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Bulletin of the American Physical Society

Minimalist approach to the classification of symmetry protected topological phases

Zhaoxi Xiong

Published Mar 14, 2017• 0 citations
OpenAlex

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Minimalist approach to the classification of symmetry protected topological phases

Charles Zhaoxi Xiong

Abstract A number of proposals with differing predictions (e.g. Borel group cohomology, oriented cobordism, group supercohomology, spin cobordism, etc) have been made for the classification of symmetry protected topological (SPT) phases. Here we treat various proposals on an equal footing and present rigorous, general results that are independent of which proposal is correct. We do so by formulating a minimalist generalized cohomology hypothesis, which is satisfied by existing proposals and captures essential aspects of SPT classification. From this hypothesis alone, formulas relating classifications in different dimensions and/or protected by different symmetry groups are derived. Our formalism is expected to work for fermionic as well as bosonic phases, Floquet as well as stationary phases, and spatial as well as on-site symmetries. As an application, we predict that the complete classification of 3D bosonic SPT phases with space group symmetry G is , where the H 1 term classifies phases beyond the Borel group cohomology proposal.

arXiv:1701.00004•Journal of Physics A Mathematical and Theoretical•Sep 12, 2018•62 citations

Organizing symmetry-protected topological phases by layering and symmetry reduction: A minimalist perspective

Charles Zhaoxi Xiong, A. Alexandradinata

It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-reduction maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the complete interacting classification in the presence of glide is ${\mathbb{Z}}_{4}\ensuremath{\bigoplus}{\mathbb{Z}}_{2}\ensuremath{\bigoplus}{\mathbb{Z}}_{2}$ in three dimensions. In particular, the hourglass-fermion phase recently realized in the band insulator KHgSb must be robust to interactions. Generalizations to spatiotemporal glide symmetries are discussed.

arXiv:1709.06998•Physical review. B./Physical review. B•Mar 26, 2018•18 citations

Symmetry-Protected Topological Phases

Bei Zeng, Xie Chen, D. L. Zhou +1 more

Quantum science and technology•Jan 1, 2019•7 citations
Abstract A number of proposals with differing predictions (e.g. Borel group cohomology, oriented cobordism, group supercohomology, spin cobordism, etc) have been made for the classification of symmetry protected topological (SPT) phases. Here we treat various proposals on an equal footing and present rigorous, general results that are independent of which proposal is correct. We do so by formulating a minimalist generalized cohomology hypothesis, which is satisfied by existing proposals and captures essential aspects of SPT classification. From this hypothesis alone, formulas relating classifications in different dimensions and/or protected by different symmetry groups are derived. Our formalism is expected to work for fermionic as well as bosonic phases, Floquet as well as stationary phases, and spatial as well as on-site symmetries. As an application, we predict that the complete classification of 3D bosonic SPT phases with space group symmetry G is , where the H 1 term classifies phases beyond the Borel group cohomology proposal.
It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-reduction maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the complete interacting classification in the presence of glide is ${\mathbb{Z}}_{4}\ensuremath{\bigoplus}{\mathbb{Z}}_{2}\ensuremath{\bigoplus}{\mathbb{Z}}_{2}$ in three dimensions. In particular, the hourglass-fermion phase recently realized in the band insulator KHgSb must be robust to interactions. Generalizations to spatiotemporal glide symmetries are discussed.