Vacancies in solid He-4 and Bose Einstein condensation

We find that a finite concentration of vacancies induces Bose Einstein condensation (BEC) of the atoms in solid He-4 at density close to the T = 0 K melting where vacancies are delocalized. No BEC is present in the perfect crystal and in the defected solid at higher densities. These results are based on variational Monte Carlo method with shadow wave function which allows for relaxation and delocalization of vacancies, Extension of the theory at finite temperature leads to several possibilities. Depending on presence or not of ground state vacancies and on parameters like energy of formation and effective mass of a vacancy one can have a normal transition with BEC at low temperature or an inverted transition with BEC at high T. He-4 on graphite appears to be a favorable candidate for a state with BEC and spatial order.