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arXiv:cond-mat/9810197 · Physical Review A

Bose-Einstein condensation in a one-dimensional interacting system due to power-law trapping potentials

Mehmet Bayındır, B. Tanatar, Z. Gedik

Published Feb 1, 1999• 17 citations
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Abstract

We examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to confining potentials of the form ${V}_{\mathrm{ext}}{(x)=V}_{0}(|x|{/a)}^{\ensuremath{\gamma}},$ in which $\ensuremath{\gamma}<2,$ by solving the Gross-Pitaevskii equation within the semiclassical two-fluid model. The condensate fraction, chemical potential, ground state energy, and specific heat of the system are calculated for various values of interaction strengths. Our results show that a significant fraction of the particles is in the lowest energy state for a finite number of particles at low temperature, indicating a phase transition for weakly interacting systems.

DOI
10.1103/physreva.59.1468

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We examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to confining potentials of the form Vext(x)=V0(∣x∣/a)\ensuremathγ,{V}_{\mathrm{ext}}{(x)=V}_{0}(|x|{/a)}^{\ensuremath{\gamma}},Vext​(x)=V0​(∣x∣/a)\ensuremathγ, in which \ensuremathγ<2,\ensuremath{\gamma}<2,\ensuremathγ<2, by solving the Gross-Pitaevskii equation within the semiclassical two-fluid model. The condensate fraction, chemical potential, ground state energy, and specific heat of the system are calculated for various values of interaction strengths. Our results show that a significant fraction of the particles is in the lowest energy state for a finite number of particles at low temperature, indicating a phase transition for weakly interacting systems.
The phase transition of the one-dimensional Bose gas, interacting weakly with a repulsive potential and trapped in an external parabolic potential, is studied in a mean field approximation. The Bose–Einstein condensed (BEC) state is identified by the non-vanishing thermal average of a bose field. Free energies between the BEC state and normal state are compared. A crossing of the free energies occurring at low temperature suggests the first order phase transition in an interacting one-dimensional bose system. From fitting of numerical results, the relation among the transition temperature T c , the total number of bosons N and the interaction V is also provided as a simple power law, T c ∝( N 2 V ) 2/3 .
We study the Bose-Einstein condensation phenomenon in a two-dimensional (2D) system of bosons subjected to a harmonic-oscillator-type confining potential. The interaction among the 2D bosons is described by a \ensuremath{\delta} function in configuration space. Solving the Gross-Pitaevskii equation within the two-fluid model we calculate the condensate fraction, ground-state energy, and specific heat of the system. Our results indicate that interacting bosons have similar behavior to those of an ideal system for weak interactions.
Bose-Einstein condensation is known to occur in one-dimensional power-law potentials $V(x)\ensuremath{\propto}|x{|}^{\ensuremath{\eta}}$ as a true phase transition for \ensuremath{\eta}2, but only if there are no interactions between the particles. We show, via a path-integral quantum Monte Carlo scheme, that for both \ensuremath{\eta}2 and \ensuremath{\eta}=2 the spatial distribution of finite numbers of hard-core bosons suddenly becomes bimodal below a certain temperature, with a central condensate of particles distributed according to the lowest single-particle eigenstate. At still lower temperatures, the hard-core interactions cause the single-particle ground state to become a poor description for the interacting gas. If \ensuremath{\eta}2, the energy per particle undergoes a sudden decrease near the same temperature at which the bimodal distribution appears. It is found that this drop in energy disappears if the range of the interaction potential is sufficiently large.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.