Four-Body Efimov Effect for Three Fermions and a Lighter Particle

Abstract

We study three same spin state fermions of mass M interacting with a distinguishable particle of mass m in the unitary limit where the interaction has a zero range and an infinite s-wave scattering length. We predict an interval of mass ratio 13.384 < M/m < 13.607 where there exists a purely fourbody Efimov effect, leading to the occurrence of weakly bound tetramers without Efimov trimers.PACS numbers: 34.50. In a system of interacting particles, the unitary limit corresponds to a zero range s-wave interaction with infinite scattering length [1]. In particular, this excludes any finite energy two-body bound state. Interestingly, in the three-body problem, the Efimov effect may take place [2], leading to the occurrence of an infinite number of threebody bound states, with an accumulation point in the spectrum at zero energy. This effect occurs in a variety of situations, the historical one being the case of three bosons, as recently studied in a series of remarkable experiments with cold atoms close to a Feshbach resonance [3]. It can also occur in a system of two same spin state fermions of mass M and a particle of another species of mass m, in which case the fermions only interact with the third particle, with an infinite s-wave scattering length: An infinite number of arbitrarily weakly bound trimers then appears in this 2 + 1 fermionic problem if the mass ratio α = M/m is larger than α c (2; 1) ≃ 13.607 [2].The four-body problem has recently attracted a lot of interest [4]. The question of the existence of a four-body Efimov effect is however to our knowledge still open. We give a positive answer to this question, by investigating the 3 + 1 fermionic problem in the unitary limit. We explicitly solve Schrödinger's equation in the zero range model [2] and we determine the critical mass ratio to have a purely four-body Efimov effect in this system, that is without Efimov trimers.In the zero-range model, the Hamiltonian reduces to a non-interacting form, here in free spacewith m 1 = m 2 = m 3 = M and m 4 = m. The interactions are indeed replaced by contact conditions on the wavefunction, ψ(r 1 , r 2 , r 3 , r 4 ), where r i , i = 1, 2, 3 is the position of a fermion and r 4 is the position of the other species particle: At the unitary limit, for i = 1, 2, 3, there exist functions A i such thatwhen r i tends to r 4 for a fixed value of the i-4 centroid R i4 ≡ (M r i + mr 4 )/(m + M ) different from the positions of the remaining particles r k , k = i, 4. The wavefunction is also subject to the fermionic exchange symmetry with respect to the first three variables r i , i = 1, 2, 3.In what follows, we shall assume that there is no three-body Efimov effect, a condition that is satisfied by imposing M/m < α c (2; 1) ≃ 13.607. The eigenvalue problem Hψ = Eψ with the contact conditions in Eq. (2) is then separable in hyperspherical coordinates [5]. After having separated out the center of mass C of the system, one introduces the hyperradius, withm = (3M + m)/4 the average mass, and a set of here 8 hyperangles Ω whose e...

Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

“…First, atomic gases provide an ideal platform for investigating phenomena related to Efimov physics [1][2][3]. While the majority of investigations of the Efimov effect have focused on the three-body system, larger systems have attracted considerable attention recently from theoretical and experimental groups [4][5][6][7][8][9][10][11][12][13][14][15]. Second, small trapped atomic systems can be realized by loading an atomic gas into an optical lattice [16][17][18][19].…”

Integral equations for the four-body problem

“…We presented elsewhere a first application of this formalism in the limiting case of a zero-range interaction [34]. We expect that other applications will appear, in particular in the regimes where the zero-range limit with no extra three-body parameter is ill-defined, as it is the case for four bosons, or for the 3 + 1 fermionic problem with a fermion to extra particle mass ratio larger than ≃ 13.607.…”

“…This was done in Ref. [34] in the particular case of the 3+1 fermionic problem with zero-range interactions, infinite scattering length and at zero energy. We shall not pursue this general issue here.…”

“…The zero energy q → 0 and unitary 1/a = 0 limits of Eq. (86) were recently used in [34] to study the emergence of a pure four-body Efimov effect (without three-body Efimov effect) in the system of three fermions plus one extra particle, as a function of the mass ratio α.…”

“…No solution of the equation is presented here, but we presented elsewhere a first application of this formalism in the limiting case of a zero-range interaction [34] and we expect that other applications will come. In this paper, we take as specific examples first the case of four spinless bosons, in section II, and then the so-called 3 + 1 fermionic problem of three same spin state fermions of mass m interacting with a distinguishable particle of mass M , in section III.…”