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arXiv:cond-mat/0003081 · Physical Review A

Bose condensates at high angular momenta

S. Viefers, T. H. Hansson, S. M. Reimann

Published Oct 16, 2000• 80 citations
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Abstract

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 $N=4$ 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.

DOI
10.1103/physreva.62.053604

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We exploit the analogy with the quantum Hall (QH) system to study weakly interacting bosons in a harmonic trap. For a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function interaction potential the ``yrast'' states with L> N(N\ensuremath−1)L>~N(N\ensuremath{-}1)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 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 $2\ensuremath{\le}L\ensuremath{\le}N$. We confirm the structure of low- LLL states proposed by Mottelson as mainly single-particle excitations with two or three units of angular momentum.
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.
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.
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 revisit the composite fermion (CF) construction of the lowest angular momentum yrast states of rotating Bose gases with weak short range interaction. For angular momenta at and below the single vortex, L≤NL \leq NL≤N, the overlaps between these trial wave functions and the corresponding exact solutions {\it increase} with increasing system size and appear to approach unity in the thermodynamic limit. In the special case L=NL=NL=N, this remarkable behaviour was previously observed numerically. Here we present methods to address this point analytically, and find strongly suggestive evidence in favour of similar behaviour for all L≤NL \leq NL≤N. While not constituting a fully conclusive proof of the converging overlaps, our results do demonstrate a striking similarity between the analytic structure of the exact ground state wave functions at L≤NL \leq NL≤N, and that of their CF counterparts. Results are given for two different projection methods commonly used in the CF approach.