Journal
PHYSICAL REVIEW X
Volume 8, Issue 1, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevX.8.011040
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Funding
- National Science Foundation Graduate Student Research Fellowship
- Microsoft Corporation
- NSF [NSF PHY-1125915]
- University of Colorado
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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 + 1) dimensions for 227 of the 230 space groups. For the 65 space groups not containing orientation-reversing elements (Sohncke groups), there are no cobordism invariants that may contribute phases beyond group cohomology, so we conjecture that our classification is complete.
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