  We investigate the spin-orbit coupling effect in a two-dimensional Wigner
crystal. We show that sufficiently strong spin-orbit coupling and an
appropriate sign of g-factor could transform the Wigner crystal to a
topological phonon system. We demonstrate the existence of chiral phonon edge
modes in finite size samples, as well as the robustness of the modes in the
topological phase. We explore the possibility of realizing the topological
phonon system in two-dimensional Wigner crystals confined in semiconductor
quantum wells/heterostructure. We find that the spin-orbit coupling is too weak
for driving a topological phase transition in these systems. We argue that one
may look for the topological phonon system in correlated Wigner crystals with
emergent effective spin-orbit coupling.
