Analytic results on the geometric entropy for free fields

The trace of integer powers of the local density matrix rho(V) 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 integral 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 rho(V) in terms of correlators. The result is that the entropy is given in terms of a functional integral in. at 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 nonlinear differential equation of the Painleve V type.