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

T. Fukui·Y. Hatsugai
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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.

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T. Fukui, Y. Hatsugai. 2006-07-19. Topological aspects of quantum spin Hall effect in graphene: Z$_2$ topological order and spin Chern number. https://doi.org/10.1103/physrevb.75.121403

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