Determining Kitaev interaction in spin-$S$ honeycomb Mott insulators

The Kitaev interaction in a honeycomb lattice with higher-spin $S$ has been one of the central attractions, as it may offer quantum spin liquids. A microscopic theory showed that when the Hund's coupling at the transition metal generates $S> \frac{1}{2}$, the spin-orbit coupling at the heavy ligands provides a route to the Kitaev interaction. However, there have been debates over its strength compared to other symmetry-allowed interactions. Investigating the symmetry of the Hamiltonian for general $S$, we show the magnon energies at two momentum points related by a broken mirror symmetry reflect the Kitaev interaction when a magnetic field is in the mirror plane. Applying the symmetry analysis to CrI$_3$ with $S=\frac{3}{2}$ together with the available angle-dependent ferromagnetic resonance data, we estimate the Kitaev interaction out of the full Hamiltonian and find that it is sub-dominant. Our theory can be tested by inelastic neutron scattering on candidate materials under the proposed magnetic field direction, which will advance the search for general $S$ Kitaev materials.