The cohomological nature of the Fu-Kane-Mele invariant
type: Item
instance of: scholarly article
title: The cohomological nature of the Fu-Kane-Mele invariant
zbMATH Open document ID: 1390.57017
author: Giuseppe De Nittis
author: Kiyonori Gomi
published in: Journal of Geometry and Physics
publication date: 2018-04-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.09421
review text: 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.
reviewed by: Piotr Garbaczewski
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 81V70
Mathematics Subject Classification ID: 81T45
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 6858411
zbMATH Keywords: quaternionic vector bundle
zbMATH Keywords: symplectic vector bundle
zbMATH Keywords: symplectic bundle
zbMATH Keywords: FKMM (Furuta-Kametani-Matsue-Minami) invariant
zbMATH Keywords: FKM (Fu-Kane-Mele) index
zbMATH Keywords: topological insulators
zbMATH Keywords: topological quantum systems
zbMATH Keywords: TQS
zbMATH Keywords: quantum spin Hall effect
zbMATH Keywords: geometric phases
MaRDI profile type: Publication
OpenAlex ID: W2962830818
arXiv ID: 1603.09421
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DOI: 10.1016/J.GEOMPHYS.2017.10.007
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