Topology of the valley-Chern effect

The quantum valley-Hall effect (QVHE) manifests in both classical and quantum materials as the emergence of pairs of quasichiral bands along certain interfaces. This bulk-boundary principle is well understood in the limit where the valleys result from a slight splitting of Dirac singularities. However, using a versatile experimental platform based on magnetically coupled spinners, we demonstrate that this regime is not suitable for metamaterial applications due to the delocalization of the interface modes. We also find that a strong splitting of the Dirac singularities washes away the QVHE. We then propose that the enlargement of the bulk gap to be accompanied by a Berry curvature engineering that keeps it localized near the valleys. This is a new regime, entirely outside the umbrella of Dirac physics, which we call the valley-Chern effect (VCE). By establishing an exact relation between VCE and quantum spin-Hall effect, we demonstrate a robust bulk-boundary principle, which could be the foundation of a new wave of applications of topological metamaterials.