Symmetry-protected Interface Modes Bifurcated from Double Dirac Cones

We rigorously prove the existence of interface modes in a sharp interface model, which bifurcate from the double Dirac cone as a consequence of the band inversion induced by super-symmetry breaking. The exact number of interface modes are determined. The proof is based on a discrete version of the layer-potential framework. Moreover, we prove that such interface modes are symmetry-protected against perturbations that respect the reflection symmetry. This finding has significant implications for the design of robust waveguides and sensors in photonics and acoustics, particularly in the context of topological metamaterials where defects can be intentionally introduced without compromising functionality.