3D: numerics for 4th order term

The general features of 4th order vertex in 3d are similar to 2D. Just like in 2D crossing the line of $2k_F \cos \alpha \approx Q$ leads to interaction switch from attraction for larger $Q$ to repulsion at smaller values.

3D plot of the vertex and the line cuts are shown in the attached figures.

In a range of $Q$, at low enough temperatures, there are well-defined minima in the vertex. However, in that regime, the vertex is attractive for all angles, signifying instability.

The situation can be remedied by introducing a separate repulsion source, which is momentum independent, as one coming from fully local 4th order term. The denominator of the Free energy for a variational state then acquires a stabilizing contribution $[1-1/(2N)]C$, where $C$ parametrizes the strength of local repulsion and $N$ is the number of ordered $\pm q$ pairs.

In addition to the coplanar 4th order vertex, there is also a non-coplanar one, which included for the case of FCC. It can always be chosen attractive, and hence matters.