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

Breaking inversion symmetry in chiral graphene systems, \textit{e.g.}, by applying a perpendicular electric field in chirally-stacked rhombohedral multilayer graphene or by introducing staggered sublattice potentials in monolayer graphene, opens up a bulk band gap that harbors a quantum valley-Hall state. When the gap size is allowed to vary and changes sign in space, a topologically-confined one-dimensional (1D) zero-line mode (ZLM) is formed along the zero lines of the local gap. Here we show that gapless ZLM with distinguishable valley degrees of freedom K and K$'$ exist for every propagation angle except for the armchair direction that exactly superpose the valleys. We further analyze the role of different geometries of top-bottom gated device setups that can be realized in experiments, discuss the effects of their edge misalignment, and analyze three common forms of topological defects that could influence the 1D ZLM transport properties in actual devices.