Unraveling the Orbital Physics in a Canonical Orbital System ${\mathrm{KCuF}}_3$

We explore the existence of the collective orbital excitations, orbitons, in the canonical orbital system ${\mathrm{KCuF}}_3$ using the Cu $L_3$-edge resonant inelastic x-ray scattering. We show that the nondispersive high-energy peaks result from the ${\mathrm{Cu}}^{2+}dd$ orbital excitations. These high-energy modes display good agreement with the ab initio quantum chemistry calculation, indicating that the $dd$ excitations are highly localized. At the same time, the low-energy excitations present clear dispersion. They match extremely well with the two-spinon continuum following the comparison with Müller ansatz calculations. The localized $dd$ excitations and the observation of the strongly dispersive magnetic excitations suggest that the orbiton dispersion is below the resolution detection limit. Our results can reconcile with the strong local Jahn-Teller effect in ${\mathrm{KCuF}}_3$, which predominantly drives orbital ordering.

Figure 1

Scattering geometry and overview of RIXS spectra. (a) Cu $L_3$-XAS spectra of ${\mathrm{KCuF}}_3$ collected with the partial fluorescence yield. (b) A sketch of the experimental geometry. Light blue arrows represent the incident ($k_i$) and scattered ($k_f$) x-rays, while the double arrows (green: $\pi$, purple: $\sigma$) are for the polarizations of incident x-rays. Red arrows indicate the momentum transfer and the corresponding projection parallel and perpendicular to the sample surface. Crystal axes are represented by black arrows. (c) A fitting example of the $dd$ excitations. The purple dotted line is an experimental spectrum, and the gray solid line represents the total fit. The inset depicts the energy splitting of $3d$ orbitals.

Figure 2

Angular dependence of $dd$ excitations. (a) Experimental results $\sigma$- (purple) and $\pi$ polarized (green) incident x-rays. The vertical dashed lines depict the averaged peak values of fitted $dd$ excitations. (b) Calculated angular-dependent spectra from the $\mathrm{MRCI}+\mathrm{SOC}$. The vertical dashed lines show the calculated values of $dd$ excitations.

Figure 3

Relative intensity variations as a function of $\theta$ angle for different $dd$ excitations. (a) is the experimental results; (b) is the calculated results. The first data point of the $d_{y^2}$ orbital is normalized to a fixed intensity.

Figure 4

Low-energy excitations of ${\mathrm{KCuF}}_3$ revealed by the Cu $L_3$ edge RIXS. (a) and (b) are color maps of the low-energy RIXS spectra from the $\sigma$ and $\pi$ polarizations, respectively. (c) is the Müller ansatz calculated results. The white dashed lines are the lower and upper boundaries of the two-spinon continuum. The thin white line is the zero energy reference. (d) The comparison of RIXS spectra at $q_c=0.5\mathrm{r}.\mathrm{l}.\mathrm{u}.$ The dashed line marks the longitudinal magnetic mode observed in INS [42]. (e) Temperature-dependent RIXS spectra at $q_c=0.5\mathrm{r}.\mathrm{l}.\mathrm{u}.$ for different polarizations.