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Stacking and the triviality of invertible phases

Sven Bachmann, Alan Getz, Pieter Naaijkens, Naomi Wray

TL;DR

This work analyzes how superselection sectors behave under stacking of 2D quantum spin systems within the operator-algebraic SSC framework. It proves that irreducible sectors of a stacked system decompose as products of sectors from each layer, and that the corresponding braided W*-categories are equivalent to the Deligne product of the layer categories. A central consequence is that invertible states have trivial topological content, with the Type I property and approximate split property providing the necessary structure to factor sectors. The authors also realize stacking as a categorical product, establishing a braided-W*-category equivalence between the stacked-sector category and the Deligne product of the component categories, and illustrate the framework with quantum doubles, Morita-equivalent Levin–Wen models, and symmetry-enriched toric code examples, linking Drinfeld centers and Morita theory to sector content.

Abstract

We study the superselection sectors of two quantum lattice systems stacked onto each other in the operator algebraic framework. We show in particular that all irreducible sectors of a stacked system are unitarily equivalent to a product of irreducible sectors of the factors. This naturally leads to a faithful functor between the categories for each system and the category of the stacked system. We construct an intermediate `product' category which we then show is equivalent to the stacked system category. As a consequence, the sectors associated with an invertible state are trivial, namely, invertible states support no anyonic quasi-particles.

Stacking and the triviality of invertible phases

TL;DR

This work analyzes how superselection sectors behave under stacking of 2D quantum spin systems within the operator-algebraic SSC framework. It proves that irreducible sectors of a stacked system decompose as products of sectors from each layer, and that the corresponding braided W*-categories are equivalent to the Deligne product of the layer categories. A central consequence is that invertible states have trivial topological content, with the Type I property and approximate split property providing the necessary structure to factor sectors. The authors also realize stacking as a categorical product, establishing a braided-W*-category equivalence between the stacked-sector category and the Deligne product of the component categories, and illustrate the framework with quantum doubles, Morita-equivalent Levin–Wen models, and symmetry-enriched toric code examples, linking Drinfeld centers and Morita theory to sector content.

Abstract

We study the superselection sectors of two quantum lattice systems stacked onto each other in the operator algebraic framework. We show in particular that all irreducible sectors of a stacked system are unitarily equivalent to a product of irreducible sectors of the factors. This naturally leads to a faithful functor between the categories for each system and the category of the stacked system. We construct an intermediate `product' category which we then show is equivalent to the stacked system category. As a consequence, the sectors associated with an invertible state are trivial, namely, invertible states support no anyonic quasi-particles.

Paper Structure

This paper contains 17 sections, 39 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting
  3. Algebras, states, and representations
  4. Stacking
  5. Invertible states
  6. Superselection sectors of stacked systems
  7. Well-definedness of the sector theory for stacked systems
  8. Tensor decomposition of the stacked sectors
  9. The superselection sectors of a stacked system
  10. Type I property
  11. Topological entanglement entropy and the approximate split property
  12. Invertible states
  13. Stacking as a categorical product
  14. Examples
  15. Stacking quantum double models
  16. ...and 2 more sections

Key Result

Theorem 1

If $\rho_1$ and $\rho_2$ are representations of $\mathfrak{A}_1$ and $\mathfrak{A}_2$ that satisfy the superselection criterion for $\pi_1$ and $\pi_2$, then $\rho_1 \otimes_{\mathrm{s}} \rho_2$ satisfies the superselection criterion for $\pi_1 \otimes_{\mathrm{s}} \pi_2$.

Figures (1)

  • Figure 1: The entanglement entropy and the mutual information.

Theorems & Definitions (99)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.4: Approximate Haag duality, ogata_derivation_2021
  • Remark 2.6
  • Definition 2.7: naaijkens2022split
  • Theorem 2.8
  • ...and 89 more