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Cited by 3,322Topological Materials and PhenomenaAdvanced Condensed Matter PhysicsQuantum many-body systems

Classification of topological insulators and superconductors in three spatial dimensions

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Schnyder, Andreas P., Ryu, Shinsei, Furusaki, Akira, Ludwig, Andreas W. W.·2008·Physical Review B
DOI 10.1103/physrevb.78.195125
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Abstract

We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced ${\mathbb{Z}}_{2}$ topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. In these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a two-dimensional surface, they support a number (which may be an arbitrary nonvanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin-rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral $({p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y})$-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to ``weak pairing'') and topologically trivial (analogous to ``strong pairing'') 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.

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  • In the odd-parity case, the material will be the first known realization of a topological superconductor.

    “In the second, odd-parity case, the material will also be the first known realization of a new state of matter specifically termed a 'topological superconductor' 9,11,20 .”

    From Observation of topological order in a superconducting doped topological insulator (2010) · cites this paper as [11]

  • In contrast to graphene, the remarkable property of topological edge states is that their dispersion is characterized by an odd number of nondegenerate Dirac cones.

    “In contrast to graphene, which has four Dirac cones (2 doubly degenerate cones at the K and K 0 points in momentum space) [9], the remarkable property of topological edge states is that their dispersion is characterized by an odd number of nondegenerate Dirac cones.”

    From Observation of Time-Reversal-Protected Single-Dirac-Cone Topological-Insulator States in Bi2Te3 and Sb2Te3 (2009) · cites this paper as [11]

  • Experimental confirmation that θ=π may be easier in topological antiferromagnets than in conventional topological insulators, using techniques previously used to extract θ in Cr2O3.

    “In the topological antiferromagnets, some surfaces have a gapped state just from the material's own time-reversal-breaking, which suggests that experimental confirmation that θ = π , which has not yet occurred, may be easier in these materials, using the same techniques previously used to extract θ in Cr 2 O 3 .”

    From Antiferromagnetic topological insulators (2010) · cites this paper as [18]

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Research works printed but not linked: 5

  1. 85 Alexei Kitaev, in the proceedings of the Mesoscopic And Strongly Correlated Electron Systems conference (916 July 2000, Chernogolovka, Moscow Region, Russia), arXiv:cond-mat/0010440 . · arXiv
  2. 95 M. V. Berry, Proc. R. Soc. London Ser. A 392 , 45 (1984).
  3. 105 See for example, Gerald V. Dunne, Les Houches Lectures 1998, arXiv:hep-th/9902115 . · arXiv
  4. 112 P. W. Brouwer, A. Furusaki, C. Mudry, and S. Ryu, BUTSURI 60 , 935 (2005); arXiv:cond-mat/0511622 . · arXiv
  5. 115 D. A. Ivanov, arXiv:cond-mat/9911147 ; D. A. Ivanov, J. Math. Phys. 43 , 126 (2002). · arXiv

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Authors
Schnyder, A. P., Ryu, S., Furusaki, A., & Ludwig, A. W. W.
Affiliations
  • University of California, Santa Barbara (US)
  • RIKEN (JP)
ORCID
Venue
Physical Review B
Publisher
American Physical Society (APS)
Published
2008-11-26
Volume
78
Issue
19
ISSN
  • 1098-0121 (ISSN-L)
  • 1550-235X
DOI
10.1103/physrevb.78.195125
OpenAlex
W2003397144
Cited by
3,322
References
114
Citations by year
2026: 119 · 2025: 189 · 2024: 198 · 2023: 213 · 2022: 226 · 2021: 232 · 2020: 306 · 2019: 254 · 2018: 234 · 2017: 260 · 2016: 216 · 2015: 206 · 2014: 188 · 2013: 178 · 2012: 128
FWCI
33.22
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review B (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
topological insulators, topological superconductors, three-dimensional topological phases, Altland-Zirnbauer symmetry classes, Z2 topological insulator, winding number, surface modes, Majorana fermions, Dirac fermions, Anderson localization, U(1) gauge field

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