Mean-field analysis of pairing in asymmetric Fermi systems at finite temperature

We study the phase diagram of a two component Fermi system with a weak attractive interaction. Our analysis includes the leading order Hartree energy shifts and pairing correlations at finite temperature and chemical potential difference between the two fermion species. We show that in an asymmetric system, the Hartree shift to the single particle energies is important for the phase competition between normal and superfluid phase and can change the phase transition curve qualitatively. At large chemical potential asymmetry we find that a somewhat fragile superfluid state can be favored due to finite-temperature effects. We also investigate the transition between the normal phase and an inhomogeneous superfluid phase to study how gradient instabilities evolve with temperature and asymmetry. Finally, we adopt our analysis to study the density profiles of similar asymmetric Fermi systems that are being observed in cold-atom experiments.