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 calculate the universal spectrum of trimer and tetramer states in heteronuclear mixtures of ultracold atoms with different masses in the vicinity of the heavy-light dimer threshold. To extract the energies, we solve the three- and four-body problem for simple two- and three-body potentials tuned to the universal region using the Gaussian expansion method. We focus on the case of one light particle of mass
m and two or three heavy bosons of mass
M with resonant heavy-light interactions. We find that trimer and tetramer cross into the heavy-light dimer threshold at almost the same point and that as the mass ratio
M/m decreases, the distance between the thresholds for trimer and tetramer states becomes smaller. We also comment on the possibility of observing exotic three-body states consisting of a dimer and two atoms in this region and compare with previous work.
The occurrence of a new limit cycle in few-body physics, expressing a universal scaling function relating the binding energies of two successive tetramer states, is revealed by considering a renormalized zero-range two-body interaction in bound state of four identical bosons. The tetramer energy spectrum is obtained by adding a boson to an Efimov bound state with energy B3 in the unitary limit (for zero two-body binding energy or infinite two-body scattering length). Each excited Nth tetramer energy B(N)4 is shown to slide along a scaling function as a short-range four-body scale is changed, emerging from the 3+1 threshold for a universal ratio B4(N)/B3 = 4.6, which does not depend on N. The new scale can also be revealed by a resonance in the atom-trimer recombination process.
The ground-state properties of
3He and
4He trimers and tetramers in infinite and restricted two-dimensional and quasi-two-dimensional (2D) space are studied. Using Monte Carlo calculation in two stages successively, simple variational Monte Carlo and diffusion Monte Carlo, we find that all trimers (except one of
3He with spin 3/2) and tetramers are bound. In infinite 2D space the binding energy of
4He trimer is $\ensuremath{-}183(1) \mathrm{mK}$ and of tetramer $\ensuremath{-}435(1) \mathrm{mK}.$ In the same environment for
3He, using Jastrow-Feenberg-Bruch wave functions trimer (spin 1/2) is bound with $\ensuremath{-}0.013(1) \mathrm{mK}$ and tetramer with $\ensuremath{-}0.021(1) \mathrm{mK}.$ Employing in the same procedure ``dimerlike'' wave functions for
3He molecules in infinite 2D space it is discovered that trimer prefers the structure of one dimer and one separate particle while a composition of two separate dimers is preferable for tetramer. In this case binding energies are $\ensuremath{-}0.020(1) \mathrm{mK}$ for trimer and $\ensuremath{-}0.040(1) \mathrm{mK}$ for tetramer. The binding energies of mixed molecules in infinite 2D space are $\ensuremath{-}74.3(4)$ mK ${(}^{3}\mathrm{He}{\ensuremath{-}}^{4}{\mathrm{He}}_{2});$ $\ensuremath{-}14(1)$ mK ${(}^{3}{\mathrm{He}}_{2}{\ensuremath{-}}^{4}\mathrm{He});$ $\ensuremath{-}254(4)$ mK ${(}^{3}\mathrm{He}{\ensuremath{-}}^{4}{\mathrm{He}}_{3});$ $\ensuremath{-}118(2)$ mK ${(}^{3}{\mathrm{He}}_{2}{\ensuremath{-}}^{4}{\mathrm{He}}_{2});$ $\ensuremath{-}11(1)$ mK ${(}^{3}{\mathrm{He}}_{3}{\ensuremath{-}}^{4}\mathrm{He}).$ In holding potentials with well-adjusted parameters there is a drastic increase of binding for trimer and tetramer in both sorts of helium atoms.
In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the use of the Gaussian expansion method for few-body systems, the energy levels and spatial structures of the
4He trimer and tetramer ground and excited states using the LM2M2 potential, which has a very strong short-range repulsion. In this work, we calculate the same quantities using the currently most accurate
4He-
4He potential [Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)] that includes the adiabatic, relativistic, QED, and residual retardation corrections. Contributions of the corrections to the tetramer ground-(excited-) state energy $\ensuremath{-}573.90\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}132.70)$ mK are respectively $\ensuremath{-}4.13\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}1.52)$ mK,
+9.37(+3.48) mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and
+0.16(+0.07) mK. Further including other realistic
4He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states,
B3(0),B3(1),B4(0), and
B4(1), respectively. We found that the four kinds of the binding energies for the different potentials exhibit perfect linear correlations between any two of them over the range of binding energies relevant for
4He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for
4He clusters, proposed in the previous work, predicts a simple interaction-independent relation
B2B4(1)=B2B3(0)+32, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with
B2 being the dimer binding energy.