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

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