Separable interactions and liquid 3He: II. Extrema of Landau expansion and phase diagram in the weak-coupling limit

In a previous paper we derived a Landau expansion starting from an exactly solvable model for a system of fermions with an l = 1 pairing interaction in the presence of a magnetic field. The Landau expansion, which can be used to study the phases of liquid ³He, is a complicated function of 9 complex variables, in which it is not obvious a priori that the field dependence of the coefficients of the fourth-order terms can be neglected. In the present paper the extrema of the Landau expansion are analyzed in some detail with the weak-coupling values of the coefficients. The absolute minimum of the Landau expansion can be found by minimizing a three-parameter function Φ_BW, the minimalization of which leads to three possible phases, the A1-phase, the ABM-phase (or two-dimensional 2D-phase) and the proper generalization of the BW-phase in the presence of a magnetic field. The phase diagram is compared with the one given by Ambegaokar and Mermin.