Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles

Abstract

We present a diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length a B in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length a F , and the already known result a B = 0.60 a F is recovered exactly. Then we make use of our approach to obtain results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound-state energies for three bbb and four bbbb resonantly interacting bosons in two dimensions. We consider also the case of a resonant interaction between fermions and bosons, and we obtain the exact bound-state energies of two bosons plus one fermion bbf, two bosons plus two fermions bf ↑ bf ↓ , and three bosons plus one fermion bbbf.

BCS-BEC crossover in bilayers of cold fermionic polar molecules

“…We note that dipolar interactions are different from short-or zero-range interactions. Zero-range interactions in 2D always allow a two-body dimer bound state [29], and a state of four bosons will also be bound [34]. This result is different from that in 3D, where the interaction must be sufficiently attractive to produce bound states.…”

Thermodynamics of Ultracold Fermi Gases

“…Interactions between dimers are described by a dimer-dimer scattering length a dd 0.60 a, given by the exact resolution of the 4-fermion scattering problem [16,209]. However, it is not clear to which extent the many-body ground state is merely equal to the equation of state of n d = n/2 bosons with short-range interactions described by a scattering length a dd .…”

Three-boson bound states in two dimensions

“…In the literature, three-boson problems in two dimensions are only solved for the case with the same kind of interaction (either repulsive or attractive) between bosons, 11,[13][14][15]27 To address the stability of the two dimensional system, we need to take into account both repulsive and attractive interactions, and for all possible scattering channels. We then consider three interacting bosons in two dimensions.…”