Comparison of |Q|=1 and |Q|=2 gauge-field configurations on the lattice four-torus

TL;DR

This study investigates the instability of |Q|=1 gauge-field configurations on the lattice 4-torus for SU(3), identifying early signals of their decay and contrasting them with stable |Q|=2 configurations, using advanced error control techniques.

Contribution

It extends previous SU(2) work to SU(3), identifying global instability signals of |Q|=1 configurations on the lattice 4-torus with improved discretization error control.

Findings

|Q|=1 configurations show early instability signals before instanton shrinkage.

These signals are absent in |Q|=2 configurations, indicating a global instability.

Discretization errors are minimized, and topological charge fluctuations are observed.

Abstract

It is known that exactly self-dual gauge-field configurations with topological charge |Q|=1 cannot exist on the untwisted continuum 4-torus. We explore the manifestation of this remarkable fact on the lattice 4-torus for SU(3) using advanced techniques for controlling lattice discretization errors, extending earlier work of De Forcrand et. al. for SU(2). We identify three distinct signals for the instability of |Q|=1 configurations, and show that these manifest themselves early in the cooling process, long before the would-be instanton has shrunk to a size comparable to the lattice discretization threshold. These signals do not appear for our |Q|=2 configurations. This indicates that these signals reflect the truly global nature of the instability, rather than local discretization effects. Monte-Carlo generated SU(3) gauge field configurations are cooled to the self-dual limit using an O(a^4)-improved gauge action chosen to have small but positive O(a^6) errors. This choice prevents lattice discretization errors from destroying instantons provided their size exceeds the dislocation threshold of the cooling algorithm. Lattice discretization errors are evaluated by comparing the O(a^4)-improved gauge-field action with an O(a^4)-improved action constructed from the square of an O(a^4)-improved lattice field-strength tensor, thus having different O(a^6) discretization errors. The number of action-density peaks, the instanton size and the topological charge of configurations is monitored. We observe a fluctuation in the total topological charge of |Q|=1 configurations, and demonstrate that the onset of this unusual behavior corresponds with the disappearance of multiple-peaks in the action density. At the same time discretization errors are minimal.