Fig. 1. Typical structures for realizing Weyl points in photonic crystals^[48]. (a) Unit cell of a double-gyroid photonic crystal, blue (red) corresponds to double-gyroid (single-gyroid) photonic crystal unit cell; (b) band structure along high-symmetry lines; (c) breaking inversion symmetry by introducing air spheres in specific single-gyroid structures, resulting in two pairs of Weyl points in the first Brillouin zone; (d) breaking of time-reversal symmetry by applying an external magnetic field, leading to one pair of Weyl points in the first Brillouin zone

Fig. 2. Non-metallic materials for constructing Weyl metamaterials with broken space inversion symmetry^[57-59]. (a) Double-gyroid photonic crystal; (b) two pairs of Weyl points are present in the first Brillouin zone, distributed along the high-symmetry lines $\Gamma ‒H$ and $\Gamma ‒N$; (c) experimental measurement structure of Weyl point structure along $\Gamma ‒H$ direction; (d) schematic of stacked woodpile structure; (e) band structure along the high-symmetry direction $Y‒\Gamma ‒X$, with Weyl points at $\Gamma$ marked by blue circles; (f) experimentally measured reflection spectrum near the Weyl points; (g) helical waveguide array; (h) dispersion plot in the $\mathrm{\delta }{k}_x‒\mathrm{\delta }{k}_z$ plane, showing a type-Ⅱ Weyl point at $\mathrm{\delta }{k}_x=\mathrm{\delta }{k}_y=\mathrm{0}$; (i) experimental measurement result of the output intensity at $a=\mathrm{27}\mathrm{\mu }\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}\lambda =\mathrm{1525}\mathrm{n}\mathrm{m},\mathrm{a}\mathrm{n}\mathrm{d}$green dots indicating the positions of the input waveguides

Fig. 3. Construction of Weyl points using synthetic dimensions^[73]. (a) Longitudinal subwavelength grating waveguides are used to construct a synthetic dimension Weyl lattice; (b) by adjusting the structural parameters, both type-Ⅰ and type-Ⅱ Weyl points can be realized; (c) projected band structure of type-Ⅰ Weyl points in the synthetic dimension space (left) and projected band structures in two planes (right); (d) projected band structure corresponding to type-Ⅱ Weyl points; simulated light propagation and experimentally measured output intensity for (e) type-Ⅰ Weyl points and (f) type-Ⅱ Weyl points

Fig. 4. Construction of Weyl metamaterials with broken space inversion symmetry based on metallic resonant structures^[75-77]. (a) Unit cell of ideal Weyl metamaterial; (b) four Weyl points in the first Brillouin zone are located at the same frequency; (c) linear dispersion near the Weyl points characterized using transmission spectra; (d) unit cell of chiral hyperbolic metamaterial; (e) band structure along the high-symmetry line $Y‒\Gamma ‒X$, with red dots indicating type-Ⅱ Weyl points; (f) experimental observation result of type-Ⅱ Weyl points, shown as the equi-frequency contour at 5.46 GHz; (g) schematic illustration of a chiral metamaterial unit cell; (h) band structure of Weyl points with topological charge 2; (i) projected band structure along the $k_x‒k_z$ plane, showing quadratic dispersion near the Weyl points with topological charge 2

Fig. 5. Fabrication of tunable Weyl metamaterials using printed circuit board technology^[78]. (a) Top view of the ${\mathrm{C}}_{\mathrm{6}}$ rotationally symmetric unit cell structure (top; blue and red represent the upper and lower Y-shaped layers, respectively) and band structure along high-symmetry lines (bottom), showing five types of Weyl points; (b) top view of the ${\mathrm{C}}_{\mathrm{3}}$ rotationally symmetric unit cell structure and the corresponding band structure, with $k_z=\mathrm{0}$; (c) ${\mathrm{C}}_{\mathrm{2}}$ rotationally symmetric unit cell structure and the corresponding band structure, with the same high-symmetry line path as in Fig.5(b)

Fig. 6. Fermi arcs in chiral hyperbolic electromagnetic metamaterials^[76]. (a) Projection of Fermi arcs onto the surface by scanning the frequency, with red and blue indicating Weyl points and cyan representing the Fermi arcs; (b) schematic of near-field scanning setup, where the field distribution is measured by scanning a grid on the sample's upper surface; (c) real-space measurement results of surface states, showing the directional propagation of the electric field at a frequency of 5.46 GHz; (d) Fermi arc surface states in momentum space, obtained from the Fourier transform of real-space results in Fig. 6(c); (e) topologically protected surface waves propagating over steps [Backscattering-immune surface waves propagate on a three-dimensional stepped geometry. The source (red triangle) is set as x-polarization, and the normal component of the electric field is detected on all surfaces. The step width, height, and length are 104 mm (top and bottom surfaces), 60 mm, and 600 mm, respectively. The normal electric field is normalized to its maximum value]

Fig. 7. Fermi arc surface states in ideal Weyl metamaterials and grating waveguides^[59,75]. (a) Schematic of near-field scanning system setup; (b)‒(d) experimental (top row) and simulated (bottom row) Fermi arc distributions, with the center of the Brillouin zone being the light cone, and the Fermi arcs undergo relative rotation around the Weyl cones with the increasing frequency; (e) no surface states exist at $a=\mathrm{29}\mathrm{\mu }\mathrm{m}$, and the light distribution in the waveguide array is discrete, with the green dot indicating the signal input position; (f) band structure showing no surface states, with a bandgap existing between the upper and lower bulk states; (g) surface states exist at $a=\mathrm{25}\mathrm{\mu }\mathrm{m}$, and light propagates along the boundary of the waveguide array; (h) corresponding band structure

Fig. 8. Reconstruction of Fermi arcs^[81-82]. (a) Schematic of the experimental setup for measuring eigenstates around Weyl points using reflection method; (b) analytical, numerical, and experimental reflection phases and the corresponding eigen polarizations of the Jones matrix for Weyl metamaterials at different azimuthal angles of incidence with a fixed horn antenna elevation of 45°; (c) formation of helical Fermi arcs due to the reflection phase difference between Weyl metamaterials and perfect electric conductor (PEC); (d) schematic of the distribution of Weyl points and helical Fermi arcs; (e) schematic of the experimental setup for measuring electric field distribution in a cavity formed by Weyl metamaterials and PEC; (f) experimentally measured and (g) numerically simulated helical Fermi arcs, with the equi-frequency contour at 13.3 GHz; (h) construction process of twisted ideal Weyl metamaterials; (i)‒(k) distribution of Fermi arcs in momentum space (top row) and real space (bottom row) for twist angles of 15°, 45°, and 75°, respectively, and the Fermi arcs undergo reconstruction as the twist angle changes

Fig. 9. Weyl points and Fermi arcs in non-periodic systems^[92-93]. (a) Penrose tiling quasicrystal stacked along the z-axis, with the mass term varying with the z-axis position; (b) mass parameter M and Chern number C as functions of $k_z$, and Weyl points, indicated by red/blue markers, are located where the Chern number changes; (c) density of states at the boundary, where Fermi‒Bragg arcs connect Weyl points with opposite chirality; (d) structure factor in Fourier space (only the brightest spots are shown, with spot size proportional to intensity), and vertical blue lines show the projection of Bragg peaks aligning precisely with the Fermi arcs; (e)‒(l) evolution of chiral edge states at clean system as disorder strength increases, and more surface chiral edge states can be accommodated within a fixed energy window with increasing disorder (For weak disorder, zero-energy states can be connected to form a sharp Fermi arc; under strong disorder, the number of states localized near zero energy and at the surface is significantly reduced)

Fig. 10. Utilizing Fermi arcs to achieve Andreev reflection^[94]. (a) Schematic of a planar normal metal (N)‒superconductor (S) junction on the top of a Weyl semimetal and the scattering of particles at the interface, and the trajectories of electrons (solid circles) and holes (dashed circles) are represented by solid and dashed lines, respectively; (b) two regions of Andreev reflection defined by momentum $k_z$; (c) band structure at a fixed $k_z$, with red solid lines (blue dashed lines) corresponding to electron (hole) surface states; (d)‒(f) Fermi arc spectra at different azimuthal angles $\theta$ (top row) and Andreev reflection probability for different $k_z$ channels (bottom row)

Fig. 11. Realization of a Veselago lens using ideal Weyl metamaterials^[95]. (a) Schematic of Weyl metamaterial sample with the excitation source placed in air; (b) schematic of the unit cell; (c) corresponding band structure; (d) equi-frequency contour at a frequency of $\mathrm{0.935}{\omega }_{\mathrm{W}\mathrm{e}\mathrm{y}\mathrm{l}}$, with the red line representing the light cone, incident and scattered wave vectors are indicated by red and blue arrows, respectively; (e) simulation results of full-angle negative refraction, with white lines indicating the direction of light propagation