The Quantum Spin Hall Effect: Theory and Experiment

König, Markus; Buhmann, H.; Molenkamp, L. W.; Hughes, Taylor L.; Liu, Chao Xing; Qi, Xiao Liang; Zhang, Shou Cheng

doi:10.1143/jpsj.77.031007

Cited by 812 publications

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“…The bulk boundary correspondence relates these modes to a Z 2 topological invariant characterizing time-reversal invariant Bloch Hamiltonians. Signatures of these protected boundary modes have been observed in transport experiments on 2D HgCdTe quantum wells [14][15][16] and in photoemission and scanning tunnel microscope experiments on three-dimensional ͑3D͒ crystals of Bi 1−x Sb x , [17][18][19] Bi 2 Se 3 , 20 Bi 2 Te 3 , 22,23,25 and Sb 2 Te 3 . 26 Topological insulator behavior has also been predicted in other classes of materials with strong spin-orbit interactions.…”

Section: Introductionmentioning

confidence: 85%

Topological defects and gapless modes in insulators and superconductors

2010

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We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians.We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes. We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H͑k , r͒ that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes. Disciplines Physical Sciences and Mathematics | Physics

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Section: Introductionmentioning

confidence: 85%

Topological defects and gapless modes in insulators and superconductors

2010

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“…Nevertheless, all calculations that acknowledge the atomistic symmetry -as opposed to a continuum view -do give large anticrossing at the critical QW thickness. They both predict the gap of about 15 meV far exceeding the estimate of a few meV due solely to the bulk inversion asymmetry [18,19] and unambiguously indicating that the subband mixing is dominated by the interface contribution, in a crutial difference with the naive k·p model. Moreover, despite the existence of an additional inter- face bands in the pseudopotential calculation, the both models predict the same dispersion of the Dirac states formed from the E1 and HH1 subbands near the anticrossing point.…”

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confidence: 91%

“…In the k·p model, it is given by A = (P/ √ 2) f 1 (z)f 3 (z)dz +δA, where P is the Kane matrix element and δA stands for the contributions from remote bands. Finally, γ describes the coupling of E1 and HH1 states at k = 0 in zinc-blende-lattice QWs [18,19], 2|γ| ≈ 15 meV as it follows from Figs. 2 and 3 neglecting the influence of additional interface states.…”

mentioning

confidence: 99%

“…Therefore, the coupling matrix element between E1 and HH1 is nonzero, and these subbands must anticross rather than cross at zero in-plane wave vector k. As pointed out in Refs. [17][18][19][20] in the case of HgTe/CdTe QWs a reduction in symmetry leads to the splitting of states at k = 0. However, a theory identifying the source of the splitting (bulk vs inter-face) and predicting its magnitude has been lacking.…”

mentioning

confidence: 99%

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Split Dirac cones in HgTe/CdTe quantum wells due to symmetry-enforced level anticrossing at interfaces

Tarasenko¹,

Durnev²,

Nestoklon³

et al. 2015

Phys. Rev. B

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We describe the fine structure of Dirac states in HgTe/CdHgTe quantum wells of critical and close-to-critical thickness and demonstrate the formation of an anticrossing gap between the tips of the Dirac cones driven by interface inversion asymmetry. By combining symmetry analysis, atomistic calculations, and k•p theory with interface terms, we obtain a quantitative description of the energy spectrum and extract the interface mixing coefficient. The zero-magnetic-field splitting of Dirac cones can be experimentally revealed in studying magnetotransport phenomena, cyclotron resonance, Raman scattering, or THz radiation absorption.

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“…Finally, we realized an explicit construction of the edge states, following the methods discussed in [18] and [12]. In order to simplify the problem, we chose to work with a unit cell containing four sites (forming a square) rather than two.…”

mentioning

confidence: 99%

Z 2 antiferromagnetic topological insulators with broken C 4 symmetry

2017

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A two-dimensional topological insulator may arise in a centrosymmetric commensurate Néel antiferromagnet (AF), where staggered magnetization breaks both the elementary translation and time reversal, but retains their product as a symmetry. Fang et al., [Phys. Rev. B 88, 085406 (2013)] proposed an expression for a Z2 topological invariant to characterize such systems. Here, we show that this expression does not allow to detect all the existing phases if a certain lattice symmetry is lacking. We implement numerical techniques to diagnose topological phases of a toy Hamiltonian, and verify our results by computing the Chern numbers of degenerate bands, and also by explicitly constructing the edge states, thus illustrating the efficiency of the method.Physical phenomena, whose description involves topology, have been invariably attracting attention regardless of whether the word "topology" was actually used at the time: early examples involve topologically non-trivial stable defects such as dislocations in crystals, as well as vortices in superconductors and superfluids. Quantum Hall Effect and its remarkably precise conductance quantization [1] marked the advent of an entirely new class of phenomena, related not so much to the appearance in the sample of finite-size topological objects, but rather to the electron state of the entire sample changing its topology in a way, that could no longer be undone by a local perturbation. More recently, it was understood that, in fact, non-trivial topology may appear even in zero magnetic field: the fact that a commonplace band insulator may find itself in distinct electron states that cannot be continuously transformed one into another without a phase transition, came as a major surprise [2,3].These phenomena invite the question of classifying topologically distinct states of matter: labeling each state by a set of discrete indices in such a way as to have different sets for any two phases that cannot be continuously transformed one into another without the system undergoing a phase transition. In the general setting, the problem remains unsolved.In fact, open questions are present even in a noninteracting description of systems that are believed to admit a Z 2 ("even-odd") classification, and thus have only one topologically trivial and one topologically non-trivial phase, commonly called topological. Here, we address one such question, that has recently attracted attention: diagnosing the topological phase of a Z 2 insulating Néel antiferromagnet.To put the subsequent presentation in context, we recapitulate the key results for the prototypical Z 2 system: a paramagnetic topological insulator. Fu and Kane [4] have shown that the Z 2 invariant for such a system can be defined via the so-called sewing matrix w(k) mn :where the |Ψ n,k is the Bloch eigenstate of the n-th band at momentum k. The w(k) mn turns out to be of particular interest at special momenta Γ i such that −Γ i = Γ i + G, with G a reciprocal lattice vector. Such Γ i are now commonly called the "time reversal-i...

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The Quantum Spin Hall Effect: Theory and Experiment

Cited by 812 publications

References 43 publications

Topological defects and gapless modes in insulators and superconductors

Topological defects and gapless modes in insulators and superconductors

Split Dirac cones in HgTe/CdTe quantum wells due to symmetry-enforced level anticrossing at interfaces

Z 2 antiferromagnetic topological insulators with broken C 4 symmetry

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