Stability of the Fulde-Ferrell-Larkin-Ovchinnikov states in anisotropic systems and critical behavior at thermal m-axial Lifshitz points

We revisit the question concerning the stability of nonuniform superfluid states of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type to thermal and quantum fluctuations. On general grounds, we argue that the mean-field phase diagram hosting a Lifshitz point cannot be stable to fluctuations for isotropic, continuum systems, at any temperature T > 0 in any dimensionality d < 4. In contrast, in layered unidirectional systems, the lower critical dimension for the onset of FFLO-type long-range order accompanied by a Lifshitz point at T > 0 is d = 5/2. In consequence, its occurrence is excluded in d = 2, but not in d = 3. We propose a relatively simple method, based on nonperturbative renormalization group, to compute the critical exponents of the thermal m-axial Lifshitz point continuously varying m, spatial dimensionality d, and the number of order parameter components, N. We point out the possibility of a robust, fine-tuning free occurrence of a quantum Lifshitz point in the phase diagram of imbalanced Fermi mixtures.