Interchannel coupling induced gapless modes in multichannel zero-line systems

In bilayer graphene, the application of a perpendicular electric field breaks the inversion symmetry to open a bulk band gap to harbor the quantum valley Hall effect. When the field varies spatially, a topologically confined mode (also named the zero-line mode) arises along the zero-field line. In this work, we theoretically investigate the electronic transport properties of the multichannel zero-line systems. The finite-size effect in topological systems (e.g., quantum Hall effect, topological insulators) often induces a topologically trivial gap to realize a normal insulator. To our surprise, we find that the coupling between neighboring zero lines can give rise to striking electronic properties depending on the number of channels m, i.e., a trivial band gap for even m, whereas a nontrivial gapless mode for odd m. We further show that these findings apply to various ribbon orientations. A general effective model is constructed to provide a clear physical picture of the emergence of gapless modes. In the end, a gate-tunable device is proposed to function as a switch with controllable current partitions. We believe that our findings are experimentally accessible, and have potential practical applications in designing multifunctional valley-based electronics.