Generic isolated interacting quantum systems are believed to be ergodic, i.e. any simple initial state evolves to a thermal state at late-times, forming a barrier to the protection of quantum information. A sufficient condition for the thermalization of initial states in an isolated quantum system is the Eigenstate Thermalization Hypothesis (ETH). A complete breakdown of ETH is well-known in two kinds of systems: Integrable and Many-Body Localized, where none of the eigenstates satisfy it. In this dissertation, we introduce two new mechanisms of ergodicity breaking: Quantum Scars, and Hilbert Space Fragmentation. In both these cases, ETH-violating eigenstates coexist with ETH-satisfying ones, and thus the fate of an initial state under time-evolution depends on the properties of eigenstates it has weights on.

We obtain the first analytical examples of quantum scars by solving for several excited states in a family of non-integrable quantum systems in one dimension: the AKLT models. These exact eigenstates include an infinite tower of states from the ground state to the highest excited state. The states in the middle of the spectrum obey a logarithmic scaling of entanglement entropy with system size, contrary to the volume-law scaling predicted by ETH. We further show the close connections between such quantum scarred models and Parent Hamiltonians of Matrix Product States, as well as connections to the phenomenon of eta-pairing known in the context of superconductivity in Hubbard models.

Hilbert space fragmentation occurs in constrained systems with a center-of-mass or dipole moment conservation law, which naturally arises within a Landau level in quantum Hall systems or in systems subjected to a large electric field limit. We show that the Hilbert space of such systems fractures into several dynamically disconnected Krylov subspaces that are not distinguished by simple symmetries, and thus constitute a violation of conventional ETH. We show that ETH can be modified to apply to each connected subspace separately, and that large integrable and non-integrable subspaces can co-exist within the same system.