On lattice models of gapped phases with fusion category symmetries

We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum an $H$-simple left $H$-comodule algebra, where $H$ a dimensional semisimple Hopf algebra. show the actions of $\mathcal{C}$ boundary conditions these TQFTs are represented by module categories over $\mathcal{C}$. This agrees classification symmetry also find have at level lattice models hence provide realizations symmetries. As application, we discuss edge modes Hamiltonians. Finally, mention can extend to case anomalous replacing algebra pseudo-unitary connected weak