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arXiv:1405.1787 · Journal of Physics A Mathematical and Theoretical

Three resonating fermions in flatland: proof of the super Efimov effect and the exact discrete spectrum asymptotics

D. K. Gridnev

Published Nov 26, 2014• 38 citations
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Abstract

We consider the system of three nonrelativistic spinless fermions in two dimensions, which interact through spherically-symmetric pair interactions. Recently, a claim has been made by Nishida et al for the existence of the so-called super Efimov effect by Nishida et al (2013 Phys. Rev. Lett. 110 235301 ). 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 a system indeed has an infinite number of bound levels. We also prove that , where N ( E ) is the number of bound states with the energy less than . The value of this limit is exactly equal to the value derived in Nishida et al using the renormalization group approach. Our proof resolves a recent controversy about the validity of results in Nishida et al (2013 Phys. Rev. Lett. 110 235301 ).

DOI
10.1088/1751-8113/47/50/505204

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We consider the system of three nonrelativistic spinless fermions in two dimensions, which interact through spherically-symmetric pair interactions. Recently, a claim has been made by Nishida et al for the existence of the so-called super Efimov effect by Nishida et al (2013 Phys. Rev. Lett. 110 235301 ). 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 a system indeed has an infinite number of bound levels. We also prove that , where N ( E ) is the number of bound states with the energy less than . The value of this limit is exactly equal to the value derived in Nishida et al using the renormalization group approach. Our proof resolves a recent controversy about the validity of results in Nishida et al (2013 Phys. Rev. Lett. 110 235301 ).
We study a system of spinless fermions in two dimensions with a short-range interaction fine-tuned to a ppp-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 nnn. 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.
A new kind of "super-Efimov" states of binding energies scaling as ln⁡∣En∣∼−e3nπ/4\ln|E_n|\sim-e^{3nπ/4}ln∣En​∣∼−e3nπ/4 were predicted by a field theory calculation for three fermions with resonant ppp-wave interactions in two dimensions [Phys. Rev. Lett. \textbf{110}, 235301 (2013)]. However, the universality of these "super-Efimov" states has not been proved independently. In this Letter, we study the three fermion system through the hyperspherical formalism. Within the adiabatic approximation, we find that at ppp-wave resonances, the low energy physics of states of angular momentum ℓ=±1\ell=\pm1ℓ=±1 crucially depends on the value of an emergent dimensionless parameter YYY determined by the detail of the inter-particle potential. Only if YYY is exactly zero, the predicted "super-Efimov" states exist. If Y>0Y>0Y>0, the scaling of the bound states changes to ln⁡∣En∣∼−(nπ)2/2Y\ln|E_n|\sim-(nπ)^2/2Yln∣En​∣∼−(nπ)2/2Y, while there are no shallow bound states if Y<0Y<0Y<0.
We study two species of particles in two dimensions interacting by isotropic short-range potentials with the interspecies potential fine-tuned to a ppp-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.
A class of universal ``three-body'' bound states has been recently predicted theoretically for identical fermions interacting at ppp-wave resonance in two dimensions. This phenomenon is called the super Efimov effect since the binding energies of the states follow a intriguing double exponential scaling. However, experimental resolution of this scaling is expected to meet formidable challenges. In this work, we introduce a thermodynamic quantity, the three-body contact ${C}_{\ensuremath{\theta}}$, to quantify three-body correlations in a two-dimensional gas composed of the resonantly interacting fermions; the contact ${C}_{\ensuremath{\theta}}$ is the consequence of the underlying universal super Efimov effect in the many-body context. We show how ${C}_{\ensuremath{\theta}}$ affects physical observables such as the radio-frequency spectrum, the momentum distribution, and the atom loss rate. The signature of the elusive super Efimov effect can be pinned down by the detection of the three-body contact ${C}_{\ensuremath{\theta}}$ via these observables in the many-body system.
We consider a system of N pairwise interacting particles described by the Hamiltonian H, where σess(H) = [0, ∞) and none of the particle pairs has a zero energy resonance. The pair potentials are allowed to take both signs and obey certain restrictions regarding the fall off. It is proved that if N ⩾ 4 and none of the Hamiltonians corresponding to the subsystems containing N − 2 or less particles has an eigenvalue equal to zero then H has a finite number of negative energy bound states. This result provides a positive proof to a long-standing conjecture of Amado and Greenwood stating that four bosons with an empty negative continuous spectrum have at most a finite number of negative energy bound states. Additionally, we give a short proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete spectrum and pose a conjecture regarding the existence of the “true” four-body Efimov effect.
Equal mass fermionic trimers with two different spin components near the unitary limit are shown to possess a universal van der Waals bound or resonance state near sss-wave unitarity, when ppp-wave interactions are included between the particles with equal spin. Our treatment uses a single-channel Lennard-Jones interaction with long-range two-body van der Waals potentials. While it is well known that there is no true Efimov effect that would produce an infinite number of bound states in the unitary limit, we demonstrate that another type of universality emerges for the symmetry ${L}^{\mathrm{\ensuremath{\Pi}}}={1}^{\ensuremath{-}}$. The universality is a remnant of Efimov physics that exists in this system at ppp-wave unitarity, and it leads to modified threshold and scaling laws in that limit. Application of our model to the system of three lithium atoms studied experimentally by Du et al. [Phys. Rev. Lett. 102, 250402 (2009)] provides an interpretation of their measured three-body recombination loss rates.