  We use canonical formalism to study the fermion determinant in different
three dimensional abelian gauge field backgrounds that contain non-zero
magnetic and electric flux in order to understand the non-perturbative
contributions to the parity-odd and parity-even parts of the phase. We show
that a certain phase associated with free fermion propagation along a closed
path in a momentum torus is responsible for the parity anomaly in a background
with non-zero electric flux. We consider perturbations around backgrounds with
non-zero magnetic flux to understand the structure of the parity-breaking
perturbative term at finite temperature and mass.
