Time-Reversal-Symmetry-Broken Quantum Spin Hall Effect

Abstract

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...

Stabilization of the Quantum Spin Hall Effect by Designed Removal of Time-Reversal Symmetry of Edge States

“…So far, QSH effect as robust as the QH effect has been elusive. It was found recently that the nontrivial bulk band topology of the QSH systems remains intact, even when the TR symmetry is broken, [23] implying that the instability of the QSH effect is solely due to properties of the edge states.…”

“…With increasing g 0 to g 0 = |M 0 |, C ± undergo a transition from (−1, 1) to (0, 1), the latter corresponding to a QAH phase. [23,24] Next we consider a QSH sample with a strip geometry, as shown in Fig. 1.…”

Robust dual topological character with spin-valley polarization in a monolayer of the Dirac semimetal Na3Bi

“…C S provides an equivalent characterization to the Z 2 number in that for 2D time-reversal symmetric systems, the even values of C S correspond to a topologically trivial insulator state, while odd values of C S indicate the emergence of a TI phase [45][46][47]. C S is given by the difference of the Chern numbers for the spin-up (C + ) and spin-down (C − ) projected manifolds, C S = (C + − C − )/2 [45]. The σ z matrix, φ mk |σ z |φ nk , is constructed and diagonalized to distinguish the spin-up and spin-down manifolds [47].…”

“…To identify the relationship between the 2D TI and the odd number of band inversions in the Na 3 Bi monolayer, we calculate the spin Chern number C S [45][46][47], which can be directly related to the Z 2 topological invariant of the system. C S provides an equivalent characterization to the Z 2 number in that for 2D time-reversal symmetric systems, the even values of C S correspond to a topologically trivial insulator state, while odd values of C S indicate the emergence of a TI phase [45][46][47]. C S is given by the difference of the Chern numbers for the spin-up (C + ) and spin-down (C − ) projected manifolds, C S = (C + − C − )/2 [45].…”

Electronic interference transport and its electron–phonon interaction in the Sb-doped Bi2Se3nanoplates synthesized by a solvothermal method

“…This feature is consistent with the fact that the edge states in the QSH systems are gapless in the presence of the TR symmetry, and usually gapped otherwise. While the spin Chern numbers yield an equivalent description for TR-invariant systems, their robustness does not rely on any symmetries [27,28,29]. Nonzero spin Chern numbers guarantee that edge states emerge in the bulk band gap, which could be gapless or gapped, depending on the symmetry and local microscopic structures of the sample edges [30].…”