(Top) Schematic illustration of lattice bosons near unit filling. At in the Mott insulating phase (MI) when the effective wave function of each particle spreads only a finite distance . For (), the extra particles (holes) travel freely within the essentially inert background, and the state is a superfluid (SF) for arbitrarily small . (Bottom) Phase diagram, showing Mott insulating (including those with higher integer filling) and SF phases. The transitions through the tips of the Mott lobes at (and following the dashed lines at constant filling beyond them) have a special particle-hole symmetry.