Entanglement and alpha entropies for a massive scalar field in two dimensions

TL;DR

This paper derives an exact analytic expression for the trace of powers of the reduced density matrix for a massive scalar field in 1+1 dimensions, linking entanglement entropy to solutions of Painlevé V equations and relating it to sine-Gordon correlators.

Contribution

It generalizes previous methods to compute entanglement measures for massive bosons, providing exact formulas and connecting them to integrable models and universal terms in various dimensions.

Findings

Exact expression for trace of powers of reduced density matrix.

Relation of partition function to sine-Gordon correlators.

Interpolation of entropic c-function between fermionic cases.

Abstract

We find the analytic expression of the trace of powers of the reduced density matrix on an interval of length L, for a massive boson field in 1+1 dimensions. This is given exactly (except for a non universal factor) in terms of a finite sum of solutions of non linear differential equations of the Painlev\'e V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the Sine-Gordon model in agreement with a result by Delfino et al. We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c-function interpolates between the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two dimensional entropy functions.