  Superfluidity in atomic Fermi gases with population imbalance has recently
become an exciting research focus. There is considerable disagreement in the
literature about the appropriate stability conditions for states in the phase
diagram throughout the BCS to Bose-Einstein condensation (BEC) crossover. Here
we discuss these stability conditions for homogeneous polarized superfluid
phases, and compare with recent alternative proposals. The requirement of a
positive second order partial derivative of the thermodynamic potential with
respect to the fermionic excitation gap $\Delta$ (at fixed chemical potentials)
is demonstrated to be equivalent to the positive definiteness of the particle
number susceptibility matrix. In addition, we show the positivity of the
effective pair mass constitutes another nontrivial stability condition. These
conditions determine the stability of the system towards phase separation of
one form or another. We also study systematically the effects of finite
temperature and the related pseudogap on the phase diagrams defined by our
stability conditions.
