  Entanglement and the R\'enyi entropies for Dirac fermions on 2 dimensional
torus in the presence of chemical potential, current source, and topological
Wilson loop are unified in a single framework by exhausting all the ingredients
of the electromagnetic vertex operators of $\mathbb{Z}_n$ orbifold conformal
field theory. We employ different normalizations for different topological
sectors to organize various topological phase transitions in the context of
entanglement entropy. Pictorial representations for the topological transitions
are given for the $n=2$ R\'enyi entropy.
  Our analytic computations reveal numerous novelties and provide resolutions
for existing issues. We have settled to provide non-singular entanglement
entropies that are also continuous across the topological sectors.
Surprisingly, in infinite space, these entropies become exact and depend only
on the Wilson loop. On a circle, we resolve to find the entropies subtly depend
on the chemical potential at zero temperature, which is useful for probing the
ground state energy levels of quantum systems.
