Experimental Observation of Large Chern Numbers in Photonic Crystals

Despite great interest in the quantum anomalous Hall phase and its analogs, all experimental studies in electronic and bosonic systems have been limited to a Chern number of one. Here, we perform microwave transmission measurements in the bulk and at the edge of ferrimagnetic photonic crystals. Band gaps with large Chern numbers of 2, 3, and 4 are present in the experimental results, which show excellent agreement with theory. We measure the mode profiles and Fourier transform them to produce dispersion relations of the edge modes, whose number and direction match our Chern number calculations.

Figure 1

Comparison of theoretical gap map and bulk transmission to experimental transmission measurement in a 2D ferrimagnetic photonic crystal. (a) Theoretical topological gap map as a function of the magnetic field and the frequency, with each band gap labeled by its gap Chern number. The diagonal black line indicates the resonance in the effective permeability (see the Supplemental Material [17]). (b) Theoretical bulk transmission. (c) Experimental bulk transmission. (d) Experimental configuration with the lattice geometry (top metal plate removed). The antenna locations are marked with 1 and 2. (e) Simulation geometry, with the green line representing the receiving antenna and the green circle representing the transmitting antenna.

Figure 2

Experimental edge transmission measurement and Fourier transform (FT) of mode profiles along the copper boundary. (a) $S21$. (b) $S12$. The band gaps that are nontrivial have direction-dependent edge transmission because the interface of a nontrivial band gap with a trivial band gap (copper boundary) supports one-way modes. In (a) and (b) this causes the nontrivial bulk band gaps from Fig. 1 to be present in one direction (e.g., $S12$) and absent in the other (e.g., $S21$), which we highlight for the $C_{\text{gap}}=-4$ band gap with black boxes. The trivial band gaps around 4 GHz do not support one-way modes, and so they do not exhibit direction-dependent transmission. (c) Experimental FT of edge-mode profiles and the theoretical edge band structures with the edge modes in red and the bulk bands in gray. The range of wave vectors included in both plots is the same and includes only one Brillouin zone. The number of one-way edge modes in both sets of panels agrees with $|C_{\text{gap}}|$ from Fig. 1, while the sign of $C_{\text{gap}}$ is consistent with the theoretical group velocity (from the edge-mode dispersion) and the directional transmission in (a) and (b).

Figure 3

Topological one-way circuit implemented using $C_{\text{gap}}=1$ ($a=2.4\mathrm{cm}$) and $C_{\text{gap}}=2$ ($a=3.0\mathrm{cm}$) photonic crystals. (a)–(c) Transmission plots showing edge transmission between antennas at 1, 2, and 3, and antenna 4 located at the center. Shared bulk band gap for $C_{\text{gap}}=1$ and $C_{\text{gap}}=2$ crystals is highlighted in yellow. (d) Experimental configuration illustrating crystals with copper boundary ($C_{\text{gap}}=0$) on the left and antenna locations 1–4. Arrows indicate the theoretical direction and the number of the edge states at each interface. The transmission data is consistent with predicted edge state directions, which confirms that the upper crystal has $C_{\text{gap}}>1$.