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arXiv:cond-mat/0012458 · Physical Review A

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

Masahito Ueda, Tatsuya Nakajima

Published Nov 15, 2001• 6 citations
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Abstract

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.

DOI
10.1103/physreva.64.063609

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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\ElzxhL\ensuremath{\Elzxh}L\ensuremath\ElzxhL for the case of L/N\ensuremath≪1.L/N\ensuremath{\ll}1.L/N\ensuremath≪1. We demonstrate that the lowest-lying excitation spectrum is given by 27gn3(n3\ensuremath−1)/34,{27gn}_{3}{(n}_{3}\ensuremath{-}1)/34,27gn3​(n3​\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.
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 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 propose a methodology to construct excited states with a fixed angular momentum, namely, ``yrast excited states'' of finite-size one-dimensional bosonic systems with periodic boundary conditions. The excitation energies such as the first yrast excited energy are calculated through the system-size asymptotic expansion and expressed analytically by dressed energy. Interestingly, they are grouped into sets of almost degenerate energy levels. The low-lying excitation spectrum near the yrast state is consistent with the U(1)\text{U}(1)U(1) conformal field theories if the total angular momentum is given by an integral multiple of particle number; i.e., if the system is supercurrent.
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-NNN 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 investigate the low-lying excitations of a weakly interacting, harmonically trapped Bose-Einstein condensed gas under rotation, in the limit where the angular mometum L of the system is much less than the number of the atoms N in the trap. We show that in the asymptotic limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ the excitation energy, measured from the energy of the lowest state, is given by ${27N}_{3}{(N}_{3}\ensuremath{-}{1)v}_{0}/68,$ where N3{N}_{3}N3​ is the number of octupole excitations, and v0{v}_{0}v0​ is the unit of the interaction energy.