  The topological quantum states in two-dimensional (2D) materials are
fascinating subjects of research, which usually highlight electron-related
systems. In this work, we present a recipe that leads to Dirac phonon states
with quantized valley Berry phase in 2D hexagonal lattices by first-principles
calculations. We show that candidates possessing the three-fold rotational
symmetry at the corners of the hexagonal Brillouin zone host valley Dirac
phonons, which are guaranteed to remain intact with respect to perturbations.
We identify that such special topological features populated by Dirac phonons
can be realized in various 2D materials. In particular, the monolayer CrI$_3$,
an attractive 2D magnetic semiconductor with exotic applications in
spintronics, is an ideal platform to investigate nontrivial phonons in
experiments. We further confirm that the phonon Berry phase is quantized to
$\pm \pi$ at two inequivalent valleys. The phonon edge states terminated at the
projection of phonon Dirac cones are clearly visible. This work demonstrates
that 2D hexagonal lattices with attractive valley Dirac phonons will extend the
knowledge of valley physics, providing wide applications of topological
phonons.

Mesoscale and Nanoscale Physics;Materials Science