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On short interval expansion of Rényi entropy

Bin Chen, Jia-ju Zhang

TL;DR

The paper addresses the problem of computing Rényi entropy for short intervals in 2D CFTs by performing a detailed short-interval expansion of twist operators within the vacuum Verma module. They implement the operator product expansion of twist fields, deriving the coefficients of holomorphic quasiprimary operators up to level 6 and their descendants, then use these to obtain leading contributions to $S_n$ to order 6 in the small-interval expansion. In the one-interval on a cylinder, the results reproduce the universal logarithmic form with finite-size corrections up to $ ext{ell}^6$; for two disjoint intervals with small cross ratio $x$, they obtain the classical (tree), 1-loop, and leading $1/c$ (2-loop) terms, all matching gravity computations, with the 2-loop term presented as a new result. This work strengthens the AdS$_3$/CFT$_2$ dictionary by showing vacuum Verma module data suffices to reproduce bulk physics at these orders and outlines pathways to include other conformal families and higher-dimensional generalizations.

Abstract

Rényi entanglement entropy provides a new window to study the AdS/CFT correspondence. In this paper we consider the short interval expansion of Rényi entanglement entropy in two-dimensional conformal field theory. This amounts to do the operator product expansion of the twist operators. We focus on the vacuum Verma module and consider the quasiprimary operators constructed from the stress tensors. After obtaining the expansion coefficients of the twist operators to level 6 in vacuum Verma module, we compute the leading contributions to the Rényi entropy, to order 6 in the short interval expansion. In the case of one short interval on cylinder, we reproduce the first several leading contributions to the Rényi entropy. In the case of two short disjoint intervals with a small cross ratio $x$, we obtain not only the classical and 1-loop quantum contributions to the Rényi entropy to order $x^6$, both of which are in perfect match with the ones found in gravity, but also the leading $1/c$ contributions, which corresponds to 2-loop corrections in the bulk.

On short interval expansion of Rényi entropy

TL;DR

The paper addresses the problem of computing Rényi entropy for short intervals in 2D CFTs by performing a detailed short-interval expansion of twist operators within the vacuum Verma module. They implement the operator product expansion of twist fields, deriving the coefficients of holomorphic quasiprimary operators up to level 6 and their descendants, then use these to obtain leading contributions to to order 6 in the small-interval expansion. In the one-interval on a cylinder, the results reproduce the universal logarithmic form with finite-size corrections up to ; for two disjoint intervals with small cross ratio , they obtain the classical (tree), 1-loop, and leading (2-loop) terms, all matching gravity computations, with the 2-loop term presented as a new result. This work strengthens the AdS/CFT dictionary by showing vacuum Verma module data suffices to reproduce bulk physics at these orders and outlines pathways to include other conformal families and higher-dimensional generalizations.

Abstract

Rényi entanglement entropy provides a new window to study the AdS/CFT correspondence. In this paper we consider the short interval expansion of Rényi entanglement entropy in two-dimensional conformal field theory. This amounts to do the operator product expansion of the twist operators. We focus on the vacuum Verma module and consider the quasiprimary operators constructed from the stress tensors. After obtaining the expansion coefficients of the twist operators to level 6 in vacuum Verma module, we compute the leading contributions to the Rényi entropy, to order 6 in the short interval expansion. In the case of one short interval on cylinder, we reproduce the first several leading contributions to the Rényi entropy. In the case of two short disjoint intervals with a small cross ratio , we obtain not only the classical and 1-loop quantum contributions to the Rényi entropy to order , both of which are in perfect match with the ones found in gravity, but also the leading contributions, which corresponds to 2-loop corrections in the bulk.

Paper Structure

This paper contains 8 sections, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General prescriptions
  3. The coefficients $\alpha_K$ and $d_K$
  4. Applications
  5. One short interval on cylinder
  6. Two intervals on complex plane with small cross ratio
  7. Conclusion and discussion
  8. Some useful formulas

Figures (5)

  • Figure 1: (a) OPE vertex of twist operators; (b) Expectation value of an operator on $\mathcal{R}_{n,1}$.
  • Figure 2: The diagrams for OPE of twist operators
  • Figure 3: The diagrams for the expectation values of the operators on $\mathcal{R}_{n,1}$
  • Figure 4: The OPE of twist operators and the expectation values of the operators on cylinder
  • Figure 5: The diagrams for two intervals