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arXiv:cond-mat/0010457

Comment on "Strongly interacting one-dimensional Bose-Einstein condensates in harmonic traps"

Marvin D. Girardeau, Ewan M. Wright

Published Oct 27, 2000• 2 citations
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Abstract

A recent paper of Tanatar and Erkan [Phys. Rev. A 62, 053601 (2000)] discusses a density functional approach to the impenetrable point Bose gas in one dimension and an equation for the order parameter of the system, due originally to Kolomeisky et al. [Phys. Rev. Lett. 85, 1146 (2000)], is derived. We comment on the regime of validity of such a model and on qualitative differences between predictions of the density functional approach and known exact results.

DOI
10.48550/arxiv.cond-mat/0010457

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A recent paper of Tanatar and Erkan [Phys. Rev. A 62, 053601 (2000)] discusses a density functional approach to the impenetrable point Bose gas in one dimension and an equation for the order parameter of the system, due originally to Kolomeisky et al. [Phys. Rev. Lett. 85, 1146 (2000)], is derived. We comment on the regime of validity of such a model and on qualitative differences between predictions of the density functional approach and known exact results.
This is a comment on the work of Kolomeisky et al., Phys. Rev. Lett. 85, 1146 (2000). We point out that they are using the wrong form of the energy functional for one-dimensional fermions. We point out two possible forms of the energy functional, both of which can be derived from first principles but using different methods. One is obtained from the collective field theory method, while the other is derived from the extended Thomas-Fermi method. These two forms of the energy functional do not support the soliton solutions which are obtained by Kolomeisky et al.
In Jaouadi et al. [Phys. Rev. A 83, 023616 (2011)] the authors derive an analytical finite-size expansion for the Bose-Einstein condensation critical temperature of an ideal Bose gas in a generic power-law trap. In the case of a harmonic trap, this expansion adds higher-order terms to the well-known first-order correction. We point out a delicate point in connection to these results, showing that the claims of Jaouadi et al. should be treated with caution. In particular, for a harmonic trap, the given expansion yields results that, depending on what is considered to be the critical temperature of the finite system, do not generally improve on the established first-order correction. For some nonharmonic traps, the results differ at first order from other results in the literature.
We study a system of trapped bosonic particles interacting by model harmonic forces. Our model allows for detailed examination of the notion of an order parameter (a condensate wave function). By decomposing a single particle density matrix into coherent eigenmodes we study an effect of interaction on the condensate. We show that sufficiently strong interactions cause that the condensate disappears even if the whole system is in its lowest energy state. In the second part of our paper we discuss the validity of the Bogoliubov approximation by comparing its predictions with results inferred from the exactl y soluble model. In particular we examine an energy spectrum, occupation, and fluctuations of the condensate. We conclude that Bogoliubov approach gives quite accurate description of the system in the limit of weak interactions.
We study the interaction effects on the condensates by considering a model of one-dimensional bosons. The harmonic external potential allows for the formation of a condensate in these systems. Using a density-functional theory-type formalism we obtain an equation describing the condensate wave function in the limit of very strong interactions between the bosons. We then consider a model of strongly interacting two-component system of bosons and study its stability conditions. Similar to the weakly interacting case, the two-component system exhibits coexisting and segregated phases depending on system parameters.
In a recent Letter (Phys. Rev. lett. 105, 195301 (2010)), Song et al. introduced a new variational approach to treat strong attractive boson-fermion (BF) correlations in BF atomic mixtures. The proposed theory predicts a first order phase transition to a condensate of composite BF pairs with center of mass momentum Q=0 as opposed to a composite fermionic molecular Fermi gas. We show in this comment that their approach is incorrect and moreover, by resorting to an exactly solvable model we demonstrate that there cannot be more than one correlated Q=0 BF pair in complete contradiction with their conclusions.