Winding number on 3D lattice

TL;DR

This paper introduces a simple numerical method for approximating the winding number of mappings from a 3D lattice torus to the unitary group, effective even on coarse lattices, with potential for higher dimensions.

Contribution

It presents a novel discretization and gradient flow approach for computing winding numbers on 3D lattices, improving accuracy on coarse grids.

Findings

Method accurately reproduces known winding numbers

Effective on coarse lattice discretizations

Generalizable to higher-dimensional tori

Abstract

We propose a simple numerical method which computes an approximate value of the winding number of a mapping from 3D torus~$T^3$ to the unitary group~$U(N)$, when $T^3$ is approximated by discrete lattice points. Our method consists of a ``tree-level improved'' discretization of the winding number and the gradient flow associated with an ``over-improved'' lattice action. By employing a one-parameter family of mappings from $T^3$ to $SU(2)$ with known winding numbers, we demonstrate that the method works quite well even for coarse lattices, reproducing integer winding numbers in a good accuracy. Our method can trivially be generalized to the case of higher-dimensional tori.