  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.
