arXiv:cond-mat/0007004 · Physical Review A

Low-lying excitations from the yrast line of weakly interacting trapped bosons

Tatsuya Nakajima, Masahito Ueda

Published Mar 16, 2001• 15 citations

Abstract

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 $L/N$ 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.

DOI: 10.1103/physreva.63.043610

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arXiv:cond-mat/0007004 · Physical Review A

Low-lying excitations from the yrast line of weakly interacting trapped bosons

Tatsuya Nakajima, Masahito Ueda

Published Mar 16, 2001• 15 citations

arXiv HTML PDF

Abstract

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/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.

DOI: 10.1103/physreva.63.043610

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Operator-algebraic approach to the yrast spectrum of weakly interacting trapped bosons

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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 ${n}_{3}$ 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 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 ${n}_{3}$ 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.

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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 $\ensuremath{\delta}$-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.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.

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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/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}

{n}_{3}

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 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.

{N}_{3}

is the number of octupole excitations, and

{v}_{0}

is the unit of the interaction energy.

(|m|=2)

surface modes. For

L/N=1,

a vortexlike state is formed and all the atoms occupy the

m=1

state. For small negative values of $L/N\ensuremath{-}1,$ the states with

m=0

and

m=2

become populated, and for small positive values of $L/N\ensuremath{-}1,$ atoms in the states with

m=5

and

m=6

carry the additional angular momentum. In the whole region

0<~L/N<~1,

we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of

L/N.

We have also found that an array of singly quantized vortices is formed as

L/N

increases. Finally, we have gone beyond the mean-field approximation and have calculated the energy of the lowest state up to order N for small negative values of $L/N\ensuremath{-}1,$ as well as the energy of the low-lying excited states.

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.