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Periodic table for topological insulators and superconductors. (English) Zbl 1180.82221
Lebedev, Vladimir V. (ed.) et al., Advances in theoretical physics. Landau memorial conference, Chernogolokova, Russia, 22–26 June 2008. Papers based on the presentations at the international conference “Advances in theoretical physics”. Melville, NY: American Institute of Physics (AIP) (ISBN 978-0-7354-0671-1/pbk). AIP Conference Proceedings 1134, 22-30 (2009).
This paper deals with gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, \({\mathbb Z}\), or \({\mathbb Z}_2\). The author is not concerned with analytic formulas for topological numbers, but rather enumerate all possible phases. It is established that two Hamiltonians belong to the same phase if they can be continuously transformed one to the other while maintaining the energy gap or localization. The identity of a phase can be determined by some local probe. In particular, the Hamiltonian around a given point may be represented (in some non-canonical way) by a mass term that anticommutes with a certain Dirac operator. In the case of integer quantum Hall systems, the \(K\)-theoretic classification is stable to interactions, but a counterexample is also given.
For the entire collection see [Zbl 1169.81003].

82D55 Statistical mechanical studies of superconductors
15A66 Clifford algebras, spinors
55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity
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