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A topological phase transition on the edge of the 2d $\mathbb{Z}_2$ topological order

Wei-Qiang Chen, Chao-Ming Jian, Liang Kong, Yi-Zhuang You, Hao Zheng

TL;DR

This work addresses the problem of describing pure edge topological phase transitions in 2d topological orders by showing that the critical point is governed by an enriched fusion category framework. The authors develop and illustrate a concrete construction for the 2d $\mathbb{Z}_2$ topological order using the triple $(V, \mathbf{Ising} \boxtimes \overline{\mathbf{Ising}}, (\mathbf{Ising} \boxtimes \overline{\mathbf{Ising}})_A)$, and demonstrate how a gapped wall via a condensable algebra $A$ yields $Z(\mathbf{Ising})_A^0 \simeq \mathbf{Toric}$, while the gapless edge is captured by the enriched edge $(V \otimes \overline{V}, Z(\mathbf{Ising}), Z(\mathbf{Ising})_A)$. A lattice realization based on the Wen plaquette model reproduces the edge data and edge partition functions, providing a concrete check of the enriched-edge framework and its bulk–edge correspondence. The results substantiate the categorical approach to pure edge transitions and offer a tangible bridge between abstract anyon condensation, Drinfeld centers, and lattice models of the $\mathbb{Z}_2$ topological order.

Abstract

The unified mathematical theory of gapped and gapless edges of 2d topological orders was developed by two of the authors. It provides a powerful tool to study pure edge topological phase transitions on the edges of 2d topological orders (without altering the bulks). In particular, it implies that the critical points are described by enriched fusion categories. In this work, we illustrate this idea in a concrete example: the 2d $\mathbb{Z}_2$ topological order. In particular, we construct an enriched fusion category, which describes a gappable non-chiral gapless edge of the 2d $\mathbb{Z}_2$ topological order; then use an explicit lattice model construction to realize the critical point and, at the same time, all the ingredients of this enriched fusion category.

A topological phase transition on the edge of the 2d $\mathbb{Z}_2$ topological order

TL;DR

This work addresses the problem of describing pure edge topological phase transitions in 2d topological orders by showing that the critical point is governed by an enriched fusion category framework. The authors develop and illustrate a concrete construction for the 2d topological order using the triple , and demonstrate how a gapped wall via a condensable algebra yields , while the gapless edge is captured by the enriched edge . A lattice realization based on the Wen plaquette model reproduces the edge data and edge partition functions, providing a concrete check of the enriched-edge framework and its bulk–edge correspondence. The results substantiate the categorical approach to pure edge transitions and offer a tangible bridge between abstract anyon condensation, Drinfeld centers, and lattice models of the topological order.

Abstract

The unified mathematical theory of gapped and gapless edges of 2d topological orders was developed by two of the authors. It provides a powerful tool to study pure edge topological phase transitions on the edges of 2d topological orders (without altering the bulks). In particular, it implies that the critical points are described by enriched fusion categories. In this work, we illustrate this idea in a concrete example: the 2d topological order. In particular, we construct an enriched fusion category, which describes a gappable non-chiral gapless edge of the 2d topological order; then use an explicit lattice model construction to realize the critical point and, at the same time, all the ingredients of this enriched fusion category.

Paper Structure

This paper contains 14 sections, 1 theorem, 60 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Categorical Preliminaries
  3. Unitary Modular Tensor Categories
  4. Toric Code and Ising UMTC's
  5. A Gappable Gapless Edge of 2d $\mathbb{Z}_2$ topological order
  6. A Mathematical Theory of Gapless Edges
  7. A Gapped Wall between Double Ising and Toric Code
  8. A Gappable Non-chiral Gapless Edge of $\mathbb{Z}_2$ topological order
  9. The partition functions of $M_{x,y}$
  10. A Lattice Model Realization
  11. Wen Plaquette Model
  12. Majorana Representation
  13. Edge Theory and Partition Functions
  14. Pure Edge Phase Transition

Key Result

Theorem 3.10

$Z(\mathbf{Ising})_A^0 \simeq \mathbf{Toric}$ as UMTC's.

Figures (9)

  • Figure 1: The picture (a) depicts a 2d topological order $(\EuScript{C},c)$ on a 2-disk, together with a 1d gapless edge, propagating in time. When a topological bulk excitation $a\in \EuScript{C}$ is moved to the edge at $t=0$, it creates a topological edge excitation $x$ or a boundary condition $M_x$ for the OSVOA $A_x$ living on the $t>0$ part of the world line. At $t=t_1>0$, the topological edge excitation $x$ is changed to another topological edge excitation $y$. This change creates a wall $M_{x,y}$ between $A_x$ and $A_y$. The picture (b) depicts the quasi-1+1D world sheet obtained by stretching the picture (a) along the dotted arrow from $a$ to $x$.
  • Figure 2: This picture depicts a horizontal fusion of two boundary conditions, together with a horizontal fusion of $M_{x,y}$ and $M_{x',y'}$. For convenience, we abbreviate $x'\otimes x$ to $x'x$ in the picture.
  • Figure 3: This picture depicts a gapped domain wall between the double Ising and the $\mathbb{Z}_2$ topological order, and a gapless edge of the double Ising topological order.
  • Figure 4: Calculation of the partition function of $M_{x,y}$ via path integrals
  • Figure 5: Wen plaquette model on the square lattice (left) and its connection to the toric code model (right). The plaquette operator $O_p$ maps to either the vertex operator $A_v$ or the plaquette (face) operator $B_f$ via local basis transformations of $\sigma^x\leftrightarrow\sigma^z$ on the hollow sites.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 3.1
  • Example 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 8 more