Renyi entropy, stationarity, and entanglement of the conformal scalar

TL;DR

This work clarifies how Rényi entropy around $q=1$ for spherical entangling surfaces in CFTs is influenced by boundary terms in the modular Hamiltonian, resolving a mismatch for the conformal scalar by distinguishing singular versus regularized cone regularizations. It shows that $S'_{q=1}$ and $S''_{q=1}$ are governed by two- and three-point stress-tensor correlators, with $S''_{q=1}$ depending on Osborn coefficients ${\cal A},{\cal B},{\cal C}$ and Vol$(\mathbb{H}^{d-1})$, and verifies these results across holographic duals and free fields while identifying the necessary boundary contributions for scalars. The paper also resolves a coupling-dependence puzzle in ${\cal N}=4$ SYM by showing $S''_{q=1}$ can run with the 't Hooft coupling due to boundary terms, and discusses non-stationarity of the renormalized entanglement entropy for a massive scalar, highlighting IR-divergences that complicate conformal perturbation theory. Finally, it frames conical regularization versus singular cones as a fundamental choice in entanglement calculations and outlines directions for a physically preferred regularization and lattice studies.

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.