(2020) Wormholes and the Thermodynamic Arrow of Time

PHYSICAL REVIEW RESEARCH 2, 043095 (2020)

Wormholes and the thermodynamic arrow of time

Zhuo-Yu Xian1,* and Long Zhao1,2,† 1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China 2School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China

(Received 16 March 2020; accepted 11 June 2020; published 19 October 2020)

In classical thermodynamics, heat cannot spontaneously pass from a colder system to a hotter system, which is called the thermodynamic arrow of time. However, if the initial states are entangled, the direction of the thermodynamic arrow of time may not be guaranteed. Here we take the thermoﬁeld double state at 0 + 1 dimension as

the initial state and assume its gravity duality to be the eternal black hole in AdS2 space. We make the temperature difference between the two sides by changing the Hamiltonian. We turn on proper interactions between the two sides and calculate the changes in energy and entropy. The energy transfer, as well as the thermodynamic arrow of time, are mainly determined by the competition between two channels: thermal diffusion and anomalous heat ﬂow. The former is not related to the wormhole and obeys the thermodynamic arrow of time; the latter is related to the wormhole and reverses the thermodynamic arrow of time, i.e., transferring energy from the colder side to the hotter side at the cost of entanglement consumption. Finally, we ﬁnd that the thermal diffusion wins the competition, and the whole thermodynamic arrow of time has not been reversed.

DOI: 10.1103/PhysRevResearch.2.043095

I. INTRODUCTION for the two systems with different temperatures and initial correlation, the thermodynamic arrow of time can be reversed With the experimental advances in the last two decades, it for some proper interactions: heat passes from colder systems is possible to prepare and control systems with an increasing to hotter systems by consuming their correlation [3,4], which number of atoms, whose scale stretches across the gap be- is called anomalous heat flow, as illustrated in Fig. 1.The tween macroscopic thermodynamics and microcosmic quan- authors of Ref. [5] discussed the conditions for its presence. tum mechanics. Taking these two theories as cornerstones, Such a phenomenon was observed in the experiment on two quantum thermodynamics tries to extend the framework of initially correlated spins [6] and the quantum computer [7]. thermodynamics to finite-size quantum systems, nonequilib- The reversion of the thermodynamic arrow of time may con- rium dynamics, strong coupling, and quantum information flict with the original second law and urges a generalized processes [1]. second law in the presence of correlations [8]. The laws of thermodynamics constrain the evolution of This traces back to the relationship between work and a system coupled to heat baths. The second law can be de- correlation [9]. By acting unitary transformation on a state, rived when the initial correlation between the system and its one can lower the energy and extract work from the system. A heat baths is absent [2]. The correlation generated during the state from which no work can be extracted is called the passive interaction between the system and the heat baths leads to state. All the thermal states are passive [10,11]. But work can non-negative entropy production. The thermodynamic arrow be extracted from the correlation between two systems, each of time also plays a crucial role in the second law of ther- of which is locally in a thermal state with the same temper- modynamics, which refers to the phenomenon that heat will ature [12]. Then, the anomalous heat flow is a refrigeration spontaneously pass from a hotter system to a colder system. driven by the work stored in correlations [8]. Furthermore, the Similarly, it can be derived if the two systems are uncorrelated authors of Refs. [8,13,14] discussed the generalizations of the initially [3]. second law in the presence of correlation. The author of Ref. [3] argued that the second law and the Similar issues have been being studied in gravity since thermodynamic arrow of time are emergent phenomena in the two decades ago as well, while the famous AdS/CFT cor- low-entropy and low-correlation environment. Furthermore, respondence was discovered. The AdS/CFT correspondence is a realization of the holography principle, proposed by Susskind according to the surface law of black hole entropy. *[email protected] It conjectures that the degree of freedom of a gravitational †[email protected] system can be mapped to its boundary [15]. Furthermore, the AdS/CFT correspondence offers the dictionary between Published by the American Physical Society under the terms of the the quantum gravity in asymptotic anti–de Sitter (AdS) space Creative Commons Attribution 4.0 International license. Further and the conformal field theory (CFT) on its boundary [16]. distribution of this work must maintain attribution to the author(s) Notably, a large AdS-Schwarzschild black hole is dual to the and the published article’s title, journal citation, and DOI. CFT at finite temperature. The correspondence reveals that

2643-1564/2020/2(4)/043095(29) 043095-1 Published by the American Physical Society ZHUO-YU XIAN AND LONG ZHAO PHYSICAL REVIEW RESEARCH 2, 043095 (2020)

try to find similar phenomena in strongly coupled systems based on the AdS2/CFT1 correspondence and the ER = EPR conjecture. In Sec. II, we review the relation between correlation and anomalous heat flow. In Sec. III, we prepare a TFD state and the dual eternal black hole in Jackiw-Teitelboim (JT) gravity with different temperatures on each side. In Sec. IV, FIG. 1. Anomalous heat flow. we show that work can be extracted from the wormhole. In Sec. V, with proper interactions, we find two channels which mainly contribute to the energy transfer: the thermal diffusion black hole thermodynamics is not only an analogy of the via the boundary and the anomalous heat flow via the worm- classical thermodynamics but also an equivalent description. hole in the eternal black hole. The former is numerically larger Another significant development in the AdS/CFT cor- than the latter. So the total thermodynamic arrow of time has respondence is the Ryu-Takayanagi (RT) formula, which not been reversed. In Sec. VI, we study the concomitant con- conjectures that the von Neumann entropy of a subregion on sumption of entanglement. Although we focus on the eternal the boundary is proportional to the area of the minimal homo- black hole in the main text, in Appendixes B and C, similar logical surface in the bulk [17]. Furthermore, the black hole issues are investigated for a product state in JT gravity and a entropy is interpreted as the entanglement entropy between TFD or product state in the Sachdev-Ye-Kitaev (SYK) model the two field theories on the two boundaries of the eternal for comparison. black hole. The spatial region behind the two horizons is called an Einstein-Rosen bridge or a wormhole. The AdS/CFT correspondence also sheds light on the II. ENERGY TRANSFER WITH CORRELATION black hole information problem by utilizing the unitary of dual A. Thermodynamics for bisystems / boundary theories [18,19]. Before the AdS CFT correspon- Consider bisystems labeled by L and R. The state of the dence was proposed, Page embedded the unitary in the process bisystems is described by the density matrix ρ; then the local of black hole evaporation and study the entanglement between density matrices of systems L and R follow ργ = Trγ¯ ρ, where the remainder black hole and its Hawking radiation [20,21]. γ = L, R andγ ¯ is the complement of γ . The entropy of the Motivated by the Hartle-Hawking-Israel state [22] and the bisystems and local systems are defined as the von Neumann black hole complementarity [23], Maldacena conjectured that entropy SLR =−Tr[ρ ln ρ] and Sγ =−Tr[ργ ln ργ ]. The two the eternal black hole is described by the thermofield-double systems may be correlated. We use mutual information I = (TFD) state and tried to resolve part of the information loss L;R SL + SR − SLR to measure their correlation, where IL;R 0 paradox [24]. Motivated by the RT formula, Van Raamsdonk because of the subadditivity of entanglement entropy. conjectured the correspondence between geometry and entan- The local Hamiltonians of systems L and R are H and glement [25]. To resolve the firewall paradox [26], Maldacena L HR. Then the local energies are Eγ = Tr[ργ Hγ ]. We will and Susskind applied the idea to general entangled states consider the following process. Prepare an initial state ρ(t ) and raised the ER = EPR conjecture, where ER refers to the i of the bisystems at time ti. Then we turn on an interaction Einstein-Roson bridge, and EPR (Einstein-Podolsky-Rosen) H between them. After time t , the total Hamiltonian H = refers to the EPR pair, i.e., quantum entanglement [27]. I i tot HL + HR + HI is time independent. We define EI = Tr[ρHI ]. The connection between the TFD state and the eternal In this paper, we will consider a weak interaction H compared black hole helps us to understand the complicated dynamics I to the local Hamiltonian Hγ . So the local energies of each of boundary field theories from the gravity side. Quantum local systems are well defined by the expectational value chaos, which is diagonalized by out-of-time-order correlators of their local Hamiltonians. At time t f > ti, the state of the (OTOCs), can be understood from the shock wave on black bisystems becomes ρ(t ) through the unitary evolution with holes [28,29]. The teleportation protocol or Hayden-Preskill f total Hamiltonian Htot. So the entropy of the bisystems SLR protocol corresponds to constructing a traversable wormhole does not change. The changes in the local entropies and ener- in an eternal black hole [30–37]. To make the wormhole gies are Sγ = Sγ (t ) − Sγ (t ), Eγ = Eγ (t ) − Eγ (t ), and traversable, one should turn on proper interactions between f i f i EI = EI (t f ) − EI (ti ). the two sides of the eternal black hole, where the interactions After time t , the total energy is conserved, are equivalent to the measurements in the teleportation pro- i tocol. The evolution of the bisystems with interactions and EL + ER + EI = 0. (2.1) entanglement is an object of research in quantum thermodynamics. According to the first law of thermodynamics, the change in We raise the following questions in the intersection local energy can be divided into work W and heat Q. We will between quantum thermodynamics and the AdS/CFT corre- adopt the definitions of work and heat in Ref. [14], as reviewed spondence. in Appendix A. For a specific model, we can calculate them (i) Does anomalous heat flow exist in the systems which accordingly. have holographic duality? (ii) If yes, will it challenge the laws of black hole thermodynamics? B. Anomalous heat flow In previous works, anomalous heat flow was realized in We review some theoretical observations about the heat weakly coupled or weakly chaotic systems. In this paper, we transfer in the presence of correlation in Refs. [3,4]. Given

043095-2 WORMHOLES AND THE THERMODYNAMIC ARROW OF TIME PHYSICAL REVIEW RESEARCH 2, 043095 (2020) an inverse temperature βγ = 1/Tγ , one can define a lo- thermodynamics. One can extract work from correlation or cal thermal state for a local Hamiltonian, namely τγ (βγ ) = create correlation by costing work [9,12,38–40]. The trade- −1 Zγ exp(−βγ Hγ ). Now we specify the initial state ρ(ti )of off between works and correlations explains the presence of the bisystems at time ti such that the two local systems are anomalous heat flow. For a state ρ and its Hamiltonian H,the individually thermal, i.e., ργ (ti ) = τγ (βγ ). work extracted from the system after a unitary transformation Note that, given a constant β, free energy F = E − S/β is U is minimized by thermal state. As each local system is thermal = ρ − ρ † . initially, we have W Tr[ H] Tr[U U H] (2.7) For a Hamiltonian H, a state is called passive if and only if no βLEL SL,βRER SR, (2.2) positive work can be extracted from it, namely where β , are taken to be constant during the process. We L R , ∀ . will call (2.2) as energy-entropy inequalities. When the in- W 0 U (2.8) equalities are saturated, they are just the first law of entropy Furthermore, it is proved in Refs. [11,41] that a state π is of local systems. Combining the two inequalities, we have passive if and only if it is diagonal in the energy eigenbasis {| } { } (EL + ER )(βL + βR ) + (EL − ER )(βL − βR ) n and its eigenvalues pn are monotonically decreasing with increasing energy {E }, namely = + , n 2 IL;R 2( SL SR ) (2.3) π = | |, . where the last equality follows from the unitary evolution. pn n n where pn pm if En Em (2.9) n For an initial state which is a product state ρ(ti ) = ρL (ti ) ⊗ − −βH ρR(ti ), we have IL;R(ti ) = 0. So IL;R 0, meaning that the Obviously, a thermal state τ = Z 1e is a passive state of interaction may induce entanglement. If we further consider its Hamiltonian H. Furthermore, the outer product of thermal ⊗ the case of heat contact, i.e., EL + ER = 0, one can easily state τ n is also a passive state of the n copies of H for any obtain the thermodynamic arrow of time from inequality (2.3), n, while an outer product of a passive state π ⊗n may not be a (E − E )(β − β ) 0. (2.4) passive state of the n copies of H. L R L R The authors of Ref. [8] argued that, for entropy-preserving For example, we assume that system R is hotter than system evolution, the extractable work solely from the correlation L, i.e., βL >βR. From the above inequality, we know EL = between system and heat bath is bounded by the product −ER 0, which just means that heat can only pass from a of mutual information and temperature. Here we take the hotter system to a colder system. The above inequality also entangled pure state of bisystems as an example. From in- holds once EL + ER 0, which means that one cannot equalities (2.2), the work extracted is bounded by transfer energy from a colder system to a hotter system with- + / . out positive work done on the bisystems. W IL;R(TL TR ) 2 (2.10) However, the above argument fails if the two systems are The authors in Ref. [8] further explained the anomalous heat entangled initially. It is possible that the mutual information flow as the refrigeration driven by the work stored in corre- IL;R decreases. Then the left-hand side of inequality (2.3)is lations. Finally, they also modified the Clausius statement of not required to be non-negative anymore. Especially for a the second law: no process is possible whose sole result is the = = pure initial state, SL SR always holds. Then SL SR. transfer of heat from a colder system to a hotter system, where + Combining it with inequalities (2.2), we have EL ER the work potential stored in the correlations does not decrease. SL (TL + TR ). If EL + ER 0 during the process, we find IL;R = 2SL 0. So the reversal of inequality (2.4), III. A THERMOFIELD DOUBLE STATE AS (EL − ER )(βL − βR ) < 0, (2.5) AN INITIAL STATE is not forbidden anymore. Energy may pass from a colder sys- A. Balanced bisystems and holography tem to a hotter system by consuming (quantum) correlation, Consider a local Hamiltonians H of a local system, with which is called anomalous heat flow. No doubt that such a eigenbasis H|n=En|n. Further, consider two copies of the phenomenon highly depends on the state and the interaction. local system labeled by L and R with Hamiltonian The reversal of the thermodynamic arrow of time during heat contact was observed in the recent experiment on the two- H0 = HL + HR, HL = H ⊗ 1, HR = 1 ⊗ H. (3.1) spin system [6]. The time-reversal algorithms on a quantum We call it the balanced Hamiltonian since the local Hamil- computer were developed in Ref. [7]. tonians are the same. We consider a state |I= |n |n , In a word, we say that the thermodynamic arrow of time is n L R where |n is the energy eigenstate in system γ . We prepare a reversed if γ TFD state at t = 0, EL + ER 0 and (EL − ER )(βL − βR ) < 0. 1 −β / 1 −β / |β = √ H0 4| = √ | | En 2, (2.6) e I n L n Re (3.2) Z Z n C. Extract work from correlation where Z = Tre−βH . The TFD state is not an eigenstate of −iH0t No doubt that the anomalous heat flow stems from (quan- H0. We define |β(t )=e |β. Tracing out a local sys- tum) correlations, which plays an important role in quantum tem on one side, we obtain a thermal state on another side,

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−1 −βH ργ = Trγ¯ |β(t )β(t )|=Z e . Both of the inverse tem- constructed in Refs. [35,36,45]. The ways of preparing TFD peratures of local systems are β. states were discussed in Refs. [35,46]. Throughout this paper, we consider H to be a conformal Given an operator O on the local system, we deﬁne ﬁeld theory at 0 + 1 dimension. We can also consider that the T local Hamiltonian H in Eq. (3.1) is the SYK model [42–44]in OL = O ⊗ 1, OR = 1 ⊗ O, (3.3) Appendix C. The TFD state in the two-site SYK model was where OT is the transpose of O on the basis of energy eigenstates. The interaction picture for both sides is

iH0t −iH0t iHt T −iHt iH0t −iH0t iHt −iHt OL (t ) = e OLe = e O e ⊗ 1, OR(t ) = e ORe = 1 ⊗ e Oe . (3.4) Due to the entanglement in the TFD state, the operators on both side are related by β O (t )|β = O −t + i |β. (3.5) L R 2

We assume that the systems have a holographic duality of nearly AdS2. The bulk description of the TFD state is the Rindler patch in AdS2 space [47,48]. For higher-dimensional CFT, one can further consider higher-dimensional generalization in the eternal black hole in AdS space. We consider that the AdS2 holography is described by dilaton gravity φ √ √ √ √ 0 2 1 2 I0 =− dx gR + dx hK − d x gφ(R + 2) + 2 dx hφ∂ K +··· 16πGN 16πGN ∂ M √ − 2 ∂χ 2 + 2χ 2 . d x g ( i ) m i (3.6) i=1

The first term is a topological term. It does not control the dynamic and only contributes to extremal entropy S0 = φ0/4GN . The second term is called Jackiw-Teitelboim (JT) theory. After integrating out dilaton field φ, one can fix the metric to be (Euclidean) AdS2 (EAdS2). The remaining degrees of freedom are the boundary trajectory and the matter fields [48]. The dots φ φ refer to the expansion of the dilaton field from higher-dimensional reduction, which is neglectable for 0. We will consider {χ } χ → χ M free scalar fields with the same mass i in the bulk. So the theory has SO(M) symmetry of rotation i j Gij j where G ∈ SO(M). In this paper, we will frequently exchange these scalar fields, which is a subgroup of the SO(M) symmetry. The coordinates of EAdS2 and AdS2 used in this paper are dμ2 + dz2 ds2 = = sinh2 ρdϕ2 + dρ2, (3.7) z2 −dν2 + dz2 −dθ 2 + dσ 2 ds2 = =−sinh2 ρdψ2 + dρ2 = , (3.8) z2 sin2 σ where (μ, z) and (ν, z) are Poincaré coordinates, (φ,ρ) and (ψ,ρ) are Rindler coordinates, and (θ,σ) are conformal coordinates. We will first work on Euclidean time. One can impose boundary conditions φ¯ 2 1 2 ds = dτ ,φ∂ = . (3.9) ∂ 2

The cutoff is imposed to prevent dilaton ﬁeld φ from exceeding φ0.Soφ/φ¯ 0 is related to the UV cutoff of the dual boundary theory, namely φ/φ¯ 0 = α with an order 1 constant α. One can parametrize the boundary as (μ(τ ), z(τ )), where z(τ )is determined by the boundary condition z = μ (τ ) + O(3). (3.10)

From the JT term in EAdS2, one can obtain the effective action of reparametrization μ(τ ) 2π C 2 2 3 Ieff =−C dτ{μ, τ} = dτ (ε − ε ) + O(ε ), 2 0 μ (τ ) μ (τ )2 ϕ φ¯ {μ, τ} = − ,μ= tan ,ϕ= τ + ε(τ ), C = ,β= 2π, (3.11) 2 μ (τ ) 2μ (τ ) 2 8πGN where {μ, τ} is the Schwarzian derivative. Coefﬁcient C has the dimension of time. The dependence on β can be recovered by dimensional analysis. Those SL(2, R) zero modes ε = 1, eiτ , e−iτ should be excluded in the functional integrate on ε. One can obtain the correlation of reparametrization modes 1 1 ε(τ )ε(0) = − (|τ|−π )2 + (|τ|−π )sin|τ|+c + c cos τ , (3.12) 2πC 2 1 2 where arbitrary coefﬁcients {c1, c2} come from the redundancy of SL(2, R) zero modes.

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For a free scalar field χ in the bulk, the correlation of its dual operator is modulated by reparametrization modes. The sources Jˆ(μ) are related to the source J(τ )by χ ∼ z1−ΔJˆ(μ) ∼ [μ (τ )]1−ΔJˆ(μ) = 1−ΔJ(τ ) ⇒ Jˆ(μ) = μ (τ )Δ−1J(τ ). (3.13) After integrating out the bulk field χ, we obtain Δ Jˆ(μ )Jˆ(μ ) μ (τ )μ (τ ) −I = dμ dμ 1 2 = dτ dτ J(τ )J(τ ) 1 1 2 2 M 1 2 2Δ 1 2 1 2 2 (μ1 − μ2 ) [μ1(τ1) − μ2(τ2 )] J(τ )J(τ ) = τ τ 1 2 + B τ ,τ + C τ ,τ + ε3 , d 1d 2 Δ [1 ( 1 2 ) ( 1 2 ) O( )] (3.14) τ12 2 2sin 2 ε2 − ε1 B(τ1,τ2 ) =Δ ε + ε + τ , (3.15) 1 2 tan 12 2 Δ2 ε − ε 2 ε − ε 2 1 C(τ ,τ ) = ε + ε + 2 1 + Δ 1 2 − ε 2 + ε 2 , (3.16) 1 2 1 2 τ12 τ12 1 2 2 tan 2 2sin 2 2 (Δ−1/2)(Δ) τ = τ − τ ε = ε τ Δ = √ where 12 1 2, 1 ( 1). The operator O has been normalized such that the common factor D π(Δ−1/2) does not appear in the correlation function. After integrating out ε, we obtain the generating functional − J(τ1)J(τ2 ) 1 J(τ1)J(τ2 )J(τ3)J(τ4) − IM = τ τ +C τ ,τ + τ τ τ τ B τ ,τ B τ ,τ + 2 , ln e d 1d 2 τ 2Δ [1 ( 1 2 ) ] d 1d 2d 3d 4 τ 2Δ τ 2Δ ( 1 2 ) ( 3 4 ) O(C ) 12 2 12 34 2sin 2 2sin 2 2sin 2 (3.17) Δ τ −2 C(τ ,τ ) =− sin 12 −τ 2 + 2π|τ |+2(|τ |−π ) sin |τ |+2 cos τ − 2 1 2 8πC 2 12 12 12 12 12 Δ2 τ τ 2 − 2π|τ | + 12 − 2 12 12 − 2 , (3.18) τ12 τ12 4πC tan τ12 tan 2 2 ε − ...= D ... Ieff B τ ,τ B τ ,τ where the bracket SL(2,R) e and Eq. (3.12) is used. The specific expression of ( 1 2 ) ( 3 4 ) is too tedious to be shown here, which does not only depend on τ12 and τ34 for ordering τ1 >τ3 >τ2 >τ4. The real-time correlator can be obtained by Wick rotation τ → it. The two boundaries of AdS2 space satisfy the boundary conditions φ¯ φ¯ 2 1 2 2 1 2 ds =− dt ,φ∂, = , ds =− dt ,φ∂, = . (3.19) ∂,L 2 L ∂,R 2 R

The boundaries are parametrized by {νL (t ),νR(t )} in Poincaré coordinates or {ψL (t ),ψR(t )} in Rindler coordinates, whose effective action is ψ ψ I =−C dt {ν , t } − C dt {ν , t } =−C dt − coth L , t − C dt tanh R , t . (3.20) eff L L L R R R L 2 L R 2 R

The SL(2, R) symmetry of parametrization {ψL (t ),ψR(t )} is the two-point functions at tree level in real time are

ψL −ψL ψL → ψL + 0 + 1e + 2e , −ψ ψ 2 R R O (t )O (t ) = O (t )O (t ) = Tr[O(t )O(t )y ] ψR → ψR − 0 + 1e + 2e . (3.21) L 1 L 2 R 1 R 2 1 2 t −2Δ = 2 2i sinh 12 , (3.23) Without the excitation of scalar ﬁelds, those three SL(2, R) 2 charges are required to vanish for the consistency of the ef- −2Δ t1 + t2 fective action. Then the equations of motion are automatically OL (t1)OR(t2 )= Tr[O(−t1)yO(t2 )y]=2 2 cosh , |β 2 satisﬁed. Corresponding to the TFD state (t ) , the solution (3.24) is

2π 2π where, within the trace, ψ = t ,ψ= t , t = t = t. L β L R β R L R (3.22) O(t ) = eiHt Oe−iHt , (3.25) The correlator in bisystems can be obtained by doing → ± β = −1/2 −βH/2. analytical continuation t t i 2 . For example, y Z e (3.26)

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B. Unbalanced bisystems and holography Now we make the Hamiltonian of the bisystems unbalanced,

H˜0 = HL + H˜R, HL = H ⊗ 1, H˜R = 1 ⊗ λH, (3.27) where the Hamiltonian of system R is multiplied by a constant λ. We still consider the TFD state |β in Eq. (3.2) at time t = 0, which is prepared by imaginary-time evolution with H0 rather than H˜0, while |β evolves with H˜0 in − ˜ +λ |β˜ = iH0t |β=|β 1 real time, (t ) e ( 2 t ) . Tracing it by part, ρ = |β˜ β˜ |= 1 −βH = 1 −(β/λ)λH we have γ Trγ¯ (t ) (t ) Z e Z e . Thus βL = β and βR = β/λ. Without loss of generality, we set λ 1 through this paper. So system R is hotter than system L or they have the same temperature, i.e., βL βR. Effectively, multiplying the Hamiltonian of system R by λ is equivalent to redeﬁning the time of system R as t˜ = t/λ. We still consider the same operator O at t = 0. Since we ν ν λ = have changed the Hamiltonian, the time evolution of O is FIG. 2. Two boundaries ˜L (t )and˜R(t )with 2 in conformal σ,θ , changed as well. Through this paper, O˜ (t ) refers to the time coordinates ( ). Systems L R are denoted by the blue curve and the orange curve. The time parametrizations are shown by the grad- evolution of O with H˜0 and O(t ) refers to the time evolution of O with H . They are related by uation line on the curves. The dashed lines indicate the insertion of 0 interactions. iH˜0t −iH˜0t iHt T −iHt O˜ L (t ) = e OLe = e O e ⊗ 1 = OL (t ),

iH˜0t −iH˜0t iHλt −iHλt O˜ R(t ) = e ORe =1 ⊗ e Oe =OR(λt ). (3.28) Corresponding to the TFD state |β˜ (t ), the saddle-point solution of an eternal black hole with unbalanced H˜0 is For system R, the evolution with H˜0 by time t is just the evolution with H0 by time λt. Take the correlator as an example, 2π 2πλ ψ˜L = t, ψ˜R = t. (3.33) iHt1 T −iHt1 iλHt2 −iλHt2 β β O˜ L (t1)O˜ R(t2 ) =e O e ⊗ e Oe

= OL (t1)OR(λt2 ), (3.29) As expected, the black hole R is hotter than the black hole L. Although the parametrizations are unbalanced, the classical ...≡β| ...|β where for short. Then we can map the location of the boundary is balanced, as shown in Fig. 2. observables under the evolution with H˜0 to those under the The source J˜R of system R can be identiﬁed as follows, evolution with H0. This trick will be frequently used in later calculations. O (λt ) = O˜ (t ) and dtJ (t )O (t ) We can further discuss the effect on the gravity side under R R R R the change of Hamiltonian. Corresponding to the evolution with unbalanced H˜0 by time t, the conditions of the two = dtJ˜R(t )O˜ R(t ) ⇒ λJR(λt ) = J˜R(t ), (3.34) boundaries in AdS2 become φ¯ which is consistent with the correlation (3.29), since 2 1 2 =− ,φ∂, = , ds∂,L dt L Δ 2 ν (t )ν (t ) dt dt J (t )J (t ) L 1 R 2 λ2 φ/λ¯ 1 2 L 1 R 2 ν − ν 2 2 =− 2,φ = , ( L (t1) R(t2 )) ds∂,R dt ∂,R (3.30) 2 /λ Δ ν (t )ν (λt ) /λ = dt dt J˜ (t )J˜ (t ) L 1 R 2 . (3.35) where the UV cutoff of system R becomes .Letthe 1 2 L 1 R 2 (ν (t ) − ν (λt ))2 boundaries corresponding to the evolution with unbalanced L 1 R 2 ˜ {ν , ν } {ψ˜ , ψ˜ } H0 be parametrized by ˜L (t ) ˜R(t ) or L (t ) R(t ) .The Compared with dictionary (3.13), the dictionary for J˜R is evolution with unbalanced H˜0 can inherit the bulk description χ ∼ 1−Δ ˆ ν ∼ ν /λ 1−Δ ˆ ν from the evolution with H0 according to Eq. (3.28) and z JR( R ) [ ˜R(t ) ] JR( R ) = 1−Δ λ = 1−Δλ−1 ν˜L (t ) = νL (t ), ν˜R(t ) = νR(λt ), JR( t ) J˜R(t ) ψ˜ = ψ , ψ˜ = ψ λ . ⇒ ˆ ν = ν Δ−1λ−Δ ˜ , L (t ) L (t ) R(t ) R( t ) (3.31) JR( R ) ˜R(t ) JR(t ) (3.36) Then the effective action becomes χ ∼ /λ 1−Δ ˜ τ ˆ ν = ν Δ−1 ˜ rather than ( ) JR(˜R ) and JR( R ) ˜R(t ) JR(t ). C This means that the holographic dictionary of O˜ R(t )isdiffer- Ieff =−C dt{ν˜L, t} − dt{ν˜R, t} λ ent from the one of OR(t ). One may wonder whether any physics will be changed by ψ˜ C ψ˜ =−C dt − L , t − dt R , t . redeﬁning the time. Indeed, no physics has been changed so coth λ tanh 2 2 far. However, physics will be changed if we couple the two (3.32) systems.

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C. Product states as a comparison g, such that the perturbation theory holds. For unbalanced H˜0, As a comparison, we still adopt the unbalanced Hamilto- the energy changes are ˜ nian H0 in Eq. (3.27) but replace the initial state by the product = β˜ | |β˜ − state EL I (t f ) HL I (t f ) HL = E (1) + E (2) + O(g3), (4.3a) ρ = τ (β) ⊗ τ (β/λ), (3.37) L L ER = β˜ I (t f )|H˜R|β˜ I (t f ) − H˜R which is the same as the TFD state (3.2) locally. The interac- = (1) + (2) + 3 , tion picture (3.28) also holds. ER ER O(g ) (4.3b)

The Hamiltonian H of conformal field theory is gapless. EI = β˜ I (t f )|H˜I (t f )|β˜ I (t f ) − H˜I (ti ) Considering a nontrivial Hamiltonian H˜ ≡ 0, we find that 0 = (1) + (2) + 3 , product state (3.37) is a passive state if and only if λ = 1. EI EI O(g ) (4.3c) The analysis is as follows. The probability and energy of the where the superscript in E (i) denotes the energy change at |n |m p = Z Z −1 −β E + energy eigenstate L R are nm ( L R ) exp[ ( n O(gi ). After t , energy conservation holds at each order of g. E E = E + λE λ = p ∝ −βE i m )] and nm n m. When 1, nm exp( nm ), By virtue of the entanglement, the work extracted at O(g)is then ρ satisfies condition (2.9). Recall that local Hamiltonian λ> (1) (1) Hγ here is gapless. When 1, we can choose m and m W = E = g[OL (t f )OR(λt f ) − OL (ti )OR(λti )]. I such that Em − Em = δ>0, and then choose n and n such (4.4) that λδ > En − En >δ. We find Enm > En m but pnm > pn m and then ρ does not satisfy condition (2.9). According to Eq. (3.24) at tree level, sgn(W (1)) = The product of two thermal states is dual to two black −sgn(g)sgn(t f + ti ), which can be positive. As explained holes without a wormhole connecting them [25,49]. We will in Ref. [32], the interaction (4.2) imposes a potential energy ˜ = λ ∼ −ml(t ) call them disconnected black holes. In AdS2 holography, we HI (t ) g OL (t )OR( t ) ge and does work on each should consider the union of two AdS2 spaces, each of which system, where l(t ) is the distance between the location of the is controlled by a theory of JT gravity. The disconnected black two operators OL (t ) and OR(λt )inAdS2 space, and m is the χ holes are described by two Rindler patches of the two AdS2 mass of the dual scalar field . spaces, as discussed in Appendix B. The wormhole in each The first law of entropy holds at first-order perturbation; Rindler path is caused by the purification of the corresponding then thermal state and has nothing to do with the entanglement βE (1) = S(1) = S(1) = (β/λ)E (1). (4.5) between system L and system R. L L R R Combining it with Eqs. (2.1) and (4.4), we have IV. EXTRACT WORK FROM WORMHOLE 1 λ E (1) =− W (1),E (1) =− W (1), The wormhole in the eternal black hole geometrically rep- L 1 + λ R 1 + λ resents the entanglement between system L and system R, (1) (1) 1 (1) each of which is in a thermal state. From quantum thermody- S = S =− βW . (4.6) L R 1 + λ namics, as work can be extracted from correlation, it is natural to expect that work can be extracted from the wormhole as Then we find such work is extracted from correlation well. 1 1 1 (1) =− + (1) . We take the TFD state as the initial state and turn on the W IL;R (4.7) 2 βL βR interaction HI at time ti. After time ti, the total Hamiltonian is Htot = H˜0 + HI . In the interaction picture, the state at time We can alternatively calculate the change in the von Neumann t f > ti is entropy from the replica trick in Sec. VI. According to the definitions of the work and heat of the t f (1) |β˜ (t ) = T exp −i dt H˜ (t ) |β, (4.1) local system in Appendix A, we find that EI is the change I f I (1) ti in the binding energy during the process. The work W is extracted through the binding energy in Eq. (4.4). The local where H˜ (t ) is the interaction picture of H under the evolution (1) (1) I I energy change in the way of heat Eγ = Qγ , and the with H˜ and T is time ordering. 0 work done on the local system vanishes, W (1) = 0. To study the work extracted from the eternal black hole, in γ As a comparison, we try to extract work from the dis- this section, we consider the interaction connected black holes, as calculated in Appendix B.But, without a wormhole, E (1) ∝O˙(t )=0, so no work can HI = gOLOR. (4.2) be extracted at O(g). We further go to O(g2 ) and find that (2) + (2) Such an interaction is also introduced in the literature to cou- EL ER 0, so no positive work can be extracted via ple two spacetimes [49] or construct a traversable wormhole this interaction. It is expectable when λ = 1, since the discon- for g < 0[31,32,35]. For balanced H0, the energy changes at nected black holes form a passive state and the eternal black the first-order perturbation of the interaction (4.2) have been hole does not. However, when λ>1, the disconnected black calculated in Refs. [31,32]. Here we revisit them from the holes do not form a passive state anymore, as explained below viewpoint of quantum thermodynamics. Due to the consid- Eq. (3.37). The failure of extracting positive work results from eration of the weak interaction HI , we should consider a small the unsatisfactory form of the interaction.

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FIG. 3. Four diagrams contribute to Eq. (5.5): (a) disconnected-tree diagram, (b) disconnected-loop diagram, (c) t channel, and (d) u channel. The loop in (b) can appear on either side. The smooth (wavy) curve is the correlator of the scalar ﬁeld (graviton). The commutator CVWVW in Eq. (5.6b) consists of (a), (b), (c). The commutator CVWWV in Eq. (5.6b) consists of (d).

V. ANOMALOUS HEAT FLOW VIA WORMHOLE At second-order perturbation, different interactions will generate different sets of Feynman diagrams. As we will see, A. Interaction the sets of diagrams generated by interactions (5.2) and (5.3) In this section, we continue to consider the unbalanced H˜0 are contained in the set of diagrams generated by the interac- and the TFD state. As explained in the previous section, the tion (5.1). Furthermore, the set of diagrams generated by the appearance of anomalous heat flow is highly dependent on interaction (5.1) is contained in the set of diagrams generated the form of the interaction term [5]. To look for the anoma- by the interaction (4.2). We do not observe any anomalous lous heat flow in the eternal black hole, we will alternatively heat flow with interactions (4.2), (5.2), and (5.3). consider the interaction 1 HI = √ g(VLWR − WLVR ), (5.1) 2 B. Early time 2 where two scalar operators V and W are dual to two scalar We will consider the interaction (5.1) and go to O(g ). The γ fields χV = χ1 and χW = χ2 separately in the bulk for M = 2. changes in the local energies E and interaction energy EI 2 The scaling dimensions of V and W are the same. Due to from time ti to time t f at O(g )are the SO(M) symmetry of Eq. (3.6), any Green’s functions of V and W are invariant under exchange V ↔ W . Similar t f t 1 interactions appeared in the thermofield dynamics [50]. One EL ≈ dt dt KL (t, t ), may consider other kinds of interactions, such as Eq. (4.2)or ti ti K (t, t ) = (−i)3g2{[V˙ (t )W (λt ),V (t )W (λt )] H = gV W (5.2) L L R L R I L R − [V˙L (t )WR(λt ),WL (t )VR(λt )]}, (5.5a) or t f t

1 ER ≈ dt dt KR(t, t ), HI = √ g(VLWR + WLVR ). (5.3) t t 2 i i 3 2 KR(t, t ) = (−i) g λ{[VL (t )W˙R(λt ),VL (t )WR(λt )] Since V and W are dual to different scalar fields, = = −[V (t )W˙ (λt ),W (t )V (λt )]}, (5.5b) VL (t1)WR(t2 ) WL (t1)VR(t2 ) 0. So the effect of each of L R L R Eqs. (5.1), (5.2), and (5.3)atO(g) vanishes. We have to look at t f ≈ , , the contribution at O(g2 ). Actually, even if it does not vanish, EI dtKI (t f t ) e.g., for the interaction (4.2), the first law of entropy leads to ti 2 Eq. (4.5). Then we have KI (t, t ) = (−i)g {[VL (t )WR(λt ),VL (t )WR(λt )] (1) − (1) β − β =− (1) + (1) . −[VL (t )WR(λt ),WL (t )VR(λt )]}. (5.5c) sgn EL ER ( L R ) sgn EL ER (5.4) We have used the SO(M) symmetry to exchange V ↔ W . Comparing it with criterion (2.6), we find that the arrow of 2 time will not be reversed at first order. This statement holds Since all the following results are of O(g ), we have omitted (2) for other forms of interaction and higher-dimensional gener- the superscript in the E ’s. The first (second) term in each integral kernel of Eq. (5.5) comes from the diagonal (cross) alization, while it does not hold for the mixed initial state. So 2 we design the interaction (5.1) to kill the effect of first-order terms in HI . If we choose the interaction (5.2), the second perturbation. term in each integral kernel of Eq. (5.5) disappears. If we choose interaction (5.3), the second term in each integral kernel of Eq. (5.5) has an inverse sign. The following discussion corresponding to the interaction (5.1) can be easily translated 1We thank Sang Pyo Kim for pointing this out. into the case of the interactions (5.2) and (5.3).

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FIG. 4. Kernels [KL (t, t )]discon, [KR(t, t )]discon, [KI (t, t )]discon as functions of t − t and energy changes (EL )discon, (ER )discon, (EI )discon as functions of t f − ti, contributed by the disconnected-tree diagram given in Fig. 3(a),where Δ = 1/6,β = 2π,λ = 2,C = 105,and = 0.2. Actually, the contribution of the loop diagram is neglectable here.

The two terms in each integral kernel of Eq. (5.5)are shifts if we change and remain ﬁnite if we set → 0. We denoted as [Kγ (t, t )]VWVW and [Kγ (t, t )]VWWV where γ = illustrate the energy changes contributed by the disconnected L, R, I. They are related to the following two commutators: diagrams shown in Figs. 3(a) and 3(b) in Fig. 4. After the dissipation time, [E (t, t )] is saturated, while energy = , I discon CVWVW [VL (t1)WR(t2 ) VL (t3)WR(t4)] starts to diffuse from hotter system R to colder system L via

= 2iImTr[V (−t3)V (−t1)yW (t2 )W (t4)y] the channel of disconnected diagrams. Further numerical calculation shows that the energy changes contributed by these ≈ + + −2 , (CVWVW )discon (CVWVW )t-channel O(C ) disconnected diagrams never satisfy criterion (2.6). This is (5.6a) expected since these disconnected diagrams are not related to the wormhole. They can also appear in the interaction C = [V (t )W (t ),W (t )V (t )] VWWV L 1 R 2 L 3 R 4 between two disconnected black holes in Appendix B, where = 2iImTr[W (−t3)V (−t1)yW (t2 )V (t4)y] the thermodynamic arrow of time should not be reversed. ≈ (C ) + O(C−2 ), (5.6b) The connected diagram in Fig. 3(c) is called the t channel, VWWV u-channel which goes across the wormhole. The interpretation of the t { , , , } ≈ { ,λ , ,λ }, < , t1 t2 t3 t4 t t t t t t (5.6c) channel is that the energy-momentum tensors of the two scalar fields χ ,χ on each side are correlated with each others and their derivative. C is transformed into VVWW-order V W VWVW caused by the propagation of the virtual graviton through the four-point functions and C is transformed into VWVW- VWWV wormhole. The propagation of the virtual graviton just reflects order four-point functions at finite temperature. the energy correlation between the two sides of the TFD state, Those diagrams contributing to the commutators (5.5)stem H |β=H |β.Thet channel contributes to the commutator from the causal connection between time t and time t , which L R C with is shown in Fig. 3. Since there are two operators on each side, VWVW − Δ (dis)connected diagrams will (not) go across the wormhole. t t 2 (C ) =−i23 sin(2πΔ) 4sinh 13 sinh 24 The first commutator (5.6b) contain the diagrams in VWVW t-channel 2 2 Figs. 3(a), 3(b), and 3(c) up to O(C−2 ). The tree diagram in Δ2 Fig. 3(a) and its one-loop correction in Fig. 3(b) are discon- t13 t24 × t coth −2 t coth −2 . nected and do not go across the wormhole. They are due to 2πC 13 2 24 2 the propagation of the two scalar fields χV ,χW in each side (5.8) separately. So, they are functions of t13 and t24, which are summed into It also depends on t13 and t24 only, but it is free from UV divergence once Δ<1. It is always −2Δ −1 3 t13 t24 O(C ), so it is smaller than the contribution of (CVWVW )discon =−i2 sin(2πΔ) 4sinh sinh 2 2 the tree diagram in Fig. 3(a). Since t13 0, t24 0, − <Δ< / − × [1 + O(C 1)]. (5.7) and 0 1 4, we find ( i)(CVWVW )t-channel 0; This has an important contribution on the kernels at short time t − t < td /2, where td = β/2π is the dissipation time. It suffers from UV divergence once 0 <Δ. However, when Δ< 1/4, the integral (EI )discon contributed by (CVWVW )discon is

ﬁnite. The kernel [KL (t, t ) − KR(t, t )]discon turns out to be 1−4Δ O((t − t ) ) so that the integral (EL − ER )discon con- verges when Δ<1/4. We will focus on the case of 0 < Δ<1/4. Of course, one can also introduce a small separation on imaginary time to regularize the UV behavior, such as V (t1)V (t2 )→V (t1 − i)V (t2 ). It acts as the UV cutoff of the boundary theory which is dual to JT gravity and FIG. 5. Kernels [KL (t, t ) + KR(t, t )]u-channel and [KL (t, t ) − β − satisﬁes . However, the integrals ( EL ER )discon KR(t, t )]u-channel contributed by the u channel at early time, where 5 and (EI )discon are not UV sensitive. They only have some Δ = 1/6,C = 10 ,β = 2π,λ = 2, and then t∗/2 = (lnC)/2 ≈ 5.7.

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TABLE I. Comparing the diagrams in Fig. 3.

Diagrams (Dis)connected Thermodynamic arrow of time Scale of energy change tree disconnected obey 1 one loop disconnected obey C−1 t channel connected obey C−1 u channel connected reverse at early time from C−1 to C−1et∗ ∼ 1

then (EL + ER )t-channel =−(EI )t-channel 0, which The second commutator (5.6b) is an OTOC. It leads to the does not satisfy criterion (2.6). diagram in Fig. 3(d) at O(C−1), which is called the u channel. Thus, the ﬁrst commutator (5.6b)isafunctionoft13 and It plays an important role in quantum chaos and traversable t24, which describes the process without the signal passing wormholes [31,32]. Its corresponding process in the bulk the wormhole since the original wormhole in the eternal black is that two particles scatter near the horizon by exchanging hole is not traversable. The energy change contributed by the gravitons and then reach the opposite boundaries. In contrast, ﬁrst commutator (5.6b) obeys the thermodynamic arrow of without the exchange of gravitons, the diagram does not con- time. tribute to Eq. (5.6) since the wormhole is not traversable. The u channel contributes to the commutator CVWWV with

+ t + t t + t −2Δ Δ2 sinh t14 t23 t + t t + t t + t (C ) =−i23 4 cosh 2 3 cosh 1 4 2 + 14 23 tanh 2 3 tanh 1 4 . VWWV u-channel t2+t3 t1+t4 2 2 C cosh 2 cosh 2 2 2 2 (5.9)

Due to the suppression of C−1, it is visible only within both of which are negative and satisfy criterion (2.6). This the chaos region, t ≈−t and td /2 t < t∗/2, where t∗ = means that the u channel with the interaction (5.1) contributes β 2πC λ = = to anomalous heat flow, at least at early time, when λ ≈ 1. 2π ln β is the scrambling time. Specifically, let 1, t1 Our numerical calculation shows that, for general λ>1, such t2 = t > 0, and t3 = t4 = t =−t; this becomes anomalous heat flow also appears. We illustrate kernels con- 3−4Δ 2 −1 (CVWWV )u-channel =−i2 Δ C sinh(2t ), (5.10) tributed by the u channel in Fig. 5. However, in Eq. (5.5), we should sum over all the di- t / t < t∗/ which will exponentially grow at early time d 2 2. agrams. We estimate their scales, as concluded in Table I. At late time t t∗/2, the approximation in Eq. (5.9)at −1 Specially, the energy change contributed by the u channel O(C ) is unreliable, because of the strong backreaction on −1 is order C before the dissipation time td and it reaches the metric, which will be investigated in the next section. If −1 t∗ order C e ∼ 1 near the scrambling time t∗. First, the worm- t =−t λ ≈ u we let i f and 1inEq.(5.5), the channel will hole supports two channels, the t channel and the u channel. contribute exponential growth on the energy changes at early Near the scrambling time, the contribution of the u channel t / t < t∗/ time d 2 f 2, whose leading behaviors are surpasses the contribution of the t channel. So, with the inter-

2 −1 2t f action (5.1), the wormhole supports an anomalous heat ﬂow (EL + ER )u-channel =−(EI )u-channel ∼−g C e near the scrambling time. Second, since the contributions of + O((λ − 1)1), (5.11) the tree diagram and the u channel are in the same order, to determine the whole thermodynamic arrow of time, we should 2 −1 2t f 2 (EL − ER )u-channel ∼−(λ − 1)g C t f e + O((λ − 1) ), compare the contribution of the two diagrams numerically, (5.12) as shown in Fig. 6. We ﬁnd that the contribution of the tree

FIG. 6. EL ± ER contributed by different diagrams in Table I as functions of t f , where the ﬁfth line “eikonal” refers to the contribution 5 of CVWWV in Eq. (5.13). Parameters are chosen to be Δ = 1/6,C = 10 ,β = 2π,λ = 2, = 0.2, and ti =−15. So t∗ ≈ 11.5 and the early time is t f < 5.8. The contributions of the one-loop correction and the t channel are too small to be shown.

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C. Late time At late time, one should consider the higher-order C−1 correction on CVWWV beyond the u channel. By taking the limit t →∞, C →∞while keeping C−1et ﬁnite, one can calculate CVWWV in the eikonal approximation [29,48]

CVWWV =−2iVL (t1)VR(t4)WR(t2 )WL (t3)ImF, i 2Δ i F = U Δ, , , ζ 2 1 ζ + − − / 1 e(t1 t2 t3 t4 ) 2 ζ = + + , 8C cosh t2 t3 cosh t1 t4 ζ Δ 2 2 FIG. 7. ∗ as a function of . t1 ≈ t2/λ ≈ t, t3 ≈ t4/λ ≈ t , t < 0 < t, (5.13) diagram is larger than the contribution of the u channel at early where the two-point functions are given in Eq. (3.24) , , = time. So we conclude that the whole thermodynamic arrow of and the conﬂuent hypergeometric function U (a 1 x) −1 ∞ −sx sa−1 ζ> time has not been reversed at early time. (a) 0 dse (1+s)a is used. In real time, 0 always

FIG. 8. Left and middle: The integral kernels (left) [KL (t, t )]VWWV and (middle) [KR(t, t )]VWWV contributed by CVWWV as functions of

{t, t } under the interaction (5.1). The dashed curves denote (blue) t + λt = 0, (orange) λt + t = 0, and (green) ζ = ζ∗ in Eq. (5.14). Right: 5 The energy changes contributed by CVWWV as functions of t f . Parameters are Δ = 1/6,C = 10 ,β = 2π, g = 1, and (a) λ = 1.1, ti =−20, (b) λ = 2, ti =−15, or (c) λ = 5, ti =−15.

043095-11 ZHUO-YU XIAN AND LONG ZHAO PHYSICAL REVIEW RESEARCH 2, 043095 (2020) holds; then ImF > 0 always holds. Equation (5.13)isneg- exhibits the anomalous heat flow via the wormhole. Note that ligible before the early time. It agrees with the u channel in the final magnitude of the anomalous heat flow decreases Eq. (5.9) well at early time, while it decays to zero at late when λ increases from 1, which agrees with the analysis of time. So its amplitude is maximized at the timescale of the Renyi entropy in the next section. scrambling time. More precisely, ImF is maximal at ζ = ζ∗, Similar to the u channel in the previous subsection, the whose dependence on Δ is shown in Fig. 7. It corresponds to amplitude of CVWWV is order 1. Numerically, it is usually the timescale on (t, t )as smaller than the contribution of the disconnected-tree dia-

(λ+1)t gram, as shown in Fig. 6. In other words, the thermal diffusion e ≈ ζ . is stronger than the anomalous heat flow. So the total thermo- +λ λ + ∗ (5.14) 2C(et t + 1)(e t t + 1) dynamic arrow of time cannot be reversed even at late time. The time dependence of the two-point functions in Eq. (3.24) According to Appendix A, we find that the work done on also affects the time of the maximal amplitude of C . the local systems vanishes as well because one-point functions VWWV Basically, when λ ≈ 1, it is maximized at the timescale (5.14); V and three-point functions VVW vanish. So the energies when λ 1, it is maximized at the timescale which is smaller change in the way of heat. This agrees with our terminologies than (5.14). “thermal diffusion” and “anomalous heat flow.” Besides, there are two lines in the plane of (t, t ) where the Finally, in Appendix C, we show that such anomalous heat two-point functions in Eq. (5.13) are maximized respectively, flow via wormhole can be constructed in the SYK model as well since its breaking of conformal symmetry at the large-N λt + t = 0, (5.15) limit is also described by Schwarzian action at low energy.

t + λt = 0. (5.16) D. An interpretation from boundary theories The intersections of Eqs. (5.14) and (5.15) and Eqs. (5.14) According to the AdS/CFT correspondence, the dual sys- and (5.16) are approximately tem on the boundary is strongly coupled and highly chaotic. It 1 t∗ goes beyond the ordinary realization of anomalous heat flow (t, t ) = (t , −λt ), t ≈ , (5.17) a a a 1 + λ 2 in weakly coupled or weakly chaotic models [3,4,6] and is relatively difficult to be understood from boundary theories. λ t∗ (t, t ) = (t , −t /λ), t ≈ . b b b + λ (5.18) Here we try to explain. Recall that energy changes highly 1 2 rely on the form of interaction, which must be taken into The two timescales ta and tb can characterize the process of consideration. For simplicity, we assume g > 0. energy change. We first consider the interaction HI = gVLWR, so only the In Fig. 8, we show the energy changes (Eγ )VWWV disconnected diagrams are presented. We will explain the be- and their kernels [Kγ (t, t )]VWWV contributed by CVWWV in haviors of kernels [Kγ (t2, t1)]discon for γ = L, R, I,asshown Eq. (5.13) including the late time. We can consider the process in Fig. 4, which corresponds to the energy change after the in- with initially time ti −t∗ and observe the energy changes stantaneous deformation H(t ) = HI (δ(t − t1) + δ(t − t2 )). contributed by CVWWV when t f goes from ti to +∞. We find At time t1, we turn on a potential gVLWR and then inter- several periods. All the energy changes in the following list mediately turn it off. This creates an EPR pair, which is a refer to the contribution by CVWWV only. superposition of all the combinations tending to lower the (i) When t f ta, these energy changes are negligible, value of the potential gVLWR. However, it will not change since the system hardly scrambles. EL + ER, since VLWR initially vanishes on quantum average. (ii) When t f ∼ ta, the amplitude of commutator CVWWV In other words, the first-order perturbation vanishes. The EPR exponentially grows. Both (EL )VWWV and (ER )VWWV syn- pair will interact with other particles on their own sides and chronously decrease and (EI )VWWV increases. During this then dissipate when time goes forward, characterized by a time, |(EL − ER )VWWV| is still small. dissipation time td which scales as the inverse temperature (iii) When ta t f tb,(EL )VWWV keeps drop- βγ on each side. At time t2 > t1, we instantaneously turn ping, while (ER )VWWV in turn increases. Then on the potential gVLWR again. The accumulation of the ten- (EL − ER )VWWV starts to drop and system R keeps dency of the EPR pair has lowered gVLWR and leads to absorbing the energy from system L, which is quite apparent [KI (t2, t1)]discon < 0. However, when t2 − t1 td , the dying in Figs. 8(b) and 8(c).(EI )VWWV reaches its maximum in EPR pair is unable to lower gVLWR, so that gVLWR as well this period. as |[KI (t2, t1)]discon| return to zero. Meanwhile [Kγ (t2, t1)]discon (iv) When t f ∼ tb,(EL )VWWV in turn increases and is proportional to the power of gVLWR done on system γ at (ER )VWWV increases, whose speeds are low when λ 1. time t2. When t2 goes from t1 to infinity, the value gVLWR(t2 ) EI decreases. decreases first and then returns to zero, such that its power (v) When tb t f , all the energies go to some constants, done on system γ is positive first and is negative later, which because of the quasinormal decay of commutator CVWWV at agrees with the tendency of [Kγ (t2, t1)]discon. The hotter the late time [48]. Especially, (EI )VWWV is forced to vanish, system is, the faster the EPR pair decays. So [KR(t2, t1)]discon while the net energy transfer (EL − ER )VWWV reaches a becomes negative earlier than [KL (t2, t1)]discon. For the sus- negative constant. tained interaction HI = gVLWR, since (EI )discon does not During the whole process contributed by CVWWV, energy change after td , the difference between the temperatures leads transfers from colder system L to hotter system R, which to a net energy transfer from system R to system L.

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Consider a classical and highly chaotic system containing N canonical coordinates {xi}(i = 1,...,N) with zero average value xi = 0. The classical model in Ref. [32]isa good example, as discussed in Appendix D.Attimet1,we = 2δ − add an instantaneous potential H(t ) gx1 (t t1), which gives the system a momentum along the direction of reducing 2 > the fluctuation x1 .Att2 t1, we measure the fluctuation 2 { 2, 2} x2 . g x2(t2 ) x1(t1) P characterizes the change of the 2 fluctuation x2 at time t2 due to the instantaneous potential at time t1, which is the classical interpretation of Eq. (5.20) if we identify x1 = V, x2 = W . Since the system is highly FIG. 9. (a) The time contour C in the complex plane z = chaotic, the fluctuation of xi is sourced from the noises given − exp(i2πtC /β), where the complex time tC = τ + it is measured by by the other (N 1) canonical coordinates. When the fluc- evolution exp(−tC H ). The blue (orange) interval indicates the evolu- tuation of x1 is suppressed by the instantaneous potential at tion of system L (R). The black point indicates the twisted operator time t1, we expect that the fluctuation of other xi=1 will be Xn. The dashed lines indicate the coupling between the two sides suppressed later due to their interactions with x1. If we only , during [ti t f ]. The thick interval in the upper (lower) half plane focus on one of them, such as x2, the response is split into N C+ C− − = λ = { 2, 2} indicates contour ( ). We have chosen ti t f and 2in parts. So g x2(t2 ) x1(t1) P is negative and is suppressed this figure. So the real-time contour of system R is longer than the by an N−1 factor. Furthermore, since the system is chaotic, C = one of system L. (b) We unfold the time contour n for n 3 replicas such response should exponentially grow along time t2 − t1. in the complex plane z1/n = exp(i2πt /nβ). C Finally, when t2 goes to infinity, the system should reach a new equilibration and lose its memory. The final value of 2 fluctuation x2 only depends on the final equilibrium state Now√ we consider the interaction HI = g(VLWR − which is determined by the energy of the system after the WRVL )/ 2. The analysis of the contribution of the perturbation H(t ), while the energy at t2 does not change { , 2} ∝ ∂ 2 = disconnected diagrams is the same as before. We will at linear order, i.e., H gx1(t ) P t x1(t ) 0. So { 2, 2} focus on kernels [Kγ (t2, t1)]VWWV for γ = L, R, I,asshown linear response g x2(t2 ) x1(t1) P should decay to zero in Fig. 8, whose main contributions are near t2 ∼−t1 = t.We with a timescale near the inverse temperature. We summarize first analyze [KI (t, −t )]VWWV when λ = 1. We find the behavior from the classical approximation

− λ g2N 1e2 Lt , early time, , , − , − = { , − } , − , , − ≈ CVWWV (t t t t ) Tr(y W (t ) V ( t ) y[W (t ) V ( t )]) [KI (t t )]VWWV − /β (5.21) 2α c1t , , (5.19) g 1e later time i 2 [KI (t, −t )]VWWV = g CVWWV (t, t, −t, −t ) α , > h¯ where coefficients 1 c1 0 and Lyapunov exponent λ π/β 2 L 2 for highly chaotic systems. The behavior of h¯→0 g 2 2 , , − −−→− {W (t ) ,V (−t ) }P, [KL R(t t )]VWWV can be approximated by the time derivative 2 of −[KI (t, −t )]VWWV.So[KL,R(t, −t )]VWWV is negative at (5.20) early time, becomes positive at later time, and vanishes finally. The classical approximation here agrees with the results in JT where we have restoredh ¯ and taken the classical limith ¯ → 0. gravity (5.11). So CVWWV (t, t, −t, −t ) is just the cross-correlation between Now we consider the case of λ 1. The general be- {...,...} , − γ = , , fluctuation and dissipation [51]. P is the Poisson haviors of [Kγ (t t )]VWWV for L R I do not change bracket. ... is the classical ensemble average. From the much. The main difference between [KL (t, −t )]VWWV and classical limit, we know that [KI (t, −t )]VWWV characterizes [KR(t, −t )]VWWV is in their timescales. The dissipation the dissipation of fluctuation, which can be understood from time and the scrambling time of system R are brought linear response. forward due to the λ in the temporal argument of

5 FIG. 10. Yn(t, t )andYn(t, −t ) as functions of t for n = 2, 3, 4; g = 1; β = 2π; Δ = 1/6; C = 10 ; = 0.2; and λ = 1.

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Eq. (5.5). So [KR(t, −t )]VWWV becomes positive earlier change as a function of real time. For the simplicity of no- than [KL (t, −t )]VWWV, as shown in Fig. 8(a). These shorter tation, we will use imaginary-time evolution as follows: timescales advance the dissipation of fluctuation in system O(τ ) = eτH Oe−τH . (6.1) R. Recall that (EI )VWWV must return to zero finally. So the integral effect is that system R obtains energy while system L Since the bisystems are in a pure state, the entropies of loses energy. both sides are the same. So we will only calculate the entropy of system R. Consider the density matrix of system R at time VI. ENTROPY CHANGE t f , A. Replica trick ρ˜R(t f ) = TrL|β˜ I (t f )β˜ I (t f )|, (6.2) We will calculate the entropy change at time t f after we turn on an interaction HI at time ti. We will work on imaginary where |β˜ I (t f ) is given in Eq. (4.1). The Renyi entropy and time first, then Wick-rotate τ → it and obtain the entropy von Neumann entropy of system R can be written as

n ln Tr[˜ρR(t f ) ] ln Zn − n ln Z S = = , S =−Tr[ρ ˜ (t )ln˜ρ (t )] =−∂ ln Z + ln Z| → , (6.3) n 1 − n 1 − n R f R f n n n 1 where Zn is the n-replicated partition function and the twisted boundary conditions are applied on system R at time t f . We deﬁne a twist operator Xn, which cyclically permutes the replica index. We can also write the Renyi entropy of system R as

(1−n)Sn n ⊗n e = Zn/Z = Tr[ρ ˜ R(t f ) Xn] (6.4a) ⊗n ⊗n = β˜ I (t f )| (I ⊗ Xn )|β˜ I (t f ) (6.4b) ⊗n ⊗ t f † t f n ⊗n ⊗n = β| T exp −i dt H˜I (it ) (I ⊗ Xn ) T exp −i dt H˜I (it ) |β (6.4c) t t i i ⊗n ⊗n ⊗n ⊗n = β| TC exp − dτ H˜I (τ ) (I ⊗ Xn ) TC exp − dτ H˜I (τ ) |β , (6.4d) C+ C− − − + + where TC is the contour ordering. Contour C goes from iti + 0 to it f and contour C goes from it f to iti + 0 , namely − − C = {z ∈ Z|z = (iti + 0 )(1 − κ) + it f κ, κ ∈ [0, 1]}, (6.5a) + + C = {z ∈ Z|z = it f (1 − κ) + (iti + 0 )κ, κ ∈ [0, 1]}, (6.5b) ⊗n whose directions in real time are opposite. In I ⊗ Xn, I acts on the n copies of system L, denoted as systems L , and Xn acts on systems R⊗n. Equation (6.4d) can help us to write down the path integral on closed time contours. Recall that an operator of system R (L) acting on the TFD state can be transformed into a corresponding operator of system L (R) acting on the TFD state. Those ⊗n operators on the left-hand side of I ⊗ Xn can go through the channels of systems L , reach the right-hand side of Xn, and ﬁnally become the operators of systems R⊗n. Taking H = V W as an example, we have I L R n−1 n−1 ⊗n ⊗n β| TC exp − dτVL,a(τ )WR,a(λτ ) (I ⊗ Xn ) TC exp − dτVL,a(τ )WR,a(λτ ) |β (6.6a) C+ C− a=0 a=0 n−1 β β ⊗n = Tr ρ XnTC exp − dτWa(λτ )Va − − τ + dτVa − − τ Wa(−β + λτ ) (6.6b) C− 2 C+ 2 a=0 n−1 β β = TC exp − dτWa(β + λτ )Va − τ + dτVa − τ Wa(λτ ) Xn , (6.6c) C− 2 C+ 2 a=0 ⊗ n n−1 = TC exp − dτ1dτ2σ+(τ1,τ2 )Va(τ1)Wa(τ2 ) Xn , (6.6d) C a=0 ⊗n = T − , C exp ( In ) ⊗n (6.6e) ... = ρ⊗n where ⊗n Tr[ ] and β β σ+(τ1,τ2 ) = dτδ τ1 − − τ δ(τ2 − (β + λτ )) + dτδ τ1 − − τ δ(τ2 − λτ ). (6.7) C− 2 C+ 2

We let the replica index a begins at 0 for later convenience. The deformation In is nonlocal. The action is evaluated on time contour C where the imaginary time goes from 0 to β, as shown in Fig. 9(a), which is covered by an n-sheet manifold. Due to the

043095-14 WORMHOLES AND THE THERMODYNAMIC ARROW OF TIME PHYSICAL REVIEW RESEARCH 2, 043095 (2020)

− + contour ordering, the integral along C is always on the left-hand side of the integral along C . The twist operator Xn imposes the following twisted boundary condition:

− + − + Va(0 ) = Va+1(0 ), Wa(0 ) = Wa+1(0 ), a ∼ a + n. (6.8)

When n = 1, we have W (β + τ ) = W (τ ) and In = 0, which is in agreement with the unitary evolution of the TFD state. One can further unfold the n-sheet manifold in Eq. (6.6) and write the path integral into n−1 T − τ + β + λτ + 1 β − τ + τ + 1 β − τ β + λτ , Cn exp d W ((a 1) )V a d V a W (a ) (6.9) C− 2 C+ 2 a=0 nβ

... = −n −nβH ... where nβ Z Tr[e ] and

V (aβ + τ ) = Va(τ ), W (aβ + τ ) = Wa(τ ). (6.10)

The action is evaluated on the unfolded contour Cn, where ln Z = 0 under unitary evolution. In perturbation theory, it the imaginary time goes from 0 to nβ, as shown in Fig. 9(b). is reduced to the Feynman diagrams in the expansion of defor- Similar approaches appeared in Refs. [37,45,52]. mation In, like (6.9), on the saddle-point background (6.14). We have − B. Entropy at equilibrium ln Z ln e In −I S = n = = n +··· . (6.16) We will first calculate the entropy at equilibrium by using n 1 − n 1 − n 1 − n {ϕ τ } the replica trick. For the Euclidean parametrizations a( ) Recall that we have extracted work from the TFD state by with n sheets in the hyperbolic dish, the twisted boundary applying the interaction H = gO O in Sec. IV.Asawarm- conditions are I L R up, we will derive Sn at O(g). From Eq. (6.16), we have ϕ β = ϕ − , ∼ + . a( ) a+1(0 ) a a n (6.11) t f β (1) ign (S ) = dt G β + i(t + λt ) It is convenient to introduce a global parametrization f (τ ) n − n 1 n ti 2 which satisfies β − Gnβ − i(t + λt ) , (6.17) f (τ + aβ) = ϕa(τ ). (6.12) 2 β πτ −2Δ Without HI , the original replicated effective action In,0 can be n n G β (τ ) = Tr[O(τ )Oρ ] = 2 sin written as a function of f (τ ), n π nβ n−1 β nβ + −1 , <τ< β, ϕa f O(C ) 0 n (6.18) In,0 =−C dτ tan ,τ =−C dτ tan ,τ . 2 2 τ a=0 0 0 where Gnβ ( ) is the Euclidean two-point function at inverse (6.13) temperature nβ. Then the change in von Neumann entropy is The saddle-point solution is β β π λ + −2Δt f (1) 2g ( 1)t −1 S = cosh + O(C ), 2πτ λ + 1 π β f = . (6.14) ti c nβ (6.19) Then we have which agrees with Eq. (4.5) derived from the first law of en- 2π 2C 2π 2C(n + 1) tropy. We justify the consumption of entanglement in Eq. (4.7) ln Z = S + , S = S + , n 0 nβ n 0 nβ when extracting work. 4π 2C S = S + , D. The entanglement changed by anomalous heat flow 0 β (6.15) 1. Renyi entropy for the nearly extremal case, where S0 = φ0/4G is the extremal entropy contributed by the topological term in We will calculate the entropy change due to the interaction (5.1). The change at the first order vanishes, and we will Eq. (3.6)[53]. So C/S = φ/¯ 2πφ = α/2π. 0 0 consider the second-order perturbation. The insertions of two HI ’s on the n-sheet time contour happen in various ways. C. The entanglement changed by extracting work Both time-order and out-of-time-order correlation functions Things become complicated when the interaction HI is will appear. Similarly to what we did in Eq. (6.6), we can write turned on. The change in Renyi entropy Sn is proportional down the deformation In on the original replicated effective√ to the change in replicated generating functional ln Zn, since action (6.13) with the interaction HI = g(VLWR − WLVR )/ 2,

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− g n 1 −In =−√ dτ1dτ2 σ+(τ1,τ2 )[Va(τ1)Wa(τ2 ) − Wa(τ1)Va(τ2 )] − IM,a (6.20a) C 2 a=0 2 n−1 = g τ τ τ τ σ τ ,τ σ τ ,τ ϕ τ ,τ ϕ τ ,τ − ϕ τ ,τ ϕ τ ,τ d 1d 2d 3d 4 +( 1 2 ) +( 3 4 ) Gab( 1 3)Gab( 2 4 ) Gab( 1 4 )Gab( 2 3) (6.20b) 2 C a,b=0 − g2 n 1 = dτ1dτ2dτ3dτ4 σ+(τ1,τ2 )σ+(τ3,τ4 ) 2 C a,b=0 f f f f × [G (aβ + τ1, bβ + τ3)G (aβ + τ2, bβ + τ4 ) − G (aβ + τ1, bβ + τ4)G (aβ + τ2, bβ + τ3)], (6.20c) where σ+(τ1,τ2 ) is deﬁned in Eq. (6.7) and ϕ τ ,τ = T τ τ = T τ τ = f β + τ , β + τ , Gab( 1 2 ) EVa( 1)Vb( 2 ) EWa( 1)Wb( 2 ) G (a 1 b 2 ) (6.21) Δ τ τ f f ( 1) f ( 2 ) G (τ1,τ2 ) = 2 . (6.22) f (τ1 )− f (τ2 2 2sin 2 )

± IM,a is the action of the matter ﬁelds on the ath sheet. In from the integral along the contours C in Eq. (6.7).

Eq. (6.20b), we further integrate out the matter ﬁelds in IM,a The integral kernel Yn(t, t ) is real and symmetric, ϕ , = , which gives the correlation function Gab. Similarly to the Yn(t t ) Yn(t t ). It contains two kinds of four-point 2 τ τ τ τ τ >τ >τ >τ situation of energy change at O(g ), the ﬁrst (second) term function V ( 1)V ( 2 )W ( 3)W ( 4) nβ ( 1 2 3 4) 2 τ τ τ τ τ >τ >τ >τ in Eq. (6.20b) comes from the diagonal (cross) terms in HI . and V ( 1)W ( 3)V ( 2 )W ( 4) nβ ( 1 3 2 4 ). We

In Eq. (6.20c), we use the global parametrization (6.12). Ac- can group those terms in Yn(t, t ) into two parts accordingly, cording to Eq. (6.13), the reparametrization modes around the denoted as [Yn(t, t )]VVWW and [Yn(t, t )]VWVW. The former = πτ/ β saddle-point solution fc 2 n are the same as the orig- suffers from UV divergence when τ1 → τ2 or τ3 → τ4.So inal modes in Eq. (3.11) except that the inverse temperature we will introduce a small separation to regularized it, β is n now. We can integrate them out and obtain the expres- such as V (τ1 + )V (τ2 )W (τ3 + )W (τ4), while the latter is f τ ,τ f τ ,τ sion of the four-point function G ( 1 2 )G ( 3 4 ) up to free from UV divergence since the situation of τ1 → τ2 or −1 O(C ), which is the same as the original one derived from the τ3 → τ4 never happens in Eq. (6.20). generating functional (3.17) except that the inverse tempera- The four-point functions will be calculated in JT grav- ture is nβ. After integrating out the four times {τ1,τ2,τ3,τ4} ity. We will calculate VVWW by integrating out the with those delta functions, we obtain the change in Renyi reparametrization modes at O(C−1) and calculate VWVW entropy with the form by using the eikonal approximation. One can alternatively −I t f t S = n + O g4 ≈ dt dt Y t, t . n − ( ) n( ) (6.23) 1 n ti ti We have used the perturbation theory at O(g2 ) and large-C limit. The double integrals on real time come

FIG. 12. The illustration of the ﬁve regions in the (t, t ) plane. (1)

FIG. 11. [Y2(t, t )]VVWW (upper, orange) and [Y2(t, t )]VWVW UV-sensitive region. (2) Conformal region. (3) Dissipation region. (lower, blue), where g = 1,β = 2π,Δ = 1/6,C = 105, = 0.2, (4a) and (4b) Early-time chaos region. (5a) and (5b) Later-time chaos and λ = 1. region.

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5 FIG. 13. Y2(t, t )andY2(t, −t ) as functions of t,whereg = 1; β = 2π; Δ = 1/6; C = 10 ; = 1, 0.5, 0.2, 0.1; and λ = 1. calculate VWVW by integrating out the reparametrization (2) Conformal region, |t − t |ntd and |t + t | −1 modes at O(C ). The result is closed to the result from the t∗. The four-point functions are factorized into conformal two- −4Δ eikonal approximation at early time while it becomes unreli- point functions. So Yn(t, t ) ∼−|t − t | < 0. able at late time. For VWVW, we have the property (3) Dissipation region, ntd |t − t | and |tt |∼0. These four-point functions are factorized into two-point functions V (it1)W (it3)V (it2 )W (it4) nβ at ﬁnite temperature, such that the kernel is dominated −2Δ|t|/nt = + β by dissipation, Yn(t, 0) ∼−e d < 0 and Yn(0, t ) ∼ W (it4 n )V (it1)W (it3)V (it2 ) nβ − Δ| |/ −e 2 t ntd < 0. = + β , V (it4 n )W (it1)V (it3)W (it2 ) nβ (6.24) (4a) Early-time chaos region A, ntd |t| < nt∗ and t ∼ t . ∼− −1β1−4Δ 4πt The kernel contains important terms C exp nβ , where we have exchanged operators V and W according to which is contributed by the OTOCs in the ﬁrst term of the SO(M) symmetry of Eq. (3.6), while the eikonal approx- Eq. (6.20c), as illustrated in Fig. 14(a). So the Renyi entropy imation on V (t1)W (t3)V (t2 )W (t4) in (6.57) of [48] does Sn will exponentially decrease with an exponent 2πt/nβ. not maintain the above property. So we adjust the formula Such behavior appears during general interactions between accordingly as follows: the two sides of the TFD state, as long as the interaction

V (it )W (it )V (it )W (it ) β contains a term like VLWR or OLOR. 1 3 2 4 n = ξ 2Δ Δ, ,ξ , || < ∼ U (2 1 ) (4b) Early-time chaos region B, ntd t nt∗ and t V (it1)V (it2 ) nβ W (it3)W (it4) nβ − ∼− −1β1−4Δ 4πt t . The kernel is dominated by terms C exp nβ π(t3+t4−t1−t2 ) 2sinh and the Renyi entropy Sn exponentially decreases with an 1 = i nβ , π − π − (6.25) exponent 2πt/nβ as well. It is contributed by the OTOCs in ξ 8C sinh (t1 t2 ) sinh (t3 t4 ) nβ nβ the second term of Eq. (6.20c), as illustrated in Fig. 14(b).So it will not appear if we adopt H = gV W . Such decrease in where we have replaced the exponent by 2 sinh which is valid I L R + − − > + − − the entropy kernel corresponds to the anomalous heat flow, for both Re[t3 t4 t1 t2] 0 and Re[t3 t4 t1 t2] ∼− < since both of them appear near t t and source from the 0. 2 cross terms in HI . We first look at Sn for integer n 2. We first analyze (5a) Late-time chaos region A, |t| > nt∗ with t ∼ t .The the case of λ = 1. The configuration of Y (t, t )isshown n kernel saturates and goes to a constant, which is contributed in Fig. 10. We also separately draw the configurations of by both -regularized conformal two-point functions and [Yn(t, t )]VVWW and [Yn(t, t )]VVWW in Fig. 11. We can find OTOCs, as shown in Fig. 13. that finally [Yn(t, t )]VVWW goes to zero due to the quasinormal (5b) Late-time chaos region B, |t| > nt∗ with t ∼−t .Due decay of OTOCs at late time. The characteristic timescales are = β to the late-time decay of OTOCs, the kernel goes to a negative UV cutoff , dissipation time td 2π , and scrambling time β 2πC ∗ = , t 2π ln β . We divides the (t t ) plane into several regions according to these timescales, as illustrated in Fig. 12.

(1) UV-sensitive region, |t − t | and |t + t |t∗. The kernel regularized by the separation is UV sensitive. In Fig. 13, we show the dependence on the choice of the separation . We ﬁnd Yn(t, t ) > 0 when t → t after such regularization. Such positivity in the UV timescale also appears even when the wormhole is absent as in Appendix B and in the two-site SYK model in Appendix C, where the latter has a known UV completion. During the interaction between product states, the Renyi entropy keeps increasing. So the Renyi mutual information is always positive, although the positivity of Renyi mutual information does not always hold for general states [54]. In Fig. 13, we further ﬁnd that when FIG. 14. Two OTOCs in (a) Y3(t, t )and(b)Y3(t, −t )shownin

becomes smaller, the value of Y2(t, t ) in the UV timescale the (n = 3)-sheet time contour. The red points and the blue points has a positive shift. represent the operators on different types.

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5 FIG. 15. Y2(t, t )andY2(t, −t ) from eikonal approximation (6.25)forλ = 0.5, 0.8, 1, 1.2, 2; g = 1; β = 2π; Δ = 1/6; C = 10 ; = 0.2.

2 number and the suppression of |S2| near t f ∼ t∗, as shown in Fig. 15 and Fig. 16(a). n−1 n 1 In Appendix C, we calculate the entropy changes for the lim Yn(t, −t ) =− W a + β + it t→∞ n − 1 2 initial TFD state in the two-site SYK model at early time. a=1 Regions (1), (2), (3), (4a), and (4b) also appear. In region (1), β × V (aβ−it )V −it W (it ) +conjugate at the large-q limit, we can get rid of the cutoff dependence. 2 nβ In region (4b), the kernel exponentially decreases as well, 2 π which agrees with the presence of the anomalous heat flow in =− csc2 + O(C−1) (6.26) (n − 1)n 2n the SYK model. But, in region (4a), the kernel exponentially increases, which is different from the case of JT gravity. without UV sensitivity, as shown in Fig. 13. The above two kinds of exponential decrease at early time 2. von Neumann entropy rely on the existence of the wormhole. In Appendix B,were- place the eternal black hole by two disconnected black holes, To obtain the change in von Neumann entropy S,we n−1 where the wormhole is absent. We calculate the change in calculate the summation a,b=0 in Eq. (6.20) via contour Renyi entropy after turning on the interaction, where chaos integration and take the n → 1 limit. The details are given regions (4) and (5) are replaced by dissipation region (3). in Appendix E. We work on a relatively simple case: large C Δ = / We discuss the case of general λ. Yn(t, t ) for different λ and 1 2. Then VVWW is approximately factorized and are shown in Fig. 15. Yn(t, t ) reaches a constant at late time VWVW is in the forms of eikonal approximation (6.25). The which is independent from λ; Yn(t, −t ) approaches zero from summations of factorized four-point functions have analytical below at late time once λ = 1, since those four-point functions forms. contributing to Eq. (6.26) finally die out due to the mismatch For the kernel of von Neumann entropy change Y1(t, t ), between t and λt. We integrate the kernel and calculate S2 its behavior also depends on the regions (1)–(5) introduced as a function of time, as shown in Fig. 16. In the case of before. There are some differences between the Y1(t, t ) and , = / → + ti 0, S2 momentarily increases near t f = 0 due to the Yn>1(t t ) when 1 2. We can set the separation 0 , positive contribution in UV-sensitive region (1). However in but keeping Y1(t, t ) finite. In regions (2), (3), and (4), the ker- other periods of time, it generally decreases. Especially, in the nel Y1(t, t ) is positive. In region (5), the kernel Y1(t, t ) goes λ = λ> case of −nt∗ < ti 0, the negative contribution mainly comes to positive numbers when 1 and zero when 1. By from chaos regions (4a) and (5a). In the case of ti < −nt∗,the comparing the kernel for finite C with the kernel for infinite C, negative contribution mainly come from chaos regions (4) and we find that the gravitational scattering (finite-C effect) lowers (5). We also check the Renyi entropy for higher n and similar down the entanglement in chaos regions (4) and (5), as shown behaviors are observed. in Fig. 17, which agrees with the timescale of anomalous heat Recall that the anomalous heat flow in Sec. II appears flow. when t f ∼ t∗ in the case of ti < −t∗ and λ>1, basically due The change in von Neumann entropy S now becomes to [KL (t, t ) − KR(t, t )]VWWV < 0 near t ∼−t ∼ t∗. Such positive, which is different from the change in Renyi entropy. agreements between the kernels of energy and entropy on But the gravitational scattering still lowers S. Recall that the timescale and sign support our expectation that anomalous gravitational scattering mainly contributes to the anomalous heat flow appears with the consumption of entanglement. Fur- heat flow as well. So we may identify it as the entanglement thermore, when λ increases, the magnitude of the anomalous consumption related to the anomalous heat flow. heat flow become smaller as shown in Fig. 8. It agrees with the suppression of |Y (t, t )| in chaos regions (4b) and (5b) n VII. CONCLUSION AND OUTLOOK Quantum entanglement can be taken as the resource to extract work or transfer energy. In this paper, we extract the work 2 To obtain limt→∞ Y1(t, −t ), we take the n → 1 limit first and then and study the thermodynamic arrow of time in the presence take the t →−t limit. We find limt→∞ limt →−t limn→1 Yn(t, t ) = 4 of wormholes in the AdS/CFT correspondence and the SYK at the large-C limit. model, where the field theory description is strongly coupled

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5 FIG. 16. S2 as a function of t f ,whereλ = 0.5, 0.8, 1, 1.2, 2; g = 1; β = 2π; Δ = 1/6; C = 10 ; = 0.2; (a) ti =−60 or (b) ti = 0. and highly chaotic. The initial state is taken to be the eternal However, it is difficult to handle the highly nonlocal term black hole in JT gravity or the TFD state in the two-site SYK in In here. One may have to consider other interactions model. Such kind of states has two faces: it is highly entangled to maintain the solvability and the anomalous heat flow and local thermal. When some proper interactions between the simultaneously. two sides are turned on, we discover the anomalous heat flow, (iii) The possibility of reversing the total thermodynamic which transfers the energy from the colder side to the hotter arrow of time. We have scanned the reasonable parameter side, and further investigate its concomitant consumption of space satisfying <β

FIG. 17. The changes in the von Neumann entropy S and its kernel Y1(t, t ), where g = 1,β = 2π,Δ = 1/2, = 0,λ= 1.1, ti =−15, C =∞(upper, orange) or C = 105 (lower, blue).

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ACKNOWLEDGMENTS CFTR, which is dual to two disconnected black holes living in two disconnected AdS spaces [49]. Similarly, we consider We are grateful to Yu-Sen An, Rong-Gen Cai, Yiming 2 that in each space there is dilaton gravity with the JT term and Chen, Johanna Erdmenger, Song He, Shao-Kai Jian, Sang some free matter fields Pyo Kim, Hong Lu, Li Li, Yue-Zhou Li, Yi Ling, Zi-Wen Liu, Chen-Te Ma, René Meyer, Aavishkar A. Patel, Yu Tian, I0 = IL + IR, Xiao-Ning Wu, and Jiaju Zhang for helpful discussions and √ correspondence. We also thank the referee for helpful com- 1 2 Iγ =− d x gγ φγ (Rγ + 2)+2 dx hγ φγ,∂Kγ ments on the analytic continuation of the Renyi entropy. This 16πG ∂ N work is supported in part by the Natural Science Foundation of √ 2 2 2 2 China under Grants No. 11875053, No. 11847229 (Z.-Y.X.), − d x gγ (∂χγ ) + m ∂χγ ,γ= L, R, (B1) and No. 11675081 (L.Z.). Z.-Y.X. also acknowledges the support from China Postdoctoral Science Foundation under where the topology terms are omitted. As shown in the main the National Postdoctoral Program for Innovative Talents No. text, the dynamic of the system R in unbalanced H˜0 can be BX20180318. obtained by redefining the time of the system R in balanced H0. The boundary conditions are APPENDIX A: DEFINE WORK AND HEAT φ¯ 2 1 2 We will review the definitions of work and heat given in ds = dτ ,φ∂, = , ∂,L 2 L Ref. [14], where the two local systems are treated equally. One can divided the state of bisystems into a noncorrelated part and λ2 φ/λ¯ 2 2 ds = dτ ,φ∂, = . (B2) correlated part at time t, ∂,R 2 R /λ

ρ(t ) = ρL (t ) ⊗ ρR(t ) + χ(t ). (A1) Furthermore, at the low-energy limit, each Iγ with scalar The infinitesimal change in local energy can be divided into source Jγ,i is reduced to a Schwarzian action plus the gen- work and heat: erating functional dEγ (t ) = dWγ (t ) + dQγ (t ),γ= L, R. (A2) Ieff =−C dτ1{μL,τ1} − dτ1dτ2JL (τ1)JL (τ2 ) By introducing two auxiliary parameters αL and αR = 1 − αL, one can rearrange the total Hamiltonians as μ τ μ τ Δ L ( 1) L ( 2 ) = eff + eff + eff , × Htot HL (t ) HR (t ) HI (t ) (A3) [μ (τ ) − μ (τ )]2 L 1 L 2 eff = + ρ − α ρ ⊗ ρ . C Hγ (t ) Hγ Trγ¯ [ γ¯ (t )HI ] γ Tr[ L (t ) R(t )HI ] − dτ {μ ,τ } − dτ dτ J τ J τ λ 1 R 1 1 2 R( 1) R( 2 ) (A4) Δ μ (λτ )μ (λτ ) One can define the so-called binding energy according to the × R 1 R 2 . (B3) 2 correlation and the interaction [μR(λτ1) − μR(λτ2 )] = χ eff . UI Tr (t )HI (t ) (A5) Work and heat are defined as

dWL (t ) =−dWR(t ) =−αLTr[dρL (t ) ⊗ ρR(t )HI ] + α Tr[ρ (t ) ⊗ dρ (t )H ], (A6a) R L R I eff dQγ (t ) =−iTr χ(t ) Hγ (t ), HI dt, (A6b) which satisfy

dWL (t ) + dWR(t ) = 0, dQL (t ) + dQR(t ) + dUI (t ) = 0. (A7)

The two parameters αL,αR should be determined by the relation between the pseudotemperature and the equilibrium temperature in Ref. [14]. However, in the models of this paper, the term Tr[ρL (t ) ⊗ ρR(t )HI ] vanishes at the ﬁrst- and second-order perturbation. So, the effect of the two parameters FIG. 18. The time contour C in the complex plane z = and the work done on the local systems vanish. exp(i2πtC /β) for initial product state, where the complex time tC = τ + it is measured by evolution exp(−tC H ). The blue (orange) con- APPENDIX B: INTERACTION BETWEEN TWO tour indicates the evolution of system L (R). The black point indicates DISCONNECTED BLACK HOLES the twisted operator Xn. The dashed lines indicate the coupling between the two sides during [ti, t f ], where ti < 0 < t f . Thick interval + − We consider the unbalanced Hamiltonian H˜0 and prepare in the upper (lower) half plane indicates contour C (C ). We choose a product of thermal state ρ = τ (β) ⊗ τ (β/λ)inCFTL ⊗ λ = 2 here.

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When Jγ = 0, the on-shell solutions with different temperatures are KR(t, t ) =−Gβ (−iΔt )Gβ (−iΔtλ) − Gβ (iΔt )Gβ (iΔtλ). πτ πλτ (B6) μ = tan ,μ= tan . (B4) L β R β Our calculations in the main text already tell us that EL + ER 0 and EL − ER 0. The same things hap- μ → ν τ → Real time is obtained by Wick rotation i and it. pen if we alternatively√ consider interaction HI = gVLWR or So far, the LL and RR correlations here are the same as those HI = g(VLWR − WLVR )/ 2. Since VW=0, in both cases, 2 of the TFD state in the main text. we ﬁnd HI HI =g VLVLWRWR and only the diagrams of After turning on the interaction HI = gOLOR, we ﬁnd Figs. 3(a) and 3(b) are left. 1 that the O(g ) contribution vanishes because of OLOR= In the following, we will calculate the entropy changes by 0. At O(g2 ), since the four-point function is factorized using replica trick. Now the entropy change of system L may OLOROLOR=OLOLOROR, the energy changes are con- not be the same as the one of system R since the total system tributed by the diagrams given in Figs. 3(a) and 3(b), whose is not a pure state anymore. We will use the parametriza- ϕγ,a μγ, = ϕγ, ∼ ϕγ, + π kernels are tion a tan 2 and a a 2 . Similarly to the method in Sec. VI, Renyi entropy Sn,R of system R can be

KL (t, t ) =−Gβ (−iΔtλ)Gβ (−iΔt ) − Gβ (iΔtλ)Gβ (iΔt ), written as (B5) ⊗n ⊗n (1−n)Sn,R ⊗n e = Tr TC exp − dτ H˜I (τ ) (ρ ⊗ ρ) TC exp − dτ H˜I (τ ) (I ⊗ Xn ) (B7a) C− C+ ⊗n ⊗n = Tr (ρ ⊗ ρ) TC exp − dτ H˜I (τ + β) − dτ H˜I (τ ) (I ⊗ Xn ) (B7b) C− C+ ⊗n = Tr (ρ ⊗ ρ) TC exp − dτ1dτ2 σ (τ1,τ2 )VL,a(τ1)WR,a(τ2 ) (I ⊗ Xn ) , (B7c) C a where, at the last step, we consider HI = gVLWR and

σ (τ1,τ2 ) = dτδ(τ1 − (β + τ ))δ(τ2 − (β + λτ )) + dτδ(τ1 − τ )δ(τ2 − λτ ), (B8) C− C+ where contours C± are deﬁned in Eq. (6.5) and contour C goes from 0 to β containing C±, as shown in Fig. 18. The twisted bound- − − ary conditions are only applied on system R, leading to boundary conditions OL,a(β) = OL,a(0 ) and OR,a(β) = OR,a+1(0 )for any scalar operator O in the path integral of action In = In,0 + In. The undeformed replicated effective action In,0 can be written as

n−1 β n−1 β ϕL,a ϕR,a I , =−C dτ tan ,τ − C dτ tan ,τ (B9a) n 0 2 2 a=0 0 a=0 0 n−1 β nβ ϕ , f =−C dτ tan L a ,τ − C dτ tan R ,τ , (B9b) 2 2 a=0 0 0 where fR(τ + aβ) = ϕR,a(τ ). The saddle-point solution is φL,a,c = 2πτ/β, fR,c = 2πτ/nβ. The deformation on effective action up to O(g2 )is n−1 −In =−g dτ1dτ2 σ (τ1,τ2 )VL,a(τ1)WR,a(τ2 ) − IM,L,a − IM,R,a (B10a) C a=0 2 n−1 g ϕ ϕ = τ τ τ τ σ τ ,τ σ τ ,τ L,a τ ,τ R,a τ ,τ d 1d 2d 3d 4 ( 1 2 ) ( 3 4 )Gaa ( 1 3)Gaa ( 2 4 ) (B10b) 2 C a=0 2 n−1 g ϕ = τ τ τ τ σ τ ,τ σ τ ,τ L,a τ ,τ fR β + τ , β + τ d 1d 2d 3d 4 ( 1 2 ) ( 3 4 )Gaa ( 1 3)G (a 2 a 4 ) (B10c) 2 C a=0 ng2 ϕL fR = dτ1dτ2dτ3dτ4 σ (τ1,τ2 )σ (τ3,τ4 )G (τ1,τ3)G (τ2,τ4 ), (B10d) 2 C ϕ ϕ L,a R,a f where G , G , G R are deﬁned in Eqs. (6.21) and (6.22). I ,γ , is the action of the matter ﬁelds in the γ AdS space on the aa aa ϕ ϕ M a 2 L,a τ ,τ = δ L,a τ ,τ ath sheet. In Eq. (B10b), we have used Gab ( 1 2 ) abGaa ( 1 2 ), since the different sheets of system L are disconnected. In

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FIG. 19. Yn(t, t ) as a function of t − t and Sn as a function of t f − ti for the product state in JT gravity, where n = 1, 2, 3, 4; g = 1; β = 2π; Δ = 1/6; C = 105; = 0.2; and λ = 2. Solid (dashed) lines correspond to system L (R). The entropy changes for the product state in the two-site SYK model are the same.

ϕ , L a ϕL Eq. (B10d√ ), we have used the translational symmetry on τ and let Gaa (τ1,τ2 ) = G (τ1,τ2 ). For the interaction HI = g(VLWR − WLVR )/ 2, we have − g n 1 −In = √ dτ1dτ2 σ (τ1,τ2 )[VL,a(τ1)WR,a(τ2 ) − WL,a(τ1)VR,a(τ2 )] − IM,L,a − IM,R,a (B11a) C 2 a=0 2 n−1 g ϕ ϕ = τ τ τ τ σ τ ,τ σ τ ,τ L,a τ ,τ R,a τ ,τ , d 1d 2d 3d 4 ( 1 2 ) ( 3 4 )Gaa ( 1 3)Gaa ( 2 4 ) (B11b) 2 C a=0 and the ﬁnal result is the same. The calculation of Renyi entropy Sn,L of system L is paralleled to the above. ϕ Finally, the change in Renyi entropy at O(g2 ) is a double integral (6.23) as well. Since the G L and G fR in Eq. (B10d)are uncorrelated with each other, the integral kernel Yn,γ (t, t ) is factorized into n Y , (t, t ) = [G β ( + it )Gβ ( + iλt ) − G β (β − + it )Gβ ( − iλt )] + (t ↔−t ), (B12) n L n − 1 n n n Y , (t, t ) = [Gβ ( + it )G β ( + iλt ) − Gβ ( − it )G β (β − + iλt )] + (t ↔−t ),t = t − t , (B13) n R n − 1 n n where Gnβ is given in Eq. (6.18). Yn,γ (t, t ) has similar UV sensitivities to those of the kernel of the eternal black hole in Sec. VI. The formula above is analytical in n. So we can take the n → 1 limit and obtain the change in the von Neumann entropy, whose kernels are

Y1,L (t, t ) =−β[Gβ (−itλ)Gβ (−it ) + Gβ (itλ)Gβ (it )], (B14)

Y1,R(t, t ) =−β[Gβ (−it )Gβ (−itλ) + Gβ (it )Gβ (itλ)]. (B15)

The configurations of kernel Yn(t, t ) and Sn are shown in where q 4 and N is even. The effective action is [43] Fig. 19. By comparing Eq. (B5) with Eq. (B14) or comparing Fig. 4 with Fig. 19, we find that the energy-entropy inequali- I0 1 1 = ln det(∂τ − ) − dτ dτ ties (2.2) are saturated at O(g2 ) for the initial product state. N 2 2 1 2 2 J q × (τ1,τ2 )G(τ1,τ2 ) − G(τ1,τ2 ) , (C3) APPENDIX C: ENERGY AND ENTROPY q IN THE TWO-SITE SYK MODEL where G(τ ,τ ) = 1 T ψi(τ )ψi(τ ). At the limit 1. TFD state in the two-site SYK model 1 2 N i E 1 2 N βJ 1, the saddle-point solution is Gc(τ1,τ2 ) = β π τ −τ −2Δ We will investigate similar phenomena in the two-site b( J sin ( 1 2 ) ) where scaling dimension Δ = [ψi] = SYK model in parallel. The local Hamiltonian of the SYK π β 1/q and coefficient bqπ = ( 1 − Δ)tan(πΔ). The extremal q N 2 model is a random -body interaction between Majorana = α α fermions [42]: entropy is S0 0N where the constant 0 is a function of q [43]. The leading UV correction is reparametriza- τ ,τ = τ τ Δ τ , τ q/2 j j j tion G( 1 2 ) [ f ( 1) f ( 2 )] Gc( f ( 1) f ( 2 )) controlled H = i J ... ψ 1 ψ 2 ...ψ q , (C1) j1 j2 jq by the Schwarzian derivative < <...< 1 j1 j2 jq N q−1 2 − α 2 2 J (q 1)! N S J ... = (no sum), (C2) I =− dτ{ f ,τ}, (C4) j1 j2 jq qNq−1 eff J

043095-22 WORMHOLES AND THE THERMODYNAMIC ARROW OF TIME PHYSICAL REVIEW RESEARCH 2, 043095 (2020) √ (1−q)/2 T where J = qJ2 and the constant αS is determined by One can find H = H in this basis. The TFD state at t = 0is numerical calculation [43,44]. The SL(2, R) symmetry of the obtained from the Euclidean evolution of |I: + → afL b − = − / ground state is f + with ad bc 1. We can give 1 2 −β(HL +HR )/4 cfL d |β = Zβ e |I. (C9) a correspondence between the parameters in JT gravity and the SYK model by matching their effective actions and the By using Eq. (C8), the two-point function evaluated on the β|ψ j ψk |β= extremal entropy TFD state can be transformed into L (t1) R(t2 ) j k −1/2 −βH/2 iTr[yψ (−t1)yψ (t2 )], where y = Z e and Tr[O] ≡ 1 NαS αS α = Δ, = C, = . TrR[OR]. J α J π (C5) q 0 2 Similarly to what we did in the main text, we further consider the unbalanced Hamiltonian We consider two sites SYKL and SYKR which consist of {ψi } {ψi } two copies of N Majorana fermions L and R .Wefirst H˜0 = HL + H˜R, H˜R = λHR, (C10) consider the balanced Hamiltonian [35,45] on the two-site SYK model. The TFD state which evolves with − = + , = ⊗ , = ⊗ T , iH˜0t H0 HL HR HL H 1 HR 1 H (C6) H˜0 is |β˜ (t )=e |β. It is equivalent to redefining the time of SYK . where the transpose is taken on a basis {|}. Those coupling R Jj j ... j in HL and HR are the same. So the disorderings of 1 2 q 2. Anomalous heat flow SYKL and SYKR are correlated. Given such a basis {|},we construe a state To construe the anomalous heat flow in the two-site SYK model, we turn on the following interaction after t , | = | | , i I L R (C7) N j k HI = i g jkψ ψ , (C11) − | = L R which satisfies (HL HR ) I 0 automatically. We specify j,k=1 the basis by imposing [45] = , = −2 2 δ δ + δ δ . g jk gkj g jkglm N g ( jl km jm kl ) (C12) ψ j + iψ j |I = 0, j = 1, 2,...,N. (C8) L R At the large-N limit, the energy changes at t f are

t f t = − 3 2 −2 ψ˙ j ψk λ ,ψj ψk λ + ψ˙ j ψk λ ,ψk ψ j λ EL ( i) g N dt dt L (t ) R( t ) L (t ) R( t ) L (t ) R( t ) L (t ) R( t ) ti ti jk t f t 2 −2 j j k k k j k j = 2g N Im dt dt Tr[ψ (−t )ψ˙ (−t )yψ (λt )ψ (λt )y] + Tr[ψ (−t )ψ˙ (−t f )yψ (λt f )ψ (λt )y], (C13a) ti ti jk t f t = − 3 2 −2λ ψ j ψ˙ k λ ,ψj ψk λ + ψ j ψ˙ k λ ,ψk ψ j λ ER ( i) g N dt dt L (t ) R( t ) L (t ) R( t ) L (t ) R( t ) L (t ) R( t ) ti ti jk t f t = 2g2N−2λIm dt dt Tr[ψ j (−t )ψ j (−t )yψ˙ k (λt )ψk (λt )y] + Tr[ψk (−t )ψ j (−t )yψ˙ k (λt )ψ j (λt )y], (C13b) ti ti jk t f =−3 2 −2 ψ j ψk λ ,ψj ψk λ + ψ j ψk λ ,ψk ψ j λ EI i g N dt L (t f ) R( t f ) L (t ) R( t ) L (t f ) R( t f ) L (t ) R( t ) ti jk t f 2 −2 j j k k k j k j = 2g N Im dtTr[ψ (−t )ψ (−t f )yψ (λt f )ψ (λt )y] + Tr[ψ (−t )ψ (−t f )yψ (λt f )ψ (λt )y], (C13c) jk ti

j itHγ j −itHγ j itH j −itH 3 where ψγ (t ) = e ψγ e (γ = L, R), ψ (t ) = e ψ e . The above equations have the same form as Eq. (5.5). The disconnected four-point functions are determined by the conformal symmetry. The leading contributions of the connected four- point functions at early time are determined by the Schwarzian action (C4). So the energy changes at early time here are the same as those in JT gravity. The anomalous heat ﬂow in the SYK model is contributed by the second term in Eq. (C13). If we consider an initial product state in the two-site SYK model and turn on the same interaction (C11), similarly to what we did in Appendix B, only the ﬁrst term in Eq. (C13)isleft. To calculate the change in Renyi entropy Sn, we apply the replica trick on system R. We will alternately use imaginary time τH −τH ψ(τ ) = e ψe below. Similarly to the case in Eq. (6.6), for the interaction (C11 ), we have − ⊗ ⊗ (1 n)Sn = β| n T − τ ψ j τ ψk λτ ⊗ T − τ ψ j τ ψk λτ |β n e C exp i d g jk L,a( ) R,a( ) (I Xn ) C exp i d g jk L,a( ) R,a( ) C+ C− a, jk a, jk (C14a)

3Exchanging the two fermions in the second term will give a minus sign.

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β β = ρ⊗n T τ ψk λτ ψ j − − τ + τ ψ j − − τ ψk −β + λτ Tr Xn C exp d g jk a ( ) a d g jk a a ( ) (C14b) C− 2 C+ 2 a, jk β β = T − τ ψ j − τ ψk β + λτ − τ ψ j − τ ψk λτ C exp d g jk a a ( ) d g jk a a ( ) Xn (C14c) C− 2 C+ 2 a, jk ⊗ n = T − τ τ σ τ ,τ ψ j τ ψk τ , C exp d 1d 2 −( 1 2 )g jk a ( 1) a ( 2 ) Xn (C14d) , C a jk ⊗n where β β σ−(τ1,τ2 ) = dτδ τ1 − − τ δ(τ2 − (β + λτ )) − dτδ τ1 − − τ δ(τ2 − λτ ). (C15) C− 2 C+ 2

In the final expression, twist operator Xn is located at a time τ∗ which is infinitesimally less than 0. It imposes the relations ψ j τ − = ψ j τ + , ∀ = , ,..., − , ψ j = ψ j . a ( ∗ ) a+1( ∗ ) a 0 1 n 1 n 0 (C16) τ ,τ = We write down the replicated action, integrate out the disorder, and introduce the bilocal field Gab( 1 2 ) −1 T ψ j τ ψ j τ τ ,τ N j E a ( 1) b ( 2 ) and Legendre multiplier ab( 1 2 ), − ψ − , = Dψ j In[ ] = D D In[G ], Zn a e Gab abe (C17) j,a ab where ⎡ ⎤ n−1 1 / j − = τ⎣− ψ j τ ∂ ψ j τ + q 2 ψ j1 τ ψ j2 τ ...ψ q τ ⎦ In d a ( ) τ a ( ) i Jj1 j2... jq a ( ) a ( ) a ( ) C 2 a=0 j j1< j2<...< jq − τ τ σ τ ,τ ψ j τ ψk τ d 1d 2 −( 1 2 )g jk a ( 1) a ( 2 ) (C18a) C a, jk 2 1 J j j =− τψj τ ∂ ψ j τ + τ τ ψ j1 τ ψ j1 τ ψ j2 τ ψ j2 τ ...ψ q τ ψ q τ d a ( ) τ a ( ) − d 1d 2 a ( 1) b ( 2 ) a ( 1) b ( 2 ) a ( 1) b ( 2 ) 2 C 2qNq 1 C a, j ab, j1< j2<...< jq g2 + τ τ τ τ σ τ ,τ σ τ ,τ ψ j τ ψk τ ψ j τ ψk τ + ψ j τ ψk τ ψk τ ψ j τ d 1d 2d 3d 4 −( 1 2 ) −( 3 4 ) a ( 1) a ( 2 ) b ( 3) b ( 4) a ( 1) a ( 2 ) b ( 3) b ( 4) 2N2 C ab, jk (C18b) 2 N N J q = ln det(∂τ δab − ab) − dτ1dτ2 ab(τ1,τ2 )Gab(τ1,τ2 ) − Gab(τ1,τ2 ) 2 2 C q ab g2 + dτ1dτ2dτ3dτ4 σ−(τ1,τ2 )σ−(τ3,τ4 )[−Gab(τ1,τ3)Gab(τ2,τ4 ) + Gab(τ1,τ4 )Gab(τ2,τ3)] (C18c) 2 C ab

=−In,0 − In, (C18d) where In is a highly nonlocal deformation on the original bilocal action In,0. Relation (C16) leads to the twisted boundary conditions in the path integral, ψ j β = ψ j − ,ψj β =−ψ j − = , ,..., − , a ( ) a+1(0 ) n−1( ) 0 (0 )fora 0 1 n 2 (C19a) − − Gab(β,τ) = G(a+1)b(0 ,τ), G(n−1)b(β,τ) =−G0b(0 ,τ)fora = 0, 1,...,n − 2, ∀b, (C19b) − − Gab(τ,β) = Ga(b+1)(τ,0 ), Ga(n−1)(τ,β) = Ga0(τ,0 )forb = 0, 1,...,n − 2, ∀a. (C19c) Similar conditions arise in Refs. [45,55]. As we did in Eq. (6.21), we introduce a global fermion ψ j (τ ) and global bilocal ﬁelds G(τ1,τ2 ),(τ1,τ2 )by ψ j τ = ψ j β + τ , τ ,τ = β + τ , β + τ ,τ ,τ = β + τ , β + τ . a ( ) (a ) Gab( 1 2 ) G(a 1 b 2 ) ab( 1 2 ) (a 1 b 2 ) (C20) According to Eq. (C19), these global ﬁelds are continuous in τ ∈ (0, nβ) and antiperiodic, ψ j (nβ)=−ψ j (0−), G(nβ,τ)=−G(0−,τ), G(τ,nβ)=−G(τ,0− ),(nβ,τ)=−(0−,τ),(τ,nβ)=−(τ,0−). (C21)

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Then we have 2 N N J q −In,0 = ln det(∂τ − ) − dτ1dτ2 (τ1,τ2 )G(τ1,τ2 ) − G(τ1,τ2 ) , (C22a) 2 2 C q n g2 −In = dτ1dτ2dτ3dτ4 σ−(τ1,τ2 )σ−(τ3,τ4 ) 2 C ab

×[−G(aβ + τ1, bβ + τ3)G(aβ + τ2, bβ + τ4 ) + G(aβ + τ1, bβ + τ4)G(aβ + τ2, bβ + τ3)], (C22b)

where Cn is the unfolded time contour in Fig. 9 and the τ in will apply the four-point functions VVWW and VWVW C dτ goes from 0 to nβ. Equation (C22a) is just the action from JT gravity but take care of their signs when evaluating n of the SYK model at the inverse temperature nβ, with the Yn(t, t ). The parameters of the SYK model and JT gravity saddle-point solution and effective action in the same forms can be related by using relation (C5). The conﬁguration of when βJ 1. Considering the reparametrization modes in kernel Yn(t, t ) at early time is shown in Fig. 20. The kernel

Eq. (C22a), we will obtain the Schwarzian derivative as in Yn(t, t ) exponentially increase at early-time chaos region A the case of JT gravity in Eq. (6.13). If we integrate out (t ∼ t ), contributed by the first term in Eq. (C22b). The kernel the four times in Eq. (C22b), expand the product into eight Yn(t, t ) exponentially decrease at early-time chaos region B terms, and sort the time ordering of fermions, we will find (t ∼−t ), contributed by the second term in Eq. (C22b). Its that the deformation I in the SYK model is different from decrease agrees with the fact that the anomalous heat flow Eq. (6.20) in JT gravity, where some terms have inverse signs. in the SYK model is also contributed by the second term in Perturbation theory (6.16)atO(g2 ) is applicable at the large- Eq. (C13).

N limit. The kernel Yn(t, t ) consists of four-point functions For an initial product state of the two-site SYK model, we ψ jψ jψkψk ψ jψkψ jψk jk and jk , which are con- calculate the entropy changes in the way of Appendix B and trolled by effective action (C4) as well. So, at early time, we obtain − ⊗ (1 n)Sn,R = ρ ⊗ ρ nT − τ τ σ τ ,τ ψ j τ ψk τ ⊗ e Tr ( ) C exp i d 1d 2 ( 1 2 )g jk L,a( 1) R,a( 2 ) (I Xn ) (C23a) C a, jk = Dψ j − − , γ,a exp ( In,0 In ) (C23b) γ,a, j − − with boundary conditions ψL,a(β) =−ψL,a(0 )for(a = 0, 1,...,n − 1), ψR,a(β) = ψR,a+1(0 )for(a = 0, 1,...,n − 2), and − ψR,n−1(β) =−ψR,0(0 ). We can transform In,0 into the effective action (B9) and ﬁnd − =− τ τ σ τ ,τ ψ j τ ψk τ In i d 1d 2 ( 1 2 )g jk L,a( 1) R,a( 2 ) (C24) C a, jk g2 =− τ τ τ τ σ τ ,τ σ τ ,τ ψ j τ ψk τ ψ j τ ψk τ + ψ j τ ψk τ ψk τ ψ j τ d 1d 2d 3d 4 ( 1 2 ) ( 3 4 ) L,a( 1) R,a( 2 ) L,b( 3) R,b( 4) L,a( 1) R,a( 2 ) L,b( 3) R,b( 4) 2N2 C ab, jk (C25) g2 = dτ1dτ2dτ3dτ4 σ (τ1,τ2 )σ (τ3,τ4 )GL,aa(τ1,τ2 )GR,aa(τ2,τ4 ). (C26) 2 C a

The ﬁnal result is the same as deformation (B10)inJT of the SYK model is known as free fermions, we can calculate gravity. So the energy-entropy inequalities (2.2) are saturated the UV behaviors. Here, we will consider the large-q limit. as well. The Euclidean Green’s function with 1/q expansion is [43] πν β 1 2 cos ( ) G(τ ) = sgn(τ ) 1 + ln 2 +··· , 2 q πν β 1 − |τ| 3. Large-q limit and UV-sensitive region cos ( ) 2 β In the main part of this paper, we only use the correlators (C27) with nearly conformal symmetry, which is only valid in the where the logarithm is required to be of order 1 and ν(β)is IR. Due to the lack of the UV part of the boundary theory determined by which is dual to JT gravity in the bulk, we are unable to give a πν β detailed description of the change in energy and entropy in the βJ = ( ) . πν(β) (C28) UV-sensitive region. Thanks to the fact that the UV ﬁxed point cos 2

043095-25 ZHUO-YU XIAN AND LONG ZHAO PHYSICAL REVIEW RESEARCH 2, 043095 (2020)

inequalities (2.2) may not be saturated any more. We illustrate the result in Fig. 22.

APPENDIX D: CLASSICAL MODEL We will try to discuss the phenomenon analogous to anomalous heat flow in the classical model of [32]. Consider a local Hamiltonian 2 1 N N N H({x }, {p }) = p2 + x J x , (D1) i i 2 i i ai j j i a ij 2 = −1 where Jai j are random couplings satisfying Jai j N . Such model is nonintegrable and expected to exhibit classical , FIG. 20. Yn(t t ) for an initial TFD state in the two-site chaos. = ,β = π,Δ = / , = 5, = SYK model. Parameters are g 1 2 1 6 C 10 Consider two copies of phase space (xL, pL ) and (xR, pR ) 0.2,λ= 1, and n = 2, 3, 4 (from top to bottom near t = t ), which i i i i where i = 1, 2,...,N and their unbalanced Hamiltonian can be translated into the parameters of the SYK model q, J, N according to relation (C5). = + = L , L + λ R , R . H0 HL HR H xi pi H xi pi (D2) Analogous to the TFD state, we prepare an ensemble at t = 0 In (C27), the second term, as a correction, should be smaller each state of which satisfies than the first term. So we should keep the real time small L = R, L =− R, and focus on the UV-sensitive region. Then we can assume xi xi pi pi (D3) that four-point functions are factorized into the product of and distributes according to the Gibb distribution −β { },{ } two-point functions. We turn on the interaction (C11)at H( xi pi ) ρ({xi}, {pi}) ∼ e . Each state evolved by H0 time ti and calculate the changes in energy and entropy at L λ = R − , L λ =− R − will satisfy xi ( t ) xi ( t ) pi ( t ) pi ( t ). time t . f To trigger heat flow, at time ti, we turn on the interaction For an initial product state in the two-site SYK model, we = L R, use the method of Appendix B. The energy-entropy inequal- HI xi gijx j (D4) ities (2.2) are saturated. We illustrate the entropy change in ij Fig. 21. The kernels of both entropies and energies are positive where g is random coupling satisfying g =−g and within the period where approximation (C27)isvalid.When ij ij ji g g =N−2g2(δ δ − δ δ )fori = j, k = l. λ = 1, the increase in energy agrees with the fact that the ij kl ik jl il jk Similarly, by using the classical interaction picture, we product of thermal states is a passive state; the increase in can estimate the energy change at time t with second-order entropy agrees with the fact that interactions increase entan- f λ> perturbation, glement for an initial product state. When 1, as shown in Fig. 21, we find that the entropies obtained in the way of the t f t f = { , } = , , large-q limit increase as well. This agrees with the behaviors EI dt HI (t f ) HI (t ) dt KI (t f t ) t t of the Renyi entropy in the UV-sensitive region obtained in i i the way of UV regularization in Fig. 19. (D5a) For an initial TFD state in the two-site SYK model, we g2 K (t, t ) = xL (t )xR(λt ), xL (t )xR(λt ) use Eqs. (C13), (C22), and (C23) to calculate the changes in I N2 i j i j i= j energy and entropy. The energy changes are the same as the case of the product state because of the factorization of the − L R λ , L R λ xi (t )x j ( t ) x j (t )xi ( t ) (D5b) four-point functions, while the entropy changes are different = , + , , because of the initial entanglement. So the energy-entropy [KI (t t )]ijij [KI (t t )]ijji (D5c)

FIG. 21. Yn(t, t ) as a function of t − t and Sn as a function of t f − ti for an initial product state in the two-site SYK model at the large-q limit, where n = 1, 2, 3, 4; g = 1; β = 2π; J = 30; q = 24; and λ = 2. Solid (dashed) lines correspond to system L (R).

043095-26 WORMHOLES AND THE THERMODYNAMIC ARROW OF TIME PHYSICAL REVIEW RESEARCH 2, 043095 (2020)

FIG. 22. Yn(t, t ) as a function of (t, t )andSn as a function of t f for an initial TFD state in the two-site SYK model, where g = 1,β =

2π,J = 30, q = 24,λ= 2, ti =−3π,andn = 2, 3, 4 (from top to bottom near t = t ). whose form is the same as Eq. (5.5c). When t =−t and λ = For those VVWW terms which are factorized, we have 1, [KI (t, −t )]ijji can be written as an analytical expression of the summation, 2 n−1 π + 1/2 π + 1/2 g 2 2 2 (d iu) 2 (d iv) [KI (t, −t )]ijji =− {xi(t ) , x j (−t ) }P, (D6) csc csc 2N2 n n i= j d=1 πu πv π(u−v) which agrees with the classical limit (5.20) in JT gravity. = csch csch +n(coth πu−coth πv)csch , n n n (E4) APPENDIX E: VON NEUMANN ENTROPY FROM CONTOUR INTEGRATION | |, | | < 1 where Imu Imv 1. Due to the factor −n in Renyi en- − 1 According to Eq. (6.20), the kernel in Eq. (6.23)isa tropy (6.23), only the O(n 1) term in the series of Eq. (E4) summation on the two replicated indexes {a, b}. Due to the contributes to the von Neumann entropy. Those leading O(1) − terms in the series cancel each other out finally. translational invariance, only the difference b a is important, For those VWVW terms, we encounter the branch cut of the confluent hypergeometric function U (1, 1,ξ)onthe n−1 n−1 negative axis ξ<0, which is mapped to two branch cuts in , = b−a , = d , , Yn(t t ) yn (t t ) n yn (t t ) (E1) the z plane, as illustrated in Fig. 23. The contour C runs a,b=0 d=0 around these two branch cuts and the pole at the origin. We use U (1, 1,ξ + i0+) − U (1, 1,ξ + i0−) =−2iπeξ for ξ< where we have used the rotational symmetry of trace 0. However, due to the complicated dependence in Eq. (6.25), d , = d−n , . yn (t t ) yn (t t ) (E2) we have to calculate the integrals numerically and then take the n → 1 limit. Alternatively, it is more convenient to expand d , The time orderings of the terms in yn (t t ) may depend on those four-point functions about n = 1 first and then calculate the index d. We will split the summation (E1) into two parts the contour integrals. The leading O(1) terms of the series 0 , n−1 d , yn (t t ) and d=1 yn (t t ), since the time orderings of the have a period i. So the contour integrals around the branch terms in each part are found to be independent from d. Then cuts and the pole at the origin cancel. The O(n − 1) terms are we apply the factorized four-point function to VVWW and not periodic. So the contour integrals around the branch cuts the eikonal approximation to VWVW. Thesummationond can be transformed into the contour integral

n−1 d , =− −iz , yn (t t ) i dzyn (t t ) f (z) = C d 1 =− −iz , , i dzyn (t t ) f (z) (E3) C where f (z) = 1/(e2πz − 1), the contour C runs anticlock- wise around the poles z = i, 2i,...,(n − 1)i, and the contour C runs clockwise around the other poles and branch cuts in the stripe Imz ∈ (−1, n − 1]. Because of the periodic- −iz , = −i(z+in) , + ity yn (t t ) f (z) yn (t t ) f (z in), we are free to choose the location of the stripe as long as its width is ﬁxed to be in. One can check that these poles and branch cuts surrounded by C are all located in the narrower stripe Imz ∈ 1 − β + λ − β − 1 β + FIG. 23. A term V (( 2 iz) i t )W ( iz it )V ( 2 − , λ − −iz , ( 1 1). To have a relatively simple contour integral, we chose i t )W ( it ) nβ f (z)inyn (t t ) f (z)onthez plane, where Δ = 1/2 in the following analysis. Δ = 1/2,C = 1,β = 2π,λ = 1, t = π,t = π/2, and n = 4.

043095-27 ZHUO-YU XIAN AND LONG ZHAO PHYSICAL REVIEW RESEARCH 2, 043095 (2020) and the pole at the origin do not cancel. They contribute to the branch cuts in the stripe Imz ∈ (−1, 1). The treatment on the von Neumann entropy. We encounter U (2, 2,ξ) and its two contour integrals around the branch cuts is similar.

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