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Anyonic symmetry fractionalization in SET phases

Jose Garre Rubio, Yoshiko Ogata

Abstract

We consider the anyonic spin systems with a global symmetry, the so-called symmetry enriched topological (SET) phases. We introduce the phase characterizing the symmetry fractionalization of the anyons. Our assumptions on how the global symmetry acts prevents anyon permutation effects.

Anyonic symmetry fractionalization in SET phases

Abstract

We consider the anyonic spin systems with a global symmetry, the so-called symmetry enriched topological (SET) phases. We introduce the phase characterizing the symmetry fractionalization of the anyons. Our assumptions on how the global symmetry acts prevents anyon permutation effects.

Paper Structure

This paper contains 19 sections, 45 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Quantum spin systems
  3. Cones
  4. Assumptions on local von Neumann algebras for cones
  5. Bulk braided $C^*$-tensor category
  6. Group action on the ground state $\omega$
  7. Group action on anyons
  8. Main Theorem
  9. $G$-localized superselection sectors and their braidings
  10. Tensor product
  11. $\Theta(g)$ on $O_G$
  12. Braiding of ${\mathcal{E}}_G$
  13. Fractionalization
  14. Subgroup $H$ of $G$ associated with anyons
  15. Proof of Theorem \ref{['mainthm']}
  16. ...and 4 more sections

Key Result

Theorem 1.5

Consider the setting in subsection sec:qss. Assume Assumption a1 and Assumption a2. There exists a braided $C^*$-tensor category $\hat{C}$ with the following structures. The full subcategory of $\hat{C}$ consisting of objects $O_{\Lambda^{(0)}}$ forms a sub braided $C^*$-tensor category $C$ of $\hat{C}$.

Theorems & Definitions (81)

  • Theorem 1.5
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • proof
  • Lemma 1.9
  • proof
  • Lemma 1.10
  • proof
  • Theorem 1.11
  • ...and 71 more