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Anyon condensation and tensor categories

Liang Kong

TL;DR

The paper develops a model-independent, tensor-categorical framework for anyon condensation in 2d topological orders. By bootstrap analysis, it shows that a post-condensation phase  is identified with the local module category _A^{loc} for a connected commutative separable Frobenius algebra A in the pre-condensation phase , while the wall excitations form _A; 2d and 1d condensations are captured by algebras A and B, with bulk-to-wall functors that are central. It further provides a general method to recover A (and partially B) from macroscopic data, and illustrates the framework with toric code, Levin-Wen models, Kitaev quantum doubles, and condensed chiral phases, connecting to boundary-bulk relations and Witt equivalence. The work consolidates the universal, model-independent role of Frobenius algebras in describing condensations, boundaries, and domain walls, and offers a precise language for analyzing gapped boundaries and their relation to bulk phases.

Abstract

Instead of studying anyon condensation in concrete models, we take an abstract approach. Assume that a system of anyons, which form a modular tensor category D, is obtained via an anyon condensation from another system of anyons (i.e. another modular tensor category C). By a bootstrap analysis, we derive the relation between C and D from natural physical requirements. It turns out that the tensor unit of D can be identified with a connected commutative separable algebra A in C. The modular tensor category D consists of all deconfined particles and can be identified with the category of local $A$-modules in C. If this condensation occurs in a 2d region in the C-phase, then it also produces a 1d gapped domain wall between the C-phase and the D-phase. The confined and deconfined particles accumulate on the wall and form a fusion category that is precisely the category of right A-modules in C. We also consider condensations that are confined to a 1d line. We show how to determine the algebra A from physical macroscopic data. We provide examples of anyon condensation in the toric code model, Kitaev quantum double models and Levin-Wen types of lattice models and in some chiral topological phases. In the end, we briefly discuss Witt equivalence between 2d topological phases. We also attach to this paper an Erratum and Addendum to the original version of "Anyon condensation and tensor categories" published in [Nucl. Phys. B 886 (2014) 436-482].

Anyon condensation and tensor categories

TL;DR

The paper develops a model-independent, tensor-categorical framework for anyon condensation in 2d topological orders. By bootstrap analysis, it shows that a post-condensation phase  is identified with the local module category _A^{loc} for a connected commutative separable Frobenius algebra A in the pre-condensation phase , while the wall excitations form _A; 2d and 1d condensations are captured by algebras A and B, with bulk-to-wall functors that are central. It further provides a general method to recover A (and partially B) from macroscopic data, and illustrates the framework with toric code, Levin-Wen models, Kitaev quantum doubles, and condensed chiral phases, connecting to boundary-bulk relations and Witt equivalence. The work consolidates the universal, model-independent role of Frobenius algebras in describing condensations, boundaries, and domain walls, and offers a precise language for analyzing gapped boundaries and their relation to bulk phases.

Abstract

Instead of studying anyon condensation in concrete models, we take an abstract approach. Assume that a system of anyons, which form a modular tensor category D, is obtained via an anyon condensation from another system of anyons (i.e. another modular tensor category C). By a bootstrap analysis, we derive the relation between C and D from natural physical requirements. It turns out that the tensor unit of D can be identified with a connected commutative separable algebra A in C. The modular tensor category D consists of all deconfined particles and can be identified with the category of local -modules in C. If this condensation occurs in a 2d region in the C-phase, then it also produces a 1d gapped domain wall between the C-phase and the D-phase. The confined and deconfined particles accumulate on the wall and form a fusion category that is precisely the category of right A-modules in C. We also consider condensations that are confined to a 1d line. We show how to determine the algebra A from physical macroscopic data. We provide examples of anyon condensation in the toric code model, Kitaev quantum double models and Levin-Wen types of lattice models and in some chiral topological phases. In the end, we briefly discuss Witt equivalence between 2d topological phases. We also attach to this paper an Erratum and Addendum to the original version of "Anyon condensation and tensor categories" published in [Nucl. Phys. B 886 (2014) 436-482].

Paper Structure

This paper contains 25 sections, 5 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bootstrap analysis
  3. Anyons in a modular tensor category
  4. Physical data associated to a condensation
  5. Vacuum in D as an algebra in E
  6. General deconfined particles
  7. 2d Condensations
  8. 1d Condensations
  9. Determining the condensation from physical data
  10. Gapped boundaries
  11. Non-trivial D-phase
  12. Examples
  13. Toric code model
  14. Levin-Wen types of lattice models
  15. Kitaev quantum-double models
  16. ...and 10 more sections

Key Result

Corollary 2.8

A 2d-condensable algebra $A$ in $\EuScript{C}$ is automatically a boson, i.e. $\theta_A = {\rm id}_A$.

Figures (3)

  • Figure 1: The set-up of bootstrap analysis: We consider an anyon condensation occurring in a 2d topological phase described by a (unitary) MTC $\EuScript{C}$. The condensed phase is described by another MTC $\EuScript{D}$, and the gapped excitations on the domain wall form a (unitary) spherical fusion category $\EuScript{E}$.
  • Figure 3: This picture show that the left $A$-action on a particle $X$ on the wall should be compatible with the right $A$-action.
  • Figure 4: This picture illustrate the results in Theorem${}^{\mathrm{ph}}$\ref{['thm:main-2']}.

Theorems & Definitions (68)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • ...and 58 more