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

Topological photonic systems have seen significant advancements, with valley-photonics gaining attention for their broad bandwidth, scalability, and versatility across materials and frequencies. However, the topological nature of valley-photonics is still unclear. 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 some studies report half-integer valley Chern number, others show that Chern number is not quantized, challenging their status as topological invariants. Our work summarizes various valley photonic crystal designs in a continuous structural spectrum and explores their Berry curvatures, Chern numbers and angular momenta, revealing that most valley photonic crystal designs lack quantized valley Chern numbers and highlighting fine structures of local Berry curvature distribution. These findings suggest flaws in our current understanding of valley-dependent topology and that further analysis is necessary.