Until last week I had several misconceptions about unconventional [i.e., non s-wave] superconductivity due to purely electronic interactions.
I thought weak coupling approaches tend to give a clear "pairing mechanism" and the symmetry of the Cooper pairs is related to the type of fluctuations or collective mode responsible for the pairing
For example, d-wave singlet pairing tends to go with antiferromagnetic spin fluctuations and p-wave triplet tends to go with ferromagnetic spin fluctuations.
There is a very nice paper
Band structure effects on the superconductivity in Hubbard models
by Weejee Cho, Ronny Thomale, Srinivas Raghu, and Steve Kivelson
They consider a weak-coupling renormalisation group (RG) treatment of a Hubbard model with specific band structures that are varied by changing tight-binding parameters.
The relevant Feynman diagrams are below
Calling this "spin fluctuation exchange" is not clear as there is no well defined collective mode that can be thought of as the superconducting "glue." The authors state, instead "the pairing is a result of overscreening by the whole band".
The authors show/claim
1. different order parameter symmetries can emerge from the same underlying mechanism, depending on the band structure.
2. in the weak-coupling limit, it is not possible to attribute the pseudogap entirely to a non-superconducting order. [Since the cuprates are in the intermediate coupling regime, this does not preclude the real pseudogap being due to non-superconducting order].
3. "The structure of the favored superconducting gap along the Fermi surface can be inferred in large part from a catalogue of wave vectors, Q, at which the susceptibility is large....
The most important portions of the Fermi surface are either those in which this approximate nesting condition is satisfied over a substantial region of the Fermi surface, or in which the Fermi velocity is small (density of states is large)."
4. For a spatially anisotropic [nematic] band structure there is a large parameter range where is a remarkable near degeneracy of a singlet (d + s)-wave and a triplet p-wave pairing channel.
There are three more situations I would like to see this weak-couping approach and formalism applied to.
a. The Hubbard model on the anisotropic triangular lattice at half filling.
Ben Powell and I showed that within an RVB [strong coupling] theory that as the frustration t'/t changed [and the band structure and Fermi surface changed accordingly] that the superconducting pairing symmetry changed from A2 [dx^2-y^2] to A2+iA1 [d+id] to A1 [d_xy].
We interpreted this as "Symmetry of the Superconducting Order Parameter in Frustrated Systems Determined by the Spatial Anisotropy of Spin Correlations"
[The isotropic triangular lattice [t'=t] case was considered earlier with perturbative RG by Raghu, Kivelson, and Scalapino, and with functional RG by Honerkamp. Both found d+id superconductivity.]
b. The "purple bronze" Li0.9Mo6O17
Jaime Merino and I recently considered the simplest possible extended Hubbard model that might describe this quasi-one-dimensional material. It consists of ladders that are weakly coupled to one another and at quarter-filling.
An outstanding question concerns whether this model will produce the observed superconductivity, which is probably triplet.
c. The extended Hubbard model on the square lattice at one-quarter filling.
Using a slave boson approach Jaime Merino and I showed that due to charge fluctuations near the charge-ordering transition there is d_xy superconducting order.
b. and c. require the extension of the weak-coupling approach that including the nearest neighbour repulsion V. This is included in this paper using the same formalism.
I thank Srinivas Raghu for bringing this work to my attention and explaining some key details. He also provided some helpful corrections to the first version of this post.