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

## TL;DR

This paper explores a topological phase transition on the edge of 2D $ ext{Z}_2$ topological order, using enriched fusion categories to describe critical points and lattice models to realize these transitions.

## Contribution

It provides a concrete example of topological phase transition on the edge of 2D $ ext{Z}_2$ topological order using enriched fusion categories and lattice models.

## Key findings

- Constructed an enriched fusion category for a gappable non-chiral gapless edge.
- Realized the critical point and categorical ingredients via explicit lattice models.
- Illustrated the theory with a specific 2D $ ext{Z}_2$ topological order example.

## 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.