  The Renyi entropies of a massless Dirac fermion on a circle with chemical
potential are calculated analytically at nonzero temperature by using the
bosonization method. The bosonization of a massive Dirac fermion to the
sine-Gordon model lets us obtain the small mass corrections to the entropies.
We numerically compute the Renyi entropies by putting a massive fermion on the
lattice and find agreement between the analytic and numerical results. In the
presence of a mass gap, we show that corrections to Renyi and entanglement
entropies in the limit m >> T scale as exp(-m/T). We also show that when there
is ground state degeneracy in the gapless case, the limits m to zero and T to
zero do not commute.
