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A lattice model for condensation in Levin-Wen systems

Jessica Christian, David Green, Peter Huston, David Penneys

Abstract

Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the {Drinfeld} center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.

A lattice model for condensation in Levin-Wen systems

Abstract

Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the {Drinfeld} center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.
Paper Structure (25 sections, 13 theorems, 131 equations, 4 figures)

This paper contains 25 sections, 13 theorems, 131 equations, 4 figures.

Table of Contents

  1. Introduction
  2. String-net models in (2+1)D
  3. Background: The Levin-Wen system
  4. Topological Excitations from String Operators
  5. String Operators on Excited States
  6. Hopping Operators
  7. Tube Algebra Representations from Excitations
  8. Lattice model for anyon condensation
  9. Background: Condensable Algebras
  10. General Lattice Model
  11. Topological order when t=0
  12. Topological Order when t=1
  13. Tube algebras of local operators when t=1
  14. String Operators when t=1
  15. Tube algebra representations from excitations at t=1
  16. ...and 10 more sections

Key Result

Corollary 1

For any link $\ell$, vertex $v$ of $\ell$, anyon $s\in\mathop{\mathrm{Irr}}\nolimits(Z(\mathcal{X})$, and morphisms $\phi,\psi\in\bigoplus_x\mathcal{X}(s\to x)$, there is a local operator $T^s_{\ell,v}(\phi,\psi)$ such that, if $|\omega\rangle=A_\ell|\omega\rangle=B_r|\omega\rangle$ for plaquettes $ and if $t\neq s$,

Figures (4)

  • Figure 1: A path for a string operator on the hexagonal lattice.
  • Figure 2: A string operator is resolved into local operators on individual vertex Hilbert spaces. The morphism $f_i$ labels the vertex $v_i$, and the object $x_i$ labels the link $\ell_i$. The effect of applying $\pi^1_{\ell_1,v_1}$ is to ensure $x_1=1$, and the effect of applying $\pi^1_{\ell_4,v_5}$ is to ensure $x_4=1$. We represent the object 1 on the right hand side by dotted edges. The entire edge is dotted, rather than just the part parallel to the string, because a string operator ending at $\ell$ is only defined on the ground state of $A_\ell$.
  • Figure 3: A string operator defined on excited states is resolved into local operators on individual vertex Hilbert spaces, as in Figure \ref{['fig:stringExpansion']}.
  • Figure 4: Condensation morphisms between $3$ nearby copies of $A$, with time depicted in the vertical direction. Blue segments depict paths on the underlying lattice. Notice that one morphism involves braiding, depending on perspective.

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Corollary : \ref{['cor:tubeKetbra']}
  • Lemma 2.5
  • proof
  • Definition 2.6: MR1782145,MR3447719
  • Definition 2.7
  • Lemma 2.8
  • ...and 23 more