Renormalized entanglement entropy

TL;DR

The paper develops a covariant holographic renormalization scheme for entanglement entropy by area renormalization of minimal surfaces, linking boundary area divergences to covariant counterterms and deriving a renormalized entropy $S_{ren}$ that is naturally inherited from the renormalized partition function via the replica trick. It establishes exact CHM map consistency for disk regions in AdS$_4$, analyzes holographic RG flows with operator dimensions in $(\tfrac{3}{2},\tfrac{5}{2})$ where $F$ can increase, and provides explicit counterterms in terms of the flow superpotential $W(\phi)$ or analytic function $Y(\phi)$, generalizing to higher-derivative theories such as Gauss–Bonnet. The results clarify how finite entanglement terms arise, connect to the F-theorem, and show how the replica trick yields the same $S_{ren}$ as area renormalization, thereby offering a robust framework for renormalized entanglement in holographic settings and RG flows.

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 $3/2 < Δ< 5/2$ 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.

Figures (2)

• Figure 1: The entangling surface embedded into the bulk manifold.
• Figure 2: The cutoff entangling surface.