Quantifying the Exchange Interactions in the Kitaev Model System RuCl$_{3}$ by Ru $L$ edge Resonant Inelastic X-ray Scattering

$\alpha$-RuCl$_{3}$ is a major candidate for the realization of the Kitaev quantum spin liquid, but the presence of zigzag antiferromagnetic order at low temperatures indicates deviations from the Kitaev Hamiltonian. To enable a realistic description of the ground state and elementary excitations of $\alpha$-RuCl$_{3}$, we have quantitatively determined the low-energy effective Hamiltonian by carrying out a comprehensive resonant inelastic x-ray scattering (RIXS) study at the Ru $L_{3}$ absorption edge and by comparing the results to detailed model calculations. The high-energy segment of the RIXS spectrum yields the parameters characterizing the electronic structure of individual Ru$^{3+}$ ions and demonstrates that the ground-state wavefunction can be described by a spin-orbit entangled $\widetilde{S}=1/2$ pseudospin. The quasi-elastic RIXS intensity, in conjunction with numerical calculations of the equal-time pseudospin correlation functions, serves as a fingerprint of the exchange interactions between the pseudospins. In the paramagnetic state, it has a broad intensity maximum around the zone center without any local maxima at the zigzag magnetic Bragg wavevectors. This finding implies that the zigzag order is destabilized by pronounced ferromagnetic correlations, and is only slightly lower in energy than other competing phases. Theoretical evaluations based on the RIXS data show that the dominant Kitaev interaction is ferromagnetic and that the magnitudes of Heisenberg and off-diagonal couplings are about half of the Kitaev term. The pseudospin Hamiltonian determined in this way provides a solid foundation for further research on $\alpha$-RuCl$_{3}$, including the quest for a theoretical understanding of its magnetic field-induced spin-liquid behavior.