Proof of the Super Efimov Effect

Abstract

We consider the system of 3 nonrelativistic spinless fermions in two dimensions, which interact through spherically-symmetric pair interactions. Recently a claim has been made for the existence of the so-called super Efimov effect [Y. Nishida et al., Phys. Rev. Lett. 110, 235301 (2013)]. Namely, if the interactions in the system are fine-tuned to a p-wave resonance, an infinite number of bound states appears, whose negative energies are scaled according to the double exponential law. We present the mathematical proof that such system indeed has an infinite number of bound levels. We also prove that ${\lim }_{E\to 0}|\ln |\ln E|{|}^{-1}N(E)=8/(3\pi )$, where $N(E)$ is the number of bound states with the energy less than $-E<0$. The value of this limit is equal exactly to the value derived in [Y. Nishida et al.] using renormalization group approach. Our proof resolves a recent controversy about the validity of results in [Y. Nishida et al.].

References in corpus (4)

• Stability of the fermionic gases close to a p-wave Feshbach resonance
• Super Efimov effect of resonantly interacting fermions in two dimensions
• Borromean ground state of fermions in two dimensions
• Why there is no Efimov effect for four bosons and related results on the finiteness of the discrete spectrum

Cited by in corpus (22)

• Efimov Physics: a review
• Universal few-body physics and cluster formation
• Universal Properties of a Strongly Interacting Fermi Gas at a $p$-wave Resonance
• Absence of a four-body Efimov effect in the 2 + 2 fermionic problem
• Revealing the origin of super-Efimov states in the hyperspherical formalism
• Super Efimov effect for mass-imbalanced systems
• Efimov effect for a three-particle system with two identical fermions
• Three-body forces and Efimov physics in nuclei and atoms
• Universality in a one-dimensional three-body system
• Dynamic Super Efimov Effect
• Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions
• Universal three-body bound states in mixed dimensions beyond the Efimov paradigm
• Scattering Hypervolume of Fermions in Two Dimensions
• Rigorous derivation of the Efimov effect in a simple model
• The absence of the Efimov effect in systems of one- and two-dimensional particles
• On the virtual level of two-body interactions and applications to three-body systems in higher dimensions
• Window for Efimov physics for few-body systems with finite-range interactions
• Efimov Effect in Long-range Quantum Spin Chains
• Universal Bound States in Long-range Spin Chains with an Impurity
• Bound-like State in a 1D Self-Similar Delta-Barrier Array
• Efficient determination of eigenenergies and eigenstates of $N$ ($N=3$--$4$) identical 1D bosons and fermions under external harmonic confinement
• Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian