Orbital Ordering in Frustrated Jahn-Teller Systems with 90° Exchange

We show that superexchange interactions in frustrated Jahn-Teller systems with transition metal ions connected by the $90°$ metal-oxygen-metal bonds (e.g., ${\mathrm{N}\mathrm{a}\mathrm{N}\mathrm{i}\mathrm{O}}_2$, ${\mathrm{L}\mathrm{i}\mathrm{N}\mathrm{i}\mathrm{O}}_2$, and ${\mathrm{Z}\mathrm{n}\mathrm{M}\mathrm{n}}_2{\mathrm{O}}_4$) are much different from those in materials with the $180°$ bonds. In the $90°$-exchange systems spins and orbitals are decoupled: the spin exchange is much weaker than the orbital one and it is ferromagnetic for all orbital states. Though the mean-field orbital ground state is strongly degenerate, quantum orbital fluctuations select particular ferro-orbital states. We explain the orbital and magnetic ordering observed in ${\mathrm{N}\mathrm{a}\mathrm{N}\mathrm{i}\mathrm{O}}_2$ and show that ${\mathrm{L}\mathrm{i}\mathrm{N}\mathrm{i}\mathrm{O}}_2$ is not a spin-orbital liquid.