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Thermal Corrections to Rényi entropies for Free Fermions

We calculate thermal corrections to Rényi entropies for free massless fermions on a sphere. More specifically, we take a free fermion on $\mathbb{R}\times\mathbb{S}^{d-1}$ and calculate the leading thermal correction to the Rényi entropies for a cap like region with opening angle $2θ$. By expanding the density matrix in a Boltzmann sum, the problem of finding the Rényi entropies can be mapped to the problem of calculating a two point function on an $n$ sheeted cover of the sphere. We follow previous work for conformal field theories to map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the Rényi entropies.