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Classification of topological insulators and superconductors. (English) Zbl 1180.82228
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, 10-21 (2009).
Summary: An exhaustive classification scheme of topological insulators and superconductors is presented. The key property of topological insulators (superconductors) is the appearance of gapless degrees of freedom at the interface/boundary between a topologically trivial and a topologically non-trivial state. Our approach consists in reducing the problem of classifying topological insulators (superconductors) in d spatial dimensions to the problem of Anderson localization at a \((d-1)\) dimensional boundary of the system. We find that in each spatial dimension there are precisely five distinct classes of topological insulators (superconductors). The different topological sectors within a given topological insulator (superconductor) can be labeled by an integer winding number or a \(\mathbb Z_2\) quantity. One of the five topological insulators is the ‘quantum spin Hall’ (or: \(\mathbb Z_2\) topological) insulator in \(d=2\), and its generalization in \(d=3\) dimensions. For each dimension d, the five topological insulators correspond to a certain subset of five of the ten generic symmetry classes of Hamiltonians introduced more than a decade ago by Altland and Zirnbauer in the context of disordered systems (which generalizes the three well known “Wigner-Dyson” symmetry classes).
For the entire collection see [Zbl 1169.81003].

82D55 Statistical mechanical studies of superconductors
81V70 Many-body theory; quantum Hall effect