an itinerant fermion description*

We present a theory of spin waves in ferromagnetic CrI₃ monolayers based on the calculation of the renormalized spin susceptibility, computed using an extended multi-orbital Hubbard model obtained from first-principles calculations. Our theory includes the multi-orbital nature of Cr and I atoms, as well as their spin orbit coupling and yield the spin waves as poles from the spin suceptibility tensor. For monolayers, theory reproduces the spin wave dispersion measured with inelastic neutron scattering, with a gap at the Dirac point compatible with a topological origin. We have computed the spin waves for a ribbon and find in-gap chiral edge states that provide further back-up to this scenaro. Importantly, our approach goes does not require to define spin Hamiltonian, and can be applied to a wide class of magnetic 2D materials.