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arXiv:1707.05628 · Physical Review A

Saturation properties of helium drops from a leading-order description

Alejandro Kievsky, A. Polls, Bruno Juliá-Díaz, N. K. Timofeyuk

Published Oct 9, 2017• 19 citations
arXivHTMLPDF

Abstract

Saturation properties are directly linked to the short-range scale of the two-body interaction of the particles. The case of helium is special; on one hand, the two-body potential has a strong repulsion at short distances. On the other hand, the extremely weak binding of the helium dimer locates this system very close to the unitary limit allowing for a description based on an effective theory. At leading order of this theory a two- and a three-body term appear, each one characterized by a low-energy constant. In a potential model this description corresponds to a soft potential model with a two-body term purely attractive plus a three-body term purely repulsive constructed to describe the dimer and trimer binding energies. Here we analyze the capability of this model to describe the saturation properties making a direct link between the low-energy scale and the short-range correlations. We will show that the energy per particle, ${E}_{N}/N$, can be obtained with reasonable accuracy at leading order extending the validity of this approximation, characterizing universal behavior in few-boson systems close to the unitary limit, to the many-body system.

DOI
10.1103/physreva.96.040501

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Saturation properties are directly linked to the short-range scale of the two-body interaction of the particles. The case of helium is special; on one hand, the two-body potential has a strong repulsion at short distances. On the other hand, the extremely weak binding of the helium dimer locates this system very close to the unitary limit allowing for a description based on an effective theory. At leading order of this theory a two- and a three-body term appear, each one characterized by a low-energy constant. In a potential model this description corresponds to a soft potential model with a two-body term purely attractive plus a three-body term purely repulsive constructed to describe the dimer and trimer binding energies. Here we analyze the capability of this model to describe the saturation properties making a direct link between the low-energy scale and the short-range correlations. We will show that the energy per particle, EN/N{E}_{N}/NEN​/N, can be obtained with reasonable accuracy at leading order extending the validity of this approximation, characterizing universal behavior in few-boson systems close to the unitary limit, to the many-body system.
We study bosonic systems in the regime in which the two-body system has a shallow bound state or, equivalently, a large value of the two-body scattering length. Using the effective field theory framework as a guidance, we construct a series of potential terms which have a decreasing importance in the description of the binding energy of the systems. The leading-order potential terms consist in a two-body term, usually attractive, plus a three-body term, usually repulsive; this last term is required to prevent the collapse of systems with more than two particles. At this order, the parametrization of the two-body potential is done to obtain a correct description of the scattering length, which governs the dynamics in this regime, whereas the three-body term fixes a three-body datum. We investigate the role of the cutoff in the leading-order description, and we extend the exploration beyond the leading order by including the next-to-leading-order terms in both the two- and three-body potentials. We use the requirement of the stability of the NNN-body system, whose energy is variationally estimated, to introduce the three-body forces. The potential parametrization, as a function of the cutoff, is fixed to describe the energy of 4He^{4}\mathrm{He}4He clusters up to seven particles within the expected accuracy. Finally, we also explore the possibility to describe at the same time the atom-dimer scattering length.
Four-body system of 4He{}^4\mathrm{He}4He atoms: Dimer-dimer scattering
The strong short-range repulsion, characteristic to realistic interatomic potentials, complicates the description of weakly-bound few-body systems such as those of \He atoms. The present work proposes an approach for solving this problem and applies it to a realistic system of four 4He{}^4\mathrm{He}4He atoms. The potential is gradually softened such that rigorous four-body equations for bound and scattering tates can be accurately solved in the momentum-space framework, and the results are extrapolated back to the limit of the original potential. Linear correlations between three- and four body quantities are observed, and the accuracy of the procedure is improved by extrapolating in one of the three-body quantities. Results for the 4He{}^4\mathrm{He}4He tetramer ground and excited state binding energies and atom-trimer scattering agree well with at least some of earlier determinations and shed light on the existing disagreements. An additional case of the Phillips correlation line is established for the dimer-dimer scattering length. The trimer production rate via the ultracold two-dimer collisions is estimated, it exhibits significant finite-range effects despite the weak binding of the dimer.
We develop an effective field theory (EFT) for strongly interacting bosonic clusters, using 4^44He as a paradigmatic example of universality in systems with large scattering length. At leading order (LO), two- and three-body zero-range interactions are entirely determined by the dimer and trimer ground-state energies. We show that ground-state energies for up to N=15N=15N=15 particles converge to cutoff-independent limits with extrapolation coefficients of natural size. At next-to-leading order (NLO), corrections stemming from the two-body interaction range and a four-body force, calibrated to the tetramer ground-state energy, reduce cutoff sensitivity. Close agreement with results from a realistic potential is found at LO and improved at NLO, demonstrating systematic convergence with few parameters at each order. The resulting EFT is directly applicable to larger clusters and bulk helium.
In this paper, we investigate small clusters of helium atoms 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 atoms with an interparticle potential which does not present a strong repulsion at short distances. We use an attractive Gaussian potential that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the widely used He-He interactions, the Aziz and Slaman potential, called LM2M2. In systems with more than two atoms, we consider a repulsive three-body force that, by construction, reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting of the sum of a two- and three-body potential, we have calculated the spectrum of clusters formed by four, five, and six helium atoms. We have found that these systems present two bound states, one deep and one shallow, close to the threshold fixed by the energy of the $(A\ensuremath{-}1)$-atom system. Universal relations between the energies of the excited state of the AAA-atom system and the ground-state energy of the $(A\ensuremath{-}1)$-atom system are extracted, as well as the ratio between the ground state of the AAA-atom system and the ground-state energy of the trimer.
We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of the Tjon line are constructed, showing correlations between few-body binding energies and the binding energy of the trimer, for a given dimer energy. As a specific example, we implement our theory for 4He^{4}\mathrm{He}4He atomic systems and show that the results are in surprisingly good agreement with those of sophisticated $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials. Potential implications for the convergence of the EFT expansion are discussed.
In this work we investigate the structure and dynamics of small clusters of Helium atoms. We consider bound states of clusters having A = 2, 3, 4, 5, 6 atoms and continuum states in the three-atom system. Motivated by the fact that the He-He system has a very large scattering length a compared to the range r 0 of the He-He potential (r 0/a < 1/10), we propose the use of a soft-core interparticle potential. We use an attractive gaussian potential that reproduces the values of the dimer binding energy and the atomatom scattering length obtained with one of the widely used He-He interactions, the LM2M2 potential. 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 a three-body potential, we show the spectrum of the four, five, and sixparticle systems and phase-shifts and inelasticities in the three-atom system. Comparisons to calculations using realistic He-He potentials are given. In addition some universal relations are explored.