Prediction of Orbital Ordering in Single-Layered Ruthenates

The key role of the orbital degree of freedom to understanding the magnetic properties of layered ruthenates is discussed based on the 3-orbital Hubbard model coupled to lattice distortions, using numerical and mean-field techniques. In the $G$-type antiferromagnetic phase of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$, recent x-ray experiments reported 0.5 holes/site in the $d_{\mathrm{xy}}$ orbital, while $d_{\mathrm{yz}}$ and $d_{\mathrm{zx}}$ orbitals contain 1.5 holes. This unexpected $t_{2g}$ hole distribution is explained by a novel orbital ordered (OO) state, stabilized by a combination of Coulombic and lattice effects. The phase diagram suggests the possibility of large magnetoresistance effects and a new ferromagnetic OO phase in ruthenates.