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...

It is well known that the topological phenomena with fractional excitations, the fractional quantum Hall effect, will emerge when electrons move in Landau levels. Here we show the theoretical discovery of the fractional quantum Hall effect in the absence of Landau levels in an interacting fermion model. The non-interacting part of our Hamiltonian is the recently proposed topologically non-trivial flat-band model on a checkerboard lattice. In the presence of nearest-neighbouring repulsion, we find that at 1/3 filling, the Fermi-liquid state is unstable towards the fractional quantum Hall effect. At 1/5 filling, however, a next-nearest-neighbouring repulsion is needed for the occurrence of the 1/5 fractional quantum Hall effect when nearest-neighbouring repulsion is not too strong. We demonstrate the characteristic features of these novel states and determine the corresponding phase diagram.

Quantum spin Hall (QSH) state of matter is usually considered to be protected by time-reversal (TR) symmetry. We investigate the fate of the QSH effect in the presence of the Rashba spin-orbit coupling and an exchange field, which break both inversion and TR symmetries. It is found that the QSH state characterized by nonzero spin Chern numbers C± = ±1 persists when the TR symmetry is broken. A topological phase transition from the TR symmetry-broken QSH phase to a quantum anomalous Hall phase occurs at a critical exchange field, where the bulk band gap just closes. It is also shown that the transition from the TR-symmetry-broken QSH phase to an ordinary insulator state can not happen without closing the band gap. The quantum spin Hall (QSH) effect is a new topologically ordered electronic state, which occurs in insulators without a magnetic field.[1] A QSH state of matter has a bulk energy gap separating the valence and conduction bands, and a pair of gapless spin filtered edge states on the boundary. The currents carried by the edge states are dissipationless due to the protection of time reversal (TR) symmetry and immune to nonmagnetic scattering. The QSH effect was first predicted in two-dimensional (2D) models [2,3]. It was experimentally confirmed soon after, not in graphene sheets [2] but in mercury telluride (HgTe) quantum wells [3,4].Graphene hosts an interesting electronic system. Its conduction and valence bands meet at two inequivalent Dirac points. Kane and Mele proposed that the intrinsic spin-orbit coupling (SOC) would open a small band gap in the bulk and also establish spin filtered edge states that cross inside the band gap, giving rise to the QSH effect [2]. The gapless edge states in the QSH systems persist even when the electron spinŝ z conservation is destroyed in the system, e.g., by the Rashba SOC, and are robust against weak electron-electron interactions and disorder [2,5]. While the SOC strength may be too weak in pure graphene system, the Kane and Mele model captures the essential physics of a class of insulators with nontrivial band topology [6,7]. A central issue relating to the QSH effect is how to describe the topological nature of the systems. A Z 2 topological index was introduced to classify TR invariant systems [8], and a spin Chern number was also suggested to characterize the topological order [5]. The spin Chern number was originally introduced in finite-size systems by imposing spin-dependent boundary conditions [5]. Recently, based upon the noncommutative theory of Chern number [9], Prodan [10] redefined the spin Chern number in the thermodynamic limit through band projection without using any boundary conditions. It has been shown that the Z 2 invariant and spin Chern number yield equivalent description for TR invariant systems [10][11][12].The QSH effect is considered to be closely related to the TR symmetry that provides a protection for the edge states and the Z 2 invariant. An open question is whether or not we can have QSH-like phase in a system where the TR symmet...

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