1+1d SPT phases with fusion category symmetry: interface modes and non-abelian Thouless pump

We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry $\mathcal{C}$ in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the $\mathcal{C}$-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two $\mathcal{C}$-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different $\mathcal{C}$-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of $\mathcal{C}$-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a $\mathcal{C}$-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.