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Tensor Category Constructions in Topological Phases of Matter

Huston, Peter
http://orcid.org/0000-0003-3903-9482
http://rave.ohiolink.edu/etdc/view?acc_num=osu1657968388945431

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
This dissertation consists of adaptations of two papers exploring the connections between tensor categories and localized excitations in (2+1)D topologically ordered systems. Topological orders have long been studied in terms of monoidal higher categories. In particular, topological orders in (2+1)D are obtained by string-net condensation from the data of unitary fusion categories, and topological domain walls and phase transitions between (2+1)D topological orders arise from anyon condensation, which can be understood in terms of the internal Morita theories of étale algebra objects in unitary modular tensor categories. Complementing existing models for domain walls, we introduce an extension of the Levin-Wen string-net model in which tuning a parameter implements anyon condensation. We then use tube algebra techniques to verify that the expected topological order results. We also describe the effects of the phase transition on anyons and string operators of the original topological order. We study compositions of parallel topological domain walls and their decompositions into superselection sectors. Our approach uses a description of particle mobility across domain walls in terms of tunneling operators. These are formalized in a 3-category of (2+1)D topological orders with a fixed anomaly described by a unitary modular tensor category A, algebraically characterized by the 3-category of A-enriched unitary fusion categories. We describe how A-enriched unitary fusion categories determine commuting projector string-net models for the corresponding anomalous (2+1)D topological orders on the boundary of (3+1)D Walker-Wang models with trivial topological order. We then give an explicit description in terms of étale algebra objects of the local operators which determine superselection sectors of parallel domain walls, characterizing the indecomposable domain wall in each sector.
David Penneys (Advisor)
Niles Johnson (Committee Member)
Thomas Kerler (Committee Member)
229 p.
Mathematics
Quantum algebra; Topological order; Anyon condensation; Tensor categories; Modular tensor categories; Domain walls

Recommended Citations

Citations

  • Huston, P. (2022). Tensor Category Constructions in Topological Phases of Matter [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1657968388945431

    APA Style (7th edition)

  • Huston, Peter. Tensor Category Constructions in Topological Phases of Matter. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1657968388945431.

    MLA Style (8th edition)

  • Huston, Peter. "Tensor Category Constructions in Topological Phases of Matter." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1657968388945431

    Chicago Manual of Style (17th edition)

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This open access ETD is published by The Ohio State University and OhioLINK.