  It has been predicted theoretically and indirectly confirmed experimentally
that single-layer CrX$_3$ (X=Cl, Br, I) might be the prototypes of topological
magnetic insulators (TMI). In this work, by using first-principles calculations
combined with atomistic spin dynamics we provide a complete picture of the
magnetic interactions and magnetic excitations in CrX$_3$. The focus is here on
the two most important aspects for the actual realization of TMI, namely the
relativistic magnetic interactions and the finite-size (edge) effects. We
compute the full interaction tensor, which includes both Kitaev and
Dzyaloshinskii-Moriya terms, which are considered as the most likely mechanisms
for stabilizing topological magnons. First, we instigate the properties of bulk
CrI$_3$ and compare the simulated magnon spectrum with the experimental data
[Phys. Rev. X 8, 041028 (2018)]. Our results suggest that a large size of
topological gap, seen in experiment ($\approx$ 4 meV), can not be explained by
considering pair-wise spin interactions only. We identify several possible
reasons for this disagreement and suggest that a pronounced magneto-elastic
coupling should be expected in this class of materials. The magnetic
interactions in the monolayers of CrX$_3$ are also investigated. The strength
of the anisotropic interactions is shown to scale with the position of halide
atom in the Periodic Table, the heavier the element the larger is the
anisotropy. Comparing the magnons for the bulk and single-layer CrI$_3$, we
find that the size of the topological gap becomes smaller in the latter case.
Finally, we investigate finite-size effects in monolayers and demonstrate that
the anisotropic couplings between Cr atoms close to the edges are much stronger
than those in ideal periodic structure. This should have impact on the dynamics
of the magnon edge modes in this class of materials.
