  The one-dimensional metallic states that appear at the zigzag edges of
semiconducting two-dimensional transition metal di-chalcogenides (TMDCs) result
from the intrinsic electric polarization in these materials, which for D$_{3h}$
symmetry is a topological invariant. These 1D states are susceptible to
electronic and structural perturbations that triple the period along the edge.
In this paper we study possible spin density waves (SDWs) and charge-density
waves (CDWs) at the zigzag edges of {\MS}, using first-principles density
functional theory calculations. Depending on the detailed structures and
termination of the edges, we observe either combined SDW/CDWs or pure CDWs,
along with structural distortions. In all cases the driving force is the
opening of a band gap at the edge. The analysis should hold for all group VI
TMDCs with the same basic structure as $MoS_2$.
