Some results on the shape dependence of entanglement and R\'enyi entropies

TL;DR

This paper investigates how the universal part of entanglement and Renyi entropies in conformal field theories varies with the shape of the entangling region, providing explicit formulas and numerical analyses for deformed circles and ellipses.

Contribution

It derives the quadratic dependence of the universal entanglement entropy on shape deformations and conjectures the sphere minimizes this entropy across all dimensions.

Findings

Universal contribution varies quadratically with deformation amplitude.

Explicit second-order variation formula for deformed circle in 3D CFT with gravity dual.

Numerical analysis of entanglement entropy for elliptical regions.

Abstract

We study how the universal contribution to entanglement entropy in a conformal field theory depends on the entangling region. We show that for a deformed sphere the variation of the universal contribution is quadratic in the deformation amplitude. We generalize these results for R\'enyi entropies. We obtain an explicit expression for the second order variation of entanglement entropy in the case of a deformed circle in a three dimensional CFT with a gravity dual. For the same system, we also consider an elliptic entangling region and determine numerically the entanglement entropy as a function of the aspect ratio of the ellipse. Based on these three-dimensional results and Solodukhin's formula in four dimensions, we conjecture that the sphere minimizes the universal contribution to entanglement entropy in all dimensions.