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arXiv:cond-mat/0507729 · Physical Review B

Threshold behavior of bosonic two-dimensional few-body systems

D. Blume

Published Sep 19, 2005• 23 citations
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Abstract

Bosonic two-dimensional self-bound clusters consisting of $N$ atoms interacting through additive van der Waals potentials become unbound at a critical mass ${m}_{*}^{(N)}$; ${m}_{*}^{(N)}$ has been predicted to be independent of the size of the system. Furthermore, it has been predicted that the ground state energy ${E}_{N}$ of the $N$-atom system varies exponentially as the atomic mass approaches ${m}_{*}$. This paper reports accurate numerical many-body calculations that allow these predictions to be tested. We confirm the existence of a universal critical mass ${m}_{*}$, and show that the near-threshold behavior can only be described properly if a previously neglected term is included. We comment on the universality of the energy ratio ${E}_{N+1}∕{E}_{N}$ near threshold.

DOI
10.1103/physrevb.72.094510

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Bosonic two-dimensional self-bound clusters consisting of NNN atoms interacting through additive van der Waals potentials become unbound at a critical mass m∗(N){m}_{*}^{(N)}m∗(N)​; m∗(N){m}_{*}^{(N)}m∗(N)​ has been predicted to be independent of the size of the system. Furthermore, it has been predicted that the ground state energy EN{E}_{N}EN​ of the NNN-atom system varies exponentially as the atomic mass approaches m∗{m}_{*}m∗​. This paper reports accurate numerical many-body calculations that allow these predictions to be tested. We confirm the existence of a universal critical mass m∗{m}_{*}m∗​, and show that the near-threshold behavior can only be described properly if a previously neglected term is included. We comment on the universality of the energy ratio EN+1∕EN{E}_{N+1}∕{E}_{N}EN+1​∕EN​ near threshold.
We treat three-dimensional bosonic clusters with up to N=40N=40N=40 atoms, interacting additively through two-body van der Waals potentials, in the near-threshold regime. Our study includes super-borromean systems with NNN atoms for which all subsystems are unbound. We determine the energetics and structural properties such as the expectation value of the interparticle distance as a function of the coupling strength. It has been shown that the coupling strength g∗(N){g}_{*}^{(N)}g∗(N)​, for which the NNN-body system becomes unbound, is bounded by the coupling constant ${g}_{*}^{(N\ensuremath{-}1)}$, for which the next smaller system with $N\ensuremath{-}1$ atoms becomes unbound, i.e., ${g}_{*}^{(N)}\ensuremath{\geqslant}(N\ensuremath{-}1){g}_{*}^{(N\ensuremath{-}1)}∕N$. By fitting our numerically determined ground-state energies to a simple functional form with three fitting parameters, we determine the relationship between g∗(N){g}_{*}^{(N)}g∗(N)​ and ${g}_{*}^{(N\ensuremath{-}1)}$. Our trimer and tetramer energies fall on the so-called Tjon line, which has been studied in nuclear physics. We confirm the existence of generalized Tjon lines for larger clusters. Signatures of the universal behavior of weakly bound three-dimensional clusters can possibly be observed in ultracold Bose gases.
The results of accurate variational calculations for the ground-state binding energies of several NNN-boson systems in two and three dimensions are reported; the systems specifically considered are small (N=2,3,4,and5N=2, 3, 4, \mathrm{and} 5N=2,3,4,and5) clusters of 4He^{4}\mathrm{He}4He atoms interacting through pairwise Lennard-Jones potentials. Evidence is found strongly suggesting each set of systems have a common functional relationship linking the binding energies and the De Boer quantum constant near threshold for self-binding. These calculations also confirm the existence of a unique threshold value of the interaction strength for all NNN-boson systems in 2D2D2D. In addition, the linear relation between the binding energies of helium clusters present in 3D3D3D holds ture in 2D2D2D as well.
The computational physics methods of various processes have become very popular during the last several years. Our interest in his field is predominantly motivated by the expectation that it will be possible to perform a detailed comparison between experimental and theoretical results obtained in the study of clusters of a variety of sizes. Algebraic invariants are employed to deal efficiently with the many-body correlations. We study the applicability of the developed trial wave functions over some range of particle masses, from Ar to He and lighter. We compute the ground and excited states energies and various geometric sizes of small clusters as a function of a continuously varied mass parameter. For sufficiently small masses the cluster can no longer support a bound state, and we are interested in particular in the behavior of the clusters in the vicinity of the critical mass, i.e., the point at which the binding energy approaches zero. We show that the use of bosonic polynomials provides enough variational freedom to approximate ground and excited states of clusters of any mass. We discuss how the properties of clusters change with mass.
We study clusters of the type ANBM{A}_{N}{B}_{M}AN​BM​ with $N\ensuremath{\le}M\ensuremath{\le}3$ in a two-dimensional mixture of AAA and BBB bosons, with attractive ABABAB and equally repulsive AAAAAA and BBBBBB interactions. In order to check universal aspects of the problem, we choose two very different models: dipolar bosons in a bilayer geometry and particles interacting via separable Gaussian potentials. We find that all the considered clusters are bound and that their energies are universal functions of the scattering lengths aAB{a}_{AB}aAB​ and aAA=aBB{a}_{AA}={a}_{BB}aAA​=aBB​, for sufficiently large attraction-to-repulsion ratios aAB/aBB{a}_{AB}/{a}_{BB}aAB​/aBB​. When aAB/aBB{a}_{AB}/{a}_{BB}aAB​/aBB​ decreases below $\ensuremath{\approx}10$, the dimer-dimer interaction changes from attractive to repulsive and the population-balanced AABBAABBAABB and AAABBBAAABBBAAABBB clusters break into ABABAB dimers. Calculating the AAABBBAAABBBAAABBB hexamer energy just below this threshold, we find an effective three-dimer repulsion which may have important implications for the many-body problem, particularly for observing liquid and supersolid states of dipolar dimers in the bilayer geometry. The population-imbalanced ABBABBABB trimer, ABBBABBBABBB tetramer, and AABBBAABBBAABBB pentamer remain bound beyond the dimer-dimer threshold. In the dipolar model, they break up at ${a}_{AB}\ensuremath{\approx}2{a}_{BB}$, where the atom-dimer interaction switches to repulsion.
We consider a system of N nonrelativistic bosons in two dimensions, interacting weakly via a short-range attractive potential. We show that for N large, but below some critical value, the properties of the N-boson bound state are universal. In particular, the ratio of the binding energies of (N+1)- and N-boson systems, B(N+1)/B(N), approaches a finite limit, approximately 8.567, at large N. We also confirm previous results that the three-body system has exactly two bound states. We find for the ground state B(0)(3)=16.522 688(1)B2 and for the excited state B(1)(3)=1.270 409 1(1)B2.