Symmetry-based classification of Chern phases in honeycomb photonic crystals

In this work, we develop a symmetry-based classification of Chern phases in honeycomb photonic crystals, considering arbitrary nonreciprocal couplings compatible with energy conservation. Our analysis focuses on crystals formed through nonreciprocal perturbations of photonic graphene. These perturbations, which can have arbitrary spatial variations, are generally described by scalar and vector fields. Using a tight-binding model, we consider the most general nonreciprocal interactions, including gyromagnetic, pseudo-Tellegen, and moving medium responses, and examine how the corresponding nonreciprocal fields influence the crystal's topology. Our findings reveal that nonreciprocal interactions alone are insufficient to induce a topologically nontrivial phase. Instead, a nontrivial $p6m$ component in the nonreciprocal fields is required to open a band gap and achieve a nonzero Chern number. These results provide a symmetry-based roadmap for engineering photonic topological phases via nonreciprocal perturbations of photonic graphene, offering practical guidelines for designing topological phases in graphenelike photonic crystals.