$Z_2$ Topological Order and the Quantum Spin Hall Effect

TL;DR

This paper introduces a new $Z_2$ topological invariant that characterizes the quantum spin Hall phase, distinguishing it from ordinary insulators and linking it to topological order and edge state transport.

Contribution

It defines the $Z_2$ topological invariant for time reversal invariant systems and generalizes the formalism beyond simple models like graphene.

Findings

Identifies the $Z_2$ invariant as a key to classifying quantum spin Hall phases.

Establishes the $Z_2$ order in a two-band graphene model.

Proposes a framework applicable to multi-band and interacting systems.

Abstract

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel $Z_2$ topological invariant, which distinguishes it from an ordinary insulator. The $Z_2$ classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the $Z_2$ order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multi band and interacting systems.