nLab topological insulator

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Context

Solid state physics

Topological physics

Contents

Idea

A topological insulator (TI) is a quantum material in a topological phase of matter where:

  1. as in an ordinary insulator, an energy gap separates the filled valence bands from the unocuupied conduction bands of electron energies, but

  2. in addition the topological class of the valence bundle, as a vector bundle over the Brillouin torus, is non-trivial (at the given level of resolution, hence at least in some unstable classification of topological phases or, more coarsely, in the K-theory classification of topological phases of matter ).

The second condition means that a topological insulator locally (over the Brillouin torus of crystal momenta) looks like an ordinary insulator, but “globally” it is qualitatively different, in that no (small enough) continuous deformation of the parameters (couplings) of the material can turn a topological trivial insulator into a non-trivial one, nor one class of topological insulators into another (whence these are all distinct “topological phases of matter”).

Of course, real samples of quantum materials are not ideal crystals in that they typically have boundaries. But in fact, the most characteristic phenomenological property of topological insulators is thought to be that at their sample boundaries, the energy gap closes (so that there the valence bundle is no longer a well-defined subbundle of the Bloch bundle there), making their boundary a conductor in which characteristic “edge modes” of electron currents propagate (“bulk-boundary correspondence”).

Moreover, the Bloch Hamiltonians of real crystals and their accessible deformations are typically constrained to respect some of the point group symmetries GG of the space group of crystal symmetries (in addition to possible PCT symmetries such as time reversal symmetry), in which case one speaks of topological crystalline insulators, for emphasis. The non-triviality of their valence bundle is then a class in GG-equivariant homotopy theory, for instance in GG-equivariant K-theory. If the topological phase of a topological crystalline insulator is non-trivial as long as such point group symmetries are respected by all of its deformation, but would be trivial otherwise, then one speaks of a symmetry protected topological phase.

In experimental practice, topological insulators are often engineered by starting with a topological semimetal — where a bulk energy gap does close over higher codimension submanifolds of the Brillouin torus (“band nodes”) — and then deforming it to make the Bloch Hamiltonian pick up “mass terms” which “open the gap” at the band nodes (see there).

Examples

References

General

The term “topological insulator” originates with:

Reviews:

Monographs:

See also:

Via effective field theory:

With focus on the case protected by crystallographic group-symmetry:

Via coarse topology:

Review in the more general context of topological phases of matter

See also:

  • Liang Fu, Charles L. Kane, Topological insulators with inversion symmetry, Physical Review B 76 (4): 045302. arXiv:cond-mat/0611341 doi;

  • Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100: 096407, arXiv:0707.1692 doi

  • Jeffrey C. Y. Teo, Liang Fu, Charles L. Kane, Surface states and topological invariants in three-dimensional topological insulators: Application to Bi 1−xSb xBi_{1-x}Sb_x, Phys. Rev. B 78, 045426 (2008) doi

  • J. Kellendonk, On the C *C^\ast-algebraic approach to topological phases for insulators, arxiv/1509.06271

  • A. Kitaev, Periodic table for topological insulators and superconductors. (Advances in Theoretical Physics: Landau Memorial Conference) AIP Conference Proceedings 1134, 22-30 (2009).

Interacting topological insulators:

The topological insulator in 2D exhibiting a quantum spin Hall effect has been first proposed in

Comment by X.-G. Wen: In fact, none of the above materials have quantum spin Hall effect since the spin is not conserved due to the spin-orbital interaction that makes those materials non trivial.

  • Ricardo Kennedy, Charles Guggenheim, Homotopy theory of strong and weak topological insulators, arxiv/1409.2529

  • L. Wu et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator, Science (2016). doi

Discussion via AdS/CFT in solid state physics:

  • Georgios Linardopoulos, Modelling Topological Materials with D-branes, INPP Annual Meeting 2017 (pdf, pdf)

Higher order topological insulators (with protected corner-modes beyond the edge-modes):

Experimental realization

  • Pascal Gehring, Hadj M. Benia, Ye Weng, Robert Dinnebier, Christian R. Ast, Marko Burghard, Klaus Kern: A Natural Topological Insulator, Nano Letters 13 (2013) 1179 [arXiv:1311.6637, doi:10.1021/nl304583m3.]

  • Bo Song, Long Zhang, Chengdong He, Ting Fung Jeffrey Poon, Elnur Hajiyev, Shanchao Zhang, Xiong-Jun Liu, Gyu-Boong Jo, Observation of symmetry-protected topological band with ultracold fermions, Science Advances 4 eaao4748 (2018) [doi:10.1126/sciadv.aao4748&rbrackl

External manipulation of topological phases via strain (see also the references here at graphene):

Interacting TIs

Discussion of topological insulators with non-negligible interactions:

  • AtMa P. O. Chan, Thomas Kvorning, Shinsei Ryu, and Eduardo Fradkin, Effective hydrodynamic field theory and condensation picture of topological insulators, Phys. Rev. B 93 155122 (2016) [[doi:10.1103/PhysRevB.93.155122]]

  • Benjamin Moy, Hart Goldman, Ramanjit Sohal, Eduardo Fradkin, Theory of oblique topological insulators [[arXiv:2206.07725]]

category: physics

Last revised on March 24, 2026 at 10:29:27. See the history of this page for a list of all contributions to it.

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