Strong spin-orbit coupling effects on the Fermi surface of Sr$_{2}$RuO$_{4}$ and Sr$_{2}$RhO$_{4}$

We present a first-principle study of spin-orbit coupling effects on the Fermi surface of Sr$_{2}$RuO$_{4}$ and Sr$_{2}$RhO$_{4}$ [1]. For nearly degenerate bands, spin- orbit coupling leads to a dramatic change of the Fermi surface with respect to non-relativistic calculations; as evidenced by the comparison with experiments on Sr$_{2}$RhO$_{4}$, it cannot be disregarded. For Sr$_{2} $RuO$_{4}$, the Fermi surface modifications are more subtle but equally dramatic in the detail: spin-orbit coupling induces a strong momentum dependence, normal to the RuO$_2$ planes, for both orbital and spin character of the low-energy electronic states. These findings have profound implications for the understanding of unconventional superconductivity in Sr$_{2}$RuO$_{4}$. [1] M.W. Haverkort {\it et al.}, Phys. Rev. Lett. \textbf{101}, 026406 (2008).