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

摘要

For generic time-reversal invariant systems with spin-orbit couplings, weclarify a close relationship between the Z$_2$ topological order and the spinChern 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 gaugetransformation connects different spin Chern numbers (even integers) modulo 4,which implies that the spin Chern number and the Z$_2$ topological order yieldthe same classification. We present a method of computing spin Chern numbersand demonstrate it in single and double plane of graphene.