R\'enyi entropies and area operator from gravity with Hayward term

TL;DR

This paper explores the role of the Hayward term in holographic entanglement entropy calculations, proposing it as a fundamental part of the gravity action that simplifies replica computations and connects to the area operator.

Contribution

It demonstrates that incorporating the Hayward term into the gravity action provides a natural explanation for Re9nyi entropies and the area operator in holography, simplifying the replica method.

Findings

The Hayward term is essential in the gravitational path integral for holographic entanglement.

The Re9nyi entropies follow an area law involving minimal surfaces in the replicated spacetime.

The gravitational modular flow includes the area operator, supporting the JLMS proposal.

Abstract

In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy $\hat{S}_n$ that is related to the R\'enyi entropies $S_n$, as given by the area of a \emph{cosmic brane} minimally coupled with gravity, with a tension related to $n$ that vanishes as $n\to1$, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry. In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account. So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the R\'enyi entropies, but it is present in the own gravity action through a Hayward term. As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original R\'enyi entropies ($S_n$) related to a minimal surface in the $n$ replicated spacetime. Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.