Abstract
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 ℓ9 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.
Highlights
It was proposed in [12] to use correlation functions of twist operators to calculate the Renyi entropy in a 2D conformal field theory (CFT), i.e., the partition function of the Riemann surface resulting from the replica trick
We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of 9 for the contributions from the vacuum conformal family
We have used the operator product expansion (OPE) of the twist operators to calculate various quantities that can be used to characterize the dissimilarity of two reduced density matrices, and these quantities include the Renyi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm
Summary
In this paper we only consider the contributions from the holomorphic sector of the vacuum conformal family in a two-dimensional large central charge CFT, and the generalization to antiholomorphic sector can be figured out . We need the quasiprimary operators to level 9, i.e., T , A, B, D, E, H, I and J as shown in table 1. The definitions, normalization factors, and conformal transformations of the quasiprimary operators up to level 8, as well as some useful structure constants, can be found in [10, 16, 17, 26]. With (X Y) denoting normal ordering of two operators X and Y. Under a general conformal transformation z → f (z) it transforms as. · · · represents the omitted terms that are proportional to T , A, B, D and their derivatives
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