Microscopic Origin and Universality Classes of the Efimov Three-Body Parameter

The low-energy spectrum of three particles interacting via nearly resonant two-body interactions in the Efimov regime is set by the so-called three-body parameter. We show that the three-body parameter is essentially determined by the zero-energy two-body correlation. As a result, we identify two classes of two-body interactions for which the three-body parameter has a universal value in units of their effective range. One class involves the universality of the three-body parameter recently found in ultracold atom systems. The other is relevant to short-range interactions that can be found in nuclear physics and solid-state physics.

Figure 1

Schematic Efimov plot: three-body energy $E_3$ scaled as an inverse length as a function of the inverse scattering length $1/a$. The arrows indicate the scattering length $a_{-}$ at which an Efimov trimer state appears, and its binding wave number $\kappa$ at unitarity ($a\to \infty$), either of which serves as a measure of the three-body parameter.

Figure 2

Binding wave number $\kappa$ of the ground-state trimer at unitarity (see Fig. 1) calculated from the separable model in Eqs. (1) and (2) using the zero-energy pair wave function ${\psi }_0$, versus its exact value, for various two-body potentials. The exact values are taken from Ref. [25] for the Lennard-Jones potential (with only one bound state) and from Ref. [31] for all the other potentials. The binding wave number is expressed in units of the effective range $r_e$ of each potential, which is calculated exactly from Eq. (3). The shaded area represents the region of 10% or less deviation from the exact results.

Figure 3

Pair correlation at unitarity for potentials decaying as power laws $-1/r^n$. Top: the 8–4 potential. Bottom: the Lennard-Jones potential. In each graph, the solid curves correspond in order of opacity to potential depths supporting, respectively 1, 2, and 3 $s$-wave bound states, which are obtained by adjusting the value of $\sigma$. The dashed curve represents the universal pair correlation in Eq. (4).

Figure 4

Pair correlation at unitarity for potentials decaying faster than any power law. Top: the Pöschl-Teller potential;, bottom: the Gaussian potential. In each graph, the solid curves correspond in order of opacity to potential depths supporting respectively, 1, 2, 10, and 120 $s$-wave bound states. The dashed lines show the universal pair correlation limit in Eq. (5). The distance is scaled in units of $\frac{1}{2}{r}_e$ in the main graphs, while it is shown in unscaled units of $r_0$ in the insets.

Figure 5

Binding wave number $\kappa$ of the ground-state Efimov trimer calculated from the separable model in Eqs. (1) and (2) for pair correlations corresponding to different two-body interactions, as a function of the depth of these potentials as measured by the number of $s$-wave two-body bound states. The dashed lines indicate from top to bottom the values obtained for the universal pair correlation in Eq. (4) with $n=4$ and $n=6$, and the universal pair correlation in Eq. (5), respectively. This figure shows how the three-body parameter converges differently and to different limits depending on the class of two-body interaction.