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We investigate numerically the effect of regulating fermions in the presenceof singular background fields in three dimensions. For this, we couple freelattice fermions to a background compact U(1) gauge field consisting of amonopole-anti-monopole pair of magnetic charge $\pm Q$ separated by a distance$s$ in a periodic $L^3$ lattice, and study the low-lying eigenvalues ofdifferent lattice Dirac operators under a continuum limit defined by taking$L\to\infty$ at fixed $s/L$. As the background gauge field is parity even, welook for a two-fold degeneracy of the Dirac spectrum that is expected of acontinuum-like Dirac operator. The naive-Dirac operator exhibits such aparity-doubling, but breaks the degeneracy of the fermion-doubler modes for the$Q$ lowest eigenvalues in the continuum limit. The Wilson-Dirac operator liftsthe fermion-doublers but breaks the parity-doubling in the $Q$ lowest modeseven in the continuum limit. The overlap-Dirac operator shows parity-doublingof all the modes even at finite $L$ that is devoid of fermion-doubling, andsingles out as a properly regulated continuum Dirac operator in the presence ofsingular gauge field configurations albeit with a peculiar algorithmic issue.

A curious behavior of three-dimensional lattice Dirac operators coupled to monopole background