Adjoint zero-modes as a tool to understand the Yang-Mills vacuum

TL;DR

This paper explores how adjoint zero-modes of the Dirac operator can be used to analyze the Yang-Mills vacuum, demonstrating improved results with the overlap operator and examining effects of lattice artifacts and instanton interactions.

Contribution

It introduces an improved lattice method using the overlap operator for probing the Yang-Mills vacuum with adjoint zero-modes, and analyzes artifacts and instanton effects.

Findings

Overlap operator yields better results than Wilson-Dirac in zero-mode detection.

Lattice artifacts cause deviations from the index theorem in thermalized configurations.

Instanton-anti-instanton separation affects zero-mode properties.

Abstract

The use of adjoint (quasi) zero-modes of the Dirac operator to probe the Yangs-Mills vacuum has been recently advocated by Gonzalez-Arroyo and Kirchner. The construction relies on the use of the super-symmetric zero mode which, for classical configurations, provides a direct estimate of the gauge action density. In the lattice implementation of this idea, we show how the results improve considerably if the overlap operator is used instead of the Wilson-Dirac one. Before proceeding to the detailed study of Monte Carlo ensembles, we studied here a series of potentially complicated situations which can be encountered. In particular, we study the case of instanton anti-instanton pairs and analyse how the results depend upon separation. The effect of lattice artifacts is also of concern. Indeed, a statistical analysis of zero modes of thermalised SU(2) configurations at beta=2.57 shows a significant fraction having 4N+2 adjoint zero modes, in contradiction with the index theorem. This violation must be associated to the roughness of the lattice configurations. Indeed, we show that this situation occurs for instantons of size of the order of the lattice spacing.