On the definition of the covariant lattice Dirac operator

TL;DR

This paper introduces a lattice Dirac operator that maintains key properties of the continuum operator, such as antihermiticity and chiral invariance, by relating it to a discretized gauge covariant derivative and analyzing its properties and complexity.

Contribution

It proposes a natural lattice gauge covariant Dirac operator based on local unitary operators, connecting it to the SLAC derivative and analyzing its properties and boundary effects.

Findings

The lattice Dirac operator is antihermitian and chiral invariant in the massless limit.

It coincides with the naive gauge covariant SLAC derivative in the infinite volume limit.

It has comparable numerical complexity to smeared lattice Dirac operators.

Abstract

In the continuum the definitions of the covariant Dirac operator and of the gauge covariant derivative operator are tightly intertwined. We point out that the naive discretization of the gauge covariant derivative operator is related to the existence of local unitary operators which allow the definition of a natural lattice gauge covariant derivative. The associated lattice Dirac operator has all the properties of the classical continuum Dirac operator, in particular antihermiticy and chiral invariance in the massless limit, but is of course non-local in accordance to the Nielsen-Ninomiya theorem. We show that this lattice Dirac operator coincides in the limit of an infinite lattice volume with the naive gauge covariant generalization of the SLAC derivative, but contains non-trivial boundary terms for finite-size lattices. Its numerical complexity compares pretty well on finite lattices with smeared lattice Dirac operators.