Four-Body Efimov Effect for Three Fermions and a Lighter Particle

We study three same-spin-state fermions of mass $M$ interacting with a distinguishable particle of mass $m$ in the unitary limit where the interaction has a zero range and an infinite $s$-wave scattering length. We predict an interval of mass ratio $13.384<M/m<13.607$ where there exists a purely four-body Efimov effect, leading to the occurrence of weakly bound tetramers without Efimov trimers.

Figure 1

Minimal eigenvalues of the Hermitian operators $M_{s=0}$ in each sector of fixed parity and angular momentum $l$, $0\le l\le 6$, as functions of the mass ratio $\alpha =M/m$. Only the curve for the even sector of $l=1$ crosses zero for $\alpha <13.607$, corresponding to the occurrence of a four-body Efimov effect in that sector. The other curves all remain above zero. They strongly overlap and are barely distinguishable at the scale of the figure. The dotted line is the analytical prediction $\Lambda (k=0,\alpha )$ for the lower border of the continuum in the spectrum of $M_{s=0}$. The inset is a magnification. In the numerics, $x$ and $u$ were discretized with a step $dx=du=1/10$, and $x$ was truncated to $x_{\max }=20$.

Figure 2

In the Efimovian channel $l=1$ with even parity, modulus of the purely imaginary Efimov exponent $s=iS$ as a function of the mass ratio $\alpha =M/m$. In the numerics, $x_{\max }$ ranges from 40 to 120, $dx=1/10$, $d\theta =\pi /20$. The dashed line results from a linear fit of $|s{|}^2$ as a function of $\alpha$ in a vicinity of the critical value ${\alpha }_c(3;1)$, $|s{|}_{\mathrm{fit}}^2\simeq 2.23(\alpha -{\alpha }_c)$. The vertical dotted line indicates the $2+1$ critical value ${\alpha }_c(2;1)$.