Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the
4He atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the
s-wave two-body
t matrix to a separable form through the unitary-pole approximation and did the same for the
s-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.
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.
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.
A comparison is made between the number of Efimov States present in
4He trimers as calculated using the Faddeev equations in the Faddeev coordinate-momentum approach with estimates predicted by N=(1/\ensuremath{\pi})ln(\ensuremath{\Vert}a\ensuremath{\Vert}/
r0) using realistic phenomenological He-He potentials. It is found that there is good agreement for systems supporting few states and that agreement remains when the Efimov states are forced to disappear by mathematically altering the potentials.
We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the
4He trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the
4He trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length
rvdW, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the
4He trimer with the realistic potentials.