d-wave pairing near the transition temperature

The problem of minimizing the general fourth-order Ginzburg-Landau free energy for $d$-wave pairing in a homogeneous system is solved. The highly degenerate family of solutions arising when the fourth-order term has the elementary BCS form, is found to be associated with a point lying on the boundary of two regimes in the general parameter space. On either side of the boundary the degeneracy is reduced to the minimum required by gauge and rotational invariance, but in very different ways: on one side pairing is in a state with $L_z=2\mathrm{}$, on the other, pairing is in a state with $〈\overrightarrow{\mathrm{L}}〉=0\mathrm{}$. It is therefore essential to know the form of the corrections to the elementary BCS theory, whether or not they prove to be large.