Universal Relations for a Fermi Gas Close to a $p$-Wave Interaction Resonance

We investigate the properties of a spinless Fermi gas close to a $p$-wave interaction resonance. We show that the effects of interaction near a $p$-wave resonance are captured by two contacts, which are related to the variation of energy with the $p$-wave scattering volume $v$ and with the effective range $R$ in two adiabatic theorems. Exact pressure and virial relations are derived. We show how the two contacts determine the leading and subleading asymptotic behavior of the momentum distribution ($\sim 1/k^2$ and $\sim 1/k^4$) and how they can be measured experimentally by radio-frequency and photoassociation spectroscopies. Finally, we evaluate the two contacts at high temperature with a virial expansion.

Figure 1

Dependence of the contacts $C_v$ (the solid line) and $C_R$ (the dashed line) on $E_b/E_F$ for $T/T_F=2$ and their temperature dependences (inset) at a $p$-wave resonance from the virial expansion. We have taken $k_{F}R=1/25$.