Anomalies, gauge field topology, and the lattice

TL;DR

This paper explores defining a lattice topological charge using the fourth power of the naive Dirac operator, examining how quantum fluctuations and cooling affect the emergence and stability of topological structures.

Contribution

It introduces a local lattice measure of topological charge based on the Dirac operator and analyzes its behavior under cooling and quantum fluctuations.

Findings

Quantum fluctuations obscure topological charge in typical configurations.

Cooling reveals instanton-like structures that shrink and disappear.

Modifying the action affects the stability of topological objects.

Abstract

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally "fall through the lattice." Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.