text:
\section{Introduction}

The physics of periodically driven quantum systems has been the subject of intense study in recent years (see \cite{Bukov_2015,Eckardt:2017kqr, Weinberg_2017,Holthaus_2016} for reviews with tons of citations). Several reasons back up this interest, a very relevant one being technological in origin: the possibility of manipulating  quantum systems in a controlled way by using time-periodic external fields. This approach goes under the name of Floquet engineering \cite{Oka_2019,Rudner_2020}, following from  the Floquet theorem,  a temporal analogue of the Bloch theorem. The artificial setup involves mainly irradiating the system with a circularly polarised laser, or shaking it around. With  appropriate periodic drivings, new phases of quantum materials have been created and  non-equilibrium phenomena have emerged. Examples include light-induced superconductivity \cite{Fausti_2011,Mitrano_2016} and Floquet topological insulators \cite{Oka_2009,Kitagawa_2011,Cayssol_2112,Lindner,Rechtsman_2013,Jotzu_2014,Wang_2013}. Moving on to  (3+1) dimensions, artificial Weyl semimetals have also been reported  \cite{Zhang_2016,Hubener_2016,Bucciantini_2017}.

In this paper we study the behavior of a strongly-coupled $(2+1)$-dimensional gauge theory under the influence of a  external electric field  $\vec {\cal E}$ rotating in the $xy$ plane as
\begin{equation}
{\cal E}_x\,+\,i\,{\cal E}_y\,=\,E\,e^{i\Omega t}\,
\,.
\label{rotating_E}
\end{equation}
We will address this problem in the context of the holographic AdS/CFT correspondence \cite{Maldacena:1997re} (see \cite{AdS_CFT_reviews} for reviews),  and will follow closely the pioneering work in  Refs. \cite{Hashimoto:2016ize,Kinoshita:2017uch} in their study of $(3+1)$-dimensional systems.  In the first of these two references, the authors studied the massless D3-D7 flavour system and computed the artificially induced Hall conductivity.  In the second they extend part of the analysis to massive flavours where an interesting phase space shows up. Our paper streamlines the same analysis in (2+1) dimensions,  extending the analysis to encompass the optical and static conductivities also for massive flavours. The  $(2+1)$-dimensional theory in place is realised  as a codimension-one defect in ${\cal N}=4$ supersymmetric $SU(N_c)$ super Yang-Mills theory in four dimensions. The field content and lagrangian   was explicitly constructed in \cite{DeWolfe:2001pq}. It contains  a matter hypermultiplet living in the defect which couples to the fields of the ambient four-dimensional theory.
In String Theory, this model can be realized  as the intersection of D3- and D5-branes along $2+1$ dimensions. In this setup the  4d gauge theory on the D3-branes is holographically dual to the $AdS_5\times {\mathbb S}^5$ geometry, while the D5-branes provide $(2+1)$-dimensional flavors, {\em i.e.},\ fields living in the fundamental representation of the gauge group \cite{Karch:2002sh,Erdmenger:2002ex,Skenderis:2002vf}. We will work in the quenched  probe approximation and will hence neglect the backreaction of the  D5-branes on the $AdS_5\times {\mathbb S}^5$ geometry.
The fluctuation of the fields living on the brane are dual to the mesonic excitations of the gauge theory. In ref. \cite{Arean:2006pk} the complete analysis of these fluctations was performed for the D3-D5 system and the exact spectrum of mesons was found (see \cite{Kruczenski:2003be} for a similar analysis in the D3-D7 case).

When an external electric field acts on a medium, a vacuum polarization due to virtual charged particles (quarks) is produced.  As a consequence, an oscillating polarization current is induced. If the external field is weak enough, the medium will remain in an insulator gapped phase in which the induced current is always perpendicular to the applied field. There is no Joule heating and the system is dissipation-less.  When the external field is large enough the vacuum is unstable against the creation of quark-antiquark pairs (Schwinger effect). In the conductive  phase the current becomes dissipative, and the driving induces a  Joule heating.

For an external oscillating electric field as in (\ref{rotating_E}) the critical value $E_c$ at which the insulator-conductor transition takes place depends on the frequency $\Omega$, {\em i.e.},\  $E_c=E_c(\Omega)$ \cite{Takayoshi:2020}. Remarkably, for some values $\Omega=\Omega_c$ of the frequency, the transition occurs for vanishing external electric field, {\em i.e.},\ $E_c(\Omega_c)=0$.  At this point of the phase space the rotating current $j$ is not zero, in spite of the fact that the external driving field vanishes. Physically, for these frequencies the driving field enters in resonance with the vector meson excitations of the gauge theory. Actually, this resonating states with zero electric field and non-zero current also occur in the conductive pase in a finite range of frequencies $\Omega_c<\Omega\le \Omega_m$.   Following \cite{Kinoshita:2017uch}, where this behaviour was first found in the D3-D7 model, we will call a configuration with $E=0$ and $j\not=0$ a zero field Floquet condensate of vector mesons, or simply a vector meson  Floquet condensate. Such stable rotating state is a fixed point of time evolution, namely a non-thermal fixed point \cite{Berges:2004ce}.

In our holographic  setup the flavor degrees of freedom are modeled as the the excitations of a  probe D5-brane embedded in the $AdS_5\times {\mathbb S}^5$ ten-dimensional background. The probe brane extends along an  $AdS_4\times {\mathbb S}^2$ submanifold of the ten-dimensional background. The holographic duality in the D5-brane worldvolume relates the fields at the boundary of  $AdS_4$ to those of the dual
$(2+1)$-dimensional gauge theory. In particular, to model the rotating field (\ref{rotating_E}) we have to switch on an electric gauge field on the worldvolume attaining the value  (\ref{rotating_E})  as we approach the $AdS_4$ boundary (the subleading term near the boundary determines the current $j$). The D5-brane with this world-volume gauge field has a non-trivial profile  which can be found by solving the equations of motion of the probe brane. The leading value of the profile function at the boundary determines mass of the quarks, while the subleading term is related to the quark condensate.

A probe brane with an electric field in his worldvolume can develop an event horizon in the effective (open string) metric on its worldvolume. This is analogous, but not  the same, to the case in which the brane is embedded in a black hole background geometry  with a non-zero Hawking temperature\cite{Frolov:2006tc,Mateos:2006nu,Mateos:2007vn}. Accordingly, we can  borrow the
terminology of the non-zero temperature case and classify the configurations of the brane depending on whether or not it crosses or not the effective horizon. In the so-called Minkowski embeddings the brane reaches the origin of $AdS_4$ without developing the effective horizon. These Minkowski configurations are dual to the insulating phase of the defect gauge theory. On the contrary, in the conductive phase the induced horizon forms at a finite radial distance from the D3-brane system. The flavour mesons are deconfined and a current can set up in response to an external electric field.  At the interface between these two cases we find the critical embeddings, in which the probe  develops the effective horizon right at its IR endpoint. In this work we study both types of embeddings in order to determine the stable non-equilibrium phases of the theory. In this task we will employ both numerical and analytic techniques.

The rest of this paper is organized as follows. In section \ref{setup} we present our setup, find the equations of motion for the probe D5-brane and study the boundary behavior of the different functions and their relation to the observables of the gauge theory.  In section \ref{types_of_embeddings} we analyze the different types of embeddings and determine the IR boundary conditions that regular solutions must satisfy. We also determine in this section the effective metric and the effective Hawking temperature.  The numerical integration of the equations of motion and the structure of the phase diagram of the theory is presented in section
\ref{phase_diagram}.

When either  the mass is zero or the driving  frequency  is large, the equations of motion can be solved analytically in the linearized approximation. The exact analytical solutions are presented in section \ref{sec:analsol}. Section
\ref{Conductivities} deals with the analysis of the AC and DC conductivities of the model, which are obtained by studying the response of the brane to an additional  probe electric field which is treated in a linearized approximation. We end up with a summary and some concluding remarks in section \ref{conclu}.  The paper is supplemented with some appendices.  In appendix \ref{app_A} we give details about the derivation of the regularity conditions of the brane embeddings at the pseudo-horizon. In appendix \ref{app_B}, we find approximate  analytic solutions of the equations of motion  when the mass  is small. The different embeddings when the equations of motion are linearized are studied in
appendix \ref{linearized_embeddings}. The  exact conductivities for the massless case  are obtained  in appendix \ref{app_C}.

\section{Setup and ansatz}
\label{setup}

In order to analyze the Floquet states in the strong coupling regime, we will engineer a holographic brane setup in which two sets of branes intersect along $2+1$ dimensions.
More concretely we will consider the  intersection of two stacks of  D3- and D5-branes according to the following array
\begin{equation}
\begin{array}{ccccccccccl}
 &1&2&3& 4& 5&6 &7&8&9 &  \\
D3: & \times &\times &\times &\_ &\_ & \_&\_ &\_ &\_ &      \\
D5: &\times&\times&\_&\times&\times&\times&\_&\_&\_ &
\end{array}
\label{D3D5intersection}
\end{equation}
In (\ref{D3D5intersection}) the D3- and D5-branes share the spatial directions $1$ and $2$, whereas the direction $3$ is parallel to the D3-brane and orthogonal to the D5-brane.  The field theory dual to the brane setup (\ref{D3D5intersection})  is well-known \cite{Karch:2002sh,DeWolfe:2001pq,Erdmenger:2002ex,Skenderis:2002vf}. It consists of a supersymmetric  theory with $N_f$ matter hypermultiplets (flavors) living on the $(2+1)$-dimensional defect and coupled to the ambient ${\cal N}=4$ theory realized by the stack of $N_c$ color D3-branes.  We will adopt the approximation in which the D5-branes are probes in the geometry generated by the D3-branes, so $N_f\ll N_c$. The latter is the standard holographic dual of four-dimensional ${\cal N}=4$ super Yang-Mills, namely  $AdS_5\times {\mathbb S}^5$  with flux. Let us start by recalling this geometry
\begin{eqnarray}
&&ds^2_{10}\,=\,{\rho^2+w_1^2+w_2^2+w_3^2\over R^2}\,\Big(-dt^2+dx^2+dy^2+dz^2\Big)\,+\, \\
& \rule{0mm}{8mm} &\qquad\qquad\qquad\qquad
+\,{R^2\over \rho^2+w_1^2+w_2^2+w_3^2}\,\Big(d\rho^2+\rho^2\,d\Omega_2^2+dw_1^2+dw_2^2+dw_3^2\Big)\,\,, \nonumber
\end{eqnarray}
where $d\Omega_2^2=d\theta^2+\sin^2\theta d\phi^2$ is the metric of a unit two-sphere and $R$ is the common radius of the $AdS_5$ and ${\mathbb S}^5$ factors  (in what follows we will take $R=1$).   In order  to embed a D5-brane  wrapping a $AdS_4\times {\mathbb S}^2$ submanifold    we choose the following set of worldvolume coordinates
\begin{equation}
\xi^{a}\,=\,(t, x,y, \rho, \theta, \phi)\,\,,
\end{equation}
and the following ansatz  for the transverse scalars
\begin{equation}
z\,=\,{\rm constant}\,\,,
\qquad\qquad
w_1=w(\tau,\rho)\,\,,
\qquad\qquad
w_2\,=\,w_3\,=\,0\,\,.
\end{equation}
The induced metric on the D5-brane woldvolume now takes the form
\begin{eqnarray}
&&ds_6^2\,=\,-\Big(\rho^2+w^2\Big)\,\Bigg(1\,-\,{\dot w^2\over \big(\rho^2+w^2\big)^2}\Bigg)\,dt^2\,+\,(\rho^2+w^2)(dx^2+dy^2)\,+\,\nonumber\\\rc
&&\qquad\qquad\qquad
+{1\over \rho^2+w^2}\,\Big((1+w'^{\,2}) d\rho^2\,+\,2\,\dot w\,w'\,dt d\rho\,+\,\rho^2\,d\Omega_2^2\Big)\,\,,
\label{induced_metric}
\end{eqnarray}
where the dot denotes derivative with respect to $t$ and the prime  with respect to the holographic coordinate $
\rho$.
The main objective to study the response of this system to the driving of an external circularly polarized electric fied
\begin{equation}
 {\vec {\cal  E}}(t) =
\left( \begin{array}{c} {\cal E}_x(t)\\{\cal  E}_y(t)\end{array} \right)\, =  \left( \begin{array}{cc} \cos \Omega t  & -\sin \Omega t  \\ \sin \Omega t & \cos \Omega t \end{array} \right)  \left( \begin{array}{c} { E}_x\\{  E}_y\end{array} \right) \equiv  O(t)\vec E\,\,,
\label{rotating_electric_field}
\end{equation}
with $\vec E\,=\, {\vec {\cal  E}}(t=0)$. This will  source of a worldvolume gauge field with one-form potential
\begin{equation}
2\pi\,\alpha'\,{\cal A}\,=\,a_x(t,\rho)\,dx\,+\,a_y(t,\rho)\,dy\,\,,\label{vec_pot}
\end{equation}
whose field strength is
\begin{equation}
2\pi\,\alpha'\,{\cal F}\,=\,\dot a_x\,dt\wedge dx\,+\,a_x'\,d\rho\wedge dx\,+\,
\dot a_y\,dt\wedge dy\,+\,a_y'\,d\rho\wedge dy\,\,.
\end{equation}
The action of the probe brane is given by the DBI action
\begin{equation}
S\,= - N_f T_5\,\int d^6\xi\,\sqrt{-\det \big(g_6+2\pi\alpha'\,{\cal F}\big)}\,\,,
\label{DBI_action}
\end{equation}
where $T_5$ is the tension of the D5-brane. In the following  we will adhere to the notation and reasoning advocated in  \cite{Hashimoto:2016ize,Kinoshita:2017uch}, with slight modifications. To start with, the following switch from  vector to complex notation is useful
\begin{equation}
 E = E_x + i E_y ~~~~, ~~~~ a\,=\,a_x+ia_y\,\,.
\end{equation}

\iffalse
Using this complex notation the DBI determinant becomes
\begin{eqnarray}
&&-\det(g_6+2\pi\alpha'\,F)\,=\,{\rho^4\over (\rho^2+w^2)^2}\,\Bigg[(1+w'^{\,2})\big( (\rho^2+w^2)^2\,-\,|\dot a|^2\big)\,-\,
\dot w^2\big(1+|a'|^2\big)\,+\,\nonumber\\\rc
&&\qquad\qquad\qquad\qquad
+\,|a'|^2\, (\rho^2+w^2)^2\,+\,2\,\dot w\,w'\,{\rm Re}(\dot a\,a'^{\,*})\,-\,\big({\rm Im}(\dot a\,a'^{\,*})\big)^2\,\Bigg]\,\,, \label{actionD5}
\end{eqnarray}
\fi
Now,  computing the  determinant $-\det(g_6+2\pi\alpha'\,F)$,  the  DBI action becomes
\begin{eqnarray}
&&S \sim -  \int dt\,d\rho\,\,{\rho^2\over \rho^2+w^2}\,\,
\Bigg[(1+w'^{\,2})\big( (\rho^2+w^2)^2\,-\,|\dot a|^2\big)\,-\,
\dot w^2\big(1+|a'|^2\big)\,+\,\nonumber\\\rc
&&\qquad\qquad\qquad\qquad
+\,|a'|^2\, (\rho^2+w^2)^2\,+\,2\,\dot w\,w' (\,{\rm Re}(\dot a\,a'^{\,*})\,-\,\big({\rm Im}(\dot a\,a'^{\,*})\big)^2\,\Bigg]^{{1\over 2}}\,\,.
\label{DBI_complex}
\end{eqnarray}
Another convenient switch is to represent the complex gauge field $a$ in the rotating frame
\begin{equation}
{\cal E} = E e^{i\Omega t} ~~~~~, ~~~~ a(t,\rho)\,=\,b(t,\rho)\,e^{i(\Omega t+\chi(t,\rho))}\,\,.
\label{complexified_potential}
\end{equation}
 In the new variables, $(b, \chi)$  the DBI action (\ref{DBI_complex}) becomes
\begin{eqnarray}
&&S  \sim -  \int dt\,d\rho\,\,{\rho^2\over \rho^2+w^2}\,\,
\Bigg[(1+w'^{\,2})\big( (\rho^2+w^2)^2\,-\,\dot b^2\,-\,(\Omega+\dot\chi)^2\,b^2\big)\,-\,\nonumber\\\rc
&&\qquad\qquad
-\dot w^2\big(1+b'^{\,2}\,+\,b^2\,\chi'^{\,2}\big)\,+\,(\rho^2+w^2)^2\big(b'^{\,2}\,+\,b^2\,\chi'^{\,2}\big)\,+\,\nonumber\\
&&\qquad\qquad
+2\,\dot w\,w'\,\big(\dot b\,b'\,+\,(\Omega+\dot\chi)\,\chi'\,b^2\big)\,-\,b^2\Big(\dot b\,\chi'\,-\,(\Omega+\dot\chi)\,b'\Big)\Bigg]^{{1\over 2}}\,\,.
\end{eqnarray}
Since the action does not depend on $t$ explicitly,  it is consistent to take the functions $b$, $\chi$ and $w$ as independent of the time variable
\begin{equation}
b\,=\,b(\rho)\,\,,
\qquad\quad
\chi\,=\,\chi(\rho)\,\,,
\qquad\quad
w\,=\,w(\rho)\,\,.
\end{equation}
By removing the terms with the time derivative from the action, we can rewrite it in a much more simplified fashion as $
S\sim \int d\rho\,{\cal L}\,\,,$
where now
\begin{equation}
{\cal L}\,=\,{\rho^2\over \rho^2+w^2}\,
\sqrt{\Big( (\rho^2+w^2)^2\,-\,\Omega^2\,b^2\Big)\,\Big(
1+b'^{\,2}+w'^{\,2}\Big)\,+\,(\rho^2+w^2)^2\,b^2\,\chi'^{\,2}}\,\,.
\label{cal_L_0}
\end{equation}
We first notice that $\chi$ is a cyclic variable which means that $q$, defined as
\begin{equation}
q\,=\,\Omega\,{\partial {\cal L}\over \partial \chi'}\,=\,\Omega\,{\rho^4\over {\cal L}}\,b^2\,\chi'\,\,,
\label{q_def}
\end{equation}
is independent of $\rho$. The Euler-Lagrange equations derived from ${\cal L}$ are
\begin{eqnarray}
&&\rho(\rho^2+w^2)\,\Big[(\rho^2+w^2)^2\,-\,\Omega^2\,b^2\Big]\,b''\,=\,\nonumber\\\rc
&&\qquad
=-(1+b'^{\,2}+w'^{\,2})\Big[2(\rho^2+w^2)^3\,b'\,+\,\Omega^2\,b\Big(\rho(\rho^2+w^2)-2w^2\,b\,b'\Big)\Big]\,+\,\nonumber\\\rc
&&\qquad\qquad\qquad\qquad\qquad\qquad+
b(\rho^2+w^2)^3(\rho-2 b b')\,\chi'^{\,2}\,\,,\qquad\qquad\nonumber\\\rc
&&\rho(\rho^2+w^2)\,\Big[(\rho^2+w^2)^2\,-\,\Omega^2\,b^2\Big]\,w''\,=\,\nonumber\\\rc
&&\qquad-2\,(1+b'^{\,2}+w'^{\,2})\Big[(\rho^2+w^2)^3\,w'\,-\,\Omega^2\,b^2\,w\,(\rho+w\,w')\Big]\,-\,2\,b^2\,(\rho^2+w^2)^3\,w'\, \chi'^{\,2}\,\,,
\qquad\qquad\nonumber\\\rc
&&b\rho(\rho^2+w^2)\,\Big[(\rho^2+w^2)^2\,-\,\Omega^2\,b^2\Big]\,\chi''\,=\,-2\,\chi'\,\,\Bigg[
(\rho^2+w^2)^3\,\Big(b^3\,\chi'^{\,2}\,+\,\rho\,b'\,+\,b(1+b'^{\,2}+w'^{\,2})\Big)\,-\,\nonumber\\\rc
&&\qquad\qquad\qquad\qquad
-\Omega^2\,b^3\Big(2\rho^2+2\rho\,w\,w'\,+\,w^2(1+b'^{\,2}+w'^{\,2})\Big)\Bigg]\,\,.
\label{eoms}
\end{eqnarray}
The last equation in (\ref{eoms}) is redundant, since we can use the first integral (\ref{q_def}) to obtain the value of $\chi'$ in terms of the other functions
\begin{equation}
\chi'^{\,2}\,=\,{q^2\,\big[(\rho^2+w^2)^2\,-\,\Omega^2\,b^2\big]\over
(\rho^2+w^2)^2\,b^2\,\big[\Omega^2\,\rho^4\,b^2\,-\,q^2\big]}\,\,(1+b'^{\,2}+w'^{\,2})\,\,.
\label{chi_p}
\end{equation}
We can next use this last equation to eliminate $\chi'$ in the first two equations in (\ref{eoms}). The resulting equations for $w$ and $b$ depend on $q$ (alternatively one could Legendre transform and obtain the corresponding Routhian).
\subsection{Boundary conditions}

Let us now study the equations of motion of the gauge field at the UV $\rho\to\infty$. It is convenient to define the  new complex combination
\begin{equation}
c(\rho)\,=\,b(\rho)\,e^{i\chi(\rho)}\,\,,\label{bdec}
\end{equation}
hence $a(\rho, t) = c(\rho) e^{i\Omega t}$.
In terms of $c(\rho)$ the lagrangian density ${\cal L}$ gives
\begin{eqnarray}
{\cal L}\,\sim\,{\rho^2\over \rho^2+w^2}\,\Bigg[
(1+w'^{\,2})\,\Big[ (\rho^2+w^2)^2\,-\,\Omega^2\,|c|^2\Big]\,+\,(\rho^2+w^2)^2\,|c'|^2\,-\,\Omega^2\,
\Big[{\rm Re}\,(c\,c'^{\,*})\Big]^2\Bigg]^{{1\over 2}}\,\,,\qquad
\label{Lagrangian_complex_c}
\end{eqnarray}
and the equation of motion for $c(\rho)$ derived from ${\cal L}$ is
\begin{eqnarray}
&&\partial_{\rho}\,\Big[{\rho^4\over (\rho^2+w^2)^2\,{\cal L}}\,\Big( (\rho^2+w^2)^2\,c'\,-\,\Omega^2\,{\rm Re}\,(c\,c'^{\,*})\,c\Big)\Big]\,+\,\\
&&\qquad\qquad\qquad\qquad
+\,  \Omega^2\,{\rho^4\over (\rho^2+w^2)^2\,{\cal L}}\Big((1+w'^{\,2})\,c\,+\,{\rm Re}\,(c\,c'^{\,*})\,c'\Big)\,=\,0\,\,. \nonumber
\end{eqnarray}
Near the UV at $\rho\to\infty$ we can show  that ${\cal L}\approx \rho^2$ and the equation of motion reduces to
$\partial_{\rho}(\rho^2\,c')  = 0$, whose general solution is of the form $c(\rho) = a_1 + a_2/\rho$.
\iffalse  for $c$ becomes
\begin{equation}
\partial_{\rho}(\rho^2\,c')\,+\,{\Omega^2\over \rho^2}\,c\,=\,0\,\,,
\qquad\qquad
(\rho\to\infty)\,\,,
\end{equation}
 which can be integrated as
\begin{equation}
c\,=\,a_1\,+\,{a_2\,\Omega^2\over \rho}\,+\,\cdots\,\,.
\end{equation}
\eega
\fi
Redefining the constants $a_1$ and $a_2$, we can write asymptotically
\begin{equation}
c(\rho)\,=\,b(\rho)\,e^{i\chi(\rho)}\,=\,{i\,E\over \Omega}\,+\,{j\over \rho}\,+\,\cdots\,\,,
\label{asymp_c}
\end{equation}
where $E$ is the complexified electric field defined in (\ref{complexified_potential}), and $j$ the related  current.  Proceeding similarly, the embedding function $w(\rho)$  behaves near the UV boundary $\rho\to \infty$ as
\begin{equation}
w(\rho)\sim m+{{\cal C}\over \rho}\,+\,\cdots\,\,,
\label{asymp_w}
\end{equation}
where $m$ and ${\cal C}$ are  constants related to the quark mass and quark condensate respectively. We will actually refer to them as such. The relation between $E$, $j$, $m$ and ${\cal C}$ and the electric field ${\cal E}_{YM}(t)$, the electric current ${\cal J}_{YM}(t)$, quark mass $m_q$ and quark condensate $\langle O_m\rangle $ in the boundary theory are
\begin{eqnarray}
&&{\cal E}_{YM}(t)\,=\,\sqrt{{\lambda\over 2\pi^2}}\,e^{i\Omega\,t}\,E\,\,,
\qquad\qquad
{\cal J}_{YM}(t)\,=\,{N_f\,N_c\over \pi^2}\,e^{i\Omega\,t}\,j\,\,,\label{dictionaryE}\\
&& m_q\,=\,{\sqrt{\lambda\over 2\pi^2}}\,m\,\,,\qquad\qquad
\langle O_m\rangle\,=\,-{N_f\,N_c\over \pi^2}\,c\,\,,
\label{dictionarym}
\end{eqnarray}
where $\lambda=g^2_{YM}\,N_c$ is the 't Hooft coupling of the ${\cal N}=4$ theory.  The integration constant
$q$  defined in (\ref{q_def}) can be written in terms of the   field $c$ and $c^*$
\begin{equation}
q\,=\,i\,{\Omega\,\rho^4\over 2\,{\cal L}}\,\big(c\,c'^{\,*}\,-\,c^*\,c'\big)\,\,,
\label{q_c_c*}
\end{equation}
and can be easily related  to UV data. As $w(\rho)\sim m+{\cal C}/\rho$ when $\rho\to\infty$, it follows that ${\cal L}\approx \rho^2$ for large $\rho$. Moreover, from (\ref{asymp_c}) $c'\sim -j/\rho^2$ and, therefore
\begin{equation}
q\,=\,{\rm Re}(E\,j^*)\,=\,j_x\,E_x\,+\,j_y\,E_y\,\,,
\end{equation}
which endows  $q$ with the physical interpretation of the Joule heating. If we assume that the driving effectively heats the system we should expect that  $q\ge 0$, and from (\ref{q_def})  this implies $\Omega\,\chi{\,}'\geq 0$. We can see that this result is consistent with the  ingoing boundary condition at the pseudohorizon, which can be argued as follows.  The wavefront of the complexified gauge potential $a(t,\rho)$ in (\ref{complexified_potential}) is the surface $\chi(\rho)+\Omega\,t\,=\,{\rm constant}$ (we are in the case in which $\chi$ does not depend on $t$). By performing a generic variation of this condition,
it follows that
\begin{equation}
\delta\,\rho\,=\,-{\Omega\over \chi{\,'}}\,\delta\,t\,\,.
\label{variation_wavefront}
\end{equation}
Requiring that our gauge potential represents an infalling wave at the pseudo-horizon implies
 $\delta\rho<0$ for $\delta t>0$ which, following (\ref{variation_wavefront}), only occurs when $\Omega\,\chi{\,'}\ge 0$ and, therefore,
$q\ge 0$.

The equations of motion (\ref{eoms}) and the UV boundary conditions (\ref{asymp_c}) and  (\ref{asymp_w}) have the following scaling symmetry
\begin{eqnarray}
&&
t\to t/\alpha\,\,,\qquad
\rho\to\alpha\,\rho\,\,,\qquad
w\to \alpha\,w\,\,,\qquad
b\to\alpha b\,\,,\qquad
\chi\to\chi\,\,,\nonumber\\\rc
&&\Omega\to\alpha\,\Omega\,\,,\qquad
E\to\alpha^2\,E\,\,,\qquad
j\to\alpha^2\,j\,, \label{scaling} \\ \nonumber\\
&&m\to\alpha \,m\,\,,\qquad
{\cal C}\to\alpha^2\,{\cal C}\,\,,\qquad
q\to\alpha^4\,q\,\,.\nonumber
\end{eqnarray}
The Lagrangian and the action transform homogeneously  ${\cal L} \to \lambda^2 {\cal L}$ and $S\to \lambda^2 S$.
By choosing $\alpha=1/m$ in  (\ref{scaling}) we can make $m=1$ and deal with the remaining quantities in units of (the  appropriate powers of) $m$.

\section{Types of embeddings}
\label{types_of_embeddings}

The equations of motion written in (\ref{eoms}) are potentially singular when
\begin{equation}
b_0\,=\,{w_0^2+\rho_c^2\over \Omega}\,\,,
\label{b0_w0_rho_c}
\end{equation}
where $b_0=b(\rho=\rho_c)$ and $w_0=w(\rho=\rho_c)$.
The point $\rho=\rho_c$  where the condition (\ref{b0_w0_rho_c}) holds will be referred to as the pseudohorizon. Indeed, we will show below that $\rho=\rho_c$ is the  event horizon of the induced open string metric on the D5-brane. The IR  behaviour  of the bulk fields when approaching this pseudohorizon determines three types of embeddings (see figure \ref{embeddings}).  We can have black hole embeddings (which cross the horizon), Minkowski embeddings (which do not intersect the pseudohorizon and reach the point $\rho=0$) and critical embeddings (for which  the brane reaches the $\rho_c$ at  $\rho_c=0$).  In order to avoid the singularity at $\rho=\rho_c$  the first derivatives of  the functions $w(\rho)$, $b(\rho)$ and $\chi(\rho)$ have to be adjusted properly
for black hole and critical embeddings.

\begin{figure}[!ht]
\center
 \includegraphics[width=0.50\textwidth]{xembeddings.pdf}
  \caption{Profiles of the different types of embeddings for $E=\Omega=1$. The solid (dashed) lines correspond to black hole (Minkowski) embeddings. The dotted line represents a critical embedding. }
\label{embeddings}
\end{figure}

\subsection{Black hole embeddings}

Expanding the functions $w(\rho)$, $b(\rho)$ and $\chi(\rho)$  around $\rho=\rho_c$ as
\begin{eqnarray}
&&w(\rho)\,=\,w_0\,+\,w_1\,(\rho-\rho_c)\,+\,\cdots\,\,, \nonumber \\
\rule{0mm}{5mm}
&&b(\rho)\,=\,b_0\,+\,b_1\,(\rho-\rho_c)\,+\,\cdots\,\,,\\
\rule{0mm}{5mm}
&&\chi(\rho)\,=\,\chi_0\,+\,\chi_1\,(\rho-\rho_c)\,+\,\cdots\,\,,\nonumber
\end{eqnarray}
we obtain coefficients $w_1$, $b_1$ and $\chi_1$ whose explicit expressions are given in appendix \ref{app_A}. Notice that $\chi_1$ appears in the expression of the on-shell lagrangian density ${\cal L}_0$  evaluated at the pseudohorizon $\rho=\rho_c$
\begin{equation}
{\cal L}_0(\rho_c)\,=\,\rho_c^2\,b_0\,\chi_1\,\,.
\label{cal_L_0_chi_1}
\end{equation}
We can use this expression to obtain the Joule heating $q$ in (\ref{q_def}). Indeed as $q$ is constant, we can evaluate it at the pseudohorizon, with the result
\begin{equation}
q\,=\,\Omega\,\rho_c^2\,b_0\,\,.
\label{qJH}
\end{equation}
which only depends on $b_0$. Using (\ref{b0_w0_rho_c}),  we get
\begin{equation}
q\,=\,\rho_c^2\big(w_0^2+\rho_c^2\big)\,\,.
\label{q_w0_rho_c}
\end{equation}

\subsection{Minkowski embeddings}
In this case there is no $\rho_c$  that solves (\ref{b0_w0_rho_c}) and the brane reaches the origin $\rho=0$. We can therefore evaluate the Joule heating constant $q$ by taking $\rho=0$ in (\ref{chi_p}) and, since the right-hand side of this equation contains a $\rho^4$ factor we get $q=0$. Thus, for these  embeddings there is no net energy flux on the boundary theory. Moreover, from (\ref{chi_p}) one readily concludes that  the phase function $\chi(\rho)$ is in fact a constant. Expanding $b(\rho)$ and
$w(\rho)$ around $\rho=0$
\begin{eqnarray}
&&b(\rho)\,=\,b_0\,-\,{b_0\,\Omega^2\over 6\big(w_0^4-\Omega^2\,b_0^2\big)}\,\rho^2\,+\,\cdots\,\,,\nonumber\\\rc
&&w(\rho)\,=\,w_0\,+\,{b_0^2\,\Omega^2\over 3\,w_0\big(w_0^4\,-\,\Omega^2\,b_0^2\big)}\,\rho^2\,+\,\cdots\,\,.
\end{eqnarray}
\begin{figure}[!ht]
\center
 \includegraphics[width=0.47\textwidth]{xEvsOmegaPoints} \includegraphics[width=0.5\textwidth]{xEJOmega3d}
  \caption{Electric field of the critical embeddings versus driving frequency. The frequencies for which $E$ vanish are the critical frequencies $\Omega_c$ of the vector meson Floquet condensates. The blue dots signal the frequencies (\ref{Omega_meson}) of the vector meson spectrum  for different values of the principal quantum number $n$. On the right hand side, the three dimensional version of the left plot, focussing close to the first critical point, which stretches into a segment $\Omega/m \in ( 1.496, 1.732)$.
   }
\label{E_Omega}
\end{figure}

\subsection{Critical embeddings}
\label{Critical_bc}
When the  effective  horizon reaches the origin, $\rho_c=0$, the embedding is critical. The value of the
$b$ field at the horizon is $b_0=w_0^2/\Omega$, $\chi(\rho)$ is also here constant and we can expand
$b(\rho)$ and $w(\rho)$ around $\rho=0$, with the result
\begin{eqnarray}
&&b(\rho)\,=\,{w_0^2\over \Omega}\,-\,{\Omega\over 2\sqrt{\Omega^2+4\,w_0^2}}\,\rho\,+\,\cdots\nonumber\\\rc
&&w(\rho)\,=\,w_0\,+\,{w_0\over \sqrt{\Omega^2+4w_0^2}}\,\rho\,+\,\cdots\,.
\end{eqnarray}

\begin{figure}[!ht]
\center
 \includegraphics[width=0.5\textwidth]{xEvsOmegaLogLine}
  \caption{Logarithmic version of the plot in Fig.\,\ref{E_Omega}.    The high $\Omega/m$ fitting is derived in Sec. \ref{sec:critembed} (see eq. (\ref{ECMinkSol})). }
\label{E_Omega_log}
\end{figure}

\subsection{Effective metric}
The effective open string metric $\gamma_{ab}$ is defined as \cite{Seiberg:1999vs}
\begin{equation}
\gamma_{ab}\,=\,h_{ab}\,+\,(2\pi \alpha')\,F_{ac}\,F_{bd}\,h^{cd}\,\,,
\end{equation}
where we now denote by $h_{ab}$ the induced metric (\ref{induced_metric}). Explicitly,   for our ansatz, we have
\begin{eqnarray}
&&\gamma_{ab}\,d\xi^a\,d\xi^b=-F(\rho)\,dt^2+{1+w'^{\, 2}+|c'|^2\over \rho^2+w^2}\,d\rho^2-
{2\Omega\,{\rm Im}(c\,c'^{\,*})\over  \rho^2+w^2}\,dt\,d\rho
+\,F(\rho)\,e_+\,e_-\,+\,
\qquad\qquad\qquad
\nonumber\\\rc
&&
\qquad\qquad
+{\rho^2+w^2\over 4(1+w'^{\, 2})}\,\big(e_-\,c'+e_+\,c'^{\,*})^2\,+\
{\Omega^2\over 4(\rho^2+w^2)}\big(e_-\,c+e_+\,c^{*})^2+{\rho^2\over \rho^2+w^2}d\Omega_2^2\,\,,
\end{eqnarray}
with $F(\rho)$ given by the following function
\begin{equation}
F(\rho)\equiv {(\rho^2+w^2)^2-\Omega^2\,|c|^2\over \rho^2+w^2}\,\,,
\end{equation}
and $e_{\pm}$ are  complex 1-forms
\begin{equation}
e_{\pm}\,=\,e^{\mp\,i\Omega t}\big(dx\pm i dy\big)\,\,.
\end{equation}
The $(t,\rho)$ part of the metric gets diagonalized  by means of  the following change of coordinates
\begin{equation}
d\tau\,=\,dt\,-\,A(\rho)\,d\rho\,\,,
\qquad\qquad
d\rho_*\,=\,B(\rho)\,d\rho\,\,,
\label{tortoise_coord}
\end{equation}
where $A(\rho)$ and $B(\rho)$ are the following functions:
\begin{equation}
A(\rho)\,=\,-{\Omega\,{\rm Im}\big(c\,c'^{\,*}\big)\over
(\rho^2+w^2)^2-\Omega^2\,|c|^2}\,\,,
\qquad\qquad
B(\rho)\,=\,{(\rho^2+w^2)\,{\cal L}_0\over \rho^2\big[(\rho^2+w^2)^2-\Omega^2\,|c|^2\big]}\,\, .
\label{A_B_tortoise}
\end{equation}
 The new coordinate $\rho_*$ is just the Eddington-Finkelstein tortoise coordinate.
In these coordinates, the transformations  in (\ref{scaling}) act as a rescaling  $ (\tau, \rho_*)\to \lambda^{-1} (\tau, \rho_*)$, and  the effective metric takes the form
\begin{eqnarray}
&&\gamma_{ab}\,d\xi^a\,d\xi^b=-F(\rho)\,(\,-d\tau^2+d\rho_*^2\,)\,+\,
F(\rho)\,e_+\,e_-\,+\,{\rho^2+w^2\over 4(1+w'^{\, 2})}\,\big(e_-\,c'+e_+\,c'^{\,*})^2\,+\nonumber\\\rc
&&
\qquad\qquad\qquad\qquad
+{\Omega^2\over 4(\rho^2+w^2)}\big(e_-\,c+e_+\,c^{*})^2+{\rho^2\over \rho^2+w^2}d\Omega_2^2\,\,.
\label{open_string_metric}
\end{eqnarray}
The function $F(\rho)$ vanishes at $\rho=\rho_c$ and, therefore, this function acts as a blackening factor for the effective metric (\ref{open_string_metric}), with $\rho=\rho_c$ playing the role on an event horizon. The effective Hawking temperature $T_H$ is given by
\begin{equation}
T_H\,=\,{\kappa\over 2\pi}\,=\,-{\gamma_{tt}'\over 4\pi\,\gamma_{t\rho}}\Bigg|_{\rho=\rho_c}\,\,.
\end{equation}
Using the values of $\gamma_{tt}$ and $\gamma_{t\rho}$ found above, we get the $T_H$ in terms of near horizon data
\begin{equation}
T_H\,=\,{2(\rho_c+w\,w')-\Omega\,b'\over 2\pi\,\chi'\,b}\Bigg|_{\rho=\rho_c}\,=\,
{2(\rho_c+w_0 w_1)\,-\,\Omega\,b_1\over 2\pi\chi_1\,b_0}\,\,.
\label{TH_final}
\end{equation}

\begin{figure}[!ht]
\center
 \includegraphics[width=0.47\textwidth]{xJvsE1}
 \qquad
 \includegraphics[width=0.47\textwidth]{xJvsE2}
  \caption{Electric current versus electric field for $\Omega\le \Omega_c$ (left) and
  $\Omega>\Omega_c$ (right) around the first vector meson resonance. The solid curves correspond to the insulator (Minkowski) phase, whereas the dotted curves represent the conductive (black hole) phase. On the right curve we show that $E$ can only vanish for $j\not=0$ for Minkowski embeddings with $\Omega_c= 1.4965\leq \Omega\leq 1.732 = \Omega_{meson}$. }
\label{J_vs_E}
\end{figure}

\section{Phase diagram}
\label{phase_diagram}

In the previous section we have addressed the different types of embedding according to the IR boundary conditions. On the other hand, the phase diagram is constructed out of the UV boundary data that contain information about the sources and the responses.
The analysis of the generic case proceeds by  numerically integrating  the system of equations  (\ref{eoms}). In this section we will obtain the precise boundary of the phase space of solutions presented in Figs. \ref{E_Omega} and \ref{E_Omega_log}.   Although the qualitative results are the same as the ones found in \cite{Kinoshita:2017uch}, we need the precise values of these boundaries in order to locate interesting places where to look for conductivity properties in the next section.
As usual, integration proceeds from the IR to the UV. Indeed, enforcing regularity conditions found in section \ref{types_of_embeddings} as initial conditions at the pseudohorizon $\rho=\rho_c$ and integrating out up to the boundary yields a unique solution. For a given frequency $\Omega$ we can specify  $b_0=b(\rho=\rho_c)$ and $w_0=w(\rho=\rho_c)$  and use the values of $w_1$, $b_1$ and $\chi_1$ computed in appendix \ref{app_A}(for Minkowski embeddings  $\rho_c=0$ and $\chi(\rho)$ is constant).  From the UV behavior of the solutions $b(\rho)$, $\chi(\rho)$ and $w(\rho)$ when $\rho\to\infty$  one can  extract
$E$, $j$, $m$ and ${\cal C}$.  Due to the scaling symmetry (\ref{scaling}) these quantities are not independent. It is natural to measure all quantities in terms of the quark mass $m$. Accordingly, we will present our results for $\Omega/m$,  $|E|/m^2$, $j/m^2$ and ${\cal C}/m^2$.

\begin{figure}[!ht]
\center
 \includegraphics[width=0.47\textwidth]{xcvsE1}
 \qquad
 \includegraphics[width=0.47\textwidth]{xcvsE2}
  \caption{Quark condensate as a function of the electric field for several $\Omega$ around the first resonance. The curves on the left (right) are for $\Omega\le \Omega_c$ ($\Omega>\Omega_c$). Solid (dashed) curves correspond to Minkowski (black hole) embeddings. }
\label{c_vs_E}
\end{figure}

\begin{figure}[!ht]
\center
 \includegraphics[width=0.47\textwidth]{xqvsE1}
 \qquad
 \includegraphics[width=0.47\textwidth]{xqvsE2}
  \caption{The Joule heating $q$ as a function of the electric field $E$ for black hole embeddings and several frequencies around the first resonance. On the left (right) plots $\Omega\le \Omega_c$ ($\Omega>\Omega_c$).  }
\label{fig:q_vs_E}
\end{figure}
\begin{figure}[H]
\center
  \includegraphics[width=0.55\textwidth]{xqvsOmega}
  \caption{The Joule heating $q$ as a function of the frequency $\Omega$ for black hole embeddings and several values of the electric field.  }
\label{fig:q_vs_Omega}
\end{figure}

In figure \ref{E_Omega} we show the phase diagram in the parameter space
$(\Omega/m, |E|/m^2)$. The solid curve is the locus of critical embeddings. We can  think of it roughly as a  boundary between the conductive phase (above) made of  black hole embeddings,  and insulator phase (below) made of Minkowski embeddings\footnote{Although it is pretty clear from figure \ref{embeddings} that the critical embeddings lie in between  Minkowski and Black Hole embeddings, the spiralling multivaluedness in the vicinity of the solid line (see figure (\ref{J_vs_E}) below) implies that one can  find Black Hole embeddings closely below it, and also Minkowski embedding above.}. In figure \ref{J_vs_E} we plot the electric current versus the electric field for driving frequencies around the firs resonance. In figure \ref{c_vs_E} we represent the quark condensate for Minkowski and black hole embeddings and different driving frequencies.  We notice in Figures  \ref{E_Omega} and \ref{E_Omega_log}
 that $|E|/m^2$ has a series of maxima whose values decrease as $\Omega/m$ increases. Moreover, for some discrete values $\Omega_c/m$ of the frequency the electric field vanishes and, as shown in figure \ref{J_vs_E}, the corresponding rotating current is non-zero.

 Thus, for these frequencies we have a vector meson Floquet condensate of the type mentioned in the introduction. Actually, these $E=0$ Floquet states exist  also for Minkowski embeddings in a finite range of frequency  $\Omega_c< \Omega \le\Omega_{meson}$, where $\Omega_{meson}$ is the mass of a  vector meson in the defect theory. This mass spectra was computed in \cite{Arean:2006pk} and is given by a tower of values depending on a principal quantum number $n$

\begin{equation}
\Omega_{meson}/m\,=\,2\sqrt{
\Big(n+{1\over 2}\Big)\Big(n+{3\over 2}\Big)}\,\,,
\qquad\qquad
n\,=\,0,1,2,\cdots\,\,.
\label{Omega_meson}
\end{equation}
In section \ref{Large_Omega}  and appendix \ref{linearized_embeddings} we obtain these frequencies by means of a linear analysis of the equations of motion. The first four values of the critical resonant frequencies and of the meson masses are
\begin{eqnarray}
&&\Omega_c/m\,=\,1.4965\,,\,3.5308\,,\,5.5676\,,\,7.5851\,\cdots\,  \label{floqcondfreqs}\\ \rule{0mm}{6mm}
&&\Omega_{meson}/m\,=\,1.7320\,,\,3.8730\,,\,5.9161\,,\,7.9372\,,\cdots\,\,.  \label{mesonfreqs}
\label{Omega_c_Omega_meson}
\end{eqnarray}

In figure \ref{fig:q_vs_E} we plot the Joule heating $q$ as a function of the electric field, whereas in figure \ref{fig:q_vs_Omega} the Joule heating for black hole embeddings is plotted versus the driving frequency for several values of the electric field.

To get further insight on the structure of the phase diagram of the model, it is interesting to analyze in detail the relation between the current $j$ and the electric field $E$. Since $j$ and $E$ are vector quantities with two components in the $xy$ plane, we expect to have  a relation of the type:
\begin{equation}
\begin{pmatrix}
j_x\cr j_y
\end{pmatrix}\,=\,
\begin{pmatrix}
\gamma_{xx}&&\gamma_{xy}\cr
-\gamma_{xy}&&\gamma_{xx}
\end{pmatrix}\,
\begin{pmatrix}
E_x\cr E_y
\end{pmatrix}\,\,,
\label{gamma_def}
\end{equation}
where the form of the matrix is dictated by the rotational symmetry in the $xy$ plane.  The matrix elements $\gamma_{xx}$ and $\gamma_{xy}$ are susceptibilities, defined in analogy with the longitudinal and transverse (Hall) conductivities.
From the definitions (\ref{gamma_def}) it is immediate to find $\gamma_{xx}$ and $\gamma_{xy}$ in terms of the components of $E$ and $j$:
\begin{equation}
\gamma_{xx}\,=\,{E_x\,j_x+E_y\,j_y\over E_x^2+E_y^2}\,=\,{|j|\over |E|}\,\cos\delta\,\,,
\qquad\qquad
\gamma_{xy}\,=\,{E_y\,j_x-E_x\,j_y\over E_x^2+E_y^2}\,=\,{|j|\over |E|}\,\sin\delta\,\,,
\end{equation}
where $\delta$ is the angle formed by the vectors $(j_x, j_y)$ and $(E_x, E_y)$ in the $xy$ plane. The quantity $\gamma_{xx}$ is related to the Joule heating $q$. Indeed, one has $\gamma_{xx}=q/|E|^2$. Given this, and the fact that we have profusely  illustrated the behaviour of $q$ in Figs. \ref{fig:q_vs_E} and \ref{fig:q_vs_Omega}, we shall  focus on the study of $\gamma_{xy}$.

\begin{figure}[!ht]
\center
 \includegraphics[width=0.47\textwidth]{xgammaxyOmegaovmFrom1d3To2}
 \qquad
  \includegraphics[width=0.47\textwidth]{xgammaxyOmegaovmFrom2d3To2d7}
   \caption{The susceptibility coefficient $\gamma_{xy}$ as a function of $\rho_c$ for black hole embeddings and different driving frequencies. On the left we plot the values for frequencies close to the first critical resonant frequency $\Omega_c/m=1.49$. On the right we represent $\gamma_{xy}$ for driving frequencies around $\Omega/m=2.5$, which is frequency  of the first zero of $j$ for critical embeddings.
}
    \label{gamma_xy}
\end{figure}

For massless black hole embeddings $\gamma_{xy}$ vanishes, since in this case $j$ and $E$ are colinear, as we demonstrate analytically in section \ref{massless_embd} below. On the other hand $\gamma_{xy}\not=0$ for Minkowski embeddings since there is no Joule heating in this case and the vectors $j$ and $E$ are orthogonal. This behavior can be verified by studying $\gamma_{xy}$ for black hole embeddings as a function of $\rho_c$ (see figure \ref{gamma_xy} ). When $\rho_c$ is large enough $\gamma_{xy}\to 0$, whereas it reaches a non-vanishing value as $\rho_c\to0$. Interestingly, the sign of $\gamma_{xy}(\rho_c\to 0)$ changes at a discrete set of values of the driving frequency. We have identified these values as the critical resonant frequencies $\Omega_c$ of (\ref{Omega_c_Omega_meson}), as well as those for which the current $j$ vanishes for critical embeddings. These last frequencies with $j=0$ are very close to the maxima of the lobes in the $|E|/m^2$ versus $\Omega/m$ plot of figure \ref{E_Omega}. The first numerical values of the frequencies which make $E$ maximal and $j$ zero for critical embeddings are:
\begin{eqnarray}
&& {\Omega\over m}(j=0)\,=\,2.5080\,,\,4.5489\,,\,6.5769\,,\,8.5960\,,\cdots\,\,,\nonumber\\\rc
&& {\Omega\over m}(E\,{\rm maximal})\,=\,2.4008\,,\,4.4772\,,\,6.5270\,,\,8.5638\,,\,\cdots
\end{eqnarray}
\begin{figure}[!ht]
\center
 \includegraphics[width=0.45\textwidth]{xEJlobes}
 \qquad
  \includegraphics[width=0.45\textwidth]{xSinDelta}
   \caption{On the left we plot  $|E|/m^2$ and $j/m^2$ versus the driving frequency for critical embeddings. On the right we depict the value of $\sin\delta$  for embeddings with $\rho_c=0$.  }
    \label{E_j_lobes_sin_delta}
\end{figure}
In figure \ref{E_j_lobes_sin_delta} we plot together the $|E|/m^2$ and $j/m^2$ curves as a function of $\Omega/m$. We notice that the current curve also displays a lobe-shaped structure, displaced with respect to the $|E|/m^2$ curve, whose maxima (zeros) are very close to the zeros (maxima) of the $|E|/m^2$ curve. We have also checked that $\sin \delta(\rho_c\to 0)=\pm 1$ and changes discontinuously its sign at the discrete set of frequencies described above (see  figure \ref{E_j_lobes_sin_delta}).

\section{Analytic Solutions}\label{sec:analsol}

There are two limits where the equations are amenable to analytic solution. These are the low mass limit $m\to 0$, and the high frequency limit $\Omega \to \infty$. In the first case all embeddings are of black hole type. In the second, the three types of embeddings can occur.

\subsection{Massless embeddings }
\label{massless_embd}

Let us consider the particular case of massless black hole embeddings, which correspond to putting the embedding function $w(\rho)=0$. By inspection  one easily concludes that $w=0$ is a consistent truncation of the equations of motion in (\ref{eoms}). In this massless case,  the lagrangian density (\ref{cal_L_0})  takes the form
\begin{equation}
{\cal L}_0\,=\,\sqrt{
(\rho^4-\Omega^2\,b^2)(1+b'^{\,2})\,+\,
\rho^4\,b^2\,(\chi')^2}\,\,.
\end{equation}
As in the general case, $\chi(\rho)$ is a cyclic variable which, according to (\ref{chi_p}), is now given by
\begin{equation}
\chi'\,=\,{q\over \rho^2 b}\,
{\sqrt{
(\rho^4-\Omega^2\,b^2)(1+b'^{\,2})}\over
\sqrt{\Omega^2\,\rho^4\,b^2\,-\,q^2}}\,\,.
\label{chi_p_massless_general}
\end{equation}
Let us now eliminate $\chi(\rho)$ by introducing the Routhian ${\cal R}_0$, defined as
\begin{equation}
{\cal R}_0\,=\,\chi'\,{\partial {\cal L}_0\over \partial\,\chi'}\,-\,{\cal L}_0\,\,.
\end{equation}
After some calculation, ${\cal R}_0$ as a function of  $b(\rho)$ and $q$ becomes
\begin{equation}
{\cal R}_0\,=\,-\sqrt{
\Big(1\,-\,{q^2\over \Omega^2\,\rho^4\,b^2}\Big)\,(\rho^4-\Omega^2\,b^2)}\,\,\,
\sqrt{1+b'^{\,2}}\,\, .
\end{equation}
As usual in this flavour brane setup, demanding reality of the Routhian fixes  transport coefficients in terms of horizon data \cite{Karch:2007}. Indeed, at $\rho_c$ the two factors under the square root must change sign, and therefore vanish,   simultaneously.
This enforces $b_0= b(\rho_c)$ and $q$ to
satisfy both equations  (\ref{b0_w0_rho_c}) with $w_0=0$,  and (\ref{qJH}). Combined together, these equation entail, in the massless case
\begin{equation}
q = \Omega^2 b_0^2\, .\label{qdeb0}
\end{equation}

Let us look for solutions to the  Euler Lagrange equations  in which $b(\rho)$ is constant. Writing these equations in the following form
\begin{equation}
{\partial\over \partial b}
\Bigg[\Big(1\,-\,{q^2\over \Omega^2\,\rho^4\,b^2}\Big)\,(\rho^4-\Omega^2\,b^2)
\Bigg]\,=\,-{2\over b}\,\Big(\Omega^2\,b^2\,-\,{q^2\over \Omega^2\,b^2}\Big)\,\,,
\label{condition_constant_b}
\end{equation}
it's clear that there is a chance to have $b(\rho)$ constant  precisely if it is equal to $b_0$ related to $q$ as in (\ref{qdeb0}).
 Plugging this value of   $b=b_0$ into the right-hand side of (\ref{chi_p_massless_general}) we find
\begin{equation}
\chi'^{\,2}\,=\,{\Omega^2\over \rho^4}\,\,,
\label{chip_square}
\end{equation}
that can be integrated right away
\begin{equation}
\chi\,=\,-{\Omega\over \rho}\,+\,\chi_0\,\,,
\end{equation}
with $\chi_0$  constant. Then $c(\rho)=b e^{i\chi}$ ends up being
\begin{equation}
c(\rho)\,=\,{\rho_c^2\over \Omega}\,
e^{i\chi_0}\,e^{-i\,{\Omega\over \rho}}\,\,.
\end{equation}
The extreme simplicity of this solution must be contrasted with the need for numerical integration in the D3-D7 setup
\cite{Hashimoto:2016ize}.
By expanding  near the UV as in (\ref{asymp_c}),  the electric field and current  at the boundary are equal  and given by
\begin{equation}
E\,=\,j\,=\,-i\,\rho_c^2\,e^{i\chi_0}\,\,.
\end{equation}
Hence, in the massless case, the rotating electric field and the induced current are  aligned.
The effective Hawking temperature for this massless case can be obtained from (\ref{TH_final})
\begin{equation}
T_H\,=\,{\rho_c\over \pi\,b_0\,\chi'(\rho_c)}\,\,.
\end{equation}
Using (\ref{chip_square}) and (\ref{b0_w0_rho_c}) this simplifies into

\begin{equation}
T_H\,=\,{\rho_c\over \pi}\,=\,{q^{{1\over 4}}\over \pi}\,\,.
\end{equation}

Black hole embeddings with a small non-vanishing mass can be studied as perturbations of the exact massless solutions of this section. This analysis is carried out in detail in appendix \ref{app_B}.

\subsection{High frequency limit}
\label{Large_Omega}

If the frequency $\Omega$ is large, the boundary condition at the UV (\ref{asymp_c}) fixes that $|c(\rho\to\infty)|=E/\Omega\ll 1$ and this  enforces $c(\rho)$ to be small for all values of $\rho$.   The gauge field  can then be treated as a perturbation around the state with $c=0$ and, as we shall see, admits an analytical solution.  To start with, using the scaling symmetry (\ref{scaling}) we can choose to  linearize the  equations of motion derived from (\ref{Lagrangian_complex_c})
near the solution with $c=0$ and $w=1$. Writing  $w=1+\delta w$ we get
\begin{eqnarray}
&& c''\,+\,{2\over \rho}\, c'\,+\,{\Omega^2\over (1+\rho^2)^2}\, c\,=\,0\,\,  ,\\ \rule{0mm}{8mm}
&&\delta w''\,+\,{2\over \rho}\,\delta w'\,=\,{2\Omega^2\over (1+\rho^2)^3}\,| c|^2\,\,.
\label{c_deltaw_eqs}
\end{eqnarray}
where $c$ and $\delta w$ are small.  Notice that, although at linear order the equations decouple, as in \cite{Kinoshita:2017uch} we have included the lowest order contribution  to the coupling of modes, ${\cal O}(|\delta c|^2)$, in the equation of $\delta w$.
The general solution of the equation of $\delta c$ is
\begin{equation}
 c(\rho)\,=\,c_1\,g_1(\rho)\,+\,c_2\,g_2(\rho)\,\,,
\label{general_c_linear}
\end{equation}
where $c_1$ and $c_2$ are constants and $g_1(\rho)$ and $g_2(\rho)$ are the functions
\begin{eqnarray}
&&g_1(\rho)\,=\,{\sqrt{1+\rho^2}\over \rho}\,\sin\Big(\sqrt{1+\Omega^2}\,\arctan(\rho)\Big)\,\,,\nonumber\\\rc
&&g_2(\rho)\,=\,{\sqrt{1+\rho^2}\over \rho}\,\cos\Big(\sqrt{1+\Omega^2}\,\arctan(\rho)\Big)\,\,.
\end{eqnarray}
with asymptotic  behaviour   as $\rho\to\infty$ given by
\begin{eqnarray}
&&g_1(\rho)\,=\,\sin\Big({\pi\over 2}\,\sqrt{1+\Omega^2}\Big)\,-\,
\sqrt{1+\Omega^2}\,\cos\Big({\pi\over 2}\,\sqrt{1+\Omega^2}\Big)\,{1\over \rho}\,+\,\cdots\,\,,\nonumber\\\rc
&&g_2(\rho)\,= \,\cos\Big({\pi\over 2}\,\sqrt{1+\Omega^2}\Big)\,+\,
\sqrt{1+\Omega^2}\,\sin\Big({\pi\over 2}\,\sqrt{1+\Omega^2}\Big)\,{1\over \rho}\,+\,\cdots\,\,.
\label{g12_UV}
\end{eqnarray}
Similarly  near $\rho=0$,  $g_1$ and $g_2$ behave as
\begin{eqnarray}
&&g_1(\rho)\,=\,\sqrt{1+\Omega^2}\,-\,{\Omega^2\,\sqrt{1+\Omega^2}\over 6}\,\rho^2\,+\,\cdots\nonumber\\\rc
&&g_2(\rho)\,=\,{1\over \rho}\,-\,{\Omega^2\over 2}\,\rho\,+\,\cdots
\label{g12_IR}
\end{eqnarray}

At the linearized level the Joule heating $q$ of Eq. (\ref{q_c_c*}) is proportional to the Wronskian of $ c$ and $ c^{*}$
\begin{equation}
q\,=\,i\,{\Omega\,\rho^2\over 2}\,\Big( c\, c'^{\,*}\,-\, c^{*}\, c'\Big)\,\,.
\label{q_linear}
\end{equation}
Since $q$ is independent of $\rho$, we could compute it by using the expansions (\ref{g12_UV}) around $\rho=\infty$ or (\ref{g12_IR}) around $\rho=0$. The result one obtains is the same and given by
\begin{equation}
q\,=\,\Omega\,\sqrt{1+\Omega^2}\,\,{\rm Im}\,(c_1\,c_2^{*})\,\,.
\end{equation}

The solution for $w(\rho)$ can be written as
\begin{equation}
\delta w(\rho)\,=\,-{2\Omega^2\over \rho}\int_{0}^{\rho}\,{s^2\over (s^2+1)^{3}}\,| c(s)|^2\,-\,
2\Omega^2\,\int_{\rho}^{\infty}\,{s\over (s^2+1)^{3}}\,| c(s)|^2\,+\,c_3\,\,,
\label{delta_w_linear}
\end{equation}
with $c_3$ being a new integration constant.

Since we obtained the general solution of our equations, we can now analyze the different types of embeddings. The
details of this study are deferred to appendix \ref{linearized_embeddings}. Let us summarize here the results that we obtain there. (i) Regularity of the  Minkowski embeddings at $\rho=0$ enforces that  the constant $c_2$ in (\ref{general_c_linear}) has to vanish.  (ii) By imposing the additional condition $E=0$ to these Minkowski solutions, we obtain the resonant frequencies (\ref{Omega_meson}), corresponding to the meson masses of the D3-D5 model.  (iii) To construct black hole embeddings we have to build up a pseudohorizon in our linearized theory and impose the corresponding infalling boundary conditions. Although the linear approximation breaks down for the critical embeddings, it turns out that  one can use the linear Minkowski solution to represent the critical configuration in the UV region of large $\rho$ and use this result to represent features of the phase diagram, such as the envelope curve of figure
\ref{E_Omega_log}.

\section{Conductivities}
\label{Conductivities}

One of the main uses of the Floquet driving  is the artificial engineering of materials exhibiting transport phenomena with topological character. In this sense, the obtention of a Hall conductivity without a magnetic field  is an important hallmark. In the present context, this has been termed photovoltaic Hall effect\cite{Oka_2009}  where it's origin has  been shown to be  topological, like the Thouless pumping. The time-reversal symmetry breaking and hence, the nontrivial band topology,  is induced  by the circularly polarized light.  This proposal has been the subject of both theoretical and experimental studies  during the last decade \cite{Morimoto:2010, Ikebe:2010, Wang:2013, Kumar:2016, McIver:2020}.
In \cite{Hashimoto:2016ize} the photovoltaic Hall conductivity for massless carriers was obtained in the D3-D7 model,   and speculated to behave consistently with the picture of a strongly coupled version of a  (3+1) Weyl semimetal.
Our findings show similarities and  departures from their results. For example, the massless case will be shown to be  trivial. However, in the massive case, we will indeed find an intricate Hall effect.

Following the proposal in \cite{Oka_2009b} we proceed to study the response of our system to a small additional electric field on the boundary pointing in a fixed direction. From this response we will be able to extract the AC and DC conductivities.
We will closely mimic the strategy in \cite{Hashimoto:2016ize} whose main steps  will be reviewed and extended here for completeness, as our case involves  three coupled field perturbations.

For the analysis it is most convenient to stick to the cartesian basis used in (\ref{rotating_electric_field}).
  On top of this background  field, the idea is to place another   fixed direction electric field
\begin{equation}
 {\vec  {\cal E}}(t) = O(t)\vec E + \vec \epsilon(t)   \label{perturbed_E}\, ,
 \end{equation}
    with $|\vec \epsilon| \ll  |\vec E|$, and a  harmonic (AC) time dependence
 \begin{equation}
   \vec \epsilon(t) = \vec\epsilon\,\,e^{-i\omega\,t}\ \, . \label{perturbed_E2}
 \end{equation}
Our task is to extract the response of the system out of the non equilibrium steady state (NESS) solutions spelled out in the previous sections. In particular,  the currents will also suffer a perturbation $\vec {\cal  J}(t) \to  O(t)\vec j + \delta \vec  j(t)$ and one of our targets will be to extract the {\em effective} conductivity response matrix $\mbox{\boldmath{$\sigma$}}$, such that $\delta \vec j(t) = \mbox{\boldmath{$\sigma$}} \cdot  \vec \epsilon(t)$.

From (\ref{perturbed_E}) we expect a perturbation of the bulk gauge field, which we also write now in cartesian components, $\vec a(t,\rho) \to \vec a(t,\rho) + \delta \vec a(t,\rho)$.
The idea is now to refine the ansatz, taking into account  the new  driving  data (\ref{perturbed_E}) and (\ref{perturbed_E2}).
 First of all, let us reconsider the boundary conditions of the unperturbed gauge field, $\vec a(t,\rho)$ (\ref{vec_pot}),  now in cartesian coordinates.  The near boundary expansion will be
\begin{eqnarray}
\vec a(t,\rho) &=&
-\int^t\vec {\cal E}(t) dt \, + \frac{\vec{\cal J}}{\rho} + ... \nonumber\\
&=& \,-{1\over \Omega}\, O(t)\,
\mbox{\boldmath{$\varepsilon$}}\,\vec E\, + \frac{\vec{ \cal J}}{\rho} + ... \label{boundcona}
\end{eqnarray}
where  $\mbox{\boldmath{$\varepsilon$}}$ stands for  the  $2\times 2$ antisymmetric matrix
\begin{equation}
\mbox{\boldmath{$\varepsilon$}}\,=\,
\left( \begin{array}{cc}0 & 1 \\ -1& 0\end{array} \right)\,\,.
\end{equation}
The leading term matches the vector potential of  the rotating electric field through   $\vec {\cal E}(t)\,=\,-\partial_t\, \vec a(t,\rho=\infty)$.
This easily follows from the relation   $\partial_t\,O\,\mbox{\boldmath{$\varepsilon$}}=-\Omega\,O$.
The boundary condition (\ref{boundcona}) suggests   the following ansatz for the bulk solution
\begin{equation}
\vec a\,(t,\rho)\,=\,O(t)\,\vec c\,(\rho)\,\,.
\end{equation}
which is equivalent to (\ref{complexified_potential}) and (\ref{bdec}) in cartesian basis.
Near the boundary
\begin{equation}
\vec c(\rho)\approx \vec c_0\,+\,{\vec c_1\over \rho}\,+\,...\,\, ~
\end{equation}
with\footnote{This replaces (\ref{asymp_c}) with $j = c_{1x}  + i c_{1y}$}
\begin{equation}
\vec c_0\,=\,-{1\over \Omega}\mbox{\boldmath{$\varepsilon$}}\,\vec E\, ~~~,~~~
\vec{\cal J}(t)\,=\,O(t)\,\vec c_1\,\,.
\label{cejota}
\end{equation}
Considering the perturbations,  now the bulk gauge potential $\vec a+\delta \vec a$   has to match the full electric field (\ref{perturbed_E}) at the boundary, $-\partial_t (\vec a +\delta a) = \vec{\cal E}(t)$
\begin{equation}
\vec a(t,\rho=\infty)\,+\delta\vec a(t,\rho=\infty)
\,=\,-{1\over \Omega}\, O(t)\,
\mbox{\boldmath{$\varepsilon$}}\,\vec E\,-\,{i\over \omega}\,
\vec\epsilon\,\,e^{-i\omega\,t}\,\,. \label{pertura}
\end{equation}
Induced by this new boundary conditions, the ansatz will assume that the static field, $c(\rho)$, develops a time dependent perturbation
\begin{equation}
\vec a\,(t,\rho)\,+\delta\vec a\,(t,\rho)
=\,O(t)\,\Big(\vec c\,(\rho)+\delta \vec c\,(t,\rho)\Big)\,\,. \label{defdeltac}
\end{equation}
When the flavour branes are massive, these fluctuations couple to a perturbation of the brane embedding functions $w(\rho) \to w(\rho) + \delta w(t,\rho)$.  In this case we will be dealing with a 3 component vector of fluctuations
$\delta \vec \xi(t,\rho) = (\delta c_x,\delta c_y,\delta \omega)$.
The perturbed  equations of motion, linear in  $\delta \vec \xi$, will assume the following form,
in terms of the $(\tau, \rho_*)$ coordinates  (see Eq. (\ref{tortoise_coord}))
\begin{equation}
\big(\partial^2_{\tau}\,-\,\partial_{\rho_*}^2\,+\,{\bf A}(\rho)\,\partial_{\tau}\,+\,{\bf B}(\rho)\,\partial_{\rho_*}\,+\,
{\bf C}(\rho)\,\big)\,\delta\vec \xi\,=\,0\,\, . \label{pertequ}
\end{equation}
Here, ${\bf A}$,  ${\bf B}$ and  ${\bf C}$ are $3\times 3$ matrices depending on the radial coordinate $\rho(\rho^*)$ and, parametrically, on the rotating frequency $\Omega$.
At the pseudo-horizon $\rho=\rho_c\,  (\rho^* \to -\infty)$ one finds
\begin{equation}
{\bf A}(\rho=\rho_c)\,=\,-{\bf B}(\rho=\rho_c)\,\equiv\,{\bf A}_c\,
\qquad\qquad
{\bf C}(\rho=\rho_c)\,=\,0\,\,.
\end{equation}
Thus, in this limit, $\rho^*\to -\infty$, the fluctuation equations become
\begin{equation}
\big(\partial^2_{\tau}\,-\,\partial_{\rho_*}^2\big)\delta\vec \xi\,+\,{\bf A}_c\,(\partial_{\tau}\,-\,\partial_{\rho_*})\delta\vec \xi\,=\,0\,\,,
\end{equation}
whose general solution takes  the form
\begin{equation}
\delta\vec \xi\,=\,\vec f (\tau+\rho_*)\,+\,e^{-{\bf A}_c\,\rho_*}\,\vec g (\tau-\rho_*)\,\,,
\end{equation}
with $\vec f$  and $\vec g$ arbitrary vector functions.  Imposing that  $\vec g=0$ selects the ingoing wave boundary condition at the pseudohorizon.

Let us now work out the boundary conditions for the fluctuations. First, we want to keep the mass of the flavor brane fixed. After inserting (\ref{defdeltac}) into (\ref{pertura}) and multiplying by ${\cal O}^{-1}$,  the boundary conditions for $\delta \vec \xi(t,\infty)$ turn out to be
\begin{equation}
\delta\vec c\,(t,\rho=\infty)\,=\,
-{i\over \omega}\,O(-t)\,e^{-i\omega\,t}\,\vec \epsilon\,\, ~~~, ~~~\delta \omega(t,\rho=\infty) = 0\, .
\label{UV_bc_deltac}
\end{equation}
Following \cite{Hashimoto:2016ize}, we expand the matrix ${\cal O}(t)$, defined in (\ref{rotating_electric_field}) as
\begin{equation}
 O(t) = {\bf M}_+ e^{i\Omega t} + {\bf M}_-  e^{-i\Omega t}\,\,, \label{odete}
\end{equation}
where ${\bf M}_{\pm} $ are orthogonal projectors given by
\begin{equation}
{\bf M}_{\pm} = \frac{1}{2} \left( \begin{array}{cc} 1 & \pm i \\ \mp i & 1 \end{array} \right) \,\, .
\label{M_pm_def}
\end{equation}
Finally, defininig $
\omega_{\pm}\,=\,\omega\,\pm\,\Omega\,\,,
$
the boundary  UV condition  (\ref{UV_bc_deltac}) of $\delta\vec c$ can be written as
\begin{equation}
\delta\vec c (t,\rho=\infty)\,=\,-{i\over \omega}\,\Big({\bf M}_+ e^{-i\omega_+ t} \,+\,{\bf M}_-e^{-i\omega_- t} \Big)\,
\vec \epsilon\,\, .
\label{delta_c_bd}
\end{equation}
It is natural to assume that from these boundary condutions the  bulk fluctuations  $\delta\vec c\,(t,\rho)$ will oscillate with frequencies
\begin{equation}
\delta\vec c\,(t,\rho)\,=\,\vec\beta_+(\rho)\,e^{-i\omega_+ t}\,+\,\vec\beta_-(\rho)\,e^{-i\omega_- t}\,\,.
\label{timedepbeta}
\end{equation}
As a consequence of the coupling in the linearized  equations, we will assume likewise that the embedding fluctuations will resonate with the same frecuencies
\begin{equation}
\delta w(t,\rho) = \gamma_+(\rho) e^{-i\omega_+ t} + \gamma_- (\rho)e^{-i\omega_- t} \, .
\end{equation}
In summary, setting $\delta \vec \xi(t,\rho) = \vec \xi_\pm(\rho) e^{-i\omega_\pm t}$,  the 3 component vectors $\vec\xi_{\pm}(\rho)= (\beta_{\pm,x} ,\beta_{\pm,y} ,\gamma_\pm )$ will satisfy the linearized ordinary differential system
\begin{equation}
\left[\frac{d^2}{d\rho_*^2} - {\bf B}(\rho) \frac{d}{d\rho^*} + \omega_\pm^2 + i\omega_\pm {\bf A}(\rho) - {\bf C}(\rho) \right] \vec\xi_{\pm} = 0\, .
\label{difeqsys}
\end{equation}
From (\ref{UV_bc_deltac}), the brane fluctuations have  vanishing boundary source $
\gamma_\pm(\infty) = 0 \, .
$
Inserting  the  boundary expansions for the fields in \eqref{timedepbeta}
\begin{equation}
\delta\vec c(t,\rho)\approx \delta\vec c^{(0)}(t)\,+\,{\delta\vec  c^{(1)}(t)\over \rho}\,+\,...
~~~~~, ~~~~~ \vec \beta_\pm (\rho) = \vec \beta^{(0)} + \frac{\vec\beta^{(1)}}{\rho} + ...
\end{equation}
and comparing we get
\begin{equation}
\delta \vec c^{(0)}(t)\,=\,\vec\beta^{(0)}_+\,e^{-i\omega_+ t}\,+\,\vec\beta^{(0)}_-\,e^{-i\omega_- t}\, ~~~\,,
\qquad
\delta \vec c^{(1)}(t)\,=\,\vec\beta^{(1)}_+\,e^{-i\omega_+ t}\,+\,\vec\beta^{(1)}_-\,e^{-i\omega_- t}\,\,.
\end{equation}
Comparing the first of these equations with (\ref{delta_c_bd}) we conclude that
\begin{equation}
\vec\beta^{(0)}_{\pm}\,=\,-{i\over \omega}\,{\bf M}_{\pm}\,\vec \epsilon\,\,.
\label{beta_0_M_epsilon}
\end{equation}

The subleading vectors $\vec\beta^{(1)}_{\pm}$  determine the variation of the current $\delta \vec {\cal J}(t) = O(t)\,\delta \vec c^{(1)}(t) $.
Using (\ref{cejota}), (\ref{odete}) and (\ref{delta_c_bd})
\begin{eqnarray}
\delta \vec{\cal J}(t)\,= \,e^{-i\omega\,t}\,
\Big({\bf M}_+\,\vec\beta^{(1)}_+\,+\,{\bf M}_-\,\vec\beta^{(1)}_-\,+\,
{\bf M}_+\,\vec\beta^{(1)}_-\,e^{2\,i\Omega\,t}\,+\,{\bf M}_-\,\vec\beta^{(1)}_+\,e^{-2\,i\Omega\,t}
\Big)\,\,.
\label{delta_J_beta1}
 \rule{0mm}{7mm}
\end{eqnarray}
As usual in the AdS/CFT program, given a regular  solution, the vectors $\vec \beta^{(1)}_{\pm}$ and $\vec \beta^{(0)}_{\pm}$ will no more be independent. From the linearity of the equations of motion it follows that this relation is also linear
\begin{equation}
\vec \beta^{(1)}_{\pm}\,=\,{\bold X_{\pm}}\,\vec \beta^{(0)}_{\pm}\, =
\,-{i\over \omega}\,{\bold X_{\pm}}\,{\bold M_{\pm}}\,\vec \epsilon \, ,
\label{X_pm}
\end{equation}
where ${\bold X_{\pm}}$ are $2\times 2$ matrices to be computed and  (\ref{beta_0_M_epsilon}) has been used.
Plugging this into (\ref{delta_J_beta1}) allows to find a relation between the current  $\delta{\cal J}$ and the applied electric field $\vec \epsilon$
which defines $\mbox{\boldmath{$\sigma$}}$, $\mbox{\boldmath{$\sigma$}}_+$ and $\mbox{\boldmath{$\sigma$}}_-$ as the $2\times 2$ conductivity matrices of the three current modes, one  with frequency $\omega$ and two  with $\omega\pm 2\Omega$ (the heterodyning mixing modes \cite{OcaBuccian})
\begin{equation}
\delta \vec{\cal J} =
 \,\Big[
\mbox{\boldmath{$\sigma$}}(\omega)\,e^{-i\omega\,t}\,+\,\mbox{\boldmath{$\sigma$}}^+(\omega)\,
e^{-i(\omega+2\Omega)\,t}\,+\,\mbox{\boldmath{$\sigma$}}^-(\omega)\,
e^{-i(\omega-2\Omega)\,t}\Big]\,\vec\epsilon\,\,,
\label{delta_J_epsilon}
\end{equation}
where
\begin{eqnarray}
 \mbox{\boldmath{$\sigma$}}(\omega) &=&-\frac{ i}{\omega}   \left( {\bf M}_+ {\bf X}_+ {\bf M}_+   + {\bf M}_-  {\bf X}_- {\bf M}_- \right)\,   \nonumber\\
 \mbox{\boldmath{$\sigma$}}^+(\omega) &=& -\frac{ i}{\omega}  \, {\bf M}_- {\bf X}_- {\bf M}_+ \,\,  \label{conduct} \\
 \mbox{\boldmath{$\sigma$}}^-(\omega) &=&-\frac{ i}{\omega} \,  {\bf M}_+  {\bf X}_+ {\bf M}_- \,\,. \nonumber
\end{eqnarray}

The procedure to compute ${\bold X}_\pm$ starts by solving the differential equation system (\ref{difeqsys}) from the pseudohorizon $\rho_* \to -\infty$
to the AdS boundary at $\rho_*=0$. At the horizon we impose the ingoing boundary condition $\vec \xi_\pm(\rho_*) \simeq \vec c \, e^{-i\omega_\pm \rho_*}$, where $\vec c$ are constant vectors $(1,0,0), (0,1,0)$ and $(0,0,1)$. Due to linearity of the equations  we obtain in this way two complex $3\times 3$ matrices $P_\pm$ and $Q_\pm$ such that $\vec\xi_\pm^{(0)} = P_\pm \vec c_\pm$, and  $\vec\xi_\pm^{(1)} = Q_\pm \vec c_\pm$. In this way we can solve  for $\vec \xi^{(1)}_\pm = Q_\pm P^{-1}_\pm \vec\xi_\pm^{(0)} $. Since we will be setting the source for the embedding fluctuations to zero, i.e. $\xi^{(0)}_{\pm,3} \equiv  \gamma_\pm^{(0)} = 0$ and
we only care about $\xi^{(1)}_{\pm,\, 1,2}$, the searched for matrices ${\bf X}_\pm $ in (\ref{X_pm}) will be given by the $2\times 2$ submatrix of the product $Q_\pm P^{-1}_\pm$
$$
{\bf X}_{\pm\, ij} =  [Q_\pm P^{-1}_\pm]_{ij=1,2} \, .
$$
From them, the conductivities are extracted using (\ref{conduct}).
\begin{figure}[!ht]
\center
\includegraphics[width=0.47\textwidth]{XOmovm1d500Rexxplot.pdf} ~ \includegraphics[width=0.48\textwidth]{XOmovm1d500Imxxplot.pdf} \\
\includegraphics[width=0.47\textwidth]{XOmovm1d500Rexyplot.pdf} ~ \includegraphics[width=0.47\textwidth]{XOmovm1d500Imxyplot.pdf} \\
\caption{AC conductivities for fixed value of $\Omega/m = 1.496$ and different values of $|E|/m^2$. They correspond to the vertically aligned dots in Fig.\,\ref{fig:PhSpMap} with the same color code.    In the insets, the curves at low values of $|E|/m^2\to 0$ deep into the critical wedge, where they develop peak resonances.  The value $\omega/m = \Omega/m = 1.496$ is signalled with a dashed vertical line. }
\label{fig:AC_conductivities}
\end{figure}

\subsection{AC conductivities}\label{sec:ACcond}

The results for the photovoltaic optical Hall conductivity, $\sigma_{xy}$, as well as for the photovoltaic optical  absortion spectrum, $\sigma_{xx}$,  can be seen in Fig.\,\ref{fig:AC_conductivities}.
We have plotted everything in units of the carrier mass $m$.\footnote{Numerically we have performed the integration for 2 different sets of  values of $(E,\Omega,m)$ in order to verify the scaling symmetry of Eq. (\ref{scaling}) }\, The driving laser  has been set to  the first resonant frequency  $\Omega/m=1.496$  for a discrete set of values of  $|E|/m^2$ (see the coloured dots along the vertical dashed line in Fig.\,\ref{fig:PhSpMap}).

The  behaviour in all components of $\mbox{\boldmath{$\sigma$}}$  shows oscillations whose amplitude grows  for decreasing  $|E|/m^2$. All conductivities oscillate around $0$ except for Re$(\sigma_{xx})$ which  does it around a saturation value  $ 1$.

The departure in (2+1) dimensions,  from the analog (3+1) dimensional holographic system is significant (see Fig.\,7 and 8 in  \cite{Hashimoto:2016ize}).
In both cases, in the large frequency limit,  the conductivity  becomes proportional to the identity matrix  $\mbox{\boldmath{$\sigma$}} (\omega\to \infty) = c\, \mathbb{I}$. In the D3/D7 case,  $c$ is a complex linearly growing function of  $\omega$, whereas in our case $c$ is a constant  (equal to 1 in our normalization in this section, see later however).  This asymptotic  behaviour is also attained in the limit of high  $|E|/m^2\gg 1$, as controlled by the massless limit $m\to 0$ which is analytically solvable.
Not only the amplitude of the maxima, but also their movement is different. In the D3/D7 case their positions shift to lower frequency for decreasing external field. Here we see that they tend to approach the critical $\omega=1.496$.

As $|E|/m^2$ becomes smaller, strong oscillations develop, with the most prominent one sharpening around the resonance value for the optical and driving frequencies $\omega = \Omega$ (signalled with a dotted vertical line in all the plots). This feature bears  resemblance with the one obtained in \cite{Oka_2009b} (Fig 1b), although the behaviour of the peak amplitude with the intensity is the opposite.

In the limit $|E|/m^2 \to 0$ these oscillations become very sharp  peaks as shown in the insets in Figs.\ref{fig:AC_conductivities}. As we move down towards $E/m^2\to 0$ the shape and  positions of these peaks change rapidly. This can be appreaciated in the insets, where the curves for $E/m^2 = 0.0175$ and $0.001$ look decorrelated. This last magenta curve shows a subseries of  peaks at frequencies   $\omega=1.496, 3.38, 4.87,...$ which have some overlap with the vector meson Floquet condensate spectrum  $\omega=1.496, 3.53, 5.56,...$ at shifted values.

  The fact that $\mbox{\boldmath{$\sigma$}}$ is dimensionless  in (2+1)  dimensions eases the comparison with results in the literature.  For the Hall component, $\sigma_{xy}(\omega)$ there is a striking similarity with the plot  in Figs.\,8  of \cite{Deghani2015}, where also the resonance peak is visible as well as secondary peaks.
     Importantly, in the limit $E\to 0$ the photoelectric Hall  conductivity does not vanish. This signals spontaneous time reversal symmetry breaking and might be a consequence of the underlying vector meson Floquet condensate state.\footnote{We thank T. Oka for this observation.}

The enhancement of the oscillations  and peak formation observed in Fig.\ref{fig:AC_conductivities} is a universal feature as we approach  the line of critical embeddings.   We have followed the movement of the peaks for a sequence of configurations very closely above the critical line, for background frequencies in the range $\Omega/m \in (1.3, 2.32)$ (see Fig.\,\ref{E_Omega}).
The resulting curves are plotted  in Fig.\,\ref{fig:criticalcond}. A remarkable fact is  the presence of one peak whose position barely changes and is confined to  the interval $\omega/m \in  (1.4965, 1.73) $, i.e. among the vector meson Floquet condensate frequency and the meson mass resonance (see Fig.\,\ref{E_Omega} and eqs. \eqref{floqcondfreqs}\eqref{mesonfreqs}).
Very slightly above the critical line there are two sidewise peaks which  appear within a  short interval in $E/m^2$. They all can be seen also in Fig.\,\ref{fig:criticalcond}. The remarkable fact about these lateral peaks is that  they shift position with the driving in such a way that their mean value coincides with the background frequency $\Omega$, while their splitting is almost constant $\Delta \omega /m \sim 1.7$ . This curious result is very neatly seen on embeddings  to the right of the vector meson Floquet condensate, i.e. in the segment $\Omega/m \in(1.496,2.32)$ and, very much damped, nearby  to its  left. The blue curve at $\Omega/m=1.496$ is the same one that can be observed in the insets of Fig.\,\ref{fig:AC_conductivities}, which has $E/m^2=0.0175$. At this frequency $\Omega=\Omega_c$, there are embedding solutions down to $E=0$ and this gives the magenta curves in those insets at values of $E/m^2 = 0.001$ and even lower.

We also computed the heterodyning conductivities $\mbox{\boldmath{$\sigma$}}^{\pm}(\omega)$.  They bear the same features as for the normal conductivity, and the reader can found further information in Appendix
\ref{app_E}. Roughly stated, the curves for these exotic conductivities share, each of them, half of the peaks visible in $\mbox{\boldmath{$\sigma$}}(\omega)$.

\begin{figure}[!ht]
\center
\includegraphics[width=0.47\textwidth]{XRexxCritplot.pdf} ~~~ \includegraphics[width=0.47\textwidth]{XImxxCritplot.pdf} \\
\includegraphics[width=0.47\textwidth]{XRexyCritplot.pdf} ~ \includegraphics[width=0.47\textwidth]{XImxyCritplot.pdf} \\
\caption{Peaks in the AC conductivities closely above along the critical line. The exact position of these curves in phase space can be seen in Fig.\,\ref{fig:PhSpMap} as coloured circles. The vertical dashed lines signal the vector meson  Floquet condensate and the  meson  mass frequencies respectively.  The central peaks bunch in this region whereas the two lateral peaks move with $\omega$ while keeping their separation almost constant, $\Delta \omega /m \sim 1.7$.
}
\label{fig:criticalcond}
\end{figure}

\begin{figure}[!ht]
\center
 \includegraphics[width=0.5\textwidth]{PhaseSpaceLines.pdf}
\caption{ The dots and circles signal the points where the AC conductivities in Figs.\,\ref{fig:AC_conductivities} and \ref{fig:criticalcond} have been plotted respectively.  Also shown are  the lines with either $\Omega/m$ or  $|E|/m^2$ fixed, used to plot the DC conductivities in Figures .\,\ref{fig:DC_Eovm2} and \ref{fig:DC_Omovm}.  In all cases, the same color code has been used.  }
\label{fig:PhSpMap}
\end{figure}

\begin{figure}[!ht]
\center
\includegraphics[width=0.43\textwidth]{XRexxEDCplot.pdf} ~ \includegraphics[width=0.45\textwidth]{XRexyEDCplot.pdf}
\caption{DC Conductivities as a function of $\Omega/m$ for various values of $|E|/m^2$.    }
\label{fig:DC_Eovm2}
\end{figure}
\begin{figure}[!ht]
\center
\includegraphics[width=0.43\textwidth]{XRexxOmDCplot.pdf} ~ \includegraphics[width=0.45\textwidth]{XRexyOmDCplot.pdf}
\caption{DC Conductivities as a function of $|E|/m^2$ for various values of $\Omega/m$.    }
\label{fig:DC_Omovm}
\end{figure}

\subsection{DC conductivities}
The zero frequency limit  $\omega\to 0$  is formally singular  (c.f.(\ref{UV_bc_deltac})). However the  numerical value of the AC conductivity is well behaved and stable down  to values of $\omega/\Omega$ well below $10^{-3}$. Hence we will use this value  to define our DC conductivity. The authors of   \cite{Hashimoto:2016ize} performed
 a numerical analysis in the time domain  and showed consistent  agreement between the two prescriptions.  From the plots in Figs.\,\ref{fig:AC_conductivities}, it is apparent that the imaginary parts vanish in the limit $\omega\to 0$. Hence
 the DC conductivities are real, as follows from the relation  $\mbox{\boldmath{$\sigma$}}(-\omega)=\mbox{\boldmath{$\sigma$}}(\omega)^*$.

 Figs.\ref{fig:DC_Eovm2} and  \ref{fig:DC_Omovm}  summarize our numerical results.
 In Fig.\,\ref{fig:DC_Eovm2} we exhibit  $\mbox{\boldmath{$\sigma$}}(\omega=0)$ for varying $\Omega/m$  along lines of constant $|E|/m^2$.
For high enough $E/m^2$ the limit $\Omega\to 0$ stays in the conducting phase and the direct conductivity $\sigma_{xx}$ has the correct static limit obtained in \cite{Karch:2007} where it becomes a constant that only depends on $m$, and saturates to a maximum value equal to one  (see section \ref{sec:massless} below for the correct normalization).

In Fig.\,\ref{fig:DC_Omovm}  we plot $\mbox{\boldmath{$\sigma$}}(\omega=0)$, varying $|E|/m^2$  along lines of constant $\Omega/m$.
 Our results should be compared with the condensed matter calculation in \cite{Oka_2009} (see the discussion after Fig.2). For the Hall conductivity we see qualitative agreement in that, in absolute value,  $|\sigma_{xy}|$ first increases with $|E|/m^2$ and then decreases. For high $\Omega/m (\sim 3.53)$ we find values where this conductivity has zeros, this being  a remarkable feature.   For $\sigma_{xx}$ we find   a monotonic growth that  reaches a  limiting value for high $|E|/m^2$. In contrast, authors of \cite{Oka_2009} (see Fig.2b) obtained an intermediate  valley  which they proposed  to be a reflection of the  gap opening at the Dirac points. If so, our model is too crude to capture this feature.

The comparison with the (3+1) case  is complicated because the authors of  \cite{Hashimoto:2016ize} only considered the massless case, which in our setup turns out to be analytic and trivial.   Indeed, taking  $(|E|/m^2,\Omega/m)\to \infty$
the Hall conductivity vanishes whereas the DC conductivity asymptotes to a constant value.  One important difference is the sign of $\sigma_{xy}$ which in our  case comes out negative, hence opposite to the one in the (3+1) dimensional setup. This is so, despite the fact that we have been careful in keeping  all the conventions the same, and the sign in $\sigma_{xx}$ is positive as well.\footnote{yet, notice a  change of sign for very low values of $|E|/m^2$ approaching the Floquet condensates. It would be interesting to know if the configuration is unstable in this region and would decay to Minkowski embeddings.  }

The region in phase space where a significant photoelectric Hall conductivity  is obtained is rather small. It happens for values of the parameters in the region of  $\Omega/m \leq 2$ and also small  $|E|/m^2  \leq 0.6$. This corresponds roughly to  the close vicinity of the first dome to the left of  the critical  value $\Omega_c = 1.496$ in Fig.\,\ref{fig:PhSpMap}.

\subsection{Massless limit}\label{sec:massless}

When the unperturbed embedding of the probe brane in massless, {\em i.e.},\ when $w=0$, the equations for $\delta w$ and $\delta \vec c$ decouple and we can take safely
$\delta w=0$. The unperturbed configuration in this case is the analytic solution discussed in section \ref{massless_embd}. The equations of motion for $\delta c_x$ and $\delta c_y$ greatly simplify for this solution. Amazingly, the general solution of these coupled  equations can be obtained analytically, as shown in detail in appendix \ref{app_C}. Using these results one can extract the exact value of the conductivity matrices, which turn out to be
\begin{equation}
\mbox{\boldmath{$\sigma$}}(\omega)\,= c\,{\mathbb I}\,\,,
\qquad\qquad
\mbox{\boldmath{$\sigma$}}^+(\omega)\,=\,\mbox{\boldmath{$\sigma$}}^-(\omega)\,=\,0\,\,.
\label{massless_cond}
\end{equation}
with $c=1$.
As mentioned before, this analytical result controls the limits in the previous sections where  $|E|/m^2$ and $\Omega/m$ both become very large.

Of course, the coefficient $c$  in Eq. \eqref{massless_cond} is not equal to 1.  Using \eqref{dictionaryE} it is easy to derive the correctly normalised physical conductivity, relating the the physical electric field and the physical current, as
$$
\mbox{\boldmath{$\sigma$}}(\omega)\,=\frac{\sqrt{2}N_f N_c}{\pi\sqrt{\lambda}}~{\mathbb I} \, .
$$
Up to a convention factor of $\sqrt{2}$ this result matches the one obtained in \cite{Karch:2007}, see eq.\,(5.8),  were the observation is made that this conductivity is nothing else than $1/g^2$ in terms of the effective defect field theory gauge coupling on the D5-brane. This normalisation should affect all  the conductivities obtained previously in this paper.

\section{Summary and outlook}
\label{conclu}

In this paper we employed holographic techniques to analyze the response of a (2+1)-dimensional  gauge theory to an external rotating electric field. The system can be in two phases (insulator and conductor) depending on the type of embedding of the probe brane.  In agreement with the (3+1)-dimensional results of \cite{Hashimoto:2016ize,Kinoshita:2017uch} we have found that for certain frequencies the driving field resonates with the vector meson excitations of the gauge theory and gives rise to zero field Floquet resonances. The lobbed structure shown in Fig.\,\ref{E_Omega} may be a universal feature.

Transport properties are an important benchmark, and they exhibit substantial differences in the two dimensionalities.  In  (2+1) dimensions, the massless case turns out to be trivial. In the massive situation, however, the conductivity tensor exhibits a rich structure. We have investigated the AC and DC conductivities, paying special attention to the first resonant frequency $\Omega = \Omega_c = 1.4965$ which is close to the meson mass frequency. In this situation the system creates efficiently charged carriers and holes. In experimental setups, the asymmetry in the electron-hole pairs for different chiralities gives rise to the so called ``optically induced valley polarization". This might be the closest physical realization to our setup at the resonant point.
Weather the double peak structure observed in Fig: \ref{fig:criticalcond}  may have some meaning in this direction is something that deserves further scrutiny. Otherwise, it would probably mean this model is too crude to capture the chiral asymmetry in the splitting of the Dirac point.

Many distinguishing features are present for configurations approaching the critical line separating the conducting from the isolating phase. The strong enhancement of the oscillations deforming into sharp peaks  is one such. These peaks seem to point to bound states that are shifted from the meson masses. At this point, a very relevant question is that of stability of the embeddings. The exact point where, say, Black Hole embeddings decay into Minkowski embeddings and viceversa is a problem that can be addressed in equilbrium thermodynamics, and has a practical answer in terms of the equal area law. Here however, the similar tools are not available from a would be periodic thermodynamics \cite{Kohn14}, and this certainly an important aspect that will need further study in the future.

Let us now discuss possible related lines of  research for the future. An interesting avenue would be to complete the phase diagram of the D3-D5 model by studying the system at (real) non-zero temperature and/or finite chemical potential. Moreover, we have treated the system in the quenched approximation. Dealing with the backreaction of the D5-branes is, in general, a tough problem. However, we could adopt some approximation, as it is done in the smearing approach  (see \cite{Nunez:2010sf} for a review). Interestingly, the smeared backreacted D3-D5 background has been obtained in \cite{Conde:2016hbg}
(see also \cite{Penin:2017lqt,Jokela:2019tsb,Gran:2019djz}) and one could try to generalize these results to the case in which an external rotating electric field is acting on the flavor brane.

The comparison of our results with the D3-D7 model points towards some universality in the phase space. To strengthen this point,  a natural strategy is to extend the study to other brane setups. Most prominently, the (2+1) dimensional ABJM model is an example of superconformal QFT with a well stablished gravity dual \cite{Aharony:2008ug}.  The flavor branes in this case are D6-branes \cite{Hohenegger:2009as,Gaiotto:2009tk} (see \cite{Jokela:2012dw} for an analysis of the thermodynamics of the flavor branes in ABJM). The ABJM supergravity background has a non-trivial topological structure wich allows to switch on fluxes of the gauge fields and realize the quantum Hall effect \cite{Bea:2014yda}. This suggests that it could be the right place to find non-trivial topological effects induced by an external driving in an holographic setup like the ones we are missing here.

Another direction worth pushing is  to consider other types of drivings. The brane picture of gauge theories allow for several possibilities. For example, instead of an electric field we could consider a magnetic field, or let the brane oscillate periodically. Actually, this last scenario has been considered  in \cite{Hoyos:2011us} in connection with the realization of the chiral magnetic effect in holography. It would be good to revisit this setup and search for possible stable non-equilibrium phases and condensates. The natural expectation is to observe resonances with the other type of mesons of the theory, like scalar mesons.

\vspace{1cm}
\noindent

\section*{Acknowledgements}
We are  indebted to Keiju Murata for many patient clarifications about his work. Also we would like to thank  Yago Bea and Javier Tarr\'io for discussions and an early  collaboration in the initial stages of this project. Special thanks also go to Takaaki Ishii inspiring conversations, and  to Takashi Oka for  insightful remarks on the physical interpretation of our results.

This work was supported by grants FPA2014-52218-P  and FPA2017-84436-P from Ministerio de Economia y Competitividad, by  Xunta de Galicia ED431C 2017/07, by FEDER and by Grant Mar\'\iota a de Maeztu Unit of Excellence MDM-2016-0692.  The work of A.G. has been supported by the Xunta de Galicia action
"Axudas de apoio \'a etapa predoutoral".

redpajama_set_name: RedPajamaArXiv


text:
Pope Francis: the Torah is a manifestation of God's love for man
Pope Francis receives a new annotated edition of the Torah at the Vatican Feb. 23
Vatican City, Feb 23, 2017 / 07:33 am (CNA/EWTN News).- On Thursday Pope Francis welcomed longtime friend and Rabbi Abraham Skorka to the Vatican for the presentation of a new version of the Torah, which he said is a sign of the love God shows to man in both words and gestures.
The Torah "manifests the paternal and visceral love of God, a love shown in words and concrete gestures, a love that becomes covenant," the Pope said during the Feb. 23 audience, adding that the word covenant in itself "is resonant with associations that bring us together."
He noted how St. John Paul II in his speech for the 25th anniversary of the Second Vatican Council document "Nostra Aetate," which marked a milestone in improving Catholic-Jewish relations, called the Torah "the living teaching of the living God." "God is the greatest and most faithful covenantal partner," he said, noting that not only did God call Abraham to form a people that would become "a blessing for all peoples of the earth," but he still desires world in which men and women "are bound to him and as a result live in harmony among themselves and with creation."
At a time when things people say often "lead to tragic division and rivalry, these divine words of covenant open before all of us paths of goodness to walk together," he said. The publication of the new edition of the Torah, he said, "is itself the fruit of a 'covenant' between persons of different nationalities, ages and religious confessions, who joined in this common effort."
Pope Francis spoke to Skorka — a longtime friend from his time in Buenos Aires — and the delegation of Jewish leaders that came with him to present a new, annotated edition of the Torah complete with colorful illustrations. The Torah refers to Jewish Written Law and traditionally consists of the first five books of the Old Testament.
Calling Skorka both a "brother and friend," Francis voiced his gratitude to the delegation for the "thoughtful gesture" of coming to the Vatican to present the Torah, which is "the Lord's gift, his revelation, his word." He noted that the "fraternal and institutional dialogue" between Christians and Jews is now "well-established and effective," and continues to be strengthened and carried forward through various encounters and collaborations.
Turning to the text of the new Torah itself, he said the editor's note inside emphasizes the "dialogical approach" that Catholics and Jews have regarding their relations, and communicates "a cultural vision of openness, mutual respect and peace that accords with the spiritual message of the Torah."
Those who designed the new edition, he said, paid special attention to both the important literary aspects of the text, as well as the colorful illustrations that now accompany it, adding "further value" to what was already there. "Every edition of sacred Scripture, however, possesses a spiritual value that infinitely surpasses its material value," he said, and prayed that God would bless all those who contributed to the new edition, as well as those present for the encounter.
The presentation of the Torah was the latest sign of collaboration between Jews and Catholics, falling just days after a new joint-exhibit of the Menorah was presented by the Vatican Museums and the Jewish Museum of Rome. The exhibit, titled "Menorah: Worship, History, Legend," will be shown simultaneously at both the Jewish Museum as well as the Braccio di Carlo Magno Museum in the Vatican, located under the left colonnade in St. Peter's Square. It will run May 15-July 23 and marks the first time such a joint-exhibit has been done. Pieces featured will include roughly 130 artifacts, including Menorah from different periods and depictions of them in paintings, sarcophagi, sculptures and medieval and Renaissance drawings and manuscripts.
St. Joseph, the ultimate man's man by Robert Brennan May 25, 2021
Pope Francis: Ask God for the strength to love your enemies by Courtney Mares February 23, 2020
Perseverance is a reflection of God's love, Pope Francis says by Hannah Brockhaus November 13, 2022

redpajama_set_name: RedPajamaCommonCrawl


Posted: 2016-06-28 07:15:00	by Metro Networks, Inc.
(Undated) -- WalletHub's list of cities that are the best places for Fourth of July celebrations ranks Seattle in the top spot. The consumer website compared the 100 largest U.S. cities, based on how well they balance holiday fun and what it costs. Portland ranks fifth overall on the WalletHub list of Best Cities for 4th of July Celebrations. The Rose City ranks only 63th in the category of "Entertainment & Food," but ranks in the top ten in the categories of "Attractions & Activities" and "Fourth of July Weather Forecast."


A pantry in the kitchen can be a huge benefit to any cook and organization is critical to being able to find what you are looking for. Even if you have a closet in the kitchen that isn't meant to initially be a pantry, you can take a few items and change it into a nice storage area for all your kitchen products. And the good news is you don't even need a closet…just a little extra space.
So what can be stored in a kitchen pantry?
While you can store literally anything in a pantry, food, groceries and small kitchen appliances are what is typically stored there. There are many options available to you that should fit whatever your needs may be….
If you aren't lucky enough to have a deep pantry, there are some very quick yet effective options. A metal shelving system such as ones made by Seville can immediately give you some storage space for food and dry storage, as well as small lesser used appliances. While not much in the way of visual aesthetics, this type of shelf is easy to assemble and can be utilitarian in the way in can hold virtually anything.
Make sure to use the space on the pantry door!
Another way to utilize the space in a pantry is to use something like FreedomRail's over the door pantry kit. This system basically installs a thin grid in the middle of the door in which you can attack shelves and other things to. You can even buy attachments for a garbage can shelf!
For spices and other things, you could add a lazy susan to a pantry cabinet as well.
Build a pantry without a closet!
One of the bigger problems with pantry closets is that they tend to get unorganized pretty quickly. For small items such as spices, a lazy susan can not only save space but will give things a definite place.
Another option if you don't have a closet pantry is to actually build one or have one build yourself. The good news is that all you need is a large piece of furniture like an armoire and then simply gut the inside. There are pantry kits that you can then buy to add shelving to it. This can not only give you extra space but can also be a centerpiece in the kitchen or dining area!


The words "long awaited" are often applied to long winded marketing-speak-riddled descriptions of various follow-ups to pieces of writing, films or other forms of serialisable entertainment. The term usually implies that somewhere vast hoards of ravenous fans are chomping at the bit, queuing for months on end waiting for some product or other to be ready for their growling appetites. So to call "Frank Billions, Chapter Two" long-awaited, while technically accurate, might be construed as misleading, were it to be revealed that the number of individuals who were actually waiting for it is quite small. Very small actually, especially when you take out the various, somewhat lofty, "future" projections of the series' fan-base. Still, a little bit of marketing hype never hurt anyone, and I'm sure the Internet Police have better things to do than monitor the odd bit of connotative inaccuracy (or overly self-indulgent prologue for that matter). So I'll do as I please and wear the consequences like a cheap summer hat.
For those who missed the opening chapter, it can be found here.
The sign out the front of the Asteroid Deli and Cooperage boasted the best damn sandwiches this side of Brendon's Comet. The sign was true enough, although it was more of a slight on the other sandwich joints than the quality of Asteroids'.
Still, you didn't show your face in this system unless you had the gut to back it up. Like I said, I hadn't been here in years, but as I walked through the green saloon doors I realised that old Jordan Asteroid never was one for change. The paintwork, the people, even parts of the salad bar looked like they hadn't moved since the last time I was in.
I looked at the old man behind the counter. Time had not been kind to Joran Asteroid. Not that time is ever particularly pleasant to the best of us, but in Jordan Asteroid's case, time must have held some longstanding grudge. The last time I saw him he was missing an ear and three fingers. He was now down an eye and a few teeth. His missing eye was covered with a black patch and his bad hand had been replaced by a laser hook. It gave him the appearance of a space-pirate, or it would have done, had he not been wearing ugg boots and a fez. His smile seemed genial, if a little toothless. Evidently the "howdy" had been genuine and the "stranger", merely jovially ironic.
Time had not been kind to Jordan Asteroid.
"It's been a while" he said, nodding at either me or the yoghurt stand, it was hard to tell with one of his eyes missing and the other lazier than an East Galaxy Council worker. It didn't matter. I got his gist, and I hadn't come for yoghurt.
Like I said, some things never change. You don't get something for nothing in this neck of the galaxy. I tossed a coin down on the counter. "Give me a turkey on wholegrain." The old man's lazy eye swivelled.
"He said he needed to go into hiding for a while. I saw him talking to Mad-Duck." Mad-Duck Mallard was the go-to man for every dirty two-bit crook in these parts. He could get you anything you wanted, for the right price.
I was about to ask Jordan where I could find Mad-Duck when a kid in a suit about three sizes too big muscles past and orders two sandwiches and a barrel to go. He must have been twenty-five years old, but swanned in the place like he was the King of Jupiter.
"Wait your turn, kid." I said and turned back to Jordan. The kid hitched his jacket up to reveal a quantum six-shooter in his shoulder holster.
"How about you mind your own business, Old-Timer," he smirked. At one time I would have knocked some sense into a guy like that. I wasn't afraid of his shooter either. I could have knocked him out cold before his finger touched the trigger. But these days there's no point. A punk kid like that never learns. It aint worth a stinging fist, not even on a smirk like the one he was giving me.
"Maybe Starmatt can send him one of his freight runs." I glanced down at the pile of newspapers on the counter. Captain Starmatt's latest deep space incident had made front page news for the past week. One of his biggest freight ships had struck a rogue comet halfway to the Archenar system.
"Don't believe everything you read in the papers." Asteroid looked as though he was attempting a conspiratorial wink. He gave up after a few minutes and resorted instead to a tap on the side of his nose.
Everyone who's anyone had always known that Captain Starmatt was as crooked as a spacewalk, but his freight business was supposed to be legit, a front for his smuggling rackets, his counterfeit money rackets and his overpriced tennis racket rackets. If Asteroid smelled a rat then I wanted to have a whiff too.
"Are you telling me," I began, but before I could finish there was a small tremor and Jordan's wife Elfour beamed into the room.
Elfour Asteroid was the kind of woman that gave formidable women a bad name. Her old man was a general in the Imperial Galactic Navy and her mother was a stage ventriloquist. Elfour knew what she wanted and nothing short of a mutiny or dummy malfunction would get in her way. She had a voice like a fog horn and a glare that could take the blush out of a red dwarf from about four light years away. Spiral-Arm Magazine had voted her "Top Woman Not To Mess With" every year for the last twenty-four years, ever since she objected to an an unfavourable article about her father's post-war antics. She retaliated with a particularly pithy letter to the editor, a fairly venomous PR campaign against the publication and an all-out nuclear assault on the moon which housed the printing presses and complaints department of the magazine. Her "three pronged strategy", as she called it, did the trick and the magazine printed a retraction the following issue.
"Hello Frank, you're not here on business, I hope." Elfour had a chip on shoulder ever since a case Marty and I were working on ended in a shoot-out at the Deli. The damage had been paid for but Elfour didn't like us sticking our noses into her joint unless it was for some chow.
"Tricks are just fine. We finally made some money from the cooperage this week." Barrel sales had been in decline for years, and the Asteroid Delicatessen and Cooperage had felt the full brunt of the downturn.
Jordan attempted another conspiratorial nose-tap, this time with his other hand, forgetting about the laser hook. The smell of singed nasal hair and cauterised flesh filled the room, but it wasn't the only thing I smelled. Old Jordan Asteroid was never one for insurance, and he certainly didn't need to send thirty thousand barrels to the Archenar system. This was a scam, but it wasn't the only one in this mixed up dirty neck of the galaxy. And unless Martin was mixed up in insurance fraud, I couldn't give three hoots about what went on between Asteroid and Starmatt.
Thanks for reading. I promise Chapter Three will be here in less than two years.
F*ckin' awesome issues here. I am very satisfied to look your post. Thank you a lot and i am looking forward to contact you. Will you please drop me a mail?


The 11 day KeaRider Down Under motorcycle tour through Australia will take you through the states of Victoria and New South Wales. You will get to know the vibrant cities Melbourne and Sydney and discover special landscapes such as the Great Ocean Road on the south coast of Australia, the Murray River and the wine-growing areas of Routherglen. On endless highways you can speed along. Don't forget in Australia there is only left-handed traffic! Other highlights on this road trip are an optional harbor cruise in Sydney with dinner and a ride with the steamboat on the Murray River. Finally, full of new impressions, you arrive back in Melbourne.
After an individual journey, your motorcycle tour begins in the beautiful metropolis of Melbourne in the south of Australia, where you head off to the direction of Great Ocean Road and see among other things the Twelve Apostles.
On the Great Ocean Road you will head west and enjoy the unique nature of this coastal road. Maybe even see whales in the wild!
Today we head inland and visit the gold mining town Souvereign Hill which end of the 19th century is a reminiscent of the gold rush. Accommodation is in Ballarat.
Relaxed cruising on the highway towards the Murray River is on the agenda - Australia is big and so are the distances. In the late afternoon, you will reach Echuca.
This morning, you can take a boat trip on Australia's longest river, the Murray River before you cross over the Lake Mulwala to Routherglen, where Australia's delicious wines can be found.
On this day, Canberra is on the program, the modern capital, which was built halfway between Melbourne and Sydney.
From Canberra you can cruise to the freeway to Sydney, Australia's largest and most beautiful city. On a harbor cruise, you can see the famous opera and the Sydney Harbour Bridge up close.
This day is at your free disposal. We will gladly help in buying tickets for the Opera or proposals for other activities.
Along the coast, you will return via the Princess Highway 1 on the long journey back to Melbourne. Visit the Bendarah State forest and spend the night at Bateman's Bay.
Enjoy the picturesque rough coastal scenery on the way to Tathra before you reach the next destination of Lakes Entrance in the afternoon.
Your last stage leads you today on the Princess Highway 1 back to Melbourne and starting your individual journey back home.


My brain woke up before the rest of me. I was so tired it took me a few minutes to reach for my phone to see if school was cancelled this morning. Nope, just a two-hour delay.
As soon as Sarah got up she went outside.
She couldn't wait to go sledding.
It was 7 o'clock in the morning.
Our house sits on a hill so it's been fun for the children all these years. They can just sled from the porch steps all the way down……beginning with the oldest Jacob and ending with the youngest, Sarah.
I watched Sarah through the window. And then I went out on the cold wet porch in my socks.
I was charmed to see her attempt something her big sister Grace would have also done at age 8; she decided to scoop up kitty Walter and take him for a sled ride. After all, this is Walter's first winter. He has a lot to learn.
He learned it really fast.
Back inside, Caleb made himself bacon, cheese, and eggs. A nice proper breakfast, served on a white fiestaware plate, with a periwinkle mug of ice-cold juice on the side.
They can't cook at the same time without squabbling (G, right?) so after his big brother was done, Seth made himself bacon and eggs, too.
PS, the first photo is of my garden gnome. He can't see right now.