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Population and mass imbalance in atomic Fermi gases

We develop an accurate theory of resonantly interacting Fermi mixtures with both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass imbalances, but focus in particular on the experimentally available ${}^{6}$Li-${}^{40}$K mixture. We determine the phase diagram of the mixture for different interactions strengths that lie on the BCS side of the Feshbach resonance. We also determine the universal phase diagram at unitarity. We find for the mixtures with a sufficiently large mass imbalance, that includes the ${}^{6}$Li-${}^{40}$K mixture, a Lifshitz point in the universal phase diagram that signals an instability towards a supersolid phase.