More on Complex Sachdev-Ye-Kitaev Eternal Wormholes · 2020. 11. 30. · More on Complex Sachdev-Ye-Kitaev Eternal Wormholes Pengfei Zhang1 1California Institute of Technology, Pasadena,

More on Complex Sachdev-Ye-Kitaev Eternal Wormholes

Pengfei Zhang1

1California Institute of Technology, Pasadena, CA 91125, USA

November 30, 2020

Abstract

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

Contents

1 Introduction 2

2 Model and its Low-energy Description 32.1 Conformal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 the TFD State Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Effective Action and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 92.4 Quench Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Numerical Results 133.1 Imaginary time results and the Phase diagram . . . . . . . . . . . . . . . . . . . 133.2 Real time results and Quench dynamics . . . . . . . . . . . . . . . . . . . . . . . 15

4 Summary 18

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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 Majoranaoperators is 1/q. Such conformal symmetry is explicitly broken in the Lagrangian, which leadsto a finite Schwarzian action for reparametrization modes [3,4]. Interestingly, such Schwarzianaction matches the boundary action of the Jackiw-Teitelboim gravity in two-dimensional nearlyanti-de Sitter (AdS2) spacetime, and consequently, the SYK model is a concrete model withlow-energy holographic description. Later, the model is generalized to study various problemsin both high energy and condensed matter communities [5–22].

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 traversablewormhole in the global AdS2 spacetime with ds2 = (−dt2 +dσ2)/ sin2 σ, where two copies of theSYK model lie on two boundaries near σ = 0 and σ = π. Such a state can also be prepared asa thermofield double (TFD) state of the SYK model. At high temperatures, the coupled SYKmodel 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], andthe real-time formation of the wormhole is discussed in [26]. Experimentally, the coupled SYKis found to be related to random spin models [27], and there is also a proposal for realizing theMQ model [28] in solid-state materials. The tunneling spectroscopy [29] and the entanglemententropy [30] of the coupled SYK model has also been studied.

In this work, we study the effect of additional conservation law on the eternal traversablewormholes in the SYK-like models. To be concrete, we focus on the complex fermion versionof the coupled SYK model with U(1) symmetry. For a single SYK model, the generalizationto complex fermions where the charge is conserved has been studied in details [6–8]. There arealso recent studies [31–33] on the complex version of the coupled SYK model. However, in theseworks, the chemical potential for two copies is fixed to be the same, which, as we will see, leadsto the same wormhole state as no chemical potential term in the low-energy limit. We insteadconsider the case where the chemical potential of both systems can be tuned freely. In thelow-energy limit, we find a family of conformal solutions for wormholes parameterized by threeIR parameters t′, P and E , which comes from the two chemical potentials and one couplingstrength 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 Schwarziantheory by requiring the diffeomorphism-invariance [4, 8], where one could can the energy levelof boundary gravitons and the quench dynamics.

The paper is organized as follows: In section 2, we first describe our model and conformalsolution, where the IR parameters can be fixed through the TFD State perspective, togetherwith minimizing the effective action. In section 3, we compare our low-energy solutions to thenumerics, in both imaginary-time and real-time approach. We finally summarize our results in4.

2

2 Model and its Low-energy Description

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

H =1

4

∑ijkl

Jijklc†L,ic†L,jcL,kcL,l +

1

4

∑ijkl

Jijklc†R,ic

†R,jcR,kcR,l

− µL∑i

(c†L,icL,i −

1

2

)− µR

∑i

(c†R,icR,i −

1

2

)− µc

∑i

(c†R,icL,i + c†L,icR,i

).

(1)

Here the random coefficients Jijkl take distribution

Jijkl = 0, |Jijkl|2 = 2J2/N3. (2)

We have added different chemical potential term µL/µR for the left/right copy of the complexSYK model, with an additional directly hopping term µc. Without loss of generality, we haveset the hopping amplitude to be real. For later convenience, we introduce the averaged µ andthe relative chemical potential µ as

µL − µR2

= µ,µL + µR

2= µ. (3)

Before analyzing the model, we add a few comments:

1. Although we introduced the model with two copies of the complex SYK4 system for sim-plicity, it is straightforward to generalize all discussions in the paper for other even finiteq ≤ 4. Generally, the random interaction strength of the right copy should contain anadditional 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 1.

2. Without the hopping term, the model contains two copies of the complex SYK model,and the symmetry is U(1) × U(1), which is broken to a single U(1) by µc. The ideawhy here we still introduce two different chemical potential terms is that for eternalwormholes, we are focusing on the limit of small µc → 0. In this limit, the breaking ofthe relative phase transformation cL,i → cL,ie

iα, cR,i → cR,ie−iα is weak and there exists a

corresponding pseudo Goldstone mode. Therefore we expect a relative chemical potentialcan play important roles in the low-energy limit.

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 ↑ and ↓. In such models, therelative chemical potential µ corresponds to a magnetic field in the z direction, whilethe coupling term µc represents a magnetic field in the x direction. By a particle-holemapping, this can also be understood as a spinful SYK model with a proximity pairingterm.

1We thank Yingfei Gu for the discussion of the large-q limit of a single complex SYK model.

3

2.1 Conformal Solution

We consider the model in the imaginary-time formalism. We focus on the large-N limit. Un-der the replica diagonal assumption, after introducing the bilocal fields G(τ, τ ′), Σ(τ, τ ′) andintegrating out fermions, the partition function becomes:

S[G,Σ]

N=− Tr [log(σαβ − Σαβ)]−

∫dτdτ ′

[Gαβ(τ, τ ′)Σβα(τ ′, τ) +

J2

4G2αβ(τ, τ ′)G2

βα(τ ′, τ)

]=− Tr

[log(−Σαβ)

]−∫dτdτ ′

[Gαβ(τ, τ ′)Σβα(τ ′, τ) +

J2

4G2αβ(τ, τ ′)G2

βα(τ ′, τ)

]−∫dτdτ ′ Gαβ(τ, τ ′)σβα(τ ′, τ).

(4)Here α, β = L,R labels two copies of the SYK model, Gαβ(τ, τ ′) =

∑i cα,i(τ)cβ,i(τ

′)/N is the

fluctuation of fermion bilinear. We have defined Σ = Σ + σ, with the UV perturbation [4, 8]

σ(τ, τ ′) =

(δ′(τ − τ ′)− µLδ(τ − τ ′) −µcδ(τ − τ ′)

−µcδ(τ − τ ′) δ′(τ − τ ′)− µRδ(τ − τ ′)

). (5)

Without σ, there is an emergent conformal symmetry of bilocal fields:

Gαβ(τ, τ ′)→ ϕ′α(τ)14ϕ′β(τ ′)

14 eiλα(τ)−iλβ(τ ′)Gαβ (ϕα(τ), ϕβ(τ ′)) ,

Σαβ(τ, τ ′)→ ϕ′α(τ)34ϕ′β(τ ′)

34 eiλα(τ)−iλβ(τ ′)Σαβ (ϕα(τ), ϕβ(τ ′)) ,

(6)

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:(G−1(ω)

)αβ

= −Σαβ(ω) = σαβ(ω)− Σαβ(ω), Σαβ(τ) = −J2Gαβ(τ)2Gβα(−τ). (7)

In the low-energy limit J |τ | 1, we hope to neglect σ and consider the zero-temperaturesolution of ∑

γ

∫ ∞−∞

dτ ′′Gαγ(τ − τ ′′)J2Gγβ(τ ′′ − τ ′)2Gβγ(τ′ − τ ′′) = δαβδ(τ − τ ′). (8)

The validity of (8) needs further explanations: When µc = 0 and µ = 0, the system repre-sents decoupled complex SYK models with opposite chemical potentials ±µ with the scalingdimension of fermion operators 1/4. Consequently, the perturbation from left-right coupling isrelevant and the system should be gapped. Consequently, to neglect the UV symmetry breakingterm σ, we should require the solution of (8) leads to a gap much larger than the bare gap µc.As we will see later, this requires µ/J 1 and µc/J 1 [5], while µ/J can be O(1).

Interestingly, there exists a family of conformal solutions with three IR parameters E , P andt′. This is what we expect since there are three UV parameters µL, µR, and µc in σ. Explicitly,

4

IR parameters t′ P ERelations t′ = b2/3J2/3µ

2/3c

2α2/3S

P = µ QL = − θπ− 1

4sin(2θ), e2πE = tan(π/4 + θ)

Table 1: A summary of relations between the IR parameters and UV parameters/physicalobservables in the eternal wormhole phase.

we have

GLL(τ) = b× sgn(τ)eπEsgn(τ)+Pτ(

t′

2 sinh(t′|τ |/2)

)1/2

,

GRR(τ) = b× sgn(τ)e−πEsgn(τ)+Pτ(

t′

2 sinh(t′|τ |/2)

)1/2

,

GLR(τ) = GRL(τ) = −b× ePτ(

t′

2 cosh(t′|τ |/2)

)1/2

.

(9)

Here the overall coefficient b satisfies 4πJ2 cosh(2πE)b4 = 1. GLR(τ) = GRL(τ) is a consequenceof the time-reversal symmetry. To make the Green’s function well-defined as |τ | → ∞, werequire that P < t′/4. In the following discussion, we will always neglect terms ∼ t′2. For|τ |t′ 1, GLL and GRR reduce to the zero-temperature Green’s function of decoupled complexSYK model with asymmetry parameters ±E [7, 8]:

GLL(τ) = −GRR(−τ) = beπEsgn(τ) sgn(τ)

|τ |1/2. (10)

On the contrary, when both µ = 0 and µ = 0, we should get back to the Majorana wormholesolution [5], which corresponds to E = P = 0. When we turn on µ with µ = 0, the ensemble ofthe system is still invariant under a particle-hole transformation cL,i → c†R,i. This is compatiblewith having a non-vanishing E . We also have the gap of the fermionic excitations Eg = t′/4.On the other hand, the existence of P breaks this particle-hole symmetry, which corresponds tothe existence of µ. The relation between the IR parameters and the UV parameters or physicalobservables are discussed in the next few subsections, while the results are summarized in theTable 1.

In frequency space, (9) becomes

GLL(ω + iP) = b× 2√

2π3/2sech(2πω/t′)(i sinh(πω/t′) cosh(πE) + cosh(πω/t′) sinh(πE))

Γ(

34− iω

t′

)Γ(iωt′

+ 34

) ,

GRR(ω + iP) = b× 2√

2π3/2sech(2πω/t′)(i sinh(πω/t′) cosh(πE)− cosh(πω/t′) sinh(πE))

Γ(

34− iω

t′

)Γ(iωt′

+ 34

) ,

GLR(ω + iP) = GRL(ω + iP) = −b×Γ(

14

)Γ(

34

)Γ(

14− iω

t′

)Γ(iωt′

+ 14

)√

2t′π3/2.

(11)Here the Gαβ is analytic on the stripe with |Im(ω + iP)| < t′/4, as anticipated from the real-time expressions. There is a similar expression for the self energy Σαβ, and it is straightforwardto check they satisfies the Schwinger-Dyson equation (8).

Here we present another method which allow us to derive (9) from the single SYK con-formal solution (10). For conciseness, let us introduce G0(τ, τ ′) = b sgn(τ − τ ′)/|τ − τ ′|1/2

5

and Σ0(τ, τ ′) = b3 sgn(τ − τ ′)/|τ − τ ′|3/2. It is known that (10) satisfies the self-consistentequation [7]:

− J2

∫ ∞−∞

dt3 eπE(sgn(t13)+sgn(t32))G0(t1, t3)Σ0(t3, t2) = δ(t1 − t2). (12)

Here τij = τi − τj. Now we split the integral of t3 into (−∞, 0) and (0,∞), which gives

− J2

(∫ 0

−∞dt3 +

∫ ∞0

dt3

)eπE(sgn(t13)+sgn(t32))G0(t1, t3)Σ0(t3, t2) = δ(t1 − t2). (13)

As an example, we consider t1 > 0 and t2 > 0. We define new time variable τ as

t =

et

′τ t > 0,

−et′τ t < 0.(14)

Here τ ∈ (−∞,∞). This change of variable can be motivated from the gravity picture, whereone can map the Poincare plane to the gAdS2 stripe. This gives

δ(τ1 − τ2)

∣∣∣∣dτ1

dt1

∣∣∣∣1/4 ∣∣∣∣dτ2

dt2

∣∣∣∣3/4 =

− J2

∫ ∞−∞

dτ3

∣∣∣∣dt3dτ3

∣∣∣∣ eπEsgn(τ13)G0(et′τ1 , et

′τ3)eπEsgn(τ32)Σ0(et′τ3 , et

′τ2)

− J2

∫ ∞−∞

dτ3

∣∣∣∣dt3dτ3

∣∣∣∣G0(et′τ1 ,−et′τ3)Σ0(−et′τ3 , et′τ2).

(15)

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

δ(τ1 − τ2) =

− J2

∫ ∞−∞

dτ3 eπEsgn(τ13)+Pτ13

∣∣∣∣dt1dτ1

dt3dτ3

∣∣∣∣1/4G0(et′τ1 , et

′τ3)︸︷︷︸GLL

eπEsgn(τ32)+Pτ32

∣∣∣∣dt3dτ3

dt2dτ2

∣∣∣∣3/4 Σ0(et′τ3 , et

′τ2)︸︷︷︸ΣLL

− J2

∫ ∞−∞

dτ3 ePτ13

∣∣∣∣dt1dτ1

dt3dτ3

∣∣∣∣1/4G0(et′τ1 ,−et′τ3)︸︷︷︸

−GLR

ePτ32

∣∣∣∣dt3dτ3

dt2dτ2

∣∣∣∣3/4 Σ0(−et′τ3 , et′τ2)︸︷︷︸−ΣRL

,

(16)where we have used the (imaginary version of) U(1) conformal symmetry (6) to introduce P ,and identified the Gαβ and Σαβ, which match the definition (9). This equation is exactly (8)with α = β = L. Other components correspond to taking t1 and t2 in different regions (t1 ≶ 0and t2 ≶ 0).

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

Q = QL +QR =

⟨c†L,icL,i −

1

2

⟩+

⟨c†R,icR,i −

1

2

⟩= −GLL(0−)−GRR(0−)− 1. (17)

6

Using the fact that Gαα(0+)−Gαα(0−) = 1, we have

Q = −tr[G(0+) +G(0−)]

2=

∫dτ tr [σ(τ)G(−τ)] τ = −

∫ ∞−∞

dτ tr[Σ(τ)G(−τ)

]τ. (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 inthe long time limit, and there is no need to introduce additional regulators.

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

Q = − 1

2πi

∫ ∞−∞

dω ∂ωtr [logG(ω)] = − 1

2πi

(∫|ω|<Λ

dω +

∫|ω|>Λ

dω

)∂ωtr [logG(ω)] , (19)

where we cut the integral into two parts with Λ ∼ J t′. In the second part of the integral,we are considering the frequency regime where the Green’s function takes the same form as asingle complex SYK model. Consequently, the result matches the corresponding calculation ofthe single complex model [8]:

− 1

2πi

∫|ω|>Λ

dω ∂ωtr [logG(ω)] =1

2πi(−2iθ + 2iθ) = 0, (20)

where θ is related to E as e2πE = tan(π/4 + θ). For the integral over small frequency, we coulddirectly use conformal solutions (11):

− 1

2πi

∫ Λ

−Λ

dω ∂ωtr [logG(ω)] = − 1

2πi

∫ Λ

−Λ

dz ∂ztr [logG(z + iP)] +O((P/J)2) ≈ 0. (21)

We have deformed the contour to absorb the presence of P , and after the deformation, theintegral becomes zero. We also neglect high-order terms in P/J (or t′/J). Finally, we find thespecific charge Q = 0 for any valid wormhole solution (9). This is consistent with the TFDperspective 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 ofcoupling two 1+1-D CFTs with different chiralities [34], where the ground state is argued tobe a TFD state. In [5], authors find the coupled SYK model show similar behaviors with theground state being the TFD state. From the gravity perspective, this is dual to the vacuum ofthe 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 jus-tification will be given in the next section. To construct the TFD state, we firstly consider

the ground state of the relevant hopping term −µc∑

i

(c†R,icL,i + c†L,icR,i

), which is simply the

maximally entangled state

|I〉 = ⊗i1√2

(|0〉L,i|1〉R,i + |1〉L,i|0〉R,i) , (22)

7

TFD parameters βeff µ′ µ′

Relations t′βeff = 2π Redundant µ− µ′ = t′E

Table 2: A summary of relations between the TFD parameters, IR parameters and parametersin the Hamiltonian (1).

with all symmetric modes cL,i + cR,i being occupied. The state satisfies

cL,i|I〉 = cR,i|I〉, c†L,i|I〉 = −c†R,i|I〉. (23)

To take other terms in the Hamiltonian into account, we assume that to the leading orderof small µc/J , the ground state of (1) is a TFD state with an effective Hamiltonian Heff:

|G〉 =1√Z

exp(−βeffHeff/4)|I〉, (24)

where Zeff is the normalization factor and Heff is given by

Heff =1

4

∑ijkl

Jijklc†L,ic†L,jcL,kcL,l +

1

4

∑ijkl

Jijklc†R,ic

†R,jcR,kcR,l

− µ′L∑i

c†L,icL,i − µ′R

∑i

c†R,icR,i.(25)

In general, we should allow (µ′L, µ′R) being different from (µL, µR). We can then define µ′ and

µ′ similar to (3). Since we are focusing on the small t′ and thus small µ limit, we expect µ′ ∝ µsince they both correspond to the particle-hole symmetry breaking. Moreover, since both µand βeff is particle-hole symmetric, we expect their µ dependence is at least second-order, whichcan be safely neglected. For µc = 0, we expect |G〉 to be a specific pure state within the groundstate sector of H, which is two decoupled complex SYK models. Consequently, we expectβeff → ∞ and µ′ = µ for µc → 0, if the coupled ground state is in the same charge sector asthe decoupled one. The precise relations are given in Table 2, which will be proved in the nexttwo subsections.

We can directly use |G〉 to determine E and P . We first notice that

c†L,icL,i|I〉 = (1− c†R,icR,i)|I〉. (26)

Consequently, we have (c†L,icL,i + c†R,icR,i

)|G〉 = |G〉. (27)

As a result, we know Q = 0, consistent with the result in the last subsection, and the µ′ term,and thus the µ term, would not change the ground state |G(µ)〉 = |G(0)〉. As a result, weexpect the only effect of µ in terms of the Green’s function is:

Gαβ(τ) = 〈G| cα,i(τ)c†β,i |G〉 = 〈G| eτH(µ)cα,ie−τH(µ)c†β,i |G〉

= eµτ 〈G| eτH(0)cα,ie−τH(0)c†β,i |G〉 = eµτGαβ(τ)µ=0

(28)

This suggests that we could directly identify P = µ.

8

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

Heff|EPR〉 =

(1

2

∑ijkl

Jijklc†L,ic†L,jcL,kcL,l − 2µ′

∑i

c†L,icL,i − µ′RN

)|EPR〉 ≡ 2HL|EPR〉

=

(1

2

∑ijkl

Jijklc†R,ic

†R,jcR,kcR,l + 2µ′

∑i

c†R,icR,i − µ′LN

)|EPR〉 ≡ 2HR|EPR〉.

(29)

Consequently, if we look at a single side, the system is in the thermal ensemble at inversetemperature βeff with chemical potential ±µ′. When we evolve the system with H, the Greensfunction GLL takes the form of (9). On the other hand, we could consider evolving the systemwith only H0, which does not include the left-right coupling term in (1), this leads to

GLL(τ) = 〈G| eτH0cL,ie−τH0c†L,i |G〉 = eµτ+δµτ 〈G| eτHLcL,ie−τHLc

†L,i |G〉 = eµτ+δµτGβeff,Es(µ′)

(τ),(30)

Where δµ = µ − µ′ is from the difference between Heff and H0. Here the subscript in Es(µ′)reminds us that it is defined for a single complex SYK model. Gβ,E(τ) is the finite temperatureGreen’s function for a single complex SYK model:

Gβ,E(τ) = b× e2πE(1/2−τ/β)

(π

β sin(πτ/β)

)1/2

, for 0 < τ < β. (31)

Since we focus on the limit where µc J which results in t′ J , we expect there is aseparation of time scale: when 1/J τ 1/t′, there should be no difference between GLL(τ)and GLL(τ). Matching the expression in this limit, we find their asymmetric factors should beidentified E = Es(µ′). This function between µ′ and E contains the UV data and no analyticalformula is available, while E can be related to the charge [7, 8]:

QL = − θπ− 1

4sin(2θ). (32)

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

S(ρL) = S0(QL) + 4π2 αSβeffJ

, (33)

where S0(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 actiondescribes the coupling between the UV source σ and IR fluctuations, which corresponds to thelast term in (4). To derive the effective action, we first determine the IR fields. Separating out

9

the U(1) and reparametrization modes, we consider the quasi-solutions (for τ1 > τ2):

GLL(τ1, τ2) = b× eπE+P(τ1−τ2)

(ϕ′L(τ1)ϕ′L(τ2)

4 sinh2(ϕL(τ1)− ϕL(τ2)/2)

)1/4

ei(λL(τ1)−λ(τ2)),

GRR(τ1, τ2) = b× e−πE+P(τ1−τ2)

(ϕ′R(τ1)ϕ′R(τ2)

4 sinh2(ϕR(τ1)− ϕR(τ2)/2)

)1/4

ei(λR(τ1)−λR(τ2)),

GLR(τ1, τ2) = −b× eP(τ1−τ2)

(ϕ′L(τ1)ϕ′R(τ2)

4 cosh2(ϕL(τ1)− ϕR(τ2)/2)

)1/4

ei(λL(τ1)−λR(τ2)),

GRL(τ1, τ2) = −b× eP(τ1−τ2)

(ϕ′R(τ1)ϕ′L(τ2)

4 cosh2(ϕR(τ1)− ϕL(τ2)/2)

)1/4

ei(λR(τ1)−λL(τ2)),

(34)

where we have defined λα(τ)− iPτ = λα(τ)− iPϕα(τ).We first consider the diagonal terms in σ, which are coupled to GLL and GRR. The IR

fields can be identified by expanding the quasi-solution around the intermediate asymptoticexpression for 1/J |τ1 − τ2| 1/t′. We have:

GLL(τ1, τ2) = G∞,E(τ1 − τ2)(1 + AL(τ+)(τ1 − τ2) +BL(τ+)(τ1 − τ2)2 + ...),

GRR(τ1, τ2) = G∞,−E(τ1 − τ2)(1 + AR(τ+)(τ1 − τ2) +BR(τ+)(τ1 − τ2)2 + ...).(35)

with τ+ = τ1+τ22

and

Aα(τ) = iλ′α(τ) + P , Bα(τ) =1

24Sch

(tanh

ϕα(τ)

2, τ

)+Aα(τ)2

2. (36)

Here the Schwarzian derivative is defined as Sch (y(τ), τ) ≡ y′′′

y′− 3

2

(y′′

y′

)2

. The general diffeo-

morphism and gauge invariance [8] suggest the first contribution to the effective action takesthe form

S(1)eff

N=− αS

J

∫dτ

[Sch

(tanh

ϕL(τ)

2, τ

)+ Sch

(tanh

ϕR(τ)

2, τ

)]+

∫dτ [f(µL − AL) + f(µR − AR)]

=− αSJ

∫dτ

[Sch

(tanh

ϕL(τ)

2, τ

)+ Sch

(tanh

ϕR(τ)

2, τ

)]+

∫dτ[f(µ− iλ′L(τ)

)+ f

(−µ− iλ′R(τ)

)].

(37)

Here we have used the fact that P = µ. The result has no dependence on µ, and this implies Q =0, making the substitution of P consistent. The function f(µ) represents the UV contributionto the free energy, which should match that of a single complex SYK model and have thesymmetry f(µ) = f(−µ). Expanding λα to the second-order gives

S(1)eff

N=− αS

J

∫dτ

[Sch

(tanh

ϕL(τ)

2, τ

)+ Sch

(tanh

ϕR(τ)

2, τ

)]+

∫dτ

[2f (µ) +

K

2λ′L(τ)2 +

K

2λ′R(τ)2

].

(38)

10

Here we define K = −f ′′(µ) = −f ′′(−µ) and assume the winding number of λα is zero for thesaddle points.

We then consider off-diagonal terms in σ, which is coupled to GLR and GRL. Differentfrom the diagonal terms where the IR fields are determined through expanding the quasi-solution around G∞,E(τ1 − τ2), the off-diagonal components of G it self can be considered asIR contributions. This can be understood from the fact that GLR in (9) can be viewed asintroducing a real-time separation δτ = iπ/t′ to the GLL. Consequently, we just add thecontribution from µc directly as:

S(2)eff

N= −2bµc

∫dτ

(ϕ′R(τ)ϕ′L(τ)

4 cosh2(ϕR(τ)− ϕL(τ)/2)

)1/4

cos(λR(τ)− λL(τ)

). (39)

The total effective action is obtained by summing up diagonal and off-diagonal contributionsSeff = S

(1)eff + S

(2)eff .

We first consider the thermal equilibrium state. The conformal solution (9) corresponds tosetting ϕα(τ) = t′τ and λα(τ) = 0. This gives the grand potential ΩWH:

ΩWH

N= Mint′

2f(µ) +

αSt′2

J−√

2bµc(t′)1/2

= 2f(µ)− 3

4

(Jb4µ4

c

αS

)1/3

, (40)

with the solution t′ = b2/3J2/3µ2/3c

2α2/3S

, where the E dependence comes form b. It is straightforward

to check (9) then satisfies the saddle-point equations for ϕα and λα, as well as the conditionthat all SL(2,R) charges should vanish [5].

The charge of the left system is given by taking derivative with µL. This gives −QL = f ′(µ),meaning the relation between µ and QL is the same as the thermodynamics of a single SYK atT = 0. It is known that for a single SYK model at finite temperature, the chemical potentialshifts as µs(0) = µs(T ) + 2πEs/β [8]. This gives the relation between µ′ and µ as

µ = µ′ + 2πE/βeff. (41)

We can also study the fluctuation of low-energy modes around the saddle point. The van-ishing of SL(2,R) charges makes it possible to choose ϕL(τ) = ϕR(τ) = ϕ(τ). For convenience,we further introduce ϕ′(τ) = eφ(τ) with the saddle point at φ(τ) = log(t′). Expanding theeffective action to the quadratic order gives

δSeff

N=

∫dτ

[K

2λ′c(τ)2 +

K

2λ′r(τ)2 +

√2bµct

′ 12λ2

r(τ) +αSJδφ′(τ)2 +

3b4/3µ4/3c J1/3

8α1/3S

δφ(τ)2

].

(42)Here we have defined λc = λL+λR√

2and λr = λL−λR√

2. We find there is a gapped graviton mode

with frequency ωgravity =√

38

(bJµcαS

)2/3

, a relative phase mode λr with frequency ωU(1) =√

2b2/3J1/6µ2/3c√

K(αS)1/6 , and a gapless field λc. These modes can be excited in certain quench dynam-

ics, similar to the oscillation behavior observed in [27]. In addition to thermodynamics, theentanglement entropy of the wormhole phase (33) can also be computed using the effective

11

action following the derivation in [35], since λα = 0 still satisfies the saddle-point equation afterintroducing replicas.

For large µ/t′ and moderate µ/J , the wormhole solution (9) breaks down, and the systemtransits into the black hole solution. The transition point is determined by comparing the freeenergy. In the black hole phase, we expect the free energy takes the form:

ΩBH

N= f(µL) + f(µR) +O(µ2

c) ≈ f(µ+ µ) + f(µ− µ) ≈ 2f(µ)− µ2K. (43)

The critical µ is then determined as Min

b2/3J2/3µ

2/3c

8α2/3S

,√

3b2/3J1/6µ2/3c

2√KαS1/6

, both of which proportional

to µ2/3c . This also justifies the neglect of higher order terms ∼ µ2

c .

2.4 Quench Dynamics

We still need to relate the TFD parameter βeff to parameters in the Hamiltonian. Followingthe idea in [5], this can be determined by considering the quench dynamics for turning off µc.

We firstly transform the effective action into the real-time with τ → iu and ϕ → iϕ.Explicitly, we have

Seff

N=

∫du

[−2αS

JSch

(tan

ϕ(u)

2, u

)+K

2λ′L(u)2 +

K

2λ′R(u)2

]+√

2b

∫du µcθ(−u)ϕ′(u)1/2 cos

(λR(u)− λL(u)

).

(44)

At u < 0, the system is in thermal equilibrium with ϕ(u) = t′u and λα(u) = 0. Foru > 0, by solving the saddle-point equations and requiring the smoothness for fields φ(u) =log(ϕ(u)), λα(u) and their first-order derivatives, we have for u > 0

ϕ(u) = 2 arctan

(tanh

t′u

2

), λα(u) = 0. (45)

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

iG>LL(u1, u2) ≡

⟨cL,i(u1)c†L,i(u2)

⟩= b× eπE+iP(u1−u2)

ϕ′(u1)ϕ′(u2)

4 sin2(iϕ(u1)−ϕ(u2)

2+ 0+

)1/4

(46)

When u1 > 0 and u2 > 0, this gives

iG>LL(u1, u2) = beπE+iPu12

(t′

2 sinh (t′|u12|/2)

)1/2

e−iπ4

sgn(u12). (47)

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

iGLL(u) = 〈G| eiH0ucL,ie−iH0uc†L,i |G〉 = beπE+iPu12

(π

βeff sinh (π|u12|/βeff)

)1/2

e−iπ4

sgn(u12),

(48)where we have used (41). The equivalence between (47) and (48) gives t′βeff = 2π.

12

+

+

+

-

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-------

/

()

-

/

()

(b)<latexit sha1_base64="b/cMj/3MakpfvQl5kysPOUFaeE8=">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</latexit>

(a)<latexit sha1_base64="7BY/x+VUns3FYonIA1k2e3ngYrc=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkWom5KooMuimy4r2gfUIsl0WkPTJEwmSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/SQMUuk4rzlrbn5hcSm/XFhZXVvfKG5uNdM4E4w3WBzGou17KQ+DiDdkIEPeTgT3Rn7IW/7wTMVbt1ykQRxdynHCuyNvEAX9gHmSqIuyt39dLDkVRy97FrgGlGBWPS6+4Ao9xGDIMAJHBEk4hIeUng5cOEiI62JCnCAU6DjHPQqkzSiLU4ZH7JC+A9p1DBvRXnmmWs3olJBeQUobe6SJKU8QVqfZOp5pZ8X+5j3RnupuY/r7xmtErMQNsX/pppn/1alaJPo40TUEVFOiGVUdMy6Z7oq6uf2lKkkOCXEK9yguCDOtnPbZ1ppU16566+n4m85UrNozk5vhXd2SBuz+HOcsaB5U3MPKwflRqXpqRp3HDnZRpnkeo4oa6miQ9wCPeMKzVbMiK7PuPlOtnNFs49uyHj4AQmyPzw==</latexit>

Figure 1: (a). The imaginary-time Green’s function for βJ = 120, µ/J = 0.05, µ/J = 0.1, andµc/J = 0.05. (b). A zoom in of (a) near 1/t′ τ β/2 or 1/t′ β − τ β/2. We can seethe numerical results are consistent with (9).



/=/=/=

/

/

/=/=/=

/

/

Figure 2: A comparison between the numerics and (50) for (a). P(µ) with different µ/J , and(b). Eg(µ) for different µ. In all cases we fix βJ = 120 and µc/J = 0.05. The dots are numericaldata and the dashed line represents the prediction of (50).

3 Numerical Results

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

3.1 Imaginary time results and the Phase diagram

In the imaginary-time approach, we adapt the standard algorithm [5] to solve the saddle-pointequation (7) for Gαβ and Σαβ. We work at large but finite temperature βJ 1 and small butfinite µc/J .

Typical Green’s functions in the wormhole phase are shown in Figure 1 (a), where we chooseβJ = 120, µ/J = 0.05, µ/J = 0.1, and µc/J = 0.05. The Green’s functions decay exponentiallyfor 1/t′ τ β/2 or 1/t′ β − τ β/2. We can approximately identify the numerical

13



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0.1

0.2

0.3

0.4

0.5

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0

0.1

0.2

0.3

0.4

0.5

/=/=/=

/

/=/=/=

/

Figure 3: The phase diagram of the model considered in this paper with βJ = 120 andµc/J = 0.05. Here we focus on moderate µ/J , with the wormhole phase, the two black holephase, and the single black hole phase. We show the density plot for Q in (a) and for QL in(b). Multiply saddle points exist in the dotted region. We also plot corresponding 1D curveswith fixed µ/J in (c) and (d).

solution at 0 < τ < β/2 with the zero-temperature Green’s function G+αβ(τ) ≡ |Gαβ(τ)|,

and the numerical solution at 0 < β − τ < β/2 with the zero-temperature Green’s functionG−αβ(τ) ≡ |Gαβ(τ − β)|. For 1/t′ τ β/2, the conformal solution predicts

logG+LL(τ) ∼ c0 + πE − (Eg − P)τ, logG−LL(τ) ∼ c0 − πE − (Eg + P)τ,

logG+RR(τ) ∼ c0 − πE − (Eg − P)τ, logG−RR(τ) ∼ c0 + πE − (Eg + P)τ,

logG+LR(τ) ∼ c0 − (Eg − P)τ, logG−LR(τ) ∼ c0 − (Eg + P)τ,

(49)

where c0 is come numerical factor and we have defined the averaged gap Eg = t′/4. In Figure 1(b), we show the numerical results, which takes exactly the same form as conformal solutions.

We then compare the numerical fitted P and Eg with the low-energy prediction summarizedin Table 1, where

P = µ, Eg(µ) = Eg(0)/ cosh(2πE)1/6. (50)

The result is shown in Figure 2 for βJ = 120 and µc/J = 0.05. We find the results match well,with a small discrepancy possibly from the fact that µc/J is not small enough.

14



- - ----

>()

- -

-

-

>()

Figure 4: The real-time Green’s functions G>(t) in thermal equilibrium within the wormholephase. We choose βJ = 40, µc/J = 0.075, µ/J = 0.1 and µ/J = 0.

We then consider large µ and explore the phase diagram for the Hamiltonian (1). For fixedµ, we perform the numerical iteration firstly from small µ to large µ, and then from large µback to small µ. We also perform similar steps for different µ with fixed µ. Near first-ordertransition points, this can give different saddle points for the same µ.

The results for βJ = 120 and µc/J = 0.05 are shown in Figure 3. At small µ/J , the systemis in the wormhole phase, with specific charge Q = 0, while QL can change when tunningµ/J . For moderate µ/J , if we consider larger µ/J , the system transits into the two black holephase. In this phase, we can approximate the solution as two separate complex SYK modelswith chemical potential µL and µR. In this case, both QL and Q change freely when tunningparameters. If we further consider a larger µ or µ, where µL becomes larger than a threshold∼ 0.25, it is known that a single complex SYK model would transit into a (nearly) fully occupiedphase with QL = 1/2 [36–38]. We call this phase the (single) black hole phase since the rightsystem 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 µL and µR become large and the systemhas QL = QR = 1/2.

3.2 Real time results and Quench dynamics

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

The quench dynamics of SYK-like models can be studied using the Keldysh approach [39–43]. We firstly define several real-time Green’s functions G> and G<:

G>αβ(u1, u2) = −i

⟨cα,i(u1)c†β,i(u2)

⟩, G<

αβ(u1, u2) = i⟨c†β,i(u2)cα,i(u1)

⟩. (51)

Here we have copied the definition (46) for completeness. The retarded and advanced Green’sfunctions GR and GA then read

GRαβ(u1, u2) = θ(u12)

(G>αβ(u1, u2)−G<

αβ(u1, u2)),

GAαβ(u1, u2) = θ(−u12)

(G<αβ(u1, u2)−G>

αβ(u1, u2)).

(52)

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ImG>LL

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ImG>LR

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- - - -

-

-

-

>(+

)



>(//

)

Figure 5: The quench dynamics when turning off µc with βJ = 40, µc/J = 0.075 and µ/J = 0.We plot ImG>

LL(u1, u2) in (a) and ImG>LR(u1, u2) in (b). When u1 > 0 and u2 > 0, we find

G>LL(u1, u2) only depends on u1 − u2, while G>

LR(u1, u2) only depends on u1 + u2. We also plotG>LL(t+ t0, t0) and G>

LR(t/2, t/2) explicitly in (c) and (d).

For the Hamiltonian (1), the self-energy Σ> and Σ< in real-time can be obtained by the ana-lytically continuation of (7) in the time domain:

Σ>αβ(u1, u2) = J2G>

αβ(u1, u2)2G<βα(u2, u1),

Σ<αβ(u1, u2) = J2G<

αβ(u1, u2)2G>βα(u2, u1),

(53)

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

ΣRαβ(u1, u2) = θ(u12)

(Σ>αβ(u1, u2)− Σ<

αβ(u1, u2)),

ΣAαβ(u1, u2) = θ(−u12)

(Σ<αβ(u1, u2)− Σ>

αβ(u1, u2)).

(54)

Defining µ(u1) = µI + µσz + µc(u1)σx, we have matrix equations

i∂u1GR/A(u1, u2) + µ(u1)GR/A(u1, u2)−

∫du3ΣR/A(u1, u3)GR/A(u3, u2) = δ(u12)I,

−i∂u2GR/A(u1, u2) +GR/A(u1, u2)µ(u2)−

∫du3G

R/A(u1, u3)ΣR/A(u3, u2) = δ(u12)I.

(55)

16





- -

/

(

)

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-

/

()

- -

/

[

>]

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-

-

-

/

[

>]

Figure 6: (a). The spectral function ρLL(ω) in the long time limit for turning off µc. (b). Thequantum distribution F (ω) in the long time limit for turning off µc. The dashed line representsthe fitting with F (ω) = tanh (3.95(ω/J − 0.029)), which gives a final state with βeffJ = 7.9 andµ′/J = 0.129. We then compare the thermal real-time Green’s function and the quench resultin (c) and (d), with only a phase difference due to δµ, as predicted by (48).

To study the quench protocol in section 2.4, we should prepare a system at thermal equilib-rium in the wormhole phase. All Green’s functions and self-energies in thermal ensemble onlydepend on the time difference. The inverse temperature β enters the self-consistent equationthrough the fluctuation-dissipation theorem:

G>αβ(ω) = −iραβ(ω)nF (−ω, β), G<

αβ(ω) = iραβ(ω)nF (ω, β). (56)

with the spectral function matrix ραβ(ω) defined as

ραβ(ω) = − 1

2πi

(GRαβ(ω)−GA

αβ(ω))

= − 1

2πi

(GRαβ(ω)−GR

βα(ω)∗). (57)

Solving (53), (54), (55), (56) and (57) iteratively with µc(u1) = µc gives the spectral functionand real-time Green’s functions. In the numerics, we choose βJ = 40, µc/J = 0.075, µ/J = 0.1and µ/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 µc(u) = µcθ(−u). The evolution of Green’s

functions is usually determined using the Kadanoff-Baym equation, which takes a similar form

17

as (55):

i∂1G≷(u1, u2) + µ(u1)G≷(u1, u2) =

∫du3

(ΣR(u1, u3)G≷(u3, u2) + Σ≷(u1, u3)GA(u3, u2)

),

−i∂2G≷(u1, u2) +G≷(u1, u2)µ(u2) =

∫du3

(GR(u1, u3)Σ≷(u3, u2) +G≷(u1, u3)ΣA(u3, u2)

).

(58)Importantly, the causality is preserved explicitly due to the presence of retarded and advancedcomponents on the R.H.S., and the derivative of the Green’s functions at time (u1, u2) onlydepends on Green’s functions G≷(u3, u4) with u3, u4 < Maxu1, u2. The initial condition for(58) is given by the thermal equilibrium result

G≷(u1, u2) = G≷(u1 − u2), for u1 < 0 and u2 < 0. (59)

We then evolve the Green’s function step by step using the standard numerical method bydiscretizing G≷(u1, u2) into large matrices [39–43].

The numerical results for G>(u1, u2) are shown in Figure 5. As we turn off the couplingbetween two systems, their correlation GLR decays exponentially. More interestingly, we seethat G>

LL(u1, u2) only depends on u1 − u2, while G>LR(u1, u2) only depends on u1 + u2 when

u1 > 0 and u2 > 0. This is consistent with what one expects for the evolution of the TFDstate.

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 thenumerical results G≷(u1 − u2) after the quench in the long-time limit. We show the long-timespectral function ρLL in Figure 6 (a). The fluctuation-dissipation theorem predicts

− G>LL(ω) +G<

LL(ω)

G>LL(ω)−G<

LL(ω)= tanh

βeff(ω + δµ)

2. (60)

We then determine βeff and δµ by fitting, as shown in Figure 6 (b). This gives βeffJ = 7.9 andµ′/J = 0.129.

Using this result, we can perform an independent numerical study by solving a single modelat corresponding parameters. In Figure 6 (c-d), we compare the single SYK numerics with theresult of quench dynamics. We find the magnitude of GLL matches to high accuracy, whilea phase difference exists due to the presence of δµ. As expected from (48), in the quenchdynamics, the phase is almost constant in the long-time limit, while approaching −π/4 or−3π/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 E , P and t′. Wedetermine the IR parameter, as shown in Table 1, by combining understandings from the TFDand 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.

18

A few possible extensions of the current work are as follows: One can ask whether we couldwrite out a gravity theory with the same effective action Seff. This should correspond to addingU(1) gauge fields [44] to the JT gravity with coupling between two boundaries [5]. It is alsointeresting to consider two copies of higher-dimensional models, etc. 1+1-D, and adding acoupling term. This can lead to higher-dimensional wormholes.

Acknowledgments

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

References

[1] A. Kitaev, Hidden correlations in the hawking radiation and thermal noise, in Talk givenat the Fundamental Physics Prize Symposium, vol. 10, 2014.

[2] S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum heisenbergmagnet, Physical review letters 70 (1993) 3339.

[3] J. Maldacena and D. Stanford, Remarks on the sachdev-ye-kitaev model, Physical ReviewD 94 (2016) 106002.

[4] A. Kitaev and S. J. Suh, The soft mode in the sachdev-ye-kitaev model and its gravitydual, Journal of High Energy Physics 2018 (2018) 183.

[5] J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv preprintarXiv:1804.00491 (2018) .

[6] S. Sachdev, Bekenstein-hawking entropy and strange metals, Physical Review X 5 (2015)041025.

[7] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectrictransport in disordered metals without quasiparticles: The sachdev-ye-kitaev models andholography, Physical Review B 95 (2017) 155131.

[8] Y. Gu, A. Kitaev, S. Sachdev and G. Tarnopolsky, Notes on the complexsachdev-ye-kitaev model, Journal of High Energy Physics 2020 (2020) 1–74.

[9] N. V. Gnezdilov, J. A. Hutasoit and C. W. J. Beenakker, Low-high voltage duality intunneling spectroscopy of the sachdev-ye-kitaev model, Physical Review B 98 (Aug, 2018).

[10] Y. Chen, H. Zhai and P. Zhang, Tunable quantum chaos in the sachdev-ye-kitaev modelcoupled to a thermal bath, Journal of High Energy Physics 2017 (2017) 150.

[11] A. Kruchkov, A. A. Patel, P. Kim and S. Sachdev, Thermoelectric power ofsachdev-ye-kitaev islands: Probing bekenstein-hawking entropy in quantum matterexperiments, Physical Review B 101 (May, 2020) .

19

https://doi.org/10.1103/physrevb.98.081413



[12] A. Altland, D. Bagrets and A. Kamenev, Sachdev-ye-kitaev non-fermi-liquid correlationsin nanoscopic quantum transport, Physical Review Letters 123 (Nov, 2019) .

[13] O. Can, E. M. Nica and M. Franz, Charge transport in graphene-based mesoscopicrealizations of sachdev-ye-kitaev models, Physical Review B 99 (2019) 045419.

[14] S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transitionfrom fast to slow scrambling, Physical Review B 95 (2017) 134302.

[15] X. Chen, R. Fan, Y. Chen, H. Zhai and P. Zhang, Competition between chaotic andnonchaotic phases in a quadratically coupled sachdev-ye-kitaev model, Physical reviewletters 119 (2017) 207603.

[16] P. Zhang, Dispersive sachdev-ye-kitaev model: Band structure and quantum chaos,Physical Review B 96 (2017) 205138.

[17] S.-K. Jian and H. Yao, Solvable syk models in higher dimensions: a new type ofmany-body localization transition, arXiv preprint arXiv:1703.02051 (2017) .

[18] Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalizedsachdev-ye-kitaev models, Journal of High Energy Physics 2017 (2017) 125.

[19] D. Khveshchenko, One syk set, arXiv preprint arXiv:1912.05691 (2019) .

[20] D. V. Khveshchenko, Connecting the syk dots, Condensed Matter 5 (2020) 37.

[21] I. R. Klebanov, A. Milekhin, G. Tarnopolsky and W. Zhao, Spontaneous breaking of u(1)symmetry in coupled complex syk models, arXiv preprint arXiv:2006.07317 (2020) .

[22] J. Kim, I. R. Klebanov, G. Tarnopolsky and W. Zhao, Symmetry breaking in coupled sykor tensor models, Physical Review X 9 (2019) 021043.

[23] P. Gao, D. L. Jafferis and A. C. Wall, Traversable wormholes via a double tracedeformation, Journal of High Energy Physics 2017 (2017) 151.

[24] X.-L. Qi and P. Zhang, The coupled syk model at finite temperature, Journal of HighEnergy Physics 2020 (May, 2020) .

[25] S. Plugge, E. Lantagne-Hurtubise and M. Franz, Revival dynamics in a traversablewormhole, Physical Review Letters 124 (Jun, 2020) .

[26] J. Maldacena and A. Milekhin, Syk wormhole formation in real time, arXiv preprintarXiv:1912.03276 (2019) .

[27] T.-G. Zhou, L. Pan, Y. Chen, P. Zhang and H. Zhai, Disconnecting a traversablewormhole: Universal quench dynamics in random spin models, arXiv preprintarXiv:2009.00277 (2020) .

[28] E. Lantagne-Hurtubise, S. Plugge, O. Can and M. Franz, Diagnosing quantum chaos inmany-body systems using entanglement as a resource, Physical Review Research 2 (2020)013254.

20

https://doi.org/10.1103/physrevlett.123.226801

https://doi.org/10.1007/jhep05(2020)129

https://doi.org/10.1007/jhep05(2020)129

https://doi.org/10.1103/physrevlett.124.221601

[29] T.-G. Zhou and P. Zhang, Tunneling through an eternal traversable wormhole, arXivpreprint arXiv:2009.02641 (2020) .

[30] Y. Chen and P. Zhang, Entanglement entropy of two coupled syk models and eternaltraversable wormhole, Journal of High Energy Physics 2019 (2019) 33.

[31] S. Sahoo, E. Lantagne-Hurtubise, S. Plugge and M. Franz, Traversable wormhole andhawking-page transition in coupled complex syk models, arXiv preprint arXiv:2006.06019(2020) .

[32] N. Sorokhaibam, Traversable wormhole without interaction, arXiv preprintarXiv:2007.07169 (2020) .

[33] A. M. Garcıa-Garcıa, J. P. Zheng and V. Ziogas, Phase diagram of a two-site coupledcomplex sachdev-ye-kitaev model, arXiv preprint arXiv:2008.00039 (2020) .

[34] X.-L. Qi, H. Katsura and A. W. Ludwig, General relationship between the entanglementspectrum and the edge state spectrum of topological quantum states, Physical reviewletters 108 (2012) 196402.

[35] Y. Chen and P. Zhang, Entanglement Entropy of Two Coupled SYK Models and EternalTraversable Wormhole, JHEP 07 (2019) 033, [1903.10532].

[36] T. Azeyanagi, F. Ferrari and F. I. S. Massolo, Phase diagram of planar matrix quantummechanics, tensor, and sachdev-ye-kitaev models, Physical review letters 120 (2018)061602.

[37] A. A. Patel and S. Sachdev, Theory of a planckian metal, Physical review letters 123(2019) 066601.

[38] M. Tikhanovskaya, H. Guo, S. Sachdev and G. Tarnopolsky, Excitation spectra ofquantum matter without quasiparticles i: Sachdev-ye-kitaev models, arXiv preprintarXiv:2010.09742 (2020) .

[39] A. Eberlein, V. Kasper, S. Sachdev and J. Steinberg, Quantum quench of thesachdev-ye-kitaev model, Physical Review B 96 (2017) 205123.

[40] P. Zhang, Evaporation dynamics of the sachdev-ye-kitaev model, Physical Review B 100(2019) 245104.

[41] A. Almheiri, A. Milekhin and B. Swingle, Universal constraints on energy flow and sykthermalization, arXiv preprint arXiv:1912.04912 (2019) .

[42] Y. Cheipesh, A. Pavlov, V. Ohanesjan, K. Schalm and N. Gnezdilov, Quantum tunnelingdynamics in complex syk model quenched-coupled to a cool-bath, arXiv preprintarXiv:2011.05238 (2020) .

[43] C. Kuhlenkamp and M. Knap, Periodically driven sachdev-ye-kitaev models, PhysicalReview Letters 124 (2020) 106401.

21

https://doi.org/10.1007/JHEP07(2019)033

https://arxiv.org/abs/1903.10532

[44] S. Sachdev, Universal low temperature theory of charged black holes with ads2 horizons,Journal of Mathematical Physics 60 (2019) 052303.

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More on Complex Sachdev-Ye-Kitaev Eternal Wormholes · 2020. 11. 30. · More on Complex Sachdev-Ye-Kitaev Eternal Wormholes Pengfei Zhang1 1California Institute of Technology, Pasadena,