Quantum Spin Hall Effect and Topologically Invariant Chern Numbers

TL;DR

This paper provides a topological framework for understanding the quantum spin Hall effect in a honeycomb lattice, characterizing phases via Chern numbers and analyzing edge states and phase diagrams.

Contribution

It introduces a Chern number matrix to classify QSHE phases and derives a conserved spin Chern number even with disorder and Rashba coupling.

Findings

Nonzero diagonal Chern number matrix elements indicate QSHE phases

Spin Chern number remains conserved despite disorder and Rashba coupling

Numerical phase diagram of QSHE states is established

Abstract

We present a topological description of quantum spin Hall effect (QSHE) in a two-dimensional electron system on honeycomb lattice with both intrinsic and Rashba spin-orbit couplings. We show that the topology of the band insulator can be characterized by a $2\times 2$ traceless matrix of first Chern integers. The nontrivial QSHE phase is identified by the nonzero diagonal matrix elements of the Chern number matrix (CNM). A spin Chern number is derived from the CNM, which is conserved in the presence of finite disorder scattering and spin nonconserving Rashba coupling. By using the Laughlin's gedanken experiment, we numerically calculate the spin polarization and spin transfer rate of the conducting edge states, and determine a phase diagram for the QSHE.