Symmetries and Mean-Field Phases of the Extended Hubbard Model
  The two-dimensional extended Hubbard model that includes a nearest- neighbor Heisenberg interaction is studied using a mean-field theory where quasiparticles are defined by an U(8) group of canonical transformations. The theory is a generalization of the ordinary BCS theory, and Balian and Werthamer's theory of He-3 that permits both broken gauge, spin and sublattice symmetry. This allows us to investigate superconductivity, antiferromagnetic order, charge density waves and, by twisting the spin quantization axis, spiral antiferromagnetic order in the same theory. Our results for positive Hubbard U and Heisenberg exchange J suggest that antiferromagnetic ordering dominates close to half filling, while spiral states and d-wave superconductivity compete when doping is introduced. For moderate values of J, we find a phase diagram where a phase transition occurs from an antiferromagnet to a d-wave superconductor as doping is increased. A narrow region of (s+id)-wave superconductor is found for some values of J and U. 
A.B. Eriksson, T. Einarsson and S. Ostlund