Single Interval Rényi Entropy At Low Temperature

TL;DR

This paper analyzes the Rényi entropy of a single interval on a circle at finite temperature in a 2D CFT, computing a low-temperature field-theory expansion and a complementary holographic calculation in AdS$_3$/CFT$_2$. The authors expand the thermal density matrix in the vacuum Verma module up to level 4, obtaining leading thermal terms that scale as $e^{-rac{4\pi\beta}{L}}$, $e^{-rac{6\pi\beta}{L}}$, and $e^{-rac{8\pi\beta}{L}}$, and separate the Renyi entropy into a central-charge dependent tree-level part and a $c$-independent 1-loop part, with detailed holomorphic/antiholomorphic contributions. On the gravity side, they use the monodromy/Schottky framework to compute the classical contribution and a 1-loop correction, and fix interval-independent constants via a size-derivative equation; the high-temperature results are related to low-temperature ones by a modular duality, and they obtain exact agreement with the field-theory results in the large-$c$ limit. The work strengthens evidence for holographic computations of Rényi entropy in AdS$_3$/CFT$_2$, including thermal effects, and lays out a method to fix remaining constants and extend to higher orders.

Abstract

In this paper, we calculate the Rényi entropy of one single interval on a circle at finite temperature in 2D CFT. In the low temperature limit, we expand the thermal density matrix level by level in the vacuum Verma module, and calculate the first few leading terms in $e^{-π/TL}$ explicitly. On the other hand, we compute the same Rényi entropy holographically. After considering the dependence of the Rényi entropy on the temperature, we manage to fix the interval-independent constant terms in the classical part of holographic Rényi entropy. We furthermore extend the analysis in Xi Dong's paper to higher orders and find exact agreement between the results from field theory and bulk computations in the large central charge limit. Our work provides another piece of evidence to support holographic computation of Rényi entropy in AdS$_3$/CFT$_2$ correspondence, even with thermal effect.