Gauging spatial symmetries and the classification of topological crystalline phases

TL;DR

This paper develops a systematic framework for understanding and classifying topological crystalline phases protected by space-group symmetries, introducing the concept of crystalline topological liquids and proving a classification principle.

Contribution

It introduces the notion of crystalline topological liquids, proves the Crystalline Equivalence Principle, and provides a partial classification of bosonic SPT phases for most 3D space groups.

Findings

Crystalline topological liquids correspond to phases with internal symmetry groups.

The Crystalline Equivalence Principle establishes a one-to-one correspondence between crystalline and internal symmetry-protected phases.

Partial classification of bosonic SPT phases for 227 space groups using group cohomology.

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 anti-unitary symmetry in the internal symmetry group. As an example, we explicitly compute, using group cohomology, a partial classification of bosonic symmetry-protected topological (SPT) phases protected by crystalline symmetries in (3+1)-D for 227 of the 230 space groups. For the 65 space groups not containing orientation-reversing elements (Sohncke groups), there are no cobordism invariants which may contribute phases beyond group cohomology, and so we conjecture our classification is complete.