Perturbative calculations of entanglement entropy

Pouria Dadras∗ and Alexei Kitaev†

California Institute of Technology, Pasadena, CA 91125, U.S.A.

November 18, 2020

Abstract

This paper is an attempt to extend the recent understanding of the Page curve for evap-orating black holes to more general systems coupled to a heat bath. Although calculatingthe von Neumann entropy by the replica trick is usually a challenge, we have identified twosolvable cases. For the initial section of the Page curve, we sum up the perturbation series inthe system-bath coupling κ; the most interesting contribution is of order 2s, where s is thenumber of replicas. For the saturated regime, we consider the effect of an external impulseon the entropy at a later time and relate it to OTOCs. A significant simplification occurs inthe maximal chaos case such that the effect may be interpreted in terms of an intermediateobject, analogous to the branching surface of a replica wormhole.

Contents

1 Introduction 2

2 Early phase of entanglement growth 32.1 The model and general formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Short initial period vs. linear growth . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Perturbations to the saturated phase 83.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Thermodynamic response theory for the replicated system . . . . . . . . . . . . . 93.3 Branched two-point correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Branched correlator related to early-time OTOCs . . . . . . . . . . . . . . . . . . 123.5 Locking two operators in the same replica . . . . . . . . . . . . . . . . . . . . . . 14

4 Summary and discussion 15

5 Acknowledgments 16

∗pdadras@caltech.edu†kitaev@caltech.edu

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1 Introduction

While not an observable quantity, entropy is useful as an abstract measure of active degrees offreedom and correlations in the system. Quantum correlations can be elusive, particularly inblack holes, where the classical space-time picture is incomplete. There has been a long but ul-timately successful chase of correlations in the Hawking radiation. If a black hole forms from anobject in a pure quantum state and then evaporates, the resulting radiation must also be in apure state. Thus, it is strongly (albeit nonlocally) correlated. The general form of such correla-tions was predicted by Don Page [1], who considered a black hole as a generic quantum system.Still, it long remained unclear how such correlations could emerge in semiclassical gravity. Someimportant works that contributed to the solution include the Dray-’t Hooft mechanism of gravita-tional interaction between infalling matter and subsequent radiation [2, 3] and the calculation ofout-of-time-order correlators (OTOCs) in the black hole setting [4]. However, the OTOC physicsis relevant on short time scales and explains correlations that are present not in the radiationitself but relative to a purifying system [5] (that is, under the assumption that the black hole ispart of a thermofield double and that we have unrestricted access to the other part). The recentbreakthrough in understanding the correlations developing over the Page time [6, 7] required acareful formulation of the problem, which we will now summarize.

The problem at hand is a semiclassical one. We do not have a complete theory of quantumgravity, nor should it be required. When working at the semiclassical level, it is not possible toderive long-term evolution from the short-term one. Rather, one should look for a global solution,which may depend on the quantity of interest. We consider the entanglement entropy betweenthe black hole and the emitted radiation at a particular time t. So let ρ = ρ(t) be the black hole’sdensity matrix; we want to compute its von Neumann entropy, S(ρ) = −Tr(ρ ln ρ). The latter isexpressed as the s→ 1 limit of the s-Renyi entropy,

Ss(ρ) =1

1− sln Tr ρs. (1)

For integer s, the expression Tr ρs may be interpreted as the partition functions of s replicas ofthe system.

Now, it turns out that the transition from the early phase of the black hole evaporation(when the radiation is uncorrelated as the naive theory predicts) to the later phase (when theentanglement entropy equals the black hole’s coarse-grained entropy) is first order. The laterphase is described by a new type of space-time geometry, the replica wormhole [6, 7]. Althoughchoosing the correct solution of the two is a global problem, each of them can be examinedlocally. We will study some properties of both solutions for general many-body systems, wherethe geometric description is not applicable.

The von Neumann and Renyi entropies are nonlinear functions of the quantum state, whichis why they are not observables. However, the logarithmic nonlinearity is mild, such that in thethermodynamic limit, S(ρ) is determined by typical microstates that contribute to the mixed stateρ. In contrast, Renyi entropies are often dominated by a fraction of microstates of tiny overallweight. This distinction is also evident from the replica wormhole picture. The s-Renyi entropy isrelated to an s-fold cover of space-time, whose metric is different from the physical one. But whenwe analytically continue the solution in s and take s to 1, we get the standard metric with anadditional piece of data, the branching surface. Thus, the s→ 1 limit is essential for compatibility

2

with the usual (non-entropic) physics. Our main technical advance is how to take this limit insome specific cases.

2 Early phase of entanglement growth

We adopt a simpler variant of the evaporation problem, where instead of radiating energy, thesystem comes into contact with a heat bath at the same temperature. Turning the system-bathinteraction on represents a slight change in the Hamiltonian and results in a brief period of non-equilibrium dynamics. Then a steady state is achieved such that all simple correlation functionsare thermal. However, if the system’s initial state was pure (though mimicking the thermal state),its von Neumann entropy will grow at a constant rate. We focus on this regime as well as thevery beginning of quantum evolution. The entropy growth eventually saturates at the thermal(i.e. coarse-grained) entropy, but that is not captured by our method.

Our calculation is perturbative in the system-bath coupling strength κ. Note that the vonNeumann entropy has a logarithmic singularity at the unperturbed state, which is pure. This isreflected by the fact that in addition to terms of order κ2 (or any constant power of κ), terms oforder κ2s (where s is the number of replicas) play an important role.

2.1 The model and general formulas

Let us consider a quantum system (meant to represent a black hole) with some Hilbert space HBand Hamiltonian HB. For an exact analogy with the evaporation problem, we would have to picka pure state that looks like thermal to all simple measurements. Instead, we double the systemand postulate that its initial state is the thermofield double, |TFDB〉 ∈ H∗B ⊗HB. Only the rightpart is coupled to the heat bath, but we are interested in the von Neumann entropy of the doublesystem as its density matrix ρB∗B evolves in time. Likewise, the bath is also doubled, so that theinitial state of the world is

|Ψ0〉 = |TFDB〉 ⊗ |TFDb〉 ∈ H∗B ⊗HB ⊗Hb ⊗H∗b. (2)

A similar, but not identical,1 setting was used in[8, 9], where the s-Renyi entropy for integer s > 1was calculated.

The full Hamiltonian H = HB +Hb +HBb acts only on the two objects in the middle, i.e. onHB ⊗Hb. We assume that the interaction term has the form

HBb = κN∑j=1

OjBOjb (3)

with some bosonic operators OjB and Ojb. In the case of fermionic systems like the SYK model [10,

11, 12, 13], we should multiply the coupling parameter κ by i. For simplicity, we will do thecomputation for a bosonic system, but the final answer will equally be applicable to fermionicsystems.

1In Refs. [8, 9], the initial state is taken to be the thermofield double of two interacting subsystems rather thanthe product of two thermofield doubles.

3

Thus, the evolution of the world in the interaction picture takes the form

ρB∗Bbb∗(t) = U(t)|Ψ0〉〈Ψ0|U−1(t), U(t) = T(e−i

∫ t0 HBb(u)du

), (4)

where T stands for time ordering. We also assume that 〈OjB〉 = 〈Ojb〉 = 0.2 In the rest of the

section, we will compute the s-Renyi entropy of the system’s density matrix ρB∗B(t) after tracingout the bath. It is given by the perturbative expansion

ρB∗B(t) =∑n,m

(iκ)n(−iκ)m

n!m!

∫ t0

[T{OjnB (un) · · ·O

j1B (u1)}

∣∣TFDB〉〈TFDB∣∣ T̃{Oj′1B (u′1) · · ·Oj′mB (u′m)}×〈T̃{Oj

′1

b (u′1) · · ·O

j′mb (u

′m)}T{O

jnb (un) · · ·O

j1b (u1)}

〉]du du′,

(5)

where the expectation value 〈· · · 〉 is with respect to the bath’s thermal state and T̃ denotes reversetime ordering. (If u1 < · · · < un and u′1 < · · · < u′m, then the operators are already ordered.)There is also an implicit sum over repeated indices, with each index going from 1 to N . Note thatoperators with same indices have the same time argument.

Since the combinatorics might soon get complicated, let us introduce some simplifying graphicnotation:

|TFDB〉 = , 〈TFDB| = . (6)

Then each term in the expansion (5) will look like this (where T, T̃, and the indices are omitted):

O(u2)O(u1)∣∣TFDB〉 〈O(u′1)O(u2)O(u1)〉b〈TFDB∣∣O(u′1) = for u1 < u2. (7)

The diagram element in the middle is the Keldysh contour for the heat bath. It consists ofa circle at the bottom representing imaginary-time evolution and a stem corresponding to thereal-time evolution; the time goes up. For integer s, Tr (ρB∗B(t))

s can be represented by gluings such diagrams (describing different replicas of the density matrix) in the cyclic order — seefigure 1, where the replicas are depicted with different colors. The expectation values should beindependently computed for each closed contour, whether it corresponds to the system or thebath.

Let us further assume that the system-bath coupling is sufficiently weak. Then the “radiationquanta” emanating from the system are sparse, which means that dominant diagrams have atmost two operators with close times per contour. Therefore, the calculation can be done usingWick contraction. Of course, if the fields OjB, O

jb are Gaussian, then no sparseness condition is

necessary.An example of a (subleading) Wick pairing contributing to Tr (ρB∗B(t))

s is as follows:

〈O(v′1)O(v

′2)〉

b

〈O(v′1)O(v

′2)O(u2)O(u1)

〉B

〈O(u2)O(u1)

〉b

for u1 < u2, v′1 < v

′2. (8)

2In case that 〈Oj〉 6= 0, one can work with Oj − 〈Oj〉.

4

(a) (b)

Figure 1: (a) A diagram for Tr (ρB∗B(t))s, where ρB∗B(t) is given by (5) and s = 4. The thick

solid lines make up 2s Keldysh contours for the system and the bath in alternating order. Eachhalf of a contour represents a thermofield-double. Lorentzian time is directed toward the centerof the diagram. Different replicas are depicted in different colors. Wick contractions of fields aredepicted by thin solid lines, while the dotted lines represent system-bath coupling. The leadingcontributions are due to purple loops; the shorter and the longer loops correspond to equations(12) and (13), respectively. (b) The simplified diagram obtained by omitting the bath’s replicas.

It corresponds to the black loop at the bottom of figure 1 (a). In general, a Wick contractiondiagram is a disjoint union of loops that consist of alternating solid and dotted lines. Solidlines represent contractions of fields on the same contour, whereas dotted lines correspond tointeraction terms such as OjB(u)O

jb(u). Each contraction comes with a Kronecker delta identifying

the indices of the contracted fields. The result is nonzero if all the indices on each loop are thesame. Therefore, each loop with d solid line segments evaluates to Nκd multiplied by the productof two-point functions.

The diagrams for ln(Tr (ρB∗B(t))

s) are connected, i.e. contain a single loop. The factor of Nthat appears here is usual for extensive thermodynamic quantities; thus, the intensive parameteris κd. If κ is small, we should only keep loops with d = 2 contractions (in one replica) and d = 2scontractions (traversing all the replicas). Both types of loops are shown in purple in figure 1.

There are three types of two-point functions for the system,〈T{OjB(u)O

lB(v)}

〉= iGTB(u− v)δjl,

〈T̃{OjB(u)O

lB(v)}

〉= iGT̃B(u− v)δjl, (9)〈

Oj′

B (u′)OjB(u)

〉= iGB(u

′ − u)δj′j, (10)

and similarly for the bath. The expressions for closed paths will be simplified if we think of atwo-point function as the matrix element of a bilocal operator Ĝ,

〈u′|Ĝ|u〉 = ft(u′)G(u′ − u)ft(u), ft(u) ≡ θ(u)− θ(u− t), (11)

where ft(u) is a time window function that vanishes outside the interval (0, t). This way, we canextend the time domain to (−∞,∞) and avoid putting limits on the integrals. We also define thetranspose of the operator Ĝ, denoted by Ĝᵀ, with the matrix element 〈t1|Ĝᵀ|t2〉 = G(t2− t1). Note

5

that ĜT and ĜT̃ for bosons are symmetric, i.e. equal to their transpose. To illustrate this notation,

the expression (8) involves a loop with four Wick contractions, v′1b←− v′2

B←− u2b←− u1

B←− v′1; hence,the result takes the form Tr

(ĜT̃ b ◦ ĜB ◦ ĜT b ◦ Ĝ

ᵀB

).

As already mentioned, there are two types of loops that contribute to ln(Tr (ρB∗B(t))

s) toleading order. The loops of length d = 2 can themselves be of two forms, one of which appears infigure 1. They give the following contributions:

P2,T(t) = Nκ2 Tr

(ĜTB ◦ Ĝ

ᵀTb

), P2,T̃(t) = Nκ

2 Tr(ĜT̃B ◦ Ĝ

ᵀ

T̃ b

). (12)

Here we have used the fact that (±i)2 from (5) cancels i2 from (9). (The extra factors present inthe fermionic case also cancel each other.) There are also loops of length d = 2s, which we say tohave winding number 1 because they traverse all replicas. The expression for such a loop takesthe following form:

P2s(t) = Nκ2s Tr

(−ĜB ◦ Ĝ

ᵀb

)s. (13)

Since in the end we are interested in the limit s→ 1, the quantities (13) and (12) are of the sameorder in the coupling.

As an exercise, let us sum up the leading diagrams in Tr (ρB∗B(t))s — we should get the

exponential of a sum of single loops with certain coefficients. It is sufficient to only keep track ofthe system’s replica, leaving the bath implicit as in figure 1 (b). Let m1, . . . ,ms and n1, . . . , ns bethe numbers of fields in the time ordered and anti-time ordered branches of the Keldysh contours.The number of ways to break them into k loops of length 2s with winding number 1 and someloops of length 2 (with winding number 0) is given by

(k!)2s−1s∏r=1

(mrk

)(nrk

)(mr − k)!(mr−k

2

)! 2

mr−k2

(nr − k)!(nr−k

2

)! 2

nr−k2

(14)

Defining mr − k = 2pr and nr − k = 2qr, after manipulation we will get

Tr (ρB∗B(t))s =

∞∑k=0

∑p1,...,ps

∑q1,...,qs

1

k!

s∏r=1

1

(pr)!(qr)! 2pr2qr

(P2s(t)

)k (P2,T(t)

)pr(P2,T̃(t)

)qr= exp

(1

2

(P2,T(t) + P2,T̃(t)

)+ P2s(t)

) (15)Using the fact that Tr

(ĜTB ◦ ĜᵀT b

)+ Tr

(ĜT̃B ◦ Ĝ

ᵀ

T̃b

)= 2 Tr

(ĜB ◦ Ĝᵀb

), the final answer is as

follows:

ln(

Tr (ρB∗B(t))s)

= N

(sκ2 Tr

(ĜB ◦ Ĝ

ᵀb

)+ κ2s Tr

((−ĜB ◦ Ĝ

ᵀb

)s)). (16)

While the above equation was derived for bosonic systems, it is the same for fermionic systemslike the SYK model.

2.2 Short initial period vs. linear growth

Equation (16) allows one to compute the von Neumann entropy S(ρB∗B(t)) for t < tPage. Althoughthe exact answer is model-dependent, there are two universal regimes: very early times, just afterthe system-bath coupling is turned on, and intermediate times, when the entropy grows linearly.

6

Very early times: Let t� tUV such that the effect of the Hamiltonians HB and Hb is negligible.For example, tUV = J

−1 for the SYK model. More exactly, we assume that the Green functionsGB(t) and Gb(t) may be approximated by some constants. Then the expression (16) and the vonNeumann entropy take this form:

ln(

Tr (ρB∗B(t))s)≈ N

(−s cκ2t2+

(cκ2t2

)s), (17)

S(ρB∗B(t)

)≈ cκ2t2

(− ln(cκ2t2) + 1

), where c = −GB(0)Gb(0). (18)

Intermediate times: For systems with continuous excitation spectrum, connected correlatorsdecay in time. Exponential decay is typical; for example, in a conformal system at finite temper-ature, the correlator of fields with scaling dimension ∆ decays as exp

(−2π∆

βt)

if t � β∆

. Let us

assume that both GB(t) and Gb(t) decay exponentially at t� t∗.If t� t∗, then Tr

(ĜB ◦ Ĝᵀb

)can be approximated as follows. This expression is as integral over

u, u′ ∈ (0, t), but the integrand is negligible unless |u′ − u| . t∗. Therefore, we may remove thelimits on u′, and then use the Fourier transform:

Tr(ĜB ◦ Ĝ

ᵀb

)≈∫ t

0

du

∫ ∞−∞

du′GB(u′, u)Gb(u

′, u) = t

∫G̃B(ω)G̃b(−ω)

dω

2π, (19)

where G̃(ω) =∫∞−∞G(t)e

iωt dt. The second term in (16) can also be approximated in such a way.Thus,

ln(

Tr(ρ(t)B∗B)s)≈ NA(s) t, (20)

where

A(s) = sκ2∫G̃B(ω)G̃b(−ω)

dω

2π+ κ2s

∫ (−G̃B(ω)G̃b(−ω)

)s dω2π. (21)

After analytically continuing to s = 1, the von Neumann entropy will be given by

S(ρB∗B(t)

)≈ −NA′(1) t,

A′(1) = κ2∫G̃B(ω)G̃b(−ω)

(− ln

(−κ2G̃B(ω)G̃b(−ω)

)+ 1

)dω

2π.

(22)

(23)

The integrals in A(s) and A′(1) converge because a possible peak at ω = 0 is broadened tohave a width ωmin = t

−1∗ . There is also a natural UV cutoff at ωmax = t

−1UV. An interesting case

is where both the system and the bath are conformal at ω � ωmax. Let us first assume that thetemperature is zero; then G̃B(ω) ∝ ω2∆B−1 and G̃b(ω) ∝ ω2∆b−1 for ω > 0, but both G̃B and G̃bvanish at ω < 0. (Recall that these are Wightman functions.) Hence, G̃B(ω)G̃b(−ω) is zero forall ω 6= 0. At finite temperature, the integral in (23) is dominated by the region ω ∼ β−1, whereiG̃B(ω) ∼ tUV(tUV/β)2∆B−1 and iG̃b(ω) ∼ tUV(tUV/β)2∆b−1. It follows that

dS(ρB∗B(t))

N dt= −A′(1) ∼ −x lnx

β, where x = (βκ)2

(β

tUV

)−2(∆B+∆b). (24)

A good example is two SYK models at large βJ . (A bath with ∆b =12

can also be realized bya critical Majorana chain.) The Renyi entropies in this case were studied in [9] using the effective

7

Figure 2: Qualitative plot of the system’s coarse-grained entropy (dashed line) and the entangle-ment entropy (solid line) in the presence of an instantaneous perturbation.

action method, which is generally more powerful than perturbation theory. However, the analyticcontinuation to s = 1 was not obtained. The computed growth rate of the s-Renyi entropy forinteger s > 1 is consistent with our estimate,

dSs(ρB∗B(t))

N dt=A(s)

1− s∼ βκ2(βJ)−2(∆B+∆b). (25)

3 Perturbations to the saturated phase

We now consider the system at later times, such that its von Neumann entropy has reachedthe coarse-grained (thermodynamic) entropy. The entanglement entropy in this phase showsinteresting behavior under perturbations. For example, a short impulse increasing the system’senergy (similar to throwing a rock into a black hole) will cause a resurgence of entropy growth.Indeed, such an action can be described by some unitary operator V . It increases the coarse-grained entropy and effective temperature, though the true microscopic entropy does not change.Letting the system interact with the bath, we should see a behavior similar to the cusp in thePage curve. Specifically, we expect the von Neumann entropy to grow until it becomes equalto the coarse-grained entropy. Since the growth rate is constant while the perturbation can bearbitrarily weak, the resurgence can be short — just slightly longer than the scrambling time. Itwill be followed by a thermal equilibration period, when both the coarse-grained and microscopicentropies decrease, see figure 2. Thus, the Page curve cusp is accessible in this setting. However,to actually produce a cusp, the perturbation should be sufficiently strong, likely beyond the Taylorexpansion. We will study a simpler problem, calculating the effect in the lowest order.

While our formal goal is to compute the von Neumann entropy in a rather general setting, thekey result pertains to systems that saturate the chaos bound [14]. The expression obtained in thiscase admits a holographic interpretation, which will be discussed section 4.

3.1 Statement of the problem

For the study of the saturated phase, it is sufficient to consider one copy of the system and thebath rather than the thermofield double. The bath can be integrated out, giving rise to theinteraction function σ(τ1, τ2) = κ

2Gb(τ1, τ2) on the Keldysh contour. Let us simplify the modela bit and replace the interaction σ, which is constantly on, with a superoperator R acting at aspecific time. (The results we will obtain in this setting can be easily generalized to the original

8

model.) So, the exact problem involves a quantum system at thermal equilibrium subjected to asequence of two instantaneous perturbations, V and R.

Let V = e−ixX , where X is a Hermitian operator and x is a small parameter. We consider theaction of V on the thermal state ρ0 and expand the resulting density matrix ρ1(x) to the secondorder in x:

ρ1(x) = V ρ0V† ≈ ρ0 − ix[X, ρ0] + x2

(Xρ0X −

1

2

(X2ρ0 + ρ0X

2))

. (26)

In fact, a nontrivial effect will be seen in the second order, and only when combined with asubsequent interaction with the environment. The latter is described by a physically realizable(i.e. completely positive, trace-preserving) superoperator R. Suppose that R is close to the identitysuch that it can be expanded to the first order in some parameter �:

R ≈ 1 · 1 + �L, L = −i(C · 1− 1 · C) +∑j

(Aj · A†j −

1

2

(A†jAj · 1 + 1 · AjA

†j

)), (27)

where A ·B stands for the superoperator that takes ρ to AρB. The first term in L (which involvesa Hermitian operator C and act as ρ 7→ −i[C, ρ]) may be neglected because it represents aninfinitesimal unitary transformation, and thus, does not change the entropy. The sum over j(known as Lindbladian) corresponds to tracing out the environment. As will be justified later, wemay replace L with

∑j Aj · A

†j so that the final density matrix becomes

ρ(x, �) = R(ρ1(x)) = (1− ixX)ρ0(1 + ixX) + �∑j

Aj(1− ixX)ρ0(1 + ixX)A†j

+ unimportant terms.

(28)

Our goal is to compute ∂2

∂x2∂∂�S(ρ(x, �)), where S(ρ) = −Tr(ρ ln ρ).

We assume that V acts at time 0, whereas R acts at a later time t. Thus, Aj is understoodas Aj(t) = e

iH0tAj(0)e−iH0t, where Aj(0) is some simple (e.g. one- or two-body) operator. The

calculation will be done by the replica method for a general large N system in the early timeregime, i.e. before the scrambling time. However t is taken to be sufficiently large such thatOTOCs are parametrically greater than correlators with non-alternating times. Note that ρ(x, �)involves only non-alternating operators such as AjXρ0XA

†j. However, OTOCs appear due to the

use of replicas. The “unimportant terms” in (28) are exactly those that do not generate anyOTOCs.

In the next section, we study partial derivatives of S(ρ), assuming that ρ depends on param-eters in some particular way. This setting does not directly include the function ρ(x, �) given byequation (28). To cover this case, we will use a trick called “locking two operators in the samereplica”, see section 3.5.

3.2 Thermodynamic response theory for the replicated system

Let us recall the standard definition of connected correlators. We begin with the partition functionZ = TrW , where W is the imaginary-time evolution operator:

W = T exp

(−∫ β

0

H(τ) dτ

). (29)

9

Without perturbation, we have H(τ) = H0. The insertion of operators X1, . . . , Xn at timesτ1, . . . , τn is described by perturbing the Hamiltonian:

H(τ) = H0 −n∑j=1

xjδ(τ − τj)Xj, β ≥ τn ≥ · · · ≥ τ1 ≥ 0, (30)

where xj are infinitesimal numbers. We generally assume that the operators Xj are bosonic. (Ifany of them is fermionic, the corresponding variable xj should be anti-commuting.) Thus,

W (β, xn, . . . x1) = e−(β−τn)H0(1 + xnXn)e

−(τn−τn−1)H0 · · · (1 + x1X1)e−τ1H0 (31)

and Z(β, xn, . . . x1) = TrW (β, xn, . . . x1). The full correlator is simply

〈Xn(τn) · · ·X1(τ1)〉 = Z−1∂nZ

∂x1 · · · ∂xn

∣∣∣∣x1=···=xn=0

. (32)

The corresponding connected correlator is defined as follows:3

〈Xn(τn), . . . , X1(τ1)〉 =∂n lnZ

∂x1 · · · ∂xn

∣∣∣∣x1=···=xn=0

. (33)

For example, 〈X, Y 〉 = 〈XY 〉 − 〈X〉〈Y 〉 and

〈X, Y, Z〉 = 〈XY Z〉 − 〈XY 〉〈Z〉 − 〈XZ〉〈Y 〉 − 〈Y Z〉〈X〉+ 2〈X〉〈Y 〉〈Z〉. (34)

Now, let us introduce s replicas of the system, such that the partition function becomes

Z(s, β, xn, . . . , x1) = Tr(W (β, xn, . . . , x1)

)s. (35)

We may think of the parameter s as being associated with a branching operator B, which commuteswith everything. It is not defined by itself but only through its connected correlators:

〈B, Xn(τn), · · · , X1(τ1)〉 =∂n

∂x1 · · · ∂xn

∣∣∣∣x1=···=xn=0

∂ lnZ

∂s

∣∣∣∣s=1

. (36)

The branched correlator (36) is related to the entropy S = S(ρ) of the density matrix ρ = Z−1Wat s = 1 because (

∂s(lnZ))∣∣s=1

= lnZ − S. (37)Thus, the entropy derivative with respect to x1, . . . , xn is given by −〈B, Xn(τn), · · · , X1(τ1)〉 +〈Xn(τn), · · · , X1(τ1)〉. It is usually the easiest to compute the derivative of the relative entropy,S(ρ||ρ0) = Tr(ρ(ln ρ− ln ρ0)):

∂nS(ρ||ρ0)∂x1 · · · ∂xn

∣∣∣∣x1=···=xn=0

=〈(B+βH0), Xn(τn), · · · , X1(τ1)

〉−〈Xn(τn), · · · , X1(τ1)

〉=

∂n

∂x1 · · · ∂xn

∣∣∣∣x1=···=xn=0

(∂s(s−1 ln Tr (W (β/s, . . .))s

))∣∣∣s=1

.

(38)

3We use commas instead of double brackets because the usual notation (without commas) has some ambiguity.

10

For integer s, the expression Tr (W (β/s, . . .))s may be interpreted in terms of gluing s intervals oflength β/s to make a circle of length β. The operators Xn(τn), . . . , X1(τ1) are distributed alongthat circle. Thus, the number in question is, essentially, a correlation function at the given β.

An important caveat is that there is no natural definition of the full correlator 〈BA〉 as afunction of A such that one could compute 〈BY Z〉 by substituting Y Z for A. If such a function(with the usual relation to the connected correlator) existed, we would have this corollary ofequation (34): 〈B, Y, Z〉 = 〈B, Y Z〉 − 〈B, Y 〉〈Z〉 − 〈Y 〉〈B, Z〉. But this last identity is falsebecause in the expression for 〈B, Y, Z〉, the operators Y and Z can occur in different replicas, butin 〈B, Y Z〉, they cannot.

3.3 Branched two-point correlator

Suppose the ordinary correlation function 〈Y (τ), X(0)〉 is known on the imaginary axis, τ = it,and let us use its Fourier transform in t. In these terms,

〈Y (τ), X(0)〉 =∫ ∞−∞

FY,X(ω)e−ωτ dω

2π. (39)

The corresponding branched correlator is expected to have a similar form,〈B+βH0, Y (τ), X(0)

〉−〈Y (τ), X(0)

〉=

∫ ∞−∞

hY,X(ω)e−ωτ dω

2π. (40)

The goal of this section is to find the function hY,X .Let us consider the Fourier modes of the operators Y and X, for example, Yω =

∫Y (it)eiωt dt.

Their connected correlator is

〈Yω, Xω′〉 = FY,X(ω) · 2πδ(ω + ω′), (41)

and we also have

Y (τ) =

∫Yωe

−ωτ︸ ︷︷ ︸Yω(τ)

dω

2π, X(0) =

∫Xω︸︷︷︸Xω(0)

dω

2π. (42)

We now calculate the branched correlator of Yω and Xω′ , equal to hY,X(ω) · 2πδ(ω + ω′). Whenthe number of replicas s is a positive integer, each of the operators in question can be inserted inany replica, so the calculation involves a double sum. Since each replica’s length is β/s, puttingYω in the k-th replica is described by Yω(kβ/s) = Yωe

−kβω/s. With this in mind, we get:〈B+βH0, Yω, Xω′

〉−〈Yω, Xω′

〉(43)

=

(∂s

(s−1

s−1∑k=0

s−1∑l=0

T

〈Yω

(kβ

s

), X−ω′

(lβ

s

)〉))∣∣∣∣∣s=1

(44)

=

(∂s

s−1∑k=0

〈Yω

(kβ

s

), X−ω′(0)

〉)∣∣∣∣∣s=1

(45)

=〈Yω, Xω′

〉(∂s

s−1∑k=0

e−kβω/s︸ ︷︷ ︸1−u

1−u1/sfor u=e−βω

)∣∣∣∣s=1

= 2πδ(ω + ω′) · FY,X(ω)βω

eβω − 1. (46)

11

Figure 3: Graphic representation of a single term in (50). In this example, s = 4 (with the replicaslabeled 0, 1, 2, 3), k = 1, j = 0, l = 2, and t2 = t3 = 0.

Thus,

hY,X(ω) = FY,X(ω)βω

eβω − 1. (47)

3.4 Branched correlator related to early-time OTOCs

Let us recall the original problem of computing S(ρ(x, �)) with ρ(x, �) given by equation (28).In this section, we calculate an analogous branched correlator 〈B, Aj(β + it), X(β), X(0), A†k(it)〉and, more generally,〈

B, X4(β + it4), X3(β + it3), X2(it2), X1(it1)〉

for t1, t4 ≈ t, t2, t3 ≈ 0. (48)

One can eliminate β from the time arguments by cyclically permuting X4, . . . , X1. As alreadymentioned, the replica calculation involves OTOCs, which are dominant for sufficiently large t.Neglecting all terms with non-alternating times, we get:

B :=〈B, X4(β + it4), X3(β + it3), X2(it2), X1(it1)

〉=〈B, X2(it2), X1(it1), X4(it4), X3(it3)

〉≈(∂s(B+(s) + B−(s)

))∣∣∣s=1

,(49)

where

B+(s) =s−1∑k=0

k∑j=0

s−1∑l=k+1

〈X1

(it1 +

lβ

s

), X2

(it2 +

kβ

s

), X4

(it4 +

jβ

s

), X3(it3)

〉, (50)

B−(s) =s−1∑k=0

k∑j=0

s−1∑l=k+1

〈X2(it2), X1

(it1 −

jβ

s

), X3

(it3 −

kβ

s

), X4

(it4 −

lβ

s

)〉. (51)

(Equation (50) is illustrated by figure 3.) In order to make further progress, we will use thesingle-mode ansatz for early-time OTOCs [13],〈

X1(τ1), X2(τ2), X4(τ4), X3(τ3)〉≈ −C−1e−iκ(τ1+τ4−τ2−τ3−β/2)/2 ΥRX1,X4(τ1 − τ4)Υ

AX2,X3

(τ2 − τ3),(52)

combined with the Fourier representation

ΥRX1,X4(τ) =

∫Υ̃RX1,X4(ω)e

−ωτ dω

2π, ΥAX2,X3(τ) =

∫Υ̃AX2,X3(ω)e

−ωτ dω

2π. (53)

12

The result has this general form:

B = −C−1eκ(t1+t4−t2−t3)/2∫B̃(ω14, ω23) e−iω14(t1−t4)e−iω23(t2−t3)

dω232π

dω142π

,

B̃(ω14, ω23) = Υ̃RX1,X4(ω14)Υ̃AX2,X3

(ω23) f(ω14, ω23),

(54)

(55)

so the task is to compute f(ω14, ω23).First, we find the similar function f+(s;ω14, ω23) related to B+(s). Let

u = e−βω23 , v = e−βω14 , w = e−iβκ/2. (56)

Then

f+(s;ω14, ω23) =s−1∑k=0

k∑j=0

s−1∑l=k+1

uk/sv(l−j)/sw(j+l−k)/s−1/2 (57)

=w−1/2

(1− (w/v)1/s)(1− (wv)1/s)

((vw)1/s

1− u/v1− (u/v)1/s

− vw 1− u/w1− (u/w)1/s

− w2/s 1− uw1− (uw)1/s

+ v1−1/sw1+1/s1− u/v

1− (u/v)1/s

),

(58)

and hence,(∂sf+(s;ω14, ω23)

)∣∣s=1

= − 1(1− uw)(1− uw−1)

(w−1/2

1− v−1u−1+

w1/2

v−1 − u−1

)lnu

+1

(1− v−1w)(1− v−1w−1)

(w−1/2

1− uv+

w1/2

u− v

)ln v

+w−1/2(1 + uv−1)− w1/2(u+ v−1)

(1− uw)(1− uw−1)(1− v−1w)(1− v−1w−1)lnw.

(59)

The function f− is obtained from f+ by replacing w with w−1. Adding both terms together, we

get:

f(ω14, ω23) =(w−1/2 + w1/2)(u−1 − 1)(1 + v−1)

(1− uw)(1− uw−1)(1− u−1v−1)(v−1 − u−1)lnu

+(w−1/2 + w1/2)(1 + u)(1− v)

(1− v−1w)(1− v−1w−1)(1− uv)(u− v)ln v

+(w−1/2 − w1/2)(1 + u)(1 + v−1)

(1− uw)(1− uw−1)(1− v−1w)(1− v−1w−1)lnw,

(60)

where u = e−βω23 , v = e−βω14 , and w = e−iβκ/2.A great simplification occurs in the maximal chaos case:

f(ω14, ω23) =2π

(1 + e−βω23)(1 + eβω14)if κ =

2π

β. (61)

Importantly, the function f(ω14, ω23) splits into two factors. They may be interpreted in termsof interaction of the fluctuating horizon (which corresponds to B) with incoming and outgoingradiation, see section 4.

13

3.5 Locking two operators in the same replica

We now adapt the obtained result to express the entropy of the density matrix ρ(x, �). The latteris a normalized version of the operator

W (β, x, �) = (1− ixX)e−βH0(1 + ixX) + �∑j

Aj(t) (1− ixX)e−βH0(1 + ixX)A†j(t). (62)

Note that we have made the time explicit and will follow the convention that Aj = Aj(0). Toproceed, we replace the set of operators Aj with a single operator Y . This is achieved by extendingthe physical system with an auxiliary one, comprising a ground state |0〉 with zero energy and aset of excited stated state |j〉 with energy Ω. We denote the Hamiltonian of the extended systemby H(Ω) and set

Y =∑j

Aj ⊗ |0〉〈j|, Y † =∑j

A†j ⊗ |j〉〈0|. (63)

Although the transformation just described alters the operator W (β, x, �) in a nontrivial way, wewill find an agreement in the Ω → ∞ limit. For the time being, let us construct some operatorsacting on the extended system that correspond to the two terms in (62) as closely as possible:

(1− ixX)e−βH(Ω)(1 + ixX) = (1− ixX)e−βH0(1 + ixX)⊗

(|0〉〈0|+ e−βΩ

∑j

|j〉〈j|

), (64)

eβΩ Y (t) (1− ixX)e−βH(Ω)(1 + ixX)Y †(t)

=

(∑j

Aj(t) (1− ixX)e−βH0(1 + ixX)A†j(t)

)⊗ |0〉〈0|.

(65)

Now let

W (Ω, β, x, y) = (1 + eβΩ/2yY (t))(1− ixX)e−βH(Ω)(1 + ixX)(1 + eβΩ/2yY †(t)). (66)

Thenlim

Ω→∞W (Ω, β, x, y) = W (β, x, y2)⊗ |0〉〈0|, (67)

and hence,lim

Ω→∞Tr(W (Ω, β, x, y)

)s= Tr

(W (β, x, y2)

)s(68)

for any s. The last equation can be interpreted as the operators Y (t) and Y †(t) in the expansionof TrW s being locked in the same replica. We now take the s derivative of both sides at s = 1and consider the x2y2 term in the Taylor expansion. Thus, equation (68) becomes

limΩ→∞

(eβΩ(−〈B, Y (β + it), X(β), X(0), Y †(it)

〉Ω

+〈Y (β + it), X(β), X(0), Y †(it)

〉Ω

))=

1

2

(∂2

∂x2∂

∂�S(ρ(x, �))

)∣∣∣∣x=�=0

.

(69)

14

Finally, let us use the results of the previous section. Part of the correspondence is obvious:X1 = Y

†, X2 = X3 = X, and X4 = Y . In the OTOC-based approximation, the branchedcorrelator

B =〈B, Y (β + it), X(β), X(0), Y †(it)

〉Ω

=〈B, X(0), Y †(it), Y (it), X(0)

〉Ω

(70)

dominates the left-hand side of equation (69) and is given by (54), (55) with t1 = t4 = t andt2 = t3 = 0. Note that if Ω is large, then

ΥRY †,Y (τ) ≈ e−Ω(β−τ)

∑j

ΥRA†j ,Aj

(τ) (71)

by analogy with the rather obvious equation 〈Y †(τ)Y (0)〉 ≈ e−Ω(β−τ)∑

j〈A†j(τ)Aj(0)〉. Hence,

Υ̃RY †,Y (ω) ≈ e−βΩ

∑j

Υ̃RA†j ,Aj

(ω + Ω). (72)

On the other hand, the function B̃(ω14, ω23) in equation (54) may be replaced with B̃(ω14−Ω, ω23)without affecting the result.4 Combining (55) with (72), we get

eβΩB̃(ω14 − Ω, ω23) =∑j

Υ̃RA†j ,Aj

(ω14)Υ̃AX,X(ω23) f(ω14 − Ω, ω23), (73)

where the prefactor eβΩ is used to match the left-hand side of (69).Thus, we have arrived at the conclusion that the replica locking amounts to replacing the

function f(ω14, ω23) in (55) with f(−∞, ω23). Using the explicit formulas (60), (61), we get:

f(−∞, ω23) =cos βκ

4· 2βω23(1− e−βω23) + sin βκ4 · βκ(1 + e

−βω23)

(1− e−β(ω23+iκ/2))(1− e−β(ω23−iκ/2))(74)

=2π

1 + e−βω23if κ =

2π

β. (75)

4 Summary and discussion

We have computed the von Neumann entropy in certain many-body settings by combining time-dependent perturbation theory and the replica trick. In particular, we have considered a systemmade of two parts that are prepared in the thermofield double state, and then one part is coupledto a heat bath. The von Neumann entropy of the two double system grows as S(ρ(t)) ≈ at ln t−1 atvery short times and as S(ρ(t)) ≈ bt at longer times (but less than the Page time); the coefficientsa and b have been calculated. These calculations involve unusual terms, proportional to κ2s, in theperturbative expansion in the system-bath coupling strength κ, where s is the number of replicas.

In the second part of the paper, we developed a variant of thermodynamic response theory.In general, thermodynamic response is concerned with perturbing the system with operators

4This is because t1 = t4. Note, however, that the same condition was implicitly used in (65). A more general

model of system-bath coupling involves Aj(t4) and A†j(t1) so that the additional factor e

−iΩ(t1−t4) has to be added

on the left-hand side of (65). To reproduce this factor, one should replace B̃(ω14, ω23) with B̃(ω14−Ω, ω23) in (54).

15

X1(τ1), . . . , Xn(τn) taken with coefficients x1, . . . , xn. The standard theory uses the generatingfunction lnZ, whereas our modified version is based on lnZ − S. Thus, the connected correlatorturns into a so-called branched correlator,

〈B, Xn(τn), · · · , X1(τ1)〉 =∂n

∂x1 · · · ∂xn

∣∣∣∣x1=···=xn=0

(lnZ − S

). (76)

(B is not an independently defined object but rather, part of the notation.) While the calculationof branched correlators is rather involved, a simplification occurs for maximally chaotic systems.We now argue that this special case is consistent with a holographic picture in a very broad sense.

For comparison, consider a Euclidean black hole in a hyperbolic space (say, in two dimensions).The replica geometry involves an s-fold cover of both the circle and the disk it bounds, with abranching point at the center. Inserting boundary fields slightly deforms the space. In the s→ 1limit, the geometry is given by a smooth metric on the disk and the position of the branchingpoint. The branching point is a special case of a quantum extremal surface [15] (where “surface”means a codimension 2 submanifold). Its position is determined by an extremum of entropy.Instead of the entropy S, we may consider lnZ − S. Indeed, the partition function Z dependsonly on the space-time metric, which should be fixed before finding the extremal surface.

In the Lorentzian case, the branching point is described by null coordinates (u+, u−). (We setaside the ambiguity in the choice of origin due to the deformation of space-time relative to AdS2.)The entropy can be expanded to the second order in u+, u−:

S(u+, u−) = S0 + p+u+ + p−u− − Cu+u−, (77)

where p+ and p− depend on the inserted field.5 Solving the extremum problem, we get

Sext = S0 + C−1p+p−. (78)

Now, let us forget about geometry. The only property we need is that if there is large time sep-aration, then p+ and p− depend only on the fields inserted in the past and the future, respectively.Thus, the change in the entropy should factor into two quantities dependent on the correspondingfields. This is exactly what we observed in the maximal chaos case, see equation (61). We leavethe interpretation of these quantities to future research.

5 Acknowledgments

We thank Douglas Stanford, Juan Maldacena, Pengfei Zhang, and Yiming Chen for useful dis-cussions. We gratefully acknowledge the support by the Simons Foundation under grant 376205.A.K. is also supported by the Simons Foundation through the “It from Qubit” program, as wellas by the Institute of Quantum Information and Matter, a NSF Frontier center funded in part bythe Gordon and Betty Moore Foundation.

5This expression is similar to ’t Hooft’s effective action [3] for the fluctuating horizon, where p+ and p− are nullenergies. In our case, they are just abstract coefficients.

16

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1 Introduction2 Early phase of entanglement growth2.1 The model and general formulas2.2 Short initial period vs. linear growth

3 Perturbations to the saturated phase3.1 Statement of the problem3.2 Thermodynamic response theory for the replicated system3.3 Branched two-point correlator3.4 Branched correlator related to early-time OTOCs3.5 Locking two operators in the same replica

4 Summary and discussion5 Acknowledgments