Phonons and the Properties of a Bose System

Abstract

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 ${\Pi }_{i<j}f\left(r_{\mathrm{ij}}\right)$, where $\ln f\left(r\right)\mathrm{}$ has an infinite range. The other factor is assumed to have this same form but with $\ln f\left(r\right)\mathrm{}$ 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 $S\left(k\right)\mathrm{}$ is equal to $\frac{\hslash k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$, $S\left(k\right)\mathrm{}$ tends to the constant value $\frac{k_{B}T}{m{c}^2}$, where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution $n_k$ has a $k^{-1\mathrm{}}$ singularity at absolute zero and a stronger, $k^{-2\mathrm{}}$ singularity at finite temperatures. The small-$k$ behavior of both $S\left(k\right)\mathrm{}$ and $n_k$ 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.