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Classification of “quaternionic” Bloch-bundles: topological quantum systems of type AII. (English) Zbl 1326.57047
The classification of topological quantum systems has been inspired by the study of topological insulators and the quantum Hall effect in particular. According to [A. Altland and M. Zirnbauer, Phys. Rev. B 55, 1142–1161 (1997)], AI and AII classes describe quantum systems that are invariant under time-reversal symmetry, but it is the behavior of the spin that distinguishes between them. Interesting physical effects appear in class AII systems that exhibit the quantum spin Hall effect (QSHE). The principal result of L. Fu, C. L. Kane and E. J. Mele [“Topological insulators in three dimensions”, Phys. Rev. Lett. 98, 106803 (2007)] was an identification of QSHE with the existence of a topological invariant, nowadays known as the Fu-Kane-Mele index. This index characterizes a topology of Bloch energy bands for periodic systems of free fermions, that have the time reversal (TR) symmetry, in the very same way as Chern numbers if the TR symmetry is broken. Major results have been established for two-dimensional lattice systems. The main goal of the present paper is an extension of the topological classification beyond \(d=2\) (actually up to \(d=4\)). The analysis is based on the construction fo the FKMM invariant (same as the FKM, but this notation gives justice to other independent derivations) which completely classifies so-called quaternionic (actually symplectic) vector bundles in dimension \(d\leq 3\). In case of \(d=4\) the classification requires a combined use of the FKKM invariant and the second Chern class.

57R22 Topology of vector bundles and fiber bundles
57R56 Topological quantum field theories (aspects of differential topology)
58B05 Homotopy and topological questions for infinite-dimensional manifolds
55P91 Equivariant homotopy theory in algebraic topology
81V70 Many-body theory; quantum Hall effect
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
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