Transition to the Fulde-Ferrel-Larkin-Ovchinnikov planar phase: A quasiclassical investigation with Fourier expansion

We explore, in three spatial dimensions, the transition from the normal state to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases. We restrict ourselves to the case of the "planar" phase, where the order parameter depends only on a single spatial coordinate. We first show that, in the case of the simple Fulde-Ferrell phase, singularities occur at zero temperature in the free energy which prevents, at low temperature, a reliable use of an expansion in powers of the order parameter. We then introduce in the quasiclassical equations a Fourier expansion for the order parameter and the Green's functions, and we show that it converges quite rapidly to the exact solution. We finally implement numerically this method and find results in excellent agreement with the earlier work of Matsuo In particular, when the temperature is lowered from the tricritical point, the transition switches from first to second order. In the case of the first-order transition, the spatial dependence of the order parameter at the transition is found to be always very nearly a pure cosine, although the maximum of its modulus may be comparable to the one of the uniform BCS phase.