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arXiv:cond-mat/9908189 · Physical Review Letters

Yrast Line for Weakly Interacting Trapped Bosons

G. F. Bertsch, T. Papenbrock

Published Dec 27, 1999• 127 citations
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Abstract

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 $L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}N$ 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.

DOI
10.1103/physrevlett.83.5412

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We compute numerically the yrast line for harmonically trapped NNN-boson systems with a weak repulsive contact interaction, studying the transition to a vortex state as the angular momentum LLL increases and approaches NNN. There is a smooth crossover between low and high LLL with no indication of a quantum phase transition; at L=NL\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}NL=N there is a kink in the slope of the spectrum. Most strikingly, the energy and wave function appear to be analytical functions of LLL over the entire range 2\ensuremath≤L\ensuremath≤N2\ensuremath{\le}L\ensuremath{\le}N2\ensuremath≤L\ensuremath≤N. We confirm the structure of low- LLL states proposed by Mottelson as mainly single-particle excitations with two or three units of angular momentum.
Through an extensive numerical study, we find that the low-lying, quasidegenerate eigenenergies of weakly interacting trapped N bosons with total angular momentum L are given in the case of small L/NL/NL/N and sufficiently small L by $E=L\ensuremath{\Elzxh}\ensuremath{\omega}+g[N(N\ensuremath{-}L/2\ensuremath{-}1)+1.59n(n\ensuremath{-}1)/2],$ where $\ensuremath{\omega}$ is the frequency of the trapping potential and g is the strength of the repulsive contact interaction. The last term arises from the pairwise repulsive interaction among n octupole excitations and describes the lowest-lying excitation spectra from the yrast line. In this case, the quadrupole modes do not interact with themselves and, together with the octupole modes, exhaust the low-lying spectra that are separated from others by energy gaps linear in N.
We present an operator-algebraic approach to derive the low-lying quasidegenerate spectrum of weakly interacting trapped N bosons with total angular momentum $\ensuremath{\Elzxh}L$ for the case of $L/N\ensuremath{\ll}1.$ We demonstrate that the lowest-lying excitation spectrum is given by ${27gn}_{3}{(n}_{3}\ensuremath{-}1)/34,$ where g is the strength of the repulsive contact interaction and n3{n}_{3}n3​ is the number of excited octupole quanta. Our method provides constraints for these quasidegenerate many-body states and gives higher excitation energies that depend linearly on N.
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 exploit the analogy with the quantum Hall (QH) system to study weakly interacting bosons in a harmonic trap. For a $\ensuremath{\delta}$-function interaction potential the ``yrast'' states with $L>~N(N\ensuremath{-}1)$ are degenerate, and we show how this can be understood in terms of Haldane exclusion statistics. We present spectra for four and eight particles obtained by numerical and algebraic methods, and demonstrate how a more general hard-core potential lifts the degeneracies on the yrast line. The exact wave functions for N=4N=4N=4 are compared with trial states constructed from composite fermions (CF), and the possibility of using CF states to study the low L region at high N is discussed.
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.