Modeling Interactions for Resonant $p$-Wave Scattering

In view of recent experiments on ultracold polarized fermions, the zero-range potential approach is generalized to situations where two-body scattering is resonant in the $p$-wave channel. We introduce a modified scalar product which reveals a deep relation between the geometry of the Hilbert space and the interaction. This formulation is used to obtain a simple interpretation for the transfer rates between atomic and molecular states within a two branches picture of the many-body system close to resonance. At resonance, the energy of the dilute gas is found to vary linearly with density.

Figure 1

Energy per particle for a homogeneous medium composed of polarized fermions as a function of $-1/{\mathcal{V}}_s$ for $\alpha =10k_F$ (dashed line) and $\alpha ={10}^3{k}_F$ (continuous line). Inset: equation of state at resonance ($B=B_0$). In this regime, the energy per particle varies linearly with the atomic density.