# More on Complex Sachdev-Ye-Kitaev Eternal Wormholes

Pengfei Zhang  California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

In this work, we study a generalization of the coupled Sachdev-Ye-Kitaev (SYK)
model with $U(1)$ charge conservations. The model contains two copies of the
complex SYK model at different chemical potentials, coupled by a direct
hopping term. In the zero-temperature and small coupling limit with small
averaged chemical potential, the ground state is an eternal wormhole
connecting two sides, with a specific charge $Q=0$, which is equivalent to a
thermofield double state. We derive the conformal Green’s functions and
determine corresponding IR parameters. At higher chemical potential, the
system transit into the black hole phase. We further derive the Schwarzian
effective action and study its quench dynamics. Finally, we compare numerical
results with the analytical predictions.

###### Contents

  1. 1 Introduction
  2. 2 Model and its Low-energy Description
    1. 2.1 Conformal Solution
    2. 2.2 the TFD State Viewpoint
    3. 2.3 Effective Action and Thermodynamics
    4. 2.4 Quench Dynamics
  3. 3 Numerical Results
    1. 3.1 Imaginary time results and the Phase diagram
    2. 3.2 Real time results and Quench dynamics
  4. 4 Summary

##  1 Introduction

Proposed by Kitaev [1] and related to the early work of Sachdev and Ye [2],
the Sachdev-Ye-Kitaev (SYK) model [3, 4] describes $N$ Majorana fermions with
random $q$ body interactions. The SYK model can be solved in terms of the
$1/N$ expansion. In the low-temperature limit, the model shows emergent
conformal symmetry, and the scaling dimension of the Majorana operators is
$1/q$. Such conformal symmetry is explicitly broken in the Lagrangian, which
leads to a finite Schwarzian action for reparametrization modes [3, 4].
Interestingly, such Schwarzian action matches the boundary action of the
Jackiw-Teitelboim gravity in two-dimensional nearly anti-de Sitter (AdS2)
spacetime, and consequently, the SYK model is a concrete model with low-energy
holographic description. Later, the model is generalized to study various
problems in both high energy and condensed matter communities [5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].

From the gravity perspective, a generalization by Maldancena and Qi [5] is of
special interest. They consider coupling two statistically correlated copies
of the SYK model by a bilinear term, which is similar to the early proposal of
traversable wormholes by Gao, Jafferis and Wall [23]. Consequently, for small
coupling strength, the ground state is dual to an eternal traversable wormhole
in the global AdS2 spacetime with
$ds^{2}=(-dt^{2}+d\sigma^{2})/\sin^{2}\sigma$, where two copies of the SYK
model lie on two boundaries near $\sigma=0$ and $\sigma=\pi$. Such a state can
also be prepared as a thermofield double (TFD) state of the SYK model. At high
temperatures, the coupled SYK model transits into the black hole phase, where
two copies become disconnected geometrically. The finite temperature spectral
of this coupled SYK model has been studied in [24, 25], and the real-time
formation of the wormhole is discussed in [26]. Experimentally, the coupled
SYK is found to be related to random spin models [27], and there is also a
proposal for realizing the MQ model [28] in solid-state materials. The
tunneling spectroscopy [29] and the entanglement entropy [30] of the coupled
SYK model has also been studied.

In this work, we study the effect of additional conservation law on the
eternal traversable wormholes in the SYK-like models. To be concrete, we focus
on the complex fermion version of the coupled SYK model with $U(1)$ symmetry.
For a single SYK model, the generalization to complex fermions where the
charge is conserved has been studied in details [6, 7, 8]. There are also
recent studies [31, 32, 33] on the complex version of the coupled SYK model.
However, in these works, the chemical potential for two copies is fixed to be
the same, which, as we will see, leads to the same wormhole state as no
chemical potential term in the low-energy limit. We instead consider the case
where the chemical potential of both systems can be tuned freely. In the low-
energy limit, we find a family of conformal solutions for wormholes
parameterized by three IR parameters $t^{\prime}$, $\mathcal{P}$ and
$\mathcal{E}$, which comes from the two chemical potentials and one coupling
strength between two copies. At higher chemical potential, the conformal
solution breaks down, and the system transits into black hole solutions. One
can also derive the effective Schwarzian theory by requiring the
diffeomorphism-invariance [4, 8], where one could can the energy level of
boundary gravitons and the quench dynamics.

The paper is organized as follows: In section 2, we first describe our model
and conformal solution, where the IR parameters can be fixed through the TFD
State perspective, together with minimizing the effective action. In section
3, we compare our low-energy solutions to the numerics, in both imaginary-time
and real-time approach. We finally summarize our results in 4.

##  2 Model and its Low-energy Description

In this section, we present the generalized coupled SYK model with $U(1)$
symmetry and derive its low-energy behaviors. We consider two copies of the
SYK4 model with the same random interaction parameters:

$\displaystyle H=$
$\displaystyle\frac{1}{4}\sum_{ijkl}J_{ijkl}c^{\dagger}_{L,i}c^{\dagger}_{L,j}c_{L,k}c_{L,l}+\frac{1}{4}\sum_{ijkl}J_{ijkl}c^{\dagger}_{R,i}c^{\dagger}_{R,j}c_{R,k}c_{R,l}$
(1)
$\displaystyle-\mu_{L}\sum_{i}\left(c^{\dagger}_{L,i}c_{L,i}-\frac{1}{2}\right)-\mu_{R}\sum_{i}\left(c^{\dagger}_{R,i}c_{R,i}-\frac{1}{2}\right)-\mu_{c}\sum_{i}\left(c^{\dagger}_{R,i}c_{L,i}+c^{\dagger}_{L,i}c_{R,i}\right).$

Here the random coefficients $J_{ijkl}$ take distribution

$\overline{J_{ijkl}}=0,\ \ \ \ \ \ \ \ \ \ \ \
\overline{|J_{ijkl}|^{2}}=2J^{2}/N^{3}.$ (2)

We have added different chemical potential term $\mu_{L}/\mu_{R}$ for the
left/right copy of the complex SYK model, with an additional directly hopping
term $\mu_{c}$. Without loss of generality, we have set the hopping amplitude
to be real. For later convenience, we introduce the averaged $\bar{\mu}$ and
the relative chemical potential $\mu$ as

$\frac{\mu_{L}-\mu_{R}}{2}=\mu,\ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{\mu_{L}+\mu_{R}}{2}=\overline{\mu}.$ (3)

Before analyzing the model, we add a few comments:

  1. 1.

Although we introduced the model with two copies of the complex SYK4 system
for simplicity, it is straightforward to generalize all discussions in the
paper for other even finite $q\leq 4$. Generally, the random interaction
strength of the right copy should contain an additional factor of
$(-1)^{q/2}$, as in the Majorana case [5]. For non-zero chemical potential,
the large-$q$ limit becomes problematic, similar to the single SYK case 111We
thank Yingfei Gu for the discussion of the large-$q$ limit of a single complex
SYK model..

  2. 2.

Without the hopping term, the model contains two copies of the complex SYK
model, and the symmetry is $U(1)\times U(1)$, which is broken to a single
$U(1)$ by $\mu_{c}$. The idea why here we still introduce two different
chemical potential terms is that for eternal wormholes, we are focusing on the
limit of small $\mu_{c}\rightarrow 0$. In this limit, the breaking of the
relative phase transformation $c_{L,i}\rightarrow c_{L,i}e^{i\alpha},\
c_{R,i}\rightarrow c_{R,i}e^{-i\alpha}$ is weak and there exists a
corresponding pseudo Goldstone mode. Therefore we expect a relative chemical
potential can play important roles in the low-energy limit.

  3. 3.

Such a model can be understood as certain spinful SYK model or large-$M$ spin
models [27], where $L/R$ can be understood as different spin components
$\uparrow$ and $\downarrow$. In such models, the relative chemical potential
$\mu$ corresponds to a magnetic field in the $z$ direction, while the coupling
term $\mu_{c}$ represents a magnetic field in the $x$ direction. By a
particle-hole mapping, this can also be understood as a spinful SYK model with
a proximity pairing term.

###  2.1 Conformal Solution

We consider the model in the imaginary-time formalism. We focus on the
large-$N$ limit. Under the replica diagonal assumption, after introducing the
bilocal fields $G(\tau,\tau^{\prime})$, $\Sigma(\tau,\tau^{\prime})$ and
integrating out fermions, the partition function becomes:

$\displaystyle\frac{S[G,\Sigma]}{N}=$
$\displaystyle-\text{Tr}\left[\log(\sigma_{\alpha\beta}-\Sigma_{\alpha\beta})\right]-\int
d\tau
d\tau^{\prime}\left[G_{\alpha\beta}(\tau,\tau^{\prime})\Sigma_{\beta\alpha}(\tau^{\prime},\tau)+\frac{J^{2}}{4}G_{\alpha\beta}^{2}(\tau,\tau^{\prime})G_{\beta\alpha}^{2}(\tau^{\prime},\tau)\right]$
(4) $\displaystyle=$
$\displaystyle-\text{Tr}\left[\log(-\tilde{\Sigma}_{\alpha\beta})\right]-\int
d\tau
d\tau^{\prime}\left[G_{\alpha\beta}(\tau,\tau^{\prime})\tilde{\Sigma}_{\beta\alpha}(\tau^{\prime},\tau)+\frac{J^{2}}{4}G_{\alpha\beta}^{2}(\tau,\tau^{\prime})G_{\beta\alpha}^{2}(\tau^{\prime},\tau)\right]$
$\displaystyle-\int d\tau d\tau^{\prime}\
G_{\alpha\beta}(\tau,\tau^{\prime})\sigma_{\beta\alpha}(\tau^{\prime},\tau).$

Here $\alpha,\beta=L,R$ labels two copies of the SYK model,
$G_{\alpha\beta}(\tau,\tau^{\prime})=\sum_{i}c_{\alpha,i}(\tau)\bar{c}_{\beta,i}(\tau^{\prime})/N$
is the fluctuation of fermion bilinear. We have defined
$\Sigma=\tilde{\Sigma}+\sigma$, with the UV perturbation [4, 8]

$\sigma(\tau,\tau^{\prime})=\begin{pmatrix}\delta^{\prime}(\tau-\tau^{\prime})-\mu_{L}\delta(\tau-\tau^{\prime})&-\mu_{c}\delta(\tau-\tau^{\prime})\\\
-\mu_{c}\delta(\tau-\tau^{\prime})&\delta^{\prime}(\tau-\tau^{\prime})-\mu_{R}\delta(\tau-\tau^{\prime})\end{pmatrix}.$
(5)

Without $\sigma$, there is an emergent conformal symmetry of bilocal fields:

$\displaystyle G_{\alpha\beta}(\tau,\tau^{\prime})$
$\displaystyle\rightarrow\varphi_{\alpha}^{\prime}(\tau)^{\frac{1}{4}}\varphi_{\beta}^{\prime}(\tau^{\prime})^{\frac{1}{4}}e^{i\lambda_{\alpha}(\tau)-i\lambda_{\beta}(\tau^{\prime})}G_{\alpha\beta}\left(\varphi_{\alpha}(\tau),\varphi_{\beta}(\tau^{\prime})\right),$
(6) $\displaystyle\Sigma_{\alpha\beta}(\tau,\tau^{\prime})$
$\displaystyle\rightarrow\varphi_{\alpha}^{\prime}(\tau)^{\frac{3}{4}}\varphi_{\beta}^{\prime}(\tau^{\prime})^{\frac{3}{4}}e^{i\lambda_{\alpha}(\tau)-i\lambda_{\beta}(\tau^{\prime})}\Sigma_{\alpha\beta}\left(\varphi_{\alpha}(\tau),\varphi_{\beta}(\tau^{\prime})\right),$

the consequence of this emergent symmetry will be studied later.

Now, we take the saddle-point equations of (4) for bilocal fields. It
corresponds to the Schwinger-Dyson equation of fermions:

$\left(G^{-1}(\omega)\right)_{\alpha\beta}=-\tilde{\Sigma}_{\alpha\beta}(\omega)=\sigma_{\alpha\beta}(\omega)-\Sigma_{\alpha\beta}(\omega),\
\ \ \ \ \
\Sigma_{\alpha\beta}(\tau)=-J^{2}G_{\alpha\beta}(\tau)^{2}G_{\beta\alpha}(-\tau).$
(7)

In the low-energy limit $J|\tau|\gg 1$, we hope to neglect $\sigma$ and
consider the zero-temperature solution of

$\sum_{\gamma}\int_{-\infty}^{\infty}d\tau^{\prime\prime}G_{\alpha\gamma}(\tau-\tau^{\prime\prime})J^{2}G_{\gamma\beta}(\tau^{\prime\prime}-\tau^{\prime})^{2}G_{\beta\gamma}(\tau^{\prime}-\tau^{\prime\prime})=\delta_{\alpha\beta}\delta(\tau-\tau^{\prime}).$
(8)

The validity of (8) needs further explanations: When $\mu_{c}=0$ and
$\bar{\mu}=0$, the system represents decoupled complex SYK models with
opposite chemical potentials $\pm\mu$ with the scaling dimension of fermion
operators $1/4$. Consequently, the perturbation from left-right coupling is
relevant and the system should be gapped. Consequently, to neglect the UV
symmetry breaking term $\sigma$, we should require the solution of (8) leads
to a gap much larger than the bare gap $\mu_{c}$. As we will see later, this
requires $\bar{\mu}/J\ll 1$ and $\mu_{c}/J\ll 1$ [5], while $\mu/J$ can be
$O(1)$.

Interestingly, there exists a family of conformal solutions with three IR
parameters $\mathcal{E}$, $\mathcal{P}$ and $t^{\prime}$. This is what we
expect since there are three UV parameters $\mu_{L}$, $\mu_{R}$, and $\mu_{c}$
in $\sigma$. Explicitly, we have

$\displaystyle G_{LL}(\tau)$
$\displaystyle=b\times\text{sgn}(\tau)e^{\pi\mathcal{E}\text{sgn}(\tau)+\mathcal{P}\tau}\left(\frac{t^{\prime}}{2\sinh(t^{\prime}|\tau|/2)}\right)^{1/2},$
(9) $\displaystyle G_{RR}(\tau)$
$\displaystyle=b\times\text{sgn}(\tau)e^{-\pi\mathcal{E}\text{sgn}(\tau)+\mathcal{P}\tau}\left(\frac{t^{\prime}}{2\sinh(t^{\prime}|\tau|/2)}\right)^{1/2},$
$\displaystyle G_{LR}(\tau)$ $\displaystyle=G_{RL}(\tau)=-b\times
e^{\mathcal{P}\tau}\left(\frac{t^{\prime}}{2\cosh(t^{\prime}|\tau|/2)}\right)^{1/2}.$

Here the overall coefficient $b$ satisfies $4\pi
J^{2}\cosh(2\pi\mathcal{E})b^{4}=1$. $G_{LR}(\tau)=G_{RL}(\tau)$ is a
consequence of the time-reversal symmetry. To make the Green’s function well-
defined as $|\tau|\rightarrow\infty$, we require that
$\mathcal{P}<t^{\prime}/4$. In the following discussion, we will always
neglect terms $\sim t^{\prime 2}$. For $|\tau|t^{\prime}\ll 1$, $G_{LL}$ and
$G_{RR}$ reduce to the zero-temperature Green’s function of decoupled complex
SYK model with asymmetry parameters $\pm\mathcal{E}$ [7, 8]:

$G_{LL}(\tau)=-G_{RR}(-\tau)=be^{\pi\mathcal{E}\text{sgn}(\tau)}\frac{\text{sgn}(\tau)}{|\tau|^{1/2}}.$
(10)

On the contrary, when both $\mu=0$ and $\bar{\mu}=0$, we should get back to
the Majorana wormhole solution [5], which corresponds to
$\mathcal{E}=\mathcal{P}=0$. When we turn on $\mu$ with $\bar{\mu}=0$, the
ensemble of the system is still invariant under a particle-hole transformation
$c_{L,i}\rightarrow c^{\dagger}_{R,i}$. This is compatible with having a non-
vanishing $\mathcal{E}$. We also have the gap of the fermionic excitations
$E_{g}=t^{\prime}/4$. On the other hand, the existence of $\mathcal{P}$ breaks
this particle-hole symmetry, which corresponds to the existence of
$\bar{\mu}$. The relation between the IR parameters and the UV parameters or
physical observables are discussed in the next few subsections, while the
results are summarized in the Table 1.

IR parameters | $t^{\prime}$ | $\mathcal{P}$ | $\mathcal{E}$
---|---|---|---
Relations | $t^{\prime}=\frac{b^{2/3}J^{2/3}\mu_{c}^{2/3}}{2\alpha_{S}^{2/3}}$ | $\mathcal{P}=\bar{\mu}$ | $Q_{L}=-\frac{\theta}{\pi}-\frac{1}{4}\sin(2\theta),\ e^{2\pi\mathcal{E}}=\tan(\pi/4+\theta)$
Table 1: A summary of relations between the IR parameters and UV
parameters/physical observables in the eternal wormhole phase.

In frequency space, (9) becomes

$\displaystyle G_{LL}(\omega+i\mathcal{P})$
$\displaystyle=b\times\frac{2\sqrt{2}\pi^{3/2}\text{sech}(2\pi\omega/t^{\prime})(i\sinh(\pi\omega/t^{\prime})\cosh(\pi\mathcal{E})+\cosh(\pi\omega/t^{\prime})\sinh(\pi\mathcal{E}))}{\Gamma\left(\frac{3}{4}-i\frac{\omega}{t^{\prime}}\right)\Gamma\left(i\frac{\omega}{t^{\prime}}+\frac{3}{4}\right)},$
(11) $\displaystyle G_{RR}(\omega+i\mathcal{P})$
$\displaystyle=b\times\frac{2\sqrt{2}\pi^{3/2}\text{sech}(2\pi\omega/t^{\prime})(i\sinh(\pi\omega/t^{\prime})\cosh(\pi\mathcal{E})-\cosh(\pi\omega/t^{\prime})\sinh(\pi\mathcal{E}))}{\Gamma\left(\frac{3}{4}-i\frac{\omega}{t^{\prime}}\right)\Gamma\left(i\frac{\omega}{t^{\prime}}+\frac{3}{4}\right)},$
$\displaystyle G_{LR}(\omega+i\mathcal{P})$
$\displaystyle=G_{RL}(\omega+i\mathcal{P})=-b\times\frac{\Gamma\left(\frac{1}{4}\right)\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{1}{4}-i\frac{\omega}{t^{\prime}}\right)\Gamma\left(i\frac{\omega}{t^{\prime}}+\frac{1}{4}\right)}{\sqrt{2t^{\prime}}\pi^{3/2}}.$

Here the $G_{\alpha\beta}$ is analytic on the stripe with
$|\text{Im}(\omega+i\mathcal{P})|<t^{\prime}/4$, as anticipated from the real-
time expressions. There is a similar expression for the self energy
$\Sigma_{\alpha\beta}$, and it is straightforward to check they satisfies the
Schwinger-Dyson equation (8).

Here we present another method which allow us to derive (9) from the single
SYK conformal solution (10). For conciseness, let us introduce
$G_{0}(\tau,\tau^{\prime})=b\
{\text{sgn}(\tau-\tau^{\prime})}/{|\tau-\tau^{\prime}|^{1/2}}$ and
$\Sigma_{0}(\tau,\tau^{\prime})=b^{3}\
{\text{sgn}(\tau-\tau^{\prime})}/{|\tau-\tau^{\prime}|^{3/2}}$. It is known
that (10) satisfies the self-consistent equation [7]:

$-J^{2}\int^{\infty}_{-\infty}dt_{3}\
e^{\pi\mathcal{E}\left(\text{sgn}(t_{13})+\text{sgn}(t_{32})\right)}G_{0}(t_{1},t_{3})\Sigma_{0}(t_{3},t_{2})=\delta(t_{1}-t_{2}).$
(12)

Here $\tau_{ij}=\tau_{i}-\tau_{j}$. Now we split the integral of $t_{3}$ into
$(-\infty,0)$ and $(0,\infty)$, which gives

$-J^{2}\left(\int^{0}_{-\infty}dt_{3}+\int^{\infty}_{0}dt_{3}\right)e^{\pi\mathcal{E}\left(\text{sgn}(t_{13})+\text{sgn}(t_{32})\right)}G_{0}(t_{1},t_{3})\Sigma_{0}(t_{3},t_{2})=\delta(t_{1}-t_{2}).$
(13)

As an example, we consider $t_{1}>0$ and $t_{2}>0$. We define new time
variable $\tau$ as

$t=\left\\{\begin{aligned} e^{t^{\prime}\tau}&\ \ \ \ t>0,\\\
-e^{t^{\prime}\tau}&\ \ \ \ t<0.\end{aligned}\right.$ (14)

Here $\tau\in(-\infty,\infty)$. This change of variable can be motivated from
the gravity picture, where one can map the Poincare plane to the gAdS2 stripe.
This gives

$\displaystyle\delta(\tau_{1}-\tau_{2})$
$\displaystyle\left|\frac{d\tau_{1}}{dt_{1}}\right|^{1/4}\left|\frac{d\tau_{2}}{dt_{2}}\right|^{3/4}=$
(15)
$\displaystyle-J^{2}\int^{\infty}_{-\infty}d\tau_{3}\left|\frac{dt_{3}}{d\tau_{3}}\right|e^{\pi\mathcal{E}\text{sgn}(\tau_{13})}G_{0}(e^{t^{\prime}\tau_{1}},e^{t^{\prime}\tau_{3}})e^{\pi\mathcal{E}\text{sgn}(\tau_{32})}\Sigma_{0}(e^{t^{\prime}\tau_{3}},e^{t^{\prime}\tau_{2}})$
$\displaystyle-J^{2}\int^{\infty}_{-\infty}d\tau_{3}\left|\frac{dt_{3}}{d\tau_{3}}\right|G_{0}(e^{t^{\prime}\tau_{1}},-e^{t^{\prime}\tau_{3}})\Sigma_{0}(-e^{t^{\prime}\tau_{3}},e^{t^{\prime}\tau_{2}}).$

Inversing the factors on the L.H.S., we find

$\displaystyle\delta$ $\displaystyle(\tau_{1}-\tau_{2})=$ (16)
$\displaystyle-J^{2}\int^{\infty}_{-\infty}d\tau_{3}\underbrace{e^{\pi\mathcal{E}\text{sgn}(\tau_{13})+\mathcal{P}\tau_{13}}\left|\frac{dt_{1}}{d\tau_{1}}\frac{dt_{3}}{d\tau_{3}}\right|^{1/4}G_{0}(e^{t^{\prime}\tau_{1}},e^{t^{\prime}\tau_{3}})}_{G_{LL}}\underbrace{e^{\pi\mathcal{E}\text{sgn}(\tau_{32})+\mathcal{P}\tau_{32}}\left|\frac{dt_{3}}{d\tau_{3}}\frac{dt_{2}}{d\tau_{2}}\right|^{3/4}\Sigma_{0}(e^{t^{\prime}\tau_{3}},e^{t^{\prime}\tau_{2}})}_{\Sigma_{LL}}$
$\displaystyle-J^{2}\int^{\infty}_{-\infty}d\tau_{3}\underbrace{e^{\mathcal{P}\tau_{13}}\left|\frac{dt_{1}}{d\tau_{1}}\frac{dt_{3}}{d\tau_{3}}\right|^{1/4}G_{0}(e^{t^{\prime}\tau_{1}},-e^{t^{\prime}\tau_{3}})}_{-G_{LR}}\underbrace{e^{\mathcal{P}\tau_{32}}\left|\frac{dt_{3}}{d\tau_{3}}\frac{dt_{2}}{d\tau_{2}}\right|^{3/4}\Sigma_{0}(-e^{t^{\prime}\tau_{3}},e^{t^{\prime}\tau_{2}})}_{-\Sigma_{RL}},$

where we have used the (imaginary version of) $U(1)$ conformal symmetry (6) to
introduce $\mathcal{P}$, and identified the $G_{\alpha\beta}$ and
$\Sigma_{\alpha\beta}$, which match the definition (9). This equation is
exactly (8) with $\alpha=\beta=L$. Other components correspond to taking
$t_{1}$ and $t_{2}$ in different regions ($t_{1}\lessgtr 0$ and $t_{2}\lessgtr
0$).

With the conformal solution, we could also determine the conserved $U(1)$
charge, as for a single complex SYK model [8]. We define the charge per mode
$Q$ as

$Q=Q_{L}+Q_{R}=\left<c^{\dagger}_{L,i}c_{L,i}-\frac{1}{2}\right>+\left<c^{\dagger}_{R,i}c_{R,i}-\frac{1}{2}\right>=-G_{LL}(0^{-})-G_{RR}(0^{-})-1.$
(17)

Using the fact that $G_{\alpha\alpha}(0^{+})-G_{\alpha\alpha}(0^{-})=1$, we
have

$Q=-\frac{\text{tr}[G(0^{+})+G(0^{-})]}{2}=\int d\tau\
\text{tr}\left[\sigma(\tau)G(-\tau)\right]\tau=-\int^{\infty}_{-\infty}d\tau\
\text{tr}\left[\tilde{\Sigma}(\tau)G(-\tau)\right]\tau.$ (18)

This can also be derived by applying Noether’s theorem of the $U(1)$ symmetry
for the action (4). Note that different from the single complex SYK case, the
integral converges absolutely in the long time limit, and there is no need to
introduce additional regulators.

To work out the result, we change to the frequency domain:

$Q=-\frac{1}{2\pi i}\int^{\infty}_{-\infty}d\omega\
\partial_{\omega}\text{tr}\left[\log G(\omega)\right]=-\frac{1}{2\pi
i}\left(\int_{|\omega|<\Lambda}d\omega+\int_{|\omega|>\Lambda}d\omega\right)\partial_{\omega}\text{tr}\left[\log
G(\omega)\right],$ (19)

where we cut the integral into two parts with $\Lambda\sim J\gg t^{\prime}$.
In the second part of the integral, we are considering the frequency regime
where the Green’s function takes the same form as a single complex SYK model.
Consequently, the result matches the corresponding calculation of the single
complex model [8]:

$-\frac{1}{2\pi i}\int_{|\omega|>\Lambda}d\omega\
\partial_{\omega}\text{tr}\left[\log G(\omega)\right]=\frac{1}{2\pi
i}(-2i\theta+2i\theta)=0,$ (20)

where $\theta$ is related to $\mathcal{E}$ as
$e^{2\pi\mathcal{E}}=\tan(\pi/4+\theta)$. For the integral over small
frequency, we could directly use conformal solutions (11):

$-\frac{1}{2\pi i}\int_{-\Lambda}^{\Lambda}d\omega\
\partial_{\omega}\text{tr}\left[\log G(\omega)\right]=-\frac{1}{2\pi
i}\int_{-\Lambda}^{\Lambda}dz\ \partial_{z}\text{tr}\left[\log
G(z+i\mathcal{P})\right]+O((\mathcal{P}/J)^{2})\approx 0.$ (21)

We have deformed the contour to absorb the presence of $\mathcal{P}$, and
after the deformation, the integral becomes zero. We also neglect high-order
terms in $\mathcal{P}/J$ (or $t^{\prime}/J$). Finally, we find the specific
charge $Q=0$ for any valid wormhole solution (9). This is consistent with the
TFD perspective discussed in the next subsection.

###  2.2 the TFD State Viewpoint

Before the introduction of the SYK model, there is already a study on the
consequence of coupling two 1+1-D CFTs with different chiralities [34], where
the ground state is argued to be a TFD state. In [5], authors find the coupled
SYK model show similar behaviors with the ground state being the TFD state.
From the gravity perspective, this is dual to the vacuum of the gAdS2
spacetime can be prepared by the path-integral over half of a disk.

In this subsection, we assume similar ideas hold for the complex version as
well. A justification will be given in the next section. To construct the TFD
state, we firstly consider the ground state of the relevant hopping term
$-\mu_{c}\sum_{i}\left(c^{\dagger}_{R,i}c_{L,i}+c^{\dagger}_{L,i}c_{R,i}\right)$,
which is simply the maximally entangled state

$|I\rangle=\otimes_{i}\frac{1}{\sqrt{2}}\left(|0\rangle_{L,i}|1\rangle_{R,i}+|1\rangle_{L,i}|0\rangle_{R,i}\right),$
(22)

with all symmetric modes $c_{L,i}+c_{R,i}$ being occupied. The state satisfies

$c_{L,i}|I\rangle=c_{R,i}|I\rangle,\ \ \ \ \ \ \ \ \
c^{\dagger}_{L,i}|I\rangle=-c^{\dagger}_{R,i}|I\rangle.$ (23)

To take other terms in the Hamiltonian into account, we assume that to the
leading order of small $\mu_{c}/J$, the ground state of (1) is a TFD state
with an effective Hamiltonian $H_{\text{eff}}$:

$|G\rangle=\frac{1}{\sqrt{Z}}\exp(-\beta_{\text{eff}}H_{\text{eff}}/4)|I\rangle,$
(24)

where $Z_{\text{eff}}$ is the normalization factor and $H_{\text{eff}}$ is
given by

$\displaystyle H_{\text{eff}}=$
$\displaystyle\frac{1}{4}\sum_{ijkl}J_{ijkl}c^{\dagger}_{L,i}c^{\dagger}_{L,j}c_{L,k}c_{L,l}+\frac{1}{4}\sum_{ijkl}J_{ijkl}c^{\dagger}_{R,i}c^{\dagger}_{R,j}c_{R,k}c_{R,l}$
(25)
$\displaystyle-\mu_{L}^{\prime}\sum_{i}c^{\dagger}_{L,i}c_{L,i}-\mu_{R}^{\prime}\sum_{i}c^{\dagger}_{R,i}c_{R,i}.$

In general, we should allow $(\mu_{L}^{\prime},\mu_{R}^{\prime})$ being
different from $(\mu_{L},\mu_{R})$. We can then define $\mu^{\prime}$ and
$\bar{\mu}^{\prime}$ similar to (3). Since we are focusing on the small
$t^{\prime}$ and thus small $\bar{\mu}$ limit, we expect
$\bar{\mu}^{\prime}\propto\bar{\mu}$ since they both correspond to the
particle-hole symmetry breaking. Moreover, since both $\mu$ and
$\beta_{\text{eff}}$ is particle-hole symmetric, we expect their $\bar{\mu}$
dependence is at least second-order, which can be safely neglected. For
$\mu_{c}=0$, we expect $|G\rangle$ to be a specific pure state within the
ground state sector of $H$, which is two decoupled complex SYK models.
Consequently, we expect $\beta_{\text{eff}}\rightarrow\infty$ and
$\mu^{\prime}=\mu$ for $\mu_{c}\rightarrow 0$, if the coupled ground state is
in the same charge sector as the decoupled one. The precise relations are
given in Table 2, which will be proved in the next two subsections.

TFD parameters | $\beta_{\text{eff}}$ | $\bar{\mu}^{\prime}$ | $\mu^{\prime}$
---|---|---|---
Relations | $t^{\prime}\beta_{\text{eff}}=2\pi$ | Redundant | $\mu-\mu^{\prime}=t^{\prime}\mathcal{E}$
Table 2: A summary of relations between the TFD parameters, IR parameters and
parameters in the Hamiltonian (1).

We can directly use $|G\rangle$ to determine $\mathcal{E}$ and $\mathcal{P}$.
We first notice that

$c^{\dagger}_{L,i}c_{L,i}|I\rangle=(1-c^{\dagger}_{R,i}c_{R,i})|I\rangle.$
(26)

Consequently, we have

$\left(c^{\dagger}_{L,i}c_{L,i}+c^{\dagger}_{R,i}c_{R,i}\right)|G\rangle=|G\rangle.$
(27)

As a result, we know $Q=0$, consistent with the result in the last subsection,
and the $\overline{\mu}^{\prime}$ term, and thus the $\overline{\mu}$ term,
would not change the ground state $|G(\bar{\mu})\rangle=|G(0)\rangle$. As a
result, we expect the only effect of $\overline{\mu}$ in terms of the Green’s
function is:

$\displaystyle G_{\alpha\beta}(\tau)$
$\displaystyle=\left<G\right|c_{\alpha,i}(\tau)c^{\dagger}_{\beta,i}\left|G\right>=\left<G\right|e^{\tau
H(\overline{\mu})}c_{\alpha,i}e^{-\tau
H(\overline{\mu})}c^{\dagger}_{\beta,i}\left|G\right>$ (28)
$\displaystyle=e^{\overline{\mu}\tau}\left<G\right|e^{\tau
H(0)}c_{\alpha,i}e^{-\tau
H(0)}c^{\dagger}_{\beta,i}\left|G\right>=e^{\overline{\mu}\tau}G_{\alpha\beta}(\tau)_{\overline{\mu}=0}$

This suggests that we could directly identify $\mathcal{P}=\overline{\mu}$.

We then consider the reduced density matrix of a single side of the system.
Noticing that

$\displaystyle H_{\text{eff}}|\text{EPR}\rangle=$
$\displaystyle\left(\frac{1}{2}\sum_{ijkl}J_{ijkl}c^{\dagger}_{L,i}c^{\dagger}_{L,j}c_{L,k}c_{L,l}-2\mu^{\prime}\sum_{i}c^{\dagger}_{L,i}c_{L,i}-\mu_{R}^{\prime}N\right)|\text{EPR}\rangle\equiv
2H_{L}|\text{EPR}\rangle$ (29) $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\sum_{ijkl}J_{ijkl}c^{\dagger}_{R,i}c^{\dagger}_{R,j}c_{R,k}c_{R,l}+2\mu^{\prime}\sum_{i}c^{\dagger}_{R,i}c_{R,i}-\mu_{L}^{\prime}N\right)|\text{EPR}\rangle\equiv
2H_{R}|\text{EPR}\rangle.$

Consequently, if we look at a single side, the system is in the thermal
ensemble at inverse temperature $\beta_{\text{eff}}$ with chemical potential
$\pm\mu^{\prime}$. When we evolve the system with $H$, the Greens function
$G_{LL}$ takes the form of (9). On the other hand, we could consider evolving
the system with only $H_{0}$, which does not include the left-right coupling
term in (1), this leads to

$\displaystyle\tilde{G}_{LL}(\tau)$ $\displaystyle=\left<G\right|e^{\tau
H_{0}}c_{L,i}e^{-\tau
H_{0}}c^{\dagger}_{L,i}\left|G\right>=e^{\overline{\mu}\tau+\delta\mu\tau}\left<G\right|e^{\tau
H_{L}}c_{L,i}e^{-\tau
H_{L}}c^{\dagger}_{L,i}\left|G\right>=e^{\overline{\mu}\tau+\delta\mu\tau}G_{\beta_{\text{eff},\mathcal{E}_{s}(\mu^{\prime})}}(\tau),$
(30)

Where $\delta\mu=\mu-{\mu}^{\prime}$ is from the difference between
$H_{\text{eff}}$ and $H_{0}$. Here the subscript in
$\mathcal{E}_{s}(\mu^{\prime})$ reminds us that it is defined for a single
complex SYK model. $G_{\beta,\mathcal{E}}(\tau)$ is the finite temperature
Green’s function for a single complex SYK model:

$G_{\beta,\mathcal{E}}(\tau)=b\times
e^{2\pi\mathcal{E}(1/2-\tau/\beta)}\left(\frac{\pi}{\beta\sin(\pi\tau/\beta)}\right)^{1/2},\
\ \ \ \ \text{for}\ 0<\tau<\beta.$ (31)

Since we focus on the limit where $\mu_{c}\ll J$ which results in
$t^{\prime}\ll J$, we expect there is a separation of time scale: when
$1/J\ll\tau\ll 1/t^{\prime}$, there should be no difference between
$G_{LL}(\tau)$ and $\tilde{G}_{LL}(\tau)$. Matching the expression in this
limit, we find their asymmetric factors should be identified
$\mathcal{E}=\mathcal{E}_{s}(\mu^{\prime})$. This function between
$\mu^{\prime}$ and $\mathcal{E}$ contains the UV data and no analytical
formula is available, while $\mathcal{E}$ can be related to the charge [7, 8]:

$Q_{L}=-\frac{\theta}{\pi}-\frac{1}{4}\sin(2\theta).$ (32)

Finally, the entanglement entropy between two sides should match the thermal
entropy

$\mathcal{S}(\rho_{L})=\mathcal{S}_{0}(Q_{L})+4\pi^{2}\frac{\alpha_{S}}{\beta_{\text{eff}}J},$
(33)

where $\mathcal{S}_{0}(Q)$ is the zero-temperature entropy of a single complex
SYK model with charge $Q$ [8].

###  2.3 Effective Action and Thermodynamics

Now we turn to the derivation of the effective action. Using the idea in [4,
8], the effective action describes the coupling between the UV source $\sigma$
and IR fluctuations, which corresponds to the last term in (4). To derive the
effective action, we first determine the IR fields. Separating out the $U(1)$
and reparametrization modes, we consider the quasi-solutions (for
$\tau_{1}>\tau_{2}$):

$\displaystyle G_{LL}(\tau_{1},\tau_{2})$ $\displaystyle=b\times
e^{\pi\mathcal{E}+\mathcal{P}(\tau_{1}-\tau_{2})}\left(\frac{\varphi_{L}^{\prime}(\tau_{1})\varphi_{L}^{\prime}(\tau_{2})}{4\sinh^{2}(\varphi_{L}(\tau_{1})-\varphi_{L}(\tau_{2})/2)}\right)^{1/4}e^{i\left(\tilde{\lambda}_{L}(\tau_{1})-\tilde{\lambda}(\tau_{2})\right)},$
(34) $\displaystyle G_{RR}(\tau_{1},\tau_{2})$ $\displaystyle=b\times
e^{-\pi\mathcal{E}+\mathcal{P}(\tau_{1}-\tau_{2})}\left(\frac{\varphi_{R}^{\prime}(\tau_{1})\varphi_{R}^{\prime}(\tau_{2})}{4\sinh^{2}(\varphi_{R}(\tau_{1})-\varphi_{R}(\tau_{2})/2)}\right)^{1/4}e^{i\left(\tilde{\lambda}_{R}(\tau_{1})-\tilde{\lambda}_{R}(\tau_{2})\right)},$
$\displaystyle G_{LR}(\tau_{1},\tau_{2})$ $\displaystyle=-b\times
e^{\mathcal{P}(\tau_{1}-\tau_{2})}\left(\frac{\varphi_{L}^{\prime}(\tau_{1})\varphi_{R}^{\prime}(\tau_{2})}{4\cosh^{2}(\varphi_{L}(\tau_{1})-\varphi_{R}(\tau_{2})/2)}\right)^{1/4}e^{i\left(\tilde{\lambda}_{L}(\tau_{1})-\tilde{\lambda}_{R}(\tau_{2})\right)},$
$\displaystyle G_{RL}(\tau_{1},\tau_{2})$ $\displaystyle=-b\times
e^{\mathcal{P}(\tau_{1}-\tau_{2})}\left(\frac{\varphi_{R}^{\prime}(\tau_{1})\varphi_{L}^{\prime}(\tau_{2})}{4\cosh^{2}(\varphi_{R}(\tau_{1})-\varphi_{L}(\tau_{2})/2)}\right)^{1/4}e^{i\left(\tilde{\lambda}_{R}(\tau_{1})-\tilde{\lambda}_{L}(\tau_{2})\right)},$

where we have defined
$\tilde{\lambda}_{\alpha}(\tau)-i\mathcal{P}\tau=\lambda_{\alpha}(\tau)-i\mathcal{P}\varphi_{\alpha}(\tau)$.

We first consider the diagonal terms in $\sigma$, which are coupled to
$G_{LL}$ and $G_{RR}$. The IR fields can be identified by expanding the quasi-
solution around the intermediate asymptotic expression for
$1/J\ll|\tau_{1}-\tau_{2}|\ll 1/t^{\prime}$. We have:

$\displaystyle G_{LL}(\tau_{1},\tau_{2})$
$\displaystyle=G_{\infty,\mathcal{E}}(\tau_{1}-\tau_{2})(1+A_{L}(\tau_{+})(\tau_{1}-\tau_{2})+B_{L}(\tau_{+})(\tau_{1}-\tau_{2})^{2}+...),$
(35) $\displaystyle G_{RR}(\tau_{1},\tau_{2})$
$\displaystyle=G_{\infty,-\mathcal{E}}(\tau_{1}-\tau_{2})(1+A_{R}(\tau_{+})(\tau_{1}-\tau_{2})+B_{R}(\tau_{+})(\tau_{1}-\tau_{2})^{2}+...).$

with $\tau_{+}=\frac{\tau_{1}+\tau_{2}}{2}$ and

$A_{\alpha}(\tau)=i\tilde{\lambda}_{\alpha}^{\prime}(\tau)+\mathcal{P},\ \ \ \
\ \ \ \ \ \ \
B_{\alpha}(\tau)=\frac{1}{24}\text{Sch}\left(\tanh\frac{\varphi_{\alpha}(\tau)}{2},\tau\right)+\frac{A_{\alpha}(\tau)^{2}}{2}.$
(36)

Here the Schwarzian derivative is defined as
$\text{Sch}\left(y(\tau),\tau\right)\equiv\frac{y^{\prime\prime\prime}}{y^{\prime}}-\frac{3}{2}\left(\frac{y^{\prime\prime}}{y^{\prime}}\right)^{2}$.
The general diffeomorphism and gauge invariance [8] suggest the first
contribution to the effective action takes the form

$\displaystyle\frac{S^{(1)}_{\text{eff}}}{N}=$
$\displaystyle-\frac{\alpha_{S}}{J}\int
d\tau\left[\text{Sch}\left(\tanh\frac{\varphi_{L}(\tau)}{2},\tau\right)+\text{Sch}\left(\tanh\frac{\varphi_{R}(\tau)}{2},\tau\right)\right]$
(37) $\displaystyle+\int d\tau\left[f(\mu_{L}-A_{L})+f(\mu_{R}-A_{R})\right]$
$\displaystyle=$ $\displaystyle-\frac{\alpha_{S}}{J}\int
d\tau\left[\text{Sch}\left(\tanh\frac{\varphi_{L}(\tau)}{2},\tau\right)+\text{Sch}\left(\tanh\frac{\varphi_{R}(\tau)}{2},\tau\right)\right]$
$\displaystyle+\int
d\tau\left[f\left(\mu-i\tilde{\lambda}_{L}^{\prime}(\tau)\right)+f\left(-\mu-i\tilde{\lambda}_{R}^{\prime}(\tau)\right)\right].$

Here we have used the fact that $\mathcal{P}=\bar{\mu}$. The result has no
dependence on $\bar{\mu}$, and this implies $Q=0$, making the substitution of
$\mathcal{P}$ consistent. The function $f(\mu)$ represents the UV contribution
to the free energy, which should match that of a single complex SYK model and
have the symmetry $f(\mu)=f(-\mu)$. Expanding $\tilde{\lambda}_{\alpha}$ to
the second-order gives

$\displaystyle\frac{S^{(1)}_{\text{eff}}}{N}=$
$\displaystyle-\frac{\alpha_{S}}{J}\int
d\tau\left[\text{Sch}\left(\tanh\frac{\varphi_{L}(\tau)}{2},\tau\right)+\text{Sch}\left(\tanh\frac{\varphi_{R}(\tau)}{2},\tau\right)\right]$
(38) $\displaystyle+\int
d\tau\left[2f\left(\mu\right)+\frac{K}{2}\tilde{\lambda}_{L}^{\prime}(\tau)^{2}+\frac{K}{2}\tilde{\lambda}_{R}^{\prime}(\tau)^{2}\right].$

Here we define $K=-f^{\prime\prime}(\mu)=-f^{\prime\prime}(-\mu)$ and assume
the winding number of $\tilde{\lambda}_{\alpha}$ is zero for the saddle
points.

We then consider off-diagonal terms in $\sigma$, which is coupled to $G_{LR}$
and $G_{RL}$. Different from the diagonal terms where the IR fields are
determined through expanding the quasi-solution around
$G_{\infty,\mathcal{E}}(\tau_{1}-\tau_{2})$, the off-diagonal components of
$G$ it self can be considered as IR contributions. This can be understood from
the fact that $G_{LR}$ in (9) can be viewed as introducing a real-time
separation $\delta\tau=i\pi/t^{\prime}$ to the $G_{LL}$. Consequently, we just
add the contribution from $\mu_{c}$ directly as:

$\frac{S^{(2)}_{\text{eff}}}{N}=-2b\mu_{c}\int d\tau\
\left(\frac{\varphi_{R}^{\prime}(\tau)\varphi_{L}^{\prime}(\tau)}{4\cosh^{2}(\varphi_{R}(\tau)-\varphi_{L}(\tau)/2)}\right)^{1/4}\cos{\left(\tilde{\lambda}_{R}(\tau)-\tilde{\lambda}_{L}(\tau)\right)}.$
(39)

The total effective action is obtained by summing up diagonal and off-diagonal
contributions $S_{\text{eff}}=S^{(1)}_{\text{eff}}+S^{(2)}_{\text{eff}}$.

We first consider the thermal equilibrium state. The conformal solution (9)
corresponds to setting $\varphi_{\alpha}(\tau)=t^{\prime}\tau$ and
$\tilde{\lambda}_{\alpha}(\tau)=0$. This gives the grand potential
$\Omega_{\text{WH}}$:

$\frac{\Omega_{\text{WH}}}{N}=\text{Min}_{t^{\prime}}\left\\{2f(\mu)+\frac{\alpha_{S}t^{\prime
2}}{J}-\sqrt{2}b\mu_{c}(t^{\prime})^{1/2}\right\\}=2f(\mu)-\frac{3}{4}\left(\frac{Jb^{4}\mu_{c}^{4}}{\alpha_{S}}\right)^{1/3},$
(40)

with the solution
$t^{\prime}=\frac{b^{2/3}J^{2/3}\mu_{c}^{2/3}}{2\alpha_{S}^{2/3}},$ where the
$\mathcal{E}$ dependence comes form $b$. It is straightforward to check (9)
then satisfies the saddle-point equations for $\varphi_{\alpha}$ and
$\tilde{\lambda}_{\alpha}$, as well as the condition that all SL(2,R) charges
should vanish [5].

The charge of the left system is given by taking derivative with $\mu_{L}$.
This gives $-Q_{L}=f^{\prime}(\mu)$, meaning the relation between $\mu$ and
$Q_{L}$ is the same as the thermodynamics of a single SYK at $T=0$. It is
known that for a single SYK model at finite temperature, the chemical
potential shifts as $\mu_{s}(0)=\mu_{s}(T)+2\pi\mathcal{E}_{s}/\beta$ [8].
This gives the relation between $\mu^{\prime}$ and $\mu$ as

$\mu=\mu^{\prime}+2\pi\mathcal{E}/\beta_{\text{eff}}.$ (41)

We can also study the fluctuation of low-energy modes around the saddle point.
The vanishing of SL(2,R) charges makes it possible to choose
$\varphi_{L}(\tau)=\varphi_{R}(\tau)=\varphi(\tau)$. For convenience, we
further introduce $\varphi^{\prime}(\tau)=e^{\phi(\tau)}$ with the saddle
point at $\phi(\tau)=\log(t^{\prime})$. Expanding the effective action to the
quadratic order gives

$\displaystyle\frac{\delta S_{\text{eff}}}{N}=\int
d\tau\left[\frac{K}{2}\lambda_{c}^{\prime}(\tau)^{2}+\frac{K}{2}\lambda_{r}^{\prime}(\tau)^{2}+\sqrt{2}b\mu_{c}t^{\prime\frac{1}{2}}\lambda_{r}^{2}(\tau)+\frac{\alpha_{S}}{J}\delta\phi^{\prime}(\tau)^{2}+\frac{3b^{4/3}\mu_{c}^{4/3}J^{1/3}}{8\alpha_{S}^{1/3}}\delta\phi(\tau)^{2}\right].$
(42)

Here we have defined $\lambda_{c}=\frac{\lambda_{L}+\lambda_{R}}{\sqrt{2}}$
and $\lambda_{r}=\frac{\lambda_{L}-\lambda_{R}}{\sqrt{2}}$. We find there is a
gapped graviton mode with frequency
$\omega_{\text{gravity}}=\sqrt{\frac{3}{8}}\left(\frac{bJ\mu_{c}}{\alpha_{S}}\right)^{2/3}$,
a relative phase mode $\lambda_{r}$ with frequency
$\omega_{U(1)}=\frac{\sqrt{2}b^{2/3}J^{1/6}\mu_{c}^{2/3}}{\sqrt{K}(\alpha_{S})^{1/6}}$,
and a gapless field $\lambda_{c}$. These modes can be excited in certain
quench dynamics, similar to the oscillation behavior observed in [27]. In
addition to thermodynamics, the entanglement entropy of the wormhole phase
(33) can also be computed using the effective action following the derivation
in [35], since $\tilde{\lambda}_{\alpha}=0$ still satisfies the saddle-point
equation after introducing replicas.

For large $\bar{\mu}/t^{\prime}$ and moderate $\mu/J$, the wormhole solution
(9) breaks down, and the system transits into the black hole solution. The
transition point is determined by comparing the free energy. In the black hole
phase, we expect the free energy takes the form:

$\frac{\Omega_{\text{BH}}}{N}=f(\mu_{L})+f(\mu_{R})+O(\mu_{c}^{2})\approx
f(\bar{\mu}+\mu)+f(\bar{\mu}-\mu)\approx 2f(\mu)-\bar{\mu}^{2}K.$ (43)

The critical $\bar{\mu}$ is then determined as
$\text{Min}\left\\{\frac{b^{2/3}J^{2/3}\mu_{c}^{2/3}}{8\alpha_{S}^{2/3}},\frac{\sqrt{3}b^{2/3}{J}^{1/6}\mu_{c}^{2/3}}{2\sqrt{K}{\alpha_{S}}^{1/6}}\right\\}$,
both of which proportional to $\mu_{c}^{2/3}$. This also justifies the neglect
of higher order terms $\sim\mu_{c}^{2}$.

###  2.4 Quench Dynamics

We still need to relate the TFD parameter $\beta_{\text{eff}}$ to parameters
in the Hamiltonian. Following the idea in [5], this can be determined by
considering the quench dynamics for turning off $\mu_{c}$.

We firstly transform the effective action into the real-time with
$\tau\rightarrow iu$ and $\varphi\rightarrow i\varphi$. Explicitly, we have

$\displaystyle\frac{\tilde{S}_{\text{eff}}}{N}=$ $\displaystyle\int
du\left[-\frac{2\alpha_{S}}{J}\text{Sch}\left(\tan\frac{\varphi(u)}{2},u\right)+\frac{K}{2}\tilde{\lambda}_{L}^{\prime}(u)^{2}+\frac{K}{2}\tilde{\lambda}_{R}^{\prime}(u)^{2}\right]$
(44) $\displaystyle+\sqrt{2}b\int du\
\mu_{c}\theta(-u)\varphi^{\prime}(u)^{1/2}\cos{\left(\tilde{\lambda}_{R}(u)-\tilde{\lambda}_{L}(u)\right)}.$

At $u<0$, the system is in thermal equilibrium with $\varphi(u)=t^{\prime}u$
and $\tilde{\lambda}_{\alpha}(u)=0$. For $u>0$, by solving the saddle-point
equations and requiring the smoothness for fields
$\phi(u)=\log(\varphi(u)),\tilde{\lambda}_{\alpha}(u)$ and their first-order
derivatives, we have for $u>0$

$\varphi(u)=2\arctan\left(\tanh\frac{t^{\prime}u}{2}\right),\ \ \ \ \ \ \ \ \
\ \ \ \tilde{\lambda}_{\alpha}(u)=0.$ (45)

We can derive the real-time Green’s functions by analytical continuation of
(34). For example,

$iG_{LL}^{>}(u_{1},u_{2})\equiv\left<c_{L,i}(u_{1})c^{\dagger}_{L,i}(u_{2})\right>=b\times
e^{\pi\mathcal{E}+i\mathcal{P}(u_{1}-u_{2})}\left(\frac{\varphi^{\prime}(u_{1})\varphi^{\prime}(u_{2})}{4\sin^{2}\left(i\frac{\varphi(u_{1})-\varphi(u_{2})}{2}+0^{+}\right)}\right)^{1/4}$
(46)

When $u_{1}>0$ and $u_{2}>0$, this gives

$iG_{LL}^{>}(u_{1},u_{2})=be^{\pi\mathcal{E}+i\mathcal{P}u_{12}}\left(\frac{t^{\prime}}{2\sinh\left(t^{\prime}|u_{12}|/2\right)}\right)^{1/2}e^{-i\frac{\pi}{4}\text{sgn}(u_{12})}.$
(47)

We compare (47) with the expectation if we assume the ground state is a TFD.
The result is an analytical continuation of (30), which reads

$\displaystyle i\tilde{G}_{LL}(u)$
$\displaystyle=\left<G\right|e^{iH_{0}u}c_{L,i}e^{-iH_{0}u}c^{\dagger}_{L,i}\left|G\right>=be^{\pi\mathcal{E}+i\mathcal{P}u_{12}}\left(\frac{\pi}{\beta_{\text{eff}}\sinh\left(\pi|u_{12}|/\beta_{\text{eff}}\right)}\right)^{1/2}e^{-i\frac{\pi}{4}\text{sgn}(u_{12})},$
(48)

where we have used (41). The equivalence between (47) and (48) gives
$t^{\prime}\beta_{\text{eff}}=2\pi$.

##  3 Numerical Results

In this section, we present numerical results to support our low-energy
analysis. In subsection 3.1, we solve the self-consistent equation in the
imaginary time and compare the Green’s function the conformal solutions (9).
We also discuss the existence of several phases for our model (1). In
subsection 3.2, we study the quench dynamics and comparing numerics with the
prediction of the TFD state.

###  3.1 Imaginary time results and the Phase diagram

Figure 1: (a). The imaginary-time Green’s function for $\beta J=120$,
$\bar{\mu}/J=0.05$, $\mu/J=0.1$, and $\mu_{c}/J=0.05$. (b). A zoom in of (a)
near $1/t^{\prime}\ll\tau\ll\beta/2$ or $1/t^{\prime}\ll\beta-\tau\ll\beta/2$.
We can see the numerical results are consistent with (9). Figure 2: A
comparison between the numerics and (50) for (a). $\mathcal{P(\bar{\mu})}$
with different $\mu/J$, and (b). $\bar{E}_{g}(\mu)$ for different $\bar{\mu}$.
In all cases we fix $\beta J=120$ and $\mu_{c}/J=0.05$. The dots are numerical
data and the dashed line represents the prediction of (50).

In the imaginary-time approach, we adapt the standard algorithm [5] to solve
the saddle-point equation (7) for $G_{\alpha\beta}$ and
$\Sigma_{\alpha\beta}$. We work at large but finite temperature $\beta J\gg 1$
and small but finite $\mu_{c}/J$.

Typical Green’s functions in the wormhole phase are shown in Figure 1 (a),
where we choose $\beta J=120$, $\bar{\mu}/J=0.05$, $\mu/J=0.1$, and
$\mu_{c}/J=0.05$. The Green’s functions decay exponentially for
$1/t^{\prime}\ll\tau\ll\beta/2$ or $1/t^{\prime}\ll\beta-\tau\ll\beta/2$. We
can approximately identify the numerical solution at $0<\tau<\beta/2$ with the
zero-temperature Green’s function
$G^{+}_{\alpha\beta}(\tau)\equiv|G_{\alpha\beta}(\tau)|$, and the numerical
solution at $0<\beta-\tau<\beta/2$ with the zero-temperature Green’s function
$G^{-}_{\alpha\beta}(\tau)\equiv|G_{\alpha\beta}(\tau-\beta)|$. For
$1/t^{\prime}\ll\tau\ll\beta/2$, the conformal solution predicts

$\displaystyle\log G^{+}_{LL}(\tau)\sim
c_{0}+\pi\mathcal{E}-(\bar{E}_{g}-\mathcal{P})\tau,\ \ \ \ \ \ \ \ \log
G^{-}_{LL}(\tau)\sim c_{0}-\pi\mathcal{E}-(\bar{E}_{g}+\mathcal{P})\tau,$ (49)
$\displaystyle\log G^{+}_{RR}(\tau)\sim
c_{0}-\pi\mathcal{E}-(\bar{E}_{g}-\mathcal{P})\tau,\ \ \ \ \ \ \ \ \log
G^{-}_{RR}(\tau)\sim c_{0}+\pi\mathcal{E}-(\bar{E}_{g}+\mathcal{P})\tau,$
$\displaystyle\log G^{+}_{LR}(\tau)\sim c_{0}-(\bar{E}_{g}-\mathcal{P})\tau,\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \log G^{-}_{LR}(\tau)\sim
c_{0}-(\bar{E}_{g}+\mathcal{P})\tau,$

where $c_{0}$ is come numerical factor and we have defined the averaged gap
$\bar{E}_{g}=t^{\prime}/4$. In Figure 1 (b), we show the numerical results,
which takes exactly the same form as conformal solutions.

Figure 3: The phase diagram of the model considered in this paper with $\beta
J=120$ and $\mu_{c}/J=0.05$. Here we focus on moderate $\bar{\mu}/J$, with the
wormhole phase, the two black hole phase, and the single black hole phase. We
show the density plot for $Q$ in (a) and for $Q_{L}$ in (b). Multiply saddle
points exist in the dotted region. We also plot corresponding 1D curves with
fixed $\mu/J$ in (c) and (d).

We then compare the numerical fitted $\mathcal{P}$ and $\bar{E}_{g}$ with the
low-energy prediction summarized in Table 1, where

$\mathcal{P}=\bar{\mu},\ \ \ \ \ \ \ \ \ \
\bar{E}_{g}(\mu)=\bar{E}_{g}(0)/\cosh(2\pi\mathcal{E})^{1/6}.$ (50)

The result is shown in Figure 2 for $\beta J=120$ and $\mu_{c}/J=0.05$. We
find the results match well, with a small discrepancy possibly from the fact
that $\mu_{c}/J$ is not small enough.

We then consider large $\bar{\mu}$ and explore the phase diagram for the
Hamiltonian (1). For fixed $\mu$, we perform the numerical iteration firstly
from small $\bar{\mu}$ to large $\bar{\mu}$, and then from large $\bar{\mu}$
back to small $\bar{\mu}$. We also perform similar steps for different $\mu$
with fixed $\bar{\mu}$. Near first-order transition points, this can give
different saddle points for the same $\bar{\mu}$.

The results for $\beta J=120$ and $\mu_{c}/J=0.05$ are shown in Figure 3. At
small $\bar{\mu}/J$, the system is in the wormhole phase, with specific charge
$Q=0$, while $Q_{L}$ can change when tunning $\mu/J$. For moderate $\mu/J$, if
we consider larger $\bar{\mu}/J$, the system transits into the two black hole
phase. In this phase, we can approximate the solution as two separate complex
SYK models with chemical potential $\mu_{L}$ and $\mu_{R}$. In this case, both
$Q_{L}$ and $Q$ change freely when tunning parameters. If we further consider
a larger $\mu$ or $\bar{\mu}$, where $\mu_{L}$ becomes larger than a threshold
$\sim 0.25$, it is known that a single complex SYK model would transit into a
(nearly) fully occupied phase with $Q_{L}=1/2$ [36, 37, 38]. We call this
phase the (single) black hole phase since the right system can be approximated
as a single complex SYK black hole. One also expects a (nearly) fully occupied
phase for both copies to exist when both $\mu_{L}$ and $\mu_{R}$ become large
and the system has $Q_{L}=Q_{R}=1/2$.

###  3.2 Real time results and Quench dynamics

Now we turn to real-time formalism. For the Majorana coupled SYK model,
related studies have been carried out in [24, 25, 26]. We are mainly
interested in study the quench dynamics, and compare the numerical to results
in 2.4.

Figure 4: The real-time Green’s functions $G^{>}(t)$ in thermal equilibrium
within the wormhole phase. We choose $\beta J=40$, $\mu_{c}/J=0.075$,
$\mu/J=0.1$ and $\bar{\mu}/J=0$.

The quench dynamics of SYK-like models can be studied using the Keldysh
approach [39, 40, 41, 42, 43]. We firstly define several real-time Green’s
functions $G^{>}$ and $G^{<}$:

$G^{>}_{\alpha\beta}(u_{1},u_{2})=-i\left<c_{\alpha,i}(u_{1})c^{\dagger}_{\beta,i}(u_{2})\right>,\
\ \ \ \ \ \ \ \ \
G^{<}_{\alpha\beta}(u_{1},u_{2})=i\left<c^{\dagger}_{\beta,i}(u_{2})c_{\alpha,i}(u_{1})\right>.$
(51)

Here we have copied the definition (46) for completeness. The retarded and
advanced Green’s functions $G^{R}$ and $G^{A}$ then read

$\displaystyle
G^{R}_{\alpha\beta}(u_{1},u_{2})=\theta(u_{12})\left(G^{>}_{\alpha\beta}(u_{1},u_{2})-G^{<}_{\alpha\beta}(u_{1},u_{2})\right),$
(52) $\displaystyle
G^{A}_{\alpha\beta}(u_{1},u_{2})=\theta(-u_{12})\left(G^{<}_{\alpha\beta}(u_{1},u_{2})-G^{>}_{\alpha\beta}(u_{1},u_{2})\right).$

For the Hamiltonian (1), the self-energy $\Sigma^{>}$ and $\Sigma^{<}$ in
real-time can be obtained by the analytically continuation of (7) in the time
domain:

$\displaystyle\Sigma^{>}_{\alpha\beta}(u_{1},u_{2})=J^{2}G^{>}_{\alpha\beta}(u_{1},u_{2})^{2}G^{<}_{\beta\alpha}(u_{2},u_{1}),$
(53)
$\displaystyle\Sigma^{<}_{\alpha\beta}(u_{1},u_{2})=J^{2}G^{<}_{\alpha\beta}(u_{1},u_{2})^{2}G^{>}_{\beta\alpha}(u_{2},u_{1}),$

and the retarded and advanced components of the self-energy is

$\displaystyle\Sigma^{R}_{\alpha\beta}(u_{1},u_{2})=\theta(u_{12})\left(\Sigma^{>}_{\alpha\beta}(u_{1},u_{2})-\Sigma^{<}_{\alpha\beta}(u_{1},u_{2})\right),$
(54)
$\displaystyle\Sigma^{A}_{\alpha\beta}(u_{1},u_{2})=\theta(-u_{12})\left(\Sigma^{<}_{\alpha\beta}(u_{1},u_{2})-\Sigma^{>}_{\alpha\beta}(u_{1},u_{2})\right).$

Figure 5: The quench dynamics when turning off $\mu_{c}$ with $\beta J=40$,
$\mu_{c}/J=0.075$ and $\bar{\mu}/J=0$. We plot Im$G^{>}_{LL}(u_{1},u_{2})$ in
(a) and Im$G^{>}_{LR}(u_{1},u_{2})$ in (b). When $u_{1}>0$ and $u_{2}>0$, we
find $G^{>}_{LL}(u_{1},u_{2})$ only depends on $u_{1}-u_{2}$, while
$G^{>}_{LR}(u_{1},u_{2})$ only depends on $u_{1}+u_{2}$. We also plot
$G^{>}_{LL}(t+t_{0},t_{0})$ and $G^{>}_{LR}(t/2,t/2)$ explicitly in (c) and
(d).

Defining $\hat{\mu}(u_{1})=\bar{\mu}I+\mu\sigma_{z}+\mu_{c}(u_{1})\sigma_{x}$,
we have matrix equations

$\displaystyle
i\partial_{u_{1}}G^{R/A}(u_{1},u_{2})+\hat{\mu}(u_{1})G^{R/A}(u_{1},u_{2})-\int
du_{3}\Sigma^{R/A}(u_{1},u_{3})G^{R/A}(u_{3},u_{2})$
$\displaystyle=\delta({u_{12}})I,$ (55)
$\displaystyle-i\partial_{u_{2}}G^{R/A}(u_{1},u_{2})+G^{R/A}(u_{1},u_{2})\hat{\mu}(u_{2})-\int
du_{3}G^{R/A}(u_{1},u_{3})\Sigma^{R/A}(u_{3},u_{2})$
$\displaystyle=\delta({u_{12}})I.$

Figure 6: (a). The spectral function $\rho_{LL}(\omega)$ in the long time
limit for turning off $\mu_{c}$. (b). The quantum distribution $F(\omega)$ in
the long time limit for turning off $\mu_{c}$. The dashed line represents the
fitting with $F(\omega)=\tanh\left(3.95(\omega/J-0.029)\right)$, which gives a
final state with $\beta_{\text{eff}}J=7.9$ and $\mu^{\prime}/J=0.129$. We then
compare the thermal real-time Green’s function and the quench result in (c)
and (d), with only a phase difference due to $\delta\mu$, as predicted by
(48).

To study the quench protocol in section 2.4, we should prepare a system at
thermal equilibrium in the wormhole phase. All Green’s functions and self-
energies in thermal ensemble only depend on the time difference. The inverse
temperature $\beta$ enters the self-consistent equation through the
fluctuation-dissipation theorem:

$G^{>}_{\alpha\beta}(\omega)=-i\rho_{\alpha\beta}(\omega)n_{F}(-\omega,\beta),\
\ \ \ \ \ \ \ \
G^{<}_{\alpha\beta}(\omega)=i\rho_{\alpha\beta}(\omega)n_{F}(\omega,\beta).$
(56)

with the spectral function matrix $\rho_{\alpha\beta}(\omega)$ defined as

$\rho_{\alpha\beta}(\omega)=-\frac{1}{2\pi
i}\left(G^{R}_{\alpha\beta}(\omega)-G^{A}_{\alpha\beta}(\omega)\right)=-\frac{1}{2\pi
i}\left(G^{R}_{\alpha\beta}(\omega)-G^{R}_{\beta\alpha}(\omega)^{*}\right).$
(57)

Solving (53), (54), (55), (56) and (57) iteratively with
$\mu_{c}(u_{1})=\mu_{c}$ gives the spectral function and real-time Green’s
functions. In the numerics, we choose $\beta J=40$, $\mu_{c}/J=0.075$,
$\mu/J=0.1$ and $\bar{\mu}/J=0$. The results for $G^{>}_{LL}$ and $G^{>}_{LR}$
are shown in Figure 4 as an example, which show rapid oscillations as in [25,
24].

Now we consider the quench dynamics with $\mu_{c}(u)=\mu_{c}\theta(-u)$. The
evolution of Green’s functions is usually determined using the Kadanoff-Baym
equation, which takes a similar form as (55):

$\displaystyle
i\partial_{1}G^{\gtrless}(u_{1},u_{2})+\hat{\mu}(u_{1})G^{\gtrless}(u_{1},u_{2})$
$\displaystyle=\int
du_{3}\left(\Sigma^{R}(u_{1},u_{3})G^{\gtrless}(u_{3},u_{2})+\Sigma^{\gtrless}(u_{1},u_{3})G^{A}(u_{3},u_{2})\right),$
(58)
$\displaystyle-i\partial_{2}G^{\gtrless}(u_{1},u_{2})+G^{\gtrless}(u_{1},u_{2})\hat{\mu}(u_{2})$
$\displaystyle=\int
du_{3}\left(G^{R}(u_{1},u_{3})\Sigma^{\gtrless}(u_{3},u_{2})+G^{\gtrless}(u_{1},u_{3})\Sigma^{A}(u_{3},u_{2})\right).$

Importantly, the causality is preserved explicitly due to the presence of
retarded and advanced components on the R.H.S., and the derivative of the
Green’s functions at time $(u_{1},u_{2})$ only depends on Green’s functions
$G^{\gtrless}(u_{3},u_{4})$ with $u_{3},u_{4}<\text{Max}\\{u_{1},u_{2}\\}$.
The initial condition for (58) is given by the thermal equilibrium result

$G^{\gtrless}(u_{1},u_{2})=G^{\gtrless}(u_{1}-u_{2}),\ \ \ \ \ \ \ \text{for
$u_{1}<0$ and $u_{2}<0$.}$ (59)

We then evolve the Green’s function step by step using the standard numerical
method by discretizing $G^{\gtrless}(u_{1},u_{2})$ into large matrices [39,
40, 41, 42, 43].

The numerical results for $G^{>}(u_{1},u_{2})$ are shown in Figure 5. As we
turn off the coupling between two systems, their correlation $G_{LR}$ decays
exponentially. More interestingly, we see that $G^{>}_{LL}(u_{1},u_{2})$ only
depends on $u_{1}-u_{2}$, while $G^{>}_{LR}(u_{1},u_{2})$ only depends on
$u_{1}+u_{2}$ when $u_{1}>0$ and $u_{2}>0$. This is consistent with what one
expects for the evolution of the TFD state.

To compare the prediction from the TFD perspective and numerical results more
carefully, we need to extract the temperature and the chemical potential of
the final state. We use the numerical results $G^{\gtrless}(u_{1}-u_{2})$
after the quench in the long-time limit. We show the long-time spectral
function $\rho_{LL}$ in Figure 6 (a). The fluctuation-dissipation theorem
predicts

$-\frac{G^{>}_{LL}(\omega)+G^{<}_{LL}(\omega)}{G^{>}_{LL}(\omega)-G^{<}_{LL}(\omega)}=\tanh\frac{\beta_{\text{eff}}(\omega+\delta\mu)}{2}.$
(60)

We then determine $\beta_{\text{eff}}$ and $\delta\mu$ by fitting, as shown in
Figure 6 (b). This gives $\beta_{\text{eff}}J=7.9$ and $\mu^{\prime}/J=0.129$.

Using this result, we can perform an independent numerical study by solving a
single model at corresponding parameters. In Figure 6 (c-d), we compare the
single SYK numerics with the result of quench dynamics. We find the magnitude
of $G_{LL}$ matches to high accuracy, while a phase difference exists due to
the presence of $\delta\mu$. As expected from (48), in the quench dynamics,
the phase is almost constant in the long-time limit, while approaching
$-\pi/4$ or $-3\pi/4$ in the short-time limit.

##  4 Summary

We study the generalization of the coupled SYK model (1) by introducing $U(1)$
symmetry. We find the conformal solution (9) at $T=0$, which contains IR
parameters $\mathcal{E}$, $\mathcal{P}$ and $t^{\prime}$. We determine the IR
parameter, as shown in Table 1, by combining understandings from the TFD and
the effective action analysis. We also determine the TFD parameters as shown
in Table 2. We then compare our results with numerics and explore the full
phase diagram.

A few possible extensions of the current work are as follows: One can ask
whether we could write out a gravity theory with the same effective action
$S_{\text{eff}}$. This should correspond to adding $U(1)$ gauge fields [44] to
the JT gravity with coupling between two boundaries [5]. It is also
interesting to consider two copies of higher-dimensional models, etc. 1+1-D,
and adding a coupling term. This can lead to higher-dimensional wormholes.

## Acknowledgments

We especially thank Yingfei Gu for helpful discussions. P.Z. acknowledges
support from the Walter Burke Institute for Theoretical Physics at Caltech.

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## Discrete two-contour system with co-directional movement

Yashina M.V.1,2, Tatashev A.G.1,2, Fomina M.J.1
${}^{1}\leavevmode\nobreak\ $Moscow Automobile and Road Construction State
Technical University (MADI)
${}^{2}\leavevmode\nobreak\ $Moscow Technical University of Communications and
Informatics (MTUCI)
<EMAIL_ADDRESS>
Abstract. The paper studies a discrete dynamical system, which belongs to the
class of contour systems developed by A.P Buslaev. The system contains two
closed contours. There are $n$ cells and a group of particles at each contour.
This group is called a cluster. The particles of a cluster are in adjacent
cells and move simultaneously. At each discrete moment any particle moves onto
a cell in the direction of movement. There are two common points of contours
called nodes. Delays of clusters occur at the nodes. These delays are due to
that two particles cannot cross the same node simultaneously. Earlier this
system were studied under the assumptions that the number of particles does
not depend on the cluster. This paper studies a system with different lengths
of clusters. Analytical results are obtained for limit cycles of the system
and the spectrum of average cluster velocities.

## 1\. Introduction

The paper studies a discrete dynamical system, which belongs to the class of
contour systems developed by A.P Buslaev. This class was introduced to develop
network transport models such that analytical results can be developed for
these models.

In [1]–[7], analytical results have been obtained for mathematical traffic
models such that, in these models, particles moves on a one-dimensional
lattice. These models can be interpreted in terms of cellular automata, [8],
or exclusive processes, [9].

The concept of traffic model was introduced in [10]. In a discrete model, a
cluster is a group of particles located in adjacent cells and moving
simultaneously. In a continuos model, a cluster is a moving segment.

The concept of contour networks (Buslaev networks) was introduced in [11]. A
contour network contains closed contours, in which particles or clusters move
under prescribed rules. Adjacent contours contain common points called nodes.
Delays of clusters occur at nodes. The delays are due to that more than one
cluster cannot to pass through the node simultaneously. In [11]–[24]
analytical results were obtained for contour network.

In [15]–[17] a contour network with two contours and two nodes were studied.
The nodes divide each contour into two parts of different lengths. The length
of clusters is the same. Two versions of the system were considered. In one of
these versions one cluster moves counter-clockwise and the other cluster moves
clockwise (co-directional movement). In the other version both the clusters
move clockwise (one-directional movement).

This paper studies a generalization of the system, considered in [15], under
the assumption that the movement is co-directional.

## 2\. Description of system

Let the system contain two closed contours, Fig. 1. They are the contour 1 and
the contour 2. There are $n$ cells in any contour. The indices of the cells
are $0,1,2,\dots,n-1.$ At each discrete moment $t=0,1,2,\dots,$ each cluster
moves onto one cell in the direction of movement if the cells are numbered in
the direction of movement modulo $n.$ There is a cluster in contour $i$
(cluster $i),$ and there are $l_{i}<n$ particles in the cluster $i,$ $i=1,2.$
At any moment the particles of each cluster are located in adjacent cells.
There are two common points of contours. These points are called nodes. The
node 1 is located between the cells $n-1$ and 0 for each contour. The node 2
is located between the cells $d-1$ and $d,$ $d\leq n/2$ for each contour. We
say that a cluster is at the node 1 (at the node 2) if the leading cell of the
cluster is located in the cell $n-1$ (in the cell $d-1).$ We say that a
cluster (occupies a node) if the node is between two particles of this
cluster. Delays occur if two particles try to cross the node simultaneously.
If, at time $t,$ a cluster $i$ is at a node, and the cluster $j\neq i$
occupies this node, then the cluster $i$ does not move and, at time $t+1,$ its
particles are located in the cells in that the particles are located at time
$t.$ If two clusters are at the same node simultaneously, then only the
cluster 1 moves.

Figure 1: A two-contour system

The state of the system at time $t$ is a vector $x(t)=(x_{1}(t),x_{2}(t)),$
where $x_{i}(t)$ is the index of cell occupied by the leading particle of the
cluster $i,$ $i=1,2.$

The state of the system is admissible if no node is occupied by both the
clusters. The initial state $x(0)$ is prescribed. This state must be
admissible.

Without loss of generality we assume that $l_{1}\leq l_{2}.$

## 3\. Limit cycles. Average velocity of cluster. States of free movement and
collapse

The state space of the system is finite. From any moment, system states are
repeated (limit cycles).

Suppose that at time interval $(0,t)$ the cluster $i$ passes the distance
$H(t).$ The limit

$v_{i}=\lim\limits_{t\to\infty}\frac{H_{i}(t)}{t},$ (1)

is called the average velocity of the cluster $i,$ $i=1,2.$ The limit (1)
exists and is equal to $A_{i}/T,$ where $A_{i}$ is the distance such that the
cluster $i$ passes this distance on the realized limit cycle, and $T$ is the
period of cycle.

If, at any time $t\geq t_{0}$ any particle moves, then, from the moment
$t_{0},$ the system is in a state of free movement.

If at any moment $t\geq t_{0}$ no particle moves, then, from the moment
$t_{0},$ the system is in the state of collapse. If the system results in the
state of collapse, then $v_{1}=v_{2}=0.$

## 4\. Some lemmas

In this section, we shall prove some lemmas. Lemma 1. If the inequality

$l_{1}+l_{2}>n,$ $None$

hplds, then the system cannot be in a state of free movement. Proof. If, on a
limit cycle, the system is in a state of free movement, then the period is
equal to $n,$ and, on the cycle, $l_{1}+l_{2}$ particles cross each node.
However it is impossible if (2) holds. Indeed, two particles cannot cross the
same node simultaneously. Lemma 1 has been proved. Lemma 2. If, on a limit
cycle, the system is not in a state of free movement nor collapse, then the
system results on this cycle in at least one of the states $(l_{1},0),$
$(0,l_{2}),$ $(d+l_{1},d),$ $(d,d+l_{2})$ (addition modulo $n).$ Proof. Under
the assumptions of the lemma, delays of at least one cluster occur on the
limit cycle, and, at moment such that, at this moment a delay ends, the system
is at one of the states $(l_{1},0),$ $(0,l_{2}),$ $(d+l_{1},d),$ $(d,d+l_{2})$
(addition modulo $n).$ This proves Lemma 2. Lemma 3. The system can be in a
state of collapse if and only if

$l_{1}>n-d.$ $None$

Proof. Taking into account the inequalities $l_{1}\leq l_{2}$ and (3), we get
the inequality $l_{2}>n-d.$ Therefore the states $(0,d)$ and $(d,0)$ are
states of collapse.

If the system is in a state of collapse, then each cluster is at a node and
occipies the other node. However, if (3) holds, a cluster cannot be at a node
and occupy the other node simultaneously. This contradiction proves Lemma 3.

## 5\. Criterion for resulting in a state of free movement

The following theorem and Lemma 1 provide a necessary and sufficient condition
of system resulting in free movement. Theorem 1. If the inequality

$l_{1}+l_{2}\leq n,$ $None$

holds, then the system results in a state of free movement from any initial
state.

Proof. If (4) holds, then (3) does not hold, and, in accordance with Lemma 3,
the system cannot be in a state of collapse. In accordance with Lemma 2, it is
sufficient to prove that from any of the states $(l_{1},0),$ $(0,l_{2}),$
$(l_{1}+d,0),$ $(0,l_{2}+d)$ (addition modulo $n),$ the system results in a
state of free movement.

We shall consider all possible cases. 1) Suppose $l_{2}\leq d.$

$A(t_{0})=(l_{1},0).$

Then we have

$A(t_{0}+d-l_{1})=(d,d-l_{1}),\ A(t_{0}+d)=(d+l_{1},d),$
$A(t_{0}+n-l_{1})=(0,n-l_{1}),\ A(t_{0}+n)=(l_{1},0).$

Thus the states $(l_{1},0),$ $(d+l_{1},d)$ are states of free movement.

We have the following sequences of states

$A(t_{0})=(0,l_{2}),\ A(t_{0}+d-l_{2})=(d-l_{2},d),$ $A(t_{0}+d)=(d,d+l_{2}),\
A(t_{0}+n-l_{2})=(n-l_{2},0),$ $A(t_{0}+n)=(0,l_{2}).$

Hence the states $(0,l_{2}),$ $(d,d+l_{2})$ are also states of free movement.
2) Assume that $l_{1}\leq d<l_{2}<n-d,$ $l_{1}+l_{2}\leq n.$

We have

$A(t_{0})=(l_{1},0),\ A(t_{0}+d-l_{1})=(d,d-l_{1}),$ $A(t_{0}+d)=(d+l_{1},d),\
A(t_{0}+n-l_{1})=(0,n-l_{1}),$ $A(t_{0}+n)=(l_{1},0);$ $A(t_{0})=(0,l_{2}),\
A(t_{0}+d)=(d,d+l_{2}),$ $A(t_{0}+n+l_{2})=(n-l_{2},0),\
A(t_{0}+n-l_{2}+d)=(n-l_{2}+d,d),$ $A(t_{0}+n)=(0,l_{2}).$

Thus the states $(l_{1},0),$ $(d+l_{1},d),$ $(0,l_{2}),$ $(d,d+l_{2})$ are
states of free movement. 3) If $l_{1}\leq d<l_{2},$ $l_{2}\geq n-d,$
$l_{1}+l_{2}\leq n,$ then we have

$A(t_{0})=(l_{1},0),\ A(t_{0}+d-l_{1})=(d,d-l_{1}),$ $A(t_{0}+d)=(d+l_{1},d),\
A(t_{0}+n-l_{1})=(0,1-l_{1}),$ $A(t_{0}+n)=(l_{1},0);$ $A(t_{0})=(0,l_{2}),\
A(t_{0}+n-l_{2})=(n-l_{2},0),$ $A(t_{0}+d)=(d,d+l_{2}-n),\
A(t_{0}+d+n-l_{2})=(n-l_{2}+d,d),$ $A(t_{0}+n)=(0,l_{2}).$

Whence, the states $(l_{1},0),$ $(d+l_{1},d),$ $(0,l_{2}),$ $(d,d+l_{2}-n)$
are states of free movement. 4) Suppose $l_{1}>d,$ $l_{1}+l_{2}\leq 1.$ Then
we have

$A(t_{0})=(l_{1},0),\ A(t_{0}+d)=(l_{1}+d,d),$ $A(t_{0}+n-l_{1})=(0,n-l_{1}),\
A(t_{0}+n-l_{1}+d)=(d,1-l_{1}+d),$ $A(t_{0}+n)=(l_{1},0),\
A(t_{0})=(0,l_{2}),$ $A(t_{0}+d)=(d,d+l_{2}),\ A(t_{0}+n-l_{2})=(n-l_{2},0),$
$A(t_{0}+n)=(0,l_{2}).$

Thus, the states $(l_{1},0),$ $(d+l_{1},d),$ $(0,l_{2}),$ $(d,d+l_{2}),$ are
states of free movement. We have considered all possible cases under the
assumption that the inequality $l_{1}+l_{2}\leq n$ holds. Theorem 1 has been
proved.

## 6\. Behavior of the system under the conditions $l_{1}\leq n-d,$
$l_{1}+l_{2}>n$

We shall prove the theorem that describes the behavior of the system in the
case such that the system does not result in a state of free movement and
collapse. Theorem 2. Assume that

$l_{1}\leq n-d,\ \ l_{1}+l_{2}>n.$

Then, from any initial state, the system results in states of the same limit
cycle with period

$T=l_{1}+l_{2},$ $None$

and the average velocity is equal to

$v_{1}=v_{2}=\frac{n}{l_{1}+l_{2}}.$ $None$

Proof. In accordance with Lemma 2, it is sufficient to prove that, from any
initial states $(l_{1},0),$ $(0,l_{2}),$ $(l_{1}+d,0),$ $(0,l_{2}+d)$
(addition modulo 1) the system results in a state of free movement.

We shall prove all possible cases. 1) Suppose $l_{1}\leq d,$ $l_{1}+l_{2}>1.$
Then we have the following sequences of transitions

$A(t_{0})=(l_{1},0),\ A(t_{0}+d-l_{1})=(d,d-l_{1}),$ $A(t_{0}+d)=(d+l_{1},d),\
A(t_{0}+n-l_{1})=(0,n-l_{1}),$ $A(t_{0}+l_{2})=(0,l_{2}),\
A(t_{0}+n)=(n-l_{2},0),$ $A(t_{0}+l_{1}+l_{2})=(l_{1},0),\
A(t_{0})=(d,d+l_{2}-n),$ $A(t_{0}+n-l_{2})=(d+n-l_{2},d),\
A(t_{0}+l_{1})=(d+l_{1},d).$

Thus the states $(l_{1},0),$ $(0,l_{2}),$ $(d+l_{1},d)$ belong to a limit
circle with period $l_{1}+l_{2}$ and the average velocity of clusters
calculated with the formulas (5), (6), and, from the state $(d,d+l_{2}-1),$
the system results in a state of this cycle. 2) If $l_{1}>d,$ $l_{2}\leq n-d,$
$l_{1}+l_{2}>n,$ then we have

$A(t_{0})=(l_{1},0),\ A(t_{0}+d)=(d+l_{1},d),$ $A(t_{0}+n-l_{1})=(0,n-l_{1}),\
A(t_{0}+l_{2})=(0,l_{2}),$ $A(n)=(n-l_{2},0),\ A(l_{1}+l_{2})=(l_{1},0),$
$A(t_{0})=(0,d+l_{2}),\ A(t_{0}+n-d-l_{2})=(n-l_{2},0),$
$A(t_{0}+l_{1}-d)=(l_{1},0).$

Hence the states $(l_{1},0),$ $(0,l_{2}),$ $(d+l_{1},d)$ belong to a limit
cycle with the period calculated with the formula (5) and the average velocity
calculated with the formula (6), and, from the state $(d,d+l_{2}-n),$ the
system results in a state of this cycle. 3) Assume that $d<l_{1}\leq n-d,$
$l_{2}>n-d.$ In this case,

$A(t_{0})=(l_{1},0),\ A(t_{0}+d)=(d+l_{1},d),$ $A(t_{0}+n-l_{1})=(0,n-l_{1}),\
A(t_{0}+l_{2})=(0,l_{2}),$ $A(t_{0}+n)=(n-l_{2},0),\
A(t_{0}+l_{1}+l_{2})=(l_{1},0);$ $A(t_{0})=(d,d+l_{2}-n),\
A(t_{0}+n-l_{2})=(d+n-l_{2},d),$ $A(t_{0}+d)=(d+l_{1},d).$

Hence the states $(l_{1},0),$ $(0,l_{2}),$ $(d+l_{1},d)$ belong to a limit
cycle with the period and average velocity calculated with the formulas (5),
(6), and, from the state $(d,d+l_{2}-n),$ the system results in a state of
this cycle.

Thus the states $(l_{1},0),$ $(0,l_{2}),$ $(d+l_{1},d)$ belong to a limit
cycle with the period and average velocity of clusters calculated with the
formulas (5), (6), and, from the state $(d,d+l_{2}-n),$ the system results in
a state of this cycle. We have considered all possible cases under the
assumption of Theorem 2. Thus Theorem 2 has been proved.

## 7\. Criterion for resulting in a state of collapse

The following theorem provides a necessary and sufficient condition of system
resulting in free movement

Theorem 3. If (3) holds, then the system results in a state collapse from any
initial state. If (3) does not hold, then the system does not result in a
state collapse from any initial state. Proof. If (3) holds, then (2) also
holds, and, in accordance with Lemma 3, the system cannot be in a state of
free movement. Therefore, in accordance with Lemma 2, it is sufficieent to
prove that the system results in a state of collapse from any of the states
$(l_{1},0),$ $(0,l_{2}),$ $(d+l_{1}-n,0),$ $(0,d+l_{2}-n).$

We have the sequences of states

$A(t_{0})=(l_{1},0),\ A(n-l_{1})=(0,n-l_{1}),\ A(d)=(0,d);$
$A(t_{0})=(0,l_{2}),\ A(n-l_{2})=(n-l_{2},0),\ A(d)=(d,0);$
$A(t_{0})=(d+l_{1}-n,0),\ A(n-l_{1})=(d,0);$ $A(t_{0})=(0,d+l_{2}-n),\
A(n-l_{2})=(0,d).$

Since the states $(0,d),$ $(d,0)$ are states of collapce, we get Theorem 3.

Figure 2:  1) v=1, 2) $v=n/(l_{1}+l_{2})$, 3) $v=0$

## 8\. Conclusion

We have proved that, for any $l_{1},$ $l_{2},$ $d,$ the average velocity of
clusters is the same, i.e., $v_{1}=v_{2},$ and the average velocity does not
depend on the initial state. We have obtained the values of the average
velocity. The values of the average velocity can be equal to 0,
$n/(l_{1}+l_{2})$ or 1.

The areas of velocity modes depending on $l_{1}$, $l_{2}$ are marked in fig.
2.

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