Collective excitations in Na$_2$IrO$_3$

We study the collective excitations of Na$_2$IrO$_3$ in an itinerant electron approach. We consider a multi-orbital tight-binding model with the electron transfer between the Ir $5d$ states mediated via oxygen $2p$ states and the direct $d$-$d$ transfer on a honeycomb lattice. The one-electron energy as well as the ground state energy are investigated within the Hartree-Fock approximation. When the direct $d$-$d$ transfer is weak, we obtain nearly flat energy bands due to the formation of quasimolecular orbitals, and the ground state exhibits the zigzag spin order. The evaluation of the density-density correlation function within the random phase approximation shows that the collective excitations emerge as bound states. For an appropriate value of the direct $d$-$d$ transfer, some of them are concentrated in the energy region $\omega <$ 50 meV (magnetic excitations) while the others lie in the energy region $\omega >$ 350 meV (excitonic excitations). This behaviour is consistent with the resonant inelastic x-ray scattering spectra. We also show that the larger values of the direct $d$-$d$ transfer are unfavourable in order to explain the observed aspects of Na$_2$IrO$_3$ such as the ordering pattern of the ground state and the excitation spectrum. These findings may indicate that the direct $d$-$d$ transfer is suppressed by the structural distortions in the view of excitation spectroscopy, as having been pointed out in the \it{ab initio} calculation.