Entanglement Renyi entropy of two disjoint intervals for large $c$ Liouville field theory

TL;DR

This paper calculates the entanglement Renyi entropy for two disjoint intervals in large central charge Liouville field theory, considering contributions from all saddle points in the semi-classical limit, challenging previous minimal-saddle assumptions.

Contribution

It introduces a method to include all saddle points in the semi-classical calculation of entanglement entropy, using the BPZ equation and Heun functions, for large $c$ Liouville theory.

Findings

All saddle points contribute equally to the ERE in the semi-classical limit.

Numerical evaluation of ERE matches previous saddle point results for $q o 1$.

Provides ERE expressions for large but finite $c$ considering multiple saddle points.

Abstract

Entanglement entropy (EE) is a quantitative measure of the effective degrees of freedom and the correlation between the sub-systems of a physical system. Using the replica trick, we can obtain the EE by evaluating the entanglement Renyi entropy (ERE). The ERE is a $q$-analogue of the EE and expressed by the $q$ replicated partition function. In the semi-classical approximation, it is apparently easy to calculate the EE because the classical action represents the partition function by the saddle point approximation and we do not need to perform the path integral for the evaluation of the partition function. In previous studies, it has been assumed that only the minimal-valued saddle point contributes to the EE. In this paper, we propose that all the saddle points contribute equally to the EE by dealing carefully with the semi-classical limit and then the $q \to 1$ limit. For example, we numerically evaluate the ERE of two disjoint intervals for the large $c$ Liouville field theory with $q \sim 1$. We exploit the BPZ equation with the four twist operators, whose solution is given by the Heun function. We determine the ERE by tuning the behavior of the Heun function such that it becomes consistent with the geometry of the replica manifold. We find the same two saddle points as previous studies for $q \sim 1$ in the above system. Then, we provide the ERE for the large but finite $c$ and the $q \sim 1$ in case that all the saddle points contribute equally to the ERE. Based on this work, it shall be of interest to reconsider EE in other semi-classical physical systems with multiple saddle points.