Development of a two-particle self-consistent method for multi-orbital systems and its application to unconventional superconductors

We extend the two-particle self-consistent method proposed by Vilk and Tremblay (J. Phys. I France 7, 1309-1368 (1997)) to study superconductivity in multi-orbital systems. Starting with the sum rules for the spin and charge susceptibilities, we derive self-consistent equations to determine the renormalized effective interactions. We apply this method to the two-orbital $d_{x^2-y^2}$-$d_{3z^2-r^2}$ model for La$_2$CuO$_4$ and the five-orbital $d$-model for LaFeAsO. Comparing the results with those of the random phase approximation or the fluctuation exchange approximation in which vertex corrections are ignored, we discuss how the vertex corrections affect the pairing instability of La$_2$CuO$_4$ and the dominant pairing symmetry of LaFeAsO.