Investigating Localization Transitions with the Forward Approximation

In this Ph.D. thesis we review the Anderson localization problem and its relevance in the current panorama of Physics; we discuss the single- and many-body problem and the features of the localization transition. We discuss in detail the forward approximation of the locator expansion, showing how it is a powerful tool for inspecting both the single- and the many-body localization transition. We analyze its predictions in the Bethe lattice, in the hypercubic lattice, and in a many-body Heisenberg model. The approximation provides an upper bound for the transition point; this result becomes increasingly accurate as the dimensionality of the system increases. We also find that the forward approximation result can be closely approximated by a single term as long as cancellations in the full series expansion are not relevant (this happens in the single particle Anderson systems but not in the many-body case). Moreover, we study a system interacting with a mesoscopic bath which shows peculiar localization properties; this is done both analytically through the forward approximation and numerically through exact diagonalization techniques. We find that, as the coupling with the bath increases, the system goes through a crossover between two mechanisms for localization, i.e. from Anderson to Zeno localization. The stability of the localized state is a non-monotonic function of the coupling with the bath, as the increasing hybridization of the bath states allows different particle hopping processes.