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arXiv:1301.4473 · Physical Review Letters

Super Efimov Effect of Resonantly Interacting Fermions in Two Dimensions

Yusuke Nishida, Sergej Moroz, Dam Thanh Son

Published Jun 4, 2013• 79 citations
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Abstract

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.

DOI
10.1103/physrevlett.110.235301

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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 \ensuremathℓ=\ifmmode±\else\textpm\fi1\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}1\ensuremathℓ=\ifmmode±\else\textpm\fi1 and their binding energies obey a universal doubly exponential scaling E3(n)\ensuremath∝exp(\ensuremath−2e3\ensuremathπn/4+\ensuremathθ){E}_{3}^{(n)}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}2{e}^{3\ensuremath{\pi}n/4+\ensuremath{\theta}})E3(n)​\ensuremath∝exp(\ensuremath−2e3\ensuremathπn/4+\ensuremathθ) 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 \ensuremathℓ=\ifmmode±\else\textpm\fi2\ensuremath{\ell}=\ifmmode\pm\else\textpm\fi{}2\ensuremathℓ=\ifmmode±\else\textpm\fi2 four-body resonances associated with every three-body bound state at E4(n)\ensuremath∝exp(\ensuremath−2e3\ensuremathπn/4+\ensuremathθ\ensuremath−0.188){E}_{4}^{(n)}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}2{e}^{3\ensuremath{\pi}n/4+\ensuremath{\theta}\ensuremath{-}0.188})E4(n)​\ensuremath∝exp(\ensuremath−2e3\ensuremathπn/4+\ensuremathθ\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 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.
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 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 ).
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.
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.
The Efimov effect was first predicted for three particles interacting at an sss-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 AAA atoms interacting with another single BBB atom in mixed dimensions: The AAA atoms are confined in a space of dimension dA{d}_{A}dA​ and the BBB atom in a space of dimension dB{d}_{B}dB​, and there is an interspecies sss-wave interaction in a dint{d}_{\mathrm{int}}dint​-codimensional space accessible to both species. We find that when the sss-wave interaction is tuned on resonance, there emerge an infinite series of universal three-body bound states for dA,dB,dint=2,2,0{{d}_{A},{d}_{B},{d}_{\mathrm{int}}}={2,2,0}dA​,dB​,dint​=2,2,0 and 2,3,1{2,3,1}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 sss being unity for the former case and mB(2mA+mB)/(mA+mB)\sqrt{{m}_{B}(2{m}_{A}+{m}_{B})}/({m}_{A}+{m}_{B})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.