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

TL;DR

This paper studies 1+1d symmetry protected topological phases with non-invertible symmetries, revealing interface modes, algebraic structures, and a non-abelian Thouless pump, advancing the classification of such phases.

Contribution

It introduces a framework for analyzing interfaces and invariants in non-invertible symmetry SPT phases, generalizing bulk-boundary correspondence and proposing a non-abelian Thouless pump.

Findings

Degenerate interface modes between different SPT phases.

Algebraic structure of symmetry operators on interfaces.

Classification of parameterized families of SPT states.

Abstract

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.