Topological aspects of quantum spin Hall effect in graphene: Z$_2$ topological order and spin Chern number

TL;DR

This paper explores the relationship between Z2 topological order and spin Chern number in quantum spin Hall systems, demonstrating their equivalence in classifying topological phases in graphene.

Contribution

It clarifies the connection between Z2 topological order and spin Chern number, providing a method to compute the latter and applying it to graphene.

Findings

Spin Chern number and Z2 order classify topological phases equivalently.

Global gauge transformation relates different spin Chern numbers modulo 4.

Method demonstrated on single and double layer graphene.

Abstract

For generic time-reversal invariant systems with spin-orbit couplings, we clarify a close relationship between the Z$_2$ topological order and the spin Chern number proposed by Kane and Mele and by Sheng {\it et al.}, respectively, in the quantum spin Hall effect. It turns out that a global gauge transformation connects different spin Chern numbers (even integers) modulo 4, which implies that the spin Chern number and the Z$_2$ topological order yield the same classification. We present a method of computing spin Chern numbers and demonstrate it in single and double plane of graphene.