Topological aspects of the quantum spin-Hall effect in graphene: Z2 topological order and spin Chern number

Abstract

For generic time-reversal-invariant systems with spin-orbit couplings, we clarify a close relationship between the Z2 topological order and the spin Chern number (SChN) in the quantum spin-Hall effect. It turns out that a global gauge transformation connects sectors with different SChNs (even integers) modulo 4, which implies that the SChN and Z2 topological orders yield the same classification. We present a method of computing the SChN and demonstrate it in single and double planes of graphene. © 2007 The American Physical Society.