Transition to Fulde-Ferrel-Larkin-Ovchinnikov phases near the tricritical point : an analytical study

We explore analytically the nature of the transition to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases in the vicinity of the tricritical point, where these phases begin to appear. We make use of an expansion of the free energy up to an overall sixth order, both in order parameter amplitude and in wavevector. We first explore the minimization of this free energy within a subspace, made of arbitrary superpositions of plane waves with wavevectors of different orientations but same modulus. We show that the standard second order FFLO phase transition is unstable and that a first order transition occurs at higher temperature. Within this subspace we prove that it is favorable to have a real order parameter and that, among these states, those with the smallest number of plane waves are prefered. This leads to an order parameter with a $\cos({\bf q}_{0}. {\bf r})$ dependence, in agreement with preceding work. Finally we show that the order parameter at the transition is only very slightly modified by higher harmonics contributions when the constraint of working within the above subspace is released.