Locality Properties of a New Class of Lattice Dirac Operators

TL;DR

This paper investigates the locality properties of a new class of lattice Dirac operators satisfying a generalized algebraic relation, demonstrating their exponential decay and local behavior under certain conditions, which is relevant for chiral gauge theories.

Contribution

It introduces and analyzes a new class of lattice Dirac operators satisfying a generalized algebraic relation, establishing their locality and decay properties.

Findings

Operators are analytic in the Brillouin zone with a mass gap.

Exponential decay of operators in coordinate space for finite k.

Finite locality domain for gauge field strength established.

Abstract

A new class of lattice Dirac operators $D$ which satisfy the index theorem have been recently proposed on the basis of the algebraic relation $\gamma_{5}(\gamma_{5}D) + (\gamma_{5}D)\gamma_{5} = 2a^{2k+1}(\gamma_{5}D)^{2k+2}$. Here $k$ stands for a non-negative integer and $k=0$ corresponds to the ordinary Ginsparg-Wilson relation. We analyze the locality properties of Dirac operators which solve the above algebraic relation. We first show that the free fermion operator is analytic in the entire Brillouin zone for a suitable choice of parameters $m_{0}$ and $r$, and there exists a well-defined ``mass gap'' in momentum space, which in turn leads to the exponential decay of the operator in coordinate space for any finite $k$. This mass gap in the free fermion operator suggests that the operator is local for sufficiently weak background gauge fields. We in fact establish a finite locality domain of gauge field strength for $\Gamma_{5}=\gamma_{5}-(a\gamma_{5}D)^{2k+1}$ for any finite $k$, which is sufficient for the cohomological analyses of chiral gauge theory. We also present a crude estimate of the localization length defined by an exponential decay of the Dirac operator, which turns out to be much shorter than the one given by the general Legendre expansion.