Topological Insulators from Group Cohomology

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.

Hourglass fermion surface states in stacked topological insulators with nonsymmorphic symmetry

“…Recently, topological crystalline insulators are generalized further to include the nonsymmorphic symmetry, and called nonsymmorphic topological crystalline insulators [16][17][18][19][20][21][22][22][23][24][25][26][27][28][29] . A nonsymmorphic topological insulator was predicted in KHgSb by first-principles calculations 31,32 and experimentally observed by ARPES (Angle-Resolved PhotoEmisison Spectroscopy) 33 in the same material soon after. A prominent feature is that there emerges entirely a new surface state called "hourglass fermion".…”

“…The constructed effective theory is comprised of complicated forms based on three orbitals of Sb and one orbital of Hg, and is applicable only in the vicinity of the certain high-symmetry points 31,32 . In this paper, motivated by these works [31][32][33] , we propose a tight-binding model producing a hourglass fermion surface state [ Fig.1] by the stacked helical edge states with the nonsymmorphic symmetry. They are realized by the binary stacking of the QSH insulators.…”

Bulk characterization of topological crystalline insulators: Stability under interactions and relations to symmetry enriched U (1) quantum spin liquids

“…These are short-range entangled phases that are protected by crystalline symmetries: they cannot (can) be connected to a trivial insulator in the presence (absence) of the relevant crystalline (and other) symmetries [4][5][6][7][8]. These states are not only of conceptual interests, but the state of the art in synthesizing materials also offer great opportunities to realizing them experimentally [9][10][11][12][13][14][15][16][17][18][19].…”

Observation of the topological surface state in the nonsymmorphic topological insulator KHgSb

“…Recently, the nonsymmorphic-symmetry-protected topological quantum states, including topological insulators and semimetals were under intensive research [42][43][44][45][46][47][48][49][50][51][52] . Among them, KHgX (X=As, Sb, Bi) compounds were proposed as the first family of nonsymmorphic topological insulator [49][50][51] .…”

“…Among them, KHgX (X=As, Sb, Bi) compounds were proposed as the first family of nonsymmorphic topological insulator [49][50][51] . According to the ab-initio calculations 49 , KHgX are insulating in the bulk but possess robust gapless surface states (see Fig.…”