text:
\section{Introduction and Summary}

\subsection{Motivation}

The Sachdev--Ye--Kitaev (SYK) model \cite{SachdevYe,kitaev,MS,KitaevSuhFirstPaper} is a strongly interacting but yet solvable model in the large $N$ limit. At low energies, it displays an approximate conformal symmetry. In this region, the model has many features in common with nearly $AdS_2$ gravity, or Jackiw-Teitelboim (JT) gravity \cite{Jackiw,Teitelboim, AP}, coupled to matter fields. This is a simple two dimensional theory of gravity which describes some aspects of nearly extremal black holes in various dimensions.

An interesting variant is to consider a pair of identical SYK models coupled through a simple bilinear interaction \cite{MQ}, see also
\cite{Garcia-Garcia:2019poj,Kim:2019upg,GaoJafferisRecent}.
The ground state of this model has a gap, but its excitation spectrum also displays an approximate conformal symmetry. Furthermore, this ground state is close to the thermofield double state of two decoupled models. For reasons we explain below,  we call the ground state of this coupled model  ``the SYK wormhole''.

A conceptually similar state also arises when one considers two nearly extremal black holes that are relatively close, so that they are coupled. In this case, a traversable wormhole can connect the near extremal throats \cite{MMP}.
This can be effectively modeled by a nearly $AdS_2$ gravity theory
where we have  direct  interactions between the values of the bulk fields near the two boundaries \cite{GaoJafferisWall,MQ}. In other words, thinking of the Penrose diagram of $AdS_2$ as a strip, we put boundary conditions for the bulk fields that connect the two boundaries.
The two boundaries are causally connected through the bulk, so that this spacetime describes a wormhole.
 This wormhole is the lowest energy configuration of the system and it also displays the approximate $SL(2,R)$ isometries of nearly $AdS_2$.

   Given that this is a remarkable state, we are interested in knowing whether it is easy to get to it. In other words, if we start out from a general excited state of the coupled model, can we easily get to the ground state by cooling the system down? Or will the system get stuck in some other state?  At first sight the answer seems straightforward, if it is the ground state, the system will surely find it if it can shed its excess energy to the bath. On the other hand, from the gravity perspective, the process involves a topology change. Such topology change might happen via a tunneling solution, but it would be exponentially suppressed in $N$ (or the entropy of each separate black hole).

  \subsection{Wormhole formation in SYK}

  With this motivation in mind, we study this problem for the
  two coupled SYK models. We start with a relatively high temperature state of the coupled model which looks like two thermal density matrices, one for each SYK factor. Then we couple the system to a bath and study the evolution in real time by solving the large $N$ Schwinger-Dyson equations. We find that the system indeed finds the ``SYK wormhole" ground state in a time that is independent of $N$. In particular, there is no exponential suppression.
  Notice that the ability to efficiently find this ground state also makes it possible to prepare the thermofield double (TFD) state of the decoupled model, by simply switching off the interaction between the two sides \cite{MQ}, after we have found the ground state.

The approach we used is the following. The large $N$
Dyson--Schwinger equations  form a closed system for the
two-point function \cite{SachdevYe,kitaev,MS,KitaevSuhFirstPaper}.
 In the out-of-equilibrium situation that we are considering,
 these equations are commonly referred to as the Kadanoff--Baym (KB) equations.   We couple the system of two interacting SYK models to a large bath and find the real
 time dynamics using KB equations.
The problem of coupling SYK to a bath was recently studied in \cite{syk_bath} and we borrow some results from there. Also, the KB equations for a single SYK
was recently studied numerically in \cite{quench1,quench2}.
Here we study this problem solving the dynamical equations at $q=4$.
 The problem has many time scales and due to numerical limitations we could not separate them all by large amounts. However,
 our numerical results seem to confirm the picture where the system follows the microcanonical equilibrium curve.
  We now briefly review this equilibrium thermodynamics.

\subsection{Equilibrium thermodynamics}

In the canonical ensemble the system has two phases: the low temperature one corresponding to
the ground state, the SYK wormhole,  and its excitations; and a higher temperature phase which is closer to two separate thermal SYK systems.
The two phases are separated by a first order phase transition.
In the large $q$ limit,
the black hole phase and the wormhole phase are   smoothly connected by a canonically unstable  phase with negative specific heat \cite{MQ}. However,
in the microcanonical ensemble, we expect that the system  smoothly interpolates between these two phases. In other words, in the microcanonical ensemble we expect no phase transition as we lower the energy.

\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.5]{./fancy.pdf}
\minipage{0.8\textwidth}
\caption{ \small
Dots: energy vs beta obtained by numerically solving DS equations for two coupled SYK models with $J=0.5, \mu=0.0053$.
Blue dots correspond to the ``two black holes phase'', whereas green dots correspond to the ``cold wormhole phase''.
Red dashed line:  curve for the ``hot wormhole'' phase expected from a low energy analytic analysis.
The question mark ``?'' reminds us that we were not able to find it as a solution of the euclidean DS equations. }
\label{fig:energy_fancy}
\endminipage
\end{center}
\end{figure}
Figure \ref{fig:energy_fancy} shows energy vs inverse temperature $\beta$ for $q=4$. We use energy instead of free energy or entropy because   we will be dealing
mostly with Lorentzian non-equilibrium correlators numerically and it is easier to find the temperature and the energy from them.
There are three different regions. At high temperatures $T>T_{\rm 2BH}$ we have the phase we name the ``two black holes phase''.
  At low temperatures $T<T_{\rm WH}$ we have  the phase we call ``cold wormhole" phase, which can be viewed a as a wormhole with few thermal excitations.  The two phases overlap, since $T_{\rm 2BH} < T_{\rm WH}$.  In the intermediate
temperatures regime $T_{\rm 2BH} < T < T_{\rm WH}$ we also expect a canonically unstable, but microcanonically stable, phase that we call
``the hot wormhole phase''.
  As we mentioned above,
this phase can be found analytically in the large $q$ limit. It   has positive entropy but negative specific heat.
However, at finite $q$ we do not known much about this phase, since we have been unable to find it by solving the Euclidean Schwinger Dyson equations. We interpret this failure as resulting from its canonical instability. This is why we put a question mark in Figure \ref{fig:energy_fancy}.   We will find evidence for this phase through the real time evolution, since we will find that the temperature goes up as the energy monotonically goes down.  There is also analytic evidence from a low energy analysis, as we will review later.
The names ``hot'' or ``cold'' wormhole refer to how these would feel to an observer who is inside the wormhole, at its center, in a gravity picture for these configurations.
As is clear from Figure \ref{fig:energy_fancy}, there are outside temperatures where we can have both a ``cold'' and ``hot'' wormhole. When we talk about temperatures in this paper, we are always referring to the physical temperature as seen from the outside.

Figure \ref{fig:energy_fancy} also displays the critical temperature, $T_c$,   when two stable phases switch dominance in the canonical ensemble.
For $T_{\rm 2BH} < T <T_c$ two black hole phase is thermodynamically metastable but is not a global minimum for the free energy. Similarly for
$T_c < T < T_{\rm WH}$ and the cold wormhole phase. For our problem the microcanonical ensemble is more relevant. Notice that the different ``phases'' are continuously connected in the microcanonical ensemble, so they are not really sharply separated  phases.

For small values of $\mu$, the coupling between the two SYK models,
 we can make different analytic approximations for the different parts of the curve.
For the two black hole region we can start with two separate thermal SYK models and use
perturbation theory in
$\mu$.
 In this regime, the left-right correlator $G_{LR}$ is small and   of order $\mu$. The gravity picture is that we have two separate hyperbolic disks with a boundary perturbation that connects the bulk  fields on the two disks.
We find that $T_{\rm 2BH}$ is in the region where this perturbation theory breaks down.
For low temperatures the left-right correlator is of order one. We can access this regime by assuming that the system is close
to the thermofield double state. The relevant part of the dynamics is captured by the Schwarzian mode. This aspect of the dynamics is the same for the SYK model and the nearly-$AdS_2$ gravity theory  \cite{cft_breaking,Jensen:2016pah,Engelsoy:2016xyb}.
This describes both the cold wormhole and hot wormhole phases. In particular, we can see the existence of the hot wormhole phase in this approximation \cite{MQ}.
In particular,    the temperature $T_{\rm WH}$ can be found within this approximation. We review this description in Section \ref{sec:sch_low}.

\subsection{Gravity picture }

One of our motivations was to understand whether a similar wormhole formation process occurs in more general theories of nearly-$AdS_2$ gravity with matter.

With this goal in mind we will present a gravity picture for the transition we have in the SYK model.
We do not know the precise gravity dual of the SYK model. But we consider a nearly-$AdS_2$ gravity theory that has some of the same features. For questions that mainly involve the Schwarzian mode,  the SYK and
nearly-$AdS_2$ answers
match precisely \cite{cft_breaking,Jensen:2016pah,Engelsoy:2016xyb}.
However,   wormhole formation goes slightly beyond this approximation,  and we need to incorporate one important feature that is related to the origin of the ground state entropy, or ``zero temperature'' entropy,
$S_0$, of the SYK model. If we start from a phase consisting of two thermal states, then the entropy will have a large contribution of size
$2 S_0$ (plus thermal corrections). As the wormhole forms, the system should be able to shed this large entropy into the bath. In gravity this involves topology change, which would naively be exponentially suppressed. On the other hand, as we discussed above,  this happens without any such suppression in SYK.

 We can reproduce this from a gravity picture as follows. First we view the two coupled systems as a nearly-$AdS_2$ gravity theory with $N$ bulk fermion fields with Neumann boundary conditions. The two black hole phase consists of two hyperbolic disks with an interaction between the boundary values of the bulk fields.
As we lower the temperature, this interaction effectively becomes strong and the theory flows to a new IR fixed point. The new fixed point is simply a theory with different boundary conditions, namely Dirichlet boundary conditions.
This change in boundary conditions is similar to the one discussed in \cite{KlebanovWitten}.
The two disks decouple again but the boundary conditions are effectively changed from Neumann to Dirichlet. Now we use the observation in \cite{KitaevRecent},  that $e^{S_0}$ is equal to the ratio of the Neumann vs Dirichlet  disk partition functions for $N$ fermions. This means that the effective theory in the IR, with Dirichlet boundary condition has now $S_0^{eff} =0$. This means that topology change ``costs us nothing'', and we can easily transition into the wormhole phase. In fact, by a similar argument we can say that the end of the hot wormhole phase also corresponds to the region where the interactions between the two sides of the global $AdS_2$ strip produce a flow that change the boundary conditions of the fermions from Neumann to Dirichlet.

In summary, we provide a qualitative gravity mechanism for the formation of the SYK wormhole. The purpose of this explanation was to contrast SYK with what we expect in a generic gravity theory.
A generic gravity theory can have a number of fields much smaller than  $S_0$. In this case, the change in boundary conditions would not significantly change $S_0$ and it would still be difficult to change the topology.
 For this reason we could not answer the question of whether there is an
 ``easy way'' of forming the wormhole for more
general gravity theories, such as the case of four
dimensional magnetically charged wormholes in the Standard Model \cite{MMP}.

The paper is organized as follows. In Section \ref{MQModel} we review the two coupled SYK model \cite{MQ}. We   describe the perturbative approach at high temperatures, for the ``two black hole phase''. We also review the Schwarzian description of the low energy dynamics that describes the hot and cold wormhole phases.

Section \ref{RealTime} contains our real time analysis of the formation of the wormhole. We set up the coupling to the bath, we write the
Kadanoff-Baym equations (the real time Schwinger Dyson equations), and we present the result of a   numerical analysis for some particular values of the parameters.
In section \ref{sec:gravity}, we discuss
differences and similarities between SYK and nearly-$AdS_2$ (or JT) gravity and also provide the qualitative picture of the transition.
Various computational details are discussed in the appendices.

\section{The two coupled SYK model and its thermodynamics }
\label{MQModel}

\subsection{Definition and properties of the ground state}
\label{sec:mq}

Here we review properties of the two coupled SYK models introduced in \cite{MQ}.
The Hamiltonian of the model consists of two SYK terms coupled by an interaction
\begin{equation} \lambda{CoupMod}
H = H_{\rm SYK }^L + H_{\rm SYK}^R + H_{\rm int}
\end{equation}
where
 \begin{equation}
 H_{\rm SYK }^L + H_{\rm SYK}^R  =  \sum_{ 1 \le i < j < k < l \le  N }J_{ijkl} \left( \psi_i^L \psi_j^L \psi_k^L \psi_l^L +
\psi_i^R \psi_j^R \psi_k^R \psi_l^R \right)
\end{equation}
where the couplings are the same for both factors. They are Gaussian random variables with variance
 \begin{equation}
\langle J_{ijkl}^2 \rangle = \frac{3! J^2 }{N^3},~~ \ \text{no sum}
\end{equation}
There is also a generalization where we consider a $q$ fermion interaction term,
 instead of four.
The interaction term has the form
 \begin{equation}
\label{eq:hint}
H_{\rm int} = i \mu \sum_{j=1}^N \psi^L_j \psi^R_j
\end{equation}
 In the large $N$ limit,  $\mu$ and $J$ stay fixed.
We will mostly consider the case $\mu/J \ll 1$ and also consider temperatures $T/J \ll 1$. This will be true even for what we call ``high'' temperatures.

  As an aside, let us mention that we can couple the two systems by an operator of dimension $\Delta$
 \begin{equation}
\label{eq:hintoth}
H^\Delta_{\rm int} = i^F \mu (\mathcal{O}_\Delta)_L (\mathcal{O}_\Delta)_R
\end{equation}
where $F$ is the fermion number of $\mathcal{O}_\Delta$. We will mostly consider the case of $(\ref{eq:hint})$ which corresponds to $\Delta=1/4$ at low energies. However, we will
give certain estimates for generic $\Delta$.

Like a single SYK, this model, \nref{CoupMod},  is solvable in the large $N$ limit.
 We have four types of correlators: $G_{LL},G_{RR},G_{LR},G_{RL}$, each defined in Euclidean space as
  \begin{equation}
G_{ab}(\tau) = \langle T \psi_a(\tau) \psi_b(0) \rangle ~,~~~~{\rm with }~~~ a, \, b = L, \, R
\end{equation}
Since we are dealing with Majorana fermions we have
 \begin{equation}
G_{LL}(0^+)=G_{RR}(0^+)=\frac{1}{2}
\end{equation}
We have a closed system of Dyson--Schwinger
 equations
for the two point functions \cite{MQ}
\begin{align}
\label{eq:ds}
\partial_\tau G_{ab}(\tau) = \sum_{c} ~ \Sigma_{ac} * G_{cb}(\tau) + \delta_{ab} \delta(\tau) \\
\Sigma_{ab}(\tau) = J^2 G_{ab}(\tau)^3 - i \mu_{ab} \delta(\tau) ~,~~~~~{\rm no~sum}
\end{align}
where the convolution $*$ is taken along the Euclidean circle, $a,b=L,R$, and $\mu_{ab}$ is given by
 \begin{equation}
\mu_{ab}=
\begin{pmatrix}
0 & \mu \\
-\mu & 0 \\
\end{pmatrix}
\label{mu_ab}
\end{equation}
The system has a $\mathbb{Z}_4$ symmetry $ \psi_L \rightarrow -\psi_R$, $ \psi_R \rightarrow \psi_L
$.
Throughout our numerical computation we do not assume that this symmetry in unbroken.
We find that it is unbroken, since the (Euclidean) correlators we obtained obey the following relations:
\begin{equation}
G_{LL} = G_{RR} \ \text{pure real} ~,~~~~~~~
G_{LR} = -G_{RL} \ \text{pure imaginary}
\end{equation}
A convenient expression for the energy is
 \begin{equation}
\frac{E}{N}= \frac{J^2}{4} \int_0^\beta d\tau\ \left( -G_{LL}^4 - G_{RR}^4 - 2 G_{LR}^4 \right) + i \mu G_{LR}(0)
\label{eucl_energy2}
\end{equation}

The interaction term (\ref{eq:hint}) is a relevant perturbation, since for a single SYK model  the
fermion $\psi$ has dimension $\Delta = 1/4$.
Therefore at relatively high temperatures we expect that we have two weakly coupled
SYK models, whereas at low temperatures the system flows into a gapped phase with a gap
that scales as \cite{MQ}
 \begin{equation}
E_{\rm gap} \propto \mu^{2/3} J^{1/3} ~,~~~~~~~~{\rm for } ~~~{ \mu \over J } \ll 1,\ {\rm and} \
q=4,\ \Delta=\frac{1}{4}
\end{equation}
Moreover the ground state
is  close to the TFD of the two models with effective (inverse) temperature $\tilde{\beta}$:
 \begin{equation}
| \text{TFD} \rangle =  \sum_n e^{-\tilde{\beta} E_n/2} | \bar{E}_n \rangle_L \times | E_n \rangle_R ~,~~~~~~
\tilde \beta J \propto  \left( { J \over \mu } \right)^{ 1 \over 2 (1-\Delta) } ~,~~~\Delta ={1 \over q	 }
\end{equation}
The energy of the ground state, relative to the energy of the two decoupled SYK models, scales as
 \begin{equation}
 E_{\rm G} - 2 E_{0, \rm SYK}  \propto - N \frac{\mu^{4/3}}{J^{1/3}}
\end{equation}
And for general $\Delta$, $ E_G - 2 E_{\rm 0,SYK}  \propto - \mu^{\frac{1}{1-\Delta}}$

Since there is a gap and the ground state is unique,  the entropy is small in the ``cold wormhole phase".
 Whereas in the two black hole phase,  we have a big entropy $2 S_0$, where
  $S_0 \approx N \times 0.23$ is the ``zero temperature'' entropy of a single SYK model.
The transition temperature $T_c$  is estimated by
 \begin{equation}
T_c \sim - \frac{E_{\rm G}}{2 S_0}  \propto { \mu^{ 4 \over 3 } \over J^{1\over 3} } , ~~~~~~~~~~{\rm for } ~~q=4
\end{equation}

For arbitrary temperatures the Dyson-Schwinger equations can be solved numerically by starting
from $G_{LR}=G_{RL}=0, G_{LL}=G_{RR}=\half $ and then using an
iteration procedure similar to one described in \cite{MS}. After obtaining the solution for some value of temperature, we
can use it as a seed for the iteration procedure at higher/lower temperature.
Figure \ref{fig:energy_fancy} shows energy as a function of beta for particular values of parameters $J=0.5, \mu=0.0053$.

\subsection{Perturbation theory at high temperature}
\label{sec:pert}
Here we use the term ``high" temperatures for temperatures for the two black hole branch of
the diagram  $T \gg T_{\rm 2BH} $, but still $T/J \ll 1$.

For $\mu =0$ we have two copies of the conventional SYK correlators \cite{SachdevYe,kitaev}
\begin{align}
G(\tau)_{LL}=G(\tau)_{RR} = b \sgn(\tau) \frac{1}{\sqrt{J \beta}} \left( \frac{\pi^2}{\sin^2  \frac{\pi \tau}{\beta}} \right)^{1/4},
  ~~~~~~~G_{LR} =0
\label{gll_conf}
\end{align}
with $b^{4} =1/( 4 \pi)$.

 Now we turn on a small value of $\mu$ \nref{eq:hint}.
If we are at sufficiently high temperature then the coupled system is still in the
phase with two separate black holes \cite{MQ}.
Nonetheless, the correlation between left- and right- SYK is not zero.
We can try to use the conformal perturbation theory to study the system.

To linear order in $\mu$, only $G_{LR}$ Green functions receive a correction:
 \begin{equation}
\Delta G_{LR}(\tau_1-\tau_2) =  i \mu \int_0^\beta d\tau \ G_{LL}(\tau_1-\tau) G_{RR}(\tau-\tau_2)
\label{glr_int}
\end{equation}
This integral is computed analytically in Appendix \ref{app:delta_glr}.
We can compare this leading order approximation against exact numerical solution of the Dyson--Schwinger equation for $J=0.5, \mu=0.05$ and different $\beta$s,
see Figure \ref{glr_comparison}.
\begin{figure}[!ht]
\minipage{0.47\textwidth}
\includegraphics[scale=0.9]{./G_LR_beta20_mu005.pdf}
\label{glr_comparison20}
\endminipage
\minipage{0.47\textwidth}
\includegraphics[scale=0.3]{./G_LR_beta53_mu005.pdf}
\label{glr_comparison60}
\endminipage \\
\caption{\small Euclidean Green function $G_{LR}$.  The blue points correspond to the  exact solution, and the red ones  to the conformal answer (\ref{glr_conf}). Left: $\beta=20$. Right: $\beta=53$.
For this values of parameters the transition to the wormhole phase happens around $\beta_{\rm 2 BH} \sim 61$. The approximation is better for higher temperatures. }
\label{glr_comparison}
\end{figure}

Also in Appendix \ref{app:delta_glr} we computed the $\mu^2$ correction to the energy. So that at low temperatures
we have the following expression for the energy:
\begin{eqnarray}
\label{delta_E}
E&=&2E_\text{SYK} + \Delta E \\
 E_\text{SYK}& =& E_\text{0,SYK} + \frac{2 \pi^2 \alpha_S \sqrt{2}}{J \beta^2} \lambda{EnSYK} \\
 E_{\rm 0,SYK} &=& -J \times 0.04063(1), ~~~~~~~~~~~\alpha_S=0.0071 \pm 0.0004 \lambda{AlphaS} \\
\Delta E &=& \frac{1}{2\sqrt{4 \pi}} \frac{\mu^2}{J} \left( 2 \log \frac{\pi}{\beta J} -2 - c_1 \right) \nonumber, \  ~~~~~~~~~~~~c_1\ =1.66(1)
\end{eqnarray}
where $E_\text{SYK}$ is the low-temperature result for a single SYK \cite{MS} and $\Delta E$ is the leading $\mu^2$
correction derived in Appendix \ref{app:delta_glr}.
The comparison between \nref{delta_E} and the numerics is presented on Figure \ref{sch_comp}(b).

This approximation works better for high temperatures and then deteriorates close to  $T_{\rm 2BH}$, where the phase is supposed to connect with the hot wormhole phase.

 Let us find out until what temperature we can trust the perturbation series in $\mu$, \nref{eq:hint}.
The first point is that only even terms are non-zero. The term of order $\mu^{2n}$ contains
  a  $2n$-point function on the left and $2n$-point function on the right, each of these now computed in a single SYK model. We are only interested in connected correlators for computing the corrections to the free energy.
Higher point functions in SYK have two contributions: a purely conformal piece which is independent of $\beta$, up to an overall factor of
$\left( 1/\sqrt{J \beta} \right)^{4n}$ as in \nref{gll_conf},   plus contributions
from the Schwarzian which are enhanced by an extra factor of
$\beta J$. We claim that the Schwarzian contributions are in fact zero, see Appendix \ref{sch:1loop}.
 The reason behind this cancellation is the following. When we work at large $N$ we are solving the classical equations. The reparametrization mode has a solution that is time translation invariant. The two sides are coupled by convolutions of Green's functions but  this translation symmetry remains unbroken. This means that there is no source for higher Fourier components of the reparametrization mode, so that the standard thermal solution continues to be a solution.

The integrals over time give  $\beta^{2n}$. In total, we
have $\frac{\mu^{2n} \beta^{2n}}{(J \beta)^n}$.
So the expansion parameter is
\begin{equation}
\lambda{ExpPar} { \mu \beta \over \sqrt{J \beta} } ~,~~~~~{\rm for } ~~q=4 ~,~~~~{\rm or }~~~~~ { \mu \beta \over ( J \beta )^{ 2 \Delta }} ~,~~~~~~~{ \rm for ~general~}q
\end{equation}
So we can trust the above perturbative answer until temperature
 \begin{equation}
T_{\rm pert} \propto J \left( { \mu \over J } \right)^{\frac{1}{1-2 \Delta}}
\end{equation}
For $\Delta=1/4$ this scales as $\mu^2$, whereas   $T_c \propto  \mu^{4/3}$ is much larger.

We conjecture that the transition temperature $T_{\rm 2BH}$, when the two black holes phase cease to exist,   in fact coincides
with $T_{\rm pert}$, when the perturbation theory in $\mu$ breaks down
 \begin{equation}
\label{T1_est}
T_{\rm 2BH} \sim T_{\rm pert} \propto  J \left( { \mu \over J } \right)^{\frac{1}{1-2 \Delta}}
\end{equation}
We check this  prediction for $\Delta=1/4$ against the numerical phase diagram obtained
in \cite{MQ}. See Figures   \ref{mq_phase}, \ref{mt_fit}.
\begin{figure}[!h]
\centering
\includegraphics{./phase_rescaled.pdf}
\minipage{0.8\textwidth}
\caption{\small A reprint of the phase diagram obtained numerically in \cite{MQ} for $J=0.5$.
The right solid black curve indicates $T_{\rm WH}$, purple line $T_c$ and  left solid black line $T_{\rm 2BH}$.
The dashed horizontal line is at $\mu=0.05$ ,  the value of $\mu$ we will use in our real time numerical simulation.
In this case $\beta_{\rm 2 BH} \sim 61, ~\beta_c \sim 54,~ \beta_{\rm WH} \sim 49$.  }
\label{mq_phase}
\endminipage
\end{figure}

\begin{figure}[!h]
\centering
\minipage{0.49\textwidth}
\includegraphics[scale=0.5]{./T_l.pdf}
\caption{\small The fit for $T_{\rm WH}$ using the numerical data from \cite{MQ} in Figure \ref{mq_phase},
using only   data points with $\mu < 0.03$. The fit is consistent
with the analytical prediction $\mu \sim T^{3/2}$.}
\endminipage\hfill
\minipage{0.49\textwidth}
\includegraphics[scale=0.5]{./t_inst.pdf}
\caption{\small The fit for $T_{\rm 2BH}$ using the numerical data from \cite{MQ} in Figure \ref{mq_phase},
using only  data points with $\mu < 0.03$. The fit is consistent
with the analytical prediction $\mu \sim \sqrt{T}$.}
\label{mt_fit}
\endminipage\hfill
\end{figure}

One last comment on the leading result (\ref{glr_int}). In Appendix \ref{sch:1loop}, we studied
the gravitational dressing of this term, searched for instabilities that would spontaneously break the $U(1)$ time translation symmetry,  but did not find any.

\subsection{Low temperature thermodynamics using the Schwarzian }
\label{sec:sch_low}

In this section we review the results of \cite{MQ} on the Schwarzian description of the wormhole. We will see that the Schwarzian indeed admits a wormhole
solution at low temperatures. Moreover, by including the matter contribution to the partition function one is able to see two phases which join at
(inverse) temperature $\beta_{\rm WH}$. One phase has positive heat capacity and almost zero entropy. This is a cold and mostly
empty wormhole. The second phase
has higher energy longer throat and negative heat capacity. This is wormhole with extra matter excitations inside. We will see shortly that at small
enough $\mu$ this approximation predicts $\beta_{\rm WH}$ with good accuracy.

As we have mentioned before, the ground state of the system is close to the TFD state. Since we have global $AdS_2$ we have the following
$G_{LR}$ correlator in Euclidean signature:
 \begin{equation}
G_{LR} = \frac{i b}{\sqrt{2 J \cosh \left( \frac{t_L- t_R}{2} \right)}}
\end{equation}
where $t_L, t_R$ are times on left/right.

The action in the Schwarzian approximation now includes two kinetic terms\footnote{For $q=4$, $J=\sqrt{2} \mathcal{J}$ hence the extra $\sqrt{2}$.} and an interaction
\begin{eqnarray}
S &=& S_{\rm kin} + S_{\rm int}
\\
S_{\rm kin} &=& - \frac{N \sqrt{2} \alpha_S}{J} \int du \ \left( \{ \tanh \left( \frac{t_L(u)}{2} \right) , u  \}
+ \{ \tanh \left( \frac{t_R(u)}{2} \right) , u  \} \right)
\\
S_{\rm int} &=& -  N i  \mu \int du G_{LR}(u) = \frac{N \mu b}{\sqrt{2 J}} \int du \frac{( {\bf t}'_L(u) {\bf t}'_R(u) )^{1/4}}{\sqrt{\cosh \left( \frac{{\bf t}_L(u) - {\bf t}_R(u)}{2} \right)}}
\end{eqnarray}
with $\alpha_S$ in \nref{AlphaS}.
The wormhole solution is simply ${\bf t}_L = {\bf t}_R = {\bf t}' u$, where ${\bf t}'$ is a constant. The effective temperature, $\tilde{\beta}$ of the  TFD state is
given by
 \begin{equation}
\tilde{\beta} = {\bf t}' \beta
\end{equation}
Inserting into the action we get  the free energy
 \begin{equation}
 F/N = \frac{\alpha_S \sqrt{2}}{J} {\bf t}'^2 - \frac{\mu b}{\sqrt{2J}} {\bf t}'^{1/2}
\end{equation}
We should also include the contribution from matter fields in the wormhole throat. If the temperature is low enough we excite only the
lightest excitation in the bulk, which is the elementary fermion with mass $1/4$.
Its contribution to the free energy is:
 \begin{equation}
\Delta  F/N = - \frac{1}{\beta} \log \left( 1 + e^{-\tilde{\beta}/4} \right) \approx - \frac{1}{\beta} e^{-{\bf t}' \beta/4}
\end{equation}
Extremizing the free energy with respect to ${\bf t}'$, which is the same as extremizing the
full effective action,  we have the following equation to determine ${\bf t}'$ and correspondingly $\tilde{\beta}$:
 \begin{equation}
\label{saddle}
0=\frac{\partial F_{\rm tot}}{\partial {\bf t}'} = \frac{2\alpha_S \sqrt{2}}{J} {\bf t}' - \frac{\mu b}{2\sqrt{2J}} {\bf t}'^{-1/2} + \frac{1}{4} e^{-{\bf t}' \beta/4}
\end{equation}

This equation  has a solution with very small ${\bf t}'$ which we can not trust, because we were assuming
that the temperatures are low.
For large enough $\beta$ two additional solutions emerge. One of them correspond to what we call the cold wormhole and the other to the hot wormhole. The latter has negative heat capacity and can be viewed as a wormhole with more excitations in the throat.

Figure \ref{sch_comp} shows the two branches for two different values of $\mu/J$ and their comparison to the numerical solution to the DS equations.

We can calculate $T_{\rm WH}$ from this equation.  We simply need to find when these two solutions merge.
To keep the discussion general, we consider general $\Delta$, which corresponds to the case when
the two sides are coupled through the product of two operators of dimension $\Delta$. The equation for ${\bf t}'$ now reads as:
 \begin{equation}
\label{Constr}
\frac{2\alpha_S \sqrt{2}}{J} {\bf t}' - \frac{\mu b_\Delta}{2 (2J)^{2 \Delta}} {\bf t}'^{2 \Delta-1} +  \Delta e^{-{\bf t}' \beta \Delta} = 0
\end{equation}
The cold wormhole branch can be approximately found \cite{MQ} by neglecting the thermal excitations exponent in the above equation, so that ${\bf t}'$ is equal
 \begin{equation}
\label{t_prime_wh}
{\bf t}' \propto \left( \frac{\mu}{J^{2 \Delta-1}} \right)^{\frac{1}{2- 2 \Delta}}
\end{equation}
Whereas the unstable branch with excitations can be approximated by neglecting the Schwarzian kinetic term $\sim \alpha_S {\bf t}'$:
 \begin{equation}
\label{UnstT}
\frac{\mu}{J^{2 \Delta}} {\bf t}'^{2 \Delta-1} \propto e^{-{\bf t}' \beta \Delta}
\end{equation}
Plugging the ${\bf t}'$ from the first solution (\ref{t_prime_wh}) into the above equation we find the $T_{\rm WH}$:
 \begin{equation}
\label{eq:T_l}
T_{\rm WH} \propto \frac{\mu^{\frac{1}{2-2 \Delta}}}{\log J/\mu} \sim \frac{\mu^{2/3}}{\log J/\mu}, \ \text{for}\ \Delta=1/4
\end{equation}
Of course, both \nref{t_prime_wh} and \nref{UnstT} are good for
$T\ll T_{\rm WH}$. Here we presented just estimates, but it is straightforward to solve \nref{Constr} numerically, see Figure
\ref{sch_comp}.

We can also compute the energy using\footnote{When computing the derivatives one has to keep in mind that ${\bf t}'$ is a function of $\beta$.  And use  (\ref{saddle}).
}
 \begin{equation}
\label{saddle_energy}
E = F + T S = F + \beta \frac{\partial F}{\partial \beta} + \beta \frac{\partial F}{\partial {\bf t}'} \frac{d {\bf t}'}{d \beta} =2 E_{\rm 0,SYK} +  \frac{\alpha_S \sqrt{2}}{J} {\bf t}'^2 - \frac{\mu b}{\sqrt{2J}} {\bf t}'^{1/2} + \frac{{\bf t}'}{4} e^{-{\bf t}' \beta/4}
\end{equation}
where $E_{\rm 0,SYK}$ is the ground state energy of a single SYK, see \nref{AlphaS}.

\begin{figure}[!ht]
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35]{./new_check.pdf}
\end{minipage}
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35]{./new_check0053.pdf}
\end{minipage}
\begin{center}(a) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\minipage{0.8\textwidth}
\caption{\small Thermodynamics of the model for two different values of $\mu/J$. In
both $J=0.5$ and in (a) $\mu =0.05$ and in (b) $\mu = 0.0053$. The dots correspond to numerical solutions of the DS equation. The blue ones for the two black hole phase and the green ones for the cold wormhole phase.
The black line is the ground state energy of two decoupled  SYK, $2 E_{0,SYK}$ (on the left plot it lies above the visible area).
Blue curve: the perturbative result (\ref{delta_E}) for the energy.
Green curve: wormhole branch of eq. (\ref{saddle}).
Red curve: unstable branch of eq. (\ref{saddle}).
The uncertainties, represented as shaded regions, come from the uncertainties in $E_{\rm 0,SYK}$ and $\alpha_S$. We see that in (b) the agreement is very good. However, in (a) the agreement is not so good, but the qualitative form of the curve is similar, if we joint the two end points of the dotted lines. }
\label{sch_comp}
\endminipage
\end{center}
\end{figure}

In Figure \ref{sch_comp} we have compared the results computed
using eq. (\ref{saddle_energy}) with the numerical solution of the DS equation.
For the value $\mu=0.05$, which is the one we will use for the real time numerical computation, the agreement is not very good, but the qualitative form of the curve is similar, see Figure \ref{sch_comp}(a).
  This means that that $\mu$ is not low enough for an accurate   Schwarzian description. Indeed if we lower $\mu$ we get very good agreement. See Figure \ref{sch_comp}(b) for   $\mu = 0.0053$.

To summarize, for small $\mu$, we have a hierarchy of  temperatures
\begin{align}
T_{\rm WH} & \propto \frac{\mu^{\frac{1}{2-2\Delta}}}{\log J/\mu} \sim ~\frac{\mu^{2/3}}{\log J/\mu}, \ ~~~~\text{for}\ \Delta=1/4 \nonumber \\
T_c & \propto \mu^{\frac{1}{1-\Delta}} \sim \mu^{4/3} \\
T_{\rm 2BH} & \propto \mu^{\frac{1}{1-2 \Delta}} \sim \mu^{2} \nonumber
\end{align}
where the rightmost term corresponds to $\Delta =1/4$.

\section{Real time results}
\lambda{RealTime}

\subsection{Coupling to a bath}

\label{sec:bath}
In order to study the real-time formation of the wormhole, we need to cool down the system.
Before considering real-time dynamics, first we need to understand how to couple
our system to a thermal bath. Ideally we want the bath to be a large system in order to avoid back-reaction.

Generally, we can couple a system's operator $\mathcal{O}_S$ to a bath operator $\mathcal{O}_B$:
 \begin{equation}
\Delta S = i^F V \int d\tau \  \mathcal{O}_S \mathcal{O}_B
\end{equation}
where $F$ is the fermionic number of $\mathcal{O}_S$.

If $V$ is small and the bath is large we can study this interaction in the Lindblandian approximation by
considering the 1-loop result and assuming that there is no back reaction on the bath, such that
we can substitute the product $\mathcal{O}_B(t_1) \mathcal{O}_B(t_2)$ by VEV $\langle \mathcal{O}_B(t_1) \mathcal{O}_B(t_2) \rangle_B$:
 \begin{equation}
\Delta S = - V^2 \int d\tau_1 d\tau_2 \ \mathcal{O}_S(\tau_1) \mathcal{O}_S(\tau_2) \langle \mathcal{O}_B(\tau_1) \mathcal{O}_B(\tau_2) \rangle_B
\end{equation}

For our problem we have a varying temperature that sets an energy scale for the model. Specially for our numerical analysis, it is convenient to choose an interaction that is  scale invariant (at least approximately), so that the effects of coupling to the bath are independent of the temperature. Otherwise the bath might be effectively decoupling in some temperature range and the system would take long to cool down.

A natural model for the bath is
 another SYK, possibly with larger number of fermions.
We can consider the bath to be another single-SYK model with $q=q_B$ with large number of fermions $M$,
much larger than the number of fermions in our system $N$.
Recently this problem was studied in \cite{syk_bath}, we refer to this paper for details.  If we
denote the bath fermions by $\chi_\alpha$, then the coupling can involve an arbitrary number $s$ of system fermions and $p$ bath fermions.

Again, to warm-up, let us first consider the case when the system consists of a single-SYK model. The coupling between the bath and the system has
the form:
 \begin{equation}
\mathcal{L}_\text{bath-system} = V_{i_1 \dots i_s}^{\alpha_1 \dots \alpha_p} \psi_{i_1} \dots \psi_{i_s} \chi_{\alpha_1} \dots \chi_{\alpha_p}
\end{equation}
where $V$ is a random Gaussian-distibuted coupling.
By choosing its variance appropriatly, the back reaction can be made of order $N/M \ll 1$.
The Euclidean Dyson--Schwinger equations for the system
stay the same except for a correction to the self-energy: the self-energy acquires an additional term,
 \begin{equation}
\Delta \Sigma = \Sigma_B = \eta G_{B}^p G^{s-1}
\end{equation}
where $G_B$ is bath two-point function and $\eta$ is proportional to the variance of $V$.

We can get a marginal interaction with
 $s=1, p=3$,    when the bath consists of $q=4$ SYK models
 (with, say    $J_B=J_S$).
This is the bath we will use.

More precisely, we introduce two separate baths, one for each SYK factor
 \begin{equation}
\mathcal{L}_\text{bath-system} = V_{i}^{\alpha_1 \alpha_2 \alpha_3} \psi_{L,i} \chi_{\alpha_1} \chi_{\alpha_2} \chi_{\alpha_3} +
\tilde{V}_{i}^{\alpha_1 \alpha_2 \alpha_3} \psi_{R,i} \tilde{\chi}_{\alpha_1} \tilde{\chi}_{\alpha_2} \tilde{\chi}_{\alpha_3}
\end{equation}
where $V$ and $\tilde{V}$ are independent Gaussian-distributed variables. This interaction leaves $\Sigma_{LR}$ unperturbed, but the other two
self-energies have additional terms now:
 \begin{equation}
\Delta{\Sigma}_{LL} = \Delta{\Sigma}_{RR} = \Sigma_B = \eta G_B^3
\end{equation}
The above equations are written in  Euclidean signature.
We now turn to Lorentzian equations.

\subsection{Kadanoff--Baym equations}
\label{sec:kb_intro}

We now  write down the Lorentzian  time version of the DS equations. For details see Appendix \ref{ds:lorentz}.

 Here we will discuss the non-equilibrum situation following \cite{MS} and \cite{quench1}.
It is convenient to work right away on the Keldysh time contour, see   \cite{kamenev_book} for a comprehensive introduction.
We will need Lorentzian time correlators which are not time ordered. This
can be achieved by introducing a Keldysh time contour which runs from $-\infty$ to $+\infty$ and then back from $+\infty$ to $+\infty$.
First, one introduces a Wightman function
 \begin{equation}
G^>_{ab}(t^-_1,t^+_2) =
-i \langle \psi_a(t^-_1) \psi_b(t^+_2) \rangle,
\end{equation}
where $t_1,t_2$ lie on different sides of the contour. This simply means that $\psi_b(t_2)$ is always located to the right of $\psi_a(t_1)$
in the correlator, regardless of time-ordering. This is why the Keldysh contour consists of two parts.
Note the overall $i$ in front of the correlator.
Since we are dealing with simple Majorana fermions the ``lesser'' function $G^<_{ab}$ is given by:
 \begin{equation}
G^<_{ab}(t_1,t_2) = - G^>_{ba}(t_2,t_1)
\end{equation}
Also we will need retarded and advanced Green functions:
\begin{align}
G^R_{ab}(t_1,t_2) = \theta(t_1-t_2) \left( G^>_{ab}(t_1,t_2) - G^<_{ab}(t_1,t_2) \right) \\ \nonumber
G^A_{ab}(t_1,t_2) = \theta(t_2-t_1) \left( G^<_{ab}(t_1,t_2) - G^>_{ab}(t_1,t_2) \right)
\label{}
\end{align}
Dyson--Schwinger equations written on the Keldysh contour
are known as Kadanoff--Baym equations, and are useful for non-equilibrium situations.   Let us write them down explicitly for a single SYK:
\begin{align}
i \partial_{t_1} G^>(t_1,t_2) = \Sigma * G =  \int_{-\infty}^{+\infty} dt \ \left( \Sigma^R(t_1,t) G^>(t,t_2) + \Sigma^>(t_1,t) G^A(t,t_2) \right) \nonumber \\
-i \partial_{t_2} G^>(t_1,t_2) = G * \Sigma =  \int_{-\infty}^{+\infty} dt \ \left( G^R(t_1,t) \Sigma^>(t,t_2) + G^>(t_1,t) \Sigma^A(t,t_2) \right)
\label{eq:kb_syk1}
\end{align}
 where the self-energy $\Sigma^>$ is given by
 \begin{equation}
\Sigma^>(t_1,t_2) = -J^2 \left( G^>(t_1,t_2) \right)^3
\end{equation}
These equations for the complete system of two interacting SYKs and a bath are derived using the path integral in Appendix \ref{eff_action_der}.

Remember that the ``greater'' Green function $G^>(t_1^-,t_2^+)$ has time arguments lying on different sides of the Keldysh contour, this is why we do not have
a delta-function on the right hand side of \nref{eq:kb_syk1}.
The integral in the right hand side of \nref{eq:kb_syk1}, which  involves different
Green functions, is just a simple convolution $\Sigma * G$ along the Keldysh contour \cite{stefanucci2013}.
 We can show it by  writing the anti-time ordered $\Sigma = \Sigma^> - \Sigma^R$ and the time ordered
$G = G^A + G^>$.
One can easily see that equations  \nref{eq:kb_syk1}  are casual.

Let us mention one subtlety. Strictly speaking, if one starts from a thermal state, then the precise Keldysh contour involves imaginary time stip at the end of the
lower branch, at $t=0$. This time strip has length $\beta$ and prepares the thermal state. One can bypass this as follows.
First we set the coupling to the bath to zero. Then we find the real time correlators at thermal equilibrium by solving the
 real time equations imposing the appropriate relations between the Green's functions, see e.g. \nref{GpGR}. We then evolve
the system for some time $t\gg \beta$ and then we turn on the coupling to the bath.

For more details see Appendix  \ref{numerics}.

For two coupled SYK models one has to be very careful with the $\mu$ term.
To understand its form on the Keldysh contour we can go back to $G,\Sigma$ effective action derived in Appendix \ref{eff_action_der}
\begin{align}
i \frac{S}{N}=\frac{1}{2} \Tr \log \left( \omega - \Sigma_{ab}(\omega) \right) -
\sum_{ab} \int_\mathcal{C} dt_1 dt_2 \left(  \frac{J^2}{8}  G_{ab}(t_1,t_2)^4  + \frac{1}{2} \Sigma_{ab}(t_1,t_2) G_{ab}(t_1,t_2) \right) + \nonumber \\
+\frac{i \mu}{2} \int_\mathcal{C} dt \ \left( G_{LR}(t,t) -G_{RL}(t,t) \right)  - \frac{\eta}{2} \int_{\mathcal{C}} dt_1 dt_2 \left( G_{LL}(t_1,t_2) + G_{RR}(t_1,t_2) \right) G_B^3(t_1,t_2)
\label{eq:eff_action}
\end{align}
We see that the $\mu$ contribution to $\Sigma(t_1,t_2)$ is
 \begin{equation}
\Delta \Sigma_{ab}(t_1,t_2) = i \mu_{ab} \delta_\mathcal{C}(t_1-t_2)
\end{equation}
In this expression $t_1,t_2$ can be on either side of the Keldysh contour,
this is why $\Sigma$ does not have an additional index, like $>,R,A$.
Notice that the delta-function $\delta_\mathcal{C}$ is defined on the Keldysh contour as well.
It yields non-zero answer if and only if $t_1=t_2$ and $t_1,t_2$  are on the same side of the contour.
Let us compute the contribitution of this term to the convolution $\Sigma * G$ along the Keldysh contour:
 \begin{equation}
\Delta (\Sigma * G)(t_1^-,t_2^+)_{ab}  = i \mu_{ac} \int_\mathcal{C} dt \ \delta_\mathcal{C}(t_1^- - t) G_{cb}(t-t_2^+) = i \mu_{ac} G^>_{cb}(t_1^-,t_2^+)
\end{equation}

The final form of the Kadanoff--Baym equations, including the bath, is:
\begin{align}
i \partial_{t_1} G_{ab}^>(t_1,t_2) = i \mu_{ac} G^>_{cb}(t_1,t_2) + \int_{-\infty}^{+\infty} dt \ \left( \Sigma^R_{ac}(t_1,t) G^>_{cb}(t,t_2) + \Sigma^>_{ac}(t_1,t) G^A_{cb}(t,t_2) \right)  \nonumber \\
-i \partial_{t_2} G_{ab}^>(t_1,t_2) =  i \mu_{ac} G^>_{cb}(t_1,t_2) +  \int_{-\infty}^{+\infty} dt \ \left( G^R_{ac}(t_1,t) \Sigma^>_{cb}(t,t_2) + G^>_{ac}(t_1,t) \Sigma^A_{cb}(t,t_2) \right)
\label{eq:kb}
\end{align}
where $\mu_{ab}$ is defined in eq. (\ref{mu_ab}) and the self-energy is
 \begin{equation}
\label{def:sigma}
\Sigma_{ab}^>(t_1,t_2) = - J^2 \left( G_{ab}^>(t_1,t_2) \right)^3 - \eta \delta_{ab} \left( G_B^>(t_1,t_2) \right)^3
\end{equation}

\subsection{Forming the wormhole}
\label{sec:formation}
Our numerical setup for solving KB equation (\ref{eq:kb}),(\ref{def:sigma}) is presented in Appendix \ref{numerics}.
We prepare initial Lorentz Green's functions using Lorentz-time Dyson--Schwinger equations described in Appendix \ref{ds:lorentz}.
Our initial Green's functions correspond to two interacting SYK models with non-zero $\mu$ at thermal equilibrium.
In order to save   computational time,  the initial temperature is chosen close to (and slightly higher than)  the
transition temperature $T_{\rm 2BH}$.

We extract the temperature using the Fluctuation-Dissipation Theorem(FDT) for the two-point functions. Precisely, our numerical setup gives us the  $G^>$ and $G^<$ Green's functions.
At thermal equilibrium a certain combination of these Green's functions (eq. (\ref{tanh})) must be equal to $\tanh(\beta \omega/2)$ in the frequency domain.
So we take the numerically obtained
$G^{>}$ and $G^<$,  make a discrete Fourier transform and fit eq. (\ref{tanh}) with the tanh. We refer to Appendix \ref{numerics} for details about the precise choice of the
Green's functions and the frequency domain for fit.

Our benchmark parameters are $J=J_B=0.5$ (the system and the bath have the same coupling strength), $\mu=0.05$.
In this case,  from Figure \ref{mq_phase} we expect $\beta_{\rm 2 BH} \sim 61,~ \beta_c \sim 54,~ \beta_{\rm WH} \sim 49$.

We considered a few values for the system-bath coupling, $\eta=0.04,~0.02,~0.01$.
Figure \ref{comparison_full} shows the results for $\beta_{\rm bath}=80,~ \eta=0.04,~ \beta_{\rm initial}=40$ for different time steps. The energy is computed using eq. (\ref{kb_energy}).
\begin{figure}[!h]
\centering
\includegraphics[scale=0.7]{./comparison.pdf}
~~~~~~~~~~~(a) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b)
\minipage{0.8\textwidth}
\centering
\caption{ \small Numerical results for $J=0.5$ and $\mu = 0.05$, $\beta_B=80$ and
timesteps $dt=0.2,0.1$. (a) The energy as a function of time. The initial rise is due
to the fact that we are coupling the bath to the system, and this changes the energy \cite{syk_bath}.
We then see the energy decreasing monotonicallly. (b) The inverse temperature as a
function of time. We also see an initial sharp increase due to the coupling of
the bath, then we see a decrease. Then a slight increase of the temperature that
signals the phase with negative specific heat. Finally the temperature decreases again.
}
\label{comparison_full}
\endminipage
\end{figure}

We can perform another check, this time taking $\eta$ much smaller, namely $\eta=0.01$.
The result is shown on Figure \ref{check001}.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.5]{./check.pdf}
\minipage{0.8\textwidth}
\caption{\small The comparison between $E(\beta)$ in equilibrium(blue points) and the real-time evolution of the system as it cools down (solid red line). The bath parameters are
$\eta=0.01, ~\beta_{\rm bath}=80$. The initial linear ramp of the red curve occurs because we switch on the coupling with the bath linearly in time.
The dashed red line is the same as the solid line but shifted to match the final energy on the phase diagram.
The green dot shows the equilibrium value of the system+bath energy.}
\label{check001}
\endminipage
\end{figure}
A few comments are in order. As is explained in detail in \cite{syk_bath} the marginal
system-bath interaction we will renormalize $J$, making it bigger. This is why expect that the actual transition will happen
at higher $\beta$. This is indeed what we see. Moreover,  the interaction with the bath will shift the ground state energy. To compensate for
this we have shifted the energy vs beta curve in Figure \ref{check001} to match the final energy.

The red curve in Figure \ref{check001} has wild oscillations in temperature near the equilibrium for small $\eta$, see also Figures \ref{comparison_full}(b).
The reason  is the following.  Because of the numerical error there is an additional flux of energy which pushes the system out of equilibrium.
From the phase diagram (blue points) we see that the derivative $d\beta/dE$ is very large. If $\eta$ is not big enough, the relaxation time is not small enough to smooth out
these fluctuations.

The crucial question is whether we indeed have reached \textit{the} wormhole phase or not. In principle, we might have ended up in some other phase.
To verify that we have reached the wormhole we can make a precise check of the system's energy.

In the real-simulation the final value of the energy is(in units where $J=0.5$):
 \begin{equation}
E_{\rm KB} = - 0.05282(2)
\label{eta0012_energy}
\end{equation}
where the uncertainty comes from changing the size of the diagonal strip and changing the timestep. Also we can
ask ourselves how carefully is the initial
state prepared. For an exact two-point function we know that $G_{aa}(0)=-0.5i$. However, the iterations of the real-time DS equations have $G_{aa}(0)=-0.5002i$. The
error coming from this is esentially the same as in the above number.

How do we compare this result with the equilibrium phase diagram?
In fact, we can solve Euclidean Dyson--Schwinger equation for the coupled system+bath and compare the equilibrium energy. We start from the Euclidean correlators
in the wormhole phase, add coupling to the bath and solve the DS equations again. The value of the
energy we obtained this way is(again in units where $J=0.5$):
 \begin{equation}
E_{\rm DS}=-0.05276(3)
\label{eta002energy_ds}
\end{equation}
It is represented as  green dot in Figure \ref{check001}.
The uncertainty is estimated by  changing the number of discretization points and imposing different cutoffs for the iteration procedure.
We see an agreement with \nref{eta0012_energy} within one standard deviation. This suggests that we indeed reached the wormhole.

To understand what happens near the transition we can look at the maximum value of $G_{LR}$, see Figure \ref{fig:betas} (lower part). We can notice that,
during the transition through the unstable phase,  the imaginary value of the correlator (which is proportional to the anticommutator) rapidly grows,
indicating the growth in the information transmission rate.

Another thing we can see is that coupling to a bath generically thwarts the information transmission between the sides; for larger bath coupling $\eta$ the ratio
$\Im G_{LR}/\Re G_{LR}$ is smaller.

\begin{figure}[!ht]
\centering
\includegraphics[scale=0.5]{./betas.pdf}
\minipage{0.8\textwidth}
\caption{\small (Upper part) $\beta$ as a function of time for $\eta=0.02$ and $0.04$. The error bars are taken from the covariance matrix, so the meaningful quantity is the relative
size between them.
(Lower part) the maximum of value of $G_{LR}$ for the same values of $\eta$. }
\label{fig:betas}
\endminipage
\end{figure}

Figure \ref{check001} shows that the system is more or less following the thermodynamic curve. We see that the temperature and energy are smooth everywhere and the
transition goes through a phase with negative heat capacity, where the energy decreases and the temperature increases.

To check whether the system remains thermal at all times we performed an additional check. Using the fluctuation dissipation theorem (FDT)
we can find the temperature two ways: from $LL$ correlator and the  $LR$ correlator, using \nref{tanh}.
The result is shown on Figure \ref{fig:betas} (upper part). We see that $LL$ correlator is very close to thermal and the curve shows a clear period of temperature increase. In contrast,   the temperature extracted from the  $LR$ correlator has big errorbars. This means that the $LR$ correlator has larger deviations from precise thermality, and larger violations to the relation \nref{tanh}.
We think that this happens for the following reason.
We couple each SYK system to its own bath, so we have two uncorrelated baths. This introduces incoherence to the system,
which can be seen on Figure \ref{fig:betas}: $G_{LR}$ is decreases when we increase $\eta$. Moreover we are in the regime of small $\mu$,  which means that the two sides
interact weakly,  so the equilibration time for $G_{LR}$ is much bigger than for $G_{LL}$. Therefore we expect that the error bars for $LR$ temperature are big because
our system-bath coupling is too big. And, if we managed to cool the system down more slowly, then the $LR$ correlators would remain thermal. Unfortunately, with our limited computer resources   we could not go below $\eta=0.01$. So we do not have a clear evidence for this interpretation.

\subsection{Time to form the wormhole }
\label{sec:est_time}

In this subsection we provide analytic estimates for the time it takes to form the wormhole.
We will first estimate the time it takes to reach $T_{\rm 2BH}$ and then the time it takes to
reach from there to $T_{\rm WH}$.
At this point we basically have a cold wormhole, so we will consider it to be already formed.
We could also consider it formed once we reach $T_{\rm 2BH}$ and we start moving on the hot wormhole region.

In order to estimate these times we need an expression for the rate of energy
emission into the bath.
For a general coupling between a system and a large bath the energy loss rate can be written as
 (see \cite{syk_bath} for more discussion)
\begin{equation}
\label{eq:flux}
{ d E \over dt } =2 i \eta \int_{-\infty}^{+\infty} d \tilde{t} \
\partial_{\tilde{t}} G_{LL} \left( { t + \tilde{t} \over 2} , { t - \tilde{t} \over 2 } \right) G_B(\tilde{t} )^3
\end{equation}
where factor of 2 comes from having both sides of coupled to a bath. In the above integral,  we can replace the system two point function by the thermal one at the instantaneous temperature, assuming that the temperature varies slowly.
Moreover, for SYK at low temperatures
we can simply use conformal approximations for two-point functions.

If the bath temperature
is much lower, than the system's temperature we can approximate bath Green's function by the zero temperature one
\begin{equation}
G_{B} = b \frac{1}{\sqrt{J_B i \left( t- i \epsilon  \right)}}
\end{equation}

\subsubsection{Reaching $T_{\rm 2BH}$}
In the two black hole holes phase we assume that $G_{LL}$ is approximated by a single SYK thermal two-point function
 \begin{equation}
 G_{LL}(t)= b\frac{\sqrt{\pi}}{\sqrt{J \beta i \sinh \frac{\pi(t-i \epsilon)}{\beta}}}
\end{equation}
Plugging these into
(\ref{eq:flux}) we see
that the answer is determined by dimensional analysis up to an irrelevant numerical coefficient\footnote{We refer to
\cite{syk_bath} for  the numerical coefficient.}:
 \begin{equation} \lambda{EnFlux}
\frac{dE}{dt} =  - \const  \frac{\eta}{J_B^{3/2} \sqrt{J}\beta^2 }
\end{equation}
Differentiating the energy expression  (\ref{EnSYK}) with respect to the time,  and solving a simple differential
equation for $\beta(t)$, we find that it grows exponentially
 \begin{equation}
\beta(t) \sim \beta_{\rm init} \exp \left( \const \frac{\eta \sqrt{J} t}{J_B^{3/2}} \right)
\end{equation}
So that the time $\Delta t_{\rm 2BH}$ needed to go through the two black hole phase and reach $T_{\rm 2BH} \sim \mu^2/J$
depends only logarithmically on $\mu$:
 \begin{equation}
\label{t_bh}
\Delta t_{\rm 2BH} \sim \frac{J_B^{3/2}}{\sqrt{J} \eta} \log \frac{J}{\mu^2 \beta_{\rm init}}
\end{equation}

\subsubsection{Reaching $T_{\rm WH}$}

Now let us calculate time $\Delta t_{\rm WH}$ which is needed to go through the hot wormhole phase and reach $T_{\rm WH}$.
To this end we will employ some results from the end of Section \ref{sec:sch_low} about Schwarzian.
This Schwarzian approximation
breaks for very hot wormholes with temperature of order $T_{\rm 2BH}$, but holds for lower temperatures.

Suppressing the numerical
coefficients, $G_{LL}$ in this regime is given by:
 \begin{equation}
\label{G_LL_hot}
G_{LL}(t) \propto  \frac{\sqrt{{\bf t}'}}{\sqrt{J \sin \left( {\bf t}' (t-i \epsilon) \right)}}
\end{equation}
where ${\bf t}'(t)$ is determined by the solution of eq. (\ref{saddle}). The hot wormhole(unstable branch) is characterized
by having ${\bf t}'\ll 1$, so eq. (\ref{saddle}) can be
simplified by neglecting Schwarzian kinetic term (the first term in eq. (\ref{saddle}))
 \begin{equation}
\label{approx_saddle}
\exp \left( -{\bf t}' \beta/4 \right) \sim \frac{\mu}{{\bf t}'^{1/2}}
\end{equation}
This approximation breaks down near $T \sim T_{\rm WH}$, so we further assume that we use this approximation for temperatures
which are slightly below or of order $T_{\rm WH}$.

From this equation, up to a logarithmic term in $\mu$, ${\bf t}'$ and $\beta$ are related by
$ {\bf t}' \sim  1/\beta $.
Using  (\ref{approx_saddle}), the  energy (\ref{saddle_energy}) can be written as
 \begin{equation}
\label{E_hot}
E - 2E_{\rm 0,SYK} \propto  - \frac{\mu}{\sqrt{J \beta}}
\end{equation}
We see that the heat capacity is negative
 \begin{equation}
C_{\rm hot\ wormhole} = \frac{dE}{dT} = -\beta^2 \frac{dE}{d \beta} \propto  \frac{\mu \sqrt{\beta}}{\sqrt{J}}
\end{equation}
We can compare the absolute value of this expression with the heat capacity of a regular SYK
$C_{\rm SYK} \sim \frac{1}{J \beta}$
 \begin{equation}
\frac{|C_{\rm hot\ wormhole}|}{C_{\rm SYK}} \propto \mu \sqrt{J} \beta^{3/2} \propto \left( { T_{\rm WH} \over T } \right)^{3/2}
\end{equation}
This ratio is much bigger than 1 for $\beta \gg 1/(J^{1/3} \mu^{2/3}) \sim \beta_{\rm WH}$. So apart from the
region close to $T_{\rm WH}$ the hot wormhole has a large  negative heat capacity, compared to a single SYK model at the same temperature.

The energy flux can be computed using the expression (\ref{G_LL_hot}) for $G_{LL}$. The result is
again determined by scale symmetry and it is again proportional to ${\bf t}'^2 \sim 1/\beta^2$ as in \nref{EnFlux}.
However,  because of the big negative heat capacity, the time it takes to go through this region is much longer than  \nref{t_bh}.
Solving for $\beta(t)$ we get
 \begin{equation}
\beta_0^{3/2} - \beta_1^{3/2} = \frac{\eta}{\mu J_B^{3/2}}\left( t_1 - t_0 \right)
\end{equation}
In our case we start from $\beta_0 \sim \beta_{\rm 2 BH}$ and end with $\beta_1 \sim \beta_{\rm WH}$.
Since $\beta_{\rm 2 BH} \gg \beta_{\rm WH}$ the overall time length is mostly determined by the region near $T_{\rm 2BH}$:
 \begin{equation}
\label{t_wh}
\Delta t_{\rm WH} \sim \frac{J^{3/2} J_B^{3/2}}{\eta \mu^2}
\end{equation}

This timescale is much larger than $\Delta t_{\rm WH}$, \nref{t_bh},
 which scaled only logarithmically in $\mu$. Moreover, it is mostly determined
by the region near $T_{\rm 2BH}$, which is where the approximation is breaking down. So \nref{t_wh} should only be viewed as an order of magnitude estimate.

Our  numerical results for $\beta(t)$ on Figures \ref{comparison_full} and \ref{fig:betas}
seem to qualitatively support these conclusions. Notice that, as expected,  the times are inversely proportional to the coupling
to the bath $\eta$.

\section{Two coupled black holes in gravity }
\label{sec:gravity}

The low energy description of the SYK model has some features in common with certain two dimensional theories of gravity.
In this section, we   study a similar problem in a gravitational theory in order to compare to the answers we found above.

We consider a Jackiw-Teitelboim theory of gravity coupled to matter, see
\cite{cft_breaking,Jensen:2016pah,Engelsoy:2016xyb} for details.
This gravity theory describes a two dimensional black hole with an $AdS_2$ geometry. The $AdS_2$ space has a boundary.
We  consider a system containing two such black hole exteriors and we  introduce a coupling for the two dimensional matter fields propagating in the bulk. We assume that have $N$ such matter fields.
Let us say that $\chi$ is a matter field with a certain mass $m$ in the bulk and quantized with Neumann
 boundary conditions so that its dimension is $\Delta$, with $\Delta < 1/2$.
 We couple their boundary values through a term, for each field,
\begin{equation} \lambda{IntGra}
 S =  i  \tilde \mu   \int du \chi_L(u) \chi_R(u)    ~,~~~~~~~~~~    \tilde \mu =      { \mu \over J^{2\Delta } }
\end{equation}
were we imagine that $J^{-1} $ is related to a cutoff in the radial $AdS_2$ direction\footnote{With the $AdS$ metric $ds^2 = (dx^2 + dz^2)/z^2$, this is the cutoff at $z= \epsilon$, and we are defining $J = 1/\epsilon$.   } and $u$ is the physical boundary time.

\subsection{High temperature phase}

We now consider the high temperature phase where in Euclidean space we have two separate disks that are connected through
the  interaction \nref{IntGra}. Concentrating on the matter system, this interaction is easy to analyze because the full matter theory is just quadratic. In principle, we also need to consider the effects of gravity, and we will discuss them later.
This interaction, \nref{IntGra},
 leads to the Feynman diagrams in Figure \ref{fig:diagrams}(a), which can
be easily summed, as we explain below.
 Since the interaction is relevant,   it  becomes important at low temperatures. For sufficiently low temperatures,  the net effect is to change the boundary conditions for the bulk fermions $\chi$ from Neumann  to Dirichlet. Namely, at low temperatures we get   two decoupled disks with Dirichlet boundary conditions for bulk fermions.
We now discuss this more explicitly.

When $\tilde \mu=0$ we have two separate disks and the matter partition function is just given by $Z_N^2$, namely the square of the partition function of a fermion with Neumann boundary conditions.
Starting from this state we can now sum the diagrams in Figure \nref{fig:diagrams}(a). For each fermion field, we get
\begin{equation}
Z_{\mu} = Z_N^2 \exp\left\{ \half Tr[ \log (1 + i \tilde \mu G) (1-i \tilde \mu G) ] \right\} = { Z_N^2 \  \left[ \det (1 + \tilde \mu^2 G_\Delta^2 ) \right]^{\half }  }
\end{equation}
where we think of $G(u_1,u_2) \propto [\sin { u_{1} - u_2 \over 2} ]^{-2\Delta } $ as a matrix with indices $u_1$, $u_2$.
We have set $\beta = 2\pi$ for simplicity and we will restore it later.

For large $\tilde{\mu}$ we find that the partition function gets an additional factor of the determinant of $G_\Delta$.
It turns out that this produces the Dirichlet partition function \cite{KitaevRecent,Maldacena:2019cbz}
\begin{equation} \lambda{NDdet}
Z_{\tilde \mu \gg 1} = Z_N^2   \det G_\Delta    e^{ - \epsilon(\tilde \mu ) \beta } \propto Z_D^2 ~,~~~~~~~~ \epsilon(\tilde \mu ) \propto - ({\tilde \mu   })^{ 1 \over 1 - 2 \Delta }
\end{equation}
 In the last equality we neglected the energy contribution, since we will be focusing on the ground state entropy contributions.

\begin{figure}[ht!]
\centering
\includegraphics[scale=0.8]{./diagrams.pdf}
\minipage{0.8\textwidth}
\centering
\caption{\small (a) Field theory diagrams when we add an interaction term $i\tilde{\mu} \chi_L \chi_R$ at the
boundary of two disks. Here we picture  one disk and the second as the outside. So $\chi^L \chi^L$ propagators are the ones inside the disk and $\chi^R \chi^R$ are outside the disk.   (b) Diagrams in the two coupled  SYK model
that reproduce the field theory diagrams in (a). (c) An example of a diagram in the two coupled
SYK model  that is not present in field theory.}
\label{fig:diagrams}
\endminipage
\end{figure}

The conclusion is that if we start out with two disks with Neumann boundary conditions, after we turn on the relevant perturbation \nref{IntGra}, for very low temperatures we get two decoupled disks again but with Dirichlet boundary conditions. This implies that for very low temperatures,    the bulk fermion would be dual to an operator of dimension $1-\Delta$.

Restoring the factors of $\beta$,  this transition happens  at $\beta_{\rm 2 BH}$ given by
\begin{equation} \lambda{TransiT}
1 \sim  \tilde \mu \beta_{\rm 2 BH}^{ 1 - 2\Delta }
~,~~~~~~~~~~~{\rm or } ~~~~~~~ T_{\rm 2BH} \propto \left( { \mu \over J^{ 2 \Delta } } \right)^{ 1 \over 1 - 2 \Delta }
\end{equation}
For $q=4$ this reproduces \nref{T1_est}.
This is not surprising because we were summing the same type of diagrams.
However, in the gravity case these are all the diagrams, so we can study the whole flow. The new IR fixed point simply corresponds to flipping the boundary conditions to Dirichlet. So nothing too dramatic happens in the gravity solution when we go to
temperatures lower than the  temperature $T_{\rm 2BH}$ in \nref{TransiT}, except that the change in the boundary conditions will change the value of the ground state entropy.

In this discussion,  we have ignored the dynamics of gravity. In principle, we could wonder
whether we should consider non-trivial solutions of the Schwarzian theory.
If we assume that the solution is invariant under translations for each disk, then,
up to gauge symmetries of the Schwarzian theory,  the only solution is the usual one. In Appendix \ref{sch:1loop}, we examine whether  non-constant Schwarzian modes could lower the action. We find that they do not, at least in the approximation we considered.
In our analysis we assumed that gravity is classical,
 which is correct if
$\phi_r/\beta \gg 1$. Here $\phi_r$ is the JT gravity analog of the coefficient of the Schwarzian, the analog of
$N \alpha_S/{\cal J}$. We have also assumed that we have a relatively low number of bulk quantum fields so that the effects of integrating them out does not significantly change the value of $S_0$, the ground state entropy. This is   the regime where the
gravity theory is simplest. As we will discuss below,  the SYK model is different in this respect.

\subsection{Low temperature phase}

At very low temperatures  the coupling \nref{IntGra}  leads to the formation of a wormhole
\cite{MQ}.
This is identical to the small $\mu$ coupled SYK model analysis of
 section \ref{sec:sch_low}, since the effects of gravity can also be described in terms of the
Schwarzian mode.

\iffalse
Just for completeness we will repeat the analysis for general values of $\Delta$.
We find that the equation that sets ${\bf t}'$, the redshift factor between the physical time and the global $AdS_2$ times, is
\begin{equation} \lambda{Constr}
 2 { \alpha_s \over {\cal J }  }  {\bf t}'  - 2   \Delta \tilde \mu {\bf t}'^{ 2 \Delta -1} + \Delta e^{ - \Delta {\bf t}' \beta } =0  ~,~~~~~~~\phi_r = { N \alpha_S \over {\cal J } }
 \end{equation}
 where $N$ is the number of bulk fields. We have written the gravitational coupling $\phi_r$ so that it gives the same answer as the SYK model one.

 In the very low temperature phase we can neglect the last term and, ignoring numerical factors, we get  \begin{equation} \lambda{LowTso}
  {\bf t}'^{ 2 - 2\Delta } \propto { \tilde \mu J } \propto  {   \mu J^{ 1 - 2 \Delta }  }
 \end{equation}
 There is a second solution describing the canonically unstable phase where
  we can neglect the first term and we balance the last two terms to get
 \begin{equation}
 2  \tilde \mu {\bf t}'^{ 2 \Delta -1} \propto   e^{ - \Delta {\bf t}' \beta }
 \end{equation}
 Since the right hand side involves an exponential, we find that
 \begin{equation} \lambda{UnstT}
 {\bf t}' \propto T + {\rm logs } ~,~~~~~~~~~T = 1/\beta
 \end{equation}
 This solution joins with the solution \nref{LowTso} when  \nref{UnstT} crosses \nref{LowTso} or
\begin{equation}
  T_{l}   \propto  ( { \mu J^{ 1 - 2 \Delta } } )^{ 1 \over 2 - 2 \Delta }
 \end{equation}
 Of course, in this regime we need to solve \nref{Constr} with all three terms to get a smooth curve, such as the one shown in Figure
\ref{sch_comp}\footnote{ There is also a third solution of \nref{Constr} with very small ${\bf t}'$ which is outside the regime of validity of \nref{Constr}. }.
\fi

When we decrease the temperature along the negative specific heat region (the hot wormhole phase),    the wormhole is getting longer and longer. Or, more precisely, there is a larger redshift factor between the boundary and the center of the wormhole.
Then,
the interaction, which is a relevant deformation,
becomes stronger. When we considered the problem for the disks, we found that for strong interactions
we get an effective change in boundary conditions from Neumann to Dirichlet.
Here we expect the same phenomenon   when ${\bf t}'$ is becomes
\begin{equation}
 {{\bf t}'}  \propto  T_{\rm 2BH}  \propto \left( { \mu \over J^{2\Delta } } \right)^{ 1 \over 1 - 2 \Delta }
\end{equation}
where ${\bf t}'$ is the variable in \nref{saddle}, which is proportional to the value of the redshift factor at the center of the wormhole.
In other words, ${\bf t}'$ becomes of the order of the temperature $T_{\rm 2BH}$
in \nref{TransiT}. We refer to Appendix \ref{app:flow} for details.
At this value of ${\bf t}'$ the wormhole is so long that the
approximations used in deriving \nref{Constr} are no longer valid.
Interestingly, due to \nref{UnstT},
this happens also at a temperature of the order of $T_{\rm 2BH}$,  which is the temperature where the two disk solutions starts being corrected.
This might appear as a coincidence, but it is not. In the hot wormhole phase we find that the temperature sets the value of
${\bf t}'$ and thus the amount of RG flow that the relevant left-right interaction undergoes. Therefore, this interaction becomes relevant at the same place.

This statement can be further verified by checking whether the hot wormhole thermodynamic curve (red curve in figure \ref{sch_comp})
will join with the two black hole phase (blue curve in figure \ref{sch_comp}) at $T=T_{\rm 2BH}$. In the hot wormhole phase
the energy is given by  (\ref{E_hot}) and in the two black hole phase by
(\ref{delta_E}).
Indeed, the two curves join at $T \sim T_{\rm 2BH}$.

In a gravity theory with a relatively low number of fields, we expect that after $T_{\rm 2BH} $  the wormhole phase might not exist any longer.

One conclusion is that, in a general JT gravity theory plus matter,
 we do {\it not} seem to be able to easily join the   high temperature phase and the hot wormhole phase.
 This is mysterious in the gravity theory because it involves a topology change.
  Of course, the low temperature phase and the hot wormhole  phase are connected smoothly at $T_{\rm WH}$ in a region where we can trust the   wormhole analysis in the Schwarzian approximation, as discussed near \nref{Constr}.

\subsection{Comparison with the SYK model }

In the SYK model, the addition of the interaction corrects the original diagrams by inserting $\mu$ terms in the propagators. If we insert them outside the self energy correction (the $\Sigma$ bubble), then we get diagrams which are identical to the ones we discussed in gravity, see Figure \ref{fig:diagrams}(b). However, in the SYK model we can also
insert $\mu$ inside the self energy corrections, these are new diagrams that are not present in the gravity discussion,
see Figure \nref{fig:diagrams}(c).

 As we mentioned above the region of the phase diagram near temperatures $T\sim T_{\rm 2BH}$ is different in a generic JT gravity theory plus matter than in SYK.  However, we can consider the folloing gravitations model that would look more {\it qualitatively} similar to the SYK model.

First we note that the SYK ground state entropy is given by $S_0 = N s_0$, where $s_0$ is given by \cite{kitaev,KitaevRecent}
\begin{equation} \lambda{EntSYK}
 e^{s_0} =( \det G_{\Delta} )^{-\half } = { Z_N \over Z_D }   ~,~~~~~~~~~ {\rm for ~one~ Majorana~fermion}
 \end{equation}
 where the first equality follows from the usual $G,\Sigma$ action at low energies. The second equality was  mentioned   in \nref{NDdet}.
 This implies that if we want to describe the SYK model in terms of JT gravity, we should think that when the fermions have Dirichlet boundary conditions, the net entropy, or value of the topological terms in the action should be zero, $\phi_0=0$. Then the actual value of the ground state entropy of the usual, single boundary SYK  model, \nref{EntSYK}, is simply given by the contribution of changing the boundary condition for the bulk fields from Dirichlet to Neumann \cite{KitaevRecent}.

 Returning now to the coupled model and starting from the high temperature phase, we see that when we reach the
temperature $T_{\rm 2BH}$ we are changing to a Dirichlet boundary condition. This means that the total $S_0$ now becomes zero, which implies that the topology change is easy.
 Similarly, if we start from the canonically unstable wormhole phase and approach $T_{\rm 2BH}$, we also see a change in the boundary conditions so that $S_0$ again becomes zero and topology change is easy.
 So we can join the two phases with a change in topology at $T_{\rm 2BH}$. In this way we can { \it qualitatively} understand the transition. We have given evidence that this is a smooth transition in the coupled SYK model. What we are discussing here is just a cartoon for a gravity picture of what is happening.

 We also see why SYK is different than a generic JT gravity theory with a smaller number of fields. In such gravity theories the flow from Neuman to Dirichlet would not change $S_0$ by too much and the topology change remains suppressed.
 For this reason we have not been able to see a
general mechanism for the transition that would also work in
more general gravity theories, such as the Standard Model in the presence of magnetically charged black holes as discussed in
 \cite{MMP}.

\section{Conclusion}

In this paper we studied the approach to the ground state of the two coupled SYK models \cite{MQ}.
We first discussed the equilibrium thermodynamics picture.
In the microcannonical ensemble we expect a continuous picture with no phase
transition. As the energy decreases, the temperature decreases up to a value $T_{\rm 2BH}$ where
the system looks like two separate thermal SYK models with a small coupling.
At $T_{\rm 2BH}$ this coupling becomes strong and the system transitions to
a ``hot wormhole" phase with negative specific heat. Now the temperature increases
up to $T_{\rm WH}$ and then the wormhole becomes cooler and shorter, and the specific heat becomes positive again.
This whole picture can be understood using simple analytic approximations,
except for the transition region at $T\sim T_{\rm 2BH}$.

We found that the real time evolution looked as if the system is following the above equilibrium phase diagram. Unfortunately, for the parameters we could use in our numerical computation, we could not trust quantitatively the simple analytic approximations. However, these gave a qualitatively correct answer. The system remained near thermal equilibrium as it cooled down, except for some deviations in the $G_{LR}$ correlator, which we think should disappear if we were to cool more slowly.

The conclusion is that, starting with a generic state of the
two coupled SYK model, we can find the ground state by coupling the system to the bath and cooling it  down. In particular, the system does not get stuck in a metastable state. This provides a feasible way to produce a state close to the TFD.

We also computed the time to form the wormhole. Most of the time is spent near the region with $T \sim T_{\rm 2BH} $.

One of our goals was to extract some general lessons for wormhole formation in gravity. Unfortunately, the SYK model seems to be special, and its special features becomes manifest in the ease by which we can connect the two black hole phase with the hot wormhole phase near $T \sim T_{\rm 2BH}$.
These two phases do not seem to be so easy to connect in more general theories of gravity. We  qualitatively explained  why topology is  simpler in a gravity theory that is similar to the SYK model, but harder in a more general theory of gravity.

Nevertheless we cannot say how hard forming a wormhole would be in a more general theory of gravity, such as the one describing the wormholes in \cite{MMP}. It seems hard, but maybe there is an ``easy'' pathway to form it. It would be interesting to answer this question.

We expect that this article would be relevant for efforts that try to do it using the SYK model, see the proposal in  \cite{Chen_2018}, for example.

\section*{Acknowledgments}
We are indebted to Ahmed Almheiri and Brian Swingle for describing to us how to couple SYK to a bath and numerous discussion about the thermalization in SYK.
Also we are grateful to Ksenia~Bulycheva, Yiming~Chen, Alexander~Gorsky, Luca~Iliesiu, Alex~Kamenev, Alexei~Kitaev, Igor~Klebanov,
Fedor~Popov, Xiaoliang Qi, Nick~Sukhov, Grisha~Tarnopolsky and Zhenbin~Yang for discussions.
J.M. is supported in part by U.S. Department of Energy grant DE-SC0009988 and by the Simons Foundation grant 385600.

redpajama_set_name: RedPajamaArXiv


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These beliefs are typified in a three-pillar approach to student success coaching: academic, life, and career. The diversity of the college student experience remains an amalgam of distinctive personal, academic, and professional challenges, which is why it is so important to think holistically about student success. The coaching approach is driven by questions, questions that get at the heart of the barriers students face in their classes, their lives, and their careers. By focusing on the whole student, institutions can help students proactively navigate their paths to graduation.


If you need an attorney for DUI charges, turn to Michael Gruszeczki and Dustin Smith of Gruszeczki & Smith Law.
Dustin Smith grew up in the southwest suburbs and attended Lincoln- Way High School in Frankfort, Illinois. After graduation he attended the University of Illinois at Urbana/Champaign where he earned a degree in History and a minor in Anthropology. Dustin graduated college in three years and went on to attend the John Marshall Law School in Chicago, Illinois. Dustin began his career as a prosecutor in the city of Chicago. Throughout his career as a prosecutor and as a criminal defense attorney, Dustin has worked on all aspects of felony cases including grand jury indictments, bond hearings, motion practice, bench trials, jury trials, and sentencing hearings. As an experienced criminal defense attorney Dustin has defended clients accused of everything from traffic matters, DUI's, theft, burglaries, armed robbery, murder and many other crimes. Dustin has also helped clients fight civil Orders of Protection as well as domestic battery charges. Dustin's background as a former prosecutor and a criminal defense attorney gives him a unique insight into all types of criminal charges and a better understanding of each client's case.
Mike has extensive experience in trying cases whether in a bench trial or jury trial and litigating issues in motions before a judge. Mike's clients rely on his commitment to their needs and his skills as a trial attorney. He prides himself on his dedication to his clients and obtaining excellent results for them.


Olann and is hosting a #knockeralong at Mad Monk in Midleton on Wednesday, 25 October 7-10pm. Tickets are €20 each and includes a special goody bag. This event is in aid of Breast Cancer Ireland.
Thousands of women each year are diagnosed with breast cancer and have to undergo a mastectomy. Knitted Knockers is a wonderful movement of generosity and compassion that gives these women a viable alternative to silicone prosthesis.
You can download the knitting pattern for C Cup Size Knitted Knockers.
To learn more about the event, please visit the Olann and official web page.
The Knitted Knockers Ireland Facebook page is also a great resource.
Big thanks for spreading the word.
The Knitted Knockers Campaign is a great example of charity knitting and crochet at it's best. The women I've spoken to all swear by the knitted prosthesis and rate them above and beyond the medical ones in terms of comfort and appearance.
Brilliant and glad to spread the word!


text:
https://rstv.nic.in/rstvweblogin?action=logout&redirect_to=https%3A%2F%2Frstv.nic.in%2F&_wpnonce=a2e73a6e64
NNationalNNewsSSports
National / India to host 2022 AFC Women's Asian Cup
India to host 2022 AFC Women's Asian Cup
5:24 PM, 5 Jun 2020
Women's World Cup in 2020
The Asian Football Confederation has awarded the hosting rights of the 2022 Women's Asian Cup to India for the first time since 1979.
The decision was taken at the AFC Women's Football Committee meeting. In February, the AFC Women's Football Committee had recommended India to be hosts.
In a letter to the All India Football Federation, Dato Windsor John, General Secretary, AFC wrote: "The Committee awarded the hosting rights of the AFC Women's Asian Cup 2022 Finals to the All India Football Federation."
The tournament will likely be held in the second half of the year. In the 1979 edition, India had finished as runners-up.
"I need to thank the Asian Football Confederation for finding us suitable to host the AFC Women's Asian Cup in 2022," AIFF President, Praful Patel said.
"The tournament will galvanise the aspiring women players and bring in a holistic social revolution as far as women's football in the country is concerned," he added.
The tournament will feature 12 teams, expanded from the previous slot of eight teams.
India qualify directly as hosts. The event will also serve as the final qualification tournament for the 2023 FIFA Women's World Cup.
The tournament comes as the latest boost to AIFF as India are slated to host the FIFA U-17 Women's World Cup next year.
India had also hosted the AFC U-16 Championships in 2016 and the FIFA U-17 World Cup in 2017.
"The tournament will play a huge role in further popularising women's football in India. The Women's Asian Cup 2022 comes on the back of the FIFA U-17 Women's World Cup India 2020, and will help us sustain, and build on the momentum gained," AIFF General Secretary, Kushal Das said.
Asian Football Confederation, FIFA, FIFA Women's World Cup

redpajama_set_name: RedPajamaCommonCrawl


I so appreciate your visit, Elizabeth.
Don't remember seeing this one, so I'm glad you reposted! 🙂 Love that graphic at the top!
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Wonderful post Isadora! The September poem is wonderful! The song is absolutely beautiful!
Hooray for a lovely positive September!
Ah… Isadora! What a wonderful post for me to end my day with! Just what the doctor ordered, methinks. I've been (ahem) a little less than serene today, so I can now drift off and dream of serenity. Lovely. I like your September poem too… also very apt!


As a physician, pathologist, and one time sufferer, I know both professionally and personally the impact of the flu.
On a personal note, I used to tell people that I wasn't sure if I had ever had the flu. Well, let me tell you, after I contracted it last year, all doubt was removed. I had never experienced anything like that. In fact, I don't know a single soul who, after having the flu, is not one of the first people in line the next year to get the vaccination.
From a pathologist perspective, I am updated frequently on the spread of the flu, effectiveness of the current year's vaccine, and public health alerts. In the laboratory, we utilize several different test methods to determine if a patient's symptoms are due to influenza infection. My own hands have also touched the potentially devastating outcome of the virus. I once had the sad task of performing an autopsy on a previously healthy, beautiful, curly-headed toddler, who succumbed to the flu.
The flu is much more than a bad cold. It is a debilitating, potentially life threatening virus that is contracted by up to 35 million Americans a year. The Centers for Disease Control and Prevention (CDC) estimates that among those who contract the flu, up to 170,000 will be hospitalized and nearly 56,000 people will die from the flu each year.
Although everyone is susceptible to the flu, there are certain people with chronic diseases that are more likely to develop complications from the flu. These include people with diabetes, heart disease, neurologic disorders, chronic lung disease, or who have a weakened immune system from factors like HIV or chemotherapy. Additionally, the very young (less than 5 and especially less than 2 years old) and seniors (65 or older) have a higher rate of complications from the flu. Pregnant women and people living in crowded spaces, such as a nursing home or long-term care facility, are also known to experience more potential health complications from the flu.
The current flu season is one of the worst in several years. Here are four ways you can flee the flu and stay healthy.
Get the Flu Vaccination. It is not too late. Contrary to folklore, getting the flu vaccination will not give you the flu. The most common side effects include pain and swelling at the injection site, low grade fever, headache, and muscle aches. These effects can be lessened by taking an anti-inflammatory of your choice immediately prior to getting the flu shot and as indicated while symptoms persist. Unless advised by a medical professional, everyone should get a flu vaccination yearly. The groups of people who are more susceptible to complications should be especially proactive about this this life-saving vaccination. Although the flu shot is not 100 percent effective (I, myself, contracted the flu after having the vaccine last year), the benefits of the vaccine far outweigh the risks.
Wash Your Hands Frequently. The flu is spread by droplets from infected people. Sneezing, coughing, and blowing your nose releases these droplets. These are easily spread on public surfaces like doorknobs, handrails, and elevator buttons. So, you may not be in direct contact with someone who is ill, but you may be passing through places they have touched. People who have the flu become contagious even before symptoms arise. Consequently, they may be out and about enjoying the same places you are. To help stay well, keep your hands away from your mouth, eyes and nose unless they are freshly washed. Frequent hand washing with soap and warm water, especially in public places, will go a long way to keeping you safe from all cold viruses and particularly from the flu during this season. If you are unable to wash, keep hand sanitizer available and use it frequently and liberally.
Stay Away from Sick People. It goes without saying that your risk of contracting the flu increases exponentially when you remain in the company of a sick individual. Do your best to decrease contact with those who have the flu and their excretions. If you must help nurse someone back to health, be sure to minimize contact and employ vigilant hand-washing protocol. Don't share glasses, utensils, pillow cases, or cell phones. Use a disinfectant on hard surfaces in the area to clear the virus. If you happen to be the sick individual, do us all a favor and stay home while you recuperate.
Know When to Seek Help. Although there is overlap between cold and flu symptoms, it is imperative to know the differences. In addition to the cough and congestion of a cold, flu symptoms include fevers, gastrointestinal distress, and body aches. In contrast to a cold, flu symptoms have a rapid onset. If you believe you may have contracted the flu, see your health care provider as soon as possible. A simple rapid influenza diagnostic laboratory test can determine if you have the flu and which strain it is in as little as 10-15 minutes. Your health care provider can prescribe antiviral medications that can help decrease the length and severity of your symptoms and decrease the chance life threatening flu complications.
Flu season lasts for several more months. Make sure to employ these four steps to keep you and your loved ones flu free this season. An ounce of prevention is worth a pound of cure and a rapid response may save you weeks of potential illness.
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You can follow Crystal Moore, MD, PhD, FCAP at www.DrCrystalMoore.com, or on social media (@DrCrystalAMoore) for more health information and to receive a Prescription For Life (#RxForLife) to maximize your wellness in body, mind, soul, and spirit. Stay tuned for her soon to be released book, Fast Forward Through The Valley: A 40-Day Devotional to Victory.
Dr. Moore is a native of the Hampton Roads area of Virginia, where she resides with her two teenage sons.
Dr. Moore professed, even as a young child, that she wanted to be a physician. For her, medicine is not just a profession, but also a calling. Following that calling led her to pursue a dual doctorate, physician-scientist, MD/PhD degree at the Medical College of Virginia. Her Ph.D. was awarded in the Department of Biochemistry and Molecular Biophysics. She completed her residency training in Anatomic and Clinical Pathology at Duke University and is a board-certified Fellow of the College of American Pathologists.
As a sought after speaker, Dr. Moore delivers complex information in a highly understandable and remarkably humorous manner. She speaks to the total person and her message heals the body, mind, soul, and spirit. Dr. Moore believes and conveys that health is wealth-not to be taken for granted, but to be actively pursued. Accordingly, she guides individuals and audiences on a journey to wellness and abundant life.