简化密度矩阵与本征态热化假设的差异

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JHEP12(2017)073

Published for SISSA by Springer

Received: September 11, 2017

Accepted: December 4, 2017

Published: December 14, 2017

Dissimilarities of reduced density matrices and

eigenstate thermalization hypothesis

Song He,

a,b

Feng-Li Lin

and Jia-ju Zhang

d,e

Max Planck Institute for Gravitational Physics (Albert Einstein Institute),

Am M¨uhlenberg 1, 14476 Golm, Germany

CAS Key Laboratory of Theoretical Physics,

Institute of Theoretical Physics, Chinese Academy of Sciences,

55 Zhong Guan Cun East Road, Beijing 100190, China

Department of Physics, National Taiwan Normal University,

No. 88, Sec. 4, Ting-Chou Road, Taipei 11677, Taiwan

Dipartimento di Fisica, Universit`a degli Studi di Milano-Bicocca,

Piazza della Scienza 3, I-20126 Milano, Italy

INFN, Sezione di Milano-Bicocca,

Piazza della Scienza 3, I-20126 Milano, Italy

E-mail: hesong17@gmail.com, fengli.lin@gmail.com,

jiaju.zhang@mib.infn.it

Abstract: We calculate various quantities that characterize the dissimilarity of reduced

density matrices for a short interval of length ` in a two-dimensional (2D) large central

charge conformal ﬁeld theory (CFT). These quantities include the R´enyi entropy, entangle-

ment entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm

and 4-norm. We adopt the method of operator product expansion of twist operators, and

calculate the short interval expansion of these quantities up to order of `

for the contribu-

tions from the vacuum conformal family. The formal forms of these dissimilarity measures

and the derived Fisher information metric from contributions of general operators are also

given. As an application of the results, we use these dissimilarity measures to compare the

excited and thermal states, and examine the eigenstate thermalization hypothesis (ETH)

by showing how they behave in high temperature limit. This would help to understand how

ETH in 2D CFT can be deﬁned more precisely. We discuss the possibility that all the dis-

similarity measures considered here vanish when comparing the reduced density matrices

of an excited state and a generalized Gibbs ensemble thermal state. We also discuss ETH

for a microcanonical ensemble thermal state in a 2D large central charge CFT, and ﬁnd

that it is approximately satisﬁed for a small subsystem and violated for a large subsystem.

Keywords: AdS-CFT Correspondence, Conformal Field Theory, Field Theories in Lower

Dimensions

ArXiv ePrint: 1708.05090

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP12(2017)073

JHEP12(2017)073

It was proposed in [12] to use correlation functions of twist operators to calculate the

R´enyi entropy in a 2D CFT, i.e., the partition function of the Riemann surface resulting

from the replica trick. When there is no compact form for these twist-operator correlation

functions, one can use operator product expansion (OPE) of twist operators to calculate

the short interval expansion of R´enyi entropy [13–17]. Following this method, in this paper

we will calculate various quantities which are just the sums of some partition functions,

and moreover can be used to characterize the dissimilarity of the reduced density matrices

of thermal and excited states, and other states on various Riemann surfaces.

Our results can be used to examine ETH. The ETH and subsystem ETH are originally

deﬁned by comparing the highly excited state with the microcanonical ensemble thermal

state [1–4]. Motivated by [18, 19], as well as [5–8], we compare in [10] the excited state

with the canonical ensemble thermal state, and adopt the so-called weak ETH [18, 19].

In [10] the short-interval ` expansions of the entanglement entropies for the excited state

and canonical ensemble thermal state are calculated to order `

, and it was found that

their diﬀerence, which is just the relative entropy, is only suppressed by the powers of

large central charge c, instead of exponential suppression. In this paper we show that

there are similar behaviors for the Jensen-Shannon divergence and Schatten 2-norm. For

the more reﬁned consideration, one should compare the excited state with the generalized

Gibbs ensemble (GGE) thermal state [20–25]. We will discuss the possibility that all the

dissimilarities considered in this paper vanish when comparing the reduced density matrices

of an excited state and a suitably deﬁned GGE thermal state. As a by-product, we also

check ETH for the microcanonical ensemble thermal state with the dissimilarity measures

of comparing with the energy eigenstate.

The rest of this paper is arranged as follows. In section 2 we give prescriptions of the

method and show how to get the partition functions from OPE of twist operators. More-

over, in subsection 2.5 we apply the prescriptions to evaluate the R´enyi and entanglement

entropies. In section 3 we calculate the various dissimilarity measures between reduced

density matrices. In section 4 we apply our results to examine ETH for the canonical

ensemble thermal state. In section 5 we discuss the possible scenarios ETH for the GGE

thermal state. In section 6 we discuss ETH for a microcanonical ensemble thermal state

in a 2D large central charge CFT, and ﬁnd that it is approximately satisﬁed for a small

subsystem and violated for a large subsystem. We conclude with discussion in section 7.

In appendix A we calculate the relative entropy from modular Hamiltonian as a consistent

check. In appendix B we consider the contributions from general operators, and get the

formal forms of the various dissimilarity measures and the Fisher information metric. Some

lengthy and not so enlightening results in section 3 are collected in appendix C.

2 Prescriptions of the method

In this section we ﬁrst give the useful basics of the vacuum conformal family in two-

dimensional large central charge CFT and then show how we calculate the partition func-

tions on various Riemann surfaces using OPE of the twist operators.

– 2 –

JHEP12(2017)073

level operator level operator level operator level operator

2 T B, D E, H, I

T T A

A 6 T A 8 T B, T D T T T T

T T T T T AA 9 J

Table 2. The holomorphic nonidentity quasiprimary operators to be considered in this paper for

CFT

and up to level 9. We have omitted the replica indices and their constraints, which can be

easily ﬁgured out and can also be found in [26].

function of twist operators T and

T in an n-fold CFT on R [12]

= hT (`)

T (0)i

, h

= h

c(n

− 1)

24n

, (2.5)

and the n folds of the CFT, which we call CFT

, are independent except the connection

by the twist operators. In this paper we only consider Riemann surface R with trans-

lation symmetry, and so the one-point functions are all constants. Using OPE of twist

operators [13–17], we may get

hT (`)

T (0)i

hΦ

(0)i

, (2.6)

and in the summation we only need to consider the quasiprimary operators Φ

in CFT

that are the direct products of the quasiprimary operators in diﬀerent replicas of the

CFT. Only considering the contributions from the vacuum conformal family, we list the

quasiprimary operators in CFT

to level 9 in table 2. To level 8, the coeﬃcients d

can

be found in [16, 26], and using the method in [15] and (2.2), (2.3) we can easily get

= 0. (2.7)

Interestingly, there is no contribution from level 9 operators, which consist of J only.

Each of the CFT

quasiprimary operator Φ

in (2.6) has the form

,··· ,j

= X

···X

, (2.8)

with X

, X

, ···, X

being nonidentity quasiprimary operators in table 1 and there are also

some constraints for the k replica indices j

, j

, ··· , j

. We have the one-point functions

that are independent of the replica indices

hΦ

···j

= hX

···hX

, (2.9)

and so we can deﬁne b

from the OPE coeﬃcient d

···j

by summing over the replica

indices [17]

,··· ,j

···j

with some constraints for 0 ≤ j

, j

, ··· , j

≤ n − 1. (2.10)

To level 8 the form of b

can be found in [10, 17], and from (2.7) we know

= 0. (2.11)

– 4 –

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