Topological Insulators from Group Cohomology

TL;DR

This paper develops a group cohomology framework to classify topological insulators with generalized symmetries, unifying real and quasimomentum space, and introduces new topological phases with unique boundary properties.

Contribution

It extends nonsymmorphic space groups by reciprocal translations, providing a symmetry-based classification of band topology using cohomology, and introduces novel topological phases.

Findings

Classified time-reversal-invariant insulators with nonsymmorphic symmetries.

Discovered new subtopologies like glide-symmetric quantum spin Hall effect.

Proposed an atypical bulk-boundary correspondence for these insulators.

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 discovered 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.