Experimental observation of photonic nodal line degeneracies in metacrystals

Nodal line semimetals (NLS) are three-dimensional (3D) crystals that support band crossings in the form of one-dimensional rings in the Brillouin zone. In the presence of spin–orbit coupling or lowered crystal symmetry, NLS may transform into Dirac semimetals, Weyl semimetals, or 3D topological insulators. In the photonics context, despite the realization of topological phases, such as Chern insulators, topological insulators, Weyl, and Dirac degeneracies, no experimental demonstration of photonic nodal lines (NLs) has been reported so far. Here, we experimentally demonstrate NL degeneracies in microwave cut-wire metacrystals with engineered negative bulk plasma dispersion. Both the bulk and surface states of the NL metamaterial are observed through spatial Fourier transformations of the scanned near-field distributions. Furthermore, we theoretically show that the NL degeneracy can transform into two Weyl points when gyroelectric materials are incorporated into the metacrystal design. Our findings may inspire further advances in topological photonics.

Supplementary Figure 6 Metamaterial design that breaks the glide mirror symmetry by setting l2=0.8*l1, while preserving the mirror symmetry (x,y,z) (x,y,-z). The simulated band structure shows that the nodal line survives the perturbation.

Supplementary
is the substrates permittivity 4.84. 2×3 = [ 2×2 , 0]Unlike the parallel wire metacrystals, spatial dispersion of our non-symmorphic symmetry imposed metacrystal lacks a microscopic theory. We empirically let = 1 − ( 2 + 2 ) to describe the spatial dispersion which is already satisfying enough describing the core physics underlying the nodal line in this type of metacrystal.
Band structure calculated by the Hamiltonian is given in Supplementary Figure 1. Here

Supplementary Note 2. Experimental setup
For measuring bulk states in the metacrystal, a z-polarized electric dipole is placed at the centre of the bottom surface of the sample serving as the source, while another z-oriented dipole probe scans the top surface of the metacrystal to measures the field component (Supplementary Figure 2a). For the surface states a y-polarized source dipole is placed close to the centre of one edge of the top surface of the metacrystal, while the field on the top surface is scanned by a ypolarized scanning probe dipole ( Supplementary Figure 2(b)).
The source and probe antennas are connected to a Anritsu MS4647 vector network analyser. As the Probe scan the top surface of the metacrystal (step size is 4.5 mm for both x and y directions). Amplitude and phase information are collected by analysing the S-parameters, which could be further Fourier transformed into momentum space.
Correspondence between the measured spectrum and band structure of the metacrystal could be understood from the Supplementary Figure 3. As is shown in Supplementary Figure 3a, when the bulk state in metacrystal is excited and propagate to the interface to air, it experience total internal reflection and is reflected back into the metacrystal. The evanescent tail in air during the total internal reflection process could be picked up by the near field antenna. Supplementary Figure  3c illustrate an example of equi-frequency contour. In the experiment, since the metacrystal has finite thickness in the z direction, the allowed kz is discredited, as is demonstrated in Supplementary  Figure 3d, where each loop represents a cut of the equi-frequency surface (EFS) in Supplementary  Figure 3c of a definite kz. Finally, since the fields are picked up at the upper boundary, only kx ky Fourier components can be extracted, but not kz. The resultant measurement result should be the projection of the rings in Supplementary Figure 3d, as is shown in Supplementary Figure 3b. Thus the measurement results for different frequencies could provide information of the band structure, not that with definite kz, but the projection with several different kz.

Supplementary Note 3. Calculation of Berry phase of the nodal line by Wilson loop
With the Hamiltonian formalism well established, it's easy to calculate Berry phase of the nodal line by Wilson loop method. The Berry phase could be easily expressed by: Here we set the loop threading the nodal line, and its radius equals 0.1 / . As is shown in Supplementary Figure 4, the Berry phase converges very fast to π as the number of discretized points on the loop N increases. where S account for the velocity field in magnetized plasma. When external bias magnetic field is not applied, band structure is given in Supplementary Figure 4. The nodal is still at present, which agrees reasonably well with Figure 5a in the main text.
Once the magnetic field is applied, the nodal line can also degenerate into Weyl points (WP) in the Hamiltonian model, as is given in Supplementary Figure 5. The mismatch between effective Hamiltonian and the real structure, especially near resonance frequency of the magnetized plasma comes from the fact that the change of resonance of the cut-wires are not considered in the Hamiltonian model for simplicity. However, behaviour around the nodal line agrees reasonably well between them.

Supplementary Note 5. Effective Hamiltonian model of the Weyl point and nodal line in metacrystal
To construct effective Hamiltonian of the WP, we firstly express eigen states at the WP as