Fig. 1: Photonic 3D CI by cubic supercell modulation at N = N_W = 2.

Each (a, b, d, e) panel shows the crystal unit cell, the irreducible Brillouin zone (IBZ) and the band-structure (BS). Frequencies f are given in reduced units, ∣a∣ being the scale invariant lattice parameter and c the speed of light. Sectors (c) and (f) contain the topological characterization via photonic Wilson loops (WL), in the Weyl semimetallic (WS) phase and in the 3D CI phase, respectively. a Photonic crystal constructed from cylinders of radius r₀ = 0.15 and dielectric constant ε = 16 at TRS. The three lowest photonic modes display a three-fold degeneracy at R and the two lowest bands are fully degenerate in the displayed energy window. b TRS breaking implemented via a gyro-electric response with \({\eta }_{z}^{{N}_{W} = 2}=16\): the bias field is adjusted in order to split the Weyl points of approximately half the BZ, i.e. at \({k}_{z}^{\pm }=\pm \! \frac{\pi }{2| a| }\), along the \({{{{{{{\bf{S}}}}}}}}{{{{{{{\bf{R}}}}}}}}{{{{{{{{\bf{S}}}}}}}}}^{\prime}\) line where \({{{{{{{{\bf{S}}}}}}}}}^{\prime}={{{{{{{\bf{S}}}}}}}}-{{{{{{{{\bf{b}}}}}}}}}_{z}\). c Electromagnetic section Chern number calculated on 2D k_z planes normal to the magnetization (upper panel) and transverse flow of the θ_y eigenvalue of the WL matrix summed over the entire subset of ν_occ bands lying below the local gap at k_z (lower panel). The discontinuity of the section Chern number at the wavevector of each Weyl point \({k}_{z}^{\pm }\) is used a measure its topological charge (q_± = 1). d Artificial folding of the bands on a N = 2 cubic supercell: Weyl points superimpose at Z in the new BZ. e Coupling and annihilation of Weyl points through a N = 2 supercell modulation with parameter r_m = r₀/20. The amplified modulation is graphically visualized via a colorbar with r_max = r₀ + r_m and r_min = r₀ − r_m. A topological direct gap (f_g) at Z is opened, with gap-to-midgap (f_g/f_m) ratio of 1.86%. The size of gap can be appropriately tuned by choosing the value of the modulation, as in the inset. f The section Chern C_z number displays constant unit value everywhere in the BZ, establishing the system to be in the 3D CI phase. Source data are provided as a Source Data file.