Characterization of 3d topological insulators by 2d invariants

TL;DR

This paper presents a method to determine 3D topological insulator invariants by analyzing 2D slices in momentum space using a Chern number-based formula, linking it to spin Hall conductance quantization.

Contribution

It introduces a novel approach to compute 3D $Z_{2}$ invariants through 2D deformations and a Chern number formula, clarifying their physical interpretation.

Findings

The method successfully characterizes 3D topological phases.

The $Z_{2}$ invariant relates to 2D spin Hall conductance quantization.

The approach simplifies the analysis of 3D topological insulators.

Abstract

The prediction of non-trivial topological phases in Bloch insulators in three dimensions has recently been experimentally verified. Here, I provide a picture for obtaining the $Z_{2}$ invariants for a three dimensional topological insulator by deforming suitable 2d planes in momentum space and by using a formula for the 2d $Z_{2}$ invariant based on the Chern number. The physical interpretation of this formula is also clarified through the connection between this formulation of the $Z_{2}$ invariant and the quantization of spin Hall conductance in two dimensions.