  The phonon localized edge modes are systematically studied, and two
conditions are proposed for the existence of the localized edge modes: (I)
coupling between different directions ($x$, $y$ or $z$) in the interaction;
(II) different boundary conditions in three directions. The generality of these
two conditions is illustrated by different lattice structures: one-dimensional
(1D) chain, 2D square lattice, 2D graphene, 3D simple cubic lattice, 3D diamond
structure, etc; and with different potentials: valence force field model,
Brenner potential, etc.
