Phonons and the Properties of a Bose System

In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ₈<₉f (r₈₉), where lnf (r) has an infinite range. The other factor is assumed to have this same form but with lnf (r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors k, the structure factor S (k) is equal to 2mc, a result that was first derived by Feynman. At a finite temperature T, S (k) tends to the constant value k₁Tm{c^2}, where m is the mass of the particles and c the velocity of propagation of the phonons. The momentum distribution n₊ has a k^-1 singularity at absolute zero and a stronger, k^-2 singularity at finite temperatures. The small-k behavior of both S (k) and n₊ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.