Topological Kirchhoff Law and Bulk-Edge Correspondence for Valley-Chern and Spin-Valley-Chern Numbers

TL;DR

This paper explores the relationship between valley-Chern and spin-valley-Chern numbers and edge states in topological phases, demonstrating a topological Kirchhoff law at Y-junctions with potential applications in topological electronics.

Contribution

It introduces a topological Kirchhoff law for Y-junctions of topological edges and clarifies the bulk-edge correspondence for valley and spin-valley Chern numbers.

Findings

Edge states are robust despite being topologically trivial in tight-binding models.

Y-junctions satisfy a topological Kirchhoff law conserving topological charges.

The work provides a framework for future topological electronic devices.

Abstract

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. The edge state between them is topologically trivial in the tight-binding model but is shown to be as robust as the topological edge. 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.