  The interplay of spin-orbit-coupling and strong electronic correlations is
studied for the single-layer and the bilayer compound of the strontium
ruthenate Ruddlesden-Popper series by a combination of first-principles
band-structure theory with mean-field rotationally invariant slave bosons. At
equilibrium strongly renormalized (spin-orbit-split) quasiparticle bands are
traced and a thorough description of the low-energy regime for the nearly
ferromagnetic bilayer system in accordance with experimental data is presented.
The metamagnetic response of Sr$_3$Ru$_2$O$_7$ in finite magnetic field $H$ is
verified and a detailed analysis of the underlying correlated electronic
structure provided. Intriguing multi-orbital physics on both local and
itinerant level, such as e.g. competing paramagnetic and diamagnetic
contributions, is observed with important differences depending on the
magnetic-field angle $\theta$ with the crystallographic c axis.
