Bose-Einstein condensation of incommensurate solid He-4

It is pointed out that the simulation computation of energy performed so far cannot be used to decide if the ground state of solid He-4 has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). The best variational wave function, a shadow wave function, gives an incommensurate state, but the equilibrium concentration of vacancies remains to be determined. We have computed the one-body density matrix in solid He-4 for the incommensurate state by means of an exact ground state projector method in which incommensurability occurs spontaneously. We find a vacancy induced Bose-Einstein condensation of about 0.23 atoms per vacancy at a pressure of 54 bar. This means that bulk solid He-4 is supersolid at low enough temperature if the exact ground state is incommensurate.