Curious behavior of three-dimensional lattice Dirac operators coupled to a monopole background

We numerically investigate the effect of regulating fermions in the presence of singular background fields in three dimensions. For this, we couple free lattice fermions to a background compact U(1) gauge field consisting of a monopole-antimonopole pair of magnetic charge +/- Q separated by a distance s in a periodic L-3 lattice and study the low-lying eigenvalues of different lattice Dirac operators under a continuum limit defined by taking L -> infinity at fixed s/L. As the background gauge field is parity even, we look for a twofold degeneracy of the Dirac spectrum that is expected of a continuum like Dirac operator. The naive-Dirac operator exhibits such a parity doubling but breaks the degeneracy of the fermion-doubler modes for the Q lowest eigenvalues in the continuum limit. The Wilson-Dirac operator lifts the fermion doublers but breaks the parity doubling in the Q lowest modes even in the continuum limit. The overlap-Dirac operator shows parity doubling of all the modes even at finite L that is devoid of fermion doubling and is singled out as a properly regulated continuum Dirac operator in the presence of singular gauge field configurations, albeit with a peculiar algorithmic issue.