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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The spin Hall effect in a two-dimensional electron system on honeycomb lattice with both intrinsic and Rashba spin-orbit couplings is studied numerically. Integer quantized spin Hall conductance is obtained at the zero Rashba coupling limit when electron Fermi energy lies in the energy gap created by the intrinsic spin-orbit coupling, in agreement with recent theoretical prediction. While nonzero Rashba coupling destroys electron spin conservation, the spin Hall conductance is found to remain near the quantized value, being insensitive to disorder scattering, until the energy gap collapses with increasing the Rashba coupling. We further show that the charge transport through counterpropagating spin-polarized edge channels is well quantized, which is associated with a topological invariant of the system. DOI: 10.1103/PhysRevLett.95.136602 PACS numbers: 72.10.2d, 71.70.Ej, 72.25.2b, 73.43.Cd The proposals of intrinsic spin Hall effect (SHE) in a Luttinger spin-orbit (SO) coupled three-dimensional p-doped semiconductor [1] and in a Rashba SO coupled two-dimensional electron system (2DES) [2] have stimulated many subsequent research activities [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The SHE may potentially provide a purely electrical means to manipulate electron spins without use of ferromagnetic materials or a magnetic field. The SHE in these systems is dissipative because of nonzero longitudinal conductance [9] and exhibits nonuniversal behavior in the presence of disorder [5,6,[10][11][12][13][14][15][16][17], which is naturally distinct from the conventional integer quantum Hall effect (IQHE). In particular, it is found [5,6,16] that the bulk SHE in the twodimensional Rashba model may be destroyed by any weak disorder in infinite samples. It is of both fundamental and practical interest to search for nondissipative SHE with universal properties similar to the IQHE, in light of the fact that IQHE can exist in nature in the absence of magnetic field, as first predicted by Haldane [19].A class of band insulators with SO coupling was suggested as possible candidates for nondissipative SHE [9]. Interestingly, Kane and Mele proposed [20] that the intrinsic SO coupling in single-layer graphene films may give rise to an integer quantized SHE (IQSHE). The intrinsic SO coupling conserves electron spin s z . The independent subsystems of two spin directions " and # are each equivalent to Haldene's spinless IQHE model [19] on honeycomb lattice without magnetic field. They contribute quantized Hall conductances e 2 =h and ÿe 2 =h, respectively, when the electron Fermi energy lies inside the energy gap created by the SO coupling. While the charge Hall conductances cancel out, the total spin Hall conductance (SHC) is quantized to sH 2 in units of e=4. We recall that each subsystem can be classified by an integer Chern number [19], which equals the Hall conductance of the subsystem in units of e 2 =h, and is conserved without spin-mixing interactions. Upon coupling the two subsystems, only the total Cher...

Based upon Raman spin-lattice interaction, we propose a theoretical model for the phonon Hall effect in paramagnetic dielectrics, which was discovered recently in an experiment [C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phys. Rev. Lett. 95, 155901 (2005).]. The phonon Hall effect is revealed to be a phonon analogue to the anomalous Hall effect in electron systems. The thermal Hall conductivity is calculated by using the Kubo formula. Our theory reproduces the essential experimental features of the phonon Hall effect, including the sign, magnitude, and linear magnetic field dependence of the thermal Hall conductivity. DOI: 10.1103/PhysRevLett.96.155901 PACS numbers: 66.70.+f, 72.10.Bg, 72.15.Gd, 72.20.Pa When an electrical current flows through a conductor with direction perpendicular to an applied magnetic field, a transverse electrical current may be generated in the third perpendicular direction. This is well known as the Hall effect, and is due to the electromagnetic Lorentz force on the charge carriers. Two interesting variants of the conventional Hall effect are the anomalous Hall effect (AHE) in ferromagnets [1] and the spin Hall effect in nonmagnetic conductors [2], where the electron spin-orbit coupling plays an essential role. The AHE is characterized by an anomalous contribution to the Hall resistivity proportional only to the magnetization. Intuitively, one would not expect a Hall effect for phonons, which do not carry charges and do not couple to the magnetic field directly. Remarkably, by applying a magnetic field perpendicular to a heat current flowing through a sample of the paramagnetic dielectric Tb 3 Ga 5 O 12 , Strohm, Rikken, and Wyder observed very recently a temperature difference of up to 200 K between the sample edges in the third perpendicular direction [3]. The temperature difference is attributed to the phonon Hall effect (PHE) [3], which becomes another intriguing and puzzling fundamental phenomenon in solid state physics.At the experimental low temperature 5.45 K [3], excitation of optical phonons is unlikely, and thermal conduction should be carried by acoustic phonons. While Tb 3 Ga 5 O 12 is an ionic material, in a perfect lattice, each unit cell is charge neutral. In the acoustic phonon modes, each unit cell vibrates as a rigid object without relative displacements between its constituting atoms [4], and does not acquire a net Lorentz force in a magnetic field. Theoretical understanding of the physical mechanism underlying the PHE is highly desirable.In this Letter, we propose a theoretical model based upon the Raman spin-lattice interaction for the PHE. The PHE is discussed to be a phonon analogue to the AHE. The thermal Hall conductivity of the phonons in the clean limit is calculated by using the Kubo formula. The theory can explain the essential features of the experimental data for Tb 3 Ga 5 O 12 , including the sign, magnitude, and linear magnetic field dependence of the thermal Hall conductivity.We consider a sample of a paramagnetic dielectric with volume V , which...

Using the four-terminal Landauer-Büttiker formula and Green's function approach, we calculate numerically the spin-Hall conductance in a two-dimensional junction system with the Rashba spinorbit (SO) coupling and disorder. We find that the spin-Hall conductance can be much greater or smaller than the universal value e/8π, depending on the magnitude of the SO coupling, the electron Fermi energy and the disorder strength. The spin-Hall conductance does not vanish with increasing sample size for a wide range of disorder strength. Our numerical calculation reveals that a nonzero SO coupling can induce electron delocalization for disorder strength smaller than a critical value, and the nonvanishing spin-Hall effect appears mainly in the metallic regime.PACS numbers: 72.15.Gd, 71.70.Ej, 72.15.Rn The emerging field of spintronics, [1,2] which is aimed at exquisite control over the transport of electron spins in solid-state systems, has attracted much recent interest. One central issue in the field is how to effectively generate spin-polarized currents in paramagnetic semiconductors. In the past several years, many works [1,2,3,4,5] have been devoted to the study of injection of spinpolarized charge flows into the nonmagnetic semiconductors from ferromagnetic metals. Recent discovery of intrinsic spin-Hall effect in p-doped semiconductors by Murakami et al. [6] and in Rashba spin-orbit (SO) coupled two-dimensional electron system (2DES) by Sinova et al. [7] may possibly lead to a new solution to the issue. For the Rashba SO coupling model, the spin-Hall conductivity is found to have a universal value e/8π in a clean bulk sample when the two Rashba bands are both occupied, being insensitive to the SO coupling strength and electron Fermi energy [7].While the spin-Hall effect has generated much interest in the research community, [8,9,10,11,12,13,14,15,16,17] theoretical works remain highly controversial regarding its fate in the presence of disorder. Within a semiclassical treatment of disorder scattering, Burkov et al.[10] and Schliemann and Loss [11] showed that spinHall effect only survives at weak disorder. On the other hand, Inoue et al. [14] pointed out that the spin-Hall effect vanishes even for weak disorder taking into account the vertex corrections. Mishchenko et al. [15] further showed that the dc spin-Hall current vanishes in an impure bulk sample, but may exist near the boundary of a finite system. Nomura et al.[16] evaluated the Kubo formula by calculating the single-particle eigenstates in momentum space with finite momentum cutoff, and found that the spin-Hall effect does not decrease with sample size at rather weak disorder. Therefore, further investigations of disorder effect in the SO coupled 2DES are highly desirable.In this Letter, the spin-Hall conductance (SHC) in a 2DES junction with the Rashba SO coupling is studied by using the four-terminal Landauer-Büttiker (LB) formula with the aid of the Green's functions. We find that the SHC does not take the universal value, and it depends critically ...

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