Collective modes on top of the Gutzwiller approximation in Hubbard models: a novel tool for quantum correlations

We develop a quantum many-body theory of the Bose-Hubbard (BH) model based on an improved Gutzwiller scheme. Our quantum theory is a generalization of the Bogoliubov theory of weakly-interacting gases and have common features with slave boson techniques. The approach provides accurate results throughout the whole BH phase diagram, from the weakly to the strongly interacting superfluid and across the superfluid-Mott transition [1]. Specifically, we provide (1) a semi-analytical expression for the superfluid stiffness in terms of two-particle correlations between the collective modes of the system and (2) a precise estimation for density fluctuations, for which a quantitative agreement with quantum Monte Carlo data is found. The predictive power of our formalism is shown to include also non-trivial dynamical problems, as the pure dephasing of a two-level impurity in a BH environment [2]. Our description of the BH quantum correlations allows to go beyond the standard spin-boson model and, in particular, to find that the decoherence dynamics is extremely sensitive to the universality class of the superfluid-Mott transition. Finally, we explore exciting perspectives on future applications of our approach, including a recent extension to Fermi-Hubbard systems [3].