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arXiv:cond-mat/0105385 · Physical Review Letters

Comment on “Low-Dimensional Bose Liquids: Beyond the Gross-Pitaevskii Approximation”

R. K. Bhaduri, Diptiman Sen

Published May 14, 2001• 5 citations
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Abstract

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.

DOI
10.1103/physrevlett.86.4708

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Kolomeisky, Newman, Straley, and Qi Reply:

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This is a published reply to the comment of Bhaduri and Sen, Physical Review Letters 86, 4708 (2001), cond-mat/0105385, on our original work, Physical Review Letters 85, 1146 (2000), cond-mat/0002282 . In this reply we point out that our continuum theory captures the correct physics of non-interacting fermions in one dimension. In contrast, the two proposed alternative theories of Bhaduri and Sen produce physically incorrect results. We take this opportunity to point out that the abstract of cond-mat/0105385 obscures the contents of the comment and reply. The soliton solutions found in our theory are the continuum analogs of solitonic states which are known from exact studies of non-interacting fermions.

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Revista Mexicana de Física•Jan 1, 2007•0 citations
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.
A modified Gross-Pitaevskii approximation was recently introduced for bosons in dimension d ⩽2 by Kolomeisky et al (Kolomeisky E B, Newman T J, Straley J P and Qi X 2000 Phys. Rev. Lett. 85 1146). We use the density-functional approach with a sixth-degree interaction energy term in the Bose field to reproduce the stationary-frame results of Kolomeisky et al for a one-dimensional Bose-Einstein system with a repulsive interaction. We also find a soliton solution for an attractive interaction, which may be boosted to a finite velocity by a Galilean transformation. The stability of such a soliton is discussed analytically. We provide a general treatment of stationary solutions in one dimension which includes the above solutions as special cases. This treatment leads to a variety of stationary wave solutions for both attractive and repulsive interactions.
This is a published reply to the comment of Bhaduri and Sen, Physical Review Letters 86, 4708 (2001), cond-mat/0105385, on our original work, Physical Review Letters 85, 1146 (2000), cond-mat/0002282 . In this reply we point out that our continuum theory captures the correct physics of non-interacting fermions in one dimension. In contrast, the two proposed alternative theories of Bhaduri and Sen produce physically incorrect results. We take this opportunity to point out that the abstract of cond-mat/0105385 obscures the contents of the comment and reply. The soliton solutions found in our theory are the continuum analogs of solitonic states which are known from exact studies of non-interacting fermions.
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)
"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."