Unconventional Symmetries from Commutant Algebras

The study of symmetry lies at the heart of various areas of physics. In equilibrium physics, symmetries are useful in classifying phases of matter, and in non-equilibrium physics, they are necessary to understand the phenomenon of thermalization. Most symmetries conventionally studied in the literature are examples of so-called on-site unitary symmetries. While such symmetries are sufficient to explain several physical phenomena, the recent discovery of weak violations of ergodicity in non-integrable quantum many-body systems, e.g., Hilbert space fragmentation and quantum many-body scarring, has called for a generalization of the notion of symmetry. The conventional theory of thermalization in quantum many-body systems demands that any simple initial state within a given symmetry sector explores the full Hilbert space of that sector under the dynamics of a symmetric Hamiltonian. However, in quantum many-body systems exhibiting weak ergodicity breaking, the Hilbert space possesses additional unexpected dynamically disconnected subspaces that cannot be explained in terms of on-site unitary symmetries, leading to a breakdown of conventional thermalization. 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 notion of symmetry beyond on-site unitary ones. The unconventional symmetries revealed by this framework provide precise explanations for several dynamical phenomena ranging from weak ergodicity breaking to strong zero modes, allowing us to cast all of them into a single unified framework and also systematically construct local Hamiltonians that exhibit these phenomena.