**Abstract:**

The study of symmetry lies at the heart of various parts of physics. However, the symmetries conventionally studied in a lot of the literature are mostly restricted to either on-site unitary symmetries or lattice symmetries. While such symmetries are sufficient to explain several physical phenomena, the recent discoveries of weak ergodicity breaking phenomena such as Hilbert space fragmentation and quantum many-body scars have called for a reconsideration of the definition of symmetry in quantum many-body physics. In this talk, I will discuss a general mathematical framework to define symmetries based on so-called commutant algebras, which leads to a generalization of the conventional notion of symmetry and explains weak ergodicity breaking in terms of unconventional non-local symmetries. In addition, it reveals a novel characterization of symmetries as frustration-free ground states of local superoperators, which leads to efficient numerical methods for obtaining symmetries, and to insights on the nature of symmetries realizable in systems with locality. In addition, the low-energy excitations of the superoperator can be related as hydrodynamic modes of the corresponding symmetries, leading to a unified understanding of conventional and unconventional symmetries and associated hydrodynamic modes in a single framework.

Reference: https://arxiv.org/abs/2309.15167