Classification of Topological Insulators and Superconductors

Schnyder, Andreas P.; Ryu, Shinsei; Furusaki, Akira; Ludwig, Andreas

doi:10.1063/1.3149481

Cited by 369 publications

(483 citation statements)

References 6 publications

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“…Here, we caution that T and U are symmetries of the circle ðk y ∈ ½−π; π ; k ∥ Þ and preserve the momentum parameter k ∥ ; this differs from the conventional [23] time-reversal and particle-hole symmetries that typically invert momentum. To precisely state (iii), we first elaborate on how Wilson bands may be labeled by mirror eigenvalues, which we define as λ j for the reflectionM j (j ∈ fx; zg).…”

Section: A Local Rules Of the Curvesmentioning

confidence: 99%

“…The point group (D 6h ) of KHgX, defined as the quotient of its space group by translations, is generated by four spatial transformations-this typifies the complexity of most space groups. This work describes a systematic method to topologically classify space groups with similar complexity; in contrast, previous classifications [1,2,[4][5][6][7]14] (with one exception by us [8]) have expanded the Altland-Zirnbauer symmetry classes [22,23] to include only a single point-group generator. For point groups with multiple generators, different submanifolds of the Brillouin torus are invariant under different symmetries; e.g., mirror and glide planes are respectively mapped to themselves by a symmorphic reflection and a glide reflection, as illustrated in Fig.…”

Section: Introductionmentioning

confidence: 99%

“…(23). Two extensions are equivalent if there exists a group isomorphism between them; the second group cohomology [H 2 ðG ∘ ; N Þ, as further elaborated in Appendix D 2] classifies the isomorphism classes of all extensions of G ∘ by N , of which GX is one example.…”

Section: Wilsonian Time-reversal Operatormentioning

confidence: 99%

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Topological Insulators from Group Cohomology

2016

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We classify insulators by generalized symmetries that combine space-time transformations with quasimomentum translations. Our group-cohomological classification generalizes the nonsymmorphic space groups, which extend point groups by real-space translations; i.e., nonsymmorphic symmetries unavoidably translate the spatial origin by a fraction of the lattice period. Here, we further extend nonsymmorphic groups by reciprocal translations, thus placing real and quasimomentum space on equal footing. We propose that group cohomology provides a symmetry-based classification of quasimomentum manifolds, which in turn determines the band topology. In this sense, cohomology underlies band topology. Our claim is exemplified by the first theory of time-reversal-invariant insulators with nonsymmorphic spatial symmetries. These insulators may be described as "piecewise topological," in the sense that subtopologies describe the different high-symmetry submanifolds of the Brillouin zone, and the various subtopologies must be pieced together to form a globally consistent topology. The subtopologies that we discover include a glide-symmetric analog of the quantum spin Hall effect, an hourglass-flow topology (exemplified by our recently proposed KHgSb material class), and quantized non-Abelian polarizations. Our cohomological classification results in an atypical bulk-boundary correspondence for our topological insulators.

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Section: A Local Rules Of the Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological Insulators from Group Cohomology

2016

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“…Fermionic SPT states include the familiar quantum spin Hall (QSH) insulator [3,4], the three-dimensional (3d) topological insulator (TI) [5][6][7], and topological superconductors. Noninteracting fermionic SPT states have been fully classified and understood [8,9]. Unlike fermionic systems, bosonic SPT (BSPT) states require strong interaction to overcome the tendency to form Bose-Einstein condensates.…”

mentioning

confidence: 99%

Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

Zhang

You

et al. 2017

Phys. Rev. Lett.

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Bosonic symmetry protected topological (BSPT) states, the bosonic analogue of topological insulators, have attracted enormous theoretical interest in the last few years. Although BSPT states have been classified by various approaches, there is so far no successful experimental realization of any BSPT state in two or higher dimensions. In this paper, we propose that a two dimensional BSPT state with U (1) × U (1) symmetry can be realized in bilayer graphene in a magnetic field.Here the two U (1) symmetries represent total spin S z and total charge conservation respectively. The Coulomb interaction plays a central role in this proposal -it gaps out all the fermions at the boundary, so that only bosonic charge and spin degrees of freedom are gapless and protected at the edge. Based on the bosonic nature of the boundary states, we derive the bulk wave function for the bosonic charge and spin degrees of freedom, which takes exactly the same form as the desired BSPT state. We also propose that the bulk quantum phase transition between the BSPT and trivial phase, could become a "bosonic phase transition" with interactions. That is, only bosonic modes close their gap at the transition, which is fundamentally different from all the well-known topological insulator to trivial insulator transitions which occur for free fermion systems. We discuss various experimental consequences of this proposal.

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“…They have been the subject of theories for more than seven decades without an experimental signature. However recently it has been suggested that these exotic particles can be discovered in condensed matters systems, in particular in topological superconductors (TSCs) [1][2][3][4][5][6][7][8] . TSCs are superconducting systems that admit exceptional boundary states -owing to nontrivial topology associated with bulk spectra-inside the quasi-particle excitation gaps.…”

Section: Introductionmentioning

confidence: 99%

Scattering theory of chiral Majorana fermion interferometry

Li¹,

Fleury²,

Büttiker³

2012

Phys. Rev. B

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Using scattering theory, we investigate interferometers composed of chiral Majorana fermion modes coupled to normal metal leads. We advance an approach in which also the basis states in the normal leads are written in terms of Majorana modes. Thus each pair of electron-hole states is associated with a pair of Majorana modes. Only one lead Majorana mode couples to the intrinsic Majorana mode whereas its partner is completely reflected. Similarly the remaining Majorana modes are completely reflected but in general mix pair-wise. We demonstrate that the charge current can also be expressed in terms of interference between pairs of Majorana modes. These two basic facts permit a treatment and understanding of current and noise signatures of chiral Majorana fermion interferometry in an especially elegant way. As a particular example of applications, in FabryPerot-type interferometers where chiral Majorana modes form loops, resonances (anti-resonances) from such loops always lead to peaked (suppressed) Andreev differential conductances, and negative (positive) cross-correlations that originate purely from two-Majorana-fermion exchange. These investigations are intimately related to current and noise signatures of Majorana bound states.

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Classification of Topological Insulators and Superconductors

Cited by 369 publications

References 6 publications

Topological Insulators from Group Cohomology

Topological Insulators from Group Cohomology

Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

Scattering theory of chiral Majorana fermion interferometry

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