Three resonating fermions in flatland: proof of the super Efimov effect and the exact discrete spectrum asymptotics (original) (raw)
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Physical Review Letters, 2013
We study a system of spinless fermions in two dimensions with a short-range interaction finetuned to a p-wave resonance. We show that three such fermions form an infinite tower of bound states of orbital angular momentum ℓ = ±1 and their binding energies obey a universal doubly exponential scaling, E (n) 3 ∝ exp −2e 3πn/4+θ , 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 ℓ = ±2 four-body resonances associated with every three-body bound state at E (n) 4 ∝ exp −2e 3πn/4+θ−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.
Exact Solution for 1D Spin-Polarized Fermions with Resonant Interactions
Physical Review Letters, 2010
Using the asymptotic Bethe Ansatz, we obtain an exact solution of the many-body problem for 1D spin-polarized fermions with resonant p-wave interactions, taking into account the effects of both scattering volume and effective range. Under typical experimental conditions, accounting for the effective range, the properties of the system are significantly modified due to the existence of "shape" resonances. The excitation spectrum of the considered model has unexpected features, such as the inverted position of the particle-and hole-like branches at small momenta, and roton-like minima. We find that the frequency of the "breathing" mode in the harmonic trap provides an unambiguous signature of the effective range. PACS numbers: 03.75.-b, 05.30.Fk Experimental progress in the cooling and trapping of ultracold atomic gases makes it possible to investigate their properties under strong transverse confinement, when the motion of atoms is effectively one-dimensional (1D). In such experiments the 1D interaction parameters are precisely known and can be tuned using Feshbach resonances [1] or by varying the harmonic transverse confinement strength. Recent experiments [2, 3] have allowed for parameter-free comparison of 1D Bose gas properties with a theoretical description based on the exactly solvable Lieb-Liniger (LL) model [4]. The possibility to compare experimental results with the outcomes of many-body calculations revived an interest in the field of exactly solvable 1D systems: spin-1/2 fermions [5], Bose-Fermi mixtures [6, 7], and spinor bosons [8] and fermions [9].
Zero-energy bound states and resonances in three-particle systems
Journal of Physics A: Mathematical and Theoretical, 2012
We consider a three-particle system in R 3 with non-positive pairpotentials and non-negative essential spectrum. Under certain restrictions on potentials it is proved that the eigenvalues are absorbed at zero energy threshold given that there is no negative energy bound states and zero energy resonances in particle pairs. It is shown that the condition on the absence of zero energy resonances in particle pairs is essential. Namely, we prove that if at least one pair of particles has a zero energy resonance then a square integrable zero energy ground state of three particles does not exist. It is also proved that one can tune the coupling constants of pair potentials so that for any given R, ǫ > 0: (a) the bottom of the essential spectrum is at zero; (b) there is a negative energy ground state ψ(ξ) such that |ψ(ξ)| 2 d 6 ξ = 1 and |ξ|≤R |ψ(ξ)| 2 d 6 ξ < ǫ.
Journal of Mathematical Physics, 2013
We consider a system of N pairwise interacting particles described by the Hamiltonian H, where σ ess (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.
New Class of Three-Body States
Physical Review Letters, 2012
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Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics
Annales Henri Poincaré, 2004
The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice Z 3 and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with k ∈ T 3 = (−π, π] 3 the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k = 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), K ∈ T 3 being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N (0, z) of eigenvalues of H(0) lying below z < 0 the following limit exists lim z→0− N (0, z) | log | z || = U 0 with U 0 > 0. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number N (K, τess(K)) of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N (K, 0) of eigenvalues lying below zero is given.
Efimov states embedded in the three-body continuum
Physical Review A, 2008
We consider a multichannel generalization of the Fermi pseudopotential to model low-energy atom-atom interactions near a magnetically tunable Feshbach resonance, and calculate the adiabatic hyperspherical potential curves for a system of three such interacting atoms. In particular, our model suggests the existence of a series of quasi-bound Efimov states attached to excited threebody thresholds, far above open channel collision energies. We discuss the conditions under which such states may be supported, and identify which interaction parameters limit the lifetime of these states. We speculate that it may be possible to observe these states using spectroscopic methods, perhaps allowing for the measurement of multiple Efimov resonances for the first time.
Universal Fermi Gas with Two- and Three-Body Resonances
Physical Review Letters, 2008
We consider a Fermi gas with two components of different masses, with the s-wave two-body interaction tuned to unitarity. In the range of mass ratio 8.62 < M/m < 13.6, it is possible for a short-range interaction between heavy fermions to produce a resonance in a three-body channel. The resulting system is scale invariant and has universal properties, and is very strongly interacting. When M/m is slightly above the lower limit 8.62, the ground state energy of a 2:1 mixture of heavy and light fermions is less than 2% of the energy of a noninteracting gas with the same number densities. We derive exact relationships between the pressures of the unitary Fermi gases with and without three-body resonance when the mass ratio is close to the critical values of 8.62 and 13.6. Possible experimental realization with cold atoms in optical lattices is discussed.
Efimov states near a Feshbach resonance
2008
We describe three-body collisions close to a Feshbach resonance by taking into account twobody scattering processes involving both the open and the closed channel. We extract the atomdimer scattering length and the three-body recombination rate, predicting the existence at negative scattering length of a sharp minimum in the recombination losses due to the presence of a shallow bound level. We obtain very good agreement with the experimental results in atomic 133 Cs of Kraemer et al. [Nature 440, 315 (2006)], and predict the position of Efimov resonances in a gas of 39 K atoms.