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Generalized Ginzburg-Landau theory for non-uniform FFLO superconductors

We derive a generalized Ginzburg-Landau (GL) functional near the tricritical point in the (T,H)-phase diagram for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state, in 1,2, and 3 dimensions. We find that the transition from the normal to the FFLO state is of second order in 1 and 2 dimensions, and the order parameter with one-coordinate sine modulation corresponds to the lowest energy near the transition line. We also describe in the one-dimensional case the transformation of the sine modulation into the soliton-lattice state as the magnetic field decreases. In 3 dimensions however, we find that the transition into an FFLO state is of first order, and it is impossible to obtain an analytic expression for the critical temperature. In this case the generalized GL functional proposed here provides a suitable basis for a numerical study of the properties of the FFLO state, and in particular for computing the critical temperature, and for describing the transition into a uniform state.