Single and Double Orbital Excitations Probed by Resonant Inelastic X-Ray Scattering

The dispersion of the elusive elementary excitations of orbital ordered systems, orbitons, has escaped detection so far. The recent advances in resonant inelastic x-ray scattering (RIXS) techniques have made it, in principle, a powerful new probe of orbiton dynamics. We compute the detailed traces that orbitons leave in RIXS for an $e_g$ orbital ordered system, using the ultrashort core-hole lifetime expansion for RIXS. We observe that both single- and double-orbiton excitations are allowed, where the former, at lower energy, have sharper features. The rich energy- and momentum-dependent intensity variations that we observe make clear that RIXS is an ideal method to identify and map out orbiton dispersions.

Figure 1

The effect of the core hole on the orbital exchange. An x ray with energy ${\omega }_{\mathrm{in}}^0$ and momentum ${\mathbf{q}}_{\mathrm{in}}$ excites the $1s$ electron to a $4p$ state. Via an intermediate state, the system reaches a final state and the core hole decays, emitting a photon of energy ${\omega }_{\mathrm{out}}^0$ and momentum ${\mathbf{q}}_{\mathrm{out}}$. In the figure, the matrix element $⟨{\downarrow }_i{\uparrow }_j|H|{\uparrow }_i{\uparrow }_j⟩$ is considered in the presence of a core hole. There are two alternative intermediate states to reach the final state. In the upper case, the amplitude is proportional to $t^{xz}{t}^{xx}/(U+U_c)$ and in the lower case to $t^{xz}{t}^{zz}/(U-U_c)$. Adding these gives a modified exchange $J^{\prime }=J(1+\eta )$, where $\eta$ depends on $U_c$.

Figure 2

The indirect RIXS spectrum for a cut through the Brillouin zone. The lower two branches originate from single-orbiton excitations, the upper, continuous spectrum from double-orbiton scattering. Selection rules are such that at $\mathbf{q}=(0,0)$ all spectral weight vanishes and at $\mathbf{q}=(\pi ,\pi )$ only one single-orbiton branch is active.

Figure 3

Comparison of the energy integrated spectral weight of the one- and two-orbiton RIXS spectra at fixed momentum transfer $\mathbf{q}$.