Five-Body Efimov Effect and Universal Pentamer in Fermionic Mixtures

We show that four heavy fermions interacting resonantly with a lighter atom ($4+1$ system) become Efimovian at mass ratio 13.279(2), which is smaller than the corresponding $2+1$ and $3+1$ thresholds. We thus predict the five-body Efimov effect for this system in the regime where any of its subsystem is non-Efimovian. For smaller mass ratios we show the existence and calculate the energy of a universal $4+1$ pentamer state, which continues the series of the $2+1$ trimer predicted by Kartavtsev and Malykh and $3+1$ tetramer discovered by Blume. We also show that the effective-range correction for the light-heavy interaction has a strong effect on all these states and larger effective ranges increase their tendency to bind.

Figure 1

(a) The trimer (dotted red), tetramer (dashed green), and pentamer (solid blue) energies in units of the dimer energy versus $M/m$ (the inset is an enlargement of the crossing region). (b) The trimer-tetramer threshold ($\mathrm{\Delta }E=E_4-E_3$). (c) The tetramer-pentamer threshold ($\mathrm{\Delta }E=E_5-E_4$).

Figure 2

(a) $s_{N+1}^2$ versus $M/m$ close to the $N+1$-body Efimov thresholds for $N=2$ (red triangles), 3 (green circles), and 4 (blue squares) calculated by our stochastic method. The dotted red curve is exact, the dashed green curve is the result of our deterministic grid calculation based on the method of Ref. [22], and the solid blue curve is the $\text{linear}+\text{quadratic}$ fit to the data. (b) $s_{N+1}-3N/2+2$, related to the energy of the trapped unitary $N+1$ system, versus $M/m$. The color and symbol coding is the same as in (a). The inset shows the crossing region in more detail.

Figure 3

Mass ratios corresponding to the dimer-trimer (dotted red), trimer-tetramer (dashed green), and tetramer-pentamer (solid blue) crossings as a function of $r_0/a$. The solid curves are the $\text{linear}+\text{quadratic}$ fit to the data. The dash-dotted line is the ${}^{53}\mathrm{Cr}\text{−}{}^6\mathrm{Li}$ mass ratio.