Octupolar order in d-orbital Mott insulators.

We discuss the quantum magnetism of $J=2$ ions which can be realized in spin-orbit coupled oxides with $5d^2$ transition metal ions. We discuss a microscopic mechanism where the combination of orbital repulsion and antiferromagnetic spin interactions leads to ferro-octupolar coupling, potentially stabilizing ferro-octupolar order. We discuss experimental signatures of octupolar order. We uncover a a gapped and dispersive magnetic exciton in the ferro-octupolar state using a slave-boson approach, which can be observed in neutron scattering. For sufficiently strong magnetic exchange, this dispersive exciton can condense, leading to conventional type-I antiferromagnetic order which can coexist with or even preempt octupolar order. Our proposal for ferrooctupolar order provides a comprehensive understanding of thermodynamics, $\mu$SR, X-ray diffraction, and inelastic neutron scattering measurements on a range of cubic $5d^2$ double perovskite materials including Ba$_2$ZnOsO$_6$, Ba$_2$CaOsO$_6$, and Ba$_2$MgOsO$_6$. Our proposal for exciton condensation leading to type-I magnetic ordering is potentially relevant to materials such as Sr$_2$MgOsO$_6$.