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.
7 More- Received 24 November 2015
DOI:https://doi.org/10.1103/PhysRevX.6.021008
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Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
When spatial symmetries such as rotations or reflections are combined with real-space translations, the resultant group of symmetries describes crystals. A field of mathematics called group cohomology dictates that there are exactly 230 possible combinations of symmetries in three spatial dimensions. Here, we demonstrate that these combinations do not exhaust all possible symmetries in crystals.
The translational symmetry of crystals guarantees that each particle has a conserved quasimomentum. We, for the first time, combine rotations and reflections with quasimomentum translations, in addition to real-space translations. Our theory thus places real and quasimomentum space on equal footing. We then obtain a richer classification of symmetries through group cohomology.
Our expanded classification provides a unified framework to discuss insulators with topologically robust properties that are protected by space-time symmetries. We exemplify our theory with the large-gap insulator KHgSb, whose surface is uniquely characterized by unremovable fermions with an hourglass-shaped dispersion.