Super Efimov effect for mass-imbalanced systems

We study two species of particles in two dimensions interacting by isotropic short-range potentials with the interspecies potential fine-tuned to a $p$-wave resonance. Their universal low-energy physics can be extracted by analyzing a properly constructed low-energy effective field theory with the renormalization group method. Consequently, a three-body system consisting of two particles of one species and one of the other is shown to exhibit the super Efimov effect, the emergence of an infinite tower of three-body bound states with orbital angular momentum $\ell =\pm 1$ whose binding energies obey a doubly exponential scaling, when the two particles are heavier than the other by a mass ratio greater than 4.034 04 for identical bosons and 2.414 21 for identical fermions. With increasing the mass ratio, the super Efimov spectrum becomes denser which would make its experimental observation easier. We also point out that the Born-Oppenheimer approximation is incapable of reproducing the super Efimov effect, the universal low-energy asymptotic scaling of the spectrum.

Figure 1

Low-energy limit of the resonant $p$-wave scattering in two dimensions reduces to the propagation of a dimer (double line) with energy-dependent couplings (dots) [see Eq. (4)]. The solid and dashed lines represent propagators of particles of species 1 and 2, respectively.

Figure 2

Feynman diagrams to renormalize the interspecies two-body couplings (a) $g$ and (b) $v_{12}$. The intraspecies two-body couplings $v_{11}$ and $v_{22}$ are renormalized by Feynman diagrams similar to the one (b).

Figure 3

Feynman diagrams to renormalize the three-body coupling $u_1$.

Figure 4

The universal scaling factor $e^{\pi /\gamma }$ as a function of the mass ratio $m_1/m_2$ for two identical bosons (upper curve) and fermions (lower curve) with the universal exponent $\gamma$ determined in Eqs. (13) and (14), respectively. The horizontal dashed line indicates $e^{3\pi /4}\approx 10.55$ corresponding to the universal scaling factor for three identical fermions [7].

Figure 5

Feynman diagrams to renormalize the four-body couplings $w$ and $w^{\prime }$.

Figure 6

Feynman diagrams representing the two-body scattering $T$ matrix (A2).

Figure 7

Feynman diagrams representing the three-body scattering $T$ matrix (A3).