An approach is presented toward validating the assumption that the ground state of bosons with repulsive interactions at low densities is characterized by macroscopic occupation of the zero momentum level. We use a cell model which affords a simple description of the high-density region, where fluctuations in number density are small and where no single-particle level is macroscopically occupied. As the density decreases, fluctuations increase, and we reach a critical density at which the small fluctuation approximation becomes unstable with respect to plane wave states of zero momentum. At this critical density, the single-particle energy gap disappears, and the dependence of excitation energy on momentum changes from quadratic to linear, for small values of momentum.

• Received 2 August 1962

DOI:https://doi.org/10.1103/PhysRev.129.959