Gauging Spatial Symmetries and the Classification of Topological Crystalline Phases

Abstract

We put the theory of interacting topological crystalline phases on a systematic footing. These are topological phases protected by space-group symmetries. Our central tool is an elucidation of what it means to "gauge" such symmetries. We introduce the notion of a crystalline topological liquid and argue that most (and perhaps all) phases of interest are likely to satisfy this criterion. We prove a crystalline equivalence principle, which states that in Euclidean space, crystalline topological liquids with symmetry group G are in one-to-one correspondence with topological phases protected by the same symmetry G, but acting internally, where if an element of G is orientation reversing, it is realized as an antiunitary symmetry in the internal symmetry group. As an example, we explicitly compute, using group cohomology, a partial classification of bosonic symmetry-protected topological phases protected by crystalline symmetries in (3 þ 1) dimensions for 227 of the 230 space groups. For the 65 space groups not containing orientationreversing elements (Sohncke groups), there are no cobordism invariants that may contribute phases beyond group cohomology, so we conjecture that our classification is complete.

Intrinsic and emergent anomalies at deconfined critical points

“…The treatment of lattice symmetries as internal symmetries for the purpose of anomaly computation is consistent with Ref. [15], which argues that the classification of crystalline SPTs with a symmetry group G comprising both lattice and internal symmetries is identical to the classification of SPTs with a purely internal symmetry group G (see also Ref. [16]).…”

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

“…There has already been some work in this direction, and intriguing results have been found in many cases [28,60,[62][63][64][65]. It will be interesting and important to obtain systematic physical characterizations of these defects, as well as their interplay with other defects such as monopoles.…”

“…[25] and further elaborated in Ref. [28]. Such a correspondence may potentially provide a simpler way of solving a difficult problem, by relating a problem with internal symmetry to a problem with crystalline symmetry, or vice versa.…”

“…[25] and later elaborated in Ref. [28] as the "crystalline equivalence principle." Below we illustrate this correspondence.…”

“…An important question then is whether the nontrivial TCIs predicted by band theories are stable under interactions, or, more precisely, whether these band-theory-based nontrivial TCIs can be smoothly connected to a trivial insulator once strong interactions are switched on (still in the presence of the relevant symmetries). Very recently, the stability of various such TCIs under interactions has been considered and some of them are shown to be unstable, by studying the anomalous surface properties of the TCIs, studying certain dimensionally reduced versions of the TCIs, or studying the formal field theories of the TCIs [20][21][22][23][24][25][26][27][28][29][30][31]. However, an important and interesting question has remained unanswered: how to characterize nontrivial interacting TCIs directly by their bulk properties in a physical way?…”