Gauging spatial symmetries and the classification of topological crystalline phases

By Ryan Thorngren, D. Else. 2016

185 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c239bb8c9b4c469b8cd7d045d2345934220377b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. R. Ma
2. Chong Wang
3. Luka Trifunovic
4. P. Brouwer
5. E. Witten

In Materials Chemistry

1. R. Ma
2. Chong Wang
3. Luka Trifunovic
4. P. Brouwer

In Mathematical Physics

1. Luka Trifunovic
2. P. Brouwer

Citations in the corpus

• Average Symmetry-Protected Topological Phases (2022, background): “Using the crystalline equivalence principle[17], we can conclude that all the generalized LSM anomalies for other lattice symmetries[22, 58] (rotation, reflection etc.”
• Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases (2018, methodology): “When applied repeatedly, these isomorphisms can be used to relate the classification problem of d-dimensional Hamiltonians with an order-two crystalline symmetry to a zerodimensional classification problemwith an on-site symmetry [21,25], which can be solved with elementary methods.”