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arXiv:cond-mat/0409022 · Physical Review E

Bose gas beyond mean field

Philippe A. Martin, J. Piasecki

Published Jan 10, 2005• 8 citations
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Abstract

We study a homogeneous Bose gas with purely repulsive forces. Using the Kac scaling of the binary potential we derive analytically the form of the thermodynamic functions of the gas for small but finite values of the scaling parameter in the low density regime. In this way we determine dominant corrections to the mean-field theory. It turns out that repulsive forces increase the pressure at fixed density and decrease the density at given chemical potential (the temperature is kept constant). They also flatten the Bose momentum distribution. However, the present analysis cannot be extended to the region where the mean-field theory predicts the appearance of condensate.

DOI
10.1103/physreve.71.016109

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We study a homogeneous Bose gas with purely repulsive forces. Using the Kac scaling of the binary potential we derive analytically the form of the thermodynamic functions of the gas for small but finite values of the scaling parameter in the low density regime. In this way we determine dominant corrections to the mean-field theory. It turns out that repulsive forces increase the pressure at fixed density and decrease the density at given chemical potential (the temperature is kept constant). They also flatten the Bose momentum distribution. However, the present analysis cannot be extended to the region where the mean-field theory predicts the appearance of condensate.
We present the description of thermodynamic properties of a Bose gas by considering the hierarchy of equations satisfied by imaginary-time Green functions. In the case of an ideal gas the appearance of the off-diagonal long-range order (presence of condensate) is derived in a particularly simple way. The simplicity of the analysis persists at the level of the mean-field theory. Using Kac's scaling of the repulsive binary potential we then determine asymptotic corrections to the mean-field theory for thermodynamic states far from the region of the mean-field condensation. Our predictions are in accordance with rigorous results obtained previously, whereas the Hartree-Fock approximation is shown to neglect a whole class of equally important contributions.
The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction; in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.
We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number NNN of atoms, harmonic oscillator length, deformation of the trap, sss-wave scattering length) fix its thermodynamic behavior in the large NNN limit through two dimensionless scaling parameters. These are the reduced temperature t=T/Tc0t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}t=T/Tc0​ and the ratio $\ensuremath{\eta}$ between the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature Tc0{T}_{c}^{0}Tc0​ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.
The properties of the uniform Bose gas are studied within the optimized variational perturbation theory (Gaussian approximation) in a self-consistent way. It is shown that the atomic Bose-Einstein condensate with a repulsive interaction becomes unstable when the gas parameter $\ensuremath{\gamma}=\ensuremath{\rho}{a}^{3}$ exceeds a critical value ${\ensuremath{\gamma}}_{crit}\ensuremath{\approx}0.01$. The quantum corrections beyond the Bogoliubov-Popov approximation to the energy density, chemical potential, and pressure in powers of $\sqrt{\ensuremath{\gamma}}$ expansions are presented.
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.
In 1963, a simplified approach was developed to study the ground-state energy of an interacting Bose gas with a purely repulsive potential. It consists in the derivation of an equation, which is not based on perturbation theory and which gives the exact expansion of the energy at low densities. This equation is expressed directly in the thermodynamic limit and only involves functions of three variables, rather than 3N3N3N. Here, we revisit this approach, introduce two more equations, and show that these yields accurate predictions for various observables for all densities for repulsive potentials with positive Fourier transform. Specifically, in addition to the ground-state energy, we have shown that the simplified approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the equations with quantum Monte Carlo calculations for exponential interaction potentials as well as a different, finite range potential of positive type, and have found remarkable agreement. We thus show that the simplified approach provides an alternative theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which much less is known.