Analytic results on the geometric entropy for free fields

TL;DR

This paper derives an explicit analytic expression for the geometric entropy of free fields by relating functional integrals to correlator-based density matrix formulas, enabling exact calculations in 1+1 dimensions.

Contribution

It introduces a method to explicitly perform the analytic continuation for geometric entropy calculations of free fields, connecting functional integrals with correlator expressions.

Findings

Exact entanglement entropies for massive scalar and Dirac fields in 1+1 dimensions.

Expression of entropy in terms of solutions to Painleve V differential equations.

Simplification of entropy calculation via boundary conditions in flat Euclidean space.

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.