Bound states of three and four resonantly interacting particles

TL;DR

This paper introduces an exact diagrammatic method for analyzing resonant interactions among multiple particles, providing precise results for dimer-dimer scattering in 3D and new bound state energies in 2D for various bosonic and fermionic complexes.

Contribution

The paper develops a novel exact diagrammatic approach for multi-particle resonant interactions, yielding known and new results in both 3D and 2D systems.

Findings

Exact dimer-dimer scattering length in 3D matches known results.

Calculated bound state energies for 3 and 4 bosons in 2D.

Derived exact energies for mixed fermion-boson complexes.

Abstract

We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonant bound state of two fermions in a spin-singlet state, with corresponding scattering length $a_F$. Applying this approach to the calculation of the dimer-dimer scattering length $a_B$, we recover exactly the already known result $a_B=0.60 a_F$. We use the developed approach to obtain new results in 2D for fermions as well as for bosons. Namely, we calculate bound state energies for three $bbb$ and four $bbbb$ resonantly interacting bosons in 2D. For the case of resonant interaction between fermions and bosons we calculate exactly bound state energies of the following complexes: two bosons plus one fermion $bbf$, two bosons plus two fermions $bf_{\uparrow}bf_{\downarrow}$, and three bosons plus one fermion $bbbf$.