A bstract We propose weak Hopf symmetry as a general framework to explore (1+1) D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry H × Ĥ, where H is a weak Hopf algebra and Ĥ is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom (H) × Cocom (Ĥ), corresponding to the cocommutative subalgebras of H × Ĥ. An essential weak Hopf sub-symmetry is Cocom (H) × Rep (H), which, in the finite group case, reduces to the familiar symmetry G × Rep (G). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry S S via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.
Zhenzhen Jia (Mon,) studied this question.