Leading Order $k_Fa$ Corrections to the Free Energy and Phase Separation in Two-component Fermion Systems

We study phase separation in a dilute two-component Fermi system with attractive interactions as a function of the coupling strength and the polarization or number density asymmetry between the two components. In weak and strong couplings with a finite number density asymmetry, phase separation is energetically more favorable. A heterogeneous phase containing a symmetric superfluid component and an asymmetric normal phase has lower energy than a homogeneous normal phase. We show that for a small number density asymmetry, taking into consideration the leading order corrections at order $k_Fa$ of the interaction parameter, phase separation is stable against the normal phase in the whole BCS range. We investigate the consequences of the consideration of the leading order $k_Fa$ corrections to the thermodynamic potentials of the normal and BCS phase on the Chandrasekhar-Clogston limit. We have also investigated the stability of a Bose-Fermi mixture in the far-BEC limit. We find that the molecular BEC is locally stable against an external magnetic field $h$, provided $|h|$ is smaller than the pairing gap $\Delta_{gap}$.