Generation of d-wave coupling in the two-dimensional Hubbard model from   functional renormalization
  Within the two-dimensional repulsive t-t'-Hubbard model, an attractive coupling in the d-wave pairing channel is induced by antiferromagnetic fluctuations. We investigate this coupling using functional renormalization group equations. The momentum dependent d-wave coupling can be bosonized by the use of scale dependent field transformations. We propose an effective coarse grained model for the Hubbard model which is based on the exchange of antiferromagnetic and d-wave collective bosons. 
H. C. Krahl, J. A. M\"uller, C. Wetterich