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Physical Review

System of Bosons with Attractive Interactions

K. K. Singh, Satish Kumar

Published Oct 5, 1967• 12 citations
Publisher

Abstract

The problem of a system of bosons at zero temperature, interacting via a strong short-ranged repulsive interaction and a weak long-ranged attraction, is treated by the methods of quantum-field theory. It is shown that if the parameters of the attractive part of the interaction are suitably restricted, an approximation to the self-energy parts of the Green's functions, similar to that used by Beliaev for the case of purely repulsive interactions, leads to an excitation spectrum which in the limit of small momenta represents phonons. The approximation also yields an $N$-body bound state corresponding to zero pressure. Apart from minor differences, these results are in agreement with an earlier calculation of Huang.

DOI
10.1103/physrev.162.173

Related papers

System of Bosons with Attractive Interactions. II. Low-Temperature Properties

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We calculate the energy and the condensate fraction of a system of trapped bosons interacting via a short-range two-body potential with positive scattering length. The potential is attractive and has a two-body bound state. When the scattering length is small compared to the trap length the system is model independent: all potential models —attractive, repulsive and zero-range— provide similar results. When the scattering length is large the attractive model differs qualitatively from the repulsive and zero-range models. In this regime the system with attractive potential becomes independent of the scattering length, with both the energy and the condensate fraction converging towards finite constants.

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Trapped Bose gases with large positive scattering length

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arXiv:0706.3300•Europhysics Letters (EPL)•Jul 19, 2007•11 citations
The problem of a system of bosons at zero temperature, interacting via a strong short-ranged repulsive interaction and a weak long-ranged attraction, is treated by the methods of quantum-field theory. It is shown that if the parameters of the attractive part of the interaction are suitably restricted, an approximation to the self-energy parts of the Green's functions, similar to that used by Beliaev for the case of purely repulsive interactions, leads to an excitation spectrum which in the limit of small momenta represents phonons. The approximation also yields an NNN-body bound state corresponding to zero pressure. Apart from minor differences, these results are in agreement with an earlier calculation of Huang.
The work of an earlier paper on the ground-state properties of a system of bosons interacting via a strong short-ranged repulsive potential and a weak long-ranged attractive potential is extended to nonzero temperatures. The technique of temperature-dependent Green's functions is employed. It is shown that with the same restrictions on the two-body potential as imposed in the case of the ground state, it is possible to calculate in a consistent fashion the leading low-temperature contributions to the various thermodynamic functions. The results provide support for Landau's point of view for treating the system at low temperatures as a collection of noninteracting quasiparticles. The isotherms of the system at the low temperatures considered indicate a first-order transition between a phase with a condensate and one without a condensate, and also the possibility, at higher temperatures, of three phases.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k)E(\mathrm{k})E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where nnn is the density and f0{f}_{0}f0​ is the scattering length for the assumed two-body interaction between the bosons. The coefficients aaa and bbb are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient bbb is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
We calculate the energy and the condensate fraction of a system of trapped bosons interacting via a short-range two-body potential with positive scattering length. The potential is attractive and has a two-body bound state. When the scattering length is small compared to the trap length the system is model independent: all potential models —attractive, repulsive and zero-range— provide similar results. When the scattering length is large the attractive model differs qualitatively from the repulsive and zero-range models. In this regime the system with attractive potential becomes independent of the scattering length, with both the energy and the condensate fraction converging towards finite constants.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
We calculate the energy and the condensate fraction of a system of trapped bosons interacting via a short-range two-body potential with positive scattering length. The potential is attractive and has a two-body bound state. When the scattering length is small compared to the trap length the system is model independent: all potential models —attractive, repulsive and zero-range— provide similar results. When the scattering length is large the attractive model differs qualitatively from the repulsive and zero-range models. In this regime the system with attractive potential becomes independent of the scattering length, with both the energy and the condensate fraction converging towards finite constants.