Topological Summation in Lattice Gauge Theory

TL;DR

This paper investigates methods to approximate full gauge theory observables by summing over topological sectors in lattice simulations, addressing challenges of sector trapping and proposing practical solutions in a 2-flavour Schwinger model.

Contribution

It introduces techniques for estimating complete observables from limited topological sector data in lattice gauge theories with chiral fermions.

Findings

Methods for summing over topological sectors improve observable estimates.

Procedures for indirect topological susceptibility evaluation are effective.

Pilot results demonstrate feasibility in the 2-flavour Schwinger model.

Abstract

In gauge theories the field configurations often occur in distinct topological sectors. In a lattice regularised system with chiral fermions, these sectors can be defined by referring to the Atiyah-Singer Index Theorem. However, if such a model is simulated with local updates of the lattice gauge configuration, the Monte Carlo history tends to get stuck in one sector for many steps, in particular on fine lattices. Then expectation values can be measured only within specific sectors. Here we present a pilot study in the 2-flavour Schwinger model which explores methods of approximating the complete result for an observable - corresponding to a suitable sum over all sectors - based on numerical measurements in a few specific topological sectors. We also probe various procedures for an indirect evaluation of the topological susceptibility, starting from such topologically restricted measurements.