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Symmetry fractionalization, defects, and gauging of topological phases

Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang
Phys. Rev. B 100, 115147 – Published 20 September 2019

Abstract

We examine the interplay of symmetry and topological order in 2+1-dimensional topological quantum phases of matter. We present a precise definition of the topological symmetry group Aut(C), which characterizes the symmetry of the emergent topological quantum numbers of a topological phase C, and we describe its relation with the microscopic symmetry of the underlying physical system. This allows us to derive a general framework to characterize and classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are antiunitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group, extending previous results in the literature. This provides a general classification of 2+1-dimensional symmetry-enriched topological (SET) phases derived from a topological phase of matter C with on-site symmetry group G. We derive a set of data and consistency conditions, the solutions of which define the algebraic theory of the defects, known as a G-crossed braided tensor category CG×. This allows us to systematically compute many properties of these theories, such as the number of topologically distinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes, braiding exchange transformations, a generalized Verlinde formula for the defects, and modular transformations of the G-crossed extensions of topological phases. We also examine the promotion of the global on-site symmetry to a local gauge invariance (“gauging the symmetry”), wherein the extrinsic G defects are turned into deconfined quasiparticle excitations, which results in a different topological phase (CG×)G. We present systematic methods to compute the properties of (CG×)G when G is a finite group. The quantum phase transition between the topological phases (CG×)G and C can be understood to be a “gauge symmetry breaking” transition, thus shedding light on the universality class of a wide variety of topological quantum phase transitions. A number of instructive and/or physically relevant examples are studied in detail.

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  • Received 20 November 2014
  • Revised 22 August 2019

DOI:https://doi.org/10.1103/PhysRevB.100.115147

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.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Maissam Barkeshli1, Parsa Bonderson1, Meng Cheng1, and Zhenghan Wang1,2

  • 1Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
  • 2Department of Mathematics, University of California, Santa Barbara, California 93106, USA

Article Text

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Issue

Vol. 100, Iss. 11 — 15 September 2019

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