Ground-state lattice vacancies and Bose-Einstein condensation in a quantum crystal

Abstract

The quantum-lattice-gas model is used to describe the solid phase of a system consisting of Bose particles. Within mean-field theory and by using the concept of pseudospin, the Helmholtz free energy per lattice cell is obtained as a function of the fraction of lattice vacancies, the crystalline long-range order, and Bose-Einstein condensation order parameter. By minimizing the free energy a phase-transition line which separates normal solid phase and Bose-Einstein condensed solid phase (supersolid) is obtained. The transition temperature is a function of the fraction of lattice vacancies. If the fraction of lattice vacancies is exactly zero, the supersolid phase cannot exist. But if the fraction of lattice vacancies approaches zero exponentially, the supersolid phase may exist for a certain range of interparticle interactions. In the later case, the transition temperature goes to absolute zero as the fraction of lattice vacancies approaches zero. The calculation shows that a supersolid phase may exist in real solid $^{4}\mathrm{He}$. A crude estimation that the transition temperature would lie in the range between 0.01 and 0.1 K is given.