Quantum spin Hall effect in three dimensional materials: Lattice computation of Z$_2$ topological invariants and its application to Bi and Sb

TL;DR

This paper develops a lattice-based method to compute Z2 topological invariants, enabling the study of quantum spin Hall effects in 3D materials like Bi and Sb through numerical simulations.

Contribution

It introduces an efficient lattice formula for Z2 invariants and applies it to analyze topological phases in bismuth and antimony.

Findings

Successfully computed Z2 invariants for Bi and Sb.

Demonstrated quantum spin Hall effect in three-dimensional models.

Provided a practical numerical approach for topological insulators.

Abstract

We derive an efficient formula for Z$_2$ topological invariants characterizing the quantum spin Hall effect. It is defined in a lattice Brillouin zone, which enables us to implement numerical calculations for realistic models even in three dimensions. Based on this, we study the quantum spin Hall effect in Bi and Sb in quasi-two and three dimensions using a tight-binding model.