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Mobile orbitons in Ca$_2$CuO$_3$: crucial role of the Hund's exchange

We investigate the Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS) spectra of a quasi-1D antiferromagnet Ca$_2$CuO$_3$. In addition to the magnetic excitations, which are well-described by the two-spinon continuum, we observe two dispersive orbital excitations, the $3d_{xy}$ and the $3d_{yz}$ orbitons. We carry out a quantitative comparison of the RIXS spectra, obtained with two distinct incident polarizations, with a theoretical model. We show that any realistic spin-orbital model needs to include a finite, but realistic, Hund's exchange $J_H \approx 0.5$ eV. Its main effect is an increase in orbiton velocities, so that their theoretically calculated values match those observed experimentally. Even though Hund's exchange also mediates some interaction between spinon and orbiton, the picture of spin-orbit separation remains intact and describes orbiton motion in this compound.