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Topological Insulators from Group Cohomology

Author(s): Alexandradinata, A; Wang, Zhijun; Bernevig, Bogdan A.

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Abstract: We classify insulators by generalized symmetries that combine space-time transformations with quasimomentum translations. Our group-cohomological classification generalizes the nonsymmorphic space groups, which extend point groups by real-space translations; i.e., nonsymmorphic symmetries unavoidably translate the spatial origin by a fraction of the lattice period. Here, we further extend nonsymmorphic groups by reciprocal translations, thus placing real and quasimomentum space on equal footing. We propose that group cohomology provides a symmetry-based classification of quasimomentum manifolds, which in turn determines the band topology. In this sense, cohomology underlies band topology. Our claim is exemplified by the first theory of time-reversal-invariant insulators with nonsymmorphic spatial symmetries. These insulators may be described as “piecewise topological,” in the sense that subtopologies describe the different high-symmetry submanifolds of the Brillouin zone, and the various subtopologies must be pieced together to form a globally consistent topology. The subtopologies that we discover include a glide-symmetric analog of the quantum spin Hall effect, an hourglass-flow topology (exemplified by our recently proposed KHgSb material class), and quantized non-Abelian polarizations. Our cohomological classification results in an atypical bulk-boundary correspondence for our topological insulators.
Publication Date: 15-Apr-2016
Electronic Publication Date: 15-Apr-2016
Citation: Alexandradinata, A, Wang, Zhijun, Bernevig, B Andrei. (2016). Topological Insulators from Group Cohomology. PHYSICAL REVIEW X, 6 (10.1103/PhysRevX.6.021008
DOI: doi:10.1103/PhysRevX.6.021008
ISSN: 2160-3308
Type of Material: Journal Article
Journal/Proceeding Title: PHYSICAL REVIEW X
Version: Final published version. This is an open access article.

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