Generalized Ginzburg-Landau theory for superfluid ^He and application to Helical textures in ³He-A

We first discuss the generalized Eilenberger equations for superfluid ³He in the presence of a magnetic field and for uniform rotation of the system. These equations determine the space-dependent Green's function, the free energy density, and the supercurrent density at all temperatures. From these equations we derive the generalized Ginzburg-Landau series expansions in powers of spatial derivatives and the order parameter. This is in contrast to Cross' generalized Ginzburg-Landau approach, which takes into account only spatial derivatives up to second order. Explicit expressions are given for the corrections of order (1 − T/T _c)to the well-known expressions for the bending energy, magnetic anisotropy energy, and superfluid current. With the help of these expressions we calculate the stability regions of uniform and helical textures in the superflow-field phase diagram. For decreasing temperature these regions are found to become smaller. The stability region of the uniform texture vanishes at T _h ⁰ ∼- 0.88T _c], in agreement with Fetter's result.