Energy spectra of small bosonic clusters having a large two-body scattering length

TL;DR

This study investigates small bosonic clusters with up to six particles using hyperspherical harmonics, exploring their energy spectra across different scattering lengths and identifying universal energy relations.

Contribution

It introduces a model combining two- and three-body potentials to analyze energy spectra of small bosonic clusters near unitarity, revealing universal energy ratios.

Findings

Clusters have two bound states: one deep and one shallow.

Universal relations between energy levels are observed.

Energy thresholds follow predictable ratios across different cluster sizes.

Abstract

In this work we investigate small clusters of bosons using the hyperspherical harmonic basis. We consider systems with $A=2,3,4,5,6$ particles interacting through a soft inter-particle 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 $^4$He-$^4$He interactions, the LM2M2 potential. 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 region explored, we have found that these systems present two bound states, one deep and one shallow close to the $A-1$ 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 continuum using the two-body gaussian