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.

Figures (8)

• Figure 1: Illustration of structure model. Starting from a honeycomb lattice, the investigated photonic crystal hole size and shape are gradually modified, to a triangular-lattice and then back to a honeycomb-lattice. The rounding factor $f_r$ defines the rounded triangle shape in the triangular-lattice. The radius ratios $f_t$ and $f_c$ define the perturbation strength in the honeycomb lattice. $f_r$ varies in range [0,0.5] and $f_{t/c}$ varies in range [0,1]
• Figure 2: Top row shows the unit cells of some investigated VPhCs. The lattice possesses $C_{2z}$ symmetry when $f_t=1.0$, $f_r=0.5$, and $f_c=1.0$ (a-d) the band diagram of four typical VPhCs, with structural factors $f_t=0.8$, $f_t=0$, $f_r=0.4$ and $f_c=0.4$ respectively. (e-g) the normalized Berry curvatures of the first three bands for varying structural factors. The first band is in green, the second band in purple and the third band in yellow. The black dashed lines are aligned to the values of structural factors. (h-j) the first band gap (in THz unit) vs. structural factor.
• Figure 3: Berry curvature distributions for (a-c) $f_t=0.8$ Tri-HPC, (d-f) $f_r=0$ TPC and (g-i) $f_r=0.4$ TPC. The top row shows the Berry curvature for band-1, middle row for band-2 and the bottom for band-3. The corresponding unit cells are shown above each column. In each plot, the solid hexagon show the first Brillouin zone. The dashed triangles show the half-Brillouin zone centered at K point. The calculated valley Chern number are shown at the left top in each plot. The color bar shows the value of normalized Berry curvature $\Omega/a^2$.
• Figure 4: Valley Chern number evolution for varying structural parameters. (a) $C_v$ in the Tri-HPC. $f_t$ varies from 0.99 to 0 (b) $C_v$ in the TPC. $f_r$ varies from 0 to 0.49.
• Figure 5: The calculated (a-c) orbital angular momentum and (d-f) spin angular momentum of the first three bands for varying structural factors. The first band is in green, the second band in purple and the third band in yellow. The top image shows illustrations of some VPhC unit cells. The black dashed lines cut across the corresponding structural factors (x axis) in the angular momentum plot.