Origin of the Three-Body Parameter Universality in Efimov Physics

In recent years extensive theoretical and experimental studies of universal few-body physics have advanced our understanding of universal Efimov physics. Whereas theory had been the driving force behind our understanding of Efimov physics for decades, recent experiments have contributed an unexpected discovery. Specifically, measurements have found that the so-called three-body parameter determining several properties of the system is universal, even though fundamental assumptions in the theory of the Efimov effect suggest that it should be a variable property that depends on the precise details of the short-range two- and three-body interactions. The present Letter resolves this apparent contradiction by elucidating previously unanticipated implications of the two-body interactions. Our study shows that the three-body parameter universality emerges because a universal effective barrier in the three-body potentials prevents the three particles from simultaneously getting close together. Our results also show limitations on this universality, as it is more likely to occur for neutral atoms but less likely to extend to light nuclei.

Figure 1

(a) Full energy landscape for the three-body potentials at $a=\infty$ for our $v_{\lambda }^a$ model potential. (b) Effective diabatic potentials $W_{\nu }$ relevant for Efimov physics for $v_{\lambda }^a$ with an increasingly large number of bound states (${\lambda }_n^{*}$ is the value of $\lambda$ that produces $a=\infty$ and $n$ $s$-wave bound states). The $W_{\nu }$ converge to a universal potential displaying the repulsive barrier at $R\approx 2r_{\mathrm{vdW}}$ that prevents particles’ access to short distances. (c)–(e) demonstrate the suppression of the wave function inside the potential well through the channel functions ${\Phi }_{\nu }(R;\theta ,\phi )$ for $R$ fixed near the minima of the Efimov potentials in (b). (c) shows the mapping of the geometrical configurations onto the hyperangles $\theta$ and $\phi$. (d) and (e) show the channel functions, where the “distance” from the origin determines $|{\Phi }_{\nu }{|}^{1/2}$, for two distinct cases: in (d) when there is a substantial probability to find two particles inside the potential well (defined by the region containing the gray disks) and in (e) with a reduced probability—see also our discussion in Fig. 2. In (d) and (e), we used the potentials $v_{\mathrm{sch}}$ and $v_{\lambda }^a$, respectively, both with $n=3$.

Figure 2

Density plot of the three-body probability density $|{\Phi }_{\nu }(R;\theta ,\phi ){|}^2\sin 2\theta$ which determines the three particle configuration [see Fig. 1c] in the $\theta -\phi$ hyperangular plane for a fixed $R$ ($\sin 2\theta$ is the volume element). (a)–(d) show the results for an $R$ near the minima of the Efimov potentials in Fig. 1b for the first four scattering length poles of the $v_{\lambda }^a$ model as indicated. (a) shows that there is a negligible probability to find the particles at distances smaller than $r_{\mathrm{vdW}}$ (outer dashed circle) and, of course, inside the $1/r^{12}$ repulsive barrier (inner solid circle). For higher poles, i.e., as the strength of the hard-core part of the $v_{\lambda }^a$ potential decreases, the potential becomes deeper and penetration into the region $r<r_{\mathrm{vdW}}$ is now classically allowed. Nevertheless, (b)–(e) show that inside-the-well suppression still efficiently suppresses the probability to find particle pairs at distances $r<r_{\mathrm{vdW}}$, found to be in the range 2%–4%.

Figure 3

(a) Efimov potential obtained from the different two-body potential models used here. The reasonably good agreement between the results obtained using models supporting many bound states ($v_{\mathrm{sch}}$, $v_{\lambda }^a$ and $v_{\lambda }^b$) and $v_{\mathrm{vdW}}^{\mathrm{hs}}$ [obtained by replacing the deep potential well with a hard wall but having only one (zero-energy) bound state] supports our conclusion that the inside-the-well suppression of the wave function is the main physical mechanism behind the universality of the three-body effective potentials. The differences between these potentials are seen to cause differences of a few percent in the three-body parameter. (b) Comparison between the effective potential proposed by Ref. 39 (green dashed curve) and the one (red solid curve) constructed to describe our findings: $2\mu r_{\mathrm{vdW}}^2{W}_{\nu }^u(R)/{\hslash }^2\approx -(s_0^2+1/4)/X^2-b_3/X^3-b_4/X^4-b_5/X^5+b_{16}/X^{16}$, where $X=R/r_{\mathrm{vdW}}$ and $b_3=2.334$, $b_4=1.348$, $b_5=44.52$, $b_{16}=4.0\times {10}^4$.

Figure 4

Values for the three-body parameter (a) ${\kappa }_{*}$ and (b) $a_{3b}^{-}$ as functions of the number $n$ of two-body $s$-wave bound states for each of the potential models studied here. (c) Experimental values for $a_{3b}^{-}$ for ${}^{133}\mathrm{Cs}$ [3] (red: $\times$, $+$, □, and *), ${}^{39}\mathrm{K}$ [4] (magenta: ▵), ${}^7\mathrm{Li}$ [5] (blue: •) and [6, 7] (green: ▪ and ◯), ${}^6\mathrm{Li}$ [8, 9] (cyan: ▴ and ▿) and [10, 11] (brown: ▾ and ◆), and ${}^{85}\mathrm{Rb}$ [12] (black: ◆). The gray region specifies a band where there is a $\pm 15\%$ deviation from the $v_{\mathrm{vdW}}^{\mathrm{hs}}$ results. The inset of (a) shows the suppression parameter ${\xi }_p^{\mathrm{in}}$ (Eq. (S.5) in Ref. 26) which can be roughly understood as the degree of sensitivity to nonuniversal corrections. Since ${\xi }_p^{\mathrm{in}}$ is always finite—even in the large $n$ limit—nonuniversal effects associated with the details of the short-range interactions can still play an important role. One example is the large deviation in ${\kappa }_{*}$ found for the $v_{\mathrm{sch}}$ ($n=6$) model, caused by a weakly bound $g$-wave state. For $n>10$ we expect ${\kappa }_{*}$ and $a_{3b}^{-}$ to lie within the range of $\pm 15\%$ established for $n\le 10$.