The cohomological nature of the Fu-Kane-Mele invariant
The paper contributes to a topological classification of systems (TQS) with an odd time-reversal symmetry (odd TRS). In the preceding paper by the authors [Commun. Math. Phys. 339, No. 1, 1--55 (2015; Zbl 1326.57047)] a classification scheme was developed based on a cohomological description. The cohomological class called the FKMM-invariant has been shown to be characteristic to a specific category of so-called ``quaternionic'' vector bundles. The reasoning was based on the hypothesis that homotopy classes constitute the FKMM space. The restriction is identified to be strong. The main result of the present paper is the extension of the validity of the previous theory beyond the just mentioned FKMM space limitation. To this end a generalised version of the FKMM invariant is introduced which yields an extension to involutive spaces that need not be of FKMM-type. It is proved that a cohomological generalization of the Fu-Kane-Mele index is possible, which is applicable to the classification of protected phases for the much broader class of topological quantum systems, which are not necessarily related to models of topological insulators. A detailed discussion of a necessary condition for non-trivial phases is given for the case of \(N=2\) (Section 5 on prototype models of TQS). As a byproduct, a complete classification of ``quaternionic'' vector bundles is provided for a big class of low-dimensional involutive spheres and tori.
FKMM (Furuta-Kametani-Matsue-Minami) invariant
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