Shape Dependence of Entanglement Entropy in Conformal Field Theories

TL;DR

This paper investigates how the shape of entangling surfaces affects entanglement entropy in conformal field theories, revealing universal behavior linked to stress tensor correlations and confirming conjectures about corner contributions.

Contribution

It demonstrates the universality of the non-local entanglement density in CFTs and proves key conjectures relating shape dependence to stress tensor two-point functions.

Findings

The non-local part of entanglement density is proportional to the stress tensor coefficient $C_T$.

Confirmed the universality of the corner term coefficient $rac{\sigma}{C_T}=rac{\pi^2}{24}$ in 3D CFTs.

Validated the holographic Mezei formula for entanglement entropy across deformed spheres.

Abstract

We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on $\mathbb{R}^{1,d-1}$. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient $C_T$ appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient $\frac{\sigma}{C_T}=\frac{\pi^2}{24}$ in $d=3$ CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.