Renyi entropy, stationarity, and entanglement of the conformal scalar

TL;DR

This paper investigates the perturbative expansion of Renyi entropy in conformal field theories, resolving discrepancies for scalar fields by considering boundary conditions and boundary terms, and explores implications for entanglement stationarity and IR divergences.

Contribution

It clarifies the role of boundary conditions in Renyi entropy calculations, resolving conflicts for scalar fields and addressing stationarity issues in entanglement entropy.

Findings

Boundary conditions near the entangling surface are crucial for correct Renyi entropy calculations.

The resolution explains discrepancies in scalar field Renyi entropies and the non-stationarity of entanglement entropy.

IR divergences hinder accurate perturbative calculations of the renormalized entanglement entropy at small masses.

Abstract

We extend previous work on the perturbative expansion of the Renyi entropy, $S_q$, around $q=1$ for a spherical entangling surface in a general CFT. Applied to conformal scalar fields in various spacetime dimensions, the results appear to conflict with the known conformal scalar Renyi entropies. On the other hand, the perturbative results agree with known Renyi entropies in a variety of other theories, including theories of free fermions and vector fields and theories with Einstein gravity duals. We propose a resolution stemming from a careful consideration of boundary conditions near the entangling surface. This is equivalent to a proper treatment of total-derivative terms in the definition of the modular Hamiltonian. As a corollary, we are able to resolve an outstanding puzzle in the literature regarding the Renyi entropy of ${\cal N}=4$ super-Yang-Mills near $q=1$. A related puzzle regards the question of stationarity of the renormalized entanglement entropy (REE) across a circle for a (2+1)-dimensional massive scalar field. We point out that the boundary contributions to the modular Hamiltonian shed light on the previously-observed non-stationarity. Moreover, IR divergences appear in perturbation theory about the massless fixed point that inhibit our ability to reliably calculate the REE at small non-zero mass.