On the $Z_2$ classification of Quantum Spin Hall Models

Abstract

We propose an alternative formulation of the $Z_2$ topological index for quantum spin Hall systems and band insulators when time reversal invariance is not broken. The index is expressed in terms of the Chern numbers of the bands of the model, and a connection with the number of pairs of robust edge states is thus established. The alternative index is easy to compute in most cases of interest. We also discuss connections with the recently proposed spin Chern number for quantum spin Hall models.Comment: Presentation changed to improve clarity, some technical aspects of the topological arguments including material previously cited as unpublished notes have now been added as an appendi