Fine Structures of Berry Curvature and Unquantized Valley Chern Numbers in Valley Photonic Crystals

Abstract: Valley photonics has emerged as a promising platform in topological photonic systems, yet the topological nature of valley-dependent phenomena remains unsettled. Theoretically, inter-valley scattering may occur with structural imperfections, and global Chern numbers vanish due to time-reversal symmetry. As a result, valley-dependent topology is locally defined around K(K') points in the half-Brillouin zone (HBZ). While half-integer valley Chern numbers have been widely assumed, their quantization and topological validity remain controversial. Here, we systematically investigate a continuous spectrum of valley photonic crystal designs by evaluating their Berry curvatures, valley Chern numbers, and angular momenta. We show that valley Chern numbers are generically unquan-tized and instead form a continuous spectrum varying with structural parameters. We further reveal previously unexplored fine structures in the Berry curvature distribution in momentum space. The unquantized valley Chern numbers are attributed to inter- and intra-valley cancellation of Berry curvature, highlighting the absence of a protecting mechanism for quantization. Our results call for a reassessment of valley-dependent topology and provide a more rigorous framework for interpreting valley-related photonic phenomena.