Winding number on 3D lattice

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.