Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states

Weyl points, as monopoles of Berry curvature in momentum space, have captured much attention recently in various branches of physics. Realizing topological materials that exhibit such nodal points is challenging and indeed, Weyl points have been found experimentally in transition metal arsenide and phosphide and gyroid photonic crystal whose structure is complex. If realizing even the simplest type of single Weyl nodes with a topological charge of 1 is difficult, then making a real crystal carrying higher topological charges may seem more challenging. Here we design, and fabricate using planar fabrication technology, a photonic crystal possessing single Weyl points (including type-II nodes) and multiple Weyl points with topological charges of 2 and 3. We characterize this photonic crystal and find nontrivial 2D bulk band gaps for a fixed kz and the associated surface modes. The robustness of these surface states against kz-preserving scattering is experimentally observed for the first time.

propagate in the bulk of the crystal. Therefore the transmission with defect (green or violet curve) can be either larger or smaller than the one without defect (black curve) at different frequency. 9

Supplementary Note 1 The charge of double Weyl points
To see that the charge of double Weyl points is two, one can integrate the Berry flux on a closed surface which encloses the Weyl points. We can also manifest the topological charge of double Weyl point by calculating the surface dispersion in the following mathematical reconstruction.
Let us consider a tube in k-space aligned along the y direction with a fixed radius 0.5 / = π r k a (rather than k z ) in the Brillouin zone to obtain a 2D subsystem, see the blue dashed tube in to the lower band]. Therefore, if we terminate the crystal in y-direction, there should be two gapless surface states.
Supplementary Figure 2b calculates the surface dispersion with the geometry described in Fig. 5a in the frequency region of the 6th band gap. It is calculated on a closed circle (blue solid curve in Supplementary Fig 2a) in the surface Brillouin zone to be consistent with the 2D subsystem we discussed above. Two gapless surface states (propagating in clockwise direction) indeed exist in this nontrivial band gap, which implies the -2 charge of this double Weyl points.

Supplementary Note 2 Bulk transmission measurements
To experimentally confirm the existence of Weyl points, we measured angle-resolved transmission of the Weyl photonic crystal. Two samples were fabricated, one with the surface normal along the K Γ − direction (as shown in Supplementary Fig. 4b), the other along the M Γ − direction ( Supplementary Fig. 4g). Gray areas in Supplementary Fig. 4c Figure 4d shows the simulated transmission spectra with different incident angles. We can clearly see the projected linear cones at about 12.25 GHz in the transmission spectra. Apart from this, we found fairly low transmission in the simulation above 12.5 GHz when 0.1 / z k d < π , even though there are allowed bulk modes in that part of the momentum space according to the projected band. The reason is that projected band only tells whether there are states or not while transmission spectra also include the information of the coupling between modes in the photonic crystal and the source. In this specific case, the incident plane wave cannot excite the antisymmetric mode along K Γ − axis. These antisymmetric modes are even under the C 2 rotation along the y axis (the K Γ − direction), as shown below in Supplementary Note 3. When k z increases, these modes become excitable. Besides this mode coupling issue, another difference is the additional interference fringes in Supplementary Fig. 4d (for example, the low transmission around 11.5GHz and k z around 0.2 / d π ). These interference fringes are caused by multiple reflections at the two interfaces between air and photonic crystal since their effective mode impedances are different. Except for the above mentioned two discrepancies, the simulated transmission spectra are similar to the projected bulk band structure.
The corresponding measured transmission is shown in Supplementary Fig. 4e. We can see the band edge around 10.3 GHz and the linear cone around 12.2 GHz. Note that the amplitude of the measured transmission is smaller than the calculated transmission. This is due to the difference between the simulation and experimental setups. In simulation we can collect all the waves transmitting through the photonic crystal, while in our experiment, the horn antennas emit or receive EM wave with finite beam width. The beam will spread out (in both yand z-directions) during its propagation in air or the multiple reflection process in the sample.
The receiving antenna can only receive part of the transmitted beam, which leads to the overall decrease of the measured transmissions in Supplementary Fig. 4e compared with the simulated result in Supplementary Fig. 4d. We also note that the band edges in the experiment seem a little blurred. In the simulation, the incident wave can be assigned a specified z k value; while in the experiment, z k value is chosen by tilting the horn angle relative to the photonic crystal (see Supplementary Fig. 3) and this provides us incident wave with a finite spectrum range of z k .
Hence the measured spectrum loses the fine structures and appears blurred when compared with the simulated spectrum. In addition, due to the fabrication error of the sample and the deviation of the dielectric constant of PCB substrate from the standard value, the lower band edge in measured spectrum shifts 2% from that of simulated spectrum.
In contrast to the K Γ − direction, one finds that the directional band gap along M Γ − direction always opens for Supplementary Figures 4h and 4i are the corresponding projected bulk band and simulated transmission. The simulated spectrum agrees well with the projected band except that the upper passing band above 13 GHz split into three red zonal regions. These in fact correspond to three peaks in each transmission spectrum for different incident angle, due to the multiple reflections at the two interfaces between photonic crystal and air. We note that the amplitude of these fringes (difference between the peak value and deep value) above 13 GHz in Supplementary Fig. 4i is much larger than those in Supplementary Fig.   4d. This phenomenon indicates that there is a significant mode impedance mismatch between source (in air) and modes of this band. It is natural to imagine that the EM wave will bounce forward and back many times before transmitting through the photonic crystal.
Supplementary Figure 4j shows the corresponding measured transmission spectrum. We can clearly see the lower passing band from 10.3 to 11.8 GHz. However the interference fringes in the upper passing band cannot be observed. This is because at these Fabry-Perot resonant frequencies, EM waves will be bounced between the two interfaces many times and constructively interfere at the transmitted interface. In our measurement, we impinge the incident beam with finite beamwidth onto the finite size crystal. The beam will spread in y-and zdirections during the multiple reflection process inside the crystal, reach the side boundaries of the sample and leak outside. Hence we can hardly receive any signal considering the finite width of the horn as a receiver. For the same reason, the unapparent interference fringes in Supplementary Fig. 4d cannot be observed in our measured result in Supplementary Fig. 4e.
Together with the transmission spectra along K Γ − direction, we found that a 2D complete band gap opens near the Weyl point frequency of 12.25 GHz in k x -k y plane for nonzero k z and that the gap width broadens as k z increases. The calculated nonzero Chern number of this band gap indicates the existence of a chiral surface state, which was confirmed in our surface measurement.

Supplementary Note 3 Antisymmetric bulk modes of the photonic crystal
In the discussion concerning bulk transmission, we mentioned that the low transmission in Supplementary Fig. 4d, when frequency is bigger than 12.5 GHz and k z <0.1, is due to the symmetry-forbidden excitation of antisymmetric mode by external plane waves. Supplementary   Figure 5a shows the band structure along the ' K Γ − direction (k y direction), which is equivalent to the K Γ − direction due to time-reversal. Red lines highlight the antisymmetric bands. Note that this direction has a C 2 rotation symmetry about y axis. The bands with rotation eigenvalue of -1 and 1 are plotted in black and red. Supplementary Figures 5b and 5c show the E z field patterns of these bands.