Flow of unitary matrices: Real-space winding numbers in one and three dimensions

TL;DR

This paper extends the concept of the flow, a topological measure introduced by Kitaev, to three-dimensional lattices, providing practical numerical tools for systems lacking translational symmetry and connecting it to conventional winding numbers.

Contribution

It introduces a three-dimensional formula for the flow, generalizing Kitaev's winding number to non-translationally invariant systems and demonstrating its computational usefulness.

Findings

Flow is effective for numerical calculations in non-translational systems.

Derived a 3D flow formula matching traditional winding numbers in translationally invariant cases.

Extended the topological framework to three dimensions for lattice systems.

Abstract

The notion of the flow introduced by Kitaev is a manifestly topological formulation of the winding number on a real lattice. First, we show in this paper that the flow is quite useful for practical numerical computations for systems without translational invariance. Second, we extend it to three dimensions. Namely, we derive a formula of the flow on a three-dimensional lattice, which corresponds to the conventional winding number when systems have translational invariance.