Crystal field splitting and correlation effect on the electronic structure of $A_2{\rm IrO}_3$

The electronic structure of the honeycomb lattice iridates Na$_2$IrO$_3$ and Li$_2$IrO$_3$ has been investigated using resonant inelastic x-ray scattering (RIXS). Crystal-field split $d$--$d$ excitations are resolved in the high-resolution RIXS spectra. In particular, the splitting due to non-cubic crystal fields, derived from the splitting of $j_{\rm{eff}}$=3/2 states, is much smaller than the typical spin-orbit energy scale in iridates, validating the applicability of $j_{\rm{eff}}$ physics in A$_2$IrO$_3$. We also find excitonic enhancement of the particle-hole excitation gap around 0.4 eV, indicating that the nearest-neighbor Coulomb interaction could be large. These findings suggest that both Na$_2$IrO$_3$ and Li$_2$IrO$_3$ can be described as spin-orbit Mott insulators, similar to the square lattice iridate Sr$_2$IrO$_4$.