Transition metal oxides (TMOs) represent the natural playground for the investigation of the exotic phases produced by electronic correlations, which include high-temperature superconductivity, giant magnetoresistance and topological insulators. The peculiar electronic and magnetic properties of TMOs stem from the interplay between the electron-electron correlations, the crystal field and the spin-orbit coupling (SOC) of the TM d valence electrons. The balance of these energy scales significantly depends on TM element considered: moving from the 3d to the 5d row of the periodic table, the electronic correlations decrease due to the larger size of the atomic orbitals, while SOC increases as a result of the increased atomic charge.
In the present talk, I present three separate case studies of 3d, 4d and 5d TMOs which highlight the impact of the competing interactions just mentioned on the electronic and magnetic properties of the system. Concerning 5d TMOs, I address the spin-wave spectrum of the electron-doped perovskite iridate (Sr1-xLax)2IrO4, where the strong SOC of 5d electrons gives rise to a spin-orbit entangled Mott state with peculiar exchange interactions. I then examine the intermediate case of the 4d oxide Ca2RuO4: here, electronic correlations, SOC and octahedral distortions act on an equal footing to determine the TM ground state. In particular, I show how the crystal field tuning achieved by La substitution affects the electronic and magnetic properties. Finally, I discuss the magnetic ground state of a family of weak ferromagnets, where the weak SOC is responsible for the appearance of a net magnetization in the main antiferromagnetic order.