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.

💡 Research Summary

This paper develops a comprehensive framework for symmetry‑protected topological (SPT) phases in one‑dimensional bosonic systems that are protected by a finite non‑invertible (categorical) symmetry 𝒞. The authors work entirely within the matrix‑product‑state (MPS) formalism, which allows them to extract the categorical data that classify such phases directly from lattice wavefunctions.

First, the authors review the necessary mathematics of unitary fusion categories, emphasizing objects (topological lines), fusion coefficients, associators, and F‑symbols. They recall that SPT phases protected by a fusion‑category symmetry 𝒞 are classified by fiber functors F:𝒞→Vec, i.e. strong monoidal functors to the category of finite‑dimensional vector spaces. This classification is known from topological quantum field theory (TQFT) but had not been explicitly linked to microscopic lattice models.

In the second section the authors show how to read off the data of a fiber functor from a 𝒞‑symmetric injective MPS. They introduce a “non‑abelian factor system” and a mixed transfer matrix that couples two MPS tensors. By evaluating a triple inner product of infinite MPSs they obtain quantities that exactly match the F‑symbols of the underlying fiber functor. This construction mirrors the computation of defect partition functions in a TQFT with symmetry defects, thereby providing a concrete lattice realization of the abstract categorical classification. An abelianization procedure then maps the non‑abelian data to ordinary group cohomology, allowing the definition of a cohomological invariant for the phase. As an explicit example they compute the fiber functor for the G × Rep(G)‑symmetric cluster state.

The third part of the paper focuses on interfaces between two possibly different 𝒞‑SPT phases. The authors identify the algebra of symmetry operators that act on the interface, which they call the “interface algebra” 𝔄. By studying the representation theory of 𝔄 they prove two key results: (i) when the two bulk phases are distinct, 𝔄 admits no one‑dimensional (scalar) representations, implying that any interface must support a degenerate ground‑state sector. This is the categorical analogue of the bulk‑boundary correspondence (or anomaly inflow) familiar from group‑symmetry SPTs. (ii) For a self‑interface (the same bulk phase on both sides) one‑dimensional representations do exist, and they are in one‑to‑one correspondence with the automorphism group Aut(F) of the fiber functor. The authors illustrate these statements with concrete calculations for Rep(D₈)‑symmetric SPT phases, showing both non‑degenerate and degenerate interface spectra.

In the fourth section the authors consider one‑parameter families of 𝒞‑symmetric invertible states, i.e. maps S¹→{𝒞‑SPT ground states}. They analyze the gauge redundancy of the MPS tensors along the loop and define an invariant χ∈Aut(F) that measures the non‑trivial winding of the family. When χ is non‑trivial, the loop implements a “non‑abelian Thouless pump”: a local excitation carrying a non‑abelian charge (an element of Aut(F)) is pumped across the system during one period. This generalizes the classic Thouless charge pump, which only transports abelian U(1) charge. The authors construct explicit S¹‑families for Rep(G)‑symmetric states and compute the associated χ, confirming the presence of a non‑abelian pump.

Finally, the authors propose a broad conjectural classification of parameterized families of general gapped phases with finite non‑invertible symmetry, extending beyond invertible SPTs and beyond 1+1 dimensions. For a given 𝒞‑module category 𝓜 (which labels a gapped phase), they conjecture that the moduli space of 𝒞‑symmetric gapped systems in that phase is the classifying space B Fun_𝒞(𝓜,𝓜)^{inv}, where Fun_𝒞(𝓜,𝓜)^{inv} is the 2‑group of invertible 𝒞‑module functors and natural transformations. Consequently, X‑parameterized families are classified by the non‑abelian cohomology Ĥ¹(X, Fun_𝒞(𝓜,𝓜)^{inv}) in 1+1 d, and by Ĥ²(X, Fun_𝒞(𝓜,𝓜)^{inv}) in 2+1 d. They support these conjectures with several examples involving Tambara‑Yamagami categories (Rep(D₈), Rep(Q₈), Rep(H₈)) and discuss how the same structure should appear for higher‑dimensional non‑chiral phases.

In summary, the paper achieves three major advances: (1) it provides a practical MPS‑based method to extract the fiber‑functor data that classifies 𝒞‑SPT phases; (2) it uncovers a universal interface algebra whose representation theory guarantees degenerate interface modes between distinct 𝒞‑SPT phases, thereby extending bulk‑boundary correspondence to categorical symmetries; and (3) it defines a non‑abelian Thouless pump as a topological invariant of one‑parameter families, and proposes a unified cohomological classification of parameterized families of gapped phases with non‑invertible symmetries in any dimension. These results open the door to systematic construction and diagnosis of exotic topological phases protected by categorical symmetries, and suggest new experimental signatures such as degenerate boundary spectra and non‑abelian charge pumping.