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Breaking inversion symmetry in chiral graphene systems, \textit{e.g.}, byapplying a perpendicular electric field in chirally-stacked rhombohedralmultilayer graphene or by introducing staggered sublattice potentials inmonolayer graphene, opens up a bulk band gap that harbors a quantum valley-Hallstate. When the gap size is allowed to vary and changes sign in space, atopologically-confined one-dimensional (1D) zero-line mode (ZLM) is formedalong the zero lines of the local gap. Here we show that gapless ZLM withdistinguishable valley degrees of freedom K and K$'$ exist for everypropagation angle except for the armchair direction that exactly superpose thevalleys. We further analyze the role of different geometries of top-bottomgated device setups that can be realized in experiments, discuss the effects oftheir edge misalignment, and analyze three common forms of topological defectsthat could influence the 1D ZLM transport properties in actual devices.

The role of geometry and topological defects in the 1D zero-line modes of graphene