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.
Highlights
Lattice regularization of noncompact QED [1] in three dimensions is defined by a noncompact action for the gauge fields, θμðnÞ ∈ R, on the link connecting n and n þ μ, and the lattice fermions couple to Uð1Þ valued link variables, UμðnÞ 1⁄4 eiθμðnÞ
When we attempted to numerically study this theory using overlapDirac fermions, we found it be numerically formidable due to anomalously small eigenvalues of the massive Wilson-Dirac kernel that is at the core of the overlapDirac operator—to contrast, for a smooth field, one would
Find the spectrum of a massive Wilson-Dirac operator to be gapped at least by the Wilson mass. This prompted us to consider the question of what happens when the conventional lattice regulated fermions, which lead to universal results in the continuum limit over generic smooth gauge fields, are coupled to a singular gauge field from a monopole; do operations at the level of lattice spacing, such as point splitting used regularly in lattice regularization, have any effect in the presence of a Dirac string which is one lattice spacing thick? We present related numerical observations in this paper
Summary
Lattice regularization of noncompact QED [1] in three dimensions is defined by a noncompact action for the gauge fields, θμðnÞ ∈ R, on the link connecting n and n þ μ, and the lattice fermions couple to Uð1Þ valued link variables, UμðnÞ 1⁄4 eiθμðnÞ. This effect propagates itself to the nonvanishing phase of detð þ VÞ of the massless overlap fermion Notwithstanding such effects in three dimensions, we expect X to commute with X† in the continuum limit, unless the gauge field background is not smooth even in the continuum limit. The failure described in the previous paragraph prompted us to study the low-lying spectrum of the following positive definite operators constructed out of lattice operators; D= †D= for the naive-Dirac operator; X†X as a function of mw for the Wilson-Dirac operator; and of the ð1 þ VÞð1 þ V†Þ for the overlap-Dirac operator in a controlled background before proceeding to address an alternative approach to the study of compact QED. We will use a similar background with a minor change to better fit it in a periodic lattice
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