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Physical Review

Bose-Einstein Gas with Repulsive Interactions: General Theory

K. A. Brueckner, K. Sawada

Published Jun 15, 1957• 225 citations
Publisher

Abstract

The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and $a$ the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach $\frac{{q}^{2}}{2M}$ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.

DOI
10.1103/physrev.106.1117

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We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.

arXiv:cond-mat/0310460•Comptes Rendus Physique•Jan 1, 2004•34 citations
The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of (\ensuremathρa3)12{(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}(\ensuremathρa3)21​, where \ensuremathρ\ensuremath{\rho}\ensuremathρ is the density and aaa the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order (\ensuremathρa3)12{(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}(\ensuremathρa3)21​ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach q22M\frac{{q}^{2}}{2M}2Mq2​ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.
A method developed by the authors for treating Bose-Einstein particles at low temperature is specialized to the case of hard sphere interactions. It is shown that the integral equations determining the energy can in this case be replaced by an integral involving the unknown excitation spectrum. The energy is first determined as a power series in ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the hard-sphere radius. This solution is shown to diverge badly at high density. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low density and to a good approximation at high density. The excitation spectrum of the system is shown to have a peculiar nonmonotonic relation between energy and momentum very similar to that deduced by Landau from the low-temperature properties of He4{\mathrm{He}}^{4}He4.
The explicit evaluation of the ground-state energy of a Bose gas interacting through a two-body repulsive potential as an expansion essentially in powers of the density of the particles and the scattering length is given. It is shown that the problem can be treated by using diagrammatical analysis up to any orders. The energy was investigated up to the fourth order in the ordinary sense of a perturbation expansion in terms of the scattering length, but including terms up to infinite order for special classes of diagrams. The energy obtained contains in general a $\ensuremath{\Sigma}{n}^{}{C}_{n}{\ensuremath{\rho}}^{n}\mathrm{ln}\ensuremath{\rho}$ ($n\ensuremath{\geqq}2$, half-integer and integer) dependence, and the coefficient C2{C}_{2}C2​ is evaluated. A self-consistent treatment is given following this analysis and is shown to lead to the formulation of Beliaev.
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 ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where lnf(r)\mathrm{ln}f(r)lnf(r) has an infinite range. The other factor is assumed to have this same form but with lnf(r)\mathrm{ln}f(r)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 kkk, the structure factor S(k)S(k)S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature TTT, S(k)S(k)S(k) tends to the constant value kBTmc2\frac{{k}_{B}T}{m{c}^{2}}mc2kB​T​, where mmm is the mass of the particles and ccc the velocity of propagation of the phonons. The momentum distribution nk{n}_{k}nk​ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-kkk behavior of both S(k)S(k)S(k) and nk{n}_{k}nk​ 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.
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.
The properties of Bose-Einstein-condensed systems of hard spheres at high density are studied by taking into account higher-order corrections beyond the Hartree-Fock-Bogoliubov approximation. Based on a diagrammatic technique of the perturbation expansion starting from the Bogoliubov approximation, we show that the effective interactions separate into two distinct parts: the screened and condensate-mediated interactions. The latter is described by the exchange of virtual quasiparticles between noncondensate bosons via the Bose-Einstein condensate and is attractive for small energy transfer. We calculate the condensation temperature, excitation spectrum, and condensate density in the whole region of the gas parameter. It is shown that the condensate temperature decreases compared to that of an ideal Bose gas due to hard-sphere interactions. We find that the condensate-mediated attraction between noncondensate bosons leads to an enhancement of the anomalous self-energy and, as a consequence, to the emergence of a roton minimum in the excitation spectrum and to the strong suppression of the condensate fraction in the region of high density.
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.