Conformal bootstrap to R\'enyi entropy in 2D Liouville and super-Liouville CFTs

TL;DR

This paper investigates the behavior of R'enyi entanglement entropy in 2D irrational conformal field theories, specifically Liouville and super-Liouville theories, revealing divergence patterns and a new way to analyze memory effects in excited states.

Contribution

It introduces a method to analyze R'enyi entropy differences in irrational CFTs, connecting their variation to fusion matrix elements and extending understanding beyond rational CFTs.

Findings

R'enyi entropy diverges at early and late times in Liouville and super-Liouville theories.

The difference in R'enyi entropy relates to the ratio of fusion matrix elements.

The identity operator does not contribute to R'enyi entropy in these theories.

Abstract

The R\'enyi entanglement entropy (REE) of the states excited by local operators in two-dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and $\mathcal{N}=1$ super-Liouville field theory (SLFT), has been investigated. In particular, the excited states obtained by acting on the vacuum with primary operators were considered. {We start from evaluating the second REE in a compact $c=1$ free boson field theory at generic radius, which is an irrational CFT. Then we focus on the two special irrational CFTs, e.g., LFT and SLFT. In these theories, the second REE of such local excited states becomes divergent in early and late time limits. For simplicity, we study the memory effect of REE for the two classes of the local excited states in LFT and SLFT. In order to restore the quasiparticles picture, we define the difference of REE between target and reference states, which belong to the same class. The variation of the difference of REE between early and late time limits always coincides with the log of the ratio of the fusion matrix elements between target and reference states. Furthermore, the locally excited states by acting generic descendent operators on the vacuum have been also investigated. The variation of the difference of REE is the summation of the log of the ratio of the fusion matrix elements between the target and reference states, and an additional normalization factor. Since the identity operator (or vacuum state) does not live in the Hilbert space of LFT and SLFT and no discrete terms contribute to REE in the intermediate channel, the variation of the difference of REE between target and reference states is no longer the log of the quantum dimension which is shown in the 1+1-dimensional rational CFTs (RCFTs).