Building 1D lattice models with $G$-graded fusion category

We construct a family of one-dimensional (1D) quantum lattice models based on $G$-graded unitary fusion category ${\mathcal{C}}_G$. This family realize an interpolation between the anyon-chain models and edge models of 2D symmetry-protected topological states, and can be thought of as edge models of 2D symmetry-enriched topological states. The models display a set of unconventional global symmetries that are characterized by the input category ${\mathcal{C}}_G$. While spontaneous symmetry breaking is also possible, our numerical evidence shows that the category symmetry constrains the the models to the extent that the low-energy physics has a large likelihood to be gapless.