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.
Highlights
In the last years, the use of ideas and results coming from quantum information theory has provided deep insights into the properties of quantum field theory (QFT)
We compute the Renyi entropies of the massless Dirac field on the Euclidean torus 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 use the explicit form of that resolvent, which was obtained recently
Our results are different in appearance from those already existing in the literature, but they agree perfectly, as we show numerically for noninteger order and analytically for integer order
Summary
The use of ideas and results coming from quantum information theory has provided deep insights into the properties of quantum field theory (QFT). As in the case of the entanglement entropy, there is a general relation between the Renyi entropies and the correlator G for free field theories in Gaussian states We will rewrite this relation in terms of the resolvent of G, and use the resolvent computed in [30] to obtain the Renyi entropies of the massless Dirac field on the torus. The simplest known Renyi entropies (those of the massless Dirac field on the plane, or their conformal transformations to the cylinders) do not provide such an example, because they are proportional to the entanglement entropy with a constant coefficient (independent of the spatial region) and they are subadditive and strongly subadditive.
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