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Physical Review A

Binding of three identical bosons in two dimensions

Ludwig W. Bruch, John A. Tjon

Published Feb 1, 1979• 100 citations
Publisher

Abstract

Qualitative features are discussed for the binding of three identical bosons interacting through pair potentials in two dimensions. Two special cases, known to yield pathologies in three dimensions, are examined using the Faddeev equation for the bound states. The Thomas effect does not occur: in the model which is treated, the trimer binding energy is finite for a zero-range force with a finite dimer energy. The Efimov effect can only occur under more restrictive conditions than in three dimensions: the number of bound trimer states is finite at the dimer threshold for a range of potential models. The trimer ground-state energy is determined as a function of the coupling constant for a simple model, and variational results for loosely bound Lennard-Jones trimers are shown to reflect a general trend found for the model.

DOI
10.1103/physreva.19.425

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Qualitative features are discussed for the binding of three identical bosons interacting through pair potentials in two dimensions. Two special cases, known to yield pathologies in three dimensions, are examined using the Faddeev equation for the bound states. The Thomas effect does not occur: in the model which is treated, the trimer binding energy is finite for a zero-range force with a finite dimer energy. The Efimov effect can only occur under more restrictive conditions than in three dimensions: the number of bound trimer states is finite at the dimer threshold for a range of potential models. The trimer ground-state energy is determined as a function of the coupling constant for a simple model, and variational results for loosely bound Lennard-Jones trimers are shown to reflect a general trend found for the model.
We construct a simple model to calculate the trimer bound state energy ET(n){E}_{T}^{(n)}ET(n)​ and three-body recombination rate K30{K}_{3}^{0}K30​ of three identical bosons with van der Waals interaction without using any two- or three-body fitting parameter. Using this simple model, we investigate the influence of the van der Waals finite-range effect on ET(n){E}_{T}^{(n)}ET(n)​ and K30{K}_{3}^{0}K30​. Our calculation shows that the finite-range effect leads to the ground trimer state energy ET(0){E}_{T}^{(0)}ET(0)​ not crossing the atom-dimer threshold, and the scaled three-body recombination rate K30/a4{K}_{3}^{0}/{a}^{4}K30​/a4 deviating from the universal three-body theory. The results of our simple model agree within a few percent with other theoretical works with van der Waals interaction and also the experimental data.
The dimensionality of a system can fundamentally impact the behavior of interacting quantum particles. Classic examples range from the fractional quantum Hall effect to high-temperature superconductivity. As a general rule, one expects confinement to favor the binding of particles. However, attractively interacting bosons apparently defy this expectation: While three identical bosons in three dimensions can support an infinite tower of Efimov trimers, only two universal trimers exist in the two-dimensional case. Here, we reveal how these two limits are connected by investigating the problem of three identical bosons confined by a harmonic potential along one direction. We show that the confinement breaks the discrete Efimov scaling symmetry and successively destroys the weakest bound trimers. However, the deepest bound trimers persist even under strong confinement. In particular, the ground-state Efimov trimer hybridizes with the two-dimensional trimers, yielding a superposition of trimer configurations that effectively involves tunneling through a short-range repulsive barrier. Our results suggest a way to use strong confinement to engineer more stable Efimov-like trimers, which have so far proved elusive.
Within the framework of integral equations, the bound-state energies of tetramers in two spatial dimensions are determined for simple potential models. The equations are solved by using unitary pole expansions for the subamplitudes. Evidence is given for the coincidence of the thresholds of the binding of the nnn-particle system with n=2,3,and4n=2, 3, \mathrm{and} 4n=2,3,and4. For the class of potentials studied, it is found that the ratio between the tetramer and trimer binding energies is, to a good approximation, given by 2.9. Using this result, an estimate is given for the 4He^{4}\mathrm{He}4He tetramer binding energy.
We calculate the low-lying energy spectrum for three trapped identical bosons interacting via a finite-range potential. The spectrum is compared to results from the zero-range model. The thresholds for trimer binding and atom-dimer binding are extracted and related to effective range corrections. Effective range corrections to Efimov physics and Borromean binding are two aspects of the same effect, and we connect these two regions qualitatively. The Borromean window becomes slightly narrower for substantial effective ranges. The structure at the atom-dimer threshold is an atom far away from the dimer and the major energy correction is due to the change of dimer energy with effective range. This structure becomes less pronounced when the effective range increases.
We study Efimov physics for three identical bosons interacting via a pairwise square-well potential, analyze the validity of the separable approximation as a function of the interaction strength, and investigate what is needed to improve this approximation. We find separable approximations to be accurate for potentials with just one (nearly) bound dimer state. For potentials with more bound or almost bound dimer states, these states need to be included for an accurate determination of the Efimov spectrum and the corresponding three-body observables. We also show that a separable approximation is insufficient to accurately compute the trimer states for energies larger than the finite-range energy even when the two-body TTT matrix is highly separable in this energy regime. Additionally, we have analyzed three distinct expansion methods for the full potential that give exact results and thus improve on the separable approximation. With these methods, we demonstrate the necessity to include higher partial-wave components of the off-shell two-body TTT matrix in the three-body calculations. Moreover, we analyze the behavior of the Efimov states near the atom-dimer threshold and observe the formation of non-Efimovian trimer states as the potential depth is increased. Our results can help to elaborate simpler theoretical models that are capable of reproducing the correct three-body physics in atomic systems.