Two tricritical lines from a Ginzburg-Landau expansion: Application to the Larkin-Ovchinnikov-Fulde-Ferrel phase

We study the behavior of the two plane waves configuration in the Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) phase close to $T=0.\mathrm{}$ The study is performed by using a Landau-Ginzburg expansion up to the eighth order in the gap. The general study of the corresponding grand potential shows, under the assumption that the eighth term in the expansion is strictly positive, the existence of two tricritical lines. This allows to understand the existence of a second tricritical point for two antipodal plane waves in the LOFF phase and justifies why the transition becomes second order at zero temperature. The general analysis done in this paper can be applied to other cases.