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Contour-time approach to the disordered Bose-Hubbard model in the strong coupling regime

There has been considerable interest in the disordered Bose Hubbard model (BHM) in recent years, particularly in the context of thermalization and many-body localization. We develop a two-particle irreducible (2PI) 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 spatio-temporal correlations and explore their equilibrium solutions. 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 $\bf{352}$, 1547 (2016)] appear to correspond to the Mott insulator -- Bose glass phase boundary.