaging system for
image-reconstruction superresolution. Again, these studies focused on
coherent or squeezed light only.
The “Rayleigh resolution limit”
mentioned by many of these papers is a misnomer, as the resolution
limit for coherent imaging should be attributed to Abbe, while
Rayleigh's criterion is defined for two incoherent sources
<cit.> and ill-suited to coherent imaging
<cit.>. Moreover, the imaging-system eigenmodes they
studied have no relation to the spatial modes we propose for the
two-source localization problem and they did not use the more rigorous
framework of statistical parameter estimation.
We can consider the schemes proposed in
Refs. <cit.>
as another class of superresolution imaging protocols, which require
coherent or nonclassical sources and multiphoton coincidence
measurements and do not consider statistical inference. It is well
known in statistical optics that a multiphoton coincidence
measurement, such as the obsolete Hanbury Brown-Twiss interferometry,
fundamentally has a much poorer signal-to-noise ratio than amplitude
interferometry because multiphoton coincidence events are rare for
thermal optical sources <cit.>. The actual
statistical resolution of this class of protocols is thus
questionable, especially for weak optical sources, without further
proofs in the context of inference accuracy. In recent years, there
has also been significant interest in quantum lithography
<cit.> and ghost imaging
<cit.>, although their applications are
clearly different from our purpose and will not be elaborated here.
The relative neglect of incoherent sources in the quantum-imaging
literature, despite their obvious importance, may be due to a lack of
appreciation that quantum mechanics can be relevant to such highly
classical light. Our work thus showcases quantum metrology as a
powerful tool to discover the ultimate performance of sensing and
imaging even for classical sources, providing not only rigorous
quantum limits but also pleasant surprises for one of the most
important applications in optics.
§ QUANTUM OPTICS: DERIVATION OF
EQS. (<REF>)–(<REF>)
Define $\alpha = (\alpha_1,\dots,\alpha_J)^\top$ as a column vector of
complex field amplitudes for $J$ optical spatial modes on the image
plane and $\ket{\alpha}$ as a multimode coherent state with amplitude
$\alpha$. Any quantum state can be expressed as
\begin{align}
\rho &= \int D\alpha \Phi(\alpha)\ket{\alpha}\bra{\alpha},
\label{glauber}
\end{align}
where $\Phi(\alpha)$ is the Sudarshan-Glauber representation and
$D\alpha$ is an appropriate measure <cit.>. For thermal
sources, it is standard <cit.> to assume $\Phi$ to be a
zero-mean complex Gaussian given by
\begin{align}
\Phi(\alpha) &= \frac{1}{\det(\pi \Gamma)}
\exp\bk{-\alpha^\dagger \Gamma^{-1}\alpha},
\end{align}
where $\alpha^\dagger = (\alpha_1^*,\dots,\alpha_J^*)$ denotes the
complex transpose of $\alpha$,
\begin{align}
\Gamma &= \expect\bk{\alpha\alpha^\dagger}
\label{Gamma}
\end{align}
is the image-plane mutual coherence matrix, and
$\expect\Bk{f(\alpha)} \equiv \int D\alpha \Phi(\alpha) f(\alpha)$
denotes the expectation of any function $f$ with respect to the $\Phi$
distribution. Writing the coherent state in terms of a superposition
of Fock states and applying the Gaussian moment theorem <cit.>
to Eq. (<ref>), we can express $\rho$ as the incoherent
\begin{align}
\rho &= \sum_{n=0}^\infty \pi_n \rho_n,
n &\equiv \sum_j n_j,
\label{rho_full}
\end{align}
where $\pi_n$ is the probability of having $n$ total photons in the
state and $\rho_n$ is an $n$-photon multimode Fock state.
At optical frequencies or beyond, it is standard
to assume that, within the short coherence time of a source, the
average photon number arriving at the imaging device is much smaller
than $1$. We will make the same assumption for two sources, viz.,
\begin{align}
\epsilon \equiv \sum_j \trace \rho a_j^\dagger a_j =
\expect\bk{\alpha^\dagger\alpha} =\sum_j \Gamma_{jj} \ll 1,
\end{align}
where $\trace$ denotes the operator trace, $a_j$ is the annihilation
operator for the $j$th mode, and $a_j^\dagger$ is the creation
operator. For example, a star with sun-like temperature
$6000~\textrm{K}$ emits $\sim 10^{-2}$ photon on average per mode at
wavelength $500~\textrm{nm}$, while the limited fraction of the
coherence area captured by the telescope aperture further reduces the
received photon number <cit.>. In microscopy, a typical
fluorophore emits $< 10^7$ photons per second <cit.> with
coherence time $< 50~\textrm{fs}$ <cit.>, leading to
$\epsilon < 10^{-6}$ for two sources. The zero-photon probability
given by
\begin{align}
\pi_0 &= \expect\bk{e^{-\alpha^\dagger\alpha}} = 1-\epsilon + O(\epsilon^2)
\end{align}
is then the highest, the one-photon probability given by
\begin{align}
\pi_1 &= \expect\bk{e^{-\alpha^\dagger\alpha}\alpha^\dagger\alpha} = \epsilon
\end{align}
is $\epsilon$ to the first order, and the multiphoton
\begin{align}
\sum_{n=2}^\infty \pi_n &= 1-\pi_0-\pi_1 = O(\epsilon^2)
\label{pn2}
\end{align}
is in the second order, leading to Eq. (<ref>). As the vacuum
state provides no information and multiphoton events are rare, we will
focus on the one-photon state $\rho_1$. This focus also makes our
formalism applicable to inefficient single-photon emitters, which may
have non-Poissonian multiphoton statistics but rare multiphoton
events, and electron microscopy <cit.>.
The negligence of the $O(\epsilon^2)$ multiphoton probability leads to
a Poisson photon-counting distribution <cit.>, which
ignores bunching or antibunching effects but remains an excellent
empirical model for both astronomical optical sources
<cit.> and
fluorophores <cit.> by virtue of the
$\epsilon \ll 1$ condition. To quote Mandel <cit.>,
The light from these sources is always so weak that
$\bar n\xi/T \ll 1$ [$\epsilon$ in our terminology] and the
degeneracy is unlikely to be detected in measurements on a single
beam. The situation is, of course, improved when correlation
measurements are undertaken on two or more coherent beams (Hanbury
Brown and Twiss 1956), since these measurements single out the
degenerate photons (Mandel 1958). Even so it is unlikely that any
faint stars could be studied in this way.
Similarly, Goodman states that
If the count degeneracy parameter [$\epsilon$ in our terminology]
is much less than 1, it is highly probable that there will be either
zero or one counts in each separate coherence interval of the incident
classical wave. In such a case the classical intensity fluctuations
have a negligible "bunching" effect on the photo-events, for (with
high probability) the light is simply too weak to generate multiple
events in a single coherence cell. If negligible bunching of the
events takes place, the count statistics will be indistinguishable
from those produced by stabilized single-mode laser radiation, for
which no bunching occurs.
A more recent work by Zmuidzinas
<cit.> also states that
It is well established that the photon counts registered by
the detectors in an optical instrument follow statistically
independent Poisson distributions, so that the fluctuations of the
counts in different detectors are uncorrelated. To be more precise,
this situation holds for the case of thermal emission (from the
source, the atmosphere, the telescope, etc.) in which the mean
photon occupation numbers of the modes incident on the detectors are
low, $n \ll 1$ [$\epsilon$ in our terminology]. In
the high occupancy limit, $n \gg 1$, photon bunching becomes
important in that it changes the counting statistics and can
introduce correlations among the detectors. We will discuss only the
first case, $n \ll 1$, which applies to most astronomical
observations at optical and infrared wavelengths.
Define $\ket{j} = a_j^\dagger\ket{\textrm{vac}}$ as the ket with one
photon only in the $j$th mode. Consider the one-photon matrix elements
\begin{align}
\bra{j}\rho\ket{k} &= \expect\bk{e^{-\alpha^\dagger\alpha} \alpha_j \alpha_k^*}
= \Gamma_{jk} + O(\epsilon^2).
\end{align}
We can then assume, to the first order of $\epsilon$,
\begin{align}
\pi_1 &\approx \epsilon,
\rho_1 &\approx \frac{1}{\epsilon}\sum_{j,k} \Gamma_{jk} \ket{j}\bra{k}.
\label{rho1_discrete}
\end{align}
Similar approximations were also used in
Refs. <cit.>. To derive $\Gamma$, let
$\beta \equiv (\beta_1,\dots,\beta_K)^\top$ be the field amplitudes
for optical modes on the object plane, and consider the field
propagation rule $\alpha = S \beta$ for a linear optical system, where
$S$ is the field scattering matrix. The image-plane mutual coherence
$\Gamma$ is then related to the object-plane mutual coherence matrix
$\Gamma^{({\rm o})}$ by
\begin{align}
\Gamma &= S \Gamma^{({\rm o})} S^\dagger.
\label{wolf}
\end{align}
This propagation rule is a basic principle in both classical and
quantum statistical optics <cit.>.
In the paraxial regime, we can use localized wavepacket modes as a
basis <cit.>. Let $u$ be the position index
for a wavepacket mode on the one-dimensional object plane and consider
two incoherent sources with equal intensities at positions $u = u_1$
and $u = u_2$. The fields are uncorrelated at different points on the
object plane, with nonzero intensities only at the sources. Then
\begin{align}
\Gamma_{uv}^{({\rm o})} = \epsilon_0 \delta_{uv}
\bk{\delta_{u u_1}+\delta_{u u_2}},
\end{align}
where $\epsilon_0$ is the average photon number from each source. On
the image plane, the mutual coherence becomes
\begin{align}
\Gamma_{jk} = \epsilon_0\bk{S_{j u_1} S^*_{k u_1}
+ S_{j u_2}S^*_{k u_2}},
\label{Gamma2}
\end{align}
and the average photon number can be expressed as
$\epsilon = 2\epsilon_0 \eta$, where
$\eta \equiv \sum_j \abs{S_{j u_s}}^2$ is the quantum efficiency of
the imaging system and we have made the reasonable assumption that the
efficiency is the same for both
sources. Equation (<ref>) can then be expressed as
Eq. (<ref>) if we define single-photon kets
$\ket{\psi_s} \equiv \sum_j \psi(j,u_s)\ket{j}$ with normalized
wavefunctions $\psi(j,u_s) = S_{j u_s}/\sqrt{\eta}$. Assuming
image-plane wavepacket positions $x_j = x_0 + jdx$, position eigenkets
$\ket{x_j} = \ket{j}/\sqrt{dx}$, and wavefunctions
$\psi_s(x_j) = \psi(j,u_s)/\sqrt{dx}$, we arrive at Eq. (<ref>)
by taking the continuous-space limit with infinitesimal $dx$
§ QUANTUM METROLOGY:
DERIVATION OF EQS. (<REF>)
The quantum Fisher information matrix with respect to
$\rho^{\otimes M}$ proposed by Helstrom <cit.> is defined as
\begin{align}
\mathcal K_{\mu\nu}(\rho^{\otimes M}) &= M\mathcal K_{\mu\nu}(\rho) =
M \real \trace \mathcal L_\mu(\rho) \mathcal L_\nu(\rho) \rho,
\label{qfi}
\end{align}
where $\mathcal L_\mu(\rho)$ is a symmetric logarithmic derivative
(SLD) of $\rho$. Writing $\rho$ in its eigenbasis as
\begin{align}
\rho &= \sum_j D_j \ket{e_j}\bra{e_j},
\end{align}
$\mathcal L_\mu(\rho)$ can be expressed as
\begin{align}
\mathcal L_\mu(\rho) &= \sum_{j,k; D_j + D_k \neq 0}
\frac{2}{D_j+D_k}
\bra{e_j}\parti{\rho}{\theta_\mu}\ket{e_k}
\ket{e_j}\bra{e_k}.
\label{sld}
\end{align}
Given this definition and Eq. (<ref>), it can be shown that
\begin{align}
\mathcal K(\rho) &= \sum_n \pi_n \mathcal K(\rho_n) \ge \pi_1 \mathcal K(\rho_1),
\end{align}
as each $\rho_n$ is in an orthogonal subspace. Since the vacuum state
$\rho_0 = \ket{\textrm{vac}}\bra{\textrm{vac}}$ contains no
information and multiphoton events are rare, the total information
will be dominated by that from the one-photon state $\rho_1$. We will
therefore focus on the one-photon component $\pi_1\mathcal K(\rho_1)$
as a tight lower bound on the quantum information and assume in the
\begin{align}
\mathcal K(\rho) &\approx \pi_1\mathcal K(\rho_1).
\label{K1}
\end{align}
With $\pi_1 \approx \epsilon$ and the probability of multiphoton
events being $O(\epsilon^2)$ according to Eq. (<ref>), this
approximation is accurate to the first order of $\epsilon$.
To compute the quantum Fisher information matrix $\mathcal K(\rho_1)$
according to Eqs. (<ref>)–(<ref>), we first need to
diagonalize the $\rho_1$ in Eqs. (<ref>) and (<ref>), noting
that the eigenvectors should span the supports of $\rho_1$ and
$\partial\rho_1/\partial\theta_\mu$. The partial derivative of
$\rho_1$ with respect to $X_\mu$ can be expressed as
\begin{align}
\parti{\rho_1}{X_\mu} &= \parti{D_1}{X_\mu}\ket{e_1}\bra{e_1}
\nonumber\\&\quad
\Bk{D_1 \parti{\ket{e_1}}{X_\mu}\bra{e_1} +
D_2 \parti{\ket{e_2}}{X_\mu}\bra{e_2} + \textrm{H.c.}},
\label{drho}
\end{align}
where H.c. denotes the Hermitian conjugate. In addition to the
support of $\rho_1$ spanned by $\ket{e_1}$ and
$\ket{e_2}$, we also need to find more eignevectors that span the
support of $\partial\ket{e_1}/\partial X_\mu$ and
$\partial\ket{e_2}/\partial X_\mu$.
Assuming that $\psi_\mu(x) = \psi(x-X_\mu)$ and the point-spread
function $\psi(x)$ has an $x$-independent phase, we can take $\psi(x)$
to be real without loss of generality and choose the following
orthonormal set of eigenvectors:
\begin{align}
\ket{e_1} &= \frac{1}{\sqrt{2(1-\delta)}}
\bk{\ket{\psi_1} - \ket{\psi_2}},
\nonumber\\
\ket{e_2} &= \frac{1}{\sqrt{2(1+\delta)}}
\bk{\ket{\psi_1} + \ket{\psi_2}},
\nonumber\\
\ket{e_3} &= \frac{1}{c_3}
\Bk{\frac{\Delta k}{\sqrt{2}}\bk{\ket{\psi_{11}} + \ket{\psi_{22}}}
- \frac{\gamma}{\sqrt{1-\delta}}\ket{e_1}},
\nonumber\\
\ket{e_4} &= \frac{1}{c_4}
\Bk{\frac{\Delta k}{\sqrt{2}}\bk{\ket{\psi_{11}} - \ket{\psi_{22}}} +
\frac{\gamma}{\sqrt{1+\delta}}\ket{e_2}},
\end{align}
where $\Delta k^2$ and $\gamma$ are given by Eqs. (<ref>)
and (<ref>), respectively,
\begin{align}
\ket{\psi_{11}} &\equiv \frac{1}{\Delta k}\intall dx \parti{\psi(x-X_1)}{X_1}\ket{x},
\nonumber\\
\ket{\psi_{22}} &\equiv \frac{1}{\Delta k}\intall dx \parti{\psi(x-X_2)}{X_2}\ket{x},
\nonumber\\
c_3 &\equiv \bk{\Delta k^2 + b^2 - \frac{\gamma^2}{1-\delta}}^{1/2},
\nonumber\\
c_4 &\equiv \bk{\Delta k^2 - b^2 - \frac{\gamma^2}{1+\delta}}^{1/2},
\nonumber\\
b^2 &\equiv \intall dx \parti{\psi(x-X_1)}{X_1}\parti{\psi(x-X_2)}{X_2},
\nonumber\\
\delta &\equiv \intall dx \psi(x-X_1)\psi(x-X_2),
\end{align}
and the eigenvalues of $\rho_1$ are
\begin{align}
D_1 &= \frac{1-\delta}{2},
D_2 &= \frac{1+\delta}{2},
D_3 &= D_4 = 0.
\label{eigenvalues}
\end{align}
After more algebra, the SLD in Eq. (<ref>) with respect to the
derivative in Eq. (<ref>) can be expressed as
\begin{align}
\mathcal L_\mu^{(X)} &= \sum_{j,k} \mathcal L_{\mu,jk}^{(X)}\ket{e_j}\bra{e_k}
\label{LX}
\end{align}
with a real and symmetric matrix
$\mathcal L_{\mu,jk}^{(X)} = \mathcal L_{\mu,kj}^{(X)}$, the nonzero
and unique elements of which are found to be
\begin{align}
\mathcal L_{1,11}^{(X)} &= -\mathcal L_{2,11}^{(X)} = \frac{\gamma}{1-\delta},
\mathcal L_{1,12}^{(X)} &= \mathcal L_{2,12}^{(X)} =
\frac{\gamma\delta}{\sqrt{1-\delta^2}},
\nonumber\\
\mathcal L_{1,13}^{(X)} &= -\mathcal L_{2,13}^{(X)} = \frac{c_3}{\sqrt{1-\delta}},
\mathcal L_{1,14}^{(X)} &= \mathcal L_{2,14}^{(X)} = \frac{c_4}{\sqrt{1-\delta}},
\nonumber\\
\mathcal L_{1,22}^{(X)} &= -\mathcal L_{2,22}^{(X)} = \frac{-\gamma}{1+\delta},
\mathcal L_{1,23}^{(X)} &= \mathcal L_{2,23}^{(X)} = \frac{c_3}{\sqrt{1+\delta}},
\nonumber\\
\mathcal L_{1,24}^{(X)} &= -\mathcal L_{2,24}^{(X)} = \frac{c_4}{\sqrt{1+\delta}}.
\end{align}
In terms of the centroid and separation parameters given by
$\theta_1 = (X_1+X_2)/2$ and $\theta_2 = X_2 - X_1$, the SLDs become
\begin{align}
\mathcal L_1 &= \mathcal L_1^{(X)} + \mathcal L_2^{(X)},
\mathcal L_2 &= \frac{\mathcal L_2^{(X)} - \mathcal L_1^{(X)}}{2}.
\label{L12}
\end{align}
We can now substitute Eqs. (<ref>)–(<ref>) into
Eq. (<ref>) to compute the quantum Fisher information matrix
$\mathcal K(\rho_1)$. The final result, assuming
$M\pi_1 \approx M\epsilon = N$, is given by Eqs. (<ref>) with
zero off-diagonal terms.
§ UNKNOWN CENTROID
AND MISALIGNMENT
Our analysis of SPADE in Sec. <ref> and <ref>
assumes that the centroid of the two sources is known exactly and the
device is optimally aligned with the centroid. For astronomy, it is
reasonable to assume that the centroid is known accurately, as
extensive telescopic data on stellar objects should be readily
available and conventional imaging is accurate in estimating the
centroid. Even if the centroid is unknown, stellar objects usually
shine long enough for one to collect ample prior information before
aligning the SPADE device. For microscopy, however, biological
samples may drift more quickly and fluorophores can bleach, giving
little time and few photons for one to estimate both parameters. One
option, to be explored in future work, is to scan the SPADE device
across the image plane in a manner similar to the operation of a
confocal microscope <cit.>.
Another option, illustrated in Fig. <ref>, is to split
the optical field by a beam splitter, measure one output port by
direct imaging, and use the centroid estimate to align SPADE at the
other port in a hybrid scheme. As the overall optical system is linear
with photon counting, the output statistics remain Poisson for
$\epsilon \ll 1$, meaning that the statistics of the measurements are
independent and can be analyzed separately. The penalty of
beam-splitting with the classical sources is simply a reduction of
photon number at each port. With direct imaging offering little
information about $\theta_2$ when Rayleigh's criterion is violated,
the additional information offered by SPADE for a reduced photon
number can still be helpful. The outstanding issues are then the
robustness of SPADE to the misalignment due to imperfect centroid
estimation, and the overhead resources of photons needed to achieve
satisfactory alignment.
A hybrid measurement scheme that splits
the optical field by a beam splitter, measures one output port by a
photon-counting array, and use the centroid estimate
$\check\theta_1$ to align SPADE at the other port.
Let the center of a SPADE device be $\check \theta_1$ and consider
$\theta_1 \neq \check \theta_1$ due to misalignment. For a Gaussian point-spread
function and the Hermite-Gaussian-basis measurement, Eq. (<ref>)
should be generalized to
\begin{align}
P_1(q) &\approx
\frac{1}{2}\Bk{\exp(-Q_1)\frac{Q_1^q}{q!}+\exp(-Q_2)\frac{Q_2^q}{q!}},
\nonumber\\
Q_1 &\equiv \frac{1}{4\sigma^2}\bk{\check \theta_1- \theta_1 + \frac{\theta_2}{2}}^2,
\nonumber\\
Q_2 &\equiv \frac{1}{4\sigma^2}\bk{\check \theta_1 - \theta_1 - \frac{\theta_2}{2}}^2.
\end{align}
Define the level of misalignment as
\begin{align}
\xi &\equiv \frac{|\check \theta_1-\theta_1|}{\sigma}.
\label{xi}
\end{align}
We treat $\xi$ as a systematic error and $\theta_2$ as the parameter
of interest for SPADE. Figure <ref> plots the resulting
Fisher information for several levels of misalignment. It can be seen
that the information degrades with misalignment, but appreciable
enhancements over direct imaging are still present even if
$\theta_2 \ll \sigma$ and the wavefunction overlap is significant, as
long as $\xi \ll 1$. Appendix <ref> confirms this
conclusion numerically for finite photon numbers.
Fisher information for separation estimation
with SPADE with misalignment levels
$\xi = 0,0.1,\dots,0.5$ (solid curves) and direct imaging
(dash-dotted curve). The different solid curves can be
distinguished by their decreasing values with larger misalignments.
To attain a tolerable level of misalignment, $\theta_1$ first needs to
be estimated and $\check \theta_1$ should be aligned with the
estimate. With conventional imaging, the centroid estimation error is
near-optimal and on the order of $\sigma/\sqrt{N}$ in terms of the
root-mean-square value, meaning that the number of extra photons $N_1$
needed to attain $\xi$ is roughly
\begin{align}
N_1 &\sim \frac{1}{\xi^2}.
\label{N1}
\end{align}
An even more realistic analysis would take $\check \theta_1$ to be a stochastic
waveform determined by the centroid measurements and the adaptive
alignment control <cit.>.
For binary SPADE, Eqs. (<ref>) and (<ref>) should be
generalized to
\begin{align}
P_1(q= 0) &\approx \frac{1}{2} \Bk{\exp(-Q_1)+\exp(-Q_2)},
\\
P_1(q> 0) &\approx 1-\frac{1}{2}\exp(-Q_1)-\frac{1}{2}\exp(-Q_2).
\end{align}
Figure <ref> plots the Fisher information for misaligned
binary SPADE, showing a similar degradation behavior to that in
Fig. <ref> for nonzero $\xi$. Significant improvements over
direct imaging are still possible for small separations and
$\xi \ll 1$.
Fisher information for separation
estimation with binary SPADE with misalignment
levels $\xi = 0,0.1,\dots,0.5$ (solid curves) and direct imaging
(dash-dotted curve). The different solid curves can be distinguished
by their decreasing values with larger misalignments.
For two-parameter estimation, consider the hybrid scheme in
Fig. <ref>, assuming 50-50 beam-splitting and binary
SPADE for example. For simplicity, assume that the binary-SPADE output
is used only for separation estimation, such that the total
information matrix with respect to $\theta_1$ and $\theta_2$ remains
diagonal. Compared with direct imaging, the centroid information for
the hybrid scheme is halved, viz.,
\begin{align}
\mathcal J_{11}^{(\rm hybrid)} &= \frac{\mathcal J_{11}^{(\rm direct)}}{2},
\end{align}
but the separation information gained by binary SPADE can be
appreciable, with
\begin{align}
\mathcal J_{22}^{(\rm hybrid)} &=
\frac{\mathcal J_{22}^{(\rm direct)}}{2} +
\frac{\mathcal J_{22}^{(\rm b)}}{2}.
\end{align}
The net performance of the hybrid scheme can be quantified in terms of
the Cramér-Rao bounds for locating $X_1$ and $X_2$. For a diagonal
information matrix $\mathcal J$ with respect to $\theta_1$ and
$\theta_2$, the bound on the mean-square error $\Sigma^{(X)}$ of
estimating either $X_1 = \theta_1-\theta_2/2$ or
$X_2 = \theta_1+\theta_2/2$ is simply
\begin{align}
\Sigma_{ss}^{(X)} &\ge \frac{1}{\mathcal J_{11}}+
\frac{1}{4\mathcal J_{22}},
\quad
s = 1,2,
\end{align}
which demonstrates the detrimental effect of small $\mathcal J_{22}$
for localization. Figure <ref> compares the localization
bounds for the hybrid scheme and direct imaging in log-log scale. For
small separations, it can be seen that the increased separation
information in the hybrid scheme more than compensates for the reduced
centroid information and allows localization errors substantially
lower than those for direct imaging. With a higher $N_1 = N/2$, more
accurate centroid information from the imaging port can be used to
reduce the misalignment at the SPADE port, and performance converging
to the ideal $\xi = 0$ case in Fig. <ref> can be expected
for high $N$.
Cramér-Rao bounds on the mean-square
error of estimating $X_1$ or $X_2$ for a 50-50 hybrid scheme (solid)
and direct imaging (dash-dotted). Note that the log-log scale is
used here for clarity, unlike all the other plots in this paper. The
vertical axis is normalized with respect to the error of locating an
isolated source with direct imaging.
§ MONTE CARLO ANALYSIS
To confirm that the classical Cramér-Rao bounds satisfactorily
represent the actual performance of SPADE for finite photon numbers,
here we simulate the device output data numerically, apply
maximum-likelihood estimation, and investigate the resulting error.
To refine our error analysis, we condition our results on the total
number of detected photons $L$, which is obtained after an experiment,
rather than the average photon number $N$ <cit.>. It is not
difficult to show that, conditioned on $L$, the classical and quantum
Fisher information retain their expressions except that $N$ is
replaced by $L$. The error bounds become
\begin{align}
\frac{1}{\mathcal J_{22}'^{(\rm HG)}}
&\approx \frac{1}{\mathcal K_{22}'}
\approx \frac{4\sigma^2}{L}.
\label{crb_L}
\end{align}
It can also be shown that the sufficient statistic $\sum_q q m_q$ in
the maximum-likelihood estimatior for SPADE in Eq. (<ref>) is
Poisson with mean $LQ$, so it is simple to generate samples of the
maximum-likelihood estimates $\check Q_{\rm ML}$ and
$\check \theta_{2\rm ML}$ according to Eq. (<ref>).
Figure <ref> plots the simulated mean-square errors,
normalized with respect to Eq. (<ref>), for several values of
$L$. It is intriguing to see that, as $\theta_2/\sigma \to 0$, the
errors go below the bounds. This is a well known statistical
phenomenon called superefficiency <cit.>, as the
maximum-likelihood estimator here is actually biased for finite
samples, and the simple Cramér-Rao bounds considered here need not
apply. In asymptotic frequentist statistics, superefficiency is not
regarded as an important idea <cit.>, because a superefficient
estimator can beat the Cramér-Rao bound only on a set of points with
zero measure in the asymptotic limit <cit.>, suggesting
that any region of superefficiency should shrink for larger samples,
as also shown in Fig. <ref>, and its usefulness is
increasingly limited. A Bayesian version of the Cramér-Rao bound
<cit.> can also be used to bound the global or minimax error
of any biased or unbiased estimator; the Fisher information still
plays a decisive role in the Bayesian bound and its significance as a
precision measure remains strong in Bayesian and minimax statistics
Simulated mean-square errors for SPADE with
maximum-likelihood estimation, conditioned on $L$ detected photons.
Note that the vertical axis is normalized with respect to the
Cramér-Rao bounds $4\sigma^2/L$, so the plotted values are the
actual errors magnified by $L/(4\sigma^2)$. Each error is computed
by averaging $10^5$ simulations, and the lines connecting the data
points are guides for eyes.
For our present purpose, the main point of Fig. <ref> is
that the errors remain less than twice the Cramér-Rao bound at
worst and even offer the pleasant surprise of superefficiency for
small separations. The overall closeness of the errors to the
Cramér-Rao bounds supports our use of the Fisher information to
represent the performance of SPADE.
For binary SPADE, the Fisher information conditioned on $L$ has the
same form as Eq. (<ref>), and the Cramér-Rao bound can be
expressed as
\begin{align}
\frac{1}{\mathcal J_{22}'^{(\rm b)}} &\approx \frac{4\sigma^2}{L}
\frac{1-\exp(-Q)}{Q\exp(-Q)}.
\label{crb_bspade}
\end{align}
The sufficient statistic $m_0$ in $\check\theta_{2\rm ML}^{(\rm b)}$
given by Eq. (<ref>) is binomial and also simple to
generate. In case $m_0 = 0$, we set $\check\theta_2 = 2\sigma$, the
maximum of our considered range of
$\theta_2$. Figure <ref> plots the simulated
mean-square errors for binary SPADE with otherwise the same parameters
as those for Fig. <ref>. For small $\theta_2/\sigma$, the
errors follow very similar trends as their counterparts in
Fig. <ref>, and for larger $\theta_2/\sigma$ the errors
begin to follow the rising Cramér-Rao bound according to
Eq. (<ref>). This supports our use of the Fisher
information to represent the performance of binary SPADE.
Simulated mean-square errors for
binary SPADE with maximum-likelihood estimation, conditioned on $L$
detected photons. Note that the vertical axis is normalized with
respect to $4\sigma^2/L$, so the plotted values are the actual
errors magnified by $L/(4\sigma^2)$. Each error is computed by
averaging $10^5$ simulations, and the lines connecting the data
points are guides for eyes.
To investigate the effect of misalignment described in
Appendix <ref>, Fig. <ref> plots the
simulated errors for binary SPADE with a misalignment level defined in
Eq. (<ref>) given by $\xi = 0.1$. The overhead photon number
required to achieve $\xi = 0.1$ is $N_1\sim 100$ according to
Eq. (<ref>) and negligible if $L\gg N_1$. Since $\xi$ is unknown in
reality, the maximum-likelihood estimator used in the simulations
assumes zero misalignment for simplicity. Despite the model mismatch,
the errors remain close to the Cramér-Rao bound, especially for
larger $L$, and substantially below the bound for direct imaging.
Simulated mean-square errors of binary
SPADE with misalignment level $\xi = 0.1$, conditioned on $L$
detected photons. The maximum-likelihood estimator that assumes no
misalignment is used. Note that the vertical axis is normalized with
respect to $4\sigma^2/L$. Each error is computed by averaging
$10^5$ simulations, and the lines connecting the data points are
guides for eyes. The errors are substantially below the Cramér-Rao
bound for direct imaging (dash-dotted curve).
For a given $N$, $L$ has a mean $M\epsilon = N$ and standard deviation
$\sqrt{M\epsilon(1-\epsilon)} \approx \sqrt{N}$. This means that the
distribution of $L$ becomes increasingly sharp around the mean at
$L = N$ for large $N$, and we can expect the performance for a given
$L = N$ to be an increasingly accurate approximation of the average
performance in the given-$N$, random-$L$ scenario.
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1511.00268
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LIMMS/CNRS-IIS, University of Tokyo, Komaba 4-6-2 Meguro-ku, Tokyo, Japan
Institut Lumière Matière, UMR5306 Université Claude Bernard Lyon 1 - CNRS, Université de Lyon, 69622 Villeurbanne, France
LIMMS/CNRS-IIS, University of Tokyo, Komaba 4-6-2 Meguro-ku, Tokyo, Japan
Laboratoire de photonique et de nanostructures, CNRS, route de Nozay, 91460 Marcoussis, France
LIMMS/CNRS-IIS, University of Tokyo, Komaba 4-6-2 Meguro-ku, Tokyo, Japan
LIMMS/CNRS-IIS, University of Tokyo, Komaba 4-6-2 Meguro-ku, Tokyo, Japan
We report the splitting of an oscillating DNA circuit into $\sim 700$ droplets with picoliter volumes. Upon incubation at constant temperature, the droplets display sustained oscillations that can be observed for more than a day. Superimposed to the bulk behaviour, we find two intriguing new phenoma – slow desynchronization between the compartments and kinematic spatial waves – and investigate their possible origins. This approach provides a route to study the influence of small volume effects in biology, and paves the way to technological applications of compartmentalized molecular programs controlling complex dynamics.
Recent progresses in molecular engineering have allowed the construction of synthetic biochemical systems that display subtle dynamics such as oscillations or multistability <cit.>. The building of such molecular programs from scratch yields a unique glimpse of the challenges and locks that evolution had to overcome in its drift towards more and more complex life forms. Moreover, the inherent biocompatibility of these chemical circuits supports their use to monitor and control biological processes.
Compartmentalization of these man-made dynamic systems would offer tantalizing additional possibilities. Cell-sized compartments ($\sim 1-\SI{100}{\micro\metre}$) may provide a model of biotic or prebiotic organization <cit.>. For example, stochastic effects arising from low numbers of molecules in small spaces are very important in biology <cit.>, but hardly explored outside of living systems. Synchronization of many autonomous elements using quorum sensing provides another example <cit.>.
Beside these basic biological motivations, micro-compartmentalization is also the obvious way forward for the exploration of man-made molecular circuits. First, it permits the running of hundreds of circuits using volumes that would have yielded a single experiment otherwise. Such high-throughput will be required to tune the many parameters controlling the behavior of molecular assemblies. Even in the case where all the compartments possess an identical “program”, compartmentalization may yield new analytical concepts such as digital PCR <cit.>. Second, splitting - and later establishing controlled connections between - spatially distributed molecular circuits may unleash the computing power of molecular programs <cit.> by removing cross-talks and allowing the reuse of modules.
Yet, compartmentalization and long-term monitoring of biochemical reactions in micro-compartments has been difficult to achieve. Indeed, the reactions involved must run sustainably in closed systems and the potentially detrimental effects of large surface/volume ratios and leaks need to be tightly controlled.
a) Water-in-oil droplets are generated inside a PDMS chip with a flow-focusing junction. b) Droplets are then transferred to a glass chamber sealed with araldite for incubation and observation. c) The chamber is placed under a heated stage and observed by fluorescence microscopy.
While biochemical systems with a simple dynamic such as qPCR were monitored in droplets <cit.>, out-of-equilibrium biochemical circuits pose distinct and additional challenges. PCR is driven sequentially by an external operator, uses only a single enzyme and lasts rarely more than 45 minutes. By contrast, our dynamic systems display oscillatory behaviour, are fully autonomous, require 3 enzymes, and last for days.
More importantly, sustainability of oscillations depends on a much more delicate balance between reagents, buffer and temperature than PCR. Most biochemical assays typically consist in a simple relaxation toward a stable steady state, but oscillations arise from the destabilization of all steady states, which requires tight control over reaction parameters.
Here we report the compartmentalization and day-long tracking of 700 oscillators into picoliter droplets. This represents an increase of throughput by two orders of magnitudes and a decrease of volume by 4 orders of magnitudes compared to the literature on synthetic DNA systems1. We show that with an experimental setup minimizing evaporation, droplets act as independent chemical containers that satisfyingly reproduce bulk conditions. Moreover, compartmentalization also reveals two striking phenomena that may remain hidden in bulk: slow desynchronization and kinematic spatial waves.
We use here a recently reported synthetic biochemical oscillator <cit.>. While simple to prepare and well characterized, it involves subtle enzymatic kinetics, which should put a stringent test on the use of droplets as compartments. The biochemical system reproduces the ecological predator-prey mechanism: molecular preys catalyse their own replication, but also serve as fuel for the replication of their molecular predators. Both prey and predators are continuously degraded by an exonuclease. This system results in robust oscillations, which are monitored via the fluorescence intensity of a dye bound to the DNA template encoding the circuit <cit.>.
Droplets were generated by a flow-focusing junction fabricated in a PDMS chip. For each run, we generated thousands of droplets from the same reactive mix. The droplets were then transferred to a chamber, which was formed between two glass slides sealed by araldite. The chamber was kept at constant temperature (45.5) by placing it under the heated glass plate of a microscope stage equipped with a temperature controller (Fig. <ref>). A field of view contained about 1000 droplets (movies in ESI). While the oscillations lasted for 2 days, we restricted the analysis to one day in order to increase the number of trajectories successfully tracked <cit.>.
a) Microscopy images showing the first kinematic spatial wave. The arrow indicates the travelling direction (colours reflect the intensity of fluorescence). The temperature is set at 45.5. b) Top. Superimposed fluorescence traces of all successfully tracked droplets (690 traces). The dashed box indicates the peak corresponding to the wave shown in a). Bottom, bulk fluorescence of the oscillator (set at the same temperature) measured in a 20 tube in a qPCR machine. c) Wave speed as a function of the wave number, fitted with a kinematic wave model (S11, ESI). d) Kymograph showing the desynchronization of a column of droplets lying on the same initial wavefront (column indicated by a vertical line in a)).
Fig. <ref>a shows a representative field of view and time traces for the fluorescence of droplets. Most droplets are monodisperse, with a diameter of $\sim\SI{100}{\micro\metre}$ and a dispersion of $\sim15\%$. This corresponds to a volume of $\sim 500$ picoliters, which is 4 orders of magnitude smaller than bulk reaction volume (20). A few droplets are significantly larger, which may result from coalescence during the generation and transfer of droplets. The fluorescence of the majority of droplets oscillates, and the oscillation of each individual droplet is similar to the bulk control.
The collective behaviour was more surprising: spatial waves move across the chamber while slow desynchronization ultimately sets each droplet’s fluorescence on a seemingly independent trajectory (Fig. <ref>b).
Since the contents of all droplets derive from the same mix, we expect them to be initially synchronized. Indeed, noting $T_i$ the time required to reach the $i^\text{th}$ peak (Fig. <ref>a), we measure a standard deviation of 7 minutes for $T_1$. After 15 hours, the standard deviation of $T_7$ had increased to 35 minutes (25% of the mean period). At this point no coherence can be visually detected in the droplet population. We tested if this desynchronization was specific to compartmentalization or also existed in bulk. In 20 tubes, we observed a similar albeit less pronounced desynchronization (S8, ESI). This common desynchronization points to the influence of an external factor such as temperature on the period of oscillations. We characterised the sensitivity of oscillator to temperature in bulk (S9, ESI) and found that an increase of temperature of 2 lengthens the period by 25%.
Desynchronization is not spatially random, but manifests itself initially as travelling waves (movies M1 and Fig. <ref>a). One may misinterpret spatial waves as the result of diffusive transport of reagents. However, pseudo (also called kinematic) waves may appear when properties determining phases or periods vary spatially <cit.>. In view of the temperature-sensitivity of the oscillator, a gradient of temperature could not only desynchronize the droplets, but also generates kinematic waves.
The kinematic wave model predicts in particular a wave speed inversely proportional to the wave number, which we observed experimentally (Fig. <ref>c, S11 ESI). The kinematic model also implies a thermal gradient of 0.3, which agrees with the measured gradient of 0.15 (S10, ESI).
On the contrary, the experimentally observed speeds of the waves do not support a model where waves are caused by diffusion of DNA strands between droplets. This model would be consistent with a time for droplet-to-droplet diffusion of $\sim 5$ minutes (S11,ESI).
This timescale is 2 orders of magnitude larger than the diffusion of a smaller molecule (fluorescein) between droplets stabilized by a more permeable surfactant (Span-80) <cit.>. The timescale is also incompatible with the possibility of digital PCR in droplets (which lasts 30 minutes) <cit.>.
Distribution of $T_i$, the time it takes for the fluorescence of a droplet to reach its $i^\text{th}$ (extinction) peak, measured from the beginning of the recording. a) Histogram of $T_i$ for the 690 traces from Fig. <ref>b, for the first 7 peaks. b) Histogram of $T_i$ for 59 traces of a mixed-population experiment where the oscillations of one population are delayed compared to the other. The black line shows the best fit with a single (a) or two (b) Gaussian distributions to each peak.
To further test that diffusion between droplets was negligible, we prepared two populations of oscillating droplets with a delay between them. One population was pre-incubated at reaction temperature (45.5) for approximately half a period, thus allowing the oscillation to progress. The remaining population was kept at room temperature where reactions are much slower. The two populations were then mixed and incubated together on stage. If diffusion between droplets is negligible, the delay between the two populations of droplets should persist over time. The movie M2 shows the sustained presence of two populations. Fig. <ref>b shows the distribution of time-to-peak for this experiment. The initial delay is clearly observable from the shape of $T_1$, where the populations are easily distinguished. This bimodal distribution was maintained at least until $T_6$ despite the widening of both populations (overall standard deviation of 16 minutes at $T_1$ versus 51 minutes for $T_6$, compared to 7 and 29 minutes for T1 and $T_6$ (in Fig. <ref>a).
This population stability supports the functioning of each droplet as an independent reactor. The kinematic waves suggest that droplets are essentially identical but placed in a thermal gradient. Yet, some droplet-specific desynchronization is also visible in the movies (Fig. <ref>d) and we cannot rule out that intrinsic factors such as stochasticity or partitioning noise play a role in the observed dynamic. Predator-prey oscillators are nonlinear and their autocatalytic loops may amplify minute differences or fluctuations between droplets. Noise-induced stochasticity is an important determinant of biological circuits dynamic.
In conclusion, the picoliter droplet-based compartments used here satisfactorily reproduce the behaviours observed in bulk, in spite of reducing volumes by a factor of $10^4$. Therefore this approach may open the high-throughput experimental exploration <cit.> of the range of dynamics displayed by synthetic molecular programs when varying their control parameters. Another interesting direction would be to engineer a form of controlled molecular communication between the droplets in order to trigger and tune the onset of collective behaviours4. A further shrinking in size would better match the dimensions of biological cells and provides a unique platform to study the impact of small molecule numbers on the dynamics of dissipative reaction networks.
Compartmentalized oscillators also offer an in vitro model to investigate pattern-formation in biology. Kinematic waves clearly illustrate that spatio-temporal patterns - ubiquitous in biological development- need not arise from a reaction-diffusion mechanism, but may also originate from a spatial gradient of parameters. Purposely tailoring spatial gradients of parameters will offer a route to create novel classes of patterns.
This research is supported by the CNRS (France) and the JSPS (Japan). We thank V. Taly, H. Guillou and A. Padirac for advice as well as B.J. Kim and R. Ueno for using their thermal camera. During the course of our research, E. Winfree and F. Simmel kindly informed us of their ongoing work on the effect of compartmentalization and stochasticity in biochemical oscillators.
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