Quantum Spin-Hall Effect and Topologically Invariant Chern Numbers

Abstract

We present a topological description of the quantum spin-Hall effect (QSHE) in a two-dimensional electron system on a 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 2 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 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. DOI: 10.1103/PhysRevLett.97.036808 PACS numbers: 73.43.Nq, 11.15.ÿq, 72.25.ÿb Topological quantities are fundamentally important in characterizing the transverse electrical transport property in integer and fractional quantum Hall effect states [1,2] of two-dimensional (2D) electron systems. It was first revealed by Thouless et. al. [3] that each integer quantum Hall effect (IQHE) state is associated with a topologically invariant integer known as the first Chern number, which precisely equals the Hall conductance in units of e 2 =h. The exact quantization of the Hall conductance can also be formulated in terms of a 2D band-structure Berry phase [3][4][5][6], which remains an integral invariant till the band energy gap (or the mobility gap [7] in the presence of disorder) collapses.While the conventional IQHE is usually associated with strong magnetic fields, Haldane [8] has explicitly shown that it can actually occur in the absence of magnetic field in band insulators with graphenelike band structure. The onecomponent Haldane model explicitly breaks time-reversal symmetry, resulting in a condensed-matter realization of a parity symmetry anomaly with chiral edge states at the boundary of the sample. In realistic electron systems, however, the coupled spin degrees of freedom can recover the time-reversal symmetry by forming Kramers degenerate states, which belong to the universality class of zero charge Chern number as the total Berry curvature of the occupied energy band of both spins sums to zero.This class of insulators has been recently found [9,10] to possess a dissipationless quantum spin-Hall effect (QSHE) [11], which is distinct from the intrinsic spin-Hall effect in the metallic systems [12]. The QSHE has been shown to be robust against disorder scattering and other perturbation effects [9,10]. Whether there exists an underlying topological invariant ''protecting'' the QSHE is a very important issue for both fundamental understanding and potential applications of the QSHE. While the previously proposed [9,13] Z 2 classification of the QSHE suggests that the conducting edge states are protected by timereversal symmetry, it does not distinguish between two QSHE states with spin-Hall conductance (SHC) of opposite signs. Thus it remains an open issue if the QSHE state...

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