What are the rainbow and ladder approximations in a solid state physics context? All references I find talk about quarks and gluons, where I have only very limited knowledge about. 
From it's name (rainbow) I guess it applies to fermions coupled to bosons and we're interested in the single particle Green function for the fermion, and we approximate the sum over all diagrams by only summing those diagrams where no phonon lines cross (so that Feynman diagrams look like rainbows).
Is that correct? 
I also noted that all search results always talk simultaneously about the rainbow and the ladder approximation, as was also pointed out in a comment below.
 A: Your description of the rainbow and ladder approximations are correct as people use them in my experience. Although there is no necessary restriction to fermions, its just most natural. I don't know if there is much to be said in general - the terms are use in so many models in quantum field theory that you should probably look carefully at the specifics. The ladder and rainbow tend to go together, since the rainbow is just the ladder diagram where we connect two of the ends.
As you point out these approximation may come in the end from restriction on crossing of "phonon" lines. This is the case in the quantum theory of disordered metals where the "phonon" lines connect two scatterings off the same impurity. Diagrams with crossed lines are smaller by a factor which is roughly the atomic length divided by the mean distance between impurities. This is small and these diagrams are neglected. The remaining diagrams form the ladder daigram, which in this case reproduces the diffusion equation.
