RG Approach to Superconductivity in Sr$_2$RuO$_4$ : Multi-Band and Spin-Orbit Effects

We study the superconductivity pairing mechanism in Sr$_2$RuO$_4$ in the limit of small interaction by extending a renormalization group (RG) calculation developed by Raghu et al (Phys Rev B.81.224505) to include multiband and spin-orbit coupling (SOC) effects. We show these effects to be crucial to discriminate between the possible order parameters. In contrast to the usual theory of an ``active'' gamma band with a large superconducting gap and ``passive'' alpha and beta bands with smaller gaps induced by a proximity effect, we obtain pseudo-spin triplet gaps of the same order of magnitude on all three bands for a large range of interaction parameters. The inclusion of SOC in the microscopic model allows us to study ab initio the breaking of degeneracy between the different d vector orientations. Implications for experiments will be discussed.