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Anomalous Symmetry Fractionalization and Surface Topological Order

Chen, Xie and Burnell, F. J. and Vishwanath, Ashvin and Fidkowski, Lukasz (2015) Anomalous Symmetry Fractionalization and Surface Topological Order. Physical Review X, 5 (4). Art. No. 041013. ISSN 2160-3308.

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In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways, leading to a variety of symmetry-enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain “anomalous” SETs can only occur on the surface of a 3D symmetry-protected topological (SPT) phase. In this paper, we describe a procedure for determining whether a SET of a discrete, on-site, unitary symmetry group G is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions is captured by the fourth cohomology group H^4 (G,U(1)), which also precisely labels the set of 3D SPT phases, with symmetry group G. An explicit procedure for calculating the cohomology data from a SET is given, with the corresponding physical intuition explained. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3D bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid [U(1)_2] topological order with a reduced symmetry Z_2 ×Z_2 ⊂SO(3) , which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3D SPT models realizing the anomalous surface terminations and demonstrate that they are nontrivial by computing three-loop braiding statistics. Possible extensions to antiunitary symmetries are also discussed.

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Additional Information:© 2015 American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Received 17 March 2015; published 23 October 2015. We are very grateful for helpful discussions with Meng Cheng, Senthil Todadri, Ryan Thorngren, Alexei Kitaev, Parsa Bonderson, and Netanel Lindner. X. C. is supported by the Miller Institute for Basic Research in Science at UC Berkeley, the Caltech Institute for Quantum Information and Matter, and the Walter Burke Institute for Theoretical Physics. A. V. is supported by Grant No. NSF DMR 1206728, and F. J. B. is supported by Grant No. NSF DMR 1352271.
Group:IQIM, Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Miller Institute for Basic Research in ScienceUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
NSFDMR 1352271
Subject Keywords:Condensed Matter Physics, Strongly Correlated Materials
Record Number:CaltechAUTHORS:20151113-155112359
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:62103
Deposited By: Tony Diaz
Deposited On:18 Nov 2015 00:30
Last Modified:18 Nov 2015 00:30

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