Gap Nodes and Time Reversal Symmetry Breaking in Strontium Ruthenate

We study the superconducting state of Sr$_2$RuO$_4$ on the bases of a phenomenological but orbital specific description of the electron-electron attraction and a realistic quantitative account of the electronic structure in the normal state. We found that a simple model which features both `in plane' and `out of plane' coupling with strengths $U_{\parallel}=40$meV and $U_{\perp}=48$meV respectively reproduced the experimentally observed power law behaviour of the low temperature specific heat $C_v(T)$, superfluid density $n_s(T)$ and thermal conductivity in quantitative detail. Moreover, it predicts that the quasi-particle spectrum on the $\gamma$ -sheet is fully gaped and the corresponding order parameter breaks the time reversal symmetry. We have also investigated the stability of this model to inclusion of further interaction constants in particular %those which describe `proximity coupling' between orbitals contributing to the $\gamma$ sheet of the Fermi surface and the $\alpha$ and $\beta$ sheets. We found that the predictions of the model are robust under such changes. Finally, we have incorporated a description of weak disorder into the model and explored some of its consequences. For example we demonstrated that the disorder has a more significant effect on the $f$-wave component of the order parameter than on the p-wave one.