We consider a model of N two-dimensional bosons in a harmonic trap with translational and rotational invariant, weak two-particle interaction. We present in configuration space a systematical recursive method for constructing all wave functions with angular momentum L and corresponding energies and apply it to $L<~6$ for all N. The lower and the upper bounds for interaction energy are estimated. We analytically confirm the conjecture of Smith et al. that the elementary symmetric polynomial is the ground state for repulsive \ensuremath{\delta} interaction for all $N>~L$ up to $L<~6.$ Additionally, we find that there exist vanishing-energy solutions for $L>~N(N\ensuremath{-}1),$ signaling the exclusive statistics. Finally, we consider briefly the case of attractive powerlike potential ${r}^{k},$ $k>\ensuremath{-}2,$ and prove that the lowest-energy state is still the one in which the whole angular momentum is absorbed by the center-of-mass motion.
We consider a model of N two-dimensional bosons in a harmonic trap with translational and rotational invariant, weak two-particle interaction. We present in configuration space a systematical recursive method for constructing all wave functions with angular momentum L and corresponding energies and apply it to L<6 for all N. The lower and the upper bounds for interaction energy are estimated. We analytically confirm the conjecture of Smith et al. that the elementary symmetric polynomial is the ground state for repulsive \ensuremath{\delta} interaction for all N>L up to L<6. Additionally, we find that there exist vanishing-energy solutions for L>N(N\ensuremath−1), signaling the exclusive statistics. Finally, we consider briefly the case of attractive powerlike potential rk,k>\ensuremath−2, and prove that the lowest-energy state is still the one in which the whole angular momentum is absorbed by the center-of-mass motion.
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.
We consider a model of N two-dimensional bosons in a harmonic potential with weak repulsive \ensuremath{\delta}-function interactions. We show analytically that, for angular momentum L<N, the elementary symmetric polynomials of particle coordinates measured from the center of mass are exact eigenstates with energy $N(N\ensuremath{-}L/2\ensuremath{-}1).$ Extensive numerical analysis confirms that these are actually the ground states, but we are currently unable to prove this analytically. The special case L=N can be thought of as the generalization of the usual superfluid one-vortex state to Bose-Einstein condensates in a trap.
We study a class of interacting, harmonically trapped boson systems at angular momentum L. The Hamiltonian leaves a L-dimensional subspace invariant, and this permits an explicit solution of several eigenstates and energies for a wide class of two-body interactions. PACS numbers: 05.30.Jp, 03.75.Fi, 03.65.Fd, 67.40.Db Typeset using REVTEX 1 The study of vortices in Bose-Einstein condensates of trapped atomic vapors is of considerable experimental [1,2] and theoretical [3–13] interest. Present experiments and much theoretical work [3–7] focus on the Thomas-Fermi regime of short coherence length [14]. The opposite limit of perturbatively weak contact interactions is not yet realized experimentally but is of theoretical interest as well [8–13]. The case of attractive interactions was studied by Wilkin et al. [9]. Mottelson developed a theory for repulsive interactions [10], and Kavoulakis et al. [12] and Jackson et al. [13] compared mean field and exact numerical results. Exact diagonalization techniques by us showed that the ground state energy depends linearly on angular momentum and led to an analytical expression for the ground state wave function [11]. Recently, the form of this wave function and its eigenenergy were confirmed in analytic calculations [15–17]. However, we still lack insight about why certain eigenstates turn out to be simple analytical functions. We show in this paper that there is a subspace structure that explains these findings. Interestingly, this structure is not limited to the two-body contact interaction considered in previous investigations but present in a rather large class of Hamiltonians. Let us consider a system of N harmonically trapped bosons in two spatial dimensions 1 at angular momentum
We study a trapped Bose-Einstein condensate under rotation in the limit of weak, translational and rotational invariant two-particle interactions. We use the perturbation-theory approach (the large-N expansion) to calculate the ground-state energy and the excitation spectrum in the asymptotic limit where the total number of particles N goes to infinity while keeping the total angular momentum L finite. Calculating the probabilities of different configurations of angular momentum in the exact eigenstates gives us a clear view of the physical content of excitations. We briefly discuss the case of repulsive contact interaction.
We study two spinless bosons interacting via two-body Gaussian potential subjected to an externally impressed rotation about an axis confined in a harmonic trap in two-spatial dimensions. We obtain a transcendental equation for the relative angular momentum ∣m∣ state with various values of the two-body interaction range σ and the two-body interaction strength g2 to study the resulting energy spectrum and analyze the role of Hilbert space dimensions N. We compare results for both attractive and repulsive interaction for δ-function potential and Gaussian potential for various values of interaction range. We study the effects of interaction parameters and relative angular momentum on the ground state energy and its various components, namely, kinetic energy, trap potential and interaction potential. For a given ∣m∣ and non-interacting case, we observe that the ground state energy becomes independent of interaction range. However, for a given relative angular momentum and interaction strength g2>0, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength g2V(r)≤−1, ground state energy diverges to physically unacceptable negative-infinity for ∣m∣=0 state. Further, for ∣m∣=1, the ground state energy becomes independent of the interaction strength. For a ∣m∣, we present a comparative study between the Gaussian interaction potential and the δ-function potential. Further, we observe that for a given g2 and ∣m∣, for δ-function potential {\it i.e.} σ→0, to achieve the convergence of ground state energy, we require a considerably large critical Hilbert space. Whereas, in the case of Gaussian interaction potential with σ→1, the ground state energy converges for a considerably small critical Hilbert space.
We compute numerically the yrast line for harmonically trapped N-boson systems with a weak repulsive contact interaction, studying the transition to a vortex state as the angular momentum L increases and approaches N. There is a smooth crossover between low and high L with no indication of a quantum phase transition; at L=N there is a kink in the slope of the spectrum. Most strikingly, the energy and wave function appear to be analytical functions of L over the entire range $2\ensuremath{\le}L\ensuremath{\le}N$. We confirm the structure of low- L states proposed by Mottelson as mainly single-particle excitations with two or three units of angular momentum.