Transition to the Fulde-Ferrel-Larkin-Ovchinnikov phases in three   dimensions : a quasiclassical investigation at low temperature with Fourier   expansion
  We investigate, in three spatial dimensions, the transition from the normal state to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases. We make use of a Fourier expansion for the order parameter and the Green's functions to handle the quasiclassical equations in the vicinity of the transition. We show that, below the tricritical point, the transition is always first order. We find that, at the transition, the higher Fourier components in the order parameter are always essentially negligible. Below the tricritical point we have the already known result that the order parameter has a spatial dependence which is essentially $\cos({\bf q}.{\bf r})$. However when the temperature is lowered, the order parameter switches to a sum of two cosines, with equal weigths and wavevector with the same length, but orthogonal directions. Finally by further lowering the temperature, and down to T=0, one finds a another transition toward an order parameter which is the sum of three cosines with again equal weigths and orthogonal directions. Hence the structure of the order parameter gets more complex as the temperature is lowered. On the other hand the resulting critical temperatures are found to be only slightly higher than the ones corresponding to the standard second order FFLO transition. We apply our results to the specific case of ultracold Fermi gases and show that the differences in atomic populations of the two hyperfine states involved in the BCS condensation display sizeable variations when one goes from the normal state to the superfluid FFLO phases, or one FFLO phase to another. Experimentally this should allow to identify clearly the various phase transitions. 
C. Mora and R. Combescot