Ergodicity breaking and restoring in disordered quantum systems

Understanding the effects of disorder in quantum many-body systems is one of the central challenges of contemporary statistical physics. Its interplay with interactions and quantum fluctuations profoundly modifies the nature of quantum states and their dynamics, giving rise to phenomena such as localization, ergodicity breaking, and the emergence of complex collective behavior. This thesis presents new results on two classes of disordered quantum systems. The first part is devoted to the study of the many-body localization transition, a paradigmatic example of disorder-induced ergodicity breaking. The critical properties of the transition are investigated using a numerical renormalization-group approach, establishing a connection between its scaling behavior and that of the Anderson localization transition on tree-like graphs. The second part focuses on the zero-temperature quantum spin-glass phase of the two-dimensional bond-disordered Heisenberg model. Combining large-scale numerical simulations with a semiclassical expansion, evidence is provided for the existence of the spin-glass phase. The effective model governing the semiclassical fluctuations is further analyzed to characterize the low-energy excitation spectrum and its localization properties. Finally, the weak-disorder regime is investigated and its implications for the ferromagnetic phase are discussed. Overall, the results presented in this thesis contribute to the understanding of the effects of disorder in interacting quantum many-body systems, encompassing both localization phenomena and the collective behavior of disordered quantum magnets.