Keldysh Wormholes and Anomalous Relaxation in the Dissipative Sachdev-Ye-Kitaev Model Antonio M. García-García1Lucas Sá2yJacobus J.

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Keldysh Wormholes and Anomalous Relaxation in the Dissipative

Sachdev-Ye-Kitaev Model

Antonio M. García-García,1, ∗Lucas Sá,2, †Jacobus J.

M. Verbaarschot,3, ‡and Jie Ping Zheng (郑杰平)1, §

1Shanghai Center for Complex Physics, School of Physics and Astronomy,

Shanghai Jiao Tong University, Shanghai 200240, China

2CeFEMA, Instituto Superior Técnico, Universidade de Lisboa,

Av. Rovisco Pais, 1049-001 Lisboa, Portugal

3Center for Nuclear Theory and Department of Physics and Astronomy,

Stony Brook University, Stony Brook, New York 11794, USA

We study the out-of-equilibrium dynamics of a Sachdev-Ye-Kitaev (SYK) model, N

fermions with a q-body interaction of inﬁnite range, coupled to a Markovian environment.

Close to the inﬁnite-temperature steady state, the real-time Lindbladian dynamics of this

system is identical to the near-zero-temperature dynamics in Euclidean time of a two-site

non-Hermitian SYK with intersite coupling whose gravity dual has been recently related to

wormhole conﬁgurations. We show that the saddle-point equations in the real-time formu-

lation are identical to those in Euclidean time. Indeed, an explicit calculation of Green’s

functions at low temperature, numerical for q= 4 and analytical for q= 2 and large q,

illustrates this equivalence. Only for very strong coupling does the decay rate approach

the linear dependence on the coupling characteristic of a dissipation-driven approach to the

steady state. For q > 2, we identify a potential gravity dual of the real-time dissipative SYK

model: a double-trumpet conﬁguration in a near-de Sitter space in two dimensions with

matter. This conﬁguration, which we term a Keldysh wormhole, is responsible for a ﬁnite

decay rate even in the absence of coupling to the environment.

∗amgg@sjtu.edu.cn

†lucas.seara.sa@tecnico.ulisboa.pt

‡jacobus.verbaarschot@stonybrook.edu

§jpzheng@sjtu.edu.cn

arXiv:2210.01695v3 [hep-th] 4 May 2023

CONTENTS

I. Introduction 3

II. Lindblad equation and SYK Hamiltonians 5

A. The vectorized Lindblad equation 6

B. The Keldysh approach 7

C. The Euclidean approach 9

III. Comparison between the Euclidean two-site SYK and the real-time open SYK for q= 4 13

IV. Analytical comparison between the Euclidean two-site SYK and the real-time open SYK

for q= 2 18

V. Analytical comparison between the Euclidean two-site SYK and the real-time open SYK

in the large-qlimit 21

VI. Gravity dual of a dissipative SYK: Keldysh wormhole 23

VII. Conclusions 26

Acknowledgments 27

A. Choi-Jamiolkowski isomorphism 28

1. Operator reﬂection 28

2. The vectorized Lindblad operator 30

B. Details on the real-time evolution of the open SYK model 31

C. Details on the Euclidean evolution of the two-site SYK model 35

D. Relation between the Euclidean and real-time calculations 36

E. Exact ﬁnite-Nresults 37

1. Analytical ﬁnite-Nresults 37

2. Numerical ﬁnite-Nresults for q= 4 41

F. Large-qlimit in the Euclidean problem 45

References 48

I. INTRODUCTION

The existence of an environment, either thermal, quantum, or the very measurement process,

makes the time evolution of a quantum system not strictly unitary, complicating the description of

its time evolution. An elegant approach to this problem, advocated by Caldeira and Leggett [1],

is to describe the system plus environment by an Hermitian Hamiltonian. The integration of the

environmental degrees of freedom then results in nonunitary dynamics. Especially for many-body

systems, it is technically hard to tackle the resulting problem, either numerically or analytically,

because the tracing out of the environment leads to nonlocal interactions in time.

In the limit where the environment has a suﬃciently short correlation time, the Liouvillian

generator that governs the quantum time evolution of the density matrix of the reduced system can

be written in the so-called Lindblad form [2–6]. Compared to the usual von Neumann equation of

motion, the Lindblad equation contains additional terms modeled by the so-called jump or Lindblad

operators [3] that describe the coupling of the system to the environment. The advantage of this

approach is that the quantum master equation is expressed only in terms of the system degrees of

freedom, which facilitates its solution. In recent times, there has been a renewed interest in diﬀerent

aspects of the Liouvillian dynamics, especially in random systems [7–16], including the study of the

spectrum [7, 10, 11, 14], with the aim to extending the Bohigas-Giannoni-Schmit conjecture [17]

to dissipative quantum systems [18–20], the characterization of the late-time dynamics [8–10, 15]

towards a steady state [10, 16, 21, 22] or the robustness of dissipative quantum chaotic features

[23].

The Keldysh path integral [24, 25] is the standard procedure used to investigate the nonequilib-

rium time evolution of the density matrix. A key diﬀerence to the standard path integral, which

will be very important in the following, is that, in order to set up the calculation for the den-

sity matrix, it is necessary [24, 25] to consider the time evolution of a state matrix, not a state

vector. This results in two time branches, one forward and one backward in time, and therefore

to the doubling of degrees of freedom of the path integral (vectorization). In this representation,

the density matrix is a state in the doubled Hilbert space and the Liouvillian has two parts: the

ﬁrst is anti-Hermitian, corresponding to the Hermitian Hamiltonian, before any coupling to the

environment but multiplied by the imaginary unit, and its conjugate such that the whole system

is anti-Hermitian; the second contains Lindblad jump operators depending on ﬁelds living in both

copies and includes (i) an explicit coupling between the degrees of freedom of the two copies and (ii)

intrasite interactions eﬀectively modifying the (anti-)Hermitian part. Using standard ﬁeld theory

techniques, it is then possible to investigate the out-of-equilibrium time evolution of a strongly

interacting system either coupled to a bath or undergoing a continuous measurement process. If we

are interested in the steady state, and in suﬃciently long timescales, it is plausible to assume that

the initial state becomes irrelevant, leading to further simpliﬁcations, namely, a time-translational

invariant relaxation.

Intriguingly, operators similar to the vectorized Liouvillian are being employed in a completely

diﬀerent context: as non-Hermitian two-site Sachdev-Ye-Kitaev (SYK) Hamiltonians [26–34], N

Majoranas with inﬁnite range q-body interactions in zero dimensions, whose gravity dual is a

wormhole [35–37]. Earlier, a single-site SYK model was proposed as an example of the holographic

duality based on the fact that features such as the saturation of a bound on chaos [38], spectral

correlations described by random matrix theory [39, 40], and an exponential growth of the den-

sity of low-energy excitations [41–43] are shared by certain near-AdS2backgrounds [44–47] in the

low-temperature limit. It has also been noticed [35] that the gravity dual of two identical two-site

Hermitian SYKs with a weak intersite coupling is a traversable wormhole [36]. These traversable

wormholes are the dominant conﬁgurations only for suﬃciently low temperature. At higher tem-

perature, a transition to a black hole conﬁguration occurs, which can also be characterized by an

analysis of level statistics [48].

By contrast, a pair of non-Hermitian PT-symmetric SYKs with no explicit intersite coupling in

the low-temperature limit is dual to a Euclidean wormhole [37]. The presence of an explicit coupling

triggers [49] a transition from Euclidean to traversable wormholes. Both traversable and Euclidean

wormholes are characterized by a gapped ground state and a ﬁrst-order phase transition in the free

energy separating the wormhole and black hole phases. Diﬀerences include a qualitatively diﬀerent

dependence of the gap on the parameters of the model and a diﬀerent pattern of oscillations of

Green’s functions in real time directly related to diﬀerent symmetries of the solutions: U(1) for

Euclidean and SL(2, R)for traversable. For earlier work on wormholes, mostly in higher dimensions,

see [50–54].

In this paper, we show that the near-zero-temperature dynamics of a two-site SYK Hamiltonian

in Euclidean time, dual to gravitational wormholes in certain limits, and the real-time Liouvillian

describing a single-site SYK coupled to a Markovian bath close to the inﬁnite-temperature steady

state coincide. More speciﬁcally, we show that the equations of motion in the Euclidean problem

in the low-temperature limit are identical to the equations of motion in the real-time Liouvillian

evolution close to the steady state. This implies that the retarded Green’s function of the real-time

problem is identical to the equivalent one in the Euclidean problem. As a consequence, the so-called

gap in the Euclidean formulation that characterizes the wormhole phase is equal to the dominant

decay rate in the real-time Liouvillian evolution. This equivalence only holds when neglecting the

boundary conditions, which are generally diﬀerent for the two approaches. In practical terms, the

equivalence works for suﬃciently low temperatures in the Euclidean problem and for times such

that the system is suﬃciently close to the steady state in the real-time problem.

Finally, we note that the gravitational dual of a dissipative ﬁeld theory with decoherence was

investigated in Ref. [55]. We also stress that there are already several works that have studied

diﬀerent aspects of the non-Hermitian SYK model: from the use of the Lindblad formalism [56, 57]

and the investigation of the entanglement entropy growth [58, 59] and decoherence eﬀects [60–62],

to its relation to Euclidean wormholes [37, 63], replica symmetry breaking [49, 64], and a symmetry

classiﬁcation [65]. However, these studies have not investigated the mentioned intriguing relation

between wormholes in Euclidean time and strongly interacting dissipative systems in real time. We

shall see that this ﬁnding has a strong impact on important features of the dynamics, such as an

enhancement of the decay rate in the limit of very weak coupling to the environment.

The remainder of the paper is organized as follows. In the next section, we introduce the

open SYK model, which will be our main object of study, and the equations of motion in the ΣG

formulation [34] in both real and Euclidean time. In Sec. III, we calculate the long-time behavior

of the Green’s function by numerically solving the Schwinger-Dyson (SD) equations for q= 4

and ﬁnd an anomalously large decay rate (or gap) for small intersite coupling. For q= 2, the

Green’s functions can be evaluated analytically and, because the model is not quantum chaotic,

no anomalous relaxation is found, which is discussed in Sec. IV. The large-qlimit is worked out

analytically in Sec. V. The gravity dual of the SYK Hamiltonian coupled to an environment, dubbed

Keldysh wormhole, is discussed in Sec. VI and concluding remarks are made in Sec. VII. Additional

technical details are worked out in six appendices.

II. LINDBLAD EQUATION AND SYK HAMILTONIANS

We study a Hermitian SYK Hamiltonian with a q-body interaction of inﬁnite range coupled to

a Markovian environment. The SYK model is deﬁned by the Hamiltonian

HSYK =−iq/2X

i1<···<iq

Ji1···iqψi1···ψiq,(1)

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KeldyshWormholesandAnomalousRelaxationintheDissipativeSachdev-Ye-KitaevModelAntonioM.García-García,1,LucasSá,2,yJacobusJ.M.Verbaarschot,3,zandJiePingZheng(郑杰平)1,x1ShanghaiCenterforComplexPhysics,SchoolofPhysicsandAstronomy,ShanghaiJiaoTongUniversity,Shanghai200240,China2CeFEMA,InstitutoSuperiorTécn...

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Keldysh Wormholes and Anomalous Relaxation in the Dissipative Sachdev-Ye-Kitaev Model Antonio M. García-García1Lucas Sá2yJacobus J.

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