Correlations in the clean and disordered Bose Hubbard model

Ultracold atoms in optical lattices are a favorable setting for exploring out-of-equilibrium phenomena in interacting quantum systems and have attracted significant attention in recent years. A critical feature of these systems is the highly tunable nature of the experimental parameters, which allows precise control and exploration of out-of-equilibrium quantum dynamics and has paved the way for exploring a diverse set of quantum many-body systems theoretically and experimentally. The Bose-Hubbard model (BHM), the simplest model for interacting bosons on a lattice has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model. The main emphasis of this thesis is to explore how information propagates in many-body quantum systems and how the presence of disorder can impact the equilibrium and out-of-equilibrium state of these systems. We focus on the question of how the speed of information propagation depends on dimensionality and model parameters in the BHM and whether the two-particle irreducible strong coupling (2PISC) theory can match experimental observations. We compute the group and phase velocities for the spread of single-particle correlations across one, two, and three dimensions and investigate how these velocities are influenced by various factors, including the quench protocol, chemical potential, temperature, and dimensionality. We develop a two-particle irreducible strong-coupling approach to the disordered BHM that allows us to treat both equilibrium and out-of-equilibrium situations. We obtain equations of motion for spatiotemporal correlations and explore their equilibrium solutions. We employ a low-frequency approximation to deduce energy excitations and spectral weights for quasi-particle/hole excitations and then establish the phase boundary for the Mott insulator phase. We study the equilibrium phase diagram as a function of disorder strength and discuss applications of the formalism to out-of-equilibrium situations. We also note that the disorder strengths where the emergence of non-ergodic dynamics was observed in a recent experiment [Choi et al. Science $\textbf{352}$, 1547 (2016)] appear to correspond to the Mott insulator – Bose glass phase boundary. Finally, we used neural network (NN) models to improve the predictions of the 2PISC in the weak interacting limit. We train a NN with the data obtained from the 2PISC and exact diagonalization calculation for various ranges of the parameter space and compare the single-particle density matrix generated from NN with the exact diagonalization and effective theory results. We establish that this approach improves the 2PISC method results and suggest that it may be applicable to out-of-equilibrium dynamics in systems beyond the BHM.