text:
\section{The Elsevier article class}

\section{Introduction}\label{sec:intro}

Behaviour of low-dimensional systems is a prolific research field~\cite{cea_numerical_2020,amnuanpol_2021_bucklingInstability,Chen_2022_TiltElasticSheets,hanakata_thermal_2021,hanakata_anomalous_2022,Jain_2021_CompressionBucklingSheets,Poinclow_2021_BookResponse,LeDoussal_2021_BucklingTransitionField,Thibado_2020_FluctuationGraphene}. Two-dimensional materials exhibit intriguing properties~\cite{Novoselov_2004_OriginGraphene,Changgy_2008_ElasticPropertiesGraphene,Neto_2009_ElectronicPropertiesGraphene,Samy_2021_MoS2Properties},  such as flexibility, high thermal and electrical conductivity, transparency, low Joule effect, to name just a few. These relevant features  can be useful for many applications, from  realisation of microsystems for healthcare~\cite{Huang_2019_GrapheneMonitoring,zhang_2022_HealthSensor} up to energy harvesting~\cite{Thibado_2020_FluctuationGraphene,mangum_mechanisms_2021,gikunda_2022_ArrayEnergyHarvesting}. For a recent review on the effects of strains in graphene and other two dimensional materials, see Ref.~\cite{amorim_novel_2016}.

An outstanding phenomenon in the context of mechanical properties of elastic systems is the buckling transition. In structural engineering, buckling is the abrupt change of the considered structure, from a flat  to a non-zero curvature (buckled) state, under load. A prototypical example occurs when a long column, i.e. a one-dimensional system, buckles under axial compression, which is known as Euler buckling~\cite{Kelly_BucklingColumn}. Nevertheless, not only is buckling an engineering subject but also an appealing physical behaviour to be better understood from a theoretical standpoint. Moreover, the buckling phenomenon emerges in a wide range of systems, from low-dimensional systems \cite{hanakata_thermal_2021,Singh_2015_BucklingMOS2,plummer_buckling_2020,shankar_thermalized_2021} to cells~\cite{Luo_2007_BucklinginCells} or in device development~\cite{Hu_2017_BucklinginEngineer}.

Graphene sheets are not perfectly two-dimensional. In fact, they display transversal displacements (ripples) due to thermal fluctuations~\cite{meyer_structure_2007,fasolino_2007_intrinsicRipples} that stabilise the membrane.\footnote{It is important to recall that the Mermin-Wagner Theorem~\cite{MerminWagner_1966_Theorem} forbids the existence of purely two-dimensional stable crystals. Also, the instability of two-dimensional crystals can be understood within the context of Landau's theory of fluctuations~\cite{landau_statistical_2013}.} Rippling has been studied employing models where its origin and behaviour are linked with the coupling between electronic and elastic degrees of freedom, i.e. the electron-phonon coupling~\cite{cea_numerical_2020,fasolino_2007_intrinsicRipples,san-jose_electron-induced_2011,Bonilla_2012_RipplesModel,bonilla_ripples_2012,ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017}.

In slender structures such as graphene, there also appears a buckling transition from a flat (or rippled, when taking into account thermal fluctuations) state to a buckled state under mechanical load \cite{hanakata_thermal_2021,plummer_buckling_2020}. In addition, scanning tunneling microscopy (STM) experiments have shown that, even in absence of mechanical load, local heating induces a transition from a soft rippled sheet to a hard buckled graphene membrane~\cite{schoelz_graphene_2015,neek-amal_thermal_2014,lindahl_2012_BuckledMembranes,eder_2013_probing}. This transition is somehow unexpected, since the increase of temperature makes the membrane go from a less ordered state (the flat or rippled one) to a more ordered (buckled) state.

The complexity of the interactions in real low-dimensional elastic systems makes it difficult to obtain analytical results, either exactly or with controlled approximations, that improve our understanding of the observed buckling transitions. Therefore, mesoscopic models are relevant, since they contain---in a simplified way---the main ingredients of low-dimensional elastic systems and allow for a detailed analytical approach. Specifically, models based on the coupling between elastic degrees of freedom and Ising spins, termed spin-string and spin-membrane in the one-dimensional and two-dimensional cases, respectively, are able to reproduce in a  qualitative way the aforementioned transition.
Despite their simplicity, these mesoscopic models present a rich behaviour, with a complicated phase diagram that include buckled phases~\cite{ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017,ruiz-garcia_ripples_2015}.

Nevertheless, the currently available spin-string and spin-membrane models~\cite{ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017,ruiz-garcia_ripples_2015}, although giving an appealing physical picture, are not completely satisfactory from a fundamental point of view. First, some of the microscopic parameters controlling the interactions have to depend on the system size to ensure that the thermodynamic properties have the correct behaviour in the large system size limit, e.g. an intensive transition temperature and an extensive free energy. Second, they are not fully consistent with the classical theory of elasticity~\cite{landau_elasticity_1986}, since rotational symmetry is broken in absence of external forces.

In this work, we propose the simplest model that keeps the appealing physical picture of the models in Refs.~\cite{ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017,ruiz-garcia_ripples_2015} and mends the above mentioned fundamental flaws. We consider a classical elastic lattice (one or two-dimensional) characterised by the out-of-plane deformations and  internal degrees of freedom described by a pseudospin variable.\footnote{We use the term pseudospin to emphasise that it is not the real spin but a two-state variable that mimics in a simple way the range of  values taken by the internal degrees of freedom.} Elastic deformations are modelled by a harmonic interaction between the atoms that depends on the curvature of the membrane, the repulsive interaction between electrons is modelled by an antiferromagnetic interaction between pseudospins, and the electron-phonon coupling is modelled by the coupling between pseudospins and the membrane curvature. In this way, we will show that the state of the system can be completely characterised by the membrane curvature, the equilibrium value of which only depends on two parameters:  temperature $T$ and  antiferromagnetic coupling constant $J$.

The paper is organised as follows. In Sec.~\ref{sec:model}, we define the one-dimensional model and study its equilibrium behaviour, by deriving a free energy functional from the probability of a given profile. In Sec.~\ref{sec:eq-profiles}, we introduce the Euler-Lagrange equation, which provides us with the equilibrium profile, and define dimensionless variables. In Sec.~\ref{sec:buckling}, we analyse, both analytically and numerically, the emergence of buckled states and put forward the main results of our study. In Sec.~\ref{sec:spin-membrane-honeycomb}, we generalise our one-dimensional model to the two-dimensional case, focusing ourselves on the specific case of a honeycomb lattice---that of actual graphene. We present the main conclusions of our work and perspectives for future research in Sec.~\ref{sec:conclusions}. The
appendices deal with some non-essential technical details that are omitted in the main text.

\section{Rotationally invariant spin-string model}\label{sec:model}

We consider a string on a one-dimensional lattice, with lattice
constant $a$. At each site, $j=0,\ldots,N$, there is a particle of mass $m$ that is characterised
by its transversal displacement $u_{j}$,  and, in
addition, there is a
``pseudospin'' variable $\sigma_j=\pm 1$.  We introduce the following rotationally invariant
energy (or Hamiltonian) for any configuration of the elastic modes
$\bm{u}$, their associated momenta $\bm{p}$, and the pseudospins $\bm{\sigma}$,
\begin{align}
    \mathcal{H}(\bm{u},\bm{p},\bm\sigma)=
    \sum_{j=0}^N
  \bigg[ & \frac{p_j^2}{2m}+
           \frac{k}{2}\left(u_{j+1}-2u_j+u_{j-1}\right)^2-h
                  \left(u_{j+1}-2u_j+u_{j-1}\right)\sigma_j
                                           \nonumber \\
  &  + J
  \sigma_{j+1}\sigma_j\bigg].
  \label{eq:energy-rot-inv}
\end{align}
The first term on the rhs is the kinetic energy, the
second one stands for the elastic contribution to the energy---with
elastic constant $k$, the third one involves the interaction between
the elastic displacements and the pseudospins---the parameter $h$
tunes the strength thereof, and the last one represents an
antiferromagnetic nearest-neighbour interaction among the
pseudospins---with coupling constant $J$. The pseudospins do
not correspond to actual spin variables, they model in a simple way
internal degrees of freedom---as discussed below.

The above system is a mesoscopic toy model, which tries to resemble in
a simple manner some of the main features of low-dimensional elastic
systems, such as graphene. Therein, the interaction between
the elastic modes and the pseudospins would model the electron-phonon
coupling,\footnote{More specifically, the coupling between the
  transverse elastic modes and out-of-plane electrons, which can be
  thought to be in two states corresponding to the two different sides
  of the plane.} and the antiferromagnetic interaction between the
pseudospins account for the Coulomb repulsion of
electrons.\footnote{Therefore, nearest-neighbour out-of-plane
  electrons prefer to be at different sides of the plane.}  There are
two main differences with respect to other proposals in the
literature~\cite{ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017,ruiz-garcia_ripples_2015}:
(i) the elastic term is proportional to the square of the (discrete)
curvature and not to the square of the (discrete) gradient, and (ii)
the coupling between the elastic modes and the pseudospins also
involves the curvature and not the transverse displacement. The key
role played by the curvature of the system, as shown below, makes
our model consistent with the classical theory of
elasticity~\cite{landau_elasticity_1986}---at variance with
previous spin-string (or spin-membrane in the two-dimensional
case)
models~\cite{ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017,ruiz-garcia_ripples_2015}. In
particular, a flat profile, $u_{j}\equiv 0$, and a rigid rotation
thereof, $u_{j}=A j$ with constant $A$, have the same
energy in Eq.~\eqref{eq:energy-rot-inv}.\footnote{In general, any two profiles differing just by a linear function $Aj+B$ has the same energy. Note that $B$ stems from a translational offset while $Aj$ represents a small rotation, $\sin(A) \simeq A$. The small rotation approximation is implicit in the development of the elastic theory.}

Thermal equilibrium of the system at temperature $T$ is described by
the canonical distribution
$P_{\text{eq}}(\bm{u},\bm{p},\bm{\sigma})=\exp\left[-\beta
  \mathcal{H}(\bm{u},\bm{p},\bm\sigma)\right]/Z$,
where $\beta=(k_{B}T)^{-1}$, with $k_{B}$ and $Z$ being the Boltzmann
constant and the partition function, respectively. We are interested in the
equilibrium profiles of the string, so we integrate over $\bm{p}$ and
$\bm{\sigma}$ to get
\begin{equation}\label{eq:free-energy-discrete}
  P_{\text{eq}}(\bm{u})\propto e^{-\beta \mathcal{F}(\bm{u})},
\end{equation}
where the free energy of the string $\mathcal{F}(\bm{u})$ is
\begin{equation}
  \mathcal{F}(\bm{u})=\sum_{j=0}^N
  \left[\frac{k}{2}\left(u_{j+1}-2u_j+u_{j-1}\right)^2
  \right]-k_{B}T\ln Z_{\text{int}}(\bm{u}).
\end{equation}
Above, $Z_{\text{int}}(\bm{u})$ stands for the partial partition function associated
with the internal degrees of freedom, which involves a sum over all
the pseudospin configurations
\begin{equation}
  Z_{\text{int}}=\sum_{\bm{\sigma}}e^{-\beta\sum_{j=0}^{N}
    \left[-h\left(u_{j+1}-2u_j+u_{j-1}\right)\sigma_j
                                + J \sigma_{j+1}\sigma_j \right] }.
\end{equation}

Now we introduce a continuum limit by assuming that the displacements
$u_{j}$ vary slowly with $j$, i.e., $u_{j}\to u(x)$ with $x=ja$. The
total length of the system is $L=Na$.  In this way,
\begin{equation}
  u_{j+1}-2u_{j}+u_{j-1}= a^{2}\chi(x)+O(a^{4}),
\end{equation}
where we have defined the curvature of the profile $\chi(x) \equiv u''(x)$, and
\begin{subequations}
  \begin{align}
    \frac{k}{2}(u_{j+1}-2u_{j}+u_{j-1})^{2}=\frac{k_{0}}{2}\chi^{2},\\
    h (u_{j+1}-2u_{j}+u_{j-1}) \sigma_{j}=h_{0}\chi\sigma_{j}.
\end{align}
\end{subequations}
The continuum limit in the spin variables is skipped willingly, since a marginal sum over all spin configurations will be carried out in brief. Terms with subdominant powers of $a$ have been omitted, and scaled parameters for the elastic constant and the
strength of the pseudospin-elastic interaction have been introduced as
\begin{equation}\label{eq:h0-k0-scalings}
  k_{0}=k a^{4}, \qquad   h_{0}=h a^{2}.
\end{equation}
In the continuum limit, the equilibrium probability of a certain
string profile $u(x)$ becomes a functional thereof, specifically
\begin{equation}
  P_{\text{eq}}[u]\propto e^{-\beta\mathcal{F}[u]}, \quad \mathcal{F}[u]=n\int_{0}^{L}
  dx f(\chi),
\end{equation}
where $n\equiv N/L$ is the number density, and the free energy density (per particle) $f$ is given by
\begin{subequations}\label{eq:free-energy-density}
\begin{align}
  f(\chi)&=\frac{k_{0}\chi^2}{2}
  -\beta^{-1} \ln\zeta_{\text{int}}^{\text{1d}} (\beta h_0 \chi, \beta J), \\

  \zeta_{\text{int}}^{\text{1d}}(\beta h_0 \chi, \beta J )&\equiv e^{-\beta J }
  \cosh\left(\beta h_0 \chi  \right)+e^{\beta J }
                   \sqrt{1+e^{-4\beta J}\sinh^{2}\left( \beta h_0 \chi \right)},                 \label{eq:one-particle-zeta}
\end{align}
\end{subequations}
Our notation explicitly tells us that $f$ only depends on the profile
through its curvature $\chi$. The second term in $f(\chi)$ comes from
the sum of the spin variables, $\zeta_{\text{int}}^{\text{1d}}$ is the
one-particle partition function of a one-dimensional antiferromagnetic Ising chain
with coupling $J$ and external field $h_{0}\chi$---e.g.\ see
Ref.~\cite{feynman_statistical_1996}.

\section{Equilibrium profiles}\label{sec:eq-profiles}

The equilibrium profiles $u_{\text{eq}}(x)$ are those that maximise the
probability distribution $P_{\text{eq}}[u]$ or, equivalently, those that
minimise the free energy functional $\mathcal{F}[u]$. Therefore, we consider the first
variation of the free energy functional upon the change $u\to u+\delta
u$,
\begin{align}

         \delta F[u]=&\int_0^L d{x}\,  \delta u\, \frac{d^{2}}{dx^{2}}\!\pr{\pdv{f}{\chi}}
   + \corch{\delta u'\,\pdv{f}{\chi}-\delta u\, \dv{}{x}\!
                          \pr{\pdv{f}{\chi}} }_0^L.
     \label{eq:free-energy-var}
\end{align}
The expression for $\delta F[u]$ is more involved than
in the usual case where the integrand only depends on the first
derivative, but the integral and the boundary terms must vanish
separately nonetheless~\cite{gelfand_calculus_2000,lanczos_variational_1970}.\footnote{For instance, for free boundary conditions, the variation $\delta u(x)$ for $x\in(0,1)$ and
$\{\delta u(0),\delta u(1),\delta u'(0),\delta u'(1)\}$ at the
boundaries are independent.} Considering the vanishing of the integral term, one obtains the Euler-Lagrange
equation
\begin{equation}
  \left.\dv[2]{}{x}\!\pr{\pdv{f}{\chi}}\right|_{\text{eq}}=0 \implies  \left.\pdv{f}{\chi}\right|_{\text{eq}}=Ax+B,
\end{equation}
where $A$ and $B$ are arbitrary constants, to be determined by
imposing the boundary conditions. The boundary conditions depend on
the physical situation: here, for the sake of simplicity we are going
to consider that the ends of the string are fixed, so that
\begin{equation}\label{eq:fixed-ends-bc}
  u(0)=u(L)=0,
\end{equation}
but the value of $u'$ at the boundaries is free---the
so-called supported boundary
conditions~\cite{landau_elasticity_1986}. Therefore, $\pdv*{f}{\chi}$
must vanish at both boundaries for the equilibrium profile, which entails $A=B=0$.

The
equilibrium profile is thus provided by
\begin{align}\label{eq:profile-eq}
  \left.\pdv{f}{\chi}\right|_{\text{eq}}\!\!\!=0 \iff k_{0}\chi_{\text{eq}}=\left. k_B T \pdv{\ln\zeta_{\text{int}}^{\text{1d}}

  }{\chi}\right|_{\chi=\chi_\text{eq}}\!\!\!=h_{0}\frac{e^{-\frac{2J}{k_B T}}\sinh{\left(\frac{h_0 \chi_{\text{eq}}}{k_B T}\right)}}{\sqrt{1+e^{-\frac{4J}{k_B T}}\sinh^2{\left(\frac{h_0 \chi_{\text{eq}}}{k_B T}\right)}}},
\end{align}
with the boundary conditions
\eqref{eq:fixed-ends-bc}. Equation~\eqref{eq:profile-eq} is a transcendental equation for the equilibrium curvature $\chi_{\text{eq}}$, the
solution of which can be written as
$\chi_{\text{eq}}=\chi_{\text{eq}}(T,J,k_{0},h_{0})$.\footnote{Zero curvature profiles, i.e. linear profiles with $\chi=0$, always solve Eq.~\eqref{eq:profile-eq}.} The equilibrium curvature is constant, independent of position, and so is $f_{\text{eq}}=f(\chi_{\text{eq}})$. Therefore, the equilibrium free energy is $\mathcal{F}_{\text{eq}}=n Lf_{\text{eq}}=Nf_{\text{eq}}$, i.e. it is extensive---note that the constant $\chi_{\text{eq}}$ is an intensive quantity, independent of system size.

If $\chi_{\text{eq}}$ is a solution of Eq.~\eqref{eq:profile-eq}, then
$-\chi_{\text{eq}}$ is also a solution. This is a consequence of the free
energy density being an even function of $\chi$, and it is clear on a
physical basis: the specular reflection of an equilibrium profile with
respect to the $x$ axis is also an equilibrium profile, since there is
no external field. Therefore, without loss of generality, we restrict
ourselves to solutions with positive curvature in the remainder of this work.

To better characterise the equilibrium curvature $\chi_{\text{eq}}$, it is
adequate to introduce dimensionless variables: this allows us to
identify the natural units of length and energy---or temperature---in
our model system, and elucidate the dimensionless combinations of the
system parameters on which the curvature
depends. Equation~\eqref{eq:profile-eq} tells us the characteristic
length $\ell_{0}$, which in turn determines the characteristic
temperature $T_{0}$,
\begin{equation}
  \ell_{0}=\frac{k_{0}}{h_{0}}, \qquad T_{0}=\frac{h_{0}^{2}}{k_{B}k_{0}}.
\end{equation}
Then we define dimensionless displacement $u^{*}$, position
$x^{*}$, curvature $\chi^{*}$, temperature $T^{*}$, and coupling constant $J^{*}$ by
\begin{equation}\label{eq:dimensionless-variables}
  u^{*}=\frac{u}{\ell_{0}}, \quad x^{*}=\frac{x}{\ell_{0}}, \quad
  \chi^{*}=\ell_{0}\chi, \quad
  T^{*}=\frac{T}{T_{0}}, \quad J^{*}=\frac{J}{k_{B}T_{0}}.
\end{equation}
Consistently, we define $\beta^*=k_B T_0 \beta=1/T^*$,  $L^{*}=L/\ell_{0}$, $n^*=n\ell_0=N/L^*$, and the dimensionless free energy $\mathcal{F}^{*}\equiv\mathcal{F}/k_{B}T_{0}$. Then, we can write
\begin{subequations}
\begin{align}
  \mathcal{F}^{*}[u^{*}](J^{*},T^{*})&=n^*\int_{0}^{L^{*}}  dx^{*}
  f^{*}(\chi^{*};J^{*},T^{*}), \\
 f^{*}(\chi^{*};J^{*},T^{*})&\equiv \frac{1}{2}{\chi^{*}}^{2}-T^{*}\ln
                              \zeta_{\text{int}}^{\text{1d}}(\beta^* \chi^{*},\beta^* \! J^{*}),
                              \label{eq:non-dim-free-energ-dens}
\end{align}
\end{subequations}
where $f^{*}$ is the dimensionless free energy density and $\zeta_{\text{int}}^{\text{1d}}
$ is the same one-particle partition function for the pseudospins than that in Eq.~\eqref{eq:one-particle-zeta}.\footnote{Note that $\beta h_0 \chi = \beta^* \chi^*$ and $\beta J = \beta^* J^*$.}

In dimensionless variables, Eq.~\eqref{eq:profile-eq} is rewritten as $\left.\pdv{f^{*}}{\chi^{*}}\right|_{\text{eq}}=0$, i.e.
\begin{subequations}\label{eq:profile-eq-nondim}
\begin{align}
   \chi_{\text{eq}}^{*}&=\mu^*(\beta^* \chi_\text{eq}^*,\beta^* \!J^*), \\
  \mu^*(\beta^* \chi^*,\beta^* \!J^*)&\equiv \pdv{\ln\zeta_{\text{int}}^{\text{1d}}}{(\beta^* \chi^*)}=\frac{e^{-{2\beta^*\!J^{*}}}
    \sinh{\left(\beta^* \chi^*\right)}}
  {\sqrt{1+e^{-4\beta^*\!J^{*}}\sinh^2{\left(\beta^* \chi^*\right)}}},
\end{align}
\end{subequations}
where $\mu^*$ is the local magnetisation of the pseudospins.
Equation~\eqref{eq:profile-eq-nondim} implies that $\chi_{\text{eq}}^{*}$ is a certain function of $J^*$ and $T^*$, $\chi_{\text{eq}}^{*}=\chi_{\text{eq}}^{*}(J^{*},T^{*})$. Therefore, it is the dimensionless pseudospin-elastic coupling $J^{*}$
and the dimensionless temperature $T^{*}$ that control the emergence
of buckled profiles with non-zero curvature. We recall that the equilibrium
curvature is constant, independent of position, and an intensive quantity---and so is the
free energy density. Therefore,
\begin{equation}\label{eq:free-energy-equil}
  \mathcal{F}_{\text{eq}}^{*}(J^{*},T^{*})\equiv \mathcal{F}^{*}[u_{\text{eq}}^{*}](J^{*},T^{*})=N
  f^{*}(\chi_{\text{eq}}^{*};J^{*},T^{*}),
\end{equation}
the equilibrium free energy is extensive.

For the supported boundary
conditions~\eqref{eq:fixed-ends-bc} we are considering, there is a unique smooth equilibrium profile
given by
\begin{equation}\label{eq:profile-sol-supported}
  u_{\text{eq}}^{*}(x^{*};J^{*},T^{*})=
  \frac{\chi_{\text{eq}}^{*}(J^{*},T^{*})}{2}x^{*}(x^{*}-L^{*}),
\end{equation}
For completely free boundary conditions, i.e. when neither the
displacement nor its derivative is fixed at the boundary, there are
multiple equilibrium profiles that are obtained by adding $Cx^{*}+D$ to
$u_{\text{eq}}^{*}$ in Eq.~\eqref{eq:profile-sol-supported}, with $C$ and $D$
being arbitrary constants---the boundary condition
$\dv*{(\pdv*{f^{*}}{\chi^{*}})}{x^{*}}|_{x^{*}=0,L^{*}}=0$ is
fulfilled for all $(C,D)$. In particular, the flat profile, $u^{*}=0$,
and any transversal shift plus rotation thereof, $u^{*}=Cx^{*}+D$, are both
possible equilibrium profiles for free boundary conditions.

Since the free energy density only depends on the curvature, and the
equilibrium curvature is the same for both supported and completely
free boundary conditions, the analysis of buckled states that follows
is valid for both supported and free boundary conditions. Then, for
the sake of concreteness and without loss of generality, we will
employ the supported boundary condition
expression~\eqref{eq:profile-sol-supported} for the equilibrium
profile---omitting the additional linear contribution $Cx^{*}+D$ for the
free case.

In the remainder of the paper, we employ dimensionless
variables---dropping the asterisks in them not to clutter our formulae.

\section{Buckled states}\label{sec:buckling}

In this section, we analyse the emergence of buckled states, i.e. with
non-zero curvature, in our model system. First, we show how buckled
profiles bifurcate from the zero-curvature solution $\chi_{\text{eq}}^{\text{ZC}}=0$ of
Eq.~\eqref{eq:profile-eq-nondim}. Second, we consider the low temperature
limit $T\ll 1$, where the equilibrium profiles can be exactly
calculated.

\subsection{Bifurcation from the zero-curvature solution}\label{sec:bifurc-from-flat-sol}

To start with, we expand the free energy density around the
zero-curvature profile, for which
\begin{equation}
  f_{\text{ZC}}(J,T)\equiv f(\chi=0;J,T)=-T\ln\corch{2\cosh(J/T)},
\end{equation}
i.e. we consider the difference of the free energy density from the
zero-curvature value,
\begin{equation}\label{eq:Deltaf-def}
  \Delta f(\chi;J,T)\equiv f(\chi;J,T)-f_{\text{ZC}}(J,T)=\frac{1}{2}\chi^{2}-T\ln\corch{
  \frac{\zeta_{\text{int}}^{\text{1d}}( \chi/T,J/T)}{2\cosh(J/T)}}.
\end{equation}
The stability of the zero-curvature profile is determined by the sign
of the second-derivative of the free energy density at $\chi=0$,
\begin{equation}\label{eq:f2}
  f_2(J,T)\equiv \left.\pdv[2]{f}{\chi}\right|_{\chi=0}=
  1-\frac{e^{-2J/T}}{T}.
\end{equation}
The change of stability takes place at the line over which $f_{2}$
vanishes, i.e.
\begin{equation}\label{eq:bifurc-curve}
  J=J_{b}(T)=-\frac{1}{2}T\ln T,
\end{equation}
which is called the bifurcation curve. This line separates the plane
$(J,T)$ into two regions: (i) a region with $f_{2}>0$, comprising two subregions I and III, which will be discussed later,   where the
zero-curvature solution is stable and (ii) region II with $f_{2}<0$,
where the zero-curvature solution becomes unstable and other stable
solution exist as explained below. Note that for $J=0$, we have
that $f_{2}<0$ for $T<1$: the unit of temperature $T_{0}$ is thus the
transition temperature in the absence of antiferromagnetic coupling
among the pseudospins. The phase diagram of the system, to be
completed along the following sections, is presented in
Fig.~\ref{kappavstheta3}.

The deviation of the free energy density from the zero-curvature
value is provided by the Taylor expansion
\begin{align}
  \Delta f(\chi;J,T)&=\frac{f_2(J,T)}{2} \chi^2 +
\frac{f_4(J,T)}{4!} \chi^4 +
  \frac{f_6(J,T)}{6!} \chi^6+O\left(\chi^8\right),
  \label{eq:free-energy-expansion}
\end{align}
where
\begin{align}\label{eq:f4-f6}
  f_4(J,T)=\frac{e^{-6J/T}}{T^3}\!\left(3-e^{4J/T}\right) \! , \;
  f_6(J,T)=-\frac{e^{-10J/T}}{T^5}
  \!\left(e^{8J/T}-30e^{4J/T}+45\right).
\end{align}
Equation~\eqref{eq:free-energy-expansion} allows us to analyse the
emergence of buckled ($\chi\neq 0$) profiles from the zero-curvature
solution, using the Landau theory of phase transitions---with the
curvature playing the role of the order parameter.  Within this
approximation, the transcendental Euler-Lagrange
equation~\eqref{eq:profile-eq-nondim} for the curvature becomes algebraic
\begin{equation}\label{eq:profile-eq-nondim-expansion}
  \left.\pdv{\Delta f(\chi;J,T)}{\chi}\right|_{\text{eq}}=0 \iff
  f_{2}(J,T)\chi_{\text{eq}}+
  \frac{f_{4}(J,T)}{3!}\chi_{\text{eq}}^{3}+
  \frac{f_{6}(J,T)}{5!}\chi_{\text{eq}}^{5}=0 .
\end{equation}
This expansion is truncated at different orders,
the sign of the last retained coefficient $f_{n}$ must be always
positive---otherwise, the corresponding equilibrium distribution
$P_{\text{eq}}$ would be non-normalisable.

\subsubsection{ \texorpdfstring{$f_{4}>0$}{f40}: Second-order phase transition}
\label{sec:2nd-order-phase-trans}

\begin{figure} \centering
\includegraphics[width=0.70\textwidth]{BuckledPhaseDiagram_TeoNew.png}
\caption{Phase diagram of the rotationally invariant spin-string.  The plane $(J,T)$ is divided into several regions, characterised by the existing phases and their stability. In region I, only the ZC phase exists. In region II, phase ZC is unstable, and there appears a stable buckled phase B. In region III, three phases coexist: ZC, and two buckled phases, B+ and B-; the latter being unstable, whereas ZC and B+ are locally stable. Phase B+ is the most stable one in IIIa, with ZC being metastable, whereas the roles are reversed in IIIb. Point $C=(J_{c},T_{c})$ is the tricritical point. Above it, the phase transition is second-order; below, it is first-order.  The curve $J_b(T)$, given by Eq.~\eqref{eq:bifurc-curve}, is a bifurcation curve. For $T>T_c$, it defines the already mentioned second-order phase transition (black solid line). For $T<T_{c}$, it demarcates the change from region II to region III (leftmost dotted line). The curve $J_{M}(T)$ (rightmost dotted line), theoretically predicted by Eq.~\eqref{eq:limit-metastab-curve} close to the tricritical point (blue dotted line), demarcates the change from region III to region I. The curve $J_{t}(T)$ (dashed line), theoretically predicted by Eq.~\eqref{eq:first-order-curve} close to the tricritical point (red dashed line), is the first-order transition line: over it, phases B+ and ZC have the same free energy. Also plotted are the theoretical low-temperature limiting values $J_{M}^{(0)}$ and $J_{t}^{(0)}$. On the one hand, the predictions for the bifurcation curve and for $\left(J_t^{(0)},J_M^{(0)}\right)$ are exact. On the other hand, the asymptotic theoretical predictions for the curves $J_t(T)$ and $J_M(T)$ show an excellent agreement with the numerical results, within their range of validity. See Sec.~\ref{sec:num-analys} for further details in the numerical determination of the curves $J_t(T)$ and $J_M(T)$.}
\label{kappavstheta3}
\end{figure}
When $f_4>0$, the sign of $f_2$ determines the possible phases of the system.  Close to any point $(J_{b},T_{b})$ over the bifurcation curve~\eqref{eq:bifurc-curve}, $f_{2}$ is small: we explicitly indicate this by introducing a parameter $0<\varepsilon\ll 1$ such that $f_{2}=\varepsilon\varphi_{2}$, with $\varphi_{2}=O(1)$. This tells us that the separation of the point $(J,T)$ we are considering from the bifurcation curve is of order
$\varepsilon$: either $T-T_{b}=O(\varepsilon)$ or $J-J_{b}=O(\varepsilon)$.

On the one hand, if $\varphi_2>0$, the free energy density has only
one minimum at $\chi_{\text{eq}}^{\text{ZC}}=0$. On the other hand, if
$\varphi_{2}<0$, $\chi_{\text{eq}}^{\text{ZC}}=0$ turns out to be a maximum and
there appear two symmetric minima at
\begin{equation}\label{eq:chieq-2nd-order}
  \chi_{\text{eq}}^{\text{B}}=\pm\varepsilon^{1/2}\sqrt{-6\varphi_{2}/f_{4,b}},
\end{equation}
where $f_{4,b}\equiv f_{4}(J_{b},T_{b})$ is the value of the
coefficient $f_{4}$ over the bifurcation curve,
\begin{equation}
  f_{4,b}=\frac{3T_{b}^{2}-1}{T_{b}^2}.
\end{equation}
The superindex B in Eq.~\eqref{eq:chieq-2nd-order} indicates that this
solution corresponds to a buckled phase, with non-zero curvature.
Inside region II of Fig.~\ref{kappavstheta3}, the stable profile for
the string is thus buckled.

Moreover, the term we have neglected in
Eq.~\eqref{eq:profile-eq-nondim-expansion}, $f_{6}\chi_{\text{eq}}^{5}/6!$
is in fact negligible: it is of order $\varepsilon^{5/2}$, whereas the first
two terms on the rhs are of order $\varepsilon^{3/2}$---this shows the
consistency of the approximations introduced.

\subsubsection{\texorpdfstring{$f_{4}<0$}{f40} and \texorpdfstring{$f_{6}>0$}{f60}: Tricritical point and
  first-order phase transition}\label{sec:1st-order-phase-trans}

The coefficient $f_{4,b}$ over the bifurcation curve vanishes at the
tricritical point $C\equiv(J_{c},T_{c})$, given by
\begin{equation}
    T_c=\frac{1}{\sqrt{3}}, \quad J_c=J_b(T_c)=\frac{\log{3}}{4\sqrt{{3}}}.
\end{equation}
Close to the tricritical point, the approximation given by
Eq.~\eqref{eq:chieq-2nd-order} thus breaks down. In fact, below the
tricritical point---over the leftmost dotted line in
Fig.~\ref{kappavstheta3}---we have that $f_{4,b}<0$ and  the term
involving $f_{6}$ in the expansion of the free energy density must be
retained.

Close to the tricritical point, a different scaling for the solutions
of Eq.~\eqref{eq:profile-eq-nondim-expansion} is needed. We still
write $f_{2}=\varepsilon\varphi_{2}$: the terms involving $f_{2}$ and $f_{6}$
are of the same order if $\chi_{\text{eq}}=O(\varepsilon^{1/4})$. Then, the terms
involving $f_{2}$ and $f_{4,b}$ are of the same order if
$f_{4,b}=O(\varepsilon^{1/2})$---this tells us the order of the distance of the
point $(J_{b},T_{b})$ over the bifurcation curve to the tricritical point, i.e. either
$T_{b}-T_{c}=O(\varepsilon^{1/2})$ or $J_{b}-J_{c}=O(\varepsilon^{1/2})$. Then, we write
$f_{4,b}=\varepsilon^{1/2}\varphi_{4}$, $f_{6,c}=f_{6}(J_{c},T_{c})=36$, and the
dominant balance for the curvature equation in this region is
\begin{equation}\label{eq:curv-cond-tricrit}
  \varphi_{2}\Xi+\frac{1}{6}\varphi_{4}\Xi^{3}+
  \frac{1}{120}f_{6,c}\Xi^{5}=0,  \qquad \chi_{\text{eq}}\equiv\varepsilon^{1/4}\Xi.
\end{equation}
We find the zero-curvature solution $\chi_{\text{eq}}^{\text{ZC}}=0$, and two
buckled solutions
\begin{align}
  \chi_{\text{eq}}^{\text{B+}}=&\pm\varepsilon^{1/4}\sqrt{\frac{-5\varphi_4+
                    \sqrt{25\varphi_4^2-30\varphi_2f_{6,c}}}{f_{6,c}/2}}, \label{eq:chi-B+}
  \\ \chi_{\text{eq}}^{\text{B-}}=&\pm\varepsilon^{1/4}\sqrt{\frac{-5\varphi_4- \sqrt{25\varphi_4^2-30\varphi_2f_{6,c}}}{f_{6,c}/2}}.
\end{align}
The zero-curvature phase corresponds to a local minimum of the free
energy as long as $f_{2}>0$, i.e. outside the bifurcation
curve~\eqref{eq:bifurc-curve}---it becomes unstable inside it, as
already discussed. The curvatures $\chi_{\text{eq}}^{\text{B}\pm}$ correspond to
two buckled phases B$\pm$, the domain of existence and stability of
which is discussed below.

The buckled phase B+ exists as long as $f_{4,b}<0$ (or $\varphi_{4}<0$),
i.e. below the tricritical point, provided that
$5f_{4,b}^{2}-6f_{2}f_{6,c}>0$ (or
$5\varphi_4^2-6\varphi_2f_{6,c}>0$). The line
\begin{equation}\label{eq:end-metastab-v1}
  5f_{4,b}^{2}-6f_{2}f_{6,c}=0
\end{equation}
marks the end of the existence of this phase (rightmost dotted curve in Fig.~\ref{kappavstheta3}). Moreover, when traversing from right to left this line, the buckled phase pops up discontinuously with a finite curvature: specifically, from
$\chi_{\text{eq}}^{\text{ZC}}=0$ to $\chi_{\text{eq}}^{\text{B+}}=\pm \sqrt{-5f_{4,b}/18}\ne 0$.  It can be easily checked---by evaluating the
second derivative of the free energy density with respect to $\chi$---that this buckled phase is locally stable in its domain of existence.

The buckled phase B- also exists as long as $f_{4,b}<0$ (or
$\varphi_{4,b}<0$), i.e. below the tricritical point. However, not only
does it need $5f_{4}^{2}-6f_{2}f_{6,c}>0$
but also $f_{2}>0$.
This phase B- only exists between the line~\eqref{eq:end-metastab-v1}  and the part of the bifurcation curve~\eqref{eq:bifurc-curve} below the tricritical point (dotted lines in Fig.~\ref{kappavstheta3}). Phase B- is always unstable: the second derivative of the free energy density for $\chi=\chi_{\text{eq}}^{\text{B-}}$ is negative.

The curve demarcating the limit of existence of the buckled phases,
Eq.~\eqref{eq:end-metastab-v1}, can be written in a more transparent
way. Let us consider, for any $T$ close to $T_{c}$, a point
$(J,T)$ close to the bifurcation curve $(J_{b}(T),T)$ by taking into account that
\begin{equation}\label{eq:f4b-f2-tricrit}
  f_{4,b}\sim 6\sqrt{3} (T-T_{c}), \quad f_{2}\sim 2\sqrt{3} (J-J_{b}(T))
\end{equation}
in the approximation we are employing. Therefore,
Eq.~\eqref{eq:end-metastab-v1} is equivalent to
\begin{equation}\label{eq:limit-metastab-curve}
  J=J_{M}(T)\equiv J_{b}(T)+\frac{5}{12}\sqrt{3} (T-T_{c})^{2}.
\end{equation}

The analysis above entails the emergence of the region III in the
plane $(J,T)$, also depicted in Fig.~\ref{kappavstheta3}. This region
extends over the zone of the plane $(J,T)$ below the tricritical point
$C$, with its left border at the bifurcation line~\eqref{eq:bifurc-curve} and its right border
at the line~\eqref{eq:limit-metastab-curve} (both dotted lines in Fig.~\ref{kappavstheta3}). Inside region III, the three phases \text{ZC}, B+, and B- coexist. Phases B+ and \text{ZC}\ correspond to local
minima of the free energy, whereas B- always corresponds to a local
maximum. The relative stability of phases B+ and \text{ZC}\ is elucidated in
the following.

In region III, the phases \text{ZC}\ and B+ are both locally stable: one of
them corresponds to the deepest minima, being the most favourable
thermodynamic state, whereas the other one corresponds to a metastable
state. This means that the
transition changes from second-order above the tricritical point to
first-order below it, since the curvature (order parameter) of the most stable phase changes abruptly below the tricitical point. The change of stability takes place at the first-order
transition line determined by the condition $\Delta \mathcal{F}(\text{B+})=\Delta
\mathcal{F}(\text{ZC})=0$.  Close to the tricritical point, this is equivalent to
\begin{equation}
  f_{2}+\frac{1}{12}f_{4,b}(\chi_{\text{eq}}^{\text{B+}})^{2}+
\frac{1}{360}f_{6,c}(\chi_{\text{eq}}^{\text{B+}})^{4}=0.
\end{equation}
Bringing to bear Eqs.~\eqref{eq:curv-cond-tricrit} and
\eqref{eq:chi-B+}, over the first-order line one has
$8f_{2}f_{6,c}=5f_{4,b}^{2}$ that translates into
\begin{equation}\label{eq:first-order-curve}
  J=J_{t}(T)\equiv J_{b}(T)+\frac{5}{16}\sqrt{3}(T-T_{c})^{2},
\end{equation}
making use of Eq.~\eqref{eq:f4b-f2-tricrit}. The buckled phase B+ is
the most stable one in region IIIa of Fig.~\ref{kappavstheta3},
i.e. between the branch of the bifurcation line below the tricritical
point (leftmost dotted line) and the first-order line
\eqref{eq:first-order-curve} (dashed line), with the \text{ZC}\ phase
being metastable. The situation is just reversed in region IIIb between the
first-order line \eqref{eq:first-order-curve} and the curve
demarcating the limit of existence of  phases B$\pm$,
Eq.~\eqref{eq:limit-metastab-curve} (rightmost dotted line): therein, the
\text{ZC}\ phase is most stable and B+ is metastable.

\subsection{Low temperature limit}\label{sec:low-temp-limit}

Now we turn our attention to the low temperature limit $T\ll 1$. By defining $e_{\text{gs}}^{\text{1d}}$ as the energy per site in the ground state of the one-dimensional pseudospin system with external field $\chi$ and antiferromagnetic coupling $J$, we can write
\begin{equation}\label{eq:egs-def}
   \lim_{T\to 0^+}-T\ln\zeta_{\text{int}}^{\text{1d}}=e_{\text{gs}}^{\text{1d}}.
\end{equation}
The ground state energy of the pseudospin system is
\begin{equation}\label{eq:egs-expression-1d}
    e_{\text{gs}}^{\text{1d}}=-\pr{\left|\chi\right|-2J}H\pr{\left|\chi\right|-2J}-J,
\end{equation}
in which  $H(x)$ is Heaviside's step function. Equation~\eqref{eq:egs-expression-1d} is readily understood on a physical basis: for small $\left|\chi\right|$, the antiferromagnetic coupling wins and the ground state corresponds to antiferromagnetic ordering, $e_{\text{gs}}^{\text{1d}}=-J$, whereas for large $\left|\chi\right|$, the external field wins and the ground state corresponds to all spins aligned with the curvature, $e_{\text{gs}}^{\text{1d}}=-\left|\chi\right|+J$. These two expressions for $e_{\text{gs}}^{\text{1d}}$ become equal at $\left|\chi\right|=2J$, which marks the crossover between them both. Of course, $e_{\text{gs}}^{\text{1d}}$ can also be derived by taking the limit $T\to 0^+$ in Eq.~\eqref{eq:egs-def}, bringing to bear Eq.~\eqref{eq:one-particle-zeta}
for $\zeta_{\text{int}}^{\text{1d}}$.

The above discussion entails that the free energy density \eqref{eq:non-dim-free-energ-dens} simplifies to
\begin{equation}\label{eq:free-energ-dens-low-T}
    f(\chi;J,T)\sim\frac{1}{2}\chi^2
    -\pr{\left|\chi\right|-2J}H\pr{\left|\chi\right|-2J}-J, \quad T\ll 1.
\end{equation}
Also, Eq.~\eqref{eq:profile-eq-nondim} for the equilibrium curvature converts to
\begin{align}\label{eq:EL-low-T}
    \abs{\chi_{\text{eq}}}=H(\abs{\chi_{\text{eq}}}-2J), \quad
  T\ll 1,
\end{align}
Both in Eqs.~\eqref{eq:free-energ-dens-low-T} and \eqref{eq:EL-low-T},
Heaviside's step function must be understood in a ``physical way'',
since it appears here as the low temperature limit of the rhs of
Eq.~\eqref{eq:profile-eq-nondim}. Therefore, $H(x)$ comprises three
strokes: two flat strokes, equal to $0$ and $1$ for $x<0$ and $x>0$,
respectively, and a vertical stroke at $x=0$, which joins the previous
two. Logically, Eq.~\eqref{eq:EL-low-T} preserves the symmetry
$\chi_{\text{eq}}\to-\chi_{\text{eq}}$: we recall that we are restricting
ourselves to positive curvatures, so $\abs{\chi_{\text{eq}}}\to\chi_{\text{eq}}$ in
the following.

In order to study the solutions of Eq.~\eqref{eq:EL-low-T}, a
graphical method is useful. In Fig.~\ref{fig:LowTemperatures}, we plot
the functions on the lhs and the rhs of Eq.~\eqref{eq:EL-low-T} and
look for the intersection points of both functions. It is clearly seen
that $\chi_{\text{eq}}=0$ is always a solution, for all $J$: the phase ZC
survives in the low temperature limit. In addition, there appear two
buckled solutions for $2J\leq 1$, i.e. $J\leq J_{M}^{(0)}= 1/2$:
specifically, $\chi_{\text{eq}}=2J$ and $\chi_{\text{eq}}=1$. Both solutions
coalesce at $J_{M}^{(0)}$: stronger antiferromagnetic interaction destroys
the buckled phases, in which the magnetisation of the pseudospins is different from zero---a signature of ferromagnetic ordering.
\begin{figure}
    \centering
    \includegraphics[width=0.8\textwidth]{LowTemperaturesSolutions.png}
    \caption{Graphical method for solving Eq.~\eqref{eq:EL-low-T}. For each $J$, the
      intersection points of the functions $H(\chi_{\text{eq}}-2J)$ (solid lines) and $\chi_{\text{eq}}$ (dashed blue line) give the equilibrium
      curvature  in the low temperature limit. For $J > J_{M}^{(0)}=1/2$,
      for instance $J=0.55$ (green solid line), the only solution is
      $\chi_{\text{eq}}=0$, i.e. the ZC phase. For $J\leq J_{M}^{(0)}$, for instance
      $J=0.25$ (red solid line), there appear three solutions: in
      addition to the ZC phase corresponding to $\chi_{\text{eq}}=0$, we
      have two buckled phases
      corresponding to $\chi_{\text{eq}}=2J$ (B- phase) and $\chi_{\text{eq}}=1$ (B+ phase).}
    \label{fig:LowTemperatures}
\end{figure}

Let us calculate the free energy density of each of the phases, in
order to identify them. First, the free energy density of the ZC phase
is given by $f_{\text{ZC}}(J,T)\sim -J$, for $T\ll 1$, so that
\begin{equation}
  \Delta f(\chi;J,T)=
  f(\chi;J,T)-f_{\text{ZC}}(J,T)\sim
  \frac{1}{2}\chi^2
  -\pr{\left|\chi\right|-2J}H\pr{\left|\chi\right|-2J},
  \quad T\ll 1.
\end{equation}
Therefore, one obtains
\begin{align}
    \Delta f(\chi_{\text{eq}}=2J; J,T)&\sim 2J^{2}, &
    \Delta f(\chi_{\text{eq}}=1;J,T)=&\sim -\frac{1}{2}+2J.
\end{align}
On the one hand, the phase with $\chi_{\text{eq}}=2J$ has always a free
energy that is larger than that of the ZC phase---recall that the total free energy is proportional to its density as stated by
Eq.~\eqref{eq:free-energy-equil}. On the other hand, $\Delta f(\chi_{\text{eq}}=1;J,T)$ changes sign at $J=J_{t}^{(0)}=1/4$: the phase
with $\chi_{\text{eq}}=1$ has a free energy that is smaller (larger) than
that of the ZC phase for $0\leq J<J_{t}^{(0)}$
$\left(J_{t}^{(0)}<J\leq J_{M}^{(0)}\right)$. Therefore, the solution with
$\chi_{\text{eq}}=1$ is expected to correspond to the low temperature limit
of phase B+, whereas the solution with $\chi_{\text{eq}}=2J$ should
correspond to the low temperature limit of  phase B-. Also, the
points $\left(J_{t}^{(0)},0\right)$ and $\left(J_{M}^{(0)},0\right)$ should correspond to
the endpoints of the first-order line $J_{t}(T)$ and the right border line
of region III $J_{M}(T)$, respectively---we have analytical
predictions for these lines, them being rigorously valid just in the vicinity of the tricritical point,
Eqs.~\eqref{eq:first-order-curve} and
\eqref{eq:limit-metastab-curve}. These
expectations are confirmed by the numerical analysis below.

\subsection{Numerical analysis}\label{sec:num-analys}

Here, we perform a numerical analysis of the phase diagram of our model system.
We construct a
200$\times$200 mesh in the $(J,T)$ plane and find $\chi_{\text{eq}}$, by numerically solving
Eq.~\eqref{eq:profile-eq-nondim} at
each point of this mesh. At some points $(J,T)$, more than one
solution is found: note that this is expected, since in certain
regions of the plane $(J,T)$ several phases coexist. This analysis
allows us to build a numerical phase diagram for our system and
compare it with our analytical predictions above---depicted in
Fig.~\ref{kappavstheta3}.

Figure~\ref{fig:chieq-num} shows the values of $\chi_{\text{eq}}$ for the
thermodynamically stable phase---the most stable one when there is
coexistence. Therein, it is clearly observed that the transition
changes from second-order to first-order at the tricritical
point. Above the tricritical point $C$, $\chi_{\text{eq}}$ changes continously
from zero to non-zero values at the bifurcation curve
$J_{b}(T)$ (solid line). Below the tricritical point, $\chi_{\text{eq}}$ changes
discontinuously from zero to non-zero values at the first-order line
$J_{t}(T)$ (dashed line). Recall that the curvature plays the role of the order
parameter in our model. In fact, $\chi_{\text{eq}}$ equals the magnetisation
of the pseudospins, which is given by the rhs of
Eq.~\eqref{eq:profile-eq-nondim}.
\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{chi_eq_PhaseDiagram.png}
  \caption{Curvature of the most stable phase. The labelling of the
    different regions and the code of the different lines is the same
    as in Fig.~\ref{kappavstheta3}. The change of order of the
    transition, from second-order (solid line) above $C$ to first-order (dashed line) below, is
    clearly observed.}
  \label{fig:chieq-num}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{BuckledPhaseDiagram_Stable_Unstable_2_200x200.png}
    \caption{Free energy density difference $\Delta f(\chi_{\text{eq}};J,T)$ of the buckled states
      with respect to the ZC phase. The labelling of the different
      regions and the code of the different lines is the same as in
       Figs.~\ref{kappavstheta3} and Fig.~\ref{fig:chieq-num}. In the top panel, we show a
      density plot for the (at least locally) stable buckled phase: B
      in region II and B+ in region III. In the bottom panel, we show
      the density plot for the unstable buckled phase B-, which only
      exists in region III.
      }
    \label{fig:phase-diagram_num_200x200}
\end{figure}
By inserting the numerical solution for $\chi_{\text{eq}}$ into
Eq.~\eqref{eq:non-dim-free-energ-dens}, we get the values of the free
energy density over the considered mesh.  We present our results in
Fig.~\ref{fig:phase-diagram_num_200x200}, where we plot the
difference of the free energy density with respect to the ZC
profile, $\Delta f(\chi_{\text{eq}};J,T)$, as defined in
Eq.~\eqref{eq:Deltaf-def}. In the top panel, the value for the stable
buckled phase, wherever it exists, is shown---in region I, within
there is no buckled phase, a constant value zero is
plotted. Therefore, phase B is shown in region II, inside the
bifurcation curve \eqref{eq:bifurc-curve} (solid line above the tricritical point $C$, leftmost dotted line below it), and phase B+
is plotted in region III. Region III is demarcated by the bifurcation
curve, the segment of the $J$-axis between the origin and the point
$\left(J_{M}^{(0)},0\right)$, and the curve $J_{M}(T)$ that marks the limit of
existence of the phases B$\pm$ (rightmost dotted line).  In the bottom panel,
$\Delta f(\chi_{\text{eq}};J,T)$ is plotted for the unstable phase B-, which
only exists in region III---consistently, in regions I and II, a
constant value zero is plotted. The three phases, ZC, B+, and B-
coexist in region III, which is divided into two subregions by the
first-order line $J_{t}(T)$ (dashed line): IIIa, where
$\Delta f(\chi_{\text{eq}}^{\text{B+}};J,T)<0$ and the most stable
phase is B+, being ZC metastable; and IIIb, where the roles are
reversed, the most stable phase is ZC, being B+ metastable.

\section{Spin-membrane model on a honeycomb lattice}\label{sec:spin-membrane-honeycomb}

Now we move to a two-dimensional system, i.e. a spin-membrane
model. Specifically, we consider the system on a honeycomb lattice,
such as that of graphene. There is a particle---carbon atom if
thinking of graphene---at each node $(i,j)$ in a certain two-dimensional region $\Omega$.
The transverse displacement with respect to the
plane of each particle is denoted by $u_{ij}$. Analogously to the
one-dimensional spin-string analysed before, displacements parallel to
the plane are not taken into consideration. Moreover, at each site
$(i,j)$ we have a pseudospin $\sigma_{ij}=\pm 1$, which models in a
simple way other degrees of freedom---for example, the out-of-plane
electron that is not bonded.

A sketch of the lattice honeycomb lattice, with lattice constant $a$,
is shown in  Fig.~\ref{fig:HoneycombLattice}. Indices $i$ and $j$
are employed for rows and columns, respectively. Note that atoms at each
row are distributed in zigzag. There are two types of sites, e-sites
and o-sites.
For the former, the
three nearest neighbours are one above and two below; for the latter,
the three nearest neighbours are two above and one
below~\cite{ruiz-garcia_ripples_2015}.  Note that the
two-dimensional domain $\Omega$ can have an arbitrary shape, not necessarily rectangular---for
the sake of simplicity, we assume that it has no holes. This means that all
the nearest neighbours are present for the \textit{bulk} sites
shown in  Fig.~\ref{fig:HoneycombLattice}---boundary sites will be taken into account by
introducing appropriate boundary conditions over the contour $\delta\Omega$ of the region $\Omega$,
as discussed later on.

The rotationally invariant Hamiltonian is the extension of the
spin-string one, as given by Eq.~\eqref{eq:energy-rot-inv}, to the
honeycomb lattice,
\begin{align} \mathcal{H}(\hat{u},\hat{p},\hat{\sigma})&= \sum_{\abs{i-j}=\text{even}} \bigg[\frac{p_{ij}^2}{2m} +
\frac{k}{2}\left(u_{i-1,j}+u_{i,j-1}+u_{i,j+1}-3u_{ij}\right)^2 \nonumber \\
& \qquad \qquad \qquad -h
\sigma_{ij}\left(u_{i-1,j}+u_{i,j-1}+u_{i,j+1}-3u_{ij}\right) \nonumber \\
 & \qquad \qquad \qquad + \frac{J}{2} \sigma_{ij} \left(\sigma_{i-1,j}+\sigma_{i,j-1}+
\sigma_{i,j+1}\right) \bigg]
\nonumber\\
  &+\sum_{\abs{i-j}=\text{odd}}
\bigg[ \frac{p_{ij}^2}{2m} + \frac{k}{2}\left(u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-3u_{ij}\right)^2 \nonumber \\
 & \qquad \qquad \qquad -h
\sigma_{ij}\left(u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-3u_{ij}\right) \nonumber \\
 & \qquad \qquad \qquad + \frac{J}{2} \sigma_{ij} \left(\sigma_{i+1,j}+\sigma_{i,j-1}+
\sigma_{i,j+1}\right) \bigg].
\end{align}
For the sake of clarity, the Hamiltonian is split into two sums because nearest neighbours are  different for e-sites and o-sites. Within each sum on the rhs, the first term stands for the kinetic energy, the second one corresponds to the elastic contribution to the Hamiltonian, the third one describes the interaction between the transverse displacements and the pseudospins, and the fourth one provides the antiferromagnetic coupling among the pseudospins.\footnote{The coupling constant is $J$, the factor $1/2$ is needed because each
interaction is counted twice.}

Now we go to a continuum limit, by assuming that $u_{ij}$ varies
slowly with $(i,j)$. We introduce continuous spatial variables
$x$ and $y$ as depicted in Fig.~\ref{fig:HoneycombLattice},
\begin{subequations}
\begin{align} x=&i \Delta x, & y=&j \Delta y , &
\text{e-sites sublattice}, \\ x=&x_o + i \Delta x, & y=&j \Delta y, &
\text{o-sites sublattice}.
\end{align}
\end{subequations}
where we have defined $x_o=a/2$, $\Delta x=3a/2$, and $\Delta y=\sqrt{3}a/2$. All nearest neighbours of o-sites are e-sites, and vice versa---the honeycomb lattice is bipartite~\cite{newman_networks_2010}. Therefore, the term in parentheses of the elastic term for e-sites is, neglecting $O(a^3)$ terms,
\begin{align}
  u_{i+1,j}+&u_{i,j-1}+u_{i,j+1}-3u_{ij} \nonumber \\
            &=u(x+l,y)+u(x-l/2,y-a/2)+u(x-l/2,y+a/2)-3u(x,y)\nonumber \\
  &\sim \left(\sqrt{3}\frac{a}{2}\right)^{2}\chi(x,y), \qquad \chi(x,y)\equiv \nabla^{2}u(x,y).
\end{align}
where, in analogy with the one-dimensional case, we have introduced the notation $\chi$ for the curvature of the membrane. The same result holds for o-sites, $u_{i-1,j}+u_{i,j-1}+u_{i,j+1}-3u_{ij}\sim \left(\sqrt{3}a/2\right)^{2}\chi(x,y)$.   Then, we get
\begin{align}
    \frac{k}{2}(u_{i+ 1,j}+u_{i,j-1}+u_{i,j+1}-3u_{i,j})^2 &= \frac{k_0}{2}\chi^2, \\
    h(u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-3u_{i,j})\sigma_{ij}&= h_0\chi\sigma_{ij},
\end{align}
in which we have defined $k_0=k(\sqrt{3}a/2)^4$ and $h_0=h(\sqrt{3}a/2)^2$.
\begin{figure}
    \centering
    \includegraphics[width=0.66\textwidth]{HoneycombLattice_New.png}
    \caption{Sketch of the honeycomb lattice with parameter $a$. The lattice is bipartite and can be thus split into two sublattices: e-sites (blue), two nearest neighbours below and one above, and o-sites (red), two nearest neighbours above and one below. The distance between sites of the same kind corresponding to consecutive indexes are $\Delta x=3a/2$, $\Delta y=\sqrt{3}a/2$. The spacing between e- and o-sites with the same value of the index $i$ is $x_o=a/2$.}
    \label{fig:HoneycombLattice}
\end{figure}

Again, we are interested in the equilibrium profiles of the membrane. Therefore, we integrate the canonical distribution over the momenta and the pseudospin variables. Also, the sum over $(i,j)$ goes to an integral over $(x,y)$ in the continuum limit developed above, which leads to
\begin{equation}
  P_{\text{eq}}[u]\propto e^{-\beta\mathcal{F}[u]}, \quad \mathcal{F}[u]=n \int_{\Omega}
  dx \, dy \, f(\chi),
\end{equation}
where  $n=(\Delta x \Delta y)^{-1}$ is the number density
and $f$ is the free energy density per particle, given by
\begin{subequations}\label{eq:free-energy-density_2D}
\begin{align}
  f(\chi)&=\frac{k_{0}\chi^2}{2}
  -k_{B}T\ln\zeta_{\text{int}}^{\text{2d}}(\beta h_{0}\chi,\beta J),
\end{align}
\end{subequations}
$\zeta_{\text{int}}^{\text{2d}}(\beta h_{0}\chi,\beta J)$ stands for the partition
function for the antiferromagnetic Ising chain
with coupling $J$ and external field $h_{0}\chi$ in the
two-dimensional case. Unfortunately, at variance with the
one-dimensional case, there is not an analytical expression for
$\zeta_{\text{int}}^{\text{2d}}$ when $J\ne 0$.

The equilibrium profile minimises the free energy, so it is determined by the condition
$\delta F[u] = 0$. Since the free energy density only depends on the curvature $\chi$, it is convenient to define the field
\begin{equation}\label{eq:Phi-def}
    \Phi(x,y)\equiv \pdv{f}{\chi}=k_0\chi - h_0\pdv{\ln\zeta_{\text{int}}^{\text{2d}}(\beta h_{0}\chi,\beta J)}{(\beta h_0 \chi)}.
\end{equation}
Making use of this definition, the variation of the free energy can be written as
\begin{align}
    \delta F[u] =&
   \int_\Omega d{x}\, d{y} \,
   \Phi \,\nabla^2 (\delta u) \nonumber \\
   =& \int_\Omega d{x}\, d{y} \,\nabla\cdot\pr{ \Phi \, \nabla(\delta u)-\nabla \Phi \delta u}
   + \int_\Omega d{x}\, d{y} \, \nabla^2\Phi \delta u \nonumber \\
   =& \int_{\delta\Omega} ds \, \hat{\bm{n}}\cdot\pr{ \Phi \, \nabla(\delta u)-\nabla \Phi \,\delta u}
   + \int_\Omega d{x}\, d{y} \, \nabla^2\Phi \,\delta u.
   \label{eq:deltaF-2d}
\end{align}
In the contour integral along the boundary $\delta\Omega$, $ds$ is the length element and $\hat{\bm{n}}$ is the outward pointing unit normal. On the one hand, the surface integral in the second term on the rhs of Eq.~\eqref{eq:deltaF-2d} provides us with the Euler-Lagrange equation,
\begin{equation}\label{eq:LaplaceEquation}
    \nabla^2 \Phi=0,
\end{equation}
i.e. the Laplace equation for the field $\Phi(x,y)$. On the other hand, the contour integral in the first term gives us the boundary conditions: since each term must vanish separately, we have
\begin{equation}\label{eq:bc-2d-1st}
    \Phi \, \pdv{}{n}\delta u=0, \quad \delta u\, \pdv{}{n}\Phi =0,
\end{equation}
where $\partial/\partial n\equiv \hat{\bm{n}}\cdot\nabla$ is the normal derivative. For the sake of concreteness, and consistently with our analysis of the one-dimensional spin-string model, we consider supported boundary conditions: the value of the transversal displacements vanish at the boundaries,
\begin{equation}\label{eq:bc-2d-2nd}
    u(x,y)=0, \quad (x,y)\in\delta\Omega,
\end{equation}
but the value of the normal derivative $\pdv{}{n}\delta u=0$ at the boundary is free. Therefore, the coefficient of $\pdv{}{n}\delta u$ in Eq.~\eqref{eq:bc-2d-1st} has to vanish,
\begin{equation}\label{eq:bc-2d-3rd}
    \Phi(x,y)=0, \quad (x,y)\in\delta\Omega.
\end{equation}

The solution of the Laplace equation \eqref{eq:LaplaceEquation} with the homogeneous Dirichlet boundary condition \eqref{eq:bc-2d-3rd} is identically zero:
\begin{equation}\label{eq:Euler-Lagrange-2d}
    \Phi(x,y)=0, \; \forall(x,y)\in \Omega \implies
    k_0\chi_\text{eq}- h_0\pdv{\ln\zeta_{\text{int}}^{\text{2d}}(\beta h_{0}\chi_\text{eq},\beta J)}{(\beta h_0 \chi_\text{eq})}=0,
\end{equation}
where we have made  use of the definition of $\Phi$. Although we do not have an explicit expression for $\zeta_{\text{int}}^{\text{2d}}$, Eq.~\eqref{eq:Euler-Lagrange-2d} neatly tells us that the equilibrium curvature is constant, independent of $(x,y)$ over all the two-dimensional domain $\Omega$. Interestingly, the nondimensional variables introduced in the one-dimensional spin-string model, as given by Eq.~\eqref{eq:dimensionless-variables}, also work in the two-dimensional case---only now the dimensionless number density is $n^*=n\ell_0^2$. The dimensionless free energy and free energy density are
\begin{subequations}
\begin{align}
  \mathcal{F}^{*}[u^{*}](J^{*},T^{*})&=n^* \int_{\Omega}  dx^{*}\, dy^{*}\,
  f^{*}(\chi^{*};J^{*},T^{*}), \\
 f^{*}(\chi^{*};J^{*},T^{*})&\equiv \frac{1}{2}{\chi^{*}}^{2}-T^{*}\ln
                              \zeta_{\text{int}}^{\text{2d}}(\beta^* \chi^{*},\beta^* \! J^{*}),
                              \label{eq:non-dim-free-energ-dens-2d}
\end{align}
\end{subequations}
where we have taken into account that $\beta h_0\chi=\beta^*\chi^*$, $\beta J=\beta^* J^*$. In what follows, we drop again the asterisks in the dimensionless variables to simplify our notation. Equation~\eqref{eq:Euler-Lagrange-2d} is written in dimensionless variables as
\begin{equation}\label{eq:Euler-Lagrange-2d-nondim}
    \chi_\text{eq}=\mu(\beta\chi_\text{eq},\beta J), \quad \mu(\beta\chi_\text{eq},\beta J)\equiv\pdv{\ln\zeta_{\text{int}}^{\text{2d}}(\beta\chi_\text{eq},\beta J)}{(\beta \chi_\text{eq})}.
\end{equation}
Note that $\mu$ is the local magnetisation of the two-dimensional pseudospin lattice. The equilibrium free energy $\mathcal{F}_{\text{eq}}$ is given by
\begin{equation}\label{eq:free-energy-equil-2d}
  \mathcal{F}_{\text{eq}}(J,T)\equiv \mathcal{F}[u_{\text{eq}}](J,T)=N
  f(\chi_{\text{eq}};J,T),
\end{equation}
which is extensive, as expected on a physical basis.

\subsection{Buckled states}\label{eq:buckled-states-2d}

The situation in the two-dimensional spin-membrane model presents strong similarities with the one previously found in the one-dimensional spin-string system: whenever a buckled profile with $\chi_\text{eq} \ne 0$ emerges, the curvature is constant throughout the whole system and the free energy per particle equals $f(\chi_{\text{eq}};J,T)$. Moreover, the free energy density $f$ has two contributions, the first quadratic in the curvature $\chi$, stemming from the elastic contribution of the transverse displacements, and the second equal to the free energy of the pseudospin system, with external field equal to the curvature $\chi$ and a ferromagnetic coupling $J$.

To univocally determine the equilibrium profile for the membrane, we have to bring to bear the remainder of the supported boundary conditions, i.e. Eq.~\eqref{eq:bc-2d-2nd}. Therefore, we have to solve Poisson's equation, with the uniform curvature playing the role of the density and  homogeneous boundary conditions:
\begin{equation}\label{eq:Poisson-supported-bc}
    \nabla^2 u_\text{eq}(x,y)=\chi_\text{eq},\; (x,y)\in\Omega; \qquad u(x,y)=0, \; (x,y)\in\delta\Omega.
\end{equation}
The solution for the profile depends on the geometry of the problem. The simplest situation appears for a circular membrane of radius $R$: therein, the equilibrium profile only depends on the distance $r$ to the centre of the membrane and straightforward integration of the Poisson equation gives
\begin{equation}\label{eq:eq-profile-circle}
    u_\text{eq}(r)=\frac{\chi_\text{eq}}{4}(r^2-R^2).
\end{equation}
For a rectangle of sides $L_x$ and $L_y$, the solution of the Poisson equation involves an expansion in eigenfunctions---for details, see~\ref{app:profile-rectangular-domain}.

Below, we analytically investigate the behaviour of the two-dimensional spin-membrane model in some limits of interest, for which the expression for $\zeta_{\text{int}}^{\text{2d}}$ can be worked out. Specifically, two situations are considered: (i) the case $(J=0,T\ne 0)$, in which the antiferromagnetic coupling among the pseudospins disappears, and (ii) the low temperature limit $(J\ne 0,T=0)$. In the former, we show that there appears a second-order phase transition at $T=1$ (in dimensionless variables), below which the zero curvature membrane is unstable and stable buckled profiles bifurcate. In the latter, we show that the zero curvature membrane is always locally stable but membrane buckled states (one locally stable, another unstable) are present below a certain value $J_M^{(0)}$ of the antiferromagnetic coupling. Moreover, metastability is present, signaling that the transition from zero curvature to buckled states is now first-order.

\subsubsection{No antiferromagnetic coupling}\label{sec:J-zero-2d}

Let us first consider the case $J=0$. The one-particle partition $\zeta_{\text{int}}^{\text{2d}}(J=0,T)$ corresponds to that of a non-interacting Ising system in an external field $\chi$, i.e.
\begin{align}
    f(\chi;J=0,T)=\frac{1}{2}\chi^2-T \ln \corch{2\cosh\pr{\frac{\chi}{T}}}.
\end{align}
and the Euler-Lagrange for the curvature reduces to
\begin{equation}\label{eq:EL-2d-Jzero}
    \chi_{\text{eq}}(J=0,T) =\tanh\corch{\frac{\chi_{\text{eq}}(J=0,T)}{T}} .
\end{equation}
This coincides with the particularisation of Eq.~\eqref{eq:profile-eq-nondim} for $J=0$---this is logical, without antiferromagnetic interaction the transition is of mean field type for any dimension. Therefore, the ZC phase is the only one for $T>1$ and is stable, whereas it becomes unstable for $T<1$. At $T=1$, a stable buckled phase B continuously bifurcates from the ZC phase, namely
\begin{equation}
    \chi_\text{eq} \sim \sqrt{3(1-T)}, \quad  \text{for} \  T\to 1^-.
\end{equation}
This phase B is the stable one for $T<1$ and continues to exist up to $T=0$. In fact,  Eq.~\eqref{eq:EL-2d-Jzero} predicts that
\begin{equation}\label{eq:chi-J0-2d-lowT}
   \lim_{T\to 0^+}\chi_\text{eq}(J=0,T)= 1.
\end{equation}

\subsubsection{Low temperature limit}\label{sec:low-temp-2d}

Now we turn to the low temperature limit $T\to 0^+$, with $J\ne 0$. In this regime, similarly to the situation in the one-dimensional spin-string system, we have
\begin{equation}\label{eq:egs-def-2d}
   \lim_{T\to 0^+}-T\ln\zeta_{\text{int}}^{\text{2d}}=e_{\text{gs}}^{\text{2d}}
\end{equation}
where
\begin{equation}\label{eq:egs-expression-2d}
    e_{\text{gs}}^{\text{2d}}=-\pr{\left|\chi\right|-3J}H\pr{\left|\chi\right|-3J}-\frac{3J}{2}.
\end{equation}
is the energy per site in the ground state of the two-dimensional spin-membrane. Again, Eq.~\eqref{eq:egs-expression-2d}  for $e_{\text{gs}}^{\text{2d}}$ has a clear physical interpretation. On the one hand, the ground state energy corresponds to antiferromagnetic ordering for small $\chi$, $e_{\text{gs}}^{\text{2d}}=-3J/2$---in the honeycomb lattice, each pseudospin has three nearest neighbours. On the other hand, the ground state corresponds to all pseudospins aligned with the curvature for large $\chi$, $e_{\text{gs}}^{\text{2d}}=-\chi+3J/2$. The crossing between these expressions for $e_{\text{gs}}^{\text{2d}}$ takes place at $\chi=3J$, which leads to Eq.~\eqref{eq:egs-expression-2d}---see details in \ref{app:GS-2d}.

The free energy density is now
\begin{equation}\label{eq:free-energ-dens-low-T-2d}
    f(\chi;J,T)\sim\frac{1}{2}\chi^2
    -\pr{\left|\chi\right|-3J}H\pr{\left|\chi\right|-3J}-3J/2, \quad T\ll 1,
\end{equation}
and the equilibrium curvature is fixed by
\begin{align}\label{eq:EL-low-T-2d}
    \abs{\chi_{\text{eq}}}=H(\abs{\chi_{\text{eq}}}-3J), \quad
  T\ll 1.
\end{align}
We recall that, without loss of generality, we are restricting ourselves to positive curvatures in the paper, thus $\abs{\chi_{\text{eq}}}\to\chi_{\text{eq}}$ henceforth. The analysis of the low temperature limit follows along the same lines as in the one-dimensional case, so we present the results in a concise way. The ZC phase is always a solution of Eq.~\eqref{eq:EL-low-T-2d}, and two buckled solutions emerge for $3J\leq 1$, i.e. $J\leq J_{M}^{(0)}= 1/3$. The two buckled phases have curvatures $\chi_{\text{eq}}=3J$ and $\chi_{\text{eq}}=1$. For $J>J_M^{(0)}$, the antiferromagnetic interaction is so strong that prevents the emergence of buckled solutions.

To identify the phases, we evaluate the free energy density over them, the ZC phase verifies  $f_{\text{ZC}}(J,T)\sim -3J/2$, for $T\ll 1$.  Therefore
\begin{equation}
  \Delta f(\chi;J,T)=
  f(\chi;J,T)-f_{\text{ZC}}(J,T)\sim
  \frac{1}{2}\chi^2
  -\pr{\left|\chi\right|-3J}H\pr{\left|\chi\right|-3J},
  \quad T\ll 1.
\end{equation}
and
\begin{align}
    \Delta f(\chi_{\text{eq}}=3J; J,T)&\sim \frac{9}{2}J^{2}, &
    \Delta f(\chi_{\text{eq}}=1;J,T)&\sim -\frac{1}{2}+3J.
\end{align}
The phase with $\chi_{\text{eq}}=3J$ always has $\Delta f>0$. The phase with $\chi_{\text{eq}}=1$ has $\Delta f<0$ ($>0$) for $0\leq J<J_{t}^{(0)}=1/6$ $\left(J_{t}^{(0)}<J\leq J_{M}^{(0)}\right)$.  Then, the phase with $\chi_{\text{eq}}=3J$ should correspond to the (unstable) B- phase, whereas the phase with $\chi_{\text{eq}}=1$ should correspond to the (at least locally stable) B+ phase. The buckled phases disappear discontinuously at $J=J_M^{(0)}$, which means that the transition is first order.

The above identification of the phases is physically sensible, since in the limit $J\to 0$ the B+ phase for $T\ll 1$ is expected to converge to the $T\to 0^+$ limit of the B phase found for $J=0$---and, in fact, the curvature of the latter has the right behaviour, as expressed by Eq.~\eqref{eq:chi-J0-2d-lowT}. The profile of the B+ phase for a circular membrane of unit area is shown in Fig.~\ref{fig:buckled-lowT-2d}; the anagolous profile for a rectangular membrane can be found in \ref{app:profile-rectangular-domain}.
\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{SpinMembraneAbs_A=1.png}
    \caption{Low temperature stable buckled configuration for the spin-membrane model.  The plotted surface corresponds to the absolute value of Eq.~\eqref{eq:eq-profile-circle}, for a unit area circle. The curvature equals unity for all $J\leq J_M^{(0)}=1/3$, corresponding to the most stable state for $0\leq J<J_t^{(0)}=1/6$ and to a metastable state for $J_t^{(0)}<J\leq J_M^{(0)}$.}
    \label{fig:buckled-lowT-2d}
\end{figure}

\section{Conclusions}\label{sec:conclusions}

In this paper, we have introduced novel, rotationally invariant, spin-string and spin-membrane models. The main role is played by the curvature associated with the transversal displacements, which not only rules the elastic energy but also the coupling with internal degrees of freedom.  These models give an appealing physical picture of the buckling transition while keeping consistency with both thermodynamics and classical theory of elasticity, which is important from a theoretical perspective. Moreover, the equilibrium profiles are very simple, with a position-independent curvature, both for the one-dimensional and the two-dimensional lattices.

The phase diagram in the $(J,T)$ plane---$J$ being the antiferromagnetic coupling among the pseudospins and $T$ the system temperature---of the one-dimensional rotationally invariant spin-string model has been characterised analytically  in great detail. It is complex and qualitatively similar to that found in Ref.~\cite{ruiz-garcia_bifurcation_2017} and, in particular, buckled phases emerge for low enough values of the temperature and the antiferromagnetic coupling. In absence of the latter, there appears a second-order phase transition at a certain temperature, below which the zero curvature phase becomes unstable and a buckled B phase is the stable one. In the buckled phase, the pseudospins magnetisation equals the non-zero curvature and, thus, they predominantly align with the sign of the curvature. For low temperatures, the ferromagnetic ordering of the pseudospins is frustrated by the antiferromagnetic coupling, which destroys the buckled states for high enough coupling constants $J$, $J>J_M^{(0)}$. For $J\leq J_M^{(0)}$, there appears coexistence between the ZC phase and a stable B+ buckled phase, phase B+ is the most stable one and phase ZC is metastable for small $J$, $0\leq J<J_t^{(0)}$, with the situation being reversed for $J_t^{(0)}<J\leq J_M^{(0)}$. This qualitative map of phases is the key ingredient that makes this kind of models qualitatively explain the transition from a rippled to a buckled state when the temperature is increased~\cite{ruiz-garcia_stm-driven_2016,schoelz_graphene_2015}.

To be concrete, and motivated by the geometry of graphene, we have studied the spin-membrane model in a honeycomb lattice. The analytical expression of the partition function for the pseudospins is not known in the two-dimensional case and thus the phase diagram cannot be characterised as completely as in the one-dimensional case. Nevertheless, our analysis of limiting cases allows us to state that the same qualitative picture of phases applies for the two-dimensional spin-membrane model. Still, the two-dimensional values of $J_t^{(0)}$ and $J_M^{(0)}$ are smaller---in dimensionless variables---than the one-dimensional ones. This can be understood as stemming from the antiferromagnetic interaction being stronger, since the number of nearest neighbours is larger for $d=2$ than for $d=1$. It should be stressed that the main conclusions of our study of the two-dimensional case hold for a quite general lattice, as long as it does not contain triangular loops---such as a rectangular one.

Previous spin-string and spin-membrane models~\cite{ruiz-garcia_stm-driven_2016,ruiz-garcia_bifurcation_2017,ruiz-garcia_ripples_2015} had several weaknesses. First, it was necessary to assume the parameters in the system Hamiltonian to have certain scalings with system size to get consistency with thermodynamics, specifically extensive free energy and transition temperature independent of system size. Second, even assuming size-dependent microscopic parameters, the models were not consistent with classical elasticity theory: e.g. a flat but rotated string or membrane was not an acceptable equilibrium profile.

The models developed in this work mend the aforementioned weaknesses. The microscopic parameters $k$ and $h$ do not depend on system size and determine natural units for length and temperature, which we have used to introduce dimensionless variables. From them, we have naturally obtained a system-size-independent transition temperature and an extensive free energy. Moreover, the Euler-Lagrange equation is consistent with the classical theory of elasticity, with the term stemming from the internal interactions involving a biLaplacian instead of a Laplacian. These two properties hold for both the one-dimensional and two-dimensional lattices.

Also, perspectives for future work are opened. In the context of elasticity of low-dimensional systems, our work may trigger new approaches to the analysis of buckled phases in more realistic models. Also, it would be interesting to consider variants of the two-dimensional model that allow for a more detailed analytical treatment. In this regard, effective antiferromagnetic interactions for which the pseudospins partition function is exactly known~\cite{fisher_lattice_1960} deserve to be studied. Moreover, the framework developed here may be useful to look into the elastic behaviour of systems in which elastic modes are coupled to other, internal, degrees of freedom---such as biomolecules~\cite{peyrard_biophysics:_2006,prados_spin-oscillator_2012,benedito_isotensional_2018,cannizzo_thermal_2022}. A different but also interesting prospect is the investigation of dynamics, since the present study only concerns equilibrium configurations. On a physical basis, one expects the dynamics of the internal degrees of freedom to be much faster than that of the elastic modes. This separation of time scales should make it possible---e.g. making use of a Chapman-Enskog expansion~\cite{bonilla_chapman-enskog_2000,baldovin_derivation_2019}---to derive an effective stochastic dynamics, either at the Langevin or Fokker-Planck level of description,  for the elastic modes.

redpajama_set_name: RedPajamaArXiv


It's the start of the Summer holidays in the Shilts household. Hubby has 6 weeks off work, time to relax and unwind and spend some quality time with Mikey. Unfortunately I'm still at work for most of it but will have a week off in a couple of weeks followed by a couple of long weekends.
We've got quite a busy week ahead and this week marks the start of the London 2012 Olympics. We've got tickets for three football matches this week. On Thursday we head over to Coventry to watch our first Mens's Football game; Belarus versus New Zealand and then on Saturday we're back in Coventry watching 2 Women's Football matches; Japan versus Sweden and then Canada versus South Africa. I'm really looking forward to watching an Olympic event, just having the chance to say we've been is exciting.
So we've planned ahead and this week we have the following menu to look forward to..
I'm hoping we can be thrifty over the next few weeks and use up the freezer goods and contents of the cupboards. I seem to be struggling with food at the minute and never fancy what we have planned but will be on my best behaviour and try and stick to the menu.


Amy Villeneuve joins the 6 River Systems Board of Directors.
We're very excited to announce that Amy Villeneuve, a seasoned executive and former President and COO of Kiva Systems, has joined our Board of Directors.
Amy brings with her extensive experience in growing businesses, particularly in the robotics space. At Kiva Systems, she spearheaded a period of rapid growth, ultimately leading to its acquisition by Amazon for $775M in 2012. She remained with Amazon as VP of Amazon, President & COO of Kiva Systems (renamed Amazon Robotics) until 2015.
While at Amazon, Amy also left her mark as a woman leader in tech. She was selected to serve on Amazon's Women in Engineering Advisory Board. She also served as Amazon's representative to the Anita Borg Institute's Technical Executive Forum.
Both 6RS co-founders and co-CEOs Jerome Dubois and Rylan Hamilton worked closely with Amy during their tenure at Kiva Systems.
6 River Systems is the new way companies fulfill. Unlike anchored down, no-touch, steel and motors, 6 River Systems is a flexible and collaborative mobile robotics solution for the warehouses of tomorrow.


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Last update on 2/11/2016
The latest news in the History of Western Art from the Middle Ages to the 1930's
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A Sergel Terracotta for the Stockholm Museum
6/03/12 - Acquisition - Stockholm, Nationalmuseum - The Nationalmuseum in Stockholm recently acquired a terracotta by Johan Tobias Sergel : a study for the bust of Duchess Hedvig Elisabet Charlotta of Schleswig-Holstein-Gottorp (1759-1818), for which the museum already owned the plaster cast and the marble. Auctioned by Bukowskis on 7 December 2011, the work went for 15,800 € and was purchased with the help of the Axel Hirsch Foundation.
Johan Tobias Sergel (1740-1814)
Portrait of Duchess Hedwige Elisabeth Charlotte
de Schleswig-Holstein-Gottorp, c.1780
Terracotta - H. 38 cm (with the base)
Stockholm, Nationalmuseum
Photo : Bodil Karlsson Nationalmuseum
Portrait of Duchess Hedvig Elisabet Charlotta of Holstein-Gottorp, c.1780
Photo : Bukowskis
The daughter of Frederick August I, Duke of Oldenburg, Hedvig Elisabet Charlotta married the future Charles XIII of Sweden in 1774, the same king who was to later adopt Bernadotte (see article in French). His portrait is part of a series of royal likenesses which the sculptor produced after returning from Italy. He had been living in Rome since 1767 when Gustav III recalled him to work in his native country in 1778. Sergel returned with an admiration for Antiquity and the Italian Renaissance, as well as the influence of "the simple and noble taste for the antique" encouraged by Bouchardon and knowledge of French art in general, as he had also spent some time in Paris where he presented a sculpture of Othryades for his reception into the Royal Academy in 1779. The exhibition on L'Antiquité rêvée (see article) had in fact included one of his works.
After returning to Stockholm, Sergel hoped to continue developing themes from mythology and Antiquity but the king commissioned his portrait, then Gustav III's brothers demanded theirs, followed by those of the queen, the crown prince, the duchess and, finally, even the dowager queen.
This recently acquired terracotta, showing only the model's face, was in the sculptor's studio along with other studies when his estate was purchased by the government after his death, in 1815. Sergel's heirs recovered certain studies and, at that same moment, various clay likenesses fell into the hands of Carl Curman, a professor at the Royal Academy. This mask comes from that collection where other, exceptionally well preserved, pieces were also held.
In fact, the unfinished aspect of this study imparts it with a vivacity missing in the perfectly accomplished and smooth, plaster version, rather comparable to the queen's portrait : both women present fashionable hairstyle and clothing, holding their head high with a wise look. On the other hand, this terracotta portrait on a pedestal, devoid of details reflecting a certain taste or period, becomes timeless, thus evoking an air of an antique statue.
Bénédicte Bonnet Saint-Georges, lundi 12 mars 2012
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Only 19 out of 80 companies make use of the potential of the internet in interim reporting issues. - nexxar lab.
Online Interim Reporting still underdeveloped.
Whereas 56 % of all DAX 80 companies present their annual report as a full HTML-solution, the percentage shrinks to 24 % in interim reporting.
Following our benchmark-analysis, where we, among other indices, explored the conversion of online annual reports of DAX 80 companies, we looked at the situation in interim reporting.
The findings are simply disappointing. Whereas 56 % of all DAX 80 companies present their annual report as a full HTML-solution, the percentage shrinks to 24 % in interim reporting. Only 19 out of 80 companies make use of the potential of the internet in interim reporting issues. The stake of (user-unfriendly) JPG-shows (11 %) is lower than in annual reporting (26 %) either. This means that 65 % of all interim reports are provided as PDF, what adds absolutely no value to the report.
Since starting our benchmark survey in 2003 we have been noticing a clear trend towards high-quality HTML reports. Search functions and links to related content are just two of the many advantages HTML provides in comparison to PDF. We will have a closer look at the development of HTML interim reporting in the future.


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Conversations with Giants in Medicine Free access | 10.1172/JCI126050
A conversation with Mary-Claire King
Ushma S. Neill
Find articles by Neill, U. in: JCI | PubMed | Google Scholar |
Published January 2, 2019 - More info
J Clin Invest. 2019;129(1):1–3. https://doi.org/10.1172/JCI126050.
Published January 2, 2019 - Version history
Few scientists have had a more massive contribution to genetics than Mary-Claire King of the University of Washington. King (Figure 1) was the first to show that breast cancer is inherited in some families as a result of mutations in the gene that she named BRCA1. She had already changed our view of evolution when she demonstrated during her PhD work that the protein-coding DNA sequences of humans and chimpanzees are 99% identical. She's also been a tireless social justice warrior, protesting wars, fighting the patenting of genes, and pioneering research in using genomic tools to aid in the investigation of human rights abuses. A condensed version of our conversation is presented here.
Mary-Claire King in October 2018.
JCI: What were you like as a child?
Mary-Claire King: I grew up in Chicago. My Dad was born in 1890, so was almost 60 when I was born. From his early 60s, he was ill with what we ultimately realized were late-onset effects of the 1918 influenza epidemic, a severe Parkinson-like syndrome. So from the time I was about six years old, he was always home with my brother and me.
One consequence of his being home is that we could watch the Cubs and White Sox on TV. My dad had relatively little formal education, likewise my mom, but both were very bright. My dad in particular was very fond of story problems. For example, as we were watching a Cubs game, he would say, "Ernie Banks is batting .277 and he'll probably have three at bats this game. What's he going to have to bat to bring his average above .280?" I was six, so I would think and think and say, "I don't know." He would say, "You're right. What else do you need to know?" Then we'd work through what else. I needed to know how many at bats Ernie Banks had already had in the season. We worked through these really complex problems. Eventually they seemed easy. As a result, nothing about arithmetic, algebra, or story problems ever threw me.
JCI: Is that what led you to do a degree in mathematics?
King: I went to a fine public high school, where I had two wonderful female math teachers, one with a PhD; they were my first role models for any women in science. I realized in college that while I was pretty good at math, I wasn't nearly good enough to do it professionally. I went to Berkeley for graduate school thinking to study statistics, which is more applied and I found more tractable. Then I took a genetics course — my first plunge into biology. I absolutely loved it. I transferred to genetics and never looked back.
JCI: During the time you were getting a PhD, you were also quite involved with social justice causes. How did you come to start working with Ralph Nader?
King: This was the '60s in Berkeley. We were all involved in social justice causes: the civil rights movement and subsequently the anti-war movement. One day, on the way to lab, I saw a little 3 × 5 card posted on a bulletin board. It said, "Ralph Nader needs a biologist to work on the California Project. He will be coming to investigate who owns California and what they are doing with the land they own." I thought that was interesting, so I called the number and talked with Ralph and ended up working on the California Project.
After the California Project was published, Ralph offered me a job in Washington, DC. Allan Wilson, who was then a friend but not yet my advisor, convinced me not to drop out of graduate school. He said, "If you leave now, you can do good work. There's no question it's righteous work, but with only a bachelor's degree, you'll never set the agenda. If you finish your PhD, you'll have many more opportunities." I countered that none of my experiments worked. He said, "If everybody whose experiments failed stopped doing science, there would be no one left. Let's see if we can figure out a project that is experimentally not too difficult but that takes advantage of the fact that you can write equations." That became the project on human and chimpanzee evolution and the demonstration that we and chimpanzees share 99% of our coding sequences.
JCI: You moved to Chile after your PhD. What was that time like?
King: We were in Chile in the early 1970s, during the years of the Popular Unity government, the Allende period. It was eye opening to a gringa like me. Beginning in the 1960s, there was an exchange sponsored by the Ford Foundation between the University of California and the Universidad de Chile that allowed for very easy exchange of faculty, postdocs, and graduate students. My then-husband was a field ecologist; Chile was heaven-sent for his work.
I soon realized that when you teach in a place that has few specialists, you end up teaching a bit of everything. I taught genetics and statistics and evolution. But my courses didn't last long, because in September 1973 a military coup overthrew President Allende's government. The universities closed. We tried to protect our students, most successfully, some not.
JCI: Why come back to Berkeley and why do a postdoc after such a stimulating teaching experience?
King: Like many progressive young Americans, we were tempted with the idea of staying, but it soon became clear that wasn't possible. We came back to Berkeley in late 1973 because my then-husband was on the faculty in zoology. The concept of postdoc didn't quite exist then, but there were many jobs available in cancer research, because President Nixon had just launched the war on cancer. One of those jobs was at UCSF with a lovely pediatric oncologist named Nicholas Petrakis who had become interested in breast cancer.
It was clear that breast cancer had a major familial component. Herodotus, the ancient Greek historian, wrote about it. There was also excellent epidemiology beginning in the early 20th century that made it clear that critical time points in a woman's life had a major influence on her subsequent breast cancer risk: in particular her age of menarche and the age at which she had her first child. The longer the interval between menarche and first childbirth, the higher the subsequent breast cancer risk. That interval is longer in places where girls are well nourished, and therefore enter puberty fairly young, and where young women are educated, and often postpone childbearing: breast cancer is uniquely a cancer of prosperity. Overlaying that, there were families that were very severely affected with breast cancer. It seemed to me that breast cancer was likely due to some factors that were genetic and some that were not.
I decided to probe the inherited genetic component. In the 1970s, no one was postulating inherited risk for a complex trait like breast cancer. I was very lucky because the National Cancer Institute (NCI) was carrying out an epidemiological study of breast cancer risk associated with oral contraceptives. I asked if I could add some questions to the NCI questionnaire about family history, and the investigators very kindly said yes. I soon had data from 1,500 women with breast cancer, beautifully ascertained, beautifully interviewed, and 1,500 cancer-free women for comparison. The women had provided detailed information about their own breast cancers and about other cancers in their families. There was no molecular genetics in this study, of course; it was just interviews. No one was picking up a pipette, least of all me. Instead, I used the information from these women to develop a maximum likelihood model, a straightforward statistical model to test all possible explanations for clustering of breast cancer in families.
It was straightforward to show that breast cancer did cluster in families to a greater degree than one would expect by chance. But why? Was the clustering because of a polygenic effect? Was it because of a recessive genetic effect, like for Tay-Sachs or cystic fibrosis? Was it because of a dominant genetic effect like for Huntington's disease? Or was it some combination? I had enough information to be able to formally test those models. The best-fitting model, by far, was the hypothesis that the clustering in families could be explained by the existence of one or more dominantly inherited genes that would move the lifetime risk of breast cancer from 8% to 80%. The gene was hypothetical. This was only mathematics. It was just a prediction. The way it would play out in reality was unknown.
JCI: During this time, you transitioned from postdoc to faculty. You've mentioned before that you got the job via affirmative action.
King: It's absolutely true. I am a poster child for affirmative action, and I'm proud to be one. At UC Berkeley in those days, there was a rule that spouses could not be faculty members in the same department. My then-husband was in zoology, so that precluded my even being considered for a position there. My degree had been in genetics at Berkeley, and there was another exclusion that a department could not hire its own graduate students, so that precluded genetics as well. A job opened up in the School of Public Health in the field of epidemiology. I thought, shucks, geneticists and epidemiologists have a lot to say to each other.
In 1975, there were brand-new affirmative action guidelines in the University of California system that included two critical elements. One was that every search committee had to include a woman as a member. I think there was also an agreement that every search committee had to include a student. The committee for this new position satisfied both rules by including a woman student, an epidemiologist-in-training named Catherine Schaefer — who is now one of the chief epidemiologists at Kaiser Permanente. The other critical element of the affirmative action guidelines was that if a woman or a person from an underrepresented minority applied for a faculty position, that person either needed to be interviewed or a detailed explanation had to be provided indicating why they were not appropriate even for an interview. I learned later that the search committee thought it would be easier to interview me than to write the explanation.
My PhD project on human and chimpanzee evolution had just appeared on the cover of Science, so it was hard to dismiss me as a flake. My talk was scheduled for a small classroom that held maybe 30 people. It was anticipated that 20 or 30 people would attend. Of course, I had been at Berkeley my whole graduate career and had been involved in the anti-war movement and the civil rights movement and the labor organizing movement and what was then the nascent feminist movement. I had many friends all across campus. The little room was filled half an hour before my talk. People were out in the hall and on the stairwells. The elevators jammed because there were so many people trying to get in. It was just totally fabulous. My talk had to be moved to the auditorium. It went well.
After I was offered the position, the senior gentleman faculty member who had chaired the committee later said to me, "Cathy Schaefer was a bulldog, she wouldn't let go." He also said, "I just want you to know that we are scratching the bottom of the barrel in hiring you. It's very clear you don't know any epidemiology." I said, "I'm going to remind you later of your views." Indeed, when I was awarded tenure three years later (half the usual time), I reminded him. He said, "I never said that. I absolutely didn't say that. I totally wanted to hire you. Remember how I said that you did a great job and you made your case?" He had honestly forgotten.
The reality is that I never would have gotten an interview if there hadn't been an affirmative action policy. I needed a shot, and affirmative action gave it to me. So yes, I'm a child of affirmative action and enormously grateful for it.
JCI: You captured one aspect of this time in your life in a storytelling festival called The Moth. I don't want to ruin the story for those who haven't heard it yet (https://www.youtube.com/watch?v=tOP5pUIYhv4). You absolutely excel in your storytelling capacity and your ability to find humor in retelling a tale of real adversity. Have you always had this gift of communication?
King: I work very hard on my teaching. I have also spent enough of my life working in languages other than English such that I'm very conscious of how to phrase ideas and be clear, even with the limited words at my command. I've had to do this in Spanish and in French. I have enormous admiration for people who work in a second language.
JCI: Getting back to your research — after establishing your lab, you spent the next few years identifying BRCA. When you were able to reduce the space where the causative gene was to chromosome 17, did you have champagne?
King: We actually reduced the location of the gene on chromosome 17 to about a megabase. Of course, we didn't know then the size of the region we'd pinpointed. There was no physical map of the human genome. By 1990, I knew we had done the genetics properly, I knew we had done the math properly. But what I didn't know was how generalizable the result would be. Were my families unique in some way I didn't know? I was way too hesitant to have champagne.
A very fine geneticist named Gilbert Lenoir was interested in the same question and studied a different set of families in France. I sent him all the probes that I had made. He repeated exactly my experiments in his families and got the same result. When that happened, I believed it. Then we had champagne.
But we didn't know what the gene was. We had no idea of its sequence, let alone its function. But we knew it was findable. In early 1991, neither of us had the remotest idea of how exhausting it would be without a genome sequence. It stimulated a four-year race, ultimately involving hundreds of investigators and perhaps 15 sizeable groups.
JCI: Would you term this time as competitive or extremely competitive?
King: It was extremely competitive. One of my collaborators, an athletic guy in one of our sister labs, said, "I run marathons. I once thought I'd be an Olympic athlete. This is as intense as that." I never had that experience, but this was as competitive as it gets. You just had to keep your eye on the prize. Everyone was using exactly the same strategy, and it took four years. Myriad did indeed clone BRCA1 first. They made critical mistakes in their sequence — but they were the first to publish an amino acid sequence of BRCA1. Everyone involved knew what to do. Myriad cloned the gene first because they had vastly more sophisticated equipment, so could move much more quickly through thousands of DNA fragments.
JCI: Ultimately Myriad was granted a patent on BRCA, which was revoked by the Supreme Court in 2013 after the American Civil Liberties Union (ACLU) filed suit. Were you involved at all in supporting the view that genes should not be patented?
King: I was able to provide historical background and detail, to help get the facts right for a whole variety of amicus briefs. Nina Totenberg [correspondent for National Public Radio] called after the decision. She said, "The Supreme Court decision has just come out: it's 9 to 0." I said, "Which way?" She said, "Oh, in favor of the ACLU position." I said, "Wow." She said, "Surely you have a longer comment than that." I said, "I'm as high as the flag on the fourth of July." She said, "That'll do."
By the next day, BRCA1 and BRCA2 sequencing were in the public domain and commercially available. You can't help asking yourself, suppose it had been available years earlier, how many women could we have saved who died because the testing was too expensive?
JCI: How did you get started in the realm of using genomic tools to aid in the investigation of human rights abuses?
King: In the early 1980s, I was doing a sort of mini-sabbatical, commuting back forth from Berkeley to Stanford, where Luigi Luca Cavalli-Sforza very kindly agreed to teach me molecular genetics. He pointed out my work could never proceed just doing research at the mathematical level.
The Grandmothers of the Plaza de Mayo (the "Abuelas") had organized themselves in 1977 in Buenos Aires to demand the return of their children who had been murdered and of their grandchildren, about whom they had increasingly good anecdotal evidence of at least some survivors. They asked the Committee on Scientific Freedom and Responsibility of the American Association of Advancement of Science (AAAS), the people who publish Science, to recommend to them a geneticist who could help them in their search. They correctly realized that if they could identify a child on the basis of genes in the grandparents, even with the parents missing and presumed dead, then they would be able to know who that child was.
The AAAS suggested that they talk to Dr. Cavalli-Sforza, who was both a brilliant population geneticist and molecular geneticist. They asked him to work out the mathematics of grandpaternity testing. It was a simple application of Bayesian statistics. I was visiting his lab at the time, so several of us worked out the statistics. We published a paper on what we called the "index of grandpaternity," describing how to extend paternity testing to grandparents. The statistics was straightforward, but of course the approach was constrained by the biological tools of the time.
The grandmothers asked Luca to come to Buenos Aires to put grandpaternity testing into practice. Luca said to me, "I'm not doing this. You, however, are perfectly set up for it. You've taught in Spanish; you know Latin America; you obviously know the genetics. Most important, you are the age of the kidnapped daughters of these women, and your daughter Emily is the age of their grandchildren. You'd be perfect." I went in June 1984 for what I thought would be one symbolic trip of solidarity. It became a 30-year collaboration.
JCI: What do you see for the future of your lab in the next 10 years?
King: Our focus in breast cancer is to try to solve the questions facing the many remaining "unsolved" families. We think that noncoding regulatory mutations are likely to hold the answers, just like in my studies of evolution 50 years ago.
At the same time, we're working now to solve congenital disorders in real time, so that parents of severely affected children can use pregestational diagnosis to have a healthy next pregnancy. We're also increasingly interested in the genetics of severe mental illness such as schizophrenia and bipolar disorder. Genetics is a useful way of thinking, and genomics offers fabulous tools. I hope to apply them to as many complex conditions as I can.
JCI: What would you do if you were not a scientist?
King: I would open a daycare for the children of my lab where the kids would learn story problems while their parents do experiments.
Reference information: J Clin Invest. 2019;129(1):1–3. https://doi.org/10.1172/JCI126050.