Existence and Splitting of Dirac Points in Honeycomb Gyromagnetic Photonic Crystals

We establish the existence of Dirac cones and magnetically induced local band gaps for the transverse magnetic Maxwell system in a two-dimensional honeycomb array of circular yttrium iron garnet (YIG) cylinders. For fixed positive material parameters with nonzero permeability contrast, we prove that sufficiently small cylinders support a Dirac point in the nonmagnetic case κg=0: two Bloch dispersion surfaces meet at a double eigenfrequency and form a nondegenerate cone. For each fixed sufficiently small nonzero gyrotropic bias κg, we show that this degeneracy splits and a local band gap opens near the original Dirac point. Our approach combines a quasi-periodic layer-potential formulation with honeycomb symmetry and rigorously controlled finite-dimensional reductions. The central contribution is a unified derivation of the conical crossing and magnetic gap opening directly from the full Maxwell transmission problem, without assuming an effective discrete model. This establishes the transition from Dirac dispersion to a locally gapped spectrum in a small-inclusion regime with fixed material contrast.