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arXiv:1710.00626 · Few-Body Systems

Hyperspherical Harmonics Expansion on Lagrange Meshes for Bosonic Systems in One Dimension

N. K. Timofeyuk, D. Baye

Published Sep 30, 2017• 12 citations
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Abstract

A one-dimensional system of bosons interacting with contact and single-Gaussian forces is studied with an expansion in hyperspherical harmonics. The hyperradial potentials are calculated using the link between the hyperspherical harmonics and the single-particle harmonic-oscillator basis while the coupled hyperradial equations are solved with the Lagrange-mesh method. Extensions of this method are proposed to achieve good convergence with small numbers of mesh points for any truncation of hypermomentum. The convergence with hypermomentum strongly depends on the range of the two-body forces: it is very good for large ranges but deteriorates as the range decreases, being the worst for the contact interaction. In all cases, the lowest-order energy is within 4.5 $$\%$$ of the exact solution and shows the correct cubic asymptotic behaviour at large boson numbers. Details of the convergence studies are presented for 3, 5, 20 and 100 bosons. A special treatment for three bosons was found to be necessary. For single-Gaussian interactions, the convergence rate improves with increasing boson number, similar to what happens in the case of three-dimensional systems of bosons.

DOI
10.1007/s00601-017-1318-y

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A one-dimensional system of bosons interacting with contact and single-Gaussian forces is studied with an expansion in hyperspherical harmonics. The hyperradial potentials are calculated using the link between the hyperspherical harmonics and the single-particle harmonic-oscillator basis while the coupled hyperradial equations are solved with the Lagrange-mesh method. Extensions of this method are proposed to achieve good convergence with small numbers of mesh points for any truncation of hypermomentum. The convergence with hypermomentum strongly depends on the range of the two-body forces: it is very good for large ranges but deteriorates as the range decreases, being the worst for the contact interaction. In all cases, the lowest-order energy is within 4.5 %\%% of the exact solution and shows the correct cubic asymptotic behaviour at large boson numbers. Details of the convergence studies are presented for 3, 5, 20 and 100 bosons. A special treatment for three bosons was found to be necessary. For single-Gaussian interactions, the convergence rate improves with increasing boson number, similar to what happens in the case of three-dimensional systems of bosons.
A formal proof is given that for NNN-boson systems in which two-body interaction potentials are described by a single Gaussian the ratio of higher-order to the lowest-order hyper-radial potentials decreases at $N\ensuremath{\rightarrow}\ensuremath{\infty}$ as ${N}^{\ensuremath{-}1/2}$ or faster. As a result, for such potentials, the convergence of binding energies for ground and several lowest excited states, obtained in expansion of the NNN-body wave function over the hyperspherical-harmonics basis, improves with increasing number of bosons. For a phenomenological three-body repulsive potential, introduced to account for the missing hard core, the ratio of higher-order to lowest-order hyper-radial potential, corresponding to this three-body potential, also decreases as ${N}^{\ensuremath{-}1/2}$ or faster when $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Although adding the three-body contributions leads to increased influence from the total nondiagonal couplings around the node of the lowest hyper-radial potential, the arguments are given that this should not dramatically deteriorate the convergence if the range of repulsion is properly chosen. This means that the hyperspherical-harmonics expansion with soft two-body and repulsive three-body effective forces may become an attractive tool for studying the spectra of many-body systems. It is suggested that fine tuning of the three-body repulsion to reproduce the binding energies over a large region of NNN is possible. In particular, it has been shown that an NNN-independent choice for the three-body repulsion exists for which the ground-state binding energies of $N\ensuremath{\le}112$ atoms of helium, obtained in the lowest-order approximation of the hyperspherical-harmonics expansion, are close to the prediction of the Green's function Monte Carlo method with a hard-core He-He potential.
We study a system of AAA identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as AAA increases. Instead of the HH basis, here we use a new basis, called ``potential harmonics'' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A\ensuremath{-}1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the $(3A\ensuremath{-}4)$ hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers---the orbital lll, azimuthal mmm, and the grand orbital 2K+l2K+l2K+l quantum numbers for any arbitrary AAA. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any AAA. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.
In this work we investigate small clusters of bosons using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6A=2,3,4,5,6A=2,3,4,5,6 particles interacting through a soft interparticle potential. In order to make contact with a real system, we use an attractive Gaussian potential that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with one of the most widely used 4{}^{4}4He-4{}^{4}4He interactions, the LM2M2 potential of Aziz and Slaman. The intensity of the potential is varied in order to explore the clusters' spectra in different regions with large positive and large negative values of the two-body scattering length. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and three-body potential, we have calculated the spectrum of the four-, five-, and six-particle systems. In all the regions explored, we have found that these systems present two states, one deep and one shallow close to the $A\ensuremath{-}1$ threshold. Some universal relations between the energy levels are extracted; in particular, we have estimated the universal ratios between thresholds of the three-, four-, and five-particle continua using the two-body Gaussian potential. They agree with recent measurements and theoretical predictions.
We examine a system of three-bosons confined to two dimensions in the presence of a perpendicular magnetic field within the framework of the adiabatic hyperspherical method. For the case of zero-range, regularized pseudopotential interactions, we find that the system is nearly separable in hyperspherical coordinates and that, away from a set of narrow avoided crossings, the full energy eigenspectrum as a function of the two-dimensional (2D) sss-wave scattering length is well described by ignoring coupling between adiabatic hyperradial potentials. In the case of weak attractive or repulsive interactions, we find the lowest three-body energy states exhibit even-odd parity oscillations as a function of total internal 2D angular momentum and that for weak repulsive interactions, the universal lowest energy interacting state has an internal angular momentum of M=3M=3M=3. With the inclusion of repulsive higher angular momentum we surmise that the origin of a set of ``magic number'' states (states with anomalously low energy) might emerge as the result of a combination of even-odd parity oscillations and the pattern of degeneracy in the noninteracting lowest Landau level states.
The hyperspherical adiabatic expansion method is used to describe correlations in a symmetric boson system rigorously confined to two spatial dimensions. The hyperangular eigenvalue equation turns out to be almost independent of the hyperradius, whereas the solutions are strongly varying with the strength of the attractive two-body potentials. Instability is encountered in hyperangular, hyperradial, and mean-field equations for almost identical strengths inversely proportional to the particle number. The derived conditions for stability are similar to mean-field conditions and closely related to the possible occurrence of the Thomas and Efimov effects. Renormalization in mean-field calculations for two spatial dimensions is probably not needed.
We study the ground-state properties of a system of NNN harmonically trapped bosons of mass mmm interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained-variational method, we are able to semiquantitatively attain Bose-Einstein condensate ground-state energies even for gases with infinite scattering length. In the large-particle-number limit, our method provides analytical estimates for the energy per particle ${E}_{0}/N\ensuremath{\approx}2.5{N}^{1/3}\ensuremath{\hbar}\ensuremath{\omega}$ and two-body contact ${C}_{2}/N\ensuremath{\approx}16{N}^{1/6}\sqrt{m\ensuremath{\omega}/\ensuremath{\hbar}}$ for a Bose gas on resonance, where $\ensuremath{\omega}$ is the trap frequency.