A Numerical Study of Spectral Flows of Hermitian Wilson-Dirac Operator and the Index Theorem in Abelian Gauge Theories on Finite Lattices

TL;DR

This paper numerically investigates the spectral properties of the Hermitian Wilson-Dirac operator in abelian gauge theories on finite lattices, demonstrating the connection between the operator's index and topological charge.

Contribution

It provides a detailed numerical analysis of spectral flows and clarifies the spectrum structure leading to the index theorem in abelian gauge theories on finite lattices.

Findings

Index coincides with topological charge for many gauge configurations

Spectral analysis reveals characteristic structures related to the index theorem

Exact index determination possible for certain nontrivial configurations in 2D

Abstract

We investigate the index of the Neuberger's Dirac operator in abelian gauge theories on finite lattices by numerically analyzing the spectrum of the hermitian Wilson-Dirac operator for a continuous family of gauge fields connecting different topological sectors. By clarifying the characteristic structure of the spectrum leading to the index theorem we show that the index coincides to the topological charge for a wide class of gauge field configurations. We also argue that the index can be found exactly for some special but nontrivial configurations in two dimensions by directly analyzing the spectrum.