Exchange of the resonant pair of fermions between two bosons is shown to induce a nearly scale-invariant attraction, which leads to an infinite number of bound states obeying the universal scaling law of binding energies. This system constitutes a new member in the class of semisuper Efimov effect, potentially relevant to two-neutron halo nuclei as well as ultracold atoms.
Exchange of the resonant pair of fermions between two bosons is shown to induce a nearly scale-invariant attraction, which leads to an infinite number of bound states obeying the universal scaling law of binding energies. This system constitutes a new member in the class of semisuper Efimov effect, potentially relevant to two-neutron halo nuclei as well as ultracold atoms.
Wave-particle duality in quantum mechanics allows for a halo bound state whose spatial extension far exceeds a range of the interaction potential. What is even more striking is that such quantum halos can be arbitrarily large on special occasions. The two examples known so far are the Efimov effect and the super Efimov effect, which predict that spatial extensions of higher excited states grow exponentially and double exponentially, respectively. Here, we establish yet another new class of arbitrarily large quantum halos formed by spinless bosons with short-range interactions in two dimensions. When the two-body interaction is absent but the three-body interaction is resonant, four bosons exhibit an infinite tower of bound states whose spatial extensions scale as R_{n}∼e^{(πn)^{2}/27} for a large n. The emergent scaling law is universal and is termed a semisuper Efimov effect, which together with the Efimov and super Efimov effects constitutes a trio of few-body universality classes allowing for arbitrarily large quantum halos.
We study the semisuper-Efimov effect, which is found for four identical bosons with a resonant three-body interaction in 2D, in various systems. Based on solutions of bound-state and renormalization-group equations, we first demonstrate an emergence of the semisuper-Efimov effect in mass-imbalanced bosons in 2D. Compared with the Efimov and the super-Efimov effects, the mass ratio-dependent scaling parameter is unexpectedly found to take on a finite value even for extremely mass-imbalanced situations, where the mass ratio is 0 or ∞. By a renormalization-group analysis, we also show that a weak two-body interaction sustains the semisuper-Efimov effect. Finally, we liberate the universality of the semisuper-Efimov effect from 2D by showing that bosons with linear-dispersion relation support the semisuper-Efimov effect in 1D.
We study a system of spinless fermions in two dimensions with a short-range interaction fine-tuned to a p-wave resonance. We show that three such fermions form an infinite tower of bound states of orbital angular momentum $\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1$ and their binding energies obey a universal doubly exponential scaling ${E}_{3}^{(n)}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}2{e}^{3\ensuremath{\pi}n/4+\ensuremath{\theta}})$ at large n. This ``super Efimov effect'' is found by a renormalization group analysis and confirmed by solving the bound state problem. We also provide an indication that there are $\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}2$ four-body resonances associated with every three-body bound state at ${E}_{4}^{(n)}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}2{e}^{3\ensuremath{\pi}n/4+\ensuremath{\theta}\ensuremath{-}0.188})$. These universal few-body states may be observed in ultracold atom experiments and should be taken into account in future many-body studies of the system.
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 $\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1$ 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.
The Efimov effect was first predicted for three particles interacting at an s-wave resonance in three dimensions. A subsequent study showed that the same effect can be realized by considering two-body and three-body interactions in mixed dimensions. In this work, we consider the three-body problem of two bosonic A atoms interacting with another single B atom in mixed dimensions: The A atoms are confined in a space of dimension dA and the B atom in a space of dimension dB, and there is an interspecies s-wave interaction in a dint-codimensional space accessible to both species. We find that when the s-wave interaction is tuned on resonance, there emerge an infinite series of universal three-body bound states for dA,dB,dint=2,2,0 and 2,3,1. Going beyond the Efimov paradigm, the binding energies of these states follow the scaling $ln|{E}_{n}{|\ensuremath{\sim}\ensuremath{-}s(n\ensuremath{\pi}\ensuremath{-}\ensuremath{\theta})}^{2}/4$, with the scaling factor s being unity for the former case and mB(2mA+mB)/(mA+mB) for the latter. We discuss the possibility of realizing our mixed-dimensional systems in a cold-atom experiment and how the effects of these universal three-body bound states may be detected.
The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.