Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances

TL;DR

This paper introduces a gauge-invariant, efficient method for computing Chern numbers and Hall conductances on discretized Brillouin zones, accurately reproducing quantized values even with coarse discretization.

Contribution

It provides a novel gauge-invariant approach to calculate (spin) Hall conductances without gauge fixing, extending to non-Abelian Berry connections.

Findings

Accurately reproduces quantized Hall conductances on coarse grids

Provides a gauge-invariant formulation for Chern number computation

Extends methodology to non-Abelian Berry connections

Abstract

We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on a discretized Brillouin zone. It provides an efficient method of computing (spin) Hall conductances without specifying gauge-fixing conditions. We demonstrate that it correctly reproduces quantized Hall conductances even on a coarsely discretized Brillouin zone. A gauge-dependent integer-valued field, which plays a key role in the formulation, is evaluated in several gauges. An extension to the non-Abelian Berry connection is also given.