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On the structure of categorical duality operators

Corey Jones, Xinping Yang

Abstract

We systematically study categorical duality operators on spin (and anyon) chains with respect to an internal fusion category symmetry C. We parameterize duality operators on the quasi-local algebra in terms of data dependent on the associated quantum cellular automata (QCA) on the symmetric subalgebra $B$. In particular, a QCA $α$ on $B$ defines an invertible C-C bimodule category $M_α$, and the duality operators extending $α$ form a simplex, with extreme points in bijective correspondence with the simple object of $M_α$. Then we consider the structure of external symmetries generated by a family of duality operators, and show that if the UV models are all defined on tensor product Hilbert spaces, these categories necessarily flow to weakly integral fusion categories in the IR.

On the structure of categorical duality operators

Abstract

We systematically study categorical duality operators on spin (and anyon) chains with respect to an internal fusion category symmetry C. We parameterize duality operators on the quasi-local algebra in terms of data dependent on the associated quantum cellular automata (QCA) on the symmetric subalgebra . In particular, a QCA on defines an invertible C-C bimodule category , and the duality operators extending form a simplex, with extreme points in bijective correspondence with the simple object of . Then we consider the structure of external symmetries generated by a family of duality operators, and show that if the UV models are all defined on tensor product Hilbert spaces, these categories necessarily flow to weakly integral fusion categories in the IR.
Paper Structure (12 sections, 7 theorems, 50 equations)

This paper contains 12 sections, 7 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Non-local operators in infinite volume
  3. Sectorization of non-local operators
  4. Microscopic SymTFT picture
  5. Charge categories
  6. Duality operators
  7. Example: Generalized Kramers-Wannier
  8. Q-system model
  9. Duality operator
  10. UV-category of duality operators
  11. The general picture
  12. Bimodule channels and topological boundary defects

Key Result

Theorem 1.1

Suppose we have a fusion category $\mathcal{C}$ acting on the lattice, with symmetric subalgebra $B\subseteq A$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Theorem 2.2: jones2024dhrbimodulesquasilocalalgebras
  • Definition 2.3
  • Definition 2.4
  • ...and 9 more