Bose-Einstein condensation and excitations in solid He-4 with vacancies

We have studied the ground state and excited state properties of solid He-4 on the basis of the variational shadow wave function technique (SWF), which allows for relaxation and delocalisation of vacancies. We have found that a finite concentration of vacancies, if present, induces Bose-Einstein condensation (BEC) of the atoms at density close to the T = 0 K melting where vacancies are delocalised. No BEC is present in a perfect crystal or in a defected solid at higher densities. We have extended this technique to study longitudinal phonons in solid He-4 and to study the vacancy excitation at a finite momentum; we have been able to compute for the first time the vacancy excitation spectrum in solid He-4 at density close to melting. Our results give a band width of about 8 K