Prediction of Orbital Ordering in Single-Layered Ruthenates

The key role of the orbital degree of freedom to understand the magnetic properties of layered ruthenates is here discussed. In the G-type antiferromagnetic phase of Ca$_2$RuO$_4$, recent X-ray experiments reported the presence of 0.5 hole per site in the $d_{xy}$ orbital, while the $d_{\rm yz}$ and $d_{zx}$ orbitals contain 1.5 holes. This unexpected $t_{2g}$ hole distribution is explained by a novel state with orbital ordering (OO), stabilized by a combination of Coulomb interactions and lattice distortions. In addition, the rich phase diagram presented here suggests the possibility of large magnetoresistance effects, and predicts a new ferromagnetic OO phase in ruthenates.