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Classification of “real” Bloch-bundles: topological quantum systems of type AI. (English) Zbl 1316.57019
The paper is devoted to a classification of systems subject to even time reversal symmetry (+TR), denoted by AI in the Altland - Zirnbauer - Cartan (AZC) classification scheme and containing spinless or integer spin (i.e. bosonic) quantum systems, which are invariant under time inversion. The authors use the specification “real” for systems in class AI. The aim of this work consists in (i) discussions of a classification procedure for “real” vector bundles over a sufficiently general involutive space $$(X, \tau)$$ based on the analysis of the equivariant structure induced by the involution $$\tau$$, and (ii) application of such a classification scheme to the case of topological insulators, considered as special examples of topological quantum systems. It is proposed that in this case a classification exists which is finer than the usual $$K$$-theoretic description capable to take care also of possible effects due to non-stable regime. The main result of the paper is defined by a Theorem on the classification of AI topological quantum systems. The proof of the Theorem is (i) a consequence of the homotopy classification of “real” vector bundles and of the homotopy reduction, (ii) a consequence of a Theorem which fixes the stable range condition for “real” vector bundles, and (iii) a consequence of a Theorem which establishes the cohomological classification for “real” line bundles. The cohomology group that appears in (iii) is equivalent to the Borel cohomology of the space $$(X, \tau)$$ computed with respect to the local system of coefficients. In order to relate the main Theorem with the theory of condensed matter type electron systems, a proper base space is specified and a time-reversal involution $$\tau$$ by introducing free charge systems. Then, the authors provide the link between translation invariant quantum systems (with +TR symmetry) and complex vector bundles (with a “real” structure). Vector bundles over a space $$X$$ are completely classified in terms of equivalence classes of homotopic maps. Then, the role of the Brillouin zone for the classification of topological phases is clarified. The application of the main Theorem to the case of free and periodic electron systems leads to the homotopy classification of AI topological insulators and classification of invariants for AI topological insulators in dimension $$d=4$$. The authors provide a concrete recipe for the construction of all topologically non-trivial systems of type AI. This construction bases on a family of standard prototype models, which are ubiquitous in the literature on topological insulators. It is proved that these models are sufficient to realize all nonequivalent topological phases of type AI in $$d=4$$. Then, an effective computational procedure is described for determination of the associated invariants based on the notion of Brouwer degree of maps. The developed classification procedure bases on the use of the proper characteristic classes (so-called, equivalent Chern classes or mixed Chern classes). These classes are elements of an equivalent cohomology theory (Borel cohomology). The authors prove that first Chern classes classify completely “real” vector bundles up to $$d=4$$. Moreover, explicit computations are given for the related Borel cohomology groups.

##### MSC:
 57R22 Topology of vector bundles and fiber bundles 55N25 Homology with local coefficients, equivariant cohomology 53C80 Applications of global differential geometry to the sciences 19L64 Geometric applications of topological $$K$$-theory 55R25 Sphere bundles and vector bundles in algebraic topology 82B10 Quantum equilibrium statistical mechanics (general) 82D20 Statistical mechanical studies of solids
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