Figure 1: (Top) Schematic illustration of lattice bosons near unit filling. At $n=1$ in the Mott insulating phase (MI) when $(J/U)<(J/U)c$ the effective wave function of each particle spreads only a finite distance $ξ(J)$. For $n>1$ ($n<1$), the extra particles (holes) travel freely within the essentially inert background, and the state is a superfluid (SF) for arbitrarily small $|n-1|$. (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 $(J/U)c$ (and following the dashed lines at constant filling beyond them) have a special particle-hole symmetry.