R\'enyi entropies of the massless Dirac field on the torus

TL;DR

This paper calculates the Rnyi entropies of the massless Dirac field on a torus using the resolvent method, providing results consistent with previous approaches and revealing non-monotonic mutual information behavior.

Contribution

It introduces a resolvent-based method to compute Rnyi entropies for the Dirac field on the torus, differing from the replica trick and offering new insights.

Findings

Results agree with existing literature both numerically and analytically.

Rnyi mutual information can be non-positive and non-monotonic.

Method provides explicit formulas for arbitrary regions and orders.

Abstract

We compute the R\'enyi entropies of the massless Dirac field on the Euclidean torus (the Lorentzian cylinder at non-zero temperature) for arbitrary spatial regions. We do it by the resolvent method, i.e., we express the entropies in terms of the resolvent of a certain operator and then use the explicit form of that resolvent, which was obtained recently. Our results are different in appearance from those already existing in the literature (obtained via the replica trick), but they agree perfectly, as we show numerically for non-integer order and analytically for integer order. We also compute the R\'enyi mutual information, and find that, for appropriate choices of the parameters, it is non-positive and non-monotonic. This behavior is expected, but it cannot be seen with the simplest known R\'enyi entropies in quantum field theory because they are proportional to the entanglement entropy.