Coupled spin–orbital fluctuations in a three orbital model for 4d and 5d oxides with electron fillings n = 3, 4, 5—application to NaOsO₃, Ca₂RuO₄ and Sr₂IrO₄

A unified approach is presented for investigating coupled spin-orbital fluctuations within a realistic three-orbital model strongly spin-orbit systems with electron fillings $n=3,4,5$ in the $t_{2g}$ sector of $d_{yz},d_{xz},d_{xy}$ orbitals. generalized fluctuation propagator constructed which consistent self-consistent Hartree-Fock approximation where all Coulomb interaction contributions involving orbital diagonal and off-diagonal spin charge condensates are included. Besides low-energy magnon, intermediate-energy orbiton spin-orbiton, high-energy exciton modes, spectral function also shows other excitations such as Hund's coupling induced gapped magnon modes. We relate characteristic features to complex magnetic behavior resulting from interplay between electronic bands, coupling, interactions, structural distortion effects, realized compounds $\rm NaOsO_3$, Ca_2RuO_4$, Sr_2IrO_4$.