Effective potential and quantum criticality for imbalanced Fermi mixtures

We study the analytical structure of the effective action for spin- and mass-imbalanced Fermi mixtures at the onset of the superfluid state. Of our particular focus is the possibility of suppressing the tricritical temperature to zero, so that the transition remains continuous down to $T=0$ and the phase diagram hosts a quantum critical point. At mean-field level we analytically identify such a possibility in a regime of parameters in dimensionality $d=3$. In contrast, in $d=2$ we demonstrate that the occurrence of a quantum critical point is (at the mean-field level) excluded. We show that the Landau expansion of the effective potential remains well-defined in the limit $T\to 0^+$ except for a subset of model parameters which includes the standard BCS limit. We calculate the mean-field asymptotic shape of the transition line. Employing the functional renormalization group framework we go beyond the mean field theory and demonstrate the stability of the quantum critical point in $d=3$ with respect to fluctuations.