LAUSR

We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality CrI_{3} single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with K∼-5.2  meV, 25 times larger than the Heisenberg exchange J∼-0.2  meV, and responsible for opening the ∼5  meV gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy Γ∼-67.5  μeV, though small, is crucial for opening a ∼0.3  meV gap in the magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data further reveals a μeV-scale quadrupolar contribution to the S=3/2 magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher T_{C} as well as magnetically frustrated quantum spin liquids based on Kitaev physics.