Yang successfully defends thesis, “Topological phases enriched with fusion category symmetries”

August 22, 2025

On August 19,  Xinping Yang, successfully defended  the thesis, “Topological phases enriched with fusion category symmetries” (aadvisor: Meng Cheng).

Yang explained, “The central question of condensed matter physics is the study of phases or phase transitions. It was believed that Landau’s paradigm fully characterizes phases and yet the discovery of topological phases of matter reveals that it was only a partial description and awaits an alternative unified framework to classify phases. Topological phases of matter are gapped quantum phases with physical properties encoded by global topological invariants rather than local order parameters. When they are enriched with global symmetries, the physical properties can be protected by the symmetries themselves, one type of which is the symmetry-protected topological phase (SPT). Thus Landau’s paradigm relying on symmetry breaking patterns and local order parameters cannot capture these exotic phases of matter. It turns out that macroscopically the complete set of physical observables at the long wavelength limit are summarized by dualizable objects in a category of their dimension. Category theory also offers a general enough framework to describe generalized symmetries. In particular, non-invertible symmetries in 1D are characterized by (unitary) fusion categories. On the other hand, the microscopic description of a phase requires a detailed quantum spin chain defined at the infinite volume limit, which is achieved within the operator algebra framework. Combining both methods, we want to understand topological phases enriched with fusion category symmetries in the UV, including an on-site symmetry action, commuting-projector fixed point model and other non-trivial physical responses.”

Yang  will be joining the Perimeter Institute for Theoretical Physics as a  postdoctoral fellow.

Thesis abstractSymmetry 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.