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

TL;DR

This paper derives exact results for entanglement and alpha entropies of a massive Dirac field in two dimensions, connecting quantum information measures with integrable models and differential equations, and verifies findings numerically.

Contribution

It introduces a novel method to compute the density matrix trace using sine-Gordon correlators and Painlevé V equations, providing precise entanglement entropy expansions.

Findings

Exact formulas for density matrix powers using sine-Gordon correlators

Series expansions for entanglement entropy at different regimes

Numerical validation of analytical results

Abstract

We present some exact results about universal quantities derived from the local density matrix, for a free massive Dirac field in two dimensions. We first find the trace of powers of the density matrix in a novel fashion, which involves the correlators of suitable operators in the sine-Gordon model. These, in turn, can be written exactly in terms of the solutions of non-linear differential equations of the Painlev\'e V type. Equipped with the previous results, we find the leading terms for the entanglement entropy, both for short and long distances, and showing that in the intermediate regime it can be expanded in a series of multiple integrals. The previous results have been checked by direct numerical calculations on the lattice, finding perfect agreement. Finally, we comment on a possible generalization of the entanglement entropy c-theorem to the alpha-entropies.