  Magnons dominate the magnetic response of the recently discovered insulating
ferromagnetic two dimensional crystals such as CrI$_3$. Because of the
arrangement of the Cr spins in a honeycomb lattice, magnons in CrI$_3$ bear a
strong resemblance with electronic quasiparticles in graphene. Neutron
scattering experiments carried out in bulk CrI$_3$ show the existence of a gap
at the Dirac points, that has been conjectured to have a topological nature.
Here we propose a theory for magnons in ferromagnetic CrI$_3$ monolayers based
on an itinerant fermion picture, with a Hamiltonian derived from first
principles. We obtain the magnon dispersion for 2D CrI$_3$ with a gap at the
Dirac points with the same Berry curvature in both valleys. For CrI$_3$
ribbons, we find chiral in-gap edge states. Analysis of the magnon wave
functions in momentum space further confirms their topological nature.
Importantly, our approach does not require to define a spin Hamiltonian, and
can be applied to both insulating and conducting 2D materials with any type of
magnetic order.
