Local formula for the $Z_2$ invariant of topological insulators

TL;DR

This paper introduces a local formula for the $Z_2$ invariant in topological insulators that is valid without translational symmetry, enabling local computation and approximation of the invariant.

Contribution

It presents a novel local expression for the $Z_2$ invariant applicable without translational invariance and proposes an approximation method using almost commute matrices.

Findings

The local formula accurately computes the $Z_2$ invariant.

The approximation method effectively estimates the invariant using local data.

The validity of both the formula and the approximation method is rigorously proved.

Abstract

We proposed a formula for the $Z_2$ invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.