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

TL;DR

This work develops a lattice-based framework for symmetry protected topological (SPT) phases in 1+1 dimensions protected by finite non-invertible (fusion-category) symmetries. By encoding fusion-category data in MPS/MPO language, it extracts fiber functor information from ground states, analyzes the interface algebra at phase boundaries, and shows that interfaces between distinct C-SPT phases host degenerate modes, generalizing bulk-boundary correspondence. It introduces a non-abelian Thouless pump for one-parameter families of C-symmetric states, linking adiabatic evolution to automorphisms of fiber functors, and provides concrete Rep(G) and Tambara–Yamagami examples. The paper then proposes a broad conjectural framework classifying X-parameterized families of gapped phases in 1+1d and 2+1d via non-abelian Čech cohomology and higher-fiber functors, offering a unified, categorial path to understand parameterized topological phenomena beyond ordinary group symmetries. These results pave the way for systematic tensor-network realizations and extensions to higher dimensions, where similar non-abelian pumping and interface phenomena are expected to persist.

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.

Figures (4)

• Figure 1: (a) The vector space $F(x)$ represents the state space on a circle twisted by $x \in \mathcal{C}$. (b) The isomorphism $J_{x, y}$ represents the transition amplitude on a pair of pants. (c) The matrix element of $J_{x, y}$ represents the three-point function.
• Figure 2: The symmetry TFT construction of 1+1d systems with symmetry $\mathcal{C}$. The left boundary of the slab is topological and supports topological lines described by $\mathcal{C}$. On the other hand, the right boundary is arbitrary and dictates the dynamics of the 1+1d system. When the right boundary is also topological, the whole 1+1d system becomes topological.
• Figure 3: The middle region of the right boundary is modulated gradually from $\mathcal{M}(0)$ to $\mathcal{M}(2\pi)$. This modulated region becomes an invertible topological line when viewed from far away. After squashing the bulk, the invertible line on the boundary becomes an invertible interface in 1+1d. The correspondence between invertible interfaces and $S^1$-parameterized families was already observed in Sec. \ref{['sec: Interface modes']} in the case of SPT phases.
• Figure 4: An $S^2$-family is viewed as an $S^1$-family of $S^1$-families. Each loop represents an $S^1$-family embedded in the $S^2$-family. The black dot at the top represents a constant family, which we choose to be the initial and final $S^1$-families.

Theorems & Definitions (8)

• conjecture 1
• conjecture 2
• conjecture 3
• conjecture 4
• proposition 1
• proof
• proposition 2
• proof