Resonant inelastic scattering spectra at the Ir $L$-edge in Na$_2$IrO$_3$

We analyze resonant x-ray scattering (RIXS) spectra in Na$_2$IrO$_3$ on the basis of the itinerant electron picture. Employing a multi-orbital tight-binding model on a honeycomb lattice, we find that the zigzag magnetic order is the most stable with sizable energy gap in the one-electron band within the Hartree-Fock approximation. We derive the RIXS spectra, which are connected to the generalized density-density correlation function. We calculate the spectra as a function of excitation energy $\omega$, within the random phase approximation. The spectra consist of the peaks with $\omega$ $<20$ meV, and of the peaks with $0.4<\omega<0.8$ eV. The former peaks are composed of four bound states in the density-density correlation function, and may be identified as the magnetic excitations, while the latter peaks are composed of sixteen bound states below the energy continuum of individual electron-hole pair excitations, and may be identified as the excitonic excitations. The calculated spectra agree qualitatively with the recent RIXS experiment.