Spectral and dynamical signatures of localization in many-body systems

The interplay between strong interactions and disorder is at the core of one of the most exciting and active fields of modern quantum condensed matter physics. In the last two decades, attempts to characterize interacting disordered quantum systems in terms of both static and dynamic properties have been the focus of much attention, mainly due to experimental advancements in quantum simulation techniques. In this Thesis, we explore aspects of disorder-induced localization in different models and setups, ranging from strongly interacting spin chains to free models of spinful fermions. Firstly, we study the emergence of local conserved quantities (;-bits) in paradigmatic models of localization through the lens of the quench dynamics of the survival probability. We discuss the appearance of periodic cusp singularities in the unitary evolution of the survival probability as a new signature of localization, which can be exactly captured and predicted by an effective model of two-level systems undergoing collective Rabi oscillations. We then test the sensitivity of many-body localized (MBL) systems to global perturbations by studying the decay of the fidelity between quantum states in different disorder regimes. Our numerical results unveil a rich decay structure and a direct connection to spectral measures of localization, suggestive of promising directions for the experimental probing of eigenstate fractality. Lastly, we turn our attention towards localization effects in non-interacting models, investigating the stability of spin-polarized transport in a linear chain of atoms grafted onto a magnetic substrate. We demonstrate the robustness of spin-filtering effects against weak disorder, confirming the proposed system as a good candidate model for realizations of spin-selective transport devices