Xinping Yang

Symmetry and its anomaly constrain a system’s dynamics and provide a universal characterization of its behaviors. It serves as a powerful tool to understand exotic phases of matter, especially symmetry-protected topological (SPT) orders. The notion of generalized symmetries requires extending our previous understanding of topological phases to those enriched by fusion category symmetries, yet new mathematical formalism is needed to apply unitary fusion category symmetries to a microscopic system. In this talk, I will start with a model-independent description of a fusion spin chain viewed as a net of C* algebras. From a microscopic SymTFT, we extract categorical structures describing symmetry and symmetry charges. Then I will introduce a general fixed-point lattice construction of (1+1)d SPTs with unitary fusion category symmetries, realized in a tensor-product Hilbert space with an “onsite” matrix-product-operator (MPO) presentation of the Hopf C*-algebra symmetry operators. Within this construction, I will address that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase, and discuss an alternative characterization of SPT phases using the Q-system in the charge category.