Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory

We introduce weak Hopf symmetry as a tool to explore (1+1)-dimensional topological phases with non-invertible symmetries. Drawing inspiration from Symmetry Topological Field Theory (SymTFT), we construct a lattice model featuring two boundary conditions: one that encodes topological symmetry and another that governs non-topological dynamics. This cluster ladder model generalizes the well-known cluster state model. We demonstrate that the model exhibits weak Hopf symmetry, incorporating both the weak Hopf algebra and its dual. On a closed manifold, the symmetry reduces to cocommutative subalgebras of the weak Hopf algebra. Additionally, we introduce weak Hopf tensor network states to provide an exact solution for the model. As every fusion category corresponds to the representation category of some weak Hopf algebra, fusion category symmetry naturally corresponds to a subalgebra of the dual weak Hopf algebra. Consequently,the cluster ladder model offers a lattice realization of arbitrary fusion category symmetries.

💡 Research Summary

This paper introduces weak Hopf symmetry as a unifying framework for describing non‑invertible symmetries in (1+1)‑dimensional quantum systems and constructs explicit lattice models that realize such symmetries. The authors start from the well‑known Z₂ cluster state, whose symmetry can be described by Z₂ × Z₂ SPT order and the non‑invertible fusion category Rep(D₈). By generalizing the underlying algebraic structure from a group algebra to an arbitrary weak Hopf algebra H (and its dual Ĥ), they obtain a broad class of models that capture all possible fusion‑category symmetries.

The central construction is the “cluster ladder” model, a one‑dimensional chain obtained by sandwiching a two‑dimensional weak‑Hopf lattice gauge theory between two distinct boundaries: a “symmetry boundary” carrying a comodule algebra K ≅ C