  Recent experiments evidence the direct observation of spin waves in chromium
trihalides and a gap at the Dirac points of the magnon dispersion in bulk
CrI$_3$. However, the topological origin of this feature remains unclear and
its emergence at the 2D limit has not yet been proven experimentally. Herein,
we perform a fully self-consistent ab initio analysis that supports the
presence of topological magnons in chromium trihalides monolayers. Our results
confirm the existence of a gap around the K high-symmetry point in the linear
magnon dispersion of CrI$_3$, which originates as a direct consequence of
intralayer Dzyaloshinskii-Moriya (DM) interaction. In addition, our orbital
resolved analysis reveals the microscopic mechanisms that can be exploited
using strain engineering to increase the strength of the DM interaction and
thus control the gap size in CrI$_3$. This paves the way to the further
development of this family of materials as building-blocks for topological
magnonics at the limit of miniaturization.
