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arXiv:cond-mat/0108216

On "Do the attractive bosons condense?" by N. K. Wilkin, J. M. F. Gunn, R. A. Smith

Mahir Saleh Hussein, Oleg Vorov

Published Aug 14, 2001• 0 citations
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Abstract

Using Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn and Smith [1] for the ground states of N Bose atoms rotating at the angular momentum L in a harmonic atomic trap with frequency omega interacting via attractive delta^2(r) forces, are valid for a broad class of predominantly attractive interactions V(r), not necessarily attractive for any r. The sufficient condition for the interaction is that all the two-body matrix elements allowed by the conservation of angular momentum k+l = m+n, are negative. This class includes, in particular, the Gaussian attraction of arbitrary radius, -1/r - Coulomb and log(r)-Coulomb forces, as well as all the short-range R << omega^{-1/2} interactions satisfying inequality int d^2r V(r) < 0. There is no condensation at L>> 1, and the angular momentum is concentrated in the collective ``center-of-mass'' mode.

DOI
10.48550/arxiv.cond-mat/0108216

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Using Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn and Smith [1] for the ground states of N Bose atoms rotating at the angular momentum L in a harmonic atomic trap with frequency omega interacting via attractive delta^2(r) forces, are valid for a broad class of predominantly attractive interactions V(r), not necessarily attractive for any r. The sufficient condition for the interaction is that all the two-body matrix elements allowed by the conservation of angular momentum k+l = m+n, are negative. This class includes, in particular, the Gaussian attraction of arbitrary radius, -1/r - Coulomb and log(r)-Coulomb forces, as well as all the short-range R << omega^{-1/2} interactions satisfying inequality int d^2r V(r) < 0. There is no condensation at L>> 1, and the angular momentum is concentrated in the collective ``center-of-mass'' mode.
Using the Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn, and Smith [Phys. Rev. Lett. 80, 2265, (1998)] for the ground states at angular momentum LLL of NNN harmonically trapped Bose atoms, interacting via weak attractive ${\ensuremath{\delta}}^{2}(r)$ forces, are valid for a broad class of predominantly attractive interactions V(r),V(r),V(r), not necessarily attractive for any r.r.r. This class is described by sufficient conditions on the two-body matrix elements of the potential V(r).V(r).V(r). It includes, in particular, the Gaussian attraction of arbitrary radius, $\ensuremath{-}1/r$-Coulomb and log(r)\mathrm{log}(r)log(r)-Coulomb forces, as well as all the short-range interactions satisfying inequality $\ensuremath{\int}{d}^{2}\stackrel{\ensuremath{\rightarrow}}{r}V(r)<0.$ In the precollapse regime, the angular momentum LLL is concentrated in the collective ``center-of-mass'' mode, and there is no condensation at high L.L.L.
We consider the criterion for Bose-Einstein condensation for particles in a harmonic trap. For a fixed angular momentum, the lowest energy state for a cloud of bosons with attractive interactions is the ground state of the cloud with all the angular momentum in the center-of-mass motion, and the one-particle reduced density matrix generally does not have a single large eigenvalue, but a number of them, suggesting that the state is an example of a fragmented condensate [N. K. Wilkin, J. M. F. Gunn, and R. A. Smith, Phys. Rev. Lett. 80, 2265 (1998)]. We show that a convenient way to describe correlations in the system is by defining an internal one-particle reduced density matrix, in which the center-of-mass motion is eliminated, and that this has a single eigenvalue equal to the number of particles for the problem considered here. Our considerations indicate that care is necessary in formulating a criterion for Bose-Einstein condensation.
For a system of N bosons in a two-dimensional harmonic trap with frequency $\ensuremath{\omega},$ interacting via forces $V\ensuremath{\ll}\ensuremath{\Elzxh}\ensuremath{\omega},$ we develop a method to find the ground states of rotating Bose condensate as a function of two quantum numbers, the total angular momentum and the angular momentum of internal excitations (generalized yrast states). The energies of these condensed vortex states are expressed through the single two-body matrix element of interaction, V.V.V. A broad universality class of predominantly repulsive interactions for which these results hold is described by a simple integral condition on V.V.V. It includes Gaussian, $\ensuremath{\delta}$ function, and log-Coulomb forces.
Motivated by experiments on Bose atoms in traps which have attractive interactions (e.g., 7Li{}^{7}\mathrm{Li}7Li), we consider two models which may be solved exactly. We construct the ground states subject to the constraint that the system is rotating with angular momentum proportional to the number of atoms. In a conventional system this would lead to quantized vortices; here, for attractive interactions, we find that the angular momentum is absorbed by the center of mass motion. Moreover, the state is uncondensed and is an example of a ``fragmented'' condensate discussed by Nozi\`eres and Saint James. The same models with repulsive interactions are fully condensed in the thermodynamic limit.
A system of N two-dimensional weakly interacting bosons in a harmonic trap is considered. When the two-particle potential is a \ensuremath{\delta} function, Smith and Wilkin have proved analytically that the elementary symmetric polynomials of particle coordinates measured from the center of mass are exact eigenstates. In this study, we point out that their proof works equally well for an arbitrary two-particle potential that possesses translational and rotational symmetries. We find that the interaction energy associated with the eigenstate with angular momentum L is equal to $aN(N\ensuremath{-}1)/2+(b\ensuremath{-}a)NL/2,$ where a and b are the interaction energies of two bosons in the lowest-energy one-particle state with zero and one unit of angular momentum, respectively. Additionally, we study briefly the case of attractive quartic interactions. We prove rigorously that the lowest-energy state is the one in which all angular momentum is carried by the center of mass motion.
In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.