The Sign of Three: Spin/Charge Density Waves at the Boundaries of Transition Metal Dichalcogenides

One-dimensional grain boundaries of two-dimensional semiconducting {\MX} (M= Mo,W; X=S,Se) transition metal di-chalcogenides are typically metallic at room temperature. The metallicity has its origin in the lattice polarization, which for these lattices with $D_{3h}$ symmetry is a topological invariant, and leads to one-dimenional boundary states inside the band gap. For boundaries perpendicular to the polarization direction, these states are necessarily 1/3 occupied by electrons or holes, making them susceptible to a metal-insulator transition that triples the translation period. Using density-functional-theory calculations we demonstrate the emergence of combined one-dimensional spin density/charge density waves of that period at the boundary, opening up a small band gap of $\sim 0.1$ eV. This unique electronic structure allows for soliton excitations at the boundary that carry a fractional charge of $\pm 1/3\ e$.