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arXiv:cond-mat/0008330 · Physical Review A

Exact eigenstates for trapped weakly interacting bosons in two dimensions

Wen-Jui Huang

Published Dec 5, 2000• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.63.015602

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•Jan 1, 2013•1 citation
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,aN(N\ensuremath{-}1)/2+(b\ensuremath{-}a)NL/2,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,L<~N,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=NL=NL=N can be thought of as the generalization of the usual superfluid one-vortex state to Bose-Einstein condensates in a trap.
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< 6L<~6L< 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> LN>~LN> L up to L< 6.L<~6.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,{r}^{k},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.
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
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.
Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.
We study the rotational properties of a two-component Bose gas trapped in a rotating harmonic potential, effectively in two dimensions. We first consider a completely homogeneous interaction between particles, and assume the bosons to be in the lowest Landau level. We derive analytical expressions for some wave functions and energies at low angular momenta, and use these results together with exact numerical diagonalization results to test the applicability of composite fermion (CF) trial wave functions to the system. We find that a general procedure based on CF wave functions reproduces the exact ground states and many of the excited states for sufficiently low angular momenta, and give very good approximate results at angular momenta up to the two-component analogy of a single vortex. Additionally, we produce very simple CF states that have remarkably high overlaps with the lowest-lying states. Finally, we briefly discuss the case of inhomogeneous interactions, pointing out the difficulties arising from this generalization.