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A derivation of braided $C^*$-tensor categories from gapped ground states satisfying the approximate Haag duality

Yoshiko Ogata

Abstract

We derive braided $C^*$-tensor categories from gapped ground states on two-dimensional quantum spin systems satisfying some additional condition which we call the approximate Haag duality.

A derivation of braided $C^*$-tensor categories from gapped ground states satisfying the approximate Haag duality

Abstract

We derive braided -tensor categories from gapped ground states on two-dimensional quantum spin systems satisfying some additional condition which we call the approximate Haag duality.

Paper Structure

This paper contains 10 sections, 51 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Two-dimensional quantum spin systems
  3. Approximate Haag duality
  4. Main result
  5. Superselection sectors and their extensions
  6. The composition
  7. The intertwiners
  8. Braided $C^*$-tensor category
  9. Stability of the braiding structure
  10. Cones

Key Result

Proposition 1.3

Let $({\mathcal{H}},\pi_0)$ be an irreducible representation of ${\mathcal{A}}_{{\mathbb Z}^2}$ satisfying the approximate Haag duality. Then for any approximately-factorizable automorphism $\alpha$ on ${\mathcal{A}}_{{\mathbb Z}^2}$, $({\mathcal{H}},\pi_0\circ\alpha)$ also satisfies the approximate

Theorems & Definitions (109)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Theorem
  • Lemma 2.2
  • proof
  • ...and 99 more