TY - JOUR

T1 - Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

AU - He, Song

AU - Lin, Feng Li

AU - Zhang, Jia ju

N1 - Funding Information:
Article funded by SCOAP3.
Publisher Copyright:
© 2017, The Author(s).

PY - 2017/12/1

Y1 - 2017/12/1

N2 -
We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length ℓ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the Rényi entropy, entanglement 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 ℓ
9
for the contributions 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 dissimilarity 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.

AB -
We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length ℓ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the Rényi entropy, entanglement 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 ℓ
9
for the contributions 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 dissimilarity 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.

KW - AdS-CFT Correspondence

KW - Conformal Field Theory

KW - Field Theories in Lower Dimensions

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U2 - 10.1007/JHEP12(2017)073

DO - 10.1007/JHEP12(2017)073

M3 - Article

AN - SCOPUS:85038383421

VL - 2017

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 12

M1 - 73

ER -