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arXiv:nucl-th/0308003

Precise study of the Efimov three-particle spectrum and structure functions within variational approach

B.E. Grinyuk, M. V. Kuzmenko, I.V. Simenog

Published Aug 1, 2003• 0 citations
arXivHTMLPDF

Abstract

A precise study within variational approach of the basic properties of the three-particle spectrum and structure functions with Gaussian potential near the critical coupling constant of interaction where the Efimov effect takes place is carried out. A method is developed to calculate highly excited states with very small binding energies, and numerical analysis is carried out for the ground and three excited energy states. For these states, one-particle density distributions, formfactors, pair correlation functions, and momentum distributions are calculated. It is found that the second excited state has already all the basic features of a level from the infinite Efimov series. An essential asymmetry is found in the position of energy levels with respect to the critical constant point. A halo-type structure is revealed in the one-particle density distributions, and formfactors are shown to have specific dips of finite depth, the number of dips being equal to the number of the state. The behaviour of pair correlation functions and momentum distributions is studied for three-particle states.

DOI
10.48550/arxiv.nucl-th/0308003

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A precise study within variational approach of the basic properties of the three-particle spectrum and structure functions with Gaussian potential near the critical coupling constant of interaction where the Efimov effect takes place is carried out. A method is developed to calculate highly excited states with very small binding energies, and numerical analysis is carried out for the ground and three excited energy states. For these states, one-particle density distributions, formfactors, pair correlation functions, and momentum distributions are calculated. It is found that the second excited state has already all the basic features of a level from the infinite Efimov series. An essential asymmetry is found in the position of energy levels with respect to the critical constant point. A halo-type structure is revealed in the one-particle density distributions, and formfactors are shown to have specific dips of finite depth, the number of dips being equal to the number of the state. The behaviour of pair correlation functions and momentum distributions is studied for three-particle states.
Near two-body unitarity, the three-boson system is characterized by an approximate discrete scale invariance manifest in a geometric tower of bound states (the Efimov effect). In the absence of a strong four-body force, the four-boson system has two states associated with each Efimov state, one very nearly unstable, the other several times deeper. We study correlations between the excited- and ground-state properties, such as binding energies and radii, up to next-to-leading order in an effective field theory for short-range forces. We obtain the parameters in these correlations from similar correlations arising from existing precise calculations based on short-range potentials. We also derive correlations among excited-state properties that emerge from the proximity of the state to the break-up threshold into a boson and a three-boson bound state, using an effective field theory for ``halo'' states.
Universal physics manifests in low-energy phenomena over a large array of different systems from nuclear to atomic physics. Here, two interesting aspects of universality are studied using the Gaussian Expansion Method (GEM). This is a variational method that uses products of Gaussians as basis functions. Gaussian potentials are used as a short-range potential. The first aspect is the behaviour close to the dimer + atom breakup threshold of Efimov states and associated universal tetramers in ultra-cold mixtures of alkali atoms. These can be treated as bosons here. It is shown that trimer and tetramer vanish into the threshold at almost the same point. The predictions of effective Efimov states in the vicinity of the dimer + atom breakup threshold are addressed, but the results are inconclusive. The second aspect is the interplay between universal states of up to four bosons and the Coulomb interaction. This is interesting because it opens up nuclear physics to the investigation. First, the effect of the Coulomb interaction on universal states is studied in natural units. This introduces a scale for the Coulomb potential's strength relative to the strength of the short-range Gaussian potential. A generalised Efimov plot of the binding energies of states of charged bosons versus the Coulomb-modified scattering length is shown. This plot illustrates the impact of different relative strengths of the Coulomb potential on universal states. To complement this analysis, the structure is also calculated via root mean square (rms) radius calculations and contour plots. The results are then applied to the excited state of 17 F, which has a proton halo. The binding energy of this state can be reproduced after fixing the effective range of the potential to the physical value. The N α system proved more problematic. It was found that the ground state of 12 C was too deep to be described within the framework used and the highest excited state of 16 O below the 4α breakup threshold proved difficult to describe. I discuss possible reasons for this. In the last part I studied the zero-range limit in the presence of the Coulomb interaction. I show that my results for the dimer and trimer can be rescaled to coincide with the zero-range result. An extrapolation towards the zero-range limit for the tetramer ground state is also presented.
Abstract We investigate the two lowest-lying weakly bound states of N ≤ 8 bosons as functions of the strength of two-body Gaussian interactions. We observe the limit for validity of Efimov physics. We calculate energies and second radial moments as functions of scattering length. For identical bosons we find that two ( N − 1 ) -body states appear before the N -body ground states become bound. This pattern ceases to exist for N ≥ 7 where the size of the ground state becomes smaller than the range of the two-body potential. All mean-square-radii for N ≥ 4 remain finite at the threshold of zero binding, where they vary as ( N − 1 ) p with p = − 3 / 2 , − 3 for ground and excited states, respectively. Decreasing the mass of one particle we find stronger binding and smaller radii. The identical particles form a symmetric system, while the lighter particle is further away in the ground states. In the excited states we find the identical bosons either surrounded or surrounding the light particle for few or many bosons, respectively. We demonstrate that the first excited states for all strengths resemble two-body halos of one particle weakly bound to a dense N -body system for N = 3, 4. This structure ceases to exist for N ≥ 5 .
The possible presence of Efimov-type states in isotopic variants of triatomic molecules composed of one Li and two boson He atoms is investigated. Firstly, through the study of two-body interactions in the frame of the effective range theory, it is shown that, at least with the present choice of dimer potentials, the system containing 7 Li unlikely presents any Efimov state, while the 6 Li isotopic variant would be a better candidate in this regard. Then, triatomic bound states are searched through variational calculations in terms of pair inter-atomic distances and distributed Gaussian functions (DGF). The calculations were performed at several values of a parameter λ controlling the strength of the potential. Each system presents two bound levels at the “physical” situation λ = 1. By increasing lambda, the excited levels become “ghost” states as their energies are above the helium dimer energy and present, therefore, an Efimov character. Hence, we conclude that three-body (3B) predictions coming only from two-body (2B) calculations do not have, at least for the systems studied here, any great relevance. Such predictions are not in fact fulfilled for LiHe 2 , as shown by the greatly reduced relevance of isotopic effects in our actual 3B calculations.
Universal behavior has been found inside the window of Efimov physics for systems with N=4,5,6N=4,5,6N=4,5,6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\ensuremath{\le}6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
Helium trimer bound states are calculated with a variational method described in terms of pair atom coordinates and distributed Gaussian functions by assuming zero total angular momentum. With the two-body interaction potential used, a weakly bound state for the dimer and only two bound states for the trimer are found. Although strictly speaking the first excited trimer state is not an Efimov state, we show that this state presents several characteristics of the Efimov behavior which are extensively discussed in the present work. Special emphasis is placed on the main geometrical configurations contributing to the two bound trimer states, and suggestions are made on the possible ways of experimentally detecting the Efimov-type state.