Bound states of three and four resonantly interacting particles

Abstract

We present an exact 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 aB in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length aF and the already known result aB = 0.60 aF is recovered exactly. Then we make use of our approach to obtain new 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 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 .

Three-body scattering from nonperturbative flow equations

“…This problem has been formulated via a momentum space integral equation a long time ago by Skorniakov and Ter-Martirosian (STM) [1], for more recent treatments see [8,10,20,21]. The STM integral equation is derived from a consideration of possible scattering processes.…”

“…The (Minkowski) cm four-momentum of the fermion at rest reads P = (−σ A = −ǫ M /2, 0), where ǫ M is the dimensionless binding energy of the dimer -the fermion propagator is gapped on the BEC side of the resonance. Our choice of the zero is different from the one of [20,21], where the fermion energy is defined to be zero, such that the boson has negative energy ǫ M = 2σ A . Of course such a shift in the zero of energy is arbitrary and our final equations are independent of this choice.…”

“…However, for the sake of clarity and simple notation, we first simplify the above equation as appropriate for our purposes. This includes two main steps [20,21]: (i) Integrating out the loop frequency, and (ii) performing an s-wave projection, since we are only interested in low energy scattering here.…”

“…We are now in the position to directly compare this result to the STM integral equation [1,20,21]. Expressed in the above matrix notation it reads…”

Equation of state of a superfluid Fermi gas in the BCS-BEC crossover

“…We thereby find that the molecular condensate density N Our diagrammatic procedure is also different from the works in Refs. [20,22], where molecule-molecule scattering processes are considered in isolation. These calculations perturb around the free-fermion state and therefore require additional terms beyond the ladder structures in Fig.…”