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arXiv:cond-mat/0408533 · Physical Review A

Ground-state energy and depletions for a dilute binary Bose gas

André Eckardt, Christoph Weiß, Martin Holthaus

Published Oct 15, 2004• 15 citations
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Abstract

When calculating the ground-state energy of a weakly interacting Bose gas with the help of the customary contact pseudopotential, one meets an artifical ultraviolet divergence which is caused by the incorrect treatment of the true interparticle interactions at small distances. We argue that this problem can be avoided by retaining the actual, momentum-dependent interaction matrix elements, and use this insight for computing both the ground-state energy and the depletions of a binary Bose gas mixture. Even when considering the experimentally relevant case of equal masses of both species, the resulting expressions are quite involved, and they are no straightforward generalizations of the known single-species formulas. On the other hand, we demonstrate in detail how these latter formulas are recovered from our two-species results in the limit of vanishing interspecies interaction.

DOI
10.1103/physreva.70.043615

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When calculating the ground-state energy of a weakly interacting Bose gas with the help of the customary contact pseudopotential, one meets an artifical ultraviolet divergence which is caused by the incorrect treatment of the true interparticle interactions at small distances. We argue that this problem can be avoided by retaining the actual, momentum-dependent interaction matrix elements, and use this insight for computing both the ground-state energy and the depletions of a binary Bose gas mixture. Even when considering the experimentally relevant case of equal masses of both species, the resulting expressions are quite involved, and they are no straightforward generalizations of the known single-species formulas. On the other hand, we demonstrate in detail how these latter formulas are recovered from our two-species results in the limit of vanishing interspecies interaction.
The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.
I formulate a 'pseudo-paradox' in the theory of a dilute Bose gas with repulsive interactions: the standard expression for the ground state energy within the Gross–Pitaevskii (GP) approximation is lower than that in the Bogoliubov approximation, and hence, by the standard variational argument, the former should prima facie be a better approximation than the latter to the true ground state—a conclusion which is of course opposite to the established wisdom concerning this problem. It is shown that the pseudo-paradox is (unsurprisingly) resolved by a correct transcription of the two-body scattering theory to the many-body case; however, contrary to what appears to be a widespread belief, the resolution has nothing to do with any spurious ultraviolet divergences which result from the replacement of the true interatomic potential by a delta-function pseudopotential. Rather, it relates to an infrared divergence which has the consequence that (a) the most obvious form of the GP 'approximation' actually does not correspond to any well-defined ansatz for the many-body wavefunction, and (b) that the 'best shot' at such a wavefunction always produces an energy which exceeds, or at best equals, that calculated in the Bogoliubov approximation. In fact, the necessity of the latter may be seen as a consequence of the need to reduce the Fock term in the energy, which is absent in the two-particle problem but dominant in the many-body case; it does this by increasing the density correlations, at distances less than or approximately equal to the correlation length , above the value extrapolated from the two-body case. As a by-product I devise an alternative formulation of the Bogoliubov approximation which does not require the explicit replacement of the true interatomic potential by a delta-function pseudopotential.
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zeta-function methods are used to study the properties of the nonrelativistic interacting Bose gas at finite temperature and density. Results for the ground-state energy and pressure are obtained at both zero and finite temperatures. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground-state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass.
$\ensuremath{\zeta}$-function methods are used to study the properties of the nonrelativistic interacting Bose gas at finite temperature and density. Results for the ground-state energy and pressure are obtained at both zero and finite temperatures. The method used does not restrict the form of the interaction, which can be completely general. A similar procedure is then applied to evaluate the ground-state energy of a binary mixture consisting of different bosons. Analytical results are obtained in the case where the two species have the same mass.
The ground-state energy density of an interacting dilute Bose gas system is studied in the canonical transformation scheme. It is shown that the transformation scheme enables us to calculate a higher order correction of order na3 in the particle depletion and ground-state energy density of a dilute Bose gas system, which corresponds to the density fluctuation resulting from the excited states. Considering a two-body interaction only, the coefficient of the na3 term is shown to be 2(π-8/3) for the particle depletion, and 16(π-8/3) for the ground-state energy density.