Renormalized entanglement entropy

TL;DR

This paper introduces a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces, applicable in various dimensions, and explores its implications for RG flows and the F theorem.

Contribution

It develops a general renormalization procedure for holographic entanglement entropy and connects it to holographic renormalization and the F quantity in RG flows.

Findings

Renormalized entanglement entropy matches the holographically renormalized action in AdS4.

Identifies RG flows where the F quantity increases, violating the strong F theorem.

Provides a method to derive entanglement entropy counterterms from standard holographic renormalization.

Abstract

We develop a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces. The renormalized entanglement entropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and for entangling surfaces in four dimensional holographic renormalization group flows. The renormalized entanglement entropy for disk regions in $AdS_4$ spacetimes agrees precisely with the holographically renormalized action for $AdS_4$ with spherical slicing and hence with the F quantity, in accordance with the Casini-Huerta-Myers map. We present a generic class of holographic RG flows associated with deformations by operators of dimension $3/2 < \Delta < 5/2$ for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method for the entanglement entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.