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.
Highlights
Symmetry is an important feature of many physical systems
As an example of results that one can deduce from this general principle, we find that bosonic symmetry-protected topological (SPT) phases protected by orientationpreserving unitary spatial symmetry G are classified by the group cohomology Hdþ1ðG; Uð1ÞÞ since that is the classification of internal SPTs with symmetry G
We expect that a topological liquid can always be made invariant under a spatial symmetry G if we make the orientation-reversing elements of G act antiunitarily; we suggestively call this the “CPT principle.” [75]; We prove this explicitly for bosonic SPT phases in Appendix F
Summary
Symmetry is an important feature of many physical systems. Many phases of matter can be characterized in part by the way the symmetry is implemented. The main result of this paper is the following crystalline equivalence principle: The classification of crystalline topological liquids with spatial symmetry group G is the same as the classification of topological phases with internal symmetry G. As an example of results that one can deduce from this general principle, we find that bosonic SPT phases protected by orientationpreserving unitary spatial symmetry G are classified by the group cohomology Hdþ1ðG; Uð1ÞÞ since that is the classification of internal SPTs with symmetry G (see Appendix A for more details on the definition of H). Applying the principle to fermionic systems, one obtains a partial classification of fermionic SPTs protected by space-group symmetries from “group supercohomology” [22] and a complete classification of fermionic crystalline SPTs from cobordism theory [27], with some caveats We hope this paper will inspire the discovery of many curious quantum crystals
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