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

## TL;DR

This paper numerically studies three-dimensional lattice Dirac operators coupled to monopole backgrounds, revealing differences in their spectral properties and how well they preserve continuum-like features such as parity-doubling.

## Contribution

It compares naive, Wilson, and overlap lattice Dirac operators in monopole backgrounds, highlighting their spectral behaviors and suitability as continuum regulators.

## Key findings

- Naive-Dirac exhibits parity-doubling but breaks fermion-doublers degeneracy.
- Wilson-Dirac lifts fermion-doublers but breaks parity-doubling.
- Overlap-Dirac maintains parity-doubling and avoids fermion-doubling issues.

## Abstract

We investigate numerically 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-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 of different 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, we look for a two-fold 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 singles out as a properly regulated continuum Dirac operator in the presence of singular gauge field configurations albeit with a peculiar algorithmic issue.