Symmetry fractionalization, defects, and gauging of topological phases

By Maissam Barkeshli, Parsa Bonderson, M. Cheng, Zhenghan Wang. Physical review B, 2014

420 citations recorded by the source corpus.

We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the topological symmetry group, which characterizes the symmetry of the emergent topological quantum numbers of a topological phase $\mathcal{C}$, and we describe its relation with the microscopic symmetry of the underlying physical system. We derive a general framework to classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter $\mathcal{C}$ with symmetry group $G$. The algebraic theory of the defects, known as a $G$-crossed braided tensor category $\mathcal{C}_{G}^{\times}$, allows one to compute many properties, 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 symmetry to a local gauge invariance, wherein the extrinsic $G$-defects are turned into deconfined quasiparticle excitations, which results in a different topological phase $\mathcal{C}/G$. A number of instructive and/or physically relevant examples are studied in detail.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/5f7d511a64a9a071647165bad8e96b34eaed0e78

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In Atomic and Molecular Physics, and Optics

1. Clay Córdova
2. Po-Shen Hsin
3. F. Benini
4. N. Seiberg
5. Shu-Heng Shao
6. Ho Tat Lam
7. R. Ma
8. Ryan Thorngren

Citations in the corpus

• Average Symmetry-Protected Topological Phases (2022, background): “[77] Maissam Barkeshli, Parsa Bonderson, Meng Cheng,”
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019, background): “1 Iterated Group Extensions and a General Strategy for Fiber Functors A G symmetry of Turaev-Viro theory defined by a fusion category A may be described by the action of G on the anyons as well as fractionalization data which describes the fusion junctions of G domain walls [3].”
• Higher Gauging and Non-invertible Condensation Defects (2022, background): “In contrast, when k is a half-integer, they cannot be made trivial by choosing a different representative for the F- and R-symbols [77].”
• Comments on one-form global symmetries and their gauging in 3d and 4d (2018, background): “This theory was first studied in [20] and more recently in [23, 24].”
• On 2-group global symmetries and their anomalies (2018, methodology): “When such theories flow to gapped TQFTs at long distances, the 0-form symmetry still does not permute the anyons and hence, (according to a conjecture of [22]; see also Appendix D) the IR Postnikov class is trivial.”