The role of spin-orbit coupling and Hund's rule coupling on magnetic ordering, anisotropy, and excitations are investigated within a minimal three-orbital model for the $5d^3$ compound $\rm Na Os O_3$. Asymmetry between the magnetic moments for the $xy$ and $xz,yz$ orbitals, arising from the hopping asymmetry generated by the $\rm Os O_6$ octahedral tilting and rotation, together with the weak correlation effect, are shown to be crucial for the large SOC induced magnetic anisotropy and spin wave gap observed in this compound... (read more)
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