Topological Kirchhoff law and bulk-edge correspondence for valley Chern and spin-valley Chern numbers

The valley Chern and spin-valley Chern numbers are the key concepts in valleytronics. They are topological numbers in the Dirac theory but not in the tight-binding model. We analyze the bulk-edge correspondence between the two phases which have the same Chern and spin-Chern numbers but different valley Chern and spin-valley Chern numbers. Though the edge state between them is topologically trivial in the tight-binding model, it is shown to be as robust as the topological one both for zigzag and armchair edges. We construct Y-junctions made of topological edges. They satisfy the topological Kirchhoff law, where the topological charges are conserved at the junction. We may interpret a Y-junction as a scattering process of particles which have four topological numbers. It would be a milestone of future topological electronics.