We investigate the superfluid-insulator transition in the disordered two-dimensional Bose-Hubbard model through quantum Monte Carlo simulations. The Bose-Hubbard model is studied in the presence of site disorder, and the quantum critical point between the Bose glass and superfluid is determined in the grand canonical ensemble at $\ensuremath{\mu}/U=0$ (close to $\ensuremath{\rho}=0.5$), $\ensuremath{\mu}/U=0.375$ (close to $\ensuremath{\rho}=1$), and $\ensuremath{\mu}/U=1$ as well as in the canonical ensemble at $\ensuremath{\rho}=0.5$ and 1. Particular attention is paid to disorder averaging, and it is shown that a large number of disorder realizations are needed in order to obtain reliable results. Typically, more than
100000 disorder realizations were used. In the grand canonical ensemble, we find
Ztc/U=0.112(1) with $\ensuremath{\mu}/U=0.375$, significantly different from previous studies. When compared to the critical point in the absence of disorder (
Ztc/U=0.2385), this result confirms previous findings showing that disorder enlarges the superfluid region. At the critical point, we then study the dynamic conductivity.
In this dissertation, we study the superfluid-insulator quantum phase transition in disordered boson systems. Recently, there has been considerable controversy over the validity of the scaling relations of the superfluid--Bose-glass quantum phase transition in three dimensions. Results from experimental and numerical studies on disordered quantum magnets contradict the scaling relations and the associated conventional scaling hypothesis for the singular part of the free energy. We determine various critical exponents of the superfluid--Bose-glass quantum phase transition in three-dimensional disordered Bose-Hubbard model through extensive Monte Carlo simulations. Our numerical study shows the previous studies on disordered quantum magnets were performed outside the quantum critical region, and our results for the critical exponents are in perfect agreement with previous theoretical predictions. We next move to study theoretically and numerically the superfluid-insulator quantum phase transition in one-dimensional disordered boson systems. While the superfluid-insulator transition in the weak disorder limit is well understood through the perturbative renormalization group study by Giamarchi and Schulz, transitions in the strong disorder regime are beyond the reach of this method. This problem was recently attacked by Altman et al. with the real space renormalization group method. They reached the conclusion that the superfluid-insulator transition in the strong disorder regime can be explained by the Coulomb blockade physics of weak links. However, their method is actually uncontrolled. Taking account of the crucial role of the hydrodynamic renormalization of weak links, finally, Pollet et al. put forward an asymptotically exact renormalization group theory of the superfluid-insulator transition. Based on this theory, we are able to provide an accurate description of the interplay between the well-known Giamarchi-Schulz criticality and the new weak-link criticality. A significant part of the ground-state phase diagram of one-dimensional disordered Bose-Hubbard model at unit filling is also determined numerically. In particular, we established the position of the multicritical point beyond which the new weak-link criticality takes over of the Giamarchi–Schulz criticality.