Attractive Boson and the Gas-Liquid Condensation

We calculate a grand partition function of the attractive Bose gas in the infinite space within some approximations. Using the idea of the Yang-Lee zeros, it is proved that the gas-liquid condensation occurs before the conventional condition of the Bose-Einstein condensation is satisfied. Further, it is pointed out that Bosons with a zero momentum play a role of a trigger to this gas-liquid condensation. We discuss its implication to the trapped atomic gas. PACS numbers: 03.75.Fi, 05.30.Jp, 64.60.-i Typeset using REVTEX ∗e-mail address: [email protected] 1 The study of the relationship between the Bose-Einstein condensation (BEC) and the gas-liquid condensation (GLC) is a long-standing problem [1]. Recently the experimental realization of BEC in dilute atomic vapors changed this academic problem to a realistic one [2]. Normally, the BEC, a condensation into a lowest energy level at low temperature and high density, is thought to be an essentially different phenomenon from the GLC in following points: (1) The BEC is caused by the Bose statistics, not by the interparticle interaction as in the GLC. (2) The BEC is sometimes described as a condensation in momentum space, while the GLC occurs in coordinate space. An interesting point is that the reason by which we distinguish them is not so obvious as it looks. Both condensations have a similar thermodynamic manifestation as a first-order phase transition. Further, in two types of the quantum gas, the attractive Fermi gas and the attractive Bose gas, the relationship between the BEC and the GLC manifests itself quite differently. For Fermions, the circumstance with low temperature and high density stabilizes the Cooper pairs: a BEC in a general sense [3]. Because of the Fermi statistics, however, two Fermions experience a strong repulsive force in the short distance. Hence, when the attractive force is increased within the Bardeen-Cooper-Schriefer (BCS) model, the GLC is impossible, a rigorous proof of which is given recently [4]. For Bosons, they remain in the gas state at high temperature and low density, because the quantum statistics plays a minor role. At low temperature, however, the Bosons have no large positive kinetic energy to stabilize the system behavior, especially in the BEC state. Thus, the attractive force plays a dominant role, so that the compressibility is no longer positive definite. The dilute Bose system will collapse into the dense one, leading to the GLC. Conversely, when the particle density increases by the GLC, an overlapping of the wave function is likely to cause the BEC. There are some kinematical or thermodynamical evidences suggesting the instability of the attractive Bose gas. (1) The Bogoliubov model says that if the interaction between the Bosons is attractive, 2 the velocity of sound propagating on the Bose-Einstein condensate would be imaginary, corresponding to a divergence of the density fluctuation and leading to a drastic change of the whole system [5]. (2)A variational argument of the many-Boson system shows that a many-body selfbinding state (liquid) is more likely to be stable than a gas of the bound dimers [6]. (3) Recent stability analysis of the Gross-Pitaevskii equation in the trapped atomic gas with the attractive interaction indicates a collapse of the BEC at the thermal equilibrium. These features suggest that an another type of relationship between the BEC and the GLC exists in the attractive Bose gas. The GLC in the classical imperfect gas is a longstanding and difficult problem [7]. In the attractive Bose gas, however, there is a clear physical reason for the instability originated from the Bose statistics. In view of this, at least for the instability mechanism to the liquid, a model of the GLC in the attractive Bose gas would be simpler than that in the imperfect classical gas or in the attractive Fermi gas. With this in mind, it seems quite natural to consider the BEC and GLC on a common ground. In this paper, we consider a spinless Bose gas with a repulsive core represented by Hre and a weak attractive s-wave pairing interaction g(< 0) [8]: