Topological winding properties of spin edge states in Kane-Mele graphene model

TL;DR

This paper investigates the topological winding properties of spin edge states in Kane-Mele graphene, revealing how the winding numbers distinguish the quantum spin-Hall phase and relate to Z2 topological invariance.

Contribution

It derives the Harper equation for spin edge states and links the topological winding numbers to the Z2 invariant in Kane-Mele graphene.

Findings

Two pairs of gapless spin-filtered edge states exist in the QSH phase.

Winding numbers of spin edge states characterize the topological phase.

The topological invariant is related to the difference in winding numbers.

Abstract

We study the spin edge states in the quantum spin-Hall (QSH) effect on a single-atomic layer graphene ribbon system with both intrinsic and Rashba spin-orbit couplings. The Harper equation for solving the energies of the spin edge states is derived. The results show that in the QSH phase, there are always two pairs of gapless spin-filtered edge states in the bulk energy gap, corresponding to two pairs of zero points of the Bloch function on the complex-energy Riemann surface (RS). The topological aspect of the QSH phase can be distinguished by the difference of the winding numbers of the spin edge states with different polarized directions cross the holes of the RS, which is equivalent to the Z2 topological invariance proposed by Kane and Mele [Phys. Rev. Lett. 95, 146802 (2005)].