Higher order perturbation theory for anisotropic pairings in repulsive Hubbard model

We evaluate the effective pairing interaction for the two-dimensional square-lattice Hubbard model, within a weak coupling approach, specifically, by expanding perturbatively to the fourth order in the on-site repulsion U. We estimate the most probable pairing symmetry and transition temperatures, by solving gap equation numerically. We consider only the following two typical cases: (I) Near the half-filled state and with a similar band structure to high-Tc cuprates. (II) Away from the half-filled state and for the single main band γ of the triplet superconductor Sr2RuO4. In the case (I), each order of the perturbation expansion for the pairing interaction gives a similar momentum dependence, and in addition the perturbation expansion for the pairing interaction converges relatively well even for moderately strong U. On the other hand, in the case (II), the perturbation expansion does not converge well. This bad convergence of the expansion can be remedied by taking higher order corrections. Consequently, we obtain reasonably the dx2-y2-wave and highly anisotropic p-wave states for the cases (I) and (II), respectively