Universal terms for the entanglement entropy in 2+1 dimensions

TL;DR

This work establishes a universal vertex-induced logarithmic term in the entanglement entropy for polygonal regions in 2+1 dimensions and provides an exact expression for the corresponding Rényi coefficients $s_n(x)$ for a free scalar field. The authors reduce the problem to the trace anomaly of a 3D manifold with conical singularities, translating it into a Green function problem on a cut sphere and solving a nonlinear ODE system to obtain the angle-dependent coefficients. A small-angle limit connects the 2+1D vertex contribution to the 1+1D entropic c-functions, offering a bridge between higher-dimensional and lower-dimensional entropic measures and suggesting possible implications for c-theorem generalizations and AdS/CFT contexts. They validate the analytic results against lattice simulations and provide detailed appendices with the Green function derivation and lattice entropies, broadening the toolkit for universal entropy coefficients in QFT.

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.

Figures (3)

• Figure 1: The solid curves are the functions $s_2(x)$ (top) and $s_3(x)$ (bottom). The points are obtained by numerical simulations in a lattice, and correspond to, from top to bottom, $s=s_1$, $s_2$ and $s_3$ evaluated for the angles $x=\pi / 4$, $\pi /2$ and $3/4 \,\pi$.
• Figure 2: The dashed sets $A$ and $B$ are anti-starshaped, in the sense that any ray from the origin has no intersection with them or the intersection is a half line which does not contain the origin. A contraction of the space transforms them into $A^\prime$ and $B^\prime$. In the limit of an infinite contraction they are mapped to two angular sectors with common vertex.
• Figure 3: The plane angular sector of angle $x=\varphi_1-\varphi_2$ and the sphere with a cut of angle $x$