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Transformation of Coulomb interaction terms to the pseudo-orbital basisconstituted by $J=1/2$ and $3/2$ states arising from spin-orbit couplingprovides a versatile tool. This formalism is applied to investigate magneticanisotropy effects on low-energy spin-wave excitations as well as high-energyspin-orbit exciton modes in $\rm Sr_2 Ir O_4$. The Hund's coupling termexplictly yields easy-plane anisotropy, resulting in gapless (in-plane) andgapped (out-of-plane) modes, in agreement with recent resonant inelastic x-rayscattering (RIXS) measurements. The collective mode of inter-orbital,spin-flip, particle-hole excitations with appropriate interaction strengths andrenormalized spin-orbit gap yields two well-defined propagating spin-orbitexciton modes, with energy scale and dispersion in good agreement with RIXSstudies.

Intra- and inter-orbital correlated electron spin dynamics in $\rm Sr_2 Ir O_4$: spin-wave gap and spin-orbit exciton