I will provide a rigorous framework of dynamics in locally interacting systems in any dimension. It is based on pseudolocal dynamical symmetries generalising pseudolocal charges. This generalization proves sufficient to construct a theory of all sufficiently local quantum many-body dynamics in closed, open and time-dependent systems, in terms of time-dependent generalized Gibbs ensembles. These ensembles unify seemingly disparate manifestations of quantum non-ergodic dynamics including quantum many-body scars, continuous, discrete and dissipative time crystals, Hilbert space fragmentation, lattice gauge theories, and disorder-free localization. In the process novel pseudo-local classes of operators are introduced: "projected local", which are local only for some states, and "crypto-local", whose locality is not manifest in terms of any finite number of local densities. This proven theory is intuitively the rigorous algebraic counterpart of the eigenstate thermalization hypothesis and has implications for thermodynamics: quantum many-body systems, rather than merely reaching a Gibbs ensemble in the long-time limit, are always in a time-dependent generalized Gibbs ensemble for any natural initial state. Using the theory two novel types of phase transitions are introduced: 1) The "scarring phase transition" where the order parameter is the locality of the projected local quantities - for certain initial states persistent oscillations are present. 2) The "fragmentation phase transition" for which long-range order is established in an entire phase due to presence of certain non-local strings. Two prototypical, but otherwise mostly intractable, models are solved using the theory: 1) a spin 1 scarred model and 2) the t-J_z model with fragmentation.
References
1. Berislav Buca. Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems. Phys. Rev. X 13, 031013 (2023).
2. Benjamin Doyon. Thermalization and pseudolocality in extended quantum systems. Commun. Math. Phys. 351: 155-200 (2017).