Charge-Transfer Excitations in the Model Superconductor HgBa$_2$CuO$_{\bf 4+\delta}$

We report a Cu $K$-edge resonant inelastic x-ray scattering (RIXS) study of charge-transfer excitations in the 2-8 eV range in the structurally simple compound HgBa$_2$CuO$_{4+\delta}$ at optimal doping ($T_{\rm c} = 96.5$ K). The spectra exhibit a significant dependence on the incident photon energy which we carefully utilize to resolve a multiplet of weakly-dispersive ($< 0.5$ eV) electron-hole excitations, including a mode at 2 eV. The observation of this 2 eV excitation suggests the existence of a charge-transfer pseudogap deep in the superconducting phase. Quite generally, our data demonstrate the importance of exploring the incident photon energy dependence of the RIXS cross section.