The two- and three-body correlation functions of the ground state of an optically trapped ultracold spin-$\frac{1}{2}$ Fermi gas (SFG) in a tight waveguide [one-dimensional (1D) regime] are calculated in the plane of even- and odd-wave coupling constants, assuming a 1D attractive zero-range odd-wave interaction induced by a 3D $p$-wave Feshbach resonance, as well as the usual repulsive zero-range even-wave interaction stemming from 3D $s$-wave scattering. The calculations are based on the exact mapping from the SFG to a “Lieb-Liniger-Heisenberg” model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, and indicate that the SFG should be stable against three-body recombination in a large region of the coupling constant plane encompassing parts of both the ferromagnetic and antiferromagnetic phases. However, the limiting case of the fermionic Tonks-Girardeau gas, a spin-aligned 1D Fermi gas with infinitely attractive $p$-wave interactions, is unstable in this sense. Effects due to the dipolar interaction and a Zeeman term due to a resonance-generating magnetic field do not lead to shrinkage of the region of stability of the SFG.

• Received 8 May 2007

DOI:https://doi.org/10.1103/PhysRevA.76.013615