Revealing the origin of super Efimov states in the hyperspherical formalism

Super Efimov states are new kinds of universal three-body bound states predicted for three identical fermions with $p\text{-wave}$ resonant interactions in two dimensions (2D) by a recent field-theoretic calculation [Phys. Rev. Lett. 110, 235301 (2013)]. The binding energies of these states obey a dramatic double exponential scaling $E_n=E_{*}exp\left(-2e^{\pi n/s_0+\theta }\right)$ 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 $-1/4{\rho }^2-\left(s_0^2+1/4\right)/{\rho }^2{ln}^2\left(\rho \right)$ 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 $2\mathrm{D}p\text{-wave}$ scattering area is about $-42.0l_{\text{vdW}}^2$, with $l_{\text{vdW}}$ the van der Waals length.

Figure 1

Numerical results for the effective potential $U_{\mathrm{eff}}$ defined in Eq. (5) for three different two-body model potentials, from top to bottom: Lennard-Jones (LJ), Gaussian (GS), and Pöschl-Teller (PT). The red solid lines are for the first $p\text{-wave}$ resonances of the three potentials, the blue dashed-dotted ones for the second, and the green dashed ones for the third. The black dashed-dot-dotted lines are ${\rho }^2{U}_{\mathrm{eff}}+1/4=-[{(4/3)}^2+1/4]/{ln}^2(\rho /r_0)$.

Figure 2

Universal effective potential $U_{\mathrm{eff}}$ for different two-body model potentials $V_k^n$, with sharp avoid crossings manually diabatized in some cases to improve visualization. An example of the manual diabatization is shown in the inset for the model potential $V_6^2$. The sharp feature arising from an accidental crossing between two channels represented by the red dashed-dotted and the black dashed curves is manually eliminated to give the smooth green solid curve.

Figure 3

Super Efimov ground-state energies $E_g$ for different two-body model potentials $V_k^n$ by full calculations. The error bars at $n=2,3$ are the decay rates of these states into atom-dimer states. The inset shows the three-body parameters $\theta$ and $\xi$ calculated by the adiabatic approximation.

Figure 4

Suppression of the zero energy two-body wave function $u_0$ at short distances. The thick curves are the radial probability of zero energy wave functions $u_0^2\left(r\right)$ for the model potential $V_6^n$ with $n=1,2,3$; the thin curves are the effective radial potential $V_6^n+\left(3/4\right)/r^2$. The black solid, red dashed, and blue dashed-dotted curves correspond to $n=1,2,3$, respectively.

Figure 5

The binding energy of the ground super Efimov trimer state $E_g$ vs the $2\mathrm{D}p\text{-wave}$ scattering area $A$ using the Lennard-Jones potential tuned around the $n\mathrm{th} p\text{-wave}$ resonance. The error bars of $E_g$ for $n=2,3$ quantify the finite lifetime of the state due to its decaying into atom-dimer states.