Figure 2: (Top) Schematic illustration of lattice bosons near unit filling in the presence of bounded disorder. If the disorder is not too strong there is still a Mott phase with a finite energy gap $ε(J)$ for adding or removing particles. Unlike in the pure case, superfluidity is not generated immediately with the addition of particles. For sufficiently small $|n-1|$, the additional particles are Anderson localized by the residual random background potential of the effectively inert layer. The finite compressibility distinguishes this Bose glass phase from the Mott phase. The superfluid critical point $μsf(J)$ occurs only once the added particles have sufficiently smoothed the background potential that its lowest lying states become extended. (Bottom) Phase diagram, showing Mott, Bose glass, and superfluid phases. The Bose glass phase, as now proven in Ref. [3], always intervenes between the Mott and superfluid phases. Being effectively detached from the underlying lattice, the nature of the Bose glass-superfluid transition is believed to be independent of position along the line—there are no points of special symmetry.