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Anyon theory in gapped many-body systems from entanglement

Shi, Bowen
http://orcid.org/0000-0002-0689-9964
http://rave.ohiolink.edu/etdc/view?acc_num=osu1587705058308889

Abstract Details

2020, Doctor of Philosophy, Ohio State University, Physics.
In this thesis, we present a theoretical framework that can derive a general anyon theory for 2D gapped phases from an assumption on the entanglement entropy. We formulate 2D quantum states by assuming two entropic conditions on local regions, (a version of entanglement area law that we advocate). We introduce the information convex set, a set of locally indistinguishable density matrices naturally defined in our framework. We derive an isomorphism theorem and structure theorems of the information convex sets by studying the internal self-consistency. This line of derivation makes extensive usage of information-theoretic tools, e.g., strong subadditivity and the properties of quantum many-body states with conditional independence. The following properties of the anyon theory are rigorously derived from this framework. We define the superselection sectors (i.e., anyon types) and their fusion rules according to the structure of information convex sets. Antiparticles are shown to be well-defined and unique. The fusion rules are shown to satisfy a set of consistency conditions. The quantum dimension of each anyon type is defined, and we derive the well-known formula of topological entanglement entropy. We further identify unitary string operators that create anyon pairs and study the circuit depth. We define the topological $S$-matrix and show it satisfies the Verlinde formula. It follows that the mutual braiding statistics of the sectors are nontrivial (they are anyons); moreover, the underlying anyon theory is modular. Three additional things, closely related to this framework, are presented: (1) The framework on a discrete lattice; (2) A calculation of information convex set based on solvable Hamiltonians; (3) A conjecture concerning the generality of our assumptions.
Yuan-Ming Lu (Advisor)
Stuart Raby (Committee Member)
Daniel Gauthier (Committee Member)
Mohit Randeria (Committee Member)
David Penneys (Other)
142 p.
Physics; Quantum Physics; Theoretical Physics
Anyons; quantum entanglement; topological entanglement entropy; quantum Markov state; entanglement area law; topological order; fusion rules; Verlinde formula

Recommended Citations

Citations

  • Shi, B. (2020). Anyon theory in gapped many-body systems from entanglement [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587705058308889

    APA Style (7th edition)

  • Shi, Bowen. Anyon theory in gapped many-body systems from entanglement. 2020. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1587705058308889.

    MLA Style (8th edition)

  • Shi, Bowen. "Anyon theory in gapped many-body systems from entanglement." Doctoral dissertation, Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587705058308889

    Chicago Manual of Style (17th edition)

osu1587705058308889
803
© 2020, some rights reserved.Creative Commons License Anyon theory in gapped many-body systems from entanglement by Bowen Shi is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.