Studying and removing effects of fixed topology in a quantum mechanical model

TL;DR

This paper extends and tests equations designed to correct fixed topology effects in lattice QCD simulations by applying them to a solvable quantum mechanical toy model, assessing their accuracy and validity.

Contribution

It develops and applies extended equations to a simple quantum model to evaluate fixed topology correction methods in a controlled setting.

Findings

The equations accurately remove fixed topology effects within certain parameter ranges.

The model confirms the validity of saddle point approximation in this context.

Limitations of the correction methods are identified at small volumes or strong fixed topology effects.

Abstract

At small lattice spacing, or when using e.g. overlap fermions, lattice QCD simulations tend to become stuck in a single topological sector. Physical observables then differ from their full QCD counterparts by 1/V corrections, where V is the spacetime volume. Brower et al. and Aoki et al. have derived equations by means of a saddle point approximation, to determine and to remove these corrections. We extend these equations and apply them to a simple toy model, a quantum mechanical particle on a circle in a square well potential at fixed topology. This model can be solved numerically up to arbitrary precision and allows to explore effects arising due to fixed topology. We investigate the range of validity and accuracy of the above mentioned equations, to remove such fixed topology effects.