Renyi entropies of the massless Dirac field on the torus

We compute the Renyi entropies of the massless Dirac field on the Euclidean torus (the Lorentzian cylinder at nonzero 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 noninteger order and analytically for integer order. We also compute the Renyi mutual information, and find that, for appropriate choices of the parameters, it is nonpositive and nonmonotonic. This behavior is expected, but it cannot be seen with the simplest known Renyi entropies in quantum field theory because they are proportional to the entanglement entropy.