\(Z_2\) Topological Order and the Quantum Spin Hall Effect

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.