Numerical study of lattice index theorem usingimproved cooling and overlap fermions

TL;DR

This study numerically verifies the lattice index theorem using improved cooling and overlap fermions, demonstrating high accuracy in topological charge calculations and confirming the Atiyah-Singer index theorem on smooth configurations.

Contribution

It introduces an improved numerical method for calculating topological charge and verifies the index theorem with high precision on lattice configurations.

Findings

Gluon field definition of topological charge is accurate to better than 1%.

The index of the overlap fermion operator agrees with the gluon field definition.

The method provides a reliable benchmark for topological calculations on lattices.

Abstract

We investigate topological charge and the index theorem on finite lattices numerically. Using mean field improved gauge field configurations we calculate the topological charge Q using the gluon field definition with ${\cal O}(a^4)$-improved cooling and an ${\cal O}(a^4)$-improved field strength tensor $F_{\mu\nu}$. We also calculate the index of the massless overlap fermion operator by directly measuring the differences of the numbers of zero modes with left- and right--handed chiralities. For sufficiently smooth field configurations we find that the gluon field definition of the topological charge is integer to better than 1% and furthermore that this agrees with the index of the overlap Dirac operator, i.e., the Atiyah-Singer index theorem is satisfied. This establishes a benchmark for reliability when calculating lattice quantities which are very sensitive to topology.