Quantum effects in glasses at ultra-low temperatures

Glasses at ultra-low temperatures present several puzzling phenomena. A notable example is the anomalous (i.e., non-Debye) behavior of thermodynamic quantities at temperatures lower than 1 K. A comprehensive quantum theory able to explain these phenomena has not been developed so far. In this thesis, we tackle this long-standing problem with different and innovative perspectives, employing various physical models, and several analytical and numerical techniques. We mainly explore two different but complementary approaches. In the first approach, we investigate the thermodynamics of models for ultra-low temperature glasses, with particular attention to mean-field models. Specifically, exploiting hard-sphere systems and constraint satisfaction problems as a minimal model for structural glasses, we explore their jamming transition both in the classical and the quantum regime. In the second approach, we focus on finite-dimensional models. We analyze the quantum dynamics of the two-level system model for glasses and generic many-body localized systems, providing clues for the presence of a deep connection between glasses and quantum many-body localized systems. This thesis aims at estimating both qualitatively and quantitatively the effects of quantum mechanics on glasses at ultra-low temperatures. In the literature, only a few studies have considered glasses deep in their quantum regime, partly due to the analytical and computational challenges this posits. Nevertheless, this perspective promises to have wide-ranging applications. One of our ambitious goals is to take a first, substantial step to unveil the possible origin of long-standing discrepancies observed between theory and experiments in ultra-low temperature glasses. Moreover, we would like to predict the presence of new experimental regimes that might be interesting to investigate.