Analytic results on the geometric entropy for free fields

TL;DR

This work tackles the challenge of computing geometric (entanglement) entropy for free fields by deriving an explicit analytic continuation for the replica-trick expressions. It shows that the entropy can be formulated as a flat-space functional integral with a cut on the entangling region and boundary conditions, linking this to Chung-Peschel correlator data and avoiding difficult conical-angle continuations. The authors obtain universal integral representations $S = \int_1^{\infty} d\lambda \frac{2}{(\lambda-1)^2} \log Z[\lambda]$ for fermions and $S = \int_1^{\infty} d\lambda \frac{2}{(\lambda+1)^2} \log Z[-\lambda]$ for bosons, where $Z[\lambda]$ is a flat-space functional integral with a cut; a 2D example reduces to a Painlevé V equation, yielding exact entropies for a massive scalar and Dirac fermion and enabling precise entanglement-c function calculations. The results connect continuum field theory to lattice data and provide a practical route to compute entanglement in free-field theories across dimensions.

Abstract

The trace of integer powers of the local density matrix corresponding to the vacuum state reduced to a region V can be formally expressed in terms of a functional integral on a manifold with conical singularities. Recently, some progress has been made in explicitly evaluating this type of integrals for free fields. However, finding the associated geometric entropy remained in general a difficult task involving an analytic continuation in the conical angle. In this paper, we obtain this analytic continuation explicitly exploiting a relation between the functional integral formulas and the Chung-Peschel expressions for the density matrix in terms of correlators. The result is that the entropy is given in terms of a functional integral in flat Euclidean space with a cut on V where a specific boundary condition is imposed. As an example we get the exact entanglement entropies for massive scalar and Dirac free fields in 1+1 dimensions in terms of the solutions of a non linear differential equation of the Painleve V type.