Universal terms for the entanglement entropy in 2+1 dimensions

TL;DR

This paper derives a universal logarithmic term in the entanglement entropy for polygonal regions in 2+1 dimensions, linking it to trace anomalies and providing explicit formulas for free scalar fields.

Contribution

It introduces a universal logarithmic term in entanglement entropy in 2+1D, expressed via solutions to nonlinear differential equations, and connects it to trace anomalies and c-functions.

Findings

Universal logarithmic term in entanglement entropy identified.

Analytic expression for the contribution as a function of the angle.

Reduction to a 2D problem yields exact Green function and entropy expressions.

Abstract

We show that the entanglement entropy and alpha entropies corresponding to spatial polygonal sets in $(2+1)$ dimensions contain a term which scales logarithmically with the cutoff. Its coefficient is a universal quantity consisting in a sum of contributions from the individual vertices. For a free scalar field this contribution is given by the trace anomaly in a three dimensional space with conical singularities located on the boundary of a plane angular sector. We find its analytic expression as a function of the angle. This is given in terms of the solution of a set of non linear ordinary differential equations. For general free fields, we also find the small-angle limit of the logarithmic coefficient, which is related to the two dimensional entropic c-functions. The calculation involves a reduction to a two dimensional problem, and as a byproduct, we obtain the trace of the Green function for a massive scalar field in a sphere where boundary conditions are specified on a segment of a great circle. This also gives the exact expression for the entropies for a scalar field in a two dimensional de Sitter space.