On the $Z_2$ classification of Quantum Spin Hall Models

TL;DR

This paper introduces a new way to compute the $Z_2$ topological index for quantum spin Hall systems using Chern numbers, simplifying calculations and linking to edge states and spin Chern numbers.

Contribution

It presents an alternative formulation of the $Z_2$ index based on Chern numbers, connecting topological invariants with physical edge states.

Findings

The new index is easier to compute in typical cases.

It establishes a direct link between the $Z_2$ index and edge state robustness.

Connections with the spin Chern number are discussed.

Abstract

We propose an alternative formulation of the $Z_2$ topological index for quantum spin Hall systems and band insulators when time reversal invariance is not broken. The index is expressed in terms of the Chern numbers of the bands of the model, and a connection with the number of pairs of robust edge states is thus established. The alternative index is easy to compute in most cases of interest. We also discuss connections with the recently proposed spin Chern number for quantum spin Hall models.