  We study the entanglement entropy, the R\'enyi entropy, and the mutual
(R\'enyi) information of Dirac fermions on a 2 dimensional torus in the
presence of constant gauge fields. We derive their general formulas using the
equivalence between twisted boundary conditions and the background gauge
fields. Novel and interesting physical consequences have been presented in
arXiv:1705.01859. Here we provide detailed computations of the entropies and
mutual information in a low temperature limit, a large radius limit, and a high
temperature limit. The high temperature limit reveals rather different physical
properties compared to those of the low temperature one: there exist two
non-trivial limits that depend on a modulus parameter and are not smoothly
connected.
