Revealing the origin of super-Efimov states in the hyperspherical formalism

Super-Efimov states are a new kind of universal three-body bound states predicted for three identical fermions with $p$-wave resonant interactions in two dimensions by a recent field-theoretic calculation [Phys.~Rev.~Lett.~\textbf{110}, 235301 (2013)]. The binding energies of these states obey a dramatic double exponential scaling $E_n=E_*\exp(-2 e^{\pi n/s_0+\theta})$ with universal scaling $s_0=4/3$ and three-body parameters $E_*$ and $\theta$. We use the hyperspherical formalism and show that the super-Efimov states originate from an emergent effective potential $-1/4\rho^2-(s_0^2+1/4)/\rho^2\ln^2\left(\rho\right)$ at large hyperradius $\rho$. Moreover, for pairwise interparticle potentials with van der Waals tails, our numerical calculation indicates that the three-body parameters $E_*$ and $\theta$ are also universal and the ground super-Efimov state shall cross the threshold when the $2$D $p$-wave scattering area is about $-42.0\, l_\text{vdW}^2$ with $l_\text{vdW}$ the van der Waals length.