Thermality and excited state R\'enyi entropy in two-dimensional CFT

TL;DR

This paper investigates the differences in R'enyi and entanglement entropy between highly excited states and thermal states in 2D CFT, revealing that thermality emerges only at the level of entanglement entropy, not R'enyi entropy.

Contribution

It provides a detailed analysis of R'enyi entropy for excited states in 2D CFT, showing the lack of universal thermalization in R'enyi entropy despite similarities in entanglement entropy.

Findings

Entanglement entropy of excited states matches thermal states after conformal weight-temperature identification.

R'enyi entropy does not exhibit universal thermal behavior in the short-interval limit.

Thermality depends on the level of entanglement structure examined, not just on entanglement entropy.

Abstract

We evaluate one-interval R\'enyi entropy and entanglement entropy for the excited states of two-dimensional conformal field theory (CFT) on a cylinder, and examine their differences from the ones for the thermal state. We assume the interval to be short so that we can use operator product expansion (OPE) of twist operators to calculate R\'enyi entropy in terms of sum of one-point functions of OPE blocks. We find that the entanglement entropy for highly excited state and thermal state behave the same way after appropriate identification of the conformal weight of the state with the temperature. However, there exists no such universal identification for the R\'enyi entropy in the short-interval expansion. Therefore, the highly excited state does not look thermal when comparing its R\'enyi entropy to the thermal state one. As the R\'enyi entropy captures the higher moments of the reduced density matrix but the entanglement entropy only the average, our results imply that the emergence of thermality depends on how refined we look into the entanglement structure of the underlying pure excited state.