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Physical Review A

Analytic representation of the Efimov effect in the helium trimer

LAWRENCE L. JUN. LOHR, S. M. Blinder

Published Jun 2, 2004• 5 citations
Publisher

Abstract

Exact solutions for the low-temperature helium dimer and trimer, $^{4}\mathrm{He}_{2}$ and $^{4}\mathrm{He}_{3}$, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the $S$-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and ${r}_{0}$ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.

DOI
10.1103/physreva.69.064102

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Exact solutions for the low-temperature helium dimer and trimer, 4He2^{4}\mathrm{He}_{2}4He2​ and 4He3^{4}\mathrm{He}_{3}4He3​, are derived, based on our \ensuremathδ\ensuremath{\delta}\ensuremathδ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the SSS-wave alone giving an exact solution. The parameters \ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ and r0{r}_{0}r0​ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.
A mechanism of disappearance and formation of the Efimov levels of the helium ^4He trimer is studied when the force of the interatomic interaction is changed. It is shown that these levels arise from virtual levels which are in turn formed from (quasi)resonances settled on the real axis. The resonances including virtual levels are calculated by the method based on the solution of the boundary value problem, at complex energies, for the Faddeev differential equations describing the scattering processes (2+1\to 2+1; 1+1+1). All the calculations are performed with the known interatomic Aziz He-He potential HFD-B. A very strong repulsive component of this potential at short distances between helium atoms is approximated by a hard core. A special attention is paid to the substantiation of the method used for computing resonances and to the investigation of its applicability range.
The Faddeev equations for the atomic helium-trimer systems are solved numerically with high accuracy both for the most sophisticated realistic potentials available and for simple phenomenological potentials. An efficient numerical procedure is described. The large-distance asymptotic behaviour, crucial for weakly bound three-body systems, is described almost analytically for arbitrary potentials. The Efimov effect is especially considered. The geometric structures of the bound states are quantitatively investigated. The accuracy of the schematic models and previous computations is comparable, i.e. within 20% for the spatially extended states and within 40% for the smaller -trimer ground state.
The system of three 4He atoms interacting through the intermolecular potential HFDHE2 is solved exactly within the Faddeev scheme. We find that the trimer has two bound states at E=−0.11 K and E=−0.16(−2) K, where the excited state shows the property of an Efimov state. It is overrun by the two-body fragmentation threshold for an artificially enhanced overall interaction strength. Uncertainties in the potential parameters, possibly combined with solvent effects, may allow a slightly weaker force, which supports a second Efimov state. In any case dynamical effects of the Efimov states will be the same whether it lies close to E=0 on the physical or unphysical sheet.
The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: 4He3^{4}\mathrm{He}_{3}4He3​ and 4He23^{4}\mathrm{He}_{2}^{3}4He23​He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.
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.
A comparison is made between the number of Efimov States present in 4He^{4}\mathrm{He}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{r}_{0}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.