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Classification of (2+1)D invertible fermionic topological phases with symmetry

Barkeshli, Maissam and Chen, Yu-An and Hsin, Po-Shen and Manjunath, Naren (2022) Classification of (2+1)D invertible fermionic topological phases with symmetry. Physical Review B, 105 (23). Art. No. 235143. ISSN 2469-9950. doi:10.1103/physrevb.105.235143.

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We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups G_f and general values of the chiral central charge c₋. Here Gf is a central extension of a bosonic symmetry group G_b by fermion parity, (−1)^F, specified by a second cohomology class [ω₂]∈H²(G_b,ℤ₂). Our approach proceeds by gauging fermion parity and classifying the resulting G_b symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using G-crossed braided tensor categories and Spin(2c₋)₁ Chern-Simons theory coupled to a background G gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge c₋ = 0. We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of c₋ and G_f. Mathematically, our results also suggest an explicit general parameterization of deformation classes of (2+1)D invertible topological quantum field theories with G_f symmetry.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Barkeshli, Maissam0000-0002-4322-9433
Chen, Yu-An0000-0002-8810-9355
Hsin, Po-Shen0000-0002-4764-1476
Manjunath, Naren0000-0003-3403-5335
Additional Information:© 2022 American Physical Society. (Received 2 November 2021; revised 26 May 2022; accepted 27 May 2022; published 29 June 2022) We thank Anton Kapustin, Parsa Bonderson, Daniel Bulmash, and Meng Cheng for discussions. Y.-A.C. thanks Nathanan Tantivasadakarn for advice on numerical calculations of the obstruction, and Dave Aasen for conceptual discussions. M.B. is supported by NSF CAREER (DMR-1753240) and JQI-PFC-UMD. Y.-A.C. is supported by the JQI fellowship at the University of Maryland. The work of P.-S.H. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the Simons Foundation through the Simons Investigator Award. We thank Qing-Rui Wang and David Aasen for pointing out some errors in our previous presentation of the ν₃ stacking rules. An updated version of Ref. [42] proves the stacking rules for n₂ in the case of unitary symmetries, and their results agree with our (corrected) stacking rules.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
University of MarylandUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Simons FoundationUNSPECIFIED
Issue or Number:23
Record Number:CaltechAUTHORS:20220715-332439000
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:115629
Deposited By: George Porter
Deposited On:18 Jul 2022 14:43
Last Modified:18 Jul 2022 14:43

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