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Magnetic excitation in resonant inelastic x-ray scattering of Sr$_2$IrO$_4$: A localized spin picture

We study the magnetic excitations in 5d transition-metal oxide Sr$_2$IrO$_4$ on the basis of the Heisenberg model with small anisotropic terms on a square lattice. We calculate the correlation functions by using the Green's functions in the spin-wave approximation. The spin waves are split into two modes with slightly different energies due to the anisotropic terms. It is shown that the spin correlation functions of the y and z components are composed of a single peak corresponding to each mode. We analyze the process of resonant inelastic x-ray scattering (RIXS) without relying on the fast collision approximation to obtain the local scattering operator. The RIXS intensity is derived as a sum of the correlation functions of the y and z spin components. We demonstrate that the RIXS intensity as a function of energy shows two-peak structure brought about by the two modes, which could be observed in the RIXS experiment.