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

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.

arXiv comment5 pages, 3 figures

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