  We study a simple model of interacting bosons on a d-dimensional cubic
lattice whose dynamics conserves both total boson number and total boson dipole
moment. This model provides a simple framework in which several remarkable
consequences of dipole conservation can be explored. As a function of chemical
potential and hopping strength, the model can be tuned between gapped Mott
insulating phases and various types of gapless condensates. The condensed phase
realized at large hopping strengths, which we dub a Bose-Einstein insulator, is
particularly interesting: despite having a Bose condensate, it is insulating,
and despite being an insulator, it is compressible.
